| Domain | Dataset (Superscripts include the URLs to the datasets) | Modalities |
| Healthcare | MIMIC-III [35][1], MIMIC-IV [34][2] | TS, Text, Tabular |
| ICBHI [65][3], Coswara [4][4], KAUH [21][5], PTB-XL [71][6], ZuCo [14, 26][7] | TS, Text |
| Image-EEG [22][8] | TS, Image |
| Finance | FNSPID [17][9], ACL18 [84][10], CIKM18 [79][11], DOW30 [11][12] | TS, Text |
| Multi-domain | Time-MMD [53][13], TimeCAP [42][14], NewsForecast [73][15], TTC [37][16], CiK [77][17], TSQA [38][18] | TS, Text |
| Retail | VISUELLE [70][19] | TS, Image, Text |
| IoT | LEMMA-RCA [40][20] | TS, Text |
| Speech | LRS3 [1][21], VoxCeleb2 [13][22] | TS (Audio), Image |
| Traffic | NYC-taxi, NYC-bike [48][23] | ST, Text |
| Environment | Terra [10][24] | ST, Text |
",
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+ "type": "text",
+ "text": "form of time series in this survey and briefly discuss representative works within this scope.",
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+ "text": "3.2 Common Datasets and Benchmarks",
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+ "text": "Multi-modal time series datasets vary a lot and are domain-dependent, each with unique data characteristics and modalities. In Table 1 we provide representative datasets categorized by domain, along with their respective modalities:",
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+ "text": "Healthcare: In this domain, physiological signals (e.g., ECG, EEG) are extensively analyzed alongside textual data such as clinical notes, patient demographics, and tabular data including vital signs and laboratory results. Common datasets include MIMIC-III [35], a comprehensive dataset containing electronic health records (EHRs) of ICU patients with physiological measurements, clinical notes, and diagnostic information, widely used for tasks like patient monitoring, mortality prediction, and clinical decision support. MIMIC-IV [34] is an extension of MIMIC-III, which provide detailed physiological signals, clinical narratives, medication records, and demographic data from a large population of critically ill patients, frequently utilized for predictive modeling, clinical outcome analysis, and health informatics research. Other notable healthcare datasets include ICBHI [65], which contains respiratory sound recordings paired with clinical annotations for respiratory disease classification; Coswara [4], which provides respiratory audio samples and rich metadata for COVID-19 detection tasks; KAUH [21], which comprises audio records and corresponding annotations for healthcare analytics; PTB-XL [71], a large-scale ECG dataset annotated with diagnostic labels for cardiac monitoring and diagnosis; ZuCo [14, 26], which consists of simultaneous EEG and textual data from reading comprehension tasks, being useful for cognitive neuroscience studies; and Image-EEG [22], which pairs EEG signals with images of objects on a natural background, aiding studies in visual neuroscience and computer vision.",
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+ "text": "Finance: Datasets that combine time series data with financial news and reports are instrumental in financial analysis and modeling. Notable examples include ACL18 [84], CIKM18 [79], and DOW30 [11]. These datasets focus on high-trade-volume stocks from the U.S. stock markets, providing historical stock price data, such as opening, high, low, and closing prices; alongside related textual information, including tweets or financial news. Another",
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+ "text": "large-scale dataset, FNSPID [17], consists of stock prices and time-aligned financial news records, covering over 4,000 companies from 1999 to 2023.",
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+ "text": "Multi-domain: Datasets featuring general-purpose numerical time series combined with textual data are suitable for broad analytical applications. Examples include Time-MMD [53], which encompasses nine primary data domains: Agriculture, Climate, Economy, Energy, Environment, Health, Security, Social Good, and Traffic, while ensuring fine-grained alignment between time series and textual data; TimeCAP [42] compiles seven real-world time series datasets across three domains: weather, finance, and healthcare. To generate textual descriptions for each time series, a large language model (LLM) agent is employed, leveraging contextual information and domain-specific knowledge. NewsForecast [73] integrates task-specific time series data with verified public news reports across various domains, including finance, energy, traffic, and cryptocurrency; TTC [37] is a meticulously curated, time-aligned dataset designed for multimodal forecasting. It consists of paired time series and text data synchronized to timestamps, spanning two distinct domains: climate science and healthcare; CiK [77] is a dataset comprising 71 forecasting tasks across seven real-world domains. Each task necessitates the integration of both numerical data and textual information. The covered domains include Climatology, Economics, Energy, Mechanics, Public Safety, Transportation, and Retail. The TSQA [38] dataset consists of 200k question-answer pairs derived from time series data across 12 domains: healthcare, finance, energy, traffic, environment, IoT, nature, transport, human activities, machine sensors, AIOps, and the web. These QA pairs are designed to support five key tasks: forecasting, imputation, anomaly detection, classification, and open-ended reasoning.",
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+ "text": "Other domains: Beyond the previously discussed major sectors, multi-modal time series analysis extends to various other domains. In Retail, datasets such as VISUELLE [70] integrate numerical sales data with product images and textual descriptions, facilitating thorough analyses of consumer behavior and inventory management. The Internet of Things (IoT) domain benefits from datasets such as LEMMA-RCA [40], which combine time series sensor data with textual metadata, enabling enhanced monitoring and more robust and secure methodologies that ensure the high performance of modern",
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+ "type": "header",
+ "text": "Multi-modal Time Series Analysis: A Tutorial and Survey",
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+ "type": "text",
+ "text": "systems. In the Speech domain, datasets like LRS3 [1] and VoxCeleb2 [13] integrate audio recordings with corresponding visual data, supporting advancements in speech recognition and speaker identification technologies. In the Traffic domain, datasets like NYC-Taxi, NYC-Bike [48] contain spatial-temporal (ST) data alongside associated textual metadata. These integrations allow LLMs to effectively capture and utilize spatial-temporal contextual signals. In the Environment domain, Terra [10] collect 45 years of global geographic spatial-temporal data, supplemented with textual descriptions.",
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+ "text": "4 Cross-modal Interactions with Time Series",
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+ "text": "In this section, we conduct a detailed review of existing research on multi-modal time series analysis by thoroughly analyzing cross-modal interactions. We also elaborate how existing multi-modal methods are tailored for domain-specific applications in Section 5. The detailed taxonomy is provided in Table 2.",
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+ "text": "We define three fundamental types of interactions between time series and other modalities, including fusion, alignment, and transference, which occur at different stages within a framework - input, intermediate (i.e., representations or intermediate outputs), and output. The representative examples are provided in Figure 2.",
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+ "text": "4.1 Fusion",
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+ "text": "Fusion refers to the process of integrating heterogeneous modalities in a way that captures complementary information across diverse sources to improve time series modeling. To fuse multi-modal inputs, a common practice is to directly integrate time series, tabular data and texts into a unified textual prompt, then use it to query LLMs for downstream tasks. This is typically facilitated by instruction fine-tuning for task-oriented analysis [19, 24, 38, 48, 56, 73, 90]. Some works also leverage the zero-shot reasoning and inference capability of pretrained LLMs (e.g., GPT-4 and its variants) [72, 81, 89]. Recent research efforts like TaTS [47] attempt to integrate paired text embedding as an additional variable of time series for temporal modeling, yielding competitive task performance.",
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+ "text": "Most existing methods perform cross-modal fusion at the intermediate stage, such as adding and concatenating multi-modal representations, where each individual modal encoder first maps the raw data into a shared latent space. Addition combines time series and other modalities by summing up encoded representations, effectively blending shared information while preserving their interconnections in the latent space [6, 30, 51, 85, 95, 97]. On the other hand, concatenation stacks multi-modal representations along the same dimension, retaining modality-specific characteristics and allowing models to capture joint relationships between the modalities [16, 36, 36, 37]. To effectively leverage cross-modal information, existing methods often incorporate alignment designs after concatenating representations [5, 11, 12, 18, 31, 33, 41, 42, 44, 54, 55, 60, 69, 70, 74, 85, 96]. Alignment is also used in the aforementioned additions, which will be detailed in Section 4.2.",
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+ "text": "When fusion is performed at the output level, different modalities contribute separately to the final output, allowing each modality to retain its unique predictive signal [31, 41, 42, 53]. Time-MMD [53] provides a paradigm that fuses predictions from both state-of-the-art forecasters and a pretrained language model with a projection",
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+ "img_path": "images/dfc75126255eb54797d7990d9a20668b0a1b9ee7743a4acf0e3af423c98e1049.jpg",
+ "image_caption": [
+ "Figure 2: Categorization of cross-modal interaction methods and representative examples."
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+ "text": "layer, in an end-to-end manner. MOAT [41] introduces a two-stage framework for multi-modal time series forecasting. In the first stage, the model is optimized to generate forecasts from decomposed time series and text embeddings. In the second stage, an offline synthesis via MLP is applied to dynamically fuse different components, yielding the final forecast based on their relative contributions. Beyond fusing outputs from a single model, TimeCAP [42] enhances performance by combining predictions from both a multi-modal predictor and a pretrained LLM, which synergizes the gradient-based method and LLM agents reasoning on real-world contexts. Output fusion gains advantage of design flexibility and robustness, but it may not fully utilize the complementary relationship between modalities without additional countermeasures.",
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+ "text": "Cross-modal fusion relies on well-aligned multi-modal data for effective exploitation of the contextual information. However, ideally-aligned data may not be given in real-world scenarios. As such, existing methods also leverage alignment mechanisms to mitigate the challenge.",
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+ "text": "4.2 Alignment",
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+ "text": "Alignment ensures that the relationships between different modalities are preserved and semantically coherent when integrated into a unified learning framework. At the input level, we primarily refer alignment to data preprocessing techniques that aim at mitigating temporal misalignment caused by missing values, irregular sampling intervals, and differing granularities across modalities. This process is crucial for ensuring that data from multiple sources are properly synchronized before fusion, where domain knowledge is usually needed to handle such inconsistencies [10, 53, 73]. In addition, none of the existing methods we reviewed explicitly perform output alignment. However, the aforementioned output fusion can be easily adapted to alignment through the incorporation of a gating or attention mechanism that we will introduce shortly.",
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+ "text": "Alignment at the intermediate stage plays a crucial role in multimodal interactions. We first introduce the alignment of multi-modal representations, spanning a range of techniques from model component design to learning objectives. The common component designs include self-attention [5, 12, 30, 33, 41, 42, 44, 54, 55, 69], cross-attention [6, 18, 51, 60, 70, 82, 85, 98] and gating mechanisms [82, 98]. Self-attention is often used to fuse multi-modal representations. It",
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+ {
+ "type": "header",
+ "text": "Jiang, et al.",
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+ {
+ "type": "table",
+ "img_path": "images/3c674ad0e2e84929c9f56d3334e69ed89be4011b598a814f2e99ea5de4015cb1.jpg",
+ "table_caption": [
+ "Table 2: Taxonomy of representative multi-modal time series methods. Modality refers to the different data modalities involved in each method. TS represents standard time series, $ST$ denotes spatial time series. The Method column lists the techniques used for each interaction, separated by semicolons, where each interaction may include one or more techniques, separated by commas. Superscripts in the Code column include the URLs to Github repositories."
+ ],
+ "table_footnote": [],
+ "table_body": "| Method | Modality | Domain | Task | Cross-Modal Interaction | Large Model | Year | Code | |
| Stage | Fusion | Align. | Trans. | Method | |
| Time-MMD [53] | TS, Text | General | Forecasting | Output | ✓ | X | X | Addition | Multiple | 2024 | Yes[1] | |
| Wang et al. [73] | TS, Text | General | Forecasting | Input Intermediate | ✓ | X | X | Prompt Prompt; LLM Reasoning | LLaMa2 GPT-4 Turbo | 2024 | Yes[2] | |
| GPT4MTS [30] | TS, Text | General | Forecasting | Intermediate | ✓ | ✓ | X | Addition; Self-attention | GPT-2 | 2024 | No | |
| TimeCMA [51] | TS, Text | General | Forecasting | Input Intermediate | X | X | ✓ | Meta-description Addition; Cross-attention | GPT-2 | 2025 | Yes[3] | |
| MOAT [41] | TS, Text | General | Forecasting | Intermediate Output | ✓ | ✓ | X | Concat.; Self-attention Offline Synthesis (MLP) | S-Bert | 2024 | No | |
| TimeCAP [42] | TS, Text | General | Classification | Input Intermediate Output | X | X | ✓ | LLM Generation Concat.; Self-attention, Retrieval Addition | Bert, GPT-4 | 2024 | No | |
| TimeXL [31] | TS, Text | General | Classification Forecasting | Intermediate Output | ✓ | ✓ | X | Concat., Prompt; LLM Reasoning Addition | Bert, S-Bert GPT-4o | 2025 | No | |
| Hybrid-MMF [37] | TS, Text | General | Forecasting | Intermediate | ✓ | X | X | Concat. | GPT-4o | 2024 | Yes[4] | |
| Time-LLM [33] | TS, Text | General | Forecasting | Input Intermediate | X | X | ✓ | Meta-description Concat.; Self-attention | LLaMA, GPT-2 | 2024 | Yes[5] | |
| Time-VLM [98] | TS, Text, Image | General | Forecasting | Input Intermediate | X | X | ✓ | Feat. Imaging, Meta-description Addition; Gating, Cross-attention | ViLT, CLIP BLIP-2 | 2025 | No | |
| Unitime [55] | TS, Text | General | Forecasting | Input Intermediate | X | X | ✓ | Meta-description Concat.; Self-attention | GPT-2 | 2024 | Yes[6] | |
| TESSA [50] | TS, Text | General | Annotation | Intermediate | ✓ | ✓ | ✓ | Prompt; RL; LLM Generation | GPT-4o | 2024 | No | |
| InstruTime [12] | TS, Text | General | Classification | Intermediate | ✓ | ✓ | X | Concat.; Self-attention | GPT-2 | 2025 | Yes[7] | |
| MATMCD [68] | TS, Text, Graph | General | Causal Discovery | Intermediate | ✓ | ✓ | ✓ | Prompt; LLM Reasoning; Supervision | Multiple | 2025 | No | |
| STG-LLM [54] | ST, Text | General | Forecasting | Intermediate | ✓ | ✓ | X | Concat.; Self-attention | GPT-2 | 2024 | No | |
| TableTime [72] | TS, Text | General | Classification | Input | ✓ | X | ✓ | Prompt; Reformulate | Multiple | 2024 | Yes[8] | |
| ContextFormer [6] | TS, Tabular | General | Forecasting | Intermediate | ✓ | ✓ | X | Addition; Cross-attention | No | 2025 | No | |
| Time-MQA [38] | TS, Text | General | Multiple | Input | ✓ | X | X | Prompt | Multiple | 2025 | Yes[9] | |
| MAN-SF [67] | TS, Text, Graph | Finance | Classification | Intermediate | ✓ | ✓ | X | Bilinear; Graph Convolution | USE | 2020 | No | |
| Bamford et al. [3] | TS, Text, TS, Image | Finance | Retrieval | Intermediate Output | X | ✓ | X | Supervision | S-Bert | 2024 | No | |
| Chen et al. [11] | TS, Text, Graph | Finance | Classification | Intermediate | X | X | ✓ | LLM Generation Concat.; Graph Convolution | ChatGPT | 2023 | No | |
| Xie et al. [81] | TS, Text | Finance | Classification | Input | ✓ | X | X | Prompt | ChatGPT | 2023 | No | |
| Yu et al. [89] | TS, Text | Finance | Forecasting | Input | ✓ | X | X | Prompt | GPT-4, Open LLaMA | 2023 | No | |
| MedTsLLM [5] | TS, Text, Tabular | Healthcare | Multiple | Intermediate | ✓ | ✓ | X | Concat.; Self-attention | Llama2 | 2024 | Yes[10] | |
| RespLLM [95] | TS (Audio), Text | Healthcare | Classification | Intermediate | ✓ | ✓ | X | Addition, Self-attention | OpenBioLLM-8B | 2024 | No | |
| METS [45] | TS, Text | Healthcare | Classification | Output | X | ✓ | X | Contrastive | ClinicalBert | 2023 | No | |
| Wang et al. [75] | TS, Text | Healthcare | Classification | Intermediate | X | X | ✓ | Supervision | Bart, Bert, Roberta | 2021 | No | |
| EEG2TEXT [52] | TS, Text | Healthcare | Generation | Output | X | X | ✓ | Self-supervision, Supervision | Bart | 2024 | No | |
| MEDHMP [74] | TS, Text | Healthcare | Classification | Intermediate | ✓ | ✓ | X | Concat.; Self-attention, Contrastive | ClinicalT5 | 2023 | Yes[11] | |
| Deznabi et al. [16] | TS, Text | Healthcare | Classification | Intermediate | ✓ | X | X | Concat. | Bio+Clinical Bert | 2021 | Yes[12] | |
| Niu et al. [60] | TS, Text | Healthcare | Classification | Intermediate | ✓ | ✓ | X | Concat.; Cross-attention | BioBERT | 2023 | No | |
| Yang et al. [85] | TS, Text | Healthcare | Classification | Intermediate | ✓ | ✓ | X | Concat.; Addition; Gating | ClinicalBERT | 2021 | Yes[13] | |
| Liu et al. [56] | TS, Text | Healthcare | Classification Regression | Input | ✓ | X | X | Prompt | PaLM | 2023 | Yes[14] | |
| xTP-LLM [24] | ST, Text | Traffic Forecasting | Input | ✓ | X | ✓ | Prompt; Meta-description | Llama2-7B-chat | 2024 | Yes[15] | |
| UrbanGPT [48] | ST, Text | Traffic Forecasting | Input | ✓ | X | ✓ | Prompt; Meta-description | Vicuna-7B | 2024 | Yes[16] | |
| CityGPT [19] | ST, Text | Mobility | Input | ✓ | X | X | Prompt | Multiple | 2025 | Yes[17] | |
| MULAN [97] | TS, Text, Graph | IoT Causal Discovery | Intermediate | ✓ | ✓ | ✓ | Addition; Contrastive; Supervision | No | 2024 | No | |
| MIA [82] | TS, Image | IoT Anomaly Detection | Intermediate | ✓ | ✓ | X | Addition; Cross-attention, Gating | No | 2023 | No | |
| Ekambaram et al. [18] | TS, Image, Text | Retail Forecasting | Intermediate | ✓ | ✓ | X | Concat.; Self & Cross-attention | No | 2020 | Yes[18] | |
| Skenderi et al. [70] | TS, Image, Text | Retail Forecasting | Intermediate | ✓ | ✓ | X | Concat.; Cross-attention | No | 2024 | Yes[19] | |
| VIMTS [96] | ST, Image Environment | Imputation | Intermediate | ✓ | ✓ | X | Concat.; Supervision | No | 2022 | No |
| LTE [44] | ST, Text, Image Environment | Forecasting | Intermediate | ✓ | ✓ | X | Concat.; Self-attention | LLaMA-2-7B | 2024 | Yes[20] |
| AV-HubERT [69] | TS (Audio), Image Speech | Classification | Intermediate | ✓ | ✓ | X | Concat.; Self-attention | HuBert | 2022 | Yes[21] |
| SpeechGPT [90] | TS(Audio), Text | Generation intermediate | ✓ | ✓ | X | Concat.; Self-attention | LLaMA-13B | 2023 | Yes[22] | |
| LA-GCN [83] | ST, Text | Classification | Intermediate | X | ✓ | Supervision | Bert | 2023 | Yes[23] | |
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+ {
+ "type": "text",
+ "text": "enables a joint and undirected alignment across all modalities by dynamically attending to important features. Given multi-modal embeddings $E_{\\mathrm{mm}} \\in \\mathbb{R}^{n \\times d}$ , where $n$ is the total number of modality tokens and $d$ is the embedding dimension, self-attention is computed as follows:",
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+ {
+ "type": "equation",
+ "text": "\n$$\n\\operatorname {A t t e n t i o n} \\left(E _ {\\mathrm {m m}}\\right) = \\operatorname {s o f t m a x} \\left(\\frac {Q K ^ {\\top}}{\\sqrt {d _ {k}}}\\right) V\n$$\n",
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+ "type": "text",
+ "text": "where the queries $Q$ , keys $K$ , and values $V$ are linear projections of $E_{\\mathrm{mm}}$ : $Q = E_{\\mathrm{mm}}W_{Q}$ , $K = E_{\\mathrm{mm}}W_{K}$ , $V = E_{\\mathrm{mm}}W_{V}$ with learnable weights of dimensionality $d_{k}$ : $W_{Q}, W_{K}, W_{V} \\in \\mathbb{R}^{d \\times d_{k}}$ .",
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+ "type": "text",
+ "text": "In cross-attention, time series serves as the query modality to get contextualized by other modalities, providing a directed alignment that ensures auxiliary modalities contribute relevant contextual information while preserving the temporal structure of time series. Given a query embedding $E_{\\mathrm{ts}} \\in \\mathbb{R}^{n \\times d}$ and that of auxiliary modalities $E_{c} \\in \\mathbb{R}^{n \\times d}$ as keys and values:",
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+ "type": "equation",
+ "text": "\n$$\n\\text {C r o s s A t t e n t i o n} \\left(E _ {\\mathrm {t s}}, E _ {\\mathrm {c}}\\right) = \\operatorname {s o f t m a x} \\left(\\frac {Q _ {\\mathrm {t s}} K _ {\\mathrm {c}} ^ {\\top}}{\\sqrt {d _ {k}}}\\right) V _ {\\mathrm {c}}\n$$\n",
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+ "type": "text",
+ "text": "where the query, key and value are denoted as $Q_{\\mathrm{ts}} = E_{\\mathrm{ts}}W_{Q}$ , $K_{\\mathrm{c}} = E_{\\mathrm{c}}W_{K}$ , $V_{\\mathrm{c}} = E_{\\mathrm{c}}W_{V}$ . Note that existing methods adopt multi-head attentions, which is omitted here for simplicity.",
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+ "type": "text",
+ "text": "Similarly, the gating mechanism is a parametric filtering operation that explicitly regulates the influence of time series and other modalities on the fused embeddings in $E$ :",
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+ "text": "\n$$\nG = \\sigma \\left(W _ {g} \\left[ E _ {\\mathrm {t s}}; E _ {c} \\right] + b _ {g}\\right), \\quad E = G \\odot E _ {\\mathrm {t s}} + (1 - G) \\odot E _ {c}\n$$\n",
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+ "text": "where $\\sigma (\\cdot)$ denotes the sigmoid function, the learnable weight and bias are denoted as $W_{g}\\in \\mathbb{R}^{2d\\times d}$ and $b_{g}\\in \\mathbb{R}^{d}$ , respectively.",
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+ "text": "When a graph modality is available from external contexts, the underlying topological insights can be leveraged for graph-based alignment [11, 67]. Unlike the above methods that rely solely on feature interactions, it explicitly aligns multi-modal representations with relational structures through graph convolution, enabling context-aware feature propagation across modalities.",
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+ "text": "Representation alignments can also be achieved by learning objectives [3, 45, 83, 96, 97]. For example, MULAN [97] extracts modality-invariant and modality-specific representations from multi-modal time series. It employs contrastive learning to enhance cross-modal alignment by maximizing the similarity between invariant representations across modalities while minimizing the similarity between invariant and specific representations of the same modality. Moreover, Bamford et al. [3] align cross-modal representations by using the mean of uni-modal cosine similarities as the target similarity and optimizing cross-modal similarity via cross-entropy loss, which effectively connects both modalities in a shared latent space for time series retrieval tasks. In general, this branch of methods is effective as it directly integrates the alignment objective into the optimization process, ensuring that meaningful representations are explicitly learned.",
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+ "text": "Lastly, we introduce the intermediate alignment of component outputs within a framework, extending beyond representation alignment within a model. The most recent studies explore the synergy between time series models and LLM agents, leveraging the strong reasoning capabilities of pretrained LLMs to provide",
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+ "text": "contextual understanding and calibration in real-world scenarios [31, 42, 50, 68, 73]. We briefly discuss a few representative examples for demonstration. TimeCAP [42] utilizes the embedding space of a trained multi-modal encoder to retrieve in-context examples with the highest cosine similarity. These retrieved examples with ground truth labels are then fed, along with the query text, into an LLM to provide contextual guidance and improve outcome prediction. TimeXL [31] incorporates a multi-modal prototype-based encoder to generate explainable case-based rationales for both time series and texts, integrating three LLM agents, where prediction, reflection, and refinement LLMs collaborate to iteratively enhance prediction accuracy, identify textual inconsistencies or noise, and calibrate textual contexts, yielding more accurate predictions and explanations. NewsForecast [73] also employs reflection in language agents to iteratively select relevant news from a large database, enhancing alignment of textual information for text-based forecasting. Similarly, MATLAB [68] ensures alignment between statistical causal discovery on time series and LLM reasoning on textual context by leveraging iterative self-reflective tool-calling to structure textual context, which is then used to explain and refine causal constraints.",
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+ "text": "In a nutshell, alignment aims to calibrate real-world contexts and effectively capture relevant multi-modal elements for a semantically coherent time series modeling. It enhances task performance, robustness and explanation, ensuring that models leverage meaningful contextual information for improved decision-making.",
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+ "text": "4.3 Transference",
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+ "text": "Transference refers to the process of mapping between different modalities. It allows one modality to be inferred, translated, or synthesized from another. This concept plays a crucial role across different stages of multi-modal time series analysis. The input-level transference typically serves for modality augmentation. It helps introduce contextual priors, enrich training samples, and provide alternative representations. This is particularly useful in scenarios of data scarcity and imbalance. In existing literature, a common practice is to use meta information to describe the narrative of real-world contexts (e.g., domain, data statistics and granularity, variable descriptions, other co-variates, etc.) [19, 24, 33, 48, 51, 55, 72, 98] or leverage pretrained LLMs to generate fine-grained textual contexts [42] or graphs [11] for real-world time series, serving as an augmented modality. In addition to texts, time series can also be transformed into high-dimensional images via feature imaging, such as stacking the original data with frequency and periodicity features [98]. Alternatively, time series can be represented in tabular form, transforming time series analysis into a table understanding task [72]. Note that the aforementioned uni-modal methods for transforming time series into other single modalities can also be integrated into multi-modal time series frameworks [15, 27, 46, 62, 99, 100]. The exploitation of input-level transference is two-fold. First, the embedding of generated modality can serve as semantic anchors that guides time series modeling via representation alignment, improving downstream supervised tasks [33, 51, 55, 98]. Second, it provides additional contextual guidance for pretrained LLMs through input fusion and prompting [19, 24, 48, 72].",
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+ "text": "Jiang, et al.",
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+ "text": "At the intermediate [11, 50, 68, 75, 97] and output [3, 52] levels, transference are more task-oriented. The output-level transference typically refers to the end-to-end generation of new modalities, such as text-based and image-based time series retrieval, where users provide textual descriptions or sketched trends to query relevant time series data [3]. This also includes EEG-to-text conversion, enabling direct transformation from physiological signals to human-readable narratives [52].",
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+ "text": "The output of intermediate transference typically serves as an initial solution to be refined for modality generation tasks [50, 68, 97] or a medium to be inferred for predictive tasks [75], facilitating downstream reasoning and further alignment within the multimodal framework. MATMCD [68] generates an initial causal graph from time series, achieving modality transference in the intermediate level. Subsequently, it incorporates textual modality to refine the causal graph, ensuring improved alignment and interpretability. Moreover, Wang et al. [75] adopt a two-stage mechanism for sentiment classification based on EEG data, where the model first converts EEG signals into reading texts and then employs a pretrained LLM based on texts for classification, achieving impressive zero-shot results.",
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+ "text": "5 Applications of Multi-modal Time Series Analysis",
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+ "text": "In this section, we review the existing applications of multi-modal time series analysis for both standard and spatial time series, covering diverse domains such as healthcare, finance, transportation, environment, retail, and the Internet of Things (IoT).",
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+ "text": "Standard Time Series",
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+ "text": "5.1 Healthcare",
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+ "text": "Recent studies in healthcare highlight the multi-modal analysis of diverse medical data sources, such as EHRs (Electronic Health Records, containing lab values and clinical reports, etc.), audio, EEG (Electroencephalogram), ECG (Electrocardiogram), and other wearable and medical sensor recordings, for better disease diagnosis and patient monitoring. For multi-modal analysis on EHR data, a common modeling strategy involves the interaction between lab values and clinical reports, including the concatenation [16] and attention mechanisms [60, 85] on modality embeddings. Moreover, existing methods explore different modeling techniques to better exploit the clinical notes, via domain-specific text encoders (e.g., ClinicalBERT [28, 85] and BioBERT [43, 60]) and different processing strategies. For example, text embeddings can be modeled separately based on patient groups [86] or through a decaying mechanism based on time intervals [36] before interacting with time series embeddings, which leads to improved mortality prediction.",
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+ "text": "In addition to EHRs, multi-modal modeling methods have been tailored for audio [95], ECG [45], and EEG [75]. Zhang et al. [95] focus on a respiratory health classification task by integrating both audio and textual descriptions. Li et al. [45] propose a multi-modal contrastive learning framework, constructing positive and negative samples by pairing patients' report texts with corresponding ECG signals for self-supervised pretraining. The classification task is then performed by computing the cosine similarity between different text representations and the target ECG representation. Wang",
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+ "text": "et al. [75] propose a two-stage method for zero-shot EEG-based sentiment classification. First, a pretrained BART model is used for EEG-to-text decoding, followed by a trained text sentiment classifier that converts the generated text into sentiment categories.",
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+ "text": "Similarly, Liu et al. [56] fuses physiological and behavioral time-series sensor data with real-world contextual information to effectively harness LLMs for wellness assessment. By fine-tuning the models with few-shot question-answer pairs that include contextual details, they improve performance on various healthcare tasks—such as cardiac signal analysis, physical activity recognition, and calorie-burn estimation, which outperform both supervised feedforward neural networks and zero-shot LLM baselines.",
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+ "text": "5.2 Finance",
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+ "text": "Recently, multi-modal time series analysis has received increasing attention in financial applications. Yu et al. [89] and Xie et al. [81] focus on stock prediction by integrating stock price movements, company profiles, and news directly into structured LLM prompts, enabling models to perform reasoning over multiple modalities. Yu et al. [89] applies GPT-4 and Open LLaMA to forecast NASDAQ-100 stock returns through instruction-based prompting and fine-tuning, demonstrating that structured LLM-driven inference can outperform traditional econometric models. Meanwhile, Xie et al. [81] conducts a zero-shot analysis of ChatGPT's capabilities for multimodal stock movement prediction, incorporating CoT prompting to assess the impact of social media sentiment on stock trends. Chen et al.[11] and Sawhneyet al.[67] also incorporate graph structures for stock movement prediction. For instance, Chen et al.[11] uses ChatGPT to infer dynamic stock relationship graphs from news, which reflects market conditions and enhances prediction accuracy.",
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+ "text": "Beyond the predictive tasks, Bamford et al. [3] proposes a multimodal retrieval framework, where the model aligns both modalities in a shared latent space through contrastive learning. This framework allows users to search for financial time series through textual descriptions or sketched trends, offering greater flexibility. It also significantly improves retrieval speed and accuracy compared to traditional SQL-based search methods.",
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+ "text": "5.3 Others",
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+ "text": "Multi-modal time series analysis also exists in other domains, such as retail, IoT, computer vision and audio. In the retail sector, Ekambaram et al. [18] utilizes product images and textual descriptions, including attributes like color, pattern, and sleeve style, while incorporating temporal and exogenous features for new product sales forecasting. More recently, Skenderi et al. [70] integrates additional modality data, including product images and text descriptions, along with Google Trends data for sales forecasting. In IoT applications, MIA [82] enhances power transformer fault diagnosis by integrating multi-modal data, including dissolved gas analysis (DGA) and infrared images, to improve accuracy and efficiency. MULAN [97] converts log sequences into time-series data using a log-tailored language model and employs contrastive learning to leverage multi-modal data, facilitating root-cause discovery for system failures. In computer vision, LA-GCN [83] utilizes textual embeddings of joint names and action labels to generate faithful structural priors, enhancing skeleton-based action modeling and",
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+ "text": "Multi-modal Time Series Analysis: A Tutorial and Survey",
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+ "text": "improving recognition tasks. In speech applications, AV-HuBERT [69] employs a self-supervised representation learning framework to leverage correlated audio and visual information [90], while SpeechGPT [69, 90] integrates audio and text to enhance generation performance.",
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+ "text": "Spatial Time Series",
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+ "text": "5.4 Transportation and Mobility",
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+ "text": "Several recent studies on traffic prediction highlight the importance of multi-modal contexts to enhance forecasting accuracy. Guo et al. [24] transforms California traffic data into structured LLM prompts. The method uses LLaMA models and instruction fine-tuning to improve spatial-temporal learning. Meanwhile, Li et al. [48] employs a spatial-temporal dependency encoder to align numerical New York City traffic data with LLMs, incorporating weather, geographic context, and historical flow patterns to refine predictions. Similarly, Feng et al. [19] proposes CityGPT, enhancing LLMs' spatial cognition for urban reasoning, mobility prediction, and navigation by integrating urban mobility data, road networks, and human behavior through instruction tuning. These studies demonstrate that LLM-based multi-modal fusion not only enhances traffic forecasting but also improves model interpretability and adaptability across diverse urban scenarios.",
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+ "text": "5.5 Environment",
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+ "text": "Integrating multi-modal information benefits environmental studies, particularly by addressing the prevalent challenge of missing values. VIMTS [96] utilizes a structured variational approximation technique to impute missing high-dimensional modalities (stream image data) by transforming them into low-dimensional features derived from simpler, related modalities (meteorological time series records), ensuring cross-modal correlations and interpretability of the imputation process. Additionally, LITE [44] addresses the imputation of missing features through a sparse Mixture of Experts framework. It integrates and encodes various environmental variables through a unified encoder. Directed by domain-specific instructions, a language model is utilized to merge these multi-modal representations, thereby improving the accuracy of environmental spatial-temporal predictions.",
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+ "text": "6 Future Research Directions",
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+ "text": "In this section, we outline several underexplored research directions that open up opportunities for future advancements.",
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+ "text": "Reasoning with Multi-modal Time Series. Enhancing reasoning with multi-modal time series is pivotal for the development of intelligent systems. Future research should focus on creating a unified framework that can seamlessly integrate temporal reasoning with contextual understanding, enabling models to handle multiple time series tasks with interpretability. One potential path is to incorporate external knowledge bases and real-world context, such as developing a retrieval-augmented generation (RAG) [59] system, to enhance the reasoning process and allow models to make informed inferences beyond the immediate data. It is also promising to synergize time series model and language agents to provide more faithful and reliable reasoning on real-world contexts [31, 68].",
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+ "text": "The recent development of LLM reasoning models, such as chain of thoughts [76] and tree of thoughts [87], also offers potential solutions to improve reasoning quality.",
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+ "text": "Decision Making. Multi-modal time series analysis presents a promising future direction to enhance decision-making processes, which is crucial in high-stakes applications. By leveraging predictive signals and explanations from multi-modal contexts, future research can develop more adaptive, interpretable, and reliable decision-support systems, to facilitate the downstream optimization tasks such as resource allocation and risk management.",
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+ "text": "Domain Generalization. One key challenge in multi-modal time series analysis is domain generalization, which enables a model trained on one or more source domains to effectively generalize to unseen target domains, ensuring robustness against distribution shifts. In multi-modal time series, distribution shifts can be multifaceted, stemming not only from time series, but also from other modalities. Therefore, it is crucial to develop specialized domain generalization methods for effective multi-modal time series analysis, including strategies to identify and preserve domain-invariant components across modalities while capturing modality-specific variations for rapid adaptation. Additionally, disentangling the effects of each modality is essential to better understand their individual contributions and mitigate cross-modal interference.",
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+ "text": "Robustness to Missing and Noisy Modalities. Multi-modal time series analysis often frequently encounters messy real-world contexts with incomplete or noisy data. Existing methods employ an iterative context-refinement algorithm [31] that filters out less relevant information, thereby enhancing the predictive insights derived from multi-modal time series. Nonetheless, effectively dealing with missing and noisy modalities still demands further exploration. In particular, developing strategies for modality-specific imputation, noise reduction, and relevance quantification will be crucial to improving the real-world applicability of existing multi-modal time series methods.",
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+ "text": "Ethical Considerations and Bias Mitigation. In light of potential biases in multi-modal time series datasets, future research should integrate fairness-aware techniques, such as fairness constraints, counterfactual analysis, and adversarial debiasing. These methods should be combined with robust bias assessment frameworks to systematically detect and mitigate inequities, ensuring outcomes that are both equitable and socially responsible.",
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+ {
+ "type": "text",
+ "text": "7 Conclusion",
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+ "text": "In this survey, we provide a comprehensive overview of existing multi-modal time series methods. We first discuss the multi-modal time series used in existing methods. Then, we propose a taxonomy based on cross-modal interactions between time series and other modalities. The existing methods are categorized and summarized accordingly. We also discuss the real-world applications and highlight future research directions in this promising area.",
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+ "type": "text",
+ "text": "Acknowledgments",
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+ "text": "This research was supported in part by the National Science Foundation (NSF) CAREER IIS-2338878, as well as by generous research gifts from NEC Labs America Inc. and Morgan Stanley.",
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+ "type": "header",
+ "text": "Jiang, et al.",
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+ "type": "text",
+ "text": "References",
+ "text_level": 1,
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+ {
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+ "content": "Yushan Jiang\\(^{1*}\\), Kanghui Ning\\(^{1*}\\), Zijie Pan\\(^{1*}\\), Xuyang Shen\\(^{1}\\), Jingchao Ni\\(^{2}\\), Wenchao Yu\\(^{4}\\), Anderson Schneider\\(^{3}\\), Haifeng Chen\\(^{4\\dagger}\\), Yuriy Nevmyvaka\\(^{3\\dagger}\\), Dongjin Song\\(^{1\\dagger}\\)"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "| Domain | Dataset (Superscripts include the URLs to the datasets) | Modalities |
| Healthcare | MIMIC-III [35][1], MIMIC-IV [34][2] | TS, Text, Tabular |
| ICBHI [65][3], Coswara [4][4], KAUH [21][5], PTB-XL [71][6], ZuCo [14, 26][7] | TS, Text |
| Image-EEG [22][8] | TS, Image |
| Finance | FNSPID [17][9], ACL18 [84][10], CIKM18 [79][11], DOW30 [11][12] | TS, Text |
| Multi-domain | Time-MMD [53][13], TimeCAP [42][14], NewsForecast [73][15], TTC [37][16], CiK [77][17], TSQA [38][18] | TS, Text |
| Retail | VISUELLE [70][19] | TS, Image, Text |
| IoT | LEMMA-RCA [40][20] | TS, Text |
| Speech | LRS3 [1][21], VoxCeleb2 [13][22] | TS (Audio), Image |
| Traffic | NYC-taxi, NYC-bike [48][23] | ST, Text |
| Environment | Terra [10][24] | ST, Text |
"
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+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "form of time series in this survey and briefly discuss representative works within this scope."
+ },
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+ "type": "title",
+ "bbox": [
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+ "content": "3.2 Common Datasets and Benchmarks"
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+ "content": "Multi-modal time series datasets vary a lot and are domain-dependent, each with unique data characteristics and modalities. In Table 1 we provide representative datasets categorized by domain, along with their respective modalities:"
+ },
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+ "content": "Healthcare: In this domain, physiological signals (e.g., ECG, EEG) are extensively analyzed alongside textual data such as clinical notes, patient demographics, and tabular data including vital signs and laboratory results. Common datasets include MIMIC-III [35], a comprehensive dataset containing electronic health records (EHRs) of ICU patients with physiological measurements, clinical notes, and diagnostic information, widely used for tasks like patient monitoring, mortality prediction, and clinical decision support. MIMIC-IV [34] is an extension of MIMIC-III, which provide detailed physiological signals, clinical narratives, medication records, and demographic data from a large population of critically ill patients, frequently utilized for predictive modeling, clinical outcome analysis, and health informatics research. Other notable healthcare datasets include ICBHI [65], which contains respiratory sound recordings paired with clinical annotations for respiratory disease classification; Coswara [4], which provides respiratory audio samples and rich metadata for COVID-19 detection tasks; KAUH [21], which comprises audio records and corresponding annotations for healthcare analytics; PTB-XL [71], a large-scale ECG dataset annotated with diagnostic labels for cardiac monitoring and diagnosis; ZuCo [14, 26], which consists of simultaneous EEG and textual data from reading comprehension tasks, being useful for cognitive neuroscience studies; and Image-EEG [22], which pairs EEG signals with images of objects on a natural background, aiding studies in visual neuroscience and computer vision."
+ },
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+ "content": "Finance: Datasets that combine time series data with financial news and reports are instrumental in financial analysis and modeling. Notable examples include ACL18 [84], CIKM18 [79], and DOW30 [11]. These datasets focus on high-trade-volume stocks from the U.S. stock markets, providing historical stock price data, such as opening, high, low, and closing prices; alongside related textual information, including tweets or financial news. Another"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "large-scale dataset, FNSPID [17], consists of stock prices and time-aligned financial news records, covering over 4,000 companies from 1999 to 2023."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Multi-domain: Datasets featuring general-purpose numerical time series combined with textual data are suitable for broad analytical applications. Examples include Time-MMD [53], which encompasses nine primary data domains: Agriculture, Climate, Economy, Energy, Environment, Health, Security, Social Good, and Traffic, while ensuring fine-grained alignment between time series and textual data; TimeCAP [42] compiles seven real-world time series datasets across three domains: weather, finance, and healthcare. To generate textual descriptions for each time series, a large language model (LLM) agent is employed, leveraging contextual information and domain-specific knowledge. NewsForecast [73] integrates task-specific time series data with verified public news reports across various domains, including finance, energy, traffic, and cryptocurrency; TTC [37] is a meticulously curated, time-aligned dataset designed for multimodal forecasting. It consists of paired time series and text data synchronized to timestamps, spanning two distinct domains: climate science and healthcare; CiK [77] is a dataset comprising 71 forecasting tasks across seven real-world domains. Each task necessitates the integration of both numerical data and textual information. The covered domains include Climatology, Economics, Energy, Mechanics, Public Safety, Transportation, and Retail. The TSQA [38] dataset consists of 200k question-answer pairs derived from time series data across 12 domains: healthcare, finance, energy, traffic, environment, IoT, nature, transport, human activities, machine sensors, AIOps, and the web. These QA pairs are designed to support five key tasks: forecasting, imputation, anomaly detection, classification, and open-ended reasoning."
+ },
+ {
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+ "bbox": [
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+ "angle": 0,
+ "content": "Other domains: Beyond the previously discussed major sectors, multi-modal time series analysis extends to various other domains. In Retail, datasets such as VISUELLE [70] integrate numerical sales data with product images and textual descriptions, facilitating thorough analyses of consumer behavior and inventory management. The Internet of Things (IoT) domain benefits from datasets such as LEMMA-RCA [40], which combine time series sensor data with textual metadata, enabling enhanced monitoring and more robust and secure methodologies that ensure the high performance of modern"
+ }
+ ],
+ [
+ {
+ "type": "header",
+ "bbox": [
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+ ],
+ "angle": 0,
+ "content": "Jiang, et al."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.082,
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+ 0.246
+ ],
+ "angle": 0,
+ "content": "systems. In the Speech domain, datasets like LRS3 [1] and VoxCeleb2 [13] integrate audio recordings with corresponding visual data, supporting advancements in speech recognition and speaker identification technologies. In the Traffic domain, datasets like NYC-Taxi, NYC-Bike [48] contain spatial-temporal (ST) data alongside associated textual metadata. These integrations allow LLMs to effectively capture and utilize spatial-temporal contextual signals. In the Environment domain, Terra [10] collect 45 years of global geographic spatial-temporal data, supplemented with textual descriptions."
+ },
+ {
+ "type": "title",
+ "bbox": [
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+ "angle": 0,
+ "content": "4 Cross-modal Interactions with Time Series"
+ },
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+ "type": "text",
+ "bbox": [
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+ "content": "In this section, we conduct a detailed review of existing research on multi-modal time series analysis by thoroughly analyzing cross-modal interactions. We also elaborate how existing multi-modal methods are tailored for domain-specific applications in Section 5. The detailed taxonomy is provided in Table 2."
+ },
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+ "content": "We define three fundamental types of interactions between time series and other modalities, including fusion, alignment, and transference, which occur at different stages within a framework - input, intermediate (i.e., representations or intermediate outputs), and output. The representative examples are provided in Figure 2."
+ },
+ {
+ "type": "title",
+ "bbox": [
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+ "angle": 0,
+ "content": "4.1 Fusion"
+ },
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+ "angle": 0,
+ "content": "Fusion refers to the process of integrating heterogeneous modalities in a way that captures complementary information across diverse sources to improve time series modeling. To fuse multi-modal inputs, a common practice is to directly integrate time series, tabular data and texts into a unified textual prompt, then use it to query LLMs for downstream tasks. This is typically facilitated by instruction fine-tuning for task-oriented analysis [19, 24, 38, 48, 56, 73, 90]. Some works also leverage the zero-shot reasoning and inference capability of pretrained LLMs (e.g., GPT-4 and its variants) [72, 81, 89]. Recent research efforts like TaTS [47] attempt to integrate paired text embedding as an additional variable of time series for temporal modeling, yielding competitive task performance."
+ },
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+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Most existing methods perform cross-modal fusion at the intermediate stage, such as adding and concatenating multi-modal representations, where each individual modal encoder first maps the raw data into a shared latent space. Addition combines time series and other modalities by summing up encoded representations, effectively blending shared information while preserving their interconnections in the latent space [6, 30, 51, 85, 95, 97]. On the other hand, concatenation stacks multi-modal representations along the same dimension, retaining modality-specific characteristics and allowing models to capture joint relationships between the modalities [16, 36, 36, 37]. To effectively leverage cross-modal information, existing methods often incorporate alignment designs after concatenating representations [5, 11, 12, 18, 31, 33, 41, 42, 44, 54, 55, 60, 69, 70, 74, 85, 96]. Alignment is also used in the aforementioned additions, which will be detailed in Section 4.2."
+ },
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+ "content": "When fusion is performed at the output level, different modalities contribute separately to the final output, allowing each modality to retain its unique predictive signal [31, 41, 42, 53]. Time-MMD [53] provides a paradigm that fuses predictions from both state-of-the-art forecasters and a pretrained language model with a projection"
+ },
+ {
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+ "bbox": [
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+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "image_caption",
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+ "angle": 0,
+ "content": "Figure 2: Categorization of cross-modal interaction methods and representative examples."
+ },
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+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "layer, in an end-to-end manner. MOAT [41] introduces a two-stage framework for multi-modal time series forecasting. In the first stage, the model is optimized to generate forecasts from decomposed time series and text embeddings. In the second stage, an offline synthesis via MLP is applied to dynamically fuse different components, yielding the final forecast based on their relative contributions. Beyond fusing outputs from a single model, TimeCAP [42] enhances performance by combining predictions from both a multi-modal predictor and a pretrained LLM, which synergizes the gradient-based method and LLM agents reasoning on real-world contexts. Output fusion gains advantage of design flexibility and robustness, but it may not fully utilize the complementary relationship between modalities without additional countermeasures."
+ },
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+ "angle": 0,
+ "content": "Cross-modal fusion relies on well-aligned multi-modal data for effective exploitation of the contextual information. However, ideally-aligned data may not be given in real-world scenarios. As such, existing methods also leverage alignment mechanisms to mitigate the challenge."
+ },
+ {
+ "type": "title",
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+ "content": "4.2 Alignment"
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+ "angle": 0,
+ "content": "Alignment ensures that the relationships between different modalities are preserved and semantically coherent when integrated into a unified learning framework. At the input level, we primarily refer alignment to data preprocessing techniques that aim at mitigating temporal misalignment caused by missing values, irregular sampling intervals, and differing granularities across modalities. This process is crucial for ensuring that data from multiple sources are properly synchronized before fusion, where domain knowledge is usually needed to handle such inconsistencies [10, 53, 73]. In addition, none of the existing methods we reviewed explicitly perform output alignment. However, the aforementioned output fusion can be easily adapted to alignment through the incorporation of a gating or attention mechanism that we will introduce shortly."
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+ "content": "Alignment at the intermediate stage plays a crucial role in multimodal interactions. We first introduce the alignment of multi-modal representations, spanning a range of techniques from model component design to learning objectives. The common component designs include self-attention [5, 12, 30, 33, 41, 42, 44, 54, 55, 69], cross-attention [6, 18, 51, 60, 70, 82, 85, 98] and gating mechanisms [82, 98]. Self-attention is often used to fuse multi-modal representations. It"
+ }
+ ],
+ [
+ {
+ "type": "header",
+ "bbox": [
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+ "angle": 0,
+ "content": "Multi-modal Time Series Analysis: A Tutorial and Survey"
+ },
+ {
+ "type": "table_caption",
+ "bbox": [
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+ "angle": 0,
+ "content": "Table 2: Taxonomy of representative multi-modal time series methods. Modality refers to the different data modalities involved in each method. TS represents standard time series, \\(ST\\) denotes spatial time series. The Method column lists the techniques used for each interaction, separated by semicolons, where each interaction may include one or more techniques, separated by commas. Superscripts in the Code column include the URLs to Github repositories."
+ },
+ {
+ "type": "table",
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+ "angle": 0,
+ "content": "| Method | Modality | Domain | Task | Cross-Modal Interaction | Large Model | Year | Code | |
| Stage | Fusion | Align. | Trans. | Method | |
| Time-MMD [53] | TS, Text | General | Forecasting | Output | ✓ | X | X | Addition | Multiple | 2024 | Yes[1] | |
| Wang et al. [73] | TS, Text | General | Forecasting | Input Intermediate | ✓ | X | X | Prompt Prompt; LLM Reasoning | LLaMa2 GPT-4 Turbo | 2024 | Yes[2] | |
| GPT4MTS [30] | TS, Text | General | Forecasting | Intermediate | ✓ | ✓ | X | Addition; Self-attention | GPT-2 | 2024 | No | |
| TimeCMA [51] | TS, Text | General | Forecasting | Input Intermediate | X | X | ✓ | Meta-description Addition; Cross-attention | GPT-2 | 2025 | Yes[3] | |
| MOAT [41] | TS, Text | General | Forecasting | Intermediate Output | ✓ | ✓ | X | Concat.; Self-attention Offline Synthesis (MLP) | S-Bert | 2024 | No | |
| TimeCAP [42] | TS, Text | General | Classification | Input Intermediate Output | X | X | ✓ | LLM Generation Concat.; Self-attention, Retrieval Addition | Bert, GPT-4 | 2024 | No | |
| TimeXL [31] | TS, Text | General | Classification Forecasting | Intermediate Output | ✓ | ✓ | X | Concat., Prompt; LLM Reasoning Addition | Bert, S-Bert GPT-4o | 2025 | No | |
| Hybrid-MMF [37] | TS, Text | General | Forecasting | Intermediate | ✓ | X | X | Concat. | GPT-4o | 2024 | Yes[4] | |
| Time-LLM [33] | TS, Text | General | Forecasting | Input Intermediate | X | X | ✓ | Meta-description Concat.; Self-attention | LLaMA, GPT-2 | 2024 | Yes[5] | |
| Time-VLM [98] | TS, Text, Image | General | Forecasting | Input Intermediate | X | X | ✓ | Feat. Imaging, Meta-description Addition; Gating, Cross-attention | ViLT, CLIP BLIP-2 | 2025 | No | |
| Unitime [55] | TS, Text | General | Forecasting | Input Intermediate | X | X | ✓ | Meta-description Concat.; Self-attention | GPT-2 | 2024 | Yes[6] | |
| TESSA [50] | TS, Text | General | Annotation | Intermediate | ✓ | ✓ | ✓ | Prompt; RL; LLM Generation | GPT-4o | 2024 | No | |
| InstruTime [12] | TS, Text | General | Classification | Intermediate | ✓ | ✓ | X | Concat.; Self-attention | GPT-2 | 2025 | Yes[7] | |
| MATMCD [68] | TS, Text, Graph | General | Causal Discovery | Intermediate | ✓ | ✓ | ✓ | Prompt; LLM Reasoning; Supervision | Multiple | 2025 | No | |
| STG-LLM [54] | ST, Text | General | Forecasting | Intermediate | ✓ | ✓ | X | Concat.; Self-attention | GPT-2 | 2024 | No | |
| TableTime [72] | TS, Text | General | Classification | Input | ✓ | X | ✓ | Prompt; Reformulate | Multiple | 2024 | Yes[8] | |
| ContextFormer [6] | TS, Tabular | General | Forecasting | Intermediate | ✓ | ✓ | X | Addition; Cross-attention | No | 2025 | No | |
| Time-MQA [38] | TS, Text | General | Multiple | Input | ✓ | X | X | Prompt | Multiple | 2025 | Yes[9] | |
| MAN-SF [67] | TS, Text, Graph | Finance | Classification | Intermediate | ✓ | ✓ | X | Bilinear; Graph Convolution | USE | 2020 | No | |
| Bamford et al. [3] | TS, Text, TS, Image | Finance | Retrieval | Intermediate Output | X | ✓ | X | Supervision | S-Bert | 2024 | No | |
| Chen et al. [11] | TS, Text, Graph | Finance | Classification | Intermediate | X | X | ✓ | LLM Generation Concat.; Graph Convolution | ChatGPT | 2023 | No | |
| Xie et al. [81] | TS, Text | Finance | Classification | Input | ✓ | X | X | Prompt | ChatGPT | 2023 | No | |
| Yu et al. [89] | TS, Text | Finance | Forecasting | Input | ✓ | X | X | Prompt | GPT-4, Open LLaMA | 2023 | No | |
| MedTsLLM [5] | TS, Text, Tabular | Healthcare | Multiple | Intermediate | ✓ | ✓ | X | Concat.; Self-attention | Llama2 | 2024 | Yes[10] | |
| RespLLM [95] | TS (Audio), Text | Healthcare | Classification | Intermediate | ✓ | ✓ | X | Addition, Self-attention | OpenBioLLM-8B | 2024 | No | |
| METS [45] | TS, Text | Healthcare | Classification | Output | X | ✓ | X | Contrastive | ClinicalBert | 2023 | No | |
| Wang et al. [75] | TS, Text | Healthcare | Classification | Intermediate | X | X | ✓ | Supervision | Bart, Bert, Roberta | 2021 | No | |
| EEG2TEXT [52] | TS, Text | Healthcare | Generation | Output | X | X | ✓ | Self-supervision, Supervision | Bart | 2024 | No | |
| MEDHMP [74] | TS, Text | Healthcare | Classification | Intermediate | ✓ | ✓ | X | Concat.; Self-attention, Contrastive | ClinicalT5 | 2023 | Yes[11] | |
| Deznabi et al. [16] | TS, Text | Healthcare | Classification | Intermediate | ✓ | X | X | Concat. | Bio+Clinical Bert | 2021 | Yes[12] | |
| Niu et al. [60] | TS, Text | Healthcare | Classification | Intermediate | ✓ | ✓ | X | Concat.; Cross-attention | BioBERT | 2023 | No | |
| Yang et al. [85] | TS, Text | Healthcare | Classification | Intermediate | ✓ | ✓ | X | Concat.; Addition; Gating | ClinicalBERT | 2021 | Yes[13] | |
| Liu et al. [56] | TS, Text | Healthcare | Classification Regression | Input | ✓ | X | X | Prompt | PaLM | 2023 | Yes[14] | |
| xTP-LLM [24] | ST, Text | Traffic Forecasting | Input | ✓ | X | ✓ | Prompt; Meta-description | Llama2-7B-chat | 2024 | Yes[15] | |
| UrbanGPT [48] | ST, Text | Traffic Forecasting | Input | ✓ | X | ✓ | Prompt; Meta-description | Vicuna-7B | 2024 | Yes[16] | |
| CityGPT [19] | ST, Text | Mobility | Input | ✓ | X | X | Prompt | Multiple | 2025 | Yes[17] | |
| MULAN [97] | TS, Text, Graph | IoT Causal Discovery | Intermediate | ✓ | ✓ | ✓ | Addition; Contrastive; Supervision | No | 2024 | No | |
| MIA [82] | TS, Image | IoT Anomaly Detection | Intermediate | ✓ | ✓ | X | Addition; Cross-attention, Gating | No | 2023 | No | |
| Ekambaram et al. [18] | TS, Image, Text | Retail Forecasting | Intermediate | ✓ | ✓ | X | Concat.; Self & Cross-attention | No | 2020 | Yes[18] | |
| Skenderi et al. [70] | TS, Image, Text | Retail Forecasting | Intermediate | ✓ | ✓ | X | Concat.; Cross-attention | No | 2024 | Yes[19] | |
| VIMTS [96] | ST, Image Environment | Imputation | Intermediate | ✓ | ✓ | X | Concat.; Supervision | No | 2022 | No |
| LTE [44] | ST, Text, Image Environment | Forecasting | Intermediate | ✓ | ✓ | X | Concat.; Self-attention | LLaMA-2-7B | 2024 | Yes[20] |
| AV-HubERT [69] | TS (Audio), Image Speech | Classification | Intermediate | ✓ | ✓ | X | Concat.; Self-attention | HuBert | 2022 | Yes[21] |
| SpeechGPT [90] | TS(Audio), Text | Generation intermediate | ✓ | ✓ | X | Concat.; Self-attention | LLaMA-13B | 2023 | Yes[22] | |
| LA-GCN [83] | ST, Text | Classification | Intermediate | X | ✓ | Supervision | Bert | 2023 | Yes[23] | |
"
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+ [
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+ "angle": 0,
+ "content": "Jiang, et al."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "enables a joint and undirected alignment across all modalities by dynamically attending to important features. Given multi-modal embeddings \\( E_{\\mathrm{mm}} \\in \\mathbb{R}^{n \\times d} \\), where \\( n \\) is the total number of modality tokens and \\( d \\) is the embedding dimension, self-attention is computed as follows:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
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+ "angle": 0,
+ "content": "\\[\n\\operatorname {A t t e n t i o n} \\left(E _ {\\mathrm {m m}}\\right) = \\operatorname {s o f t m a x} \\left(\\frac {Q K ^ {\\top}}{\\sqrt {d _ {k}}}\\right) V\n\\]"
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+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "where the queries \\(Q\\), keys \\(K\\), and values \\(V\\) are linear projections of \\(E_{\\mathrm{mm}}\\): \\(Q = E_{\\mathrm{mm}}W_{Q}\\), \\(K = E_{\\mathrm{mm}}W_{K}\\), \\(V = E_{\\mathrm{mm}}W_{V}\\) with learnable weights of dimensionality \\(d_{k}\\): \\(W_{Q}, W_{K}, W_{V} \\in \\mathbb{R}^{d \\times d_{k}}\\)."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "In cross-attention, time series serves as the query modality to get contextualized by other modalities, providing a directed alignment that ensures auxiliary modalities contribute relevant contextual information while preserving the temporal structure of time series. Given a query embedding \\( E_{\\mathrm{ts}} \\in \\mathbb{R}^{n \\times d} \\) and that of auxiliary modalities \\( E_{c} \\in \\mathbb{R}^{n \\times d} \\) as keys and values:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
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+ "angle": 0,
+ "content": "\\[\n\\text {C r o s s A t t e n t i o n} \\left(E _ {\\mathrm {t s}}, E _ {\\mathrm {c}}\\right) = \\operatorname {s o f t m a x} \\left(\\frac {Q _ {\\mathrm {t s}} K _ {\\mathrm {c}} ^ {\\top}}{\\sqrt {d _ {k}}}\\right) V _ {\\mathrm {c}}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "where the query, key and value are denoted as \\( Q_{\\mathrm{ts}} = E_{\\mathrm{ts}}W_{Q} \\), \\( K_{\\mathrm{c}} = E_{\\mathrm{c}}W_{K} \\), \\( V_{\\mathrm{c}} = E_{\\mathrm{c}}W_{V} \\). Note that existing methods adopt multi-head attentions, which is omitted here for simplicity."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Similarly, the gating mechanism is a parametric filtering operation that explicitly regulates the influence of time series and other modalities on the fused embeddings in \\( E \\):"
+ },
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+ "angle": 0,
+ "content": "\\[\nG = \\sigma \\left(W _ {g} \\left[ E _ {\\mathrm {t s}}; E _ {c} \\right] + b _ {g}\\right), \\quad E = G \\odot E _ {\\mathrm {t s}} + (1 - G) \\odot E _ {c}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.082,
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+ "angle": 0,
+ "content": "where \\(\\sigma (\\cdot)\\) denotes the sigmoid function, the learnable weight and bias are denoted as \\(W_{g}\\in \\mathbb{R}^{2d\\times d}\\) and \\(b_{g}\\in \\mathbb{R}^{d}\\), respectively."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.082,
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+ "angle": 0,
+ "content": "When a graph modality is available from external contexts, the underlying topological insights can be leveraged for graph-based alignment [11, 67]. Unlike the above methods that rely solely on feature interactions, it explicitly aligns multi-modal representations with relational structures through graph convolution, enabling context-aware feature propagation across modalities."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Representation alignments can also be achieved by learning objectives [3, 45, 83, 96, 97]. For example, MULAN [97] extracts modality-invariant and modality-specific representations from multi-modal time series. It employs contrastive learning to enhance cross-modal alignment by maximizing the similarity between invariant representations across modalities while minimizing the similarity between invariant and specific representations of the same modality. Moreover, Bamford et al. [3] align cross-modal representations by using the mean of uni-modal cosine similarities as the target similarity and optimizing cross-modal similarity via cross-entropy loss, which effectively connects both modalities in a shared latent space for time series retrieval tasks. In general, this branch of methods is effective as it directly integrates the alignment objective into the optimization process, ensuring that meaningful representations are explicitly learned."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Lastly, we introduce the intermediate alignment of component outputs within a framework, extending beyond representation alignment within a model. The most recent studies explore the synergy between time series models and LLM agents, leveraging the strong reasoning capabilities of pretrained LLMs to provide"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "contextual understanding and calibration in real-world scenarios [31, 42, 50, 68, 73]. We briefly discuss a few representative examples for demonstration. TimeCAP [42] utilizes the embedding space of a trained multi-modal encoder to retrieve in-context examples with the highest cosine similarity. These retrieved examples with ground truth labels are then fed, along with the query text, into an LLM to provide contextual guidance and improve outcome prediction. TimeXL [31] incorporates a multi-modal prototype-based encoder to generate explainable case-based rationales for both time series and texts, integrating three LLM agents, where prediction, reflection, and refinement LLMs collaborate to iteratively enhance prediction accuracy, identify textual inconsistencies or noise, and calibrate textual contexts, yielding more accurate predictions and explanations. NewsForecast [73] also employs reflection in language agents to iteratively select relevant news from a large database, enhancing alignment of textual information for text-based forecasting. Similarly, MATLAB [68] ensures alignment between statistical causal discovery on time series and LLM reasoning on textual context by leveraging iterative self-reflective tool-calling to structure textual context, which is then used to explain and refine causal constraints."
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+ "angle": 0,
+ "content": "In a nutshell, alignment aims to calibrate real-world contexts and effectively capture relevant multi-modal elements for a semantically coherent time series modeling. It enhances task performance, robustness and explanation, ensuring that models leverage meaningful contextual information for improved decision-making."
+ },
+ {
+ "type": "title",
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+ "content": "4.3 Transference"
+ },
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+ "type": "text",
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+ "content": "Transference refers to the process of mapping between different modalities. It allows one modality to be inferred, translated, or synthesized from another. This concept plays a crucial role across different stages of multi-modal time series analysis. The input-level transference typically serves for modality augmentation. It helps introduce contextual priors, enrich training samples, and provide alternative representations. This is particularly useful in scenarios of data scarcity and imbalance. In existing literature, a common practice is to use meta information to describe the narrative of real-world contexts (e.g., domain, data statistics and granularity, variable descriptions, other co-variates, etc.) [19, 24, 33, 48, 51, 55, 72, 98] or leverage pretrained LLMs to generate fine-grained textual contexts [42] or graphs [11] for real-world time series, serving as an augmented modality. In addition to texts, time series can also be transformed into high-dimensional images via feature imaging, such as stacking the original data with frequency and periodicity features [98]. Alternatively, time series can be represented in tabular form, transforming time series analysis into a table understanding task [72]. Note that the aforementioned uni-modal methods for transforming time series into other single modalities can also be integrated into multi-modal time series frameworks [15, 27, 46, 62, 99, 100]. The exploitation of input-level transference is two-fold. First, the embedding of generated modality can serve as semantic anchors that guides time series modeling via representation alignment, improving downstream supervised tasks [33, 51, 55, 98]. Second, it provides additional contextual guidance for pretrained LLMs through input fusion and prompting [19, 24, 48, 72]."
+ }
+ ],
+ [
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+ "type": "header",
+ "bbox": [
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+ "angle": 0,
+ "content": "Multi-modal Time Series Analysis: A Tutorial and Survey"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "At the intermediate [11, 50, 68, 75, 97] and output [3, 52] levels, transference are more task-oriented. The output-level transference typically refers to the end-to-end generation of new modalities, such as text-based and image-based time series retrieval, where users provide textual descriptions or sketched trends to query relevant time series data [3]. This also includes EEG-to-text conversion, enabling direct transformation from physiological signals to human-readable narratives [52]."
+ },
+ {
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+ "bbox": [
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+ "angle": 0,
+ "content": "The output of intermediate transference typically serves as an initial solution to be refined for modality generation tasks [50, 68, 97] or a medium to be inferred for predictive tasks [75], facilitating downstream reasoning and further alignment within the multimodal framework. MATMCD [68] generates an initial causal graph from time series, achieving modality transference in the intermediate level. Subsequently, it incorporates textual modality to refine the causal graph, ensuring improved alignment and interpretability. Moreover, Wang et al. [75] adopt a two-stage mechanism for sentiment classification based on EEG data, where the model first converts EEG signals into reading texts and then employs a pretrained LLM based on texts for classification, achieving impressive zero-shot results."
+ },
+ {
+ "type": "title",
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+ "angle": 0,
+ "content": "5 Applications of Multi-modal Time Series Analysis"
+ },
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+ "content": "In this section, we review the existing applications of multi-modal time series analysis for both standard and spatial time series, covering diverse domains such as healthcare, finance, transportation, environment, retail, and the Internet of Things (IoT)."
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+ "content": "Standard Time Series"
+ },
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+ "angle": 0,
+ "content": "5.1 Healthcare"
+ },
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+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Recent studies in healthcare highlight the multi-modal analysis of diverse medical data sources, such as EHRs (Electronic Health Records, containing lab values and clinical reports, etc.), audio, EEG (Electroencephalogram), ECG (Electrocardiogram), and other wearable and medical sensor recordings, for better disease diagnosis and patient monitoring. For multi-modal analysis on EHR data, a common modeling strategy involves the interaction between lab values and clinical reports, including the concatenation [16] and attention mechanisms [60, 85] on modality embeddings. Moreover, existing methods explore different modeling techniques to better exploit the clinical notes, via domain-specific text encoders (e.g., ClinicalBERT [28, 85] and BioBERT [43, 60]) and different processing strategies. For example, text embeddings can be modeled separately based on patient groups [86] or through a decaying mechanism based on time intervals [36] before interacting with time series embeddings, which leads to improved mortality prediction."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "In addition to EHRs, multi-modal modeling methods have been tailored for audio [95], ECG [45], and EEG [75]. Zhang et al. [95] focus on a respiratory health classification task by integrating both audio and textual descriptions. Li et al. [45] propose a multi-modal contrastive learning framework, constructing positive and negative samples by pairing patients' report texts with corresponding ECG signals for self-supervised pretraining. The classification task is then performed by computing the cosine similarity between different text representations and the target ECG representation. Wang"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "et al. [75] propose a two-stage method for zero-shot EEG-based sentiment classification. First, a pretrained BART model is used for EEG-to-text decoding, followed by a trained text sentiment classifier that converts the generated text into sentiment categories."
+ },
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+ "angle": 0,
+ "content": "Similarly, Liu et al. [56] fuses physiological and behavioral time-series sensor data with real-world contextual information to effectively harness LLMs for wellness assessment. By fine-tuning the models with few-shot question-answer pairs that include contextual details, they improve performance on various healthcare tasks—such as cardiac signal analysis, physical activity recognition, and calorie-burn estimation, which outperform both supervised feedforward neural networks and zero-shot LLM baselines."
+ },
+ {
+ "type": "title",
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+ "angle": 0,
+ "content": "5.2 Finance"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Recently, multi-modal time series analysis has received increasing attention in financial applications. Yu et al. [89] and Xie et al. [81] focus on stock prediction by integrating stock price movements, company profiles, and news directly into structured LLM prompts, enabling models to perform reasoning over multiple modalities. Yu et al. [89] applies GPT-4 and Open LLaMA to forecast NASDAQ-100 stock returns through instruction-based prompting and fine-tuning, demonstrating that structured LLM-driven inference can outperform traditional econometric models. Meanwhile, Xie et al. [81] conducts a zero-shot analysis of ChatGPT's capabilities for multimodal stock movement prediction, incorporating CoT prompting to assess the impact of social media sentiment on stock trends. Chen et al.[11] and Sawhneyet al.[67] also incorporate graph structures for stock movement prediction. For instance, Chen et al.[11] uses ChatGPT to infer dynamic stock relationship graphs from news, which reflects market conditions and enhances prediction accuracy."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Beyond the predictive tasks, Bamford et al. [3] proposes a multimodal retrieval framework, where the model aligns both modalities in a shared latent space through contrastive learning. This framework allows users to search for financial time series through textual descriptions or sketched trends, offering greater flexibility. It also significantly improves retrieval speed and accuracy compared to traditional SQL-based search methods."
+ },
+ {
+ "type": "title",
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+ "angle": 0,
+ "content": "5.3 Others"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Multi-modal time series analysis also exists in other domains, such as retail, IoT, computer vision and audio. In the retail sector, Ekambaram et al. [18] utilizes product images and textual descriptions, including attributes like color, pattern, and sleeve style, while incorporating temporal and exogenous features for new product sales forecasting. More recently, Skenderi et al. [70] integrates additional modality data, including product images and text descriptions, along with Google Trends data for sales forecasting. In IoT applications, MIA [82] enhances power transformer fault diagnosis by integrating multi-modal data, including dissolved gas analysis (DGA) and infrared images, to improve accuracy and efficiency. MULAN [97] converts log sequences into time-series data using a log-tailored language model and employs contrastive learning to leverage multi-modal data, facilitating root-cause discovery for system failures. In computer vision, LA-GCN [83] utilizes textual embeddings of joint names and action labels to generate faithful structural priors, enhancing skeleton-based action modeling and"
+ }
+ ],
+ [
+ {
+ "type": "header",
+ "bbox": [
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+ "angle": 0,
+ "content": "Jiang, et al."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.086,
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+ ],
+ "angle": 0,
+ "content": "improving recognition tasks. In speech applications, AV-HuBERT [69] employs a self-supervised representation learning framework to leverage correlated audio and visual information [90], while SpeechGPT [69, 90] integrates audio and text to enhance generation performance."
+ },
+ {
+ "type": "title",
+ "bbox": [
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+ "angle": 0,
+ "content": "Spatial Time Series"
+ },
+ {
+ "type": "title",
+ "bbox": [
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+ "angle": 0,
+ "content": "5.4 Transportation and Mobility"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Several recent studies on traffic prediction highlight the importance of multi-modal contexts to enhance forecasting accuracy. Guo et al. [24] transforms California traffic data into structured LLM prompts. The method uses LLaMA models and instruction fine-tuning to improve spatial-temporal learning. Meanwhile, Li et al. [48] employs a spatial-temporal dependency encoder to align numerical New York City traffic data with LLMs, incorporating weather, geographic context, and historical flow patterns to refine predictions. Similarly, Feng et al. [19] proposes CityGPT, enhancing LLMs' spatial cognition for urban reasoning, mobility prediction, and navigation by integrating urban mobility data, road networks, and human behavior through instruction tuning. These studies demonstrate that LLM-based multi-modal fusion not only enhances traffic forecasting but also improves model interpretability and adaptability across diverse urban scenarios."
+ },
+ {
+ "type": "title",
+ "bbox": [
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+ "angle": 0,
+ "content": "5.5 Environment"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Integrating multi-modal information benefits environmental studies, particularly by addressing the prevalent challenge of missing values. VIMTS [96] utilizes a structured variational approximation technique to impute missing high-dimensional modalities (stream image data) by transforming them into low-dimensional features derived from simpler, related modalities (meteorological time series records), ensuring cross-modal correlations and interpretability of the imputation process. Additionally, LITE [44] addresses the imputation of missing features through a sparse Mixture of Experts framework. It integrates and encodes various environmental variables through a unified encoder. Directed by domain-specific instructions, a language model is utilized to merge these multi-modal representations, thereby improving the accuracy of environmental spatial-temporal predictions."
+ },
+ {
+ "type": "title",
+ "bbox": [
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+ "angle": 0,
+ "content": "6 Future Research Directions"
+ },
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+ "angle": 0,
+ "content": "In this section, we outline several underexplored research directions that open up opportunities for future advancements."
+ },
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+ "angle": 0,
+ "content": "Reasoning with Multi-modal Time Series. Enhancing reasoning with multi-modal time series is pivotal for the development of intelligent systems. Future research should focus on creating a unified framework that can seamlessly integrate temporal reasoning with contextual understanding, enabling models to handle multiple time series tasks with interpretability. One potential path is to incorporate external knowledge bases and real-world context, such as developing a retrieval-augmented generation (RAG) [59] system, to enhance the reasoning process and allow models to make informed inferences beyond the immediate data. It is also promising to synergize time series model and language agents to provide more faithful and reliable reasoning on real-world contexts [31, 68]."
+ },
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+ "angle": 0,
+ "content": "The recent development of LLM reasoning models, such as chain of thoughts [76] and tree of thoughts [87], also offers potential solutions to improve reasoning quality."
+ },
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+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Decision Making. Multi-modal time series analysis presents a promising future direction to enhance decision-making processes, which is crucial in high-stakes applications. By leveraging predictive signals and explanations from multi-modal contexts, future research can develop more adaptive, interpretable, and reliable decision-support systems, to facilitate the downstream optimization tasks such as resource allocation and risk management."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Domain Generalization. One key challenge in multi-modal time series analysis is domain generalization, which enables a model trained on one or more source domains to effectively generalize to unseen target domains, ensuring robustness against distribution shifts. In multi-modal time series, distribution shifts can be multifaceted, stemming not only from time series, but also from other modalities. Therefore, it is crucial to develop specialized domain generalization methods for effective multi-modal time series analysis, including strategies to identify and preserve domain-invariant components across modalities while capturing modality-specific variations for rapid adaptation. Additionally, disentangling the effects of each modality is essential to better understand their individual contributions and mitigate cross-modal interference."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ ],
+ "angle": 0,
+ "content": "Robustness to Missing and Noisy Modalities. Multi-modal time series analysis often frequently encounters messy real-world contexts with incomplete or noisy data. Existing methods employ an iterative context-refinement algorithm [31] that filters out less relevant information, thereby enhancing the predictive insights derived from multi-modal time series. Nonetheless, effectively dealing with missing and noisy modalities still demands further exploration. In particular, developing strategies for modality-specific imputation, noise reduction, and relevance quantification will be crucial to improving the real-world applicability of existing multi-modal time series methods."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Ethical Considerations and Bias Mitigation. In light of potential biases in multi-modal time series datasets, future research should integrate fairness-aware techniques, such as fairness constraints, counterfactual analysis, and adversarial debiasing. These methods should be combined with robust bias assessment frameworks to systematically detect and mitigate inequities, ensuring outcomes that are both equitable and socially responsible."
+ },
+ {
+ "type": "title",
+ "bbox": [
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+ "angle": 0,
+ "content": "7 Conclusion"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "In this survey, we provide a comprehensive overview of existing multi-modal time series methods. We first discuss the multi-modal time series used in existing methods. Then, we propose a taxonomy based on cross-modal interactions between time series and other modalities. The existing methods are categorized and summarized accordingly. We also discuss the real-world applications and highlight future research directions in this promising area."
+ },
+ {
+ "type": "title",
+ "bbox": [
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+ "angle": 0,
+ "content": "Acknowledgments"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.518,
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+ "angle": 0,
+ "content": "This research was supported in part by the National Science Foundation (NSF) CAREER IIS-2338878, as well as by generous research gifts from NEC Labs America Inc. and Morgan Stanley."
+ }
+ ],
+ [
+ {
+ "type": "header",
+ "bbox": [
+ 0.085,
+ 0.076,
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+ ],
+ "angle": 0,
+ "content": "Multi-modal Time Series Analysis: A Tutorial and Survey"
+ },
+ {
+ "type": "title",
+ "bbox": [
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+ "angle": 0,
+ "content": "References"
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+ {
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+ ],
+ "angle": 0,
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+# Multi-modal Time Series Analysis: A Tutorial and Survey
+
+Yushan Jiang $^{1*}$ , Kanghui Ning $^{1*}$ , Zijie Pan $^{1*}$ , Xuyang Shen $^{1}$ , Jingchao Ni $^{2}$ , Wenchao Yu $^{4}$ , Anderson Schneider $^{3}$ , Haifeng Chen $^{4\dagger}$ , Yuriy Nevmyvaka $^{3\dagger}$ , Dongjin Song $^{1\dagger}$
+
+| Domain | Dataset (Superscripts include the URLs to the datasets) | Modalities |
| Healthcare | MIMIC-III [35][1], MIMIC-IV [34][2] | TS, Text, Tabular |
| ICBHI [65][3], Coswara [4][4], KAUH [21][5], PTB-XL [71][6], ZuCo [14, 26][7] | TS, Text |
| Image-EEG [22][8] | TS, Image |
| Finance | FNSPID [17][9], ACL18 [84][10], CIKM18 [79][11], DOW30 [11][12] | TS, Text |
| Multi-domain | Time-MMD [53][13], TimeCAP [42][14], NewsForecast [73][15], TTC [37][16], CiK [77][17], TSQA [38][18] | TS, Text |
| Retail | VISUELLE [70][19] | TS, Image, Text |
| IoT | LEMMA-RCA [40][20] | TS, Text |
| Speech | LRS3 [1][21], VoxCeleb2 [13][22] | TS (Audio), Image |
| Traffic | NYC-taxi, NYC-bike [48][23] | ST, Text |
| Environment | Terra [10][24] | ST, Text |
+
+form of time series in this survey and briefly discuss representative works within this scope.
+
+# 3.2 Common Datasets and Benchmarks
+
+Multi-modal time series datasets vary a lot and are domain-dependent, each with unique data characteristics and modalities. In Table 1 we provide representative datasets categorized by domain, along with their respective modalities:
+
+Healthcare: In this domain, physiological signals (e.g., ECG, EEG) are extensively analyzed alongside textual data such as clinical notes, patient demographics, and tabular data including vital signs and laboratory results. Common datasets include MIMIC-III [35], a comprehensive dataset containing electronic health records (EHRs) of ICU patients with physiological measurements, clinical notes, and diagnostic information, widely used for tasks like patient monitoring, mortality prediction, and clinical decision support. MIMIC-IV [34] is an extension of MIMIC-III, which provide detailed physiological signals, clinical narratives, medication records, and demographic data from a large population of critically ill patients, frequently utilized for predictive modeling, clinical outcome analysis, and health informatics research. Other notable healthcare datasets include ICBHI [65], which contains respiratory sound recordings paired with clinical annotations for respiratory disease classification; Coswara [4], which provides respiratory audio samples and rich metadata for COVID-19 detection tasks; KAUH [21], which comprises audio records and corresponding annotations for healthcare analytics; PTB-XL [71], a large-scale ECG dataset annotated with diagnostic labels for cardiac monitoring and diagnosis; ZuCo [14, 26], which consists of simultaneous EEG and textual data from reading comprehension tasks, being useful for cognitive neuroscience studies; and Image-EEG [22], which pairs EEG signals with images of objects on a natural background, aiding studies in visual neuroscience and computer vision.
+
+Finance: Datasets that combine time series data with financial news and reports are instrumental in financial analysis and modeling. Notable examples include ACL18 [84], CIKM18 [79], and DOW30 [11]. These datasets focus on high-trade-volume stocks from the U.S. stock markets, providing historical stock price data, such as opening, high, low, and closing prices; alongside related textual information, including tweets or financial news. Another
+
+large-scale dataset, FNSPID [17], consists of stock prices and time-aligned financial news records, covering over 4,000 companies from 1999 to 2023.
+
+Multi-domain: Datasets featuring general-purpose numerical time series combined with textual data are suitable for broad analytical applications. Examples include Time-MMD [53], which encompasses nine primary data domains: Agriculture, Climate, Economy, Energy, Environment, Health, Security, Social Good, and Traffic, while ensuring fine-grained alignment between time series and textual data; TimeCAP [42] compiles seven real-world time series datasets across three domains: weather, finance, and healthcare. To generate textual descriptions for each time series, a large language model (LLM) agent is employed, leveraging contextual information and domain-specific knowledge. NewsForecast [73] integrates task-specific time series data with verified public news reports across various domains, including finance, energy, traffic, and cryptocurrency; TTC [37] is a meticulously curated, time-aligned dataset designed for multimodal forecasting. It consists of paired time series and text data synchronized to timestamps, spanning two distinct domains: climate science and healthcare; CiK [77] is a dataset comprising 71 forecasting tasks across seven real-world domains. Each task necessitates the integration of both numerical data and textual information. The covered domains include Climatology, Economics, Energy, Mechanics, Public Safety, Transportation, and Retail. The TSQA [38] dataset consists of 200k question-answer pairs derived from time series data across 12 domains: healthcare, finance, energy, traffic, environment, IoT, nature, transport, human activities, machine sensors, AIOps, and the web. These QA pairs are designed to support five key tasks: forecasting, imputation, anomaly detection, classification, and open-ended reasoning.
+
+Other domains: Beyond the previously discussed major sectors, multi-modal time series analysis extends to various other domains. In Retail, datasets such as VISUELLE [70] integrate numerical sales data with product images and textual descriptions, facilitating thorough analyses of consumer behavior and inventory management. The Internet of Things (IoT) domain benefits from datasets such as LEMMA-RCA [40], which combine time series sensor data with textual metadata, enabling enhanced monitoring and more robust and secure methodologies that ensure the high performance of modern
+
+systems. In the Speech domain, datasets like LRS3 [1] and VoxCeleb2 [13] integrate audio recordings with corresponding visual data, supporting advancements in speech recognition and speaker identification technologies. In the Traffic domain, datasets like NYC-Taxi, NYC-Bike [48] contain spatial-temporal (ST) data alongside associated textual metadata. These integrations allow LLMs to effectively capture and utilize spatial-temporal contextual signals. In the Environment domain, Terra [10] collect 45 years of global geographic spatial-temporal data, supplemented with textual descriptions.
+
+# 4 Cross-modal Interactions with Time Series
+
+In this section, we conduct a detailed review of existing research on multi-modal time series analysis by thoroughly analyzing cross-modal interactions. We also elaborate how existing multi-modal methods are tailored for domain-specific applications in Section 5. The detailed taxonomy is provided in Table 2.
+
+We define three fundamental types of interactions between time series and other modalities, including fusion, alignment, and transference, which occur at different stages within a framework - input, intermediate (i.e., representations or intermediate outputs), and output. The representative examples are provided in Figure 2.
+
+# 4.1 Fusion
+
+Fusion refers to the process of integrating heterogeneous modalities in a way that captures complementary information across diverse sources to improve time series modeling. To fuse multi-modal inputs, a common practice is to directly integrate time series, tabular data and texts into a unified textual prompt, then use it to query LLMs for downstream tasks. This is typically facilitated by instruction fine-tuning for task-oriented analysis [19, 24, 38, 48, 56, 73, 90]. Some works also leverage the zero-shot reasoning and inference capability of pretrained LLMs (e.g., GPT-4 and its variants) [72, 81, 89]. Recent research efforts like TaTS [47] attempt to integrate paired text embedding as an additional variable of time series for temporal modeling, yielding competitive task performance.
+
+Most existing methods perform cross-modal fusion at the intermediate stage, such as adding and concatenating multi-modal representations, where each individual modal encoder first maps the raw data into a shared latent space. Addition combines time series and other modalities by summing up encoded representations, effectively blending shared information while preserving their interconnections in the latent space [6, 30, 51, 85, 95, 97]. On the other hand, concatenation stacks multi-modal representations along the same dimension, retaining modality-specific characteristics and allowing models to capture joint relationships between the modalities [16, 36, 36, 37]. To effectively leverage cross-modal information, existing methods often incorporate alignment designs after concatenating representations [5, 11, 12, 18, 31, 33, 41, 42, 44, 54, 55, 60, 69, 70, 74, 85, 96]. Alignment is also used in the aforementioned additions, which will be detailed in Section 4.2.
+
+When fusion is performed at the output level, different modalities contribute separately to the final output, allowing each modality to retain its unique predictive signal [31, 41, 42, 53]. Time-MMD [53] provides a paradigm that fuses predictions from both state-of-the-art forecasters and a pretrained language model with a projection
+
+
+Figure 2: Categorization of cross-modal interaction methods and representative examples.
+
+layer, in an end-to-end manner. MOAT [41] introduces a two-stage framework for multi-modal time series forecasting. In the first stage, the model is optimized to generate forecasts from decomposed time series and text embeddings. In the second stage, an offline synthesis via MLP is applied to dynamically fuse different components, yielding the final forecast based on their relative contributions. Beyond fusing outputs from a single model, TimeCAP [42] enhances performance by combining predictions from both a multi-modal predictor and a pretrained LLM, which synergizes the gradient-based method and LLM agents reasoning on real-world contexts. Output fusion gains advantage of design flexibility and robustness, but it may not fully utilize the complementary relationship between modalities without additional countermeasures.
+
+Cross-modal fusion relies on well-aligned multi-modal data for effective exploitation of the contextual information. However, ideally-aligned data may not be given in real-world scenarios. As such, existing methods also leverage alignment mechanisms to mitigate the challenge.
+
+# 4.2 Alignment
+
+Alignment ensures that the relationships between different modalities are preserved and semantically coherent when integrated into a unified learning framework. At the input level, we primarily refer alignment to data preprocessing techniques that aim at mitigating temporal misalignment caused by missing values, irregular sampling intervals, and differing granularities across modalities. This process is crucial for ensuring that data from multiple sources are properly synchronized before fusion, where domain knowledge is usually needed to handle such inconsistencies [10, 53, 73]. In addition, none of the existing methods we reviewed explicitly perform output alignment. However, the aforementioned output fusion can be easily adapted to alignment through the incorporation of a gating or attention mechanism that we will introduce shortly.
+
+Alignment at the intermediate stage plays a crucial role in multimodal interactions. We first introduce the alignment of multi-modal representations, spanning a range of techniques from model component design to learning objectives. The common component designs include self-attention [5, 12, 30, 33, 41, 42, 44, 54, 55, 69], cross-attention [6, 18, 51, 60, 70, 82, 85, 98] and gating mechanisms [82, 98]. Self-attention is often used to fuse multi-modal representations. It
+
+Table 2: Taxonomy of representative multi-modal time series methods. Modality refers to the different data modalities involved in each method. TS represents standard time series, $ST$ denotes spatial time series. The Method column lists the techniques used for each interaction, separated by semicolons, where each interaction may include one or more techniques, separated by commas. Superscripts in the Code column include the URLs to Github repositories.
+
+| Method | Modality | Domain | Task | Cross-Modal Interaction | Large Model | Year | Code | |
| Stage | Fusion | Align. | Trans. | Method | |
| Time-MMD [53] | TS, Text | General | Forecasting | Output | ✓ | X | X | Addition | Multiple | 2024 | Yes[1] | |
| Wang et al. [73] | TS, Text | General | Forecasting | Input Intermediate | ✓ | X | X | Prompt Prompt; LLM Reasoning | LLaMa2 GPT-4 Turbo | 2024 | Yes[2] | |
| GPT4MTS [30] | TS, Text | General | Forecasting | Intermediate | ✓ | ✓ | X | Addition; Self-attention | GPT-2 | 2024 | No | |
| TimeCMA [51] | TS, Text | General | Forecasting | Input Intermediate | X | X | ✓ | Meta-description Addition; Cross-attention | GPT-2 | 2025 | Yes[3] | |
| MOAT [41] | TS, Text | General | Forecasting | Intermediate Output | ✓ | ✓ | X | Concat.; Self-attention Offline Synthesis (MLP) | S-Bert | 2024 | No | |
| TimeCAP [42] | TS, Text | General | Classification | Input Intermediate Output | X | X | ✓ | LLM Generation Concat.; Self-attention, Retrieval Addition | Bert, GPT-4 | 2024 | No | |
| TimeXL [31] | TS, Text | General | Classification Forecasting | Intermediate Output | ✓ | ✓ | X | Concat., Prompt; LLM Reasoning Addition | Bert, S-Bert GPT-4o | 2025 | No | |
| Hybrid-MMF [37] | TS, Text | General | Forecasting | Intermediate | ✓ | X | X | Concat. | GPT-4o | 2024 | Yes[4] | |
| Time-LLM [33] | TS, Text | General | Forecasting | Input Intermediate | X | X | ✓ | Meta-description Concat.; Self-attention | LLaMA, GPT-2 | 2024 | Yes[5] | |
| Time-VLM [98] | TS, Text, Image | General | Forecasting | Input Intermediate | X | X | ✓ | Feat. Imaging, Meta-description Addition; Gating, Cross-attention | ViLT, CLIP BLIP-2 | 2025 | No | |
| Unitime [55] | TS, Text | General | Forecasting | Input Intermediate | X | X | ✓ | Meta-description Concat.; Self-attention | GPT-2 | 2024 | Yes[6] | |
| TESSA [50] | TS, Text | General | Annotation | Intermediate | ✓ | ✓ | ✓ | Prompt; RL; LLM Generation | GPT-4o | 2024 | No | |
| InstruTime [12] | TS, Text | General | Classification | Intermediate | ✓ | ✓ | X | Concat.; Self-attention | GPT-2 | 2025 | Yes[7] | |
| MATMCD [68] | TS, Text, Graph | General | Causal Discovery | Intermediate | ✓ | ✓ | ✓ | Prompt; LLM Reasoning; Supervision | Multiple | 2025 | No | |
| STG-LLM [54] | ST, Text | General | Forecasting | Intermediate | ✓ | ✓ | X | Concat.; Self-attention | GPT-2 | 2024 | No | |
| TableTime [72] | TS, Text | General | Classification | Input | ✓ | X | ✓ | Prompt; Reformulate | Multiple | 2024 | Yes[8] | |
| ContextFormer [6] | TS, Tabular | General | Forecasting | Intermediate | ✓ | ✓ | X | Addition; Cross-attention | No | 2025 | No | |
| Time-MQA [38] | TS, Text | General | Multiple | Input | ✓ | X | X | Prompt | Multiple | 2025 | Yes[9] | |
| MAN-SF [67] | TS, Text, Graph | Finance | Classification | Intermediate | ✓ | ✓ | X | Bilinear; Graph Convolution | USE | 2020 | No | |
| Bamford et al. [3] | TS, Text, TS, Image | Finance | Retrieval | Intermediate Output | X | ✓ | X | Supervision | S-Bert | 2024 | No | |
| Chen et al. [11] | TS, Text, Graph | Finance | Classification | Intermediate | X | X | ✓ | LLM Generation Concat.; Graph Convolution | ChatGPT | 2023 | No | |
| Xie et al. [81] | TS, Text | Finance | Classification | Input | ✓ | X | X | Prompt | ChatGPT | 2023 | No | |
| Yu et al. [89] | TS, Text | Finance | Forecasting | Input | ✓ | X | X | Prompt | GPT-4, Open LLaMA | 2023 | No | |
| MedTsLLM [5] | TS, Text, Tabular | Healthcare | Multiple | Intermediate | ✓ | ✓ | X | Concat.; Self-attention | Llama2 | 2024 | Yes[10] | |
| RespLLM [95] | TS (Audio), Text | Healthcare | Classification | Intermediate | ✓ | ✓ | X | Addition, Self-attention | OpenBioLLM-8B | 2024 | No | |
| METS [45] | TS, Text | Healthcare | Classification | Output | X | ✓ | X | Contrastive | ClinicalBert | 2023 | No | |
| Wang et al. [75] | TS, Text | Healthcare | Classification | Intermediate | X | X | ✓ | Supervision | Bart, Bert, Roberta | 2021 | No | |
| EEG2TEXT [52] | TS, Text | Healthcare | Generation | Output | X | X | ✓ | Self-supervision, Supervision | Bart | 2024 | No | |
| MEDHMP [74] | TS, Text | Healthcare | Classification | Intermediate | ✓ | ✓ | X | Concat.; Self-attention, Contrastive | ClinicalT5 | 2023 | Yes[11] | |
| Deznabi et al. [16] | TS, Text | Healthcare | Classification | Intermediate | ✓ | X | X | Concat. | Bio+Clinical Bert | 2021 | Yes[12] | |
| Niu et al. [60] | TS, Text | Healthcare | Classification | Intermediate | ✓ | ✓ | X | Concat.; Cross-attention | BioBERT | 2023 | No | |
| Yang et al. [85] | TS, Text | Healthcare | Classification | Intermediate | ✓ | ✓ | X | Concat.; Addition; Gating | ClinicalBERT | 2021 | Yes[13] | |
| Liu et al. [56] | TS, Text | Healthcare | Classification Regression | Input | ✓ | X | X | Prompt | PaLM | 2023 | Yes[14] | |
| xTP-LLM [24] | ST, Text | Traffic Forecasting | Input | ✓ | X | ✓ | Prompt; Meta-description | Llama2-7B-chat | 2024 | Yes[15] | |
| UrbanGPT [48] | ST, Text | Traffic Forecasting | Input | ✓ | X | ✓ | Prompt; Meta-description | Vicuna-7B | 2024 | Yes[16] | |
| CityGPT [19] | ST, Text | Mobility | Input | ✓ | X | X | Prompt | Multiple | 2025 | Yes[17] | |
| MULAN [97] | TS, Text, Graph | IoT Causal Discovery | Intermediate | ✓ | ✓ | ✓ | Addition; Contrastive; Supervision | No | 2024 | No | |
| MIA [82] | TS, Image | IoT Anomaly Detection | Intermediate | ✓ | ✓ | X | Addition; Cross-attention, Gating | No | 2023 | No | |
| Ekambaram et al. [18] | TS, Image, Text | Retail Forecasting | Intermediate | ✓ | ✓ | X | Concat.; Self & Cross-attention | No | 2020 | Yes[18] | |
| Skenderi et al. [70] | TS, Image, Text | Retail Forecasting | Intermediate | ✓ | ✓ | X | Concat.; Cross-attention | No | 2024 | Yes[19] | |
| VIMTS [96] | ST, Image Environment | Imputation | Intermediate | ✓ | ✓ | X | Concat.; Supervision | No | 2022 | No |
| LTE [44] | ST, Text, Image Environment | Forecasting | Intermediate | ✓ | ✓ | X | Concat.; Self-attention | LLaMA-2-7B | 2024 | Yes[20] |
| AV-HubERT [69] | TS (Audio), Image Speech | Classification | Intermediate | ✓ | ✓ | X | Concat.; Self-attention | HuBert | 2022 | Yes[21] |
| SpeechGPT [90] | TS(Audio), Text | Generation intermediate | ✓ | ✓ | X | Concat.; Self-attention | LLaMA-13B | 2023 | Yes[22] | |
| LA-GCN [83] | ST, Text | Classification | Intermediate | X | ✓ | Supervision | Bert | 2023 | Yes[23] | |
+
+enables a joint and undirected alignment across all modalities by dynamically attending to important features. Given multi-modal embeddings $E_{\mathrm{mm}} \in \mathbb{R}^{n \times d}$ , where $n$ is the total number of modality tokens and $d$ is the embedding dimension, self-attention is computed as follows:
+
+$$
+\operatorname {A t t e n t i o n} \left(E _ {\mathrm {m m}}\right) = \operatorname {s o f t m a x} \left(\frac {Q K ^ {\top}}{\sqrt {d _ {k}}}\right) V
+$$
+
+where the queries $Q$ , keys $K$ , and values $V$ are linear projections of $E_{\mathrm{mm}}$ : $Q = E_{\mathrm{mm}}W_{Q}$ , $K = E_{\mathrm{mm}}W_{K}$ , $V = E_{\mathrm{mm}}W_{V}$ with learnable weights of dimensionality $d_{k}$ : $W_{Q}, W_{K}, W_{V} \in \mathbb{R}^{d \times d_{k}}$ .
+
+In cross-attention, time series serves as the query modality to get contextualized by other modalities, providing a directed alignment that ensures auxiliary modalities contribute relevant contextual information while preserving the temporal structure of time series. Given a query embedding $E_{\mathrm{ts}} \in \mathbb{R}^{n \times d}$ and that of auxiliary modalities $E_{c} \in \mathbb{R}^{n \times d}$ as keys and values:
+
+$$
+\text {C r o s s A t t e n t i o n} \left(E _ {\mathrm {t s}}, E _ {\mathrm {c}}\right) = \operatorname {s o f t m a x} \left(\frac {Q _ {\mathrm {t s}} K _ {\mathrm {c}} ^ {\top}}{\sqrt {d _ {k}}}\right) V _ {\mathrm {c}}
+$$
+
+where the query, key and value are denoted as $Q_{\mathrm{ts}} = E_{\mathrm{ts}}W_{Q}$ , $K_{\mathrm{c}} = E_{\mathrm{c}}W_{K}$ , $V_{\mathrm{c}} = E_{\mathrm{c}}W_{V}$ . Note that existing methods adopt multi-head attentions, which is omitted here for simplicity.
+
+Similarly, the gating mechanism is a parametric filtering operation that explicitly regulates the influence of time series and other modalities on the fused embeddings in $E$ :
+
+$$
+G = \sigma \left(W _ {g} \left[ E _ {\mathrm {t s}}; E _ {c} \right] + b _ {g}\right), \quad E = G \odot E _ {\mathrm {t s}} + (1 - G) \odot E _ {c}
+$$
+
+where $\sigma (\cdot)$ denotes the sigmoid function, the learnable weight and bias are denoted as $W_{g}\in \mathbb{R}^{2d\times d}$ and $b_{g}\in \mathbb{R}^{d}$ , respectively.
+
+When a graph modality is available from external contexts, the underlying topological insights can be leveraged for graph-based alignment [11, 67]. Unlike the above methods that rely solely on feature interactions, it explicitly aligns multi-modal representations with relational structures through graph convolution, enabling context-aware feature propagation across modalities.
+
+Representation alignments can also be achieved by learning objectives [3, 45, 83, 96, 97]. For example, MULAN [97] extracts modality-invariant and modality-specific representations from multi-modal time series. It employs contrastive learning to enhance cross-modal alignment by maximizing the similarity between invariant representations across modalities while minimizing the similarity between invariant and specific representations of the same modality. Moreover, Bamford et al. [3] align cross-modal representations by using the mean of uni-modal cosine similarities as the target similarity and optimizing cross-modal similarity via cross-entropy loss, which effectively connects both modalities in a shared latent space for time series retrieval tasks. In general, this branch of methods is effective as it directly integrates the alignment objective into the optimization process, ensuring that meaningful representations are explicitly learned.
+
+Lastly, we introduce the intermediate alignment of component outputs within a framework, extending beyond representation alignment within a model. The most recent studies explore the synergy between time series models and LLM agents, leveraging the strong reasoning capabilities of pretrained LLMs to provide
+
+contextual understanding and calibration in real-world scenarios [31, 42, 50, 68, 73]. We briefly discuss a few representative examples for demonstration. TimeCAP [42] utilizes the embedding space of a trained multi-modal encoder to retrieve in-context examples with the highest cosine similarity. These retrieved examples with ground truth labels are then fed, along with the query text, into an LLM to provide contextual guidance and improve outcome prediction. TimeXL [31] incorporates a multi-modal prototype-based encoder to generate explainable case-based rationales for both time series and texts, integrating three LLM agents, where prediction, reflection, and refinement LLMs collaborate to iteratively enhance prediction accuracy, identify textual inconsistencies or noise, and calibrate textual contexts, yielding more accurate predictions and explanations. NewsForecast [73] also employs reflection in language agents to iteratively select relevant news from a large database, enhancing alignment of textual information for text-based forecasting. Similarly, MATLAB [68] ensures alignment between statistical causal discovery on time series and LLM reasoning on textual context by leveraging iterative self-reflective tool-calling to structure textual context, which is then used to explain and refine causal constraints.
+
+In a nutshell, alignment aims to calibrate real-world contexts and effectively capture relevant multi-modal elements for a semantically coherent time series modeling. It enhances task performance, robustness and explanation, ensuring that models leverage meaningful contextual information for improved decision-making.
+
+# 4.3 Transference
+
+Transference refers to the process of mapping between different modalities. It allows one modality to be inferred, translated, or synthesized from another. This concept plays a crucial role across different stages of multi-modal time series analysis. The input-level transference typically serves for modality augmentation. It helps introduce contextual priors, enrich training samples, and provide alternative representations. This is particularly useful in scenarios of data scarcity and imbalance. In existing literature, a common practice is to use meta information to describe the narrative of real-world contexts (e.g., domain, data statistics and granularity, variable descriptions, other co-variates, etc.) [19, 24, 33, 48, 51, 55, 72, 98] or leverage pretrained LLMs to generate fine-grained textual contexts [42] or graphs [11] for real-world time series, serving as an augmented modality. In addition to texts, time series can also be transformed into high-dimensional images via feature imaging, such as stacking the original data with frequency and periodicity features [98]. Alternatively, time series can be represented in tabular form, transforming time series analysis into a table understanding task [72]. Note that the aforementioned uni-modal methods for transforming time series into other single modalities can also be integrated into multi-modal time series frameworks [15, 27, 46, 62, 99, 100]. The exploitation of input-level transference is two-fold. First, the embedding of generated modality can serve as semantic anchors that guides time series modeling via representation alignment, improving downstream supervised tasks [33, 51, 55, 98]. Second, it provides additional contextual guidance for pretrained LLMs through input fusion and prompting [19, 24, 48, 72].
+
+At the intermediate [11, 50, 68, 75, 97] and output [3, 52] levels, transference are more task-oriented. The output-level transference typically refers to the end-to-end generation of new modalities, such as text-based and image-based time series retrieval, where users provide textual descriptions or sketched trends to query relevant time series data [3]. This also includes EEG-to-text conversion, enabling direct transformation from physiological signals to human-readable narratives [52].
+
+The output of intermediate transference typically serves as an initial solution to be refined for modality generation tasks [50, 68, 97] or a medium to be inferred for predictive tasks [75], facilitating downstream reasoning and further alignment within the multimodal framework. MATMCD [68] generates an initial causal graph from time series, achieving modality transference in the intermediate level. Subsequently, it incorporates textual modality to refine the causal graph, ensuring improved alignment and interpretability. Moreover, Wang et al. [75] adopt a two-stage mechanism for sentiment classification based on EEG data, where the model first converts EEG signals into reading texts and then employs a pretrained LLM based on texts for classification, achieving impressive zero-shot results.
+
+# 5 Applications of Multi-modal Time Series Analysis
+
+In this section, we review the existing applications of multi-modal time series analysis for both standard and spatial time series, covering diverse domains such as healthcare, finance, transportation, environment, retail, and the Internet of Things (IoT).
+
+# Standard Time Series
+
+# 5.1 Healthcare
+
+Recent studies in healthcare highlight the multi-modal analysis of diverse medical data sources, such as EHRs (Electronic Health Records, containing lab values and clinical reports, etc.), audio, EEG (Electroencephalogram), ECG (Electrocardiogram), and other wearable and medical sensor recordings, for better disease diagnosis and patient monitoring. For multi-modal analysis on EHR data, a common modeling strategy involves the interaction between lab values and clinical reports, including the concatenation [16] and attention mechanisms [60, 85] on modality embeddings. Moreover, existing methods explore different modeling techniques to better exploit the clinical notes, via domain-specific text encoders (e.g., ClinicalBERT [28, 85] and BioBERT [43, 60]) and different processing strategies. For example, text embeddings can be modeled separately based on patient groups [86] or through a decaying mechanism based on time intervals [36] before interacting with time series embeddings, which leads to improved mortality prediction.
+
+In addition to EHRs, multi-modal modeling methods have been tailored for audio [95], ECG [45], and EEG [75]. Zhang et al. [95] focus on a respiratory health classification task by integrating both audio and textual descriptions. Li et al. [45] propose a multi-modal contrastive learning framework, constructing positive and negative samples by pairing patients' report texts with corresponding ECG signals for self-supervised pretraining. The classification task is then performed by computing the cosine similarity between different text representations and the target ECG representation. Wang
+
+et al. [75] propose a two-stage method for zero-shot EEG-based sentiment classification. First, a pretrained BART model is used for EEG-to-text decoding, followed by a trained text sentiment classifier that converts the generated text into sentiment categories.
+
+Similarly, Liu et al. [56] fuses physiological and behavioral time-series sensor data with real-world contextual information to effectively harness LLMs for wellness assessment. By fine-tuning the models with few-shot question-answer pairs that include contextual details, they improve performance on various healthcare tasks—such as cardiac signal analysis, physical activity recognition, and calorie-burn estimation, which outperform both supervised feedforward neural networks and zero-shot LLM baselines.
+
+# 5.2 Finance
+
+Recently, multi-modal time series analysis has received increasing attention in financial applications. Yu et al. [89] and Xie et al. [81] focus on stock prediction by integrating stock price movements, company profiles, and news directly into structured LLM prompts, enabling models to perform reasoning over multiple modalities. Yu et al. [89] applies GPT-4 and Open LLaMA to forecast NASDAQ-100 stock returns through instruction-based prompting and fine-tuning, demonstrating that structured LLM-driven inference can outperform traditional econometric models. Meanwhile, Xie et al. [81] conducts a zero-shot analysis of ChatGPT's capabilities for multimodal stock movement prediction, incorporating CoT prompting to assess the impact of social media sentiment on stock trends. Chen et al.[11] and Sawhneyet al.[67] also incorporate graph structures for stock movement prediction. For instance, Chen et al.[11] uses ChatGPT to infer dynamic stock relationship graphs from news, which reflects market conditions and enhances prediction accuracy.
+
+Beyond the predictive tasks, Bamford et al. [3] proposes a multimodal retrieval framework, where the model aligns both modalities in a shared latent space through contrastive learning. This framework allows users to search for financial time series through textual descriptions or sketched trends, offering greater flexibility. It also significantly improves retrieval speed and accuracy compared to traditional SQL-based search methods.
+
+# 5.3 Others
+
+Multi-modal time series analysis also exists in other domains, such as retail, IoT, computer vision and audio. In the retail sector, Ekambaram et al. [18] utilizes product images and textual descriptions, including attributes like color, pattern, and sleeve style, while incorporating temporal and exogenous features for new product sales forecasting. More recently, Skenderi et al. [70] integrates additional modality data, including product images and text descriptions, along with Google Trends data for sales forecasting. In IoT applications, MIA [82] enhances power transformer fault diagnosis by integrating multi-modal data, including dissolved gas analysis (DGA) and infrared images, to improve accuracy and efficiency. MULAN [97] converts log sequences into time-series data using a log-tailored language model and employs contrastive learning to leverage multi-modal data, facilitating root-cause discovery for system failures. In computer vision, LA-GCN [83] utilizes textual embeddings of joint names and action labels to generate faithful structural priors, enhancing skeleton-based action modeling and
+
+improving recognition tasks. In speech applications, AV-HuBERT [69] employs a self-supervised representation learning framework to leverage correlated audio and visual information [90], while SpeechGPT [69, 90] integrates audio and text to enhance generation performance.
+
+# Spatial Time Series
+
+# 5.4 Transportation and Mobility
+
+Several recent studies on traffic prediction highlight the importance of multi-modal contexts to enhance forecasting accuracy. Guo et al. [24] transforms California traffic data into structured LLM prompts. The method uses LLaMA models and instruction fine-tuning to improve spatial-temporal learning. Meanwhile, Li et al. [48] employs a spatial-temporal dependency encoder to align numerical New York City traffic data with LLMs, incorporating weather, geographic context, and historical flow patterns to refine predictions. Similarly, Feng et al. [19] proposes CityGPT, enhancing LLMs' spatial cognition for urban reasoning, mobility prediction, and navigation by integrating urban mobility data, road networks, and human behavior through instruction tuning. These studies demonstrate that LLM-based multi-modal fusion not only enhances traffic forecasting but also improves model interpretability and adaptability across diverse urban scenarios.
+
+# 5.5 Environment
+
+Integrating multi-modal information benefits environmental studies, particularly by addressing the prevalent challenge of missing values. VIMTS [96] utilizes a structured variational approximation technique to impute missing high-dimensional modalities (stream image data) by transforming them into low-dimensional features derived from simpler, related modalities (meteorological time series records), ensuring cross-modal correlations and interpretability of the imputation process. Additionally, LITE [44] addresses the imputation of missing features through a sparse Mixture of Experts framework. It integrates and encodes various environmental variables through a unified encoder. Directed by domain-specific instructions, a language model is utilized to merge these multi-modal representations, thereby improving the accuracy of environmental spatial-temporal predictions.
+
+# 6 Future Research Directions
+
+In this section, we outline several underexplored research directions that open up opportunities for future advancements.
+
+Reasoning with Multi-modal Time Series. Enhancing reasoning with multi-modal time series is pivotal for the development of intelligent systems. Future research should focus on creating a unified framework that can seamlessly integrate temporal reasoning with contextual understanding, enabling models to handle multiple time series tasks with interpretability. One potential path is to incorporate external knowledge bases and real-world context, such as developing a retrieval-augmented generation (RAG) [59] system, to enhance the reasoning process and allow models to make informed inferences beyond the immediate data. It is also promising to synergize time series model and language agents to provide more faithful and reliable reasoning on real-world contexts [31, 68].
+
+The recent development of LLM reasoning models, such as chain of thoughts [76] and tree of thoughts [87], also offers potential solutions to improve reasoning quality.
+
+Decision Making. Multi-modal time series analysis presents a promising future direction to enhance decision-making processes, which is crucial in high-stakes applications. By leveraging predictive signals and explanations from multi-modal contexts, future research can develop more adaptive, interpretable, and reliable decision-support systems, to facilitate the downstream optimization tasks such as resource allocation and risk management.
+
+Domain Generalization. One key challenge in multi-modal time series analysis is domain generalization, which enables a model trained on one or more source domains to effectively generalize to unseen target domains, ensuring robustness against distribution shifts. In multi-modal time series, distribution shifts can be multifaceted, stemming not only from time series, but also from other modalities. Therefore, it is crucial to develop specialized domain generalization methods for effective multi-modal time series analysis, including strategies to identify and preserve domain-invariant components across modalities while capturing modality-specific variations for rapid adaptation. Additionally, disentangling the effects of each modality is essential to better understand their individual contributions and mitigate cross-modal interference.
+
+Robustness to Missing and Noisy Modalities. Multi-modal time series analysis often frequently encounters messy real-world contexts with incomplete or noisy data. Existing methods employ an iterative context-refinement algorithm [31] that filters out less relevant information, thereby enhancing the predictive insights derived from multi-modal time series. Nonetheless, effectively dealing with missing and noisy modalities still demands further exploration. In particular, developing strategies for modality-specific imputation, noise reduction, and relevance quantification will be crucial to improving the real-world applicability of existing multi-modal time series methods.
+
+Ethical Considerations and Bias Mitigation. In light of potential biases in multi-modal time series datasets, future research should integrate fairness-aware techniques, such as fairness constraints, counterfactual analysis, and adversarial debiasing. These methods should be combined with robust bias assessment frameworks to systematically detect and mitigate inequities, ensuring outcomes that are both equitable and socially responsible.
+
+# 7 Conclusion
+
+In this survey, we provide a comprehensive overview of existing multi-modal time series methods. We first discuss the multi-modal time series used in existing methods. Then, we propose a taxonomy based on cross-modal interactions between time series and other modalities. The existing methods are categorized and summarized accordingly. We also discuss the real-world applications and highlight future research directions in this promising area.
+
+# Acknowledgments
+
+This research was supported in part by the National Science Foundation (NSF) CAREER IIS-2338878, as well as by generous research gifts from NEC Labs America Inc. and Morgan Stanley.
+
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| Healthcare | MIMIC-III [35][1], MIMIC-IV [34][2] | TS, Text, Tabular |
| ICBHI [65][3], Coswara [4][4], KAUH [21][5], PTB-XL [71][6], ZuCo [14, 26][7] | TS, Text |
| Image-EEG [22][8] | TS, Image |
| Finance | FNSPID [17][9], ACL18 [84][10], CIKM18 [79][11], DOW30 [11][12] | TS, Text |
| Multi-domain | Time-MMD [53][13], TimeCAP [42][14], NewsForecast [73][15], TTC [37][16], CiK [77][17], TSQA [38][18] | TS, Text |
| Retail | VISUELLE [70][19] | TS, Image, Text |
| IoT | LEMMA-RCA [40][20] | TS, Text |
| Speech | LRS3 [1][21], VoxCeleb2 [13][22] | TS (Audio), Image |
| Traffic | NYC-taxi, NYC-bike [48][23] | ST, Text |
| Environment | Terra [10][24] | ST, Text |
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| Stage | Fusion | Align. | Trans. | Method | |
| Time-MMD [53] | TS, Text | General | Forecasting | Output | ✓ | X | X | Addition | Multiple | 2024 | Yes[1] | |
| Wang et al. [73] | TS, Text | General | Forecasting | Input Intermediate | ✓ | X | X | Prompt Prompt; LLM Reasoning | LLaMa2 GPT-4 Turbo | 2024 | Yes[2] | |
| GPT4MTS [30] | TS, Text | General | Forecasting | Intermediate | ✓ | ✓ | X | Addition; Self-attention | GPT-2 | 2024 | No | |
| TimeCMA [51] | TS, Text | General | Forecasting | Input Intermediate | X | X | ✓ | Meta-description Addition; Cross-attention | GPT-2 | 2025 | Yes[3] | |
| MOAT [41] | TS, Text | General | Forecasting | Intermediate Output | ✓ | ✓ | X | Concat.; Self-attention Offline Synthesis (MLP) | S-Bert | 2024 | No | |
| TimeCAP [42] | TS, Text | General | Classification | Input Intermediate Output | X | X | ✓ | LLM Generation Concat.; Self-attention, Retrieval Addition | Bert, GPT-4 | 2024 | No | |
| TimeXL [31] | TS, Text | General | Classification Forecasting | Intermediate Output | ✓ | ✓ | X | Concat., Prompt; LLM Reasoning Addition | Bert, S-Bert GPT-4o | 2025 | No | |
| Hybrid-MMF [37] | TS, Text | General | Forecasting | Intermediate | ✓ | X | X | Concat. | GPT-4o | 2024 | Yes[4] | |
| Time-LLM [33] | TS, Text | General | Forecasting | Input Intermediate | X | X | ✓ | Meta-description Concat.; Self-attention | LLaMA, GPT-2 | 2024 | Yes[5] | |
| Time-VLM [98] | TS, Text, Image | General | Forecasting | Input Intermediate | X | X | ✓ | Feat. Imaging, Meta-description Addition; Gating, Cross-attention | ViLT, CLIP BLIP-2 | 2025 | No | |
| Unitime [55] | TS, Text | General | Forecasting | Input Intermediate | X | X | ✓ | Meta-description Concat.; Self-attention | GPT-2 | 2024 | Yes[6] | |
| TESSA [50] | TS, Text | General | Annotation | Intermediate | ✓ | ✓ | ✓ | Prompt; RL; LLM Generation | GPT-4o | 2024 | No | |
| InstruTime [12] | TS, Text | General | Classification | Intermediate | ✓ | ✓ | X | Concat.; Self-attention | GPT-2 | 2025 | Yes[7] | |
| MATMCD [68] | TS, Text, Graph | General | Causal Discovery | Intermediate | ✓ | ✓ | ✓ | Prompt; LLM Reasoning; Supervision | Multiple | 2025 | No | |
| STG-LLM [54] | ST, Text | General | Forecasting | Intermediate | ✓ | ✓ | X | Concat.; Self-attention | GPT-2 | 2024 | No | |
| TableTime [72] | TS, Text | General | Classification | Input | ✓ | X | ✓ | Prompt; Reformulate | Multiple | 2024 | Yes[8] | |
| ContextFormer [6] | TS, Tabular | General | Forecasting | Intermediate | ✓ | ✓ | X | Addition; Cross-attention | No | 2025 | No | |
| Time-MQA [38] | TS, Text | General | Multiple | Input | ✓ | X | X | Prompt | Multiple | 2025 | Yes[9] | |
| MAN-SF [67] | TS, Text, Graph | Finance | Classification | Intermediate | ✓ | ✓ | X | Bilinear; Graph Convolution | USE | 2020 | No | |
| Bamford et al. [3] | TS, Text, TS, Image | Finance | Retrieval | Intermediate Output | X | ✓ | X | Supervision | S-Bert | 2024 | No | |
| Chen et al. [11] | TS, Text, Graph | Finance | Classification | Intermediate | X | X | ✓ | LLM Generation Concat.; Graph Convolution | ChatGPT | 2023 | No | |
| Xie et al. [81] | TS, Text | Finance | Classification | Input | ✓ | X | X | Prompt | ChatGPT | 2023 | No | |
| Yu et al. [89] | TS, Text | Finance | Forecasting | Input | ✓ | X | X | Prompt | GPT-4, Open LLaMA | 2023 | No | |
| MedTsLLM [5] | TS, Text, Tabular | Healthcare | Multiple | Intermediate | ✓ | ✓ | X | Concat.; Self-attention | Llama2 | 2024 | Yes[10] | |
| RespLLM [95] | TS (Audio), Text | Healthcare | Classification | Intermediate | ✓ | ✓ | X | Addition, Self-attention | OpenBioLLM-8B | 2024 | No | |
| METS [45] | TS, Text | Healthcare | Classification | Output | X | ✓ | X | Contrastive | ClinicalBert | 2023 | No | |
| Wang et al. [75] | TS, Text | Healthcare | Classification | Intermediate | X | X | ✓ | Supervision | Bart, Bert, Roberta | 2021 | No | |
| EEG2TEXT [52] | TS, Text | Healthcare | Generation | Output | X | X | ✓ | Self-supervision, Supervision | Bart | 2024 | No | |
| MEDHMP [74] | TS, Text | Healthcare | Classification | Intermediate | ✓ | ✓ | X | Concat.; Self-attention, Contrastive | ClinicalT5 | 2023 | Yes[11] | |
| Deznabi et al. [16] | TS, Text | Healthcare | Classification | Intermediate | ✓ | X | X | Concat. | Bio+Clinical Bert | 2021 | Yes[12] | |
| Niu et al. [60] | TS, Text | Healthcare | Classification | Intermediate | ✓ | ✓ | X | Concat.; Cross-attention | BioBERT | 2023 | No | |
| Yang et al. [85] | TS, Text | Healthcare | Classification | Intermediate | ✓ | ✓ | X | Concat.; Addition; Gating | ClinicalBERT | 2021 | Yes[13] | |
| Liu et al. [56] | TS, Text | Healthcare | Classification Regression | Input | ✓ | X | X | Prompt | PaLM | 2023 | Yes[14] | |
| xTP-LLM [24] | ST, Text | Traffic Forecasting | Input | ✓ | X | ✓ | Prompt; Meta-description | Llama2-7B-chat | 2024 | Yes[15] | |
| UrbanGPT [48] | ST, Text | Traffic Forecasting | Input | ✓ | X | ✓ | Prompt; Meta-description | Vicuna-7B | 2024 | Yes[16] | |
| CityGPT [19] | ST, Text | Mobility | Input | ✓ | X | X | Prompt | Multiple | 2025 | Yes[17] | |
| MULAN [97] | TS, Text, Graph | IoT Causal Discovery | Intermediate | ✓ | ✓ | ✓ | Addition; Contrastive; Supervision | No | 2024 | No | |
| MIA [82] | TS, Image | IoT Anomaly Detection | Intermediate | ✓ | ✓ | X | Addition; Cross-attention, Gating | No | 2023 | No | |
| Ekambaram et al. [18] | TS, Image, Text | Retail Forecasting | Intermediate | ✓ | ✓ | X | Concat.; Self & Cross-attention | No | 2020 | Yes[18] | |
| Skenderi et al. [70] | TS, Image, Text | Retail Forecasting | Intermediate | ✓ | ✓ | X | Concat.; Cross-attention | No | 2024 | Yes[19] | |
| VIMTS [96] | ST, Image Environment | Imputation | Intermediate | ✓ | ✓ | X | Concat.; Supervision | No | 2022 | No |
| LTE [44] | ST, Text, Image Environment | Forecasting | Intermediate | ✓ | ✓ | X | Concat.; Self-attention | LLaMA-2-7B | 2024 | Yes[20] |
| AV-HubERT [69] | TS (Audio), Image Speech | Classification | Intermediate | ✓ | ✓ | X | Concat.; Self-attention | HuBert | 2022 | Yes[21] |
| SpeechGPT [90] | TS(Audio), Text | Generation intermediate | ✓ | ✓ | X | Concat.; Self-attention | LLaMA-13B | 2023 | Yes[22] | |
| LA-GCN [83] | ST, Text | Classification | Intermediate | X | ✓ | Supervision | Bert | 2023 | Yes[23] | |
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+ "content": "Lastly, we introduce the intermediate alignment of component outputs within a framework, extending beyond representation alignment within a model. The most recent studies explore the synergy between time series models and LLM agents, leveraging the strong reasoning capabilities of pretrained LLMs to provide"
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+ "content": "contextual understanding and calibration in real-world scenarios [31, 42, 50, 68, 73]. We briefly discuss a few representative examples for demonstration. TimeCAP [42] utilizes the embedding space of a trained multi-modal encoder to retrieve in-context examples with the highest cosine similarity. These retrieved examples with ground truth labels are then fed, along with the query text, into an LLM to provide contextual guidance and improve outcome prediction. TimeXL [31] incorporates a multi-modal prototype-based encoder to generate explainable case-based rationales for both time series and texts, integrating three LLM agents, where prediction, reflection, and refinement LLMs collaborate to iteratively enhance prediction accuracy, identify textual inconsistencies or noise, and calibrate textual contexts, yielding more accurate predictions and explanations. NewsForecast [73] also employs reflection in language agents to iteratively select relevant news from a large database, enhancing alignment of textual information for text-based forecasting. Similarly, MATLAB [68] ensures alignment between statistical causal discovery on time series and LLM reasoning on textual context by leveraging iterative self-reflective tool-calling to structure textual context, which is then used to explain and refine causal constraints."
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+ "content": "In this survey, we provide a comprehensive overview of existing multi-modal time series methods. We first discuss the multi-modal time series used in existing methods. Then, we propose a taxonomy based on cross-modal interactions between time series and other modalities. The existing methods are categorized and summarized accordingly. We also discuss the real-world applications and highlight future research directions in this promising area."
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+ "content": "This research was supported in part by the National Science Foundation (NSF) CAREER IIS-2338878, as well as by generous research gifts from NEC Labs America Inc. and Morgan Stanley."
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+ "content": "[1] Triantafyllos Afouras, Joon Son Chung, and Andrew Zisserman. 2018. LRS3-TED: a large-scale dataset for visual speech recognition. arXiv:1809.00496 [cs.CV] https://arxiv.org/abs/1809.00496"
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+ "content": "[2] Tadas Baltrusaitis, Chaitanya Ahuja, and Louis-Philippe Morency. 2018. Multimodal machine learning: A survey and taxonomy. IEEE transactions on pattern analysis and machine intelligence 41, 2 (2018), 423-443."
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diff --git a/data/2025/2503_13xxx/2503.13751/2a994aa1-2c31-48af-8c61-ad007e40c304_content_list.json b/data/2025/2503_13xxx/2503.13751/2a994aa1-2c31-48af-8c61-ad007e40c304_content_list.json
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+[
+ {
+ "type": "text",
+ "text": "Optimizing ML Training with Metagradient Descent",
+ "text_level": 1,
+ "bbox": [
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+ "page_idx": 0
+ },
+ {
+ "type": "text",
+ "text": "Logan Engstrom\\*1, Andrew Ilyas\\*2†, Benjamin Chen\\*1, Axel Feldmann\\*1, William Moses\\*3, Aleksander Madry\\*1",
+ "bbox": [
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+ "page_idx": 0
+ },
+ {
+ "type": "text",
+ "text": "*Equal contribution ${}^{1}$ MIT, ${}^{2}$ Stanford, ${}^{3}$ UIUC",
+ "bbox": [
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+ "page_idx": 0
+ },
+ {
+ "type": "text",
+ "text": "Abstract",
+ "text_level": 1,
+ "bbox": [
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+ "page_idx": 0
+ },
+ {
+ "type": "text",
+ "text": "A major challenge in training large-scale machine learning models is configuring the training process to maximize model performance, i.e., finding the best training setup from a vast design space. In this work, we unlock a gradient-based approach to this problem. We first introduce an algorithm for efficiently calculating metagradient gradients through model training at scale. We then introduce a \"smooth model training\" framework that enables effective optimization using metagradient. With metagradient descent (MGD), we greatly improve on existing dataset selection methods, outperform accuracy-degrading data poisoning attacks by an order of magnitude, and automatically find competitive learning rate schedules.",
+ "bbox": [
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+ "page_idx": 0
+ },
+ {
+ "type": "text",
+ "text": "1 Introduction",
+ "text_level": 1,
+ "bbox": [
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+ ],
+ "page_idx": 0
+ },
+ {
+ "type": "text",
+ "text": "How should I clean my data? What architecture should I use? Training large-scale (i.e., deep) machine learning models entails making many design decisions. When making such decisions, typical practice is to exhaustively search over a small set of standard options. For example, we might try a few well-known data cleaning heuristics, construct a grid over a hyperparameters, and choose the options that yield the best models. However, given that this process explores only a small part of the overall design space (e.g., one can construct $2^{n}$ possible training datasets from a pool of $n$ candidate datapoints), it is unlikely that this approach really yields the optimal training configuration.",
+ "bbox": [
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+ "page_idx": 0
+ },
+ {
+ "type": "text",
+ "text": "How can we find optimal (or at least, better) training configurations? To do so, we take the optimization perspective on designing model training. From this well-studied perspective, deciding on a training configuration—or as we will call it, a set of metaparameters—is just a high-dimensional optimization problem. The input space of this problem comprises all possible metaparameter choices, including which datapoints to train on, what model architecture to use, and how to initialize model weights. The objective function takes in a set of metaparameters, trains a machine learning model according to those metaparameters, and then returns a target metric evaluated on that model (e.g., test accuracy). From this perspective, any procedure for selecting metaparameters—including the typical practice of grid-searching over standard options—is just an optimization algorithm, whose goal is to maximize the objective function with respect to the (high-dimensional) input.",
+ "bbox": [
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+ "page_idx": 0
+ },
+ {
+ "type": "text",
+ "text": "Given that selecting metaparameters is \"just\" a high-dimensional optimization problem, a natural tool to consider is the gradient. After all, in many contexts, gradients offer a more effective approach to maximizing high-dimensional functions than grid search. Indeed, for a sufficiently \"well-behaved\" function $f(x)$ with gradient $\\nabla f(x)$ , we can optimize $f$ by iteratively updating $x$ in the direction of $\\nabla f(x)$ . This insight suggests a generic recipe for selecting metaparameters: first, make the objective differentiable with respect to the metaparameters; second, update via gradient steps.",
+ "bbox": [
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+ "page_idx": 0
+ },
+ {
+ "type": "text",
+ "text": "Now, the idea of using gradients to search for metaparameters is not new. Indeed, there is a substantial line of work that aims to optimize metaparameters (e.g., architectures, regularizers, or data augmentation schemes) with gradient-based methods [MDA15; LSY18; LVD20]. However, such methods have not managed to scale beyond relatively small settings. This state of affairs prompts our main question:",
+ "bbox": [
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+ "page_idx": 0
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+ {
+ "type": "text",
+ "text": "Can we scalably configure model training using gradient-based methods?",
+ "bbox": [
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+ "page_idx": 0
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+ {
+ "type": "aside_text",
+ "text": "arXiv:2503.13751v1 [stat.ML] 17 Mar 2025",
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+ "page_idx": 0
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+ {
+ "type": "page_footnote",
+ "text": "$^{\\dagger}$ Work done at MIT EECS. Correspondence to {engstrom,ailyas,benchen}@mit.edu.",
+ "bbox": [
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+ "text": "1",
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+ },
+ {
+ "type": "image",
+ "img_path": "images/4f160909052d60c475611d729d8627a48f2ec03ab554e430d02e6ecf55122fc3.jpg",
+ "image_caption": [
+ "Figure 1: Our proto-algorithm, metagradient descent (MGD), uses gradients to achieve state-of-the-art performance across a variety of applications, including data selection and data poisoning."
+ ],
+ "image_footnote": [],
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+ "type": "text",
+ "text": "1.1 Contributions",
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+ "page_idx": 1
+ },
+ {
+ "type": "text",
+ "text": "In this work, we answer this question in the affirmative, adding \"gradient descent on metaparameters\" to the large-scale machine learning toolkit. Along the way, we will face—and address—two main challenges.",
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+ "type": "text",
+ "text": "First, existing methods for computing metagradients do not scale. In response, we devise an algorithm, REPLAY, that can take metagradients in large-scale settings. By combining reverse-mode autodifferentiation (AD) with an efficient data structure, REPLAY can calculate exact metagradients for models with billions of parameters and thousands of training steps.",
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+ "type": "text",
+ "text": "Second, we find that metagradients of standard training routines are not necessarily helpful for optimization, which we connect to non-smoothness of the metaparameter optimization landscape. Borrowing tools from convex optimization, we devise a framework for designing \"metasmooth\" training routines that do admit helpful metagradients.",
+ "bbox": [
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+ "type": "text",
+ "text": "Addressing the challenges above unlocks a simple recipe for solving a broad range of machine learning tasks: (a) frame the task as a continuous optimization problem over metaparameters; (b) design a metasmooth training routine; (c) perform metagradient descent (MGD). Applying this recipe:",
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+ "- In the DataComp-small11 competition [GIF+24], we achieve state-of-the-art pre-training data selection for CLIP (2x larger performance improvement than the previous DataComp-small1 leader [Eco24]);",
+ "- In the context of data selection for instruction tuning (as introduced by Xia et al. [XMG+24]), we substantially improve on data selection for Gemma-2B (outperforming existing selection methods as well as full-data training);",
+ "- In the accuracy-degrading data poisoning setting (defined by Huber [Hub64] and pioneered by Lu et al. [LKY22] for deep neural networks), we improve attacks on DNNs by an order of magnitude, dropping CIFAR-10 accuracy from $92\\% \\rightarrow 78\\%$ (the best previous attack [LKY23] only reduces accuracy to $91\\%$ );",
+ "- For the task of hyperparameter optimization, we efficiently find a competitive CIFAR-10 learning rate schedule (matching the performance of a schedule found by grid search)."
+ ],
+ "bbox": [
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+ "page_idx": 1
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+ "type": "text",
+ "text": "2 Scalably computing metagradients",
+ "text_level": 1,
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+ "page_idx": 1
+ },
+ {
+ "type": "text",
+ "text": "In this section we present REPLAY, an algorithm for computing metagradients of large-scale iterative ML algorithms. We first detail the setting, then discuss existing approaches to computing metagradients, and conclude by describing REPLAY.",
+ "bbox": [
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+ "type": "image",
+ "img_path": "images/1dc990e467236d18f25ade5d515382e45fc4338efb7565274a7fd059602d5385.jpg",
+ "image_caption": [
+ "Training setup",
+ "Trained model",
+ "Observed behavior",
+ "Figure 2: An illustration of the metagradient. We embed a given aspect of the training setup (e.g., the training dataset, or optimizer hyperparameters) into a continuous metaparameter vector $z \\in \\mathbb{R}^d$ . This metaparameter defines a model $\\mathcal{A}(z)$ by way of the learning algorithm $\\mathcal{A}$ , which in turn defines an output $\\phi(z)$ . The metagradient $\\nabla_z \\phi(\\mathcal{A}(z)) \\in \\mathbb{R}^d$ is the gradient of this model output with respect to the metaparameter."
+ ],
+ "image_footnote": [],
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+ "type": "text",
+ "text": "2.1 What is a metagradient?",
+ "text_level": 1,
+ "bbox": [
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+ "page_idx": 2
+ },
+ {
+ "type": "text",
+ "text": "Training a machine learning model is a two-step process. First, we decide on a training setup—we must pick, for example, a neural network architecture, a training dataset, and an optimizer for training. Second, we apply the algorithm defined by this training setup to train a model.",
+ "bbox": [
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+ "page_idx": 2
+ },
+ {
+ "type": "text",
+ "text": "Our overall goal in this paper is to optimize model behavior as a function of the training setup (or, as we call it, the metaparameters) using gradient-based methods. To this end, we define the following notation:",
+ "bbox": [
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+ "- Let $\\mathbf{z} \\in \\mathbb{R}^n$ be a vector of continuous metaparameters representing the aspects of the training setup we aim to optimize. For example, if we only want to adjust the learning rate and weight decay of SGD then $n = 2$ . We handle discrete metaparameters (e.g., choice of training data) by finding a continuous relaxation (e.g., importance weights).",
+ "- Let $\\mathcal{A}$ be an algorithm mapping $\\mathbf{z}$ to a trained machine learning model; we assume all other aspects of the training setup outside $\\mathbf{z}$ are fixed and thus part of the algorithm $\\mathcal{A}$ .",
+ "- Finally, let $\\phi$ be an output function mapping a model $\\theta$ to a vector $\\phi(\\theta) \\in \\mathbb{R}$ . For example, $\\phi(\\theta)$ might represent the validation loss of the model $\\theta$ . We require that $\\phi$ be differentiable with respect to $\\theta$ , but otherwise make no assumptions on $\\phi$ ."
+ ],
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+ "type": "text",
+ "text": "With this notation in place, we define the training function $f \\coloneqq \\phi \\circ \\mathcal{A}$ mapping the training setup $\\mathbf{z}$ directly to the output function $\\phi$ evaluated on the corresponding model.",
+ "bbox": [
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+ "page_idx": 2
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+ "type": "text",
+ "text": "Finally, the metagradient is the gradient of the training function with respect to the metaparameters, $\\nabla_{\\mathbf{z}}f(\\mathbf{z})$ . Intuitively, the metagradient defines the \"direction of steepest ascent\" in metaparameter space.",
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+ "text": "Our focus: iterative algorithms. To efficiently compute the metagradient, we restrict our focus to cases where the algorithm $\\mathcal{A}$ is iterative, i.e., when it can be written in the form",
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+ "text": "\n$$\n\\underbrace {\\mathcal {A} (z) := \\mathbf {s} _ {T}} _ {\\text {m o d e l s t a t e a f t e r T s t e p s}}, \\quad \\text {w h e r e} \\quad \\underbrace {\\mathbf {s} _ {t + 1} : = h _ {t} (\\mathbf {s} _ {t} , \\mathbf {z})} _ {\\text {o p t i m i z e r s t e p t}}. \\tag {1}\n$$\n",
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+ "text": "Here, $\\mathbf{s}_t$ is the optimizer state at step $t$ (with $\\mathbf{s}_0$ being the initial state) and $h_t$ is the update mapping from state $t$ to state $t + 1$ . The form of (1) captures most large-scale training algorithms. For example, if the setup $\\mathbf{z} \\in \\mathbb{R}^T$ is a per-step learning rate, and the algorithm $\\mathcal{A}$ is full batch gradient descent, then each update $h_t$ is",
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+ "text": "\n$$\nh _ {t} (\\mathbf {s} _ {t}, \\mathbf {z}) := \\mathbf {s} _ {t} - z _ {t} \\nabla \\ell (\\mathbf {s} _ {t}),\n$$\n",
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+ "text": "where $z_{t}$ is the learning rate at step $t$ , $\\ell$ is the training loss, and the state $\\mathbf{s}_t$ comprises the parameters at step $t$ . For more complex algorithms like Adam [KB15], the state $\\mathbf{s}_t$ includes terms like gradient moments.",
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+ "type": "text",
+ "text": "2.2 Warmup: Metagradients via autodifferentiation",
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+ "text": "A key primitive we leverage to calculate metagradients is automatic differentiation (AD)—a standard tool for taking gradients through computer-defined functions. AD takes gradients by decomposing functions into elementary operations with known derivatives, then combining these derivatives using the chain rule. Concretely, AD operates in two passes: a \"forward pass,\" which executes the function of interest and stores intermediate products for each elementary operation; and a \"backward pass,\" which calculates the gradient by propagating chains of partial derivatives using these stored products. For the purposes of this paper, we will view AD as a black box that calculates the gradient of a many-to-one function (i.e., any $f: \\mathbb{R}^d \\to \\mathbb{R}$ ) at a given point using only a small constant factor more time than calculating the function itself (along with the space cost of storing the necessary forward-pass products).",
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+ "text": "What does this have to do with metagradients? Well, seeing as how training itself is a computer-defined function, AD is a natural tool for calculating the metagradient. The main challenge, as we discuss in the sequel, is that AD-based approaches to calculating the metagradient tend to be too resource-intensive for the large-scale machine learning algorithms we consider. In the remainder of this section we build up background before finally describing REPLAY, our algorithm for scalably computing (exact) metagradients.",
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+ "text": "Approach #1: Direct AD. The direct approach to calculating metagradients exploits the fact that nearly any learning algorithm is itself a sequence of differentiable computer-defined operations—meaning the training function $f$ is also differentiable.",
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+ "text": "However, operationalizing this observation to compute metagradients turns out to be challenging. The reason is that AD stores intermediate products for each operation. The amount of data stored thus scales with the number of operations in the function of interest. In the case of our training function $f$ , this number encompasses all the operations used to train a machine learning model. As a result, even in a toy scenario like MNIST training, computing metagradients with naive AD would require storing terabytes of data.",
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+ "text": "Approach #2: Exploiting structure with step-wise AD. A more efficient method for calculating the metagradient, step-wise AD, leverages the structure of iterative learning algorithms [Wer90; MDA15; FDF+17]. Recall from (1) that such algorithms take the form",
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+ "text": "\n$$\n\\mathcal {A} (\\mathbf {z}) := \\mathbf {s} _ {T}, \\quad \\text {w h e r e} \\quad \\mathbf {s} _ {t + 1} := h _ {t} (\\mathbf {s} _ {t}, \\mathbf {z}).\n$$\n",
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+ "text": "Algebraic manipulation (in particular, using the chain rule, the law of the total derivative, and the identity $\\mathbf{s}_t = h_{t-1}(\\mathbf{s}_{t-1}, \\mathbf{z})$ ) allows us to write the metagradient over an iterative algorithm as",
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+ {
+ "type": "equation",
+ "text": "\n$$\n\\frac {\\partial f (\\mathbf {z})}{\\partial \\mathbf {z}} = \\frac {\\partial \\phi (\\mathcal {A} (\\mathbf {z}))}{\\partial \\mathbf {z}} = \\sum_ {t = 1} ^ {T} \\underbrace {\\overbrace {\\partial \\phi (\\mathbf {s} _ {T})} ^ {A _ {t}} . \\overbrace {\\partial \\mathbf {s} _ {t}} ^ {\\partial \\phi (\\mathbf {s} _ {T})}} _ {B _ {t}}, \\tag {2}\n$$\n",
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+ "text": "where we have introduced the notation $A_{t}$ and $B_{t}$ for notational convenience. Step-wise AD computes the metagradient by calculating each term in the sum of (2) one at a time. For each term, the main challenge lies in computing $A_{t}$ , since given $A_{t}$ we can straightforwardly compute $B_{t}$ (the entire term) by differentiating through a single model update, i.e.,",
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+ "text": "\n$$\nB _ {t} := A _ {t} \\cdot \\frac {\\partial h _ {t - 1} (\\mathbf {s} _ {t - 1} , \\mathbf {z})}{\\partial \\mathbf {z}} = \\frac {\\partial (A _ {t} \\cdot h _ {t - 1} (\\mathbf {s} _ {t - 1} , \\mathbf {z}))}{\\partial \\mathbf {z}},\n$$\n",
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+ "text": "which is just a single call to our assumed \"AD oracle\" on the function $\\mathbf{z} \\mapsto A_t \\cdot h_{t-1}(\\mathbf{s}_{t-1}, \\mathbf{z})$ . Computing the $A_t$ terms is less straightforward as we need to relate $s_t$ and $s_T$ ; to do so, we exploit the recurrence",
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+ "text": "\n$$\nA _ {t} := \\frac {\\partial \\phi (\\mathbf {s} _ {T})}{\\partial \\mathbf {s} _ {t}} = \\frac {\\partial \\phi (\\mathbf {s} _ {T})}{\\partial \\mathbf {s} _ {t + 1}} \\cdot \\frac {\\partial h _ {t} (\\mathbf {s} _ {t} , \\mathbf {z})}{\\partial \\mathbf {s} _ {t}} = \\frac {\\partial \\left(A _ {t + 1} \\cdot h _ {t} (\\mathbf {s} _ {t} , \\mathbf {z})\\right)}{\\partial \\mathbf {s} _ {t}}, \\tag {3}\n$$\n",
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+ "text": "making $A_{t}$ straightforward to compute (again, a single \"AD oracle\" call) given $A_{t+1}$ . Step-wise AD exploits this fact to successively calculate the gradient with respect to each state, from state $T$ down to state 0.",
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+ "type": "page_number",
+ "text": "4",
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+ "img_path": "images/e61aefc6d8918a5c6e6ab0e0697fcbfc36486ad10f9f7a6216b3e5fefb7518f9.jpg",
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+ "Figure 3: The lazy $k$ -ary tree structure for traversing optimizer states in reverse order, with $k = 2$ . Recall that $n$ is the number of states (parameterized such that $n = T + 1$ ). Each node represents the correspondingly numbered state. We give an example of the traversal using the blue arrows in the figure, which denote the traversal path up to state $s_{\\frac{3n}{4} + 1}$ . The gray cylinders indicate the states that are stored when the traversal is at state $s_{\\frac{3n}{4} + 1}$ ; the other states are not stored at this point in the traversal. Traversing this structure requires storing $\\mathcal{O}(\\log(n))$ state and computing $\\mathcal{O}(n \\log(n))$ optimizer steps—compared to $n$ for simply training."
+ ],
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+ "text": "Bringing these ingredients together, the algorithm executes as follows. As a preprocessing step, it trains the model and stores all intermediate states $\\mathbf{s}_0,\\dots ,\\mathbf{s}_T$ . Then, the algorithm calculates and sums the terms in (2). It first computes $A_{T}\\coloneqq \\partial \\phi (\\mathbf{s}_{T}) / \\mathbf{s}_{T}$ , the gradient of the output function $\\phi$ with respect to the final state. Then, the algorithm steps through $\\mathbf{s}_{T - 1},\\ldots ,\\mathbf{s}_0$ in reverse order, calculating (a) the gradient with respect to each state $A_{t}$ (via (3)) and (b) the gradient with respect to $\\mathbf{z}$ at that step $B_{t}$ (via (2), using the previously calculated gradient with respect to that state). AD calculates both quantities--each requires differentiating over only one train step. Finally, the algorithm returns the final metagradient as the sum of the terms.",
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+ "type": "text",
+ "text": "Despite improving storage overhead compared to \"direct AD\", step-wise AD is still too space-intensive at scale. After all, this algorithm saves every optimizer state.",
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+ "type": "text",
+ "text": "2.3 REPLAY",
+ "text_level": 1,
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+ "text": "REPLAY is our algorithm for efficiently and exactly computing metagradients. It uses $\\mathcal{O}(k\\log_k(T))$ space and requires running the learning algorithm $\\mathcal{A}$ a total of $1 + \\log_{k}(T)$ times, with $k$ a user-chosen constant. The main idea is to make the space-intensive subroutine of step-wise AD—a reverse-order traversal of the optimizer states at each step—much more efficient. After all, step-wise AD stores all the states to reverse traverse them. REPLAY modifies step-wise AD to traverse states in less space by exploiting a simple observation: when training is deterministic, one can reinstantiate an optimizer state $\\mathbf{s}_t$ by \"replaying\" training from a fixed point $t' < t$ at the compute cost of $t - t'$ training steps. For example, one simple scheme saves every other state, then \"replays\" the remaining states when (reverse) traversing; this routine stores $T/2$ states but computes an extra $T/2$ model updates compared to storing all the states.",
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+ "text": "REPLAY performs a reverse-order traversal the optimizer states while balancing the compute cost of \"replaying\" training with the storage cost of saving states. We use a combination of deterministic training (fixing data ordering, data augmentation, and any other randomness in the training process) and an efficient data structure (similar to a segment tree; see Figure 3) to reverse-order traverse the optimizer states with $\\mathcal{O}(k\\log_k(T))$ space and an additional $T\\log_k(T)$ model steps.",
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+ "type": "page_number",
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+ "text": "Specifically, REPLAY recursively saves and replays training states. The algorithm splits the training trajectory into $k$ segments, performs the full training routine while saving only the start of each segment, then recurses into each segment (in reverse) to retrieve the states in reverse-order. The recursion depth bottoms out at $\\log_k(T)$ , at which point the algorithm has $k$ consecutive optimizer states in memory; the algorithm then backpropagates along this segment, before deleting all these states from memory and then reinstantating the next $k$ -length segment of optimizer states. We provide additional details on the algorithm in Appendix A.2. REPLAY unlocks computing large-scale metagradients by requiring only logarithmic storage and additional compute time.",
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+ "text": "Remark 1 (Connection to rematerialization). In a broad sense, both REPLAY and step-wise AD above can be viewed as special cases of a classical approach in AD (and computing broadly) known as rematerialization [CAC+81; BCT92; ZP00; GW08; CXZ+16]. To our knowledge, however, REPLAY is the first application of this particular rematerialization technique to the problem of computing metagradients through model training.",
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+ "text": "Remark 2 (Reversible learning). An alternative approach to calculating metagradients that does not save any state is reversible learning [MDA15], for which one can \"invert\" previous training states from future ones. We focus here on general (non-reversible) learning algorithms for two reasons: first, even simple algorithms such as SGD without momentum are non-reversible; second, reversibility in practice introduces numerical precision issues.",
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+ "type": "text",
+ "text": "3 Designing metasmooth training routines",
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+ "text": "Given a training function $f$ , REPLAY enables us to compute metagradients $\\nabla f(\\mathbf{z})$ for any setup $\\mathbf{z}$ . Can we immediately use these metagradients to optimize model training setups? The answer is (generally) no: we find that applying REPLAY to a function $f$ representing a standard model training and evaluation routine yields metagradients that are often $\\pm \\infty$ -valued and generally unhelpful for optimization. Indeed, previous work has observed similar issues optimizing over even (very) small-scale training [BSF94; Pea96; MDA15].",
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+ "text": "In this section, we show that an underlying source of the issue is the landscape of the metaparameter optimization problem. We then present a framework for modifying standard learning algorithms to admit useful metagradient, i.e., to be metasmooth. To use a familiar analogy: just as residual connections and improved initialization schemes can improve optimization in standard deep learning algorithms, our framework introduces an analogous set of modifications to enable optimization with metagradient.",
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+ "text": "3.1 The metaparameter optimization landscape",
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+ "text": "We first review the notion of smoothness from optimization theory, and then adapt it to the setting of metagradients. The resulting metasmoothness metric allows us to quantify (and later, improve) the amenability of the metaparameter optimization problem to gradient-based methods.",
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+ "text": "Smoothness. In optimization theory, the basic property of a function that controls how effectively it can be optimized with first-order methods is smoothness. Specifically, a function $f(\\mathbf{z})$ is $\\beta$ -smooth at a point $\\mathbf{z}$ if its gradient $\\nabla f$ satisfies the property that",
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+ "text": "\n$$\n\\left\\| \\nabla f (\\mathbf {z}) - \\nabla f \\left(\\mathbf {z} ^ {\\prime}\\right) \\right\\| \\leq \\beta \\cdot \\left\\| \\mathbf {z} - \\mathbf {z} ^ {\\prime} \\right\\| \\quad \\text {f o r a l l} \\mathbf {z} ^ {\\prime}, \\tag {4}\n$$\n",
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+ "text": "or in other words, if its gradient does not change too quickly around $\\mathbf{z}$ . To motivate this definition: if a function $f$ is $\\beta$ -smooth at $\\mathbf{z}$ , then a step of gradient descent with step size $1 / \\beta$ will successfully decrease the value of the function:",
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+ "text": "\n$$\nf \\left(\\mathbf {z} - \\frac {1}{\\beta} \\nabla f (\\mathbf {z})\\right) \\leq f (\\mathbf {z}) - \\frac {1}{2 \\beta} \\| \\nabla f (\\mathbf {z}) \\| ^ {2}.\n$$\n",
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+ "text": "This guarantee makes $\\beta$ -smoothness a good measure of gradient utility.",
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+ "text": "Metasmoothness. There are two main challenges in adapting the smoothness property to the metagradient setting. First, evaluating (4) requires a search over all possible $\\mathbf{z}'$ , which is infeasible. Second, even if we could exactly evaluate the left-hand side of (4), it would be difficult to disentangle non-smoothness of the training function $f$ from potential error in metagradient computation (e.g., a numerically unstable operation in REPLAY).",
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+ "text": "To sidestep these issues, we propose a metric called metasmoothness, given in Definition 1. Metasmoothness is cheap to compute—requiring only three evaluations of the training function—and does not rely on metagradient computation. For the remainder of this section, we fix a small constant $h > 0$ , and define the corresponding finite-differences estimator of the directional derivative $\\Delta_f$ as",
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+ "text": "\n$$\n\\Delta_ {f} (\\mathbf {z}; \\mathbf {v}) := \\frac {f (\\mathbf {z} + h \\mathbf {v}) - f (\\mathbf {z})}{h}.\n$$\n",
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+ "text": "Definition 1 (Metasmoothness of $f$ at $\\mathbf{z}$ towards $\\mathbf{v}$ ). Consider a training function $f$ mapping metaparameters $\\mathbf{z} \\in \\mathbb{R}^n$ to model output $f(\\mathbf{z}) \\in \\mathbb{R}$ . Given a metaparameter $\\mathbf{z}$ and a vector $\\mathbf{v} \\in \\mathbb{R}^n$ , the metasmoothness of $f$ at $\\mathbf{z}$ towards $\\mathbf{v}$ is given by",
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+ "text": "\n$$\nS _ {h, \\mathbf {v}} (f; \\mathbf {z}) := \\left| \\frac {\\Delta_ {f} (\\mathbf {z} + h \\mathbf {v}) - \\Delta_ {f} (\\mathbf {z})}{h} \\right|. \\tag {5}\n$$\n",
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+ "type": "text",
+ "text": "Definition 1 measures the rate of change of the derivative of $f(\\mathbf{z})$ in the direction of a given vector $\\mathbf{v}$ , and is therefore related to $\\beta$ -smoothness in that:",
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+ "(a) If $f$ is $\\beta$ -smooth at $\\mathbf{z}$ , then $S_{h,\\mathbf{v}}(f;\\mathbf{z}) \\leq \\beta$ for any $(h,\\mathbf{v})$ (so Definition 1 is necessary for smoothness).",
+ "(b) If $\\lim_{h\\to 0}S_{h,\\mathbf{v}}(f;\\mathbf{z})\\leq \\beta$ for all $\\mathbf{z}\\in \\mathbb{R}^n$ and $\\mathbf{v}\\in \\mathbb{S}^{n - 1}$ , then $f$ is $\\beta$ -smooth everywhere (so a global version of Definition 1 is sufficient for smoothness)."
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+ "type": "text",
+ "text": "Empirical metasmoothness. Definition 1 lets us measure the meta-smoothness of a training function $f$ at a particular metaparameter $\\mathbf{z}$ (towards a direction $\\mathbf{v}$ ). This definition, however, has two shortcomings. First, recall that the training function $f$ is a composition of a learning algorithm $\\mathcal{A}$ and an output function $\\phi$ , so the smoothness of $f$ depends on that of both $\\mathcal{A}$ and $\\phi$ (in particular, $\\frac{\\partial f}{\\partial \\mathbf{z}} = \\frac{\\partial \\phi}{\\partial \\mathcal{A}} \\cdot \\frac{\\partial \\mathcal{A}}{\\partial \\mathbf{z}}$ ). Since the output function $\\phi$ might be unknown ahead of time, we are most interested in measuring the overall metasmoothness of a learning algorithm $\\mathcal{A}$ . Second, while the result of (5) does have a concrete basis in optimization theory, it may not be easy to interpret in practice (e.g., what does $S = 200$ mean?). We address both issues simultaneously by (a) proposing an interpretable \"binarized\" version of Definition 1, and (b) studying metasmoothness in the space of model parameters $\\theta$ , instead of the output space.",
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+ "text": "Definition 2 (Empirical metasmoothness of $\\mathcal{A}$ ). Let $\\mathcal{A}$ be a learning algorithm which maps metaparameters $\\mathbf{z} \\in \\mathbb{R}^n$ to model parameters $\\theta \\in \\mathbb{R}^d$ , let $\\mathbf{z}$ be a metaparameter vector, and let $\\mathbf{v}$ be a given direction. Let $\\mathbf{d} \\in \\mathbb{R}^d$ be the per-coordinate variation in $\\theta$ , i.e.,",
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+ "type": "equation",
+ "text": "\n$$\n\\mathbf {d} = \\left| \\mathcal {A} (\\mathbf {z} + 2 h \\mathbf {v}) - \\mathcal {A} (\\mathbf {z}) \\right|\n$$\n",
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+ "text": "The empirical $(h,\\mathbf{v})$ -metasmoothness of $\\mathcal{A}$ at $\\mathbf{z}$ is given by",
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+ "text": "\n$$\n\\widehat {S} _ {h, \\mathbf {v}} (\\mathcal {A}; \\mathbf {z}) = \\operatorname {s i g n} \\left(\\Delta_ {\\mathcal {A}} (\\mathbf {z}; \\mathbf {v})\\right) ^ {\\top} \\cdot \\operatorname {d i a g} \\left(\\frac {\\mathbf {d}}{\\| \\mathbf {d} \\| _ {1}}\\right) \\cdot \\operatorname {s i g n} \\left(\\Delta_ {\\mathcal {A}} (\\mathbf {z} + h \\mathbf {v}; \\mathbf {v})\\right), \\tag {6}\n$$\n",
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+ "type": "text",
+ "text": "weights each parameter by its range.",
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+ "text": "Intuitively, (6) measures the agreement in sign between the (finite-difference approximation of the) metagradient in the direction of $\\mathbf{v}$ at $\\mathbf{z}$ and at $\\mathbf{z} + h\\mathbf{v}$ , averaged across parameter coordinates and weighted by the variation in each coordinate. Taking a weighted average of sign agreements ensures that $\\widehat{S} \\in [-1,1]$ (making it easier to interpret than Definition 1). The $\\mathrm{diag}(\\mathsf{d} / \\| \\mathsf{d}\\|_1)$ term weights each agreement proportionally to the scale of the corresponding parameter change (downweighting, e.g., coordinates $i$ that are essentially constant). Finally, observe that Definition 2 is efficient to compute in practice: it requires only three calls to the learning algorithm $\\mathcal{A}$ .",
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+ "img_path": "images/8903e4b2221d100768bfe268b5a001dd442b4e5c27be2741e58072ab69f0a150.jpg",
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+ "(a)"
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+ "Figure 4: (a) For a variety of training configurations of a ResNet-9 model, we plot metasmoothness (Def. 2) against test accuracy. Strategies such as increasing width, placing batch normalization before activations, and scaling down network outputs consistently improve metasmoothness, at a minor cost to accuracy. (b) Smoother training configurations can be optimized via metagradients more effectively. Here, as in Section 4.3, we use metagradients to gradient ascend on validation loss."
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+ "(b)"
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+ "type": "text",
+ "text": "Remark 3. Ideally, recalling the smoothness definition (4), we would evaluate metasmoothness in all possible directions $\\mathbf{v}$ and all points $\\mathbf{z}$ . Empirically, we find in the sequel (Section 3.2) that this single-direction approximation at a single point $\\mathbf{z}$ still yields a useful estimate of metasmoothness (e.g., one that correlates with metagradients utility).",
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+ "type": "text",
+ "text": "3.2 Estimating and improving metasmoothness",
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+ "text": "Having established a method for quantifying metasmoothness, we turn to the practical question: how can we design learning algorithms that are amenable to metagradient optimization? To answer this question, we introduce a straightforward framework: given a learning algorithm, explore a fixed menu of possible modifications to the training setup, and choose the combination that maximizes empirical metasmoothness. In practice, we find that this framework allows us to slightly modify learning algorithms in a way that makes them amenable to first-order methods.",
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+ "text": "As a case study, we study the task of training ResNet-9 on the CIFAR-10 dataset [Kri09]. We let the metaparameters $\\mathbf{z}$ be a perturbation to the pixels of 1000 random training images (so $\\mathbf{z} \\in \\mathbb{R}^{1000 \\times 32 \\times 32 \\times 3}$ ). We estimate the empirical metasmoothness of different learning algorithms $\\mathcal{A}$ at $\\mathbf{z} = \\mathbf{0}$ using Definition 2. Concretely, we proceed as follows for each learning algorithm $\\mathcal{A}$ :",
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+ "1. Let $\\mathbf{z}_0 = \\mathbf{0}$ be the metaparameter corresponding to the original dataset.",
+ "2. Sample a random perturbation vector $\\mathbf{v} \\sim \\mathcal{N}(0,1)$ .",
+ "3. Compute the empirical metasmoothness (6), i.e.,"
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+ "(a) Let $\\theta_0\\coloneqq \\mathcal{A}(\\mathbf{z}_0)$ , $\\theta_h\\coloneqq \\mathcal{A}(\\mathbf{z}_0 + h\\cdot \\mathbf{v})$ , and $\\theta_{2h}\\coloneqq \\mathcal{A}(\\mathbf{z}_0 + 2h\\cdot \\mathbf{v})$ be the model parameters that result from training with training dataset perturbations $\\mathbf{z}_0,\\mathbf{z}_0 + h\\mathbf{v}$ , and $\\mathbf{z}_0 + 2h\\mathbf{v}$ , respectively.",
+ "(b) Compute the approximate derivatives"
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+ "type": "equation",
+ "text": "\n$$\n\\Delta_ {\\mathcal {A}} (\\mathbf {z} _ {0}; \\mathbf {v}) = \\left(\\theta_ {h} - \\theta_ {0}\\right) / h, \\quad \\Delta_ {\\mathcal {A}} (\\mathbf {z} _ {0} + h \\mathbf {v}; \\mathbf {v}) = \\left(\\theta_ {2 h} - \\theta_ {h}\\right) / h.\n$$\n",
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+ "type": "text",
+ "text": "(c) Compute the weighting vector $\\mathbf{d} = |\\theta_{2h} - \\theta_0|$ , and compute the average metasmoothness (6), i.e.,",
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+ "type": "equation",
+ "text": "\n$$\n\\widehat {S} _ {h, \\mathbf {v}} (\\mathcal {A}; z _ {0}) = \\operatorname {s i g n} \\left(\\Delta_ {\\mathcal {A}} \\left(\\mathbf {z} _ {0} + h \\mathbf {v}; \\mathbf {v}\\right)\\right) ^ {\\top} \\cdot \\operatorname {d i a g} \\left(\\frac {\\mathbf {d}}{\\| \\mathbf {d} \\| _ {1}}\\right) \\cdot \\operatorname {s i g n} \\left(\\Delta_ {\\mathcal {A}} \\left(\\mathbf {z} _ {0}; \\mathbf {v}\\right)\\right).\n$$\n",
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+ "img_path": "images/65efa3e7039ec2bf0c17df31db0fea551abb1e8ebec13cd70da4376df689dec2.jpg",
+ "image_caption": [
+ "Figure 5: The effect of metasmoothness on the optimization landscape. Each plot above visualizes the loss landscape of a (deterministic) learning algorithm $\\mathcal{A}$ , with the $x$ - and $y$ -axes representing additive perturbations to 1000 examples in the training set and the $z$ -axis representing the resulting model's loss on the test example given in the title. In each row, the left plot is a non-smooth algorithm, and the right plot is a smooth algorithm (as per Definition 2) evaluated on the same example. Overall, empirical metasmoothness seems to strongly correlate with qualitative landscape smoothness. See Figure 12 for more examples."
+ ],
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+ "text": "Metasmooth learning algorithms. We apply the procedure above to estimate the metasmoothness of learning algorithms induced by different design choices (batch size, network width, BatchNorm placement, gradient scaling), and report the results in Figure 4 (left). On one hand, \"standard\" learning algorithms (i.e., those designed without metasmoothness in mind) are not metasmooth. On the other hand, our investigation reveals central factors driving metasmoothness. In addition to \"standard\" hyperparameters such as batch size and network width playing a role, we find that placing Batch Normalization layers prior to nonlinearities (instead of after) and scaling the final layer output are both crucial to metasmoothness. Note that the modifications we consider above are not exhaustive—see Appendix E for the full training setup.",
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+ "type": "text",
+ "text": "Finally, in Figure 5, we plot the optimization landscape of both metasmooth (right) and non-metasmooth (left) models. We find that the landscapes of metasmooth models are much smoother and—qualitatively—more straightforward to optimize.",
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+ "type": "text",
+ "text": "Metasmoothness/performance tradeoffs? Figure 4 (left) relates metasmoothness to model accuracy for the considered learning algorithms. While there is no clear trend, the top-performing learning algorithms are not always metasmooth. However, the trade-off is not too severe: the most metasmooth algorithms still achieve near-optimal accuracy. Furthermore, it is possible that with additional searching we could identify even more accurate metasmooth models. Taken together with our previous experiment, our results suggest that jointly searching over metasmoothness and model accuracy is a general recipe for designing learning algorithms that are both performant and metasmooth. Finally, as we discuss in Section 5, a fruitful avenue",
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+ "type": "text",
+ "text": "for future work may be to design metasmooth learning algorithms directly, i.e., without relying on stability heuristics or grid search.",
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+ "type": "text",
+ "text": "Does metasmoothness aid downstream optimization? Recall that our motivation for studying metasmoothness is to develop learning algorithms that we can optimize the metaparameters of via metagradients (using first-order methods). We started with the notion of $\\beta$ -smoothness from optimization theory, and we adapted it to the setting of metagradients by making a series of approximations and modifications. The final question we address is: does our final notion of metasmoothness actually predict the utility of metagradients for optimization? Figure 4 (right) demonstrates that metasmoothness strongly predicts our ability to optimize the metaparameters of a given learning algorithm. We use metagradients (computed by REPLAY) to gradient ascend on validation loss with respect to the metaparameters $\\mathbf{z}$ , and measure the change in model loss.",
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+ "type": "text",
+ "text": "4 Applications",
+ "text_level": 1,
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+ "type": "text",
+ "text": "In this section, apply metagradients to three problems in machine learning: selecting training data, poisoning training data, and searching for hyperparameters. In each setting we follow the same recipe: we frame the task as an optimization problem, modify the learning algorithm of interest to be smooth, then solve by first-order optimizing with meta-gradients—which we refer to, in a catch-all manner across algorithms, as metagradient descent (MGD). In particular: we substantially improve on existing dataset selection methods (Section 4.1, Section 4.2), perform the first effective accuracy-degrading data poisoning attack (Section 4.3), and discover one-cycle learning rate schedules with MGD (Section 4.4).",
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+ "type": "text",
+ "text": "4.1 Selecting multimodal training data",
+ "text_level": 1,
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+ "text": "Curating a training dataset from a mass of unfiltered data is a necessary and influential step in any large-scale machine learning pipeline. Deciding how to curate such a dataset is a challenging problem that has attracted substantial recent interest [FiW+22; ATS+23; EFM24; GIF+24]. In this section, we frame pre-training data selection as an optimization problem, and then solve this problem by first-order optimizing with metagradient. Applying our method to the DataComp-small benchmark [GIF+24], we greatly improve on the state-of-the-art (our improvement over state-of-the-art is roughly the same as the improvement of state-ofthe-art over training on random data).",
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+ "type": "text",
+ "text": "4.1.1 Setup",
+ "text_level": 1,
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+ "text": "The goal of dataset selection is to choose a training data subset (out of a broad pool of data) that maximizes trained machine learning model performance. Given this goal, dataset selection has a natural interpretation as a combinatorial metaparameter optimization problem. In particular, in the language of Section 2.1, for a training set of size $n$ , let",
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+ "(a) the metaparameters $\\mathbf{c} \\in \\mathcal{C} \\coloneqq \\mathbb{Z}_{\\geq 0}^{n}$ be non-negative data counts representing the number of times each training sample repeats in the training data;",
+ "(b) the algorithm $\\mathcal{A}$ be a standard large-scale learning procedure, which runs on a training set comprising $c_{i}$ copies of each sample $i$ for $i\\in [n]$ ;",
+ "(c) the output function $\\phi$ be the loss of the trained model on a target distribution $D$ ."
+ ],
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+ "type": "text",
+ "text": "Then, defining $f(\\mathbf{c}) \\coloneqq \\phi(\\mathcal{A}(\\mathbf{c}))$ (as in Section 2.1), our goal is to find the data counts $\\mathbf{c}^*$ that solve",
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+ "text": "\n$$\n\\mathbf {c} ^ {*} := \\underset {\\mathbf {c} \\in \\mathcal {C}} {\\arg \\min } f (\\mathbf {c}). \\tag {7}\n$$\n",
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+ "type": "text",
+ "text": "4.1.2 Gradient descent on training data",
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+ "text": "Metagradients let us directly minimize the target task loss (7) with respect to the choice of training data. At a high level, our algorithm operates as follows: we start with a randomly chosen set of training data, then iteratively update the dataset selection using metagradients with respect to importance weights placed on each training datapoint. The specifics of our method are in Algorithm 1; we describe its core ideas below.",
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+ "text": "Idea 1: A surrogate algorithm. We cannot use metagradients to optimize (7) directly, because the metaparameters of interest $\\mathbf{c}$ are discrete counts (and so the algorithm $\\mathcal{A}$ is non-differentiable with respect to $\\mathbf{c}$ ). To circumvent this problem, we relax $\\mathcal{A}$ : we define a surrogate algorithm $\\mathcal{A}_{\\mathbf{c}}^{\\prime}$ that takes in a continuous metaparameter $\\mathbf{z} \\in \\mathbb{R}^{n}$ , whose metagradient we can compute, then optimize using the metagradient on $\\mathcal{A}_{\\mathbf{c}}^{\\prime}$ .",
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+ "type": "text",
+ "text": "This surrogate learning algorithm $\\mathcal{A}_{\\mathrm{c}}^{\\prime}$ maps a metaparameter $\\mathbf{z} \\in \\mathbb{R}^{n}$ (representing a perturbation to training data weights) to a machine learning model. The surrogate is defined by a set of counts $\\mathbf{c} \\in \\mathbb{Z}_{+}^{n}$ , and a hyperparameter $k$ denoting a specific training iteration, both of which we bake into the surrogate algorithm itself. Given a metaparameter $\\mathbf{z} \\in \\mathbb{R}^{n}$ , the algorithm $\\mathcal{A}_{\\mathrm{c}}^{\\prime}$ trains a model \"as usual\" using the fixed counts $\\mathbf{c}$ . That is, it makes $c_{i}$ copies of each training sample $i$ , shuffles and partitions the data into batches, and then at each iteration minimizes the batch loss with a step—just as the original learning algorithm $\\mathcal{A}$ . At iteration $k$ , however, in addition to the original loss on the $k$ -th batch, the algorithm upweights each training sample $i$ according to the metaparameter $z_{i}$ . In other words, the objective at iteration $t$ of the surrogate algorithm $\\mathcal{A}_{\\mathrm{c}}^{\\prime}$ is",
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+ "type": "equation",
+ "text": "\n$$\n\\ell_ {t} ^ {\\prime} (\\theta) := \\left\\{ \\begin{array}{l l} \\sum_ {x \\in t ^ {\\text {t h}} \\text {b a t c h}} \\ell (x; \\theta) & \\text {i f} t \\neq k \\\\ \\sum_ {x \\in t ^ {\\text {t h}} \\text {b a t c h}} \\ell (x; \\theta) + \\sum_ {i = 1} ^ {n} z _ {i} \\ell (x _ {i}; \\theta) & \\text {i f} t = k \\end{array} \\right.\n$$\n",
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+ "type": "text",
+ "text": "where $\\ell (x;\\theta)$ is the training loss on example $x$",
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+ "type": "text",
+ "text": "Observe that when $\\mathbf{z} = \\mathbf{0}_n$ , the algorithm $\\mathcal{A}_{\\mathbf{c}}'$ is identical to the standard learning algorithm $\\mathcal{A}$ . And while $\\mathcal{A}$ was a function of (nondifferentiable) discrete data counts $\\mathbf{c}$ , $\\mathcal{A}_{\\mathbf{c}}'$ is differentiable with respect to its input $\\mathbf{z}$ , and so we can compute the metagradient",
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+ "type": "equation",
+ "text": "\n$$\n\\mathbf {g} := \\nabla_ {\\mathbf {z}} \\phi \\big (\\mathcal {A} _ {\\mathbf {c}} ^ {\\prime} (\\mathbf {z}) \\big) \\big | _ {\\mathbf {z} = \\mathbf {0} _ {n}}.\n$$\n",
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+ "text": "Intuitively, the entries of the metagradient $\\mathbf{g}$ capture the effect of adding an infinitesimal amount of each training sample $i$ to the training data at iteration $k$ . A positive entry $g_{i}$ indicates that adding an infinitesimal amount of sample $i$ to the training data would increase the loss, and a negative entry indicates that adding an infinitesimal amount of sample $i$ to the training data would decrease the loss; the slot at $i$ represents the (estimated) effect of adding a copy of sample $i$ to the training data at every batch containing the sample.",
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+ "type": "text",
+ "text": "Idea 2: Block coordinate descent. We then use the metagradient $\\mathbf{g}$ to iteratively update our selected dataset. We update data counts as",
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+ "type": "equation",
+ "text": "\n$$\n\\mathbf {c} \\leftarrow \\mathbf {c} - \\operatorname {s i g n} (\\mathbf {g}) \\odot \\mathbf {m}, \\quad \\mathbf {m} \\sim \\text {B e r n o u l l i} (p), \\tag {8}\n$$\n",
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+ "text": "where $p$ is a hyperparameter controlling the fraction of sample counts to update. This algorithm resembles a block coordinate descent algorithm [OR00], with the main difference being that we take signed gradient steps with step size 1 (projected onto non-negative integers) to ensure that the counts remain well-defined. As a result, $p$ implicitly controls the algorithm's step size.",
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+ "text": "Applying (8) concludes a single optimization step. By repeating this process of estimating the metagradient, updating our counts vector, then constructing a new training dataset, we iteratively improve the selected data. Pseudocode for our algorithm can be found in Algorithm 1.",
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+ "type": "text",
+ "text": "4.1.3 Results",
+ "text_level": 1,
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+ "text": "We evaluate our data selection algorithm using DataComp [GIF+24], a standardized framework for evaluating data selection methods for multimodal models. Algorithm 1 greatly improves on the state-of-the-art for the benchmark. Below, we describe the setting, outline our method, and conclude with our results.",
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+ "text": "11",
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+ "type": "code",
+ "sub_type": "algorithm",
+ "code_caption": [
+ "Algorithm 1: Dataset selection using using metagradient descent (MGD)."
+ ],
+ "code_body": "Input: initial data counts $\\mathbf{c}\\in \\mathbb{Z}_{\\geq 0}^{n}$ , learning algorithm $\\mathcal{A}$ output function Hyperparameters: step size $p$ # opt steps $T$ iteration number $k$ for $t\\gets 1$ to $T$ do \n2 $\\mathbf{z}\\gets \\mathbf{0}_n / /$ Build input to surrogate \n3 $\\mathbf{g}\\leftarrow \\frac{\\partial\\phi(\\mathcal{A}_c'(\\mathbf{z}))}{\\partial\\mathbf{z}} / /$ Calculate metagradient using REPLAY \n4 m<- sample from Bernoulli(p) // Sample indices to step on \n5 c<- c-sign(g) $\\odot$ m// Take optimization step \n6 Return c// Return final data counts",
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+ "type": "text",
+ "text": "Setting. DataComp [GIF+24] is a multimodal model training competition and benchmark for evaluating dataset selection methods. DataComp provides a fixed learning algorithm chosen in advance by the organizers and a large fixed candidate pool of internet data. The goal is to choose a subset of the candidate pool—possibly with repeated datapoints—that yields the best-performing model after training with the given learning algorithm, as measured by a predetermined set of 38 benchmarks. Given a submission subset, the mean score on the evaluation datasets for a model trained with that subset is taken as the final \"score.\" DataComp offers four separate \"scales\" requiring different amounts of compute; we focus on the small scale in this paper due to compute limitations.",
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+ "text": "Method. We select data with MGD (Algorithm 1) to minimize loss on data on a \"target set\" that is distributionally similar to the DataComp benchmark tasks, and select hyperparameters with a held-out \"validation set.\" In particular, we construct target and validation sets by taking samples from the DataComp evaluation tasks with extra samples available beyond those used in the DataComp test set (e.g., ImageNet, one of the tasks in DataComp, has a training set in addition to the test set evaluated in DataComp). See Appendix C for the exact details of the target and validation sets, the precise hyperparameters used with Algorithm 1, and a discussion on scalability (including further engineering details on executing our algorithm efficiently).",
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+ "type": "text",
+ "text": "Results. MGD greatly outperforms the current state-of-the-art: the difference in accuracy between MGD and the current best method is roughly as large as the difference between the previous state-of-the-art (EcoDatum [Eco24]) and training on randomly chosen data (cf. Figure 6). Inspecting scores over the course of the optimization in Figure 6, we find that only a few steps are necessary to outperform previous methods.",
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+ {
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+ "img_path": "images/b4f07dfbec991e71985782d6905bc8288383a71ad6a7e6b4aaf74a949165576e.jpg",
+ "image_caption": [
+ "Figure 6: MGD dataset selection greatly outperforms existing methods (improving over the previous SOTA by as much as the previous SOTA improves over no filtering at all). We compare DataComp scores for MGD (over optimization steps), training on the entire candidate pool, the best baseline originally proposed by DataComp, and the previous SOTA [Eco24]."
+ ],
+ "image_footnote": [],
+ "bbox": [
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+ {
+ "type": "table",
+ "img_path": "images/f7cdf24f42ff7038688bd05478168802f6a6fa09213f5ffad00db62ef65980fa.jpg",
+ "table_caption": [],
+ "table_footnote": [],
+ "table_body": "| Dataset | Task | Test size | Train size | Val size | Main metric | Target set |
| Caltech-101 [FFP04] | Object recognition | 6085 | 2754 | 306 | mean per class | ✓ |
| CIFAR-10 [Kri09] | Visual recognition | 10000 | 45000 | 5000 | accuracy | ✓ |
| CIFAR-100 [Kri09] | Visual recognition | 10000 | 45000 | 5000 | accuracy | ✓ |
| CLEVR Counts [JHV+17; ZPK+19] | Counting | 15000 | 65000 | 5000 | accuracy | ✓ |
| CLEVR Distance [JHV+17; ZPK+19] | Distance prediction | 15000 | 65000 | 5000 | accuracy | ✓ |
| Country211 [RKH+21; TSF+16] | Geolocation | 21100 | 37980 | 4220 | accuracy | ✓ |
| DTD [CMK+14] | Texture classification | 1880 | 3384 | 376 | accuracy | ✓ |
| EuroSAT [HBD+19; ZPK+19] | Satellite imagery recognition | 5400 | 19440 | 2160 | accuracy | ✓ |
| FGVC Aircraft [MRK+13] | Aircraft recognition | 3333 | 6001 | 666 | mean per class | ✓ |
| Food-101 [BGV14] | Food recognition | 25250 | 70750 | 5000 | accuracy | ✓ |
| GTSRB [SSS+11] | Traffic sign recognition | 12630 | 35289 | 3920 | accuracy | ✓ |
| ImageNet 1k [DDS+09] | Visual recognition | 50000 | 1276167 | 5000 | accuracy | ✓ |
| ImageNet Sketch [WGX+19] | Visual recognition | 50889 | N/A | N/A | accuracy | * |
| ImageNet V2 [RKS+19] | Visual recognition | 10000 | N/A | N/A | accuracy | * |
| ImageNet-A [HZB+19] | Visual recognition | 7500 | N/A | N/A | accuracy | * |
| ImageNet-O [HZB+19] | Visual recognition | 2000 | N/A | N/A | accuracy | * |
| ImageNet-R [HBM+20] | Visual recognition | 30000 | N/A | N/A | accuracy | * |
| KITTI distance [GLU12; ZPK+19] | Distance prediction | 711 | N/A | N/A | accuracy | † |
| MNIST [LeC98] | Digit recognition | 10000 | 55000 | 5000 | accuracy | ✓ |
| ObjectNet [BMA+19] | Visual recognition | 18574 | N/A | N/A | accuracy | * |
| Oxford Flowers-102 [NZ08] | Flower recognition | 6149 | 1836 | 204 | mean per class | ✓ |
| Oxford-IIIT Pet [PVZ+12; ZPK+19] | Pet classification | 3669 | 3312 | 368 | mean per class | ✓ |
| Pascal VOC 2007 [EVW+10] | Object recognition | 14976 | 14096 | 1566 | accuracy | ✓ |
| PatchCamelyon [VLW+18; ZPK+19] | Metastatic tissue cls. | 32768 | 289912 | 5000 | accuracy | ✓ |
| Rendered SST2 [ZPK+19] | Sentiment classification | 1821 | 7013 | 779 | accuracy | ✓ |
| RESISC45 [CHL17; ZPK+19] | Satellite imagery recognition | 6300 | 22680 | 2520 | accuracy | ✓ |
| Stanford Cars [KSD+13] | Vehicle recognition | 8041 | 7329 | 814 | accuracy | ✓ |
| STL-10 [CNL11] | Visual recognition | 8000 | 4500 | 500 | accuracy | ✓ |
| SUN-397 [XHE+10] | Scene recognition | 108753 | N/A | N/A | accuracy | ‡ |
| SVHN [NWC+11; ZPK+19] | Digit recognition | 26032 | 68257 | 5000 | accuracy | ✓ |
| iWildCam [BAC+21; KSM+20] | Animal recognition | 42791 | 147084 | 5000 | macro F1 score | ✓ |
| Camelyon17 [BGM+18; KSM+20] | Metastatic tissue cls. | 85054 | 365900 | 5000 | accuracy | ✓ |
| FMoW [CFW+18; KSM+20] | Satellite imagery recognition | 22108 | 103261 | 5000 | worst-region acc. | ✓ |
| Dollar Street [RDK+22] | Object recognition | 3503 | 13842 | 1537 | worst-income top-5 acc. | ✓ |
| GeoDE [RLZ+24] | Object recognition | 12438 | 44488 | 4943 | worst-region acc. | ✓ |
| Flickr30k [YLH+14] | Image and text retrieval | 31014 | N/A | N/A | R@1 | § |
| MSCOCO [LMB+14] | Image and text retrieval | 5000 | N/A | N/A | R@1 | § |
| WinoGAViL [BBY+22] | Commonsense association | 3563 | N/A | N/A | Jaccard score | § |
",
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+ "page_idx": 32
+ },
+ {
+ "type": "page_footnote",
+ "text": "*No train or val set exists for this dataset, so we were unable to create disjoint target and val sets.",
+ "bbox": [
+ 130,
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+ ],
+ "page_idx": 32
+ },
+ {
+ "type": "page_footnote",
+ "text": "We were unable to use this dataset due to technical difficulties.",
+ "bbox": [
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+ "page_idx": 32
+ },
+ {
+ "type": "page_footnote",
+ "text": "Both the train and val sets were used by DataComp to make their test set, so we were unable to create disjoint target and val sets.",
+ "bbox": [
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+ "page_idx": 32
+ },
+ {
+ "type": "page_footnote",
+ "text": "${}^{S}$ Retrieval tasks were not used for metagradients.",
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+ ],
+ "page_idx": 32
+ },
+ {
+ "type": "page_number",
+ "text": "33",
+ "bbox": [
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+ "page_idx": 32
+ },
+ {
+ "type": "text",
+ "text": "D Selecting IFT data",
+ "text_level": 1,
+ "bbox": [
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+ ],
+ "page_idx": 33
+ },
+ {
+ "type": "text",
+ "text": "In this section, we describe the details of the IFT setting of Xia et al. [XMG+24], as well as the details of our method.",
+ "bbox": [
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+ "page_idx": 33
+ },
+ {
+ "type": "text",
+ "text": "Setting. The setting contains a fixed data pool: instruction fine-tuning data from a data pool consisting of four combined IFT datasets (cf. Table 4 and Xia et al. [XMG+24] for more information). The goal is to select the data that yields the best possible task performance for a LoRA fine-tuning run. We adapt a LoRA to a Gemma-2B model (the pretraining-only Gemma-2B model) using the LoRA configuration from Xia et al. [XMG+24].",
+ "bbox": [
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+ "page_idx": 33
+ },
+ {
+ "type": "text",
+ "text": "Data splits. See Table 3 for a description of the available data for each task, along with the task setup details. Xia et al. [XMG+24] constructed these extra samples by drawing from the ICL samples given in the tasks originally. Note that we drop TydiQA from the original work of Xia et al. [XMG+24] as there are not enough samples to select with (there is only one from each category, for a total of 7).",
+ "bbox": [
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+ "page_idx": 33
+ },
+ {
+ "type": "text",
+ "text": "Method. We execute Algorithm 1 with $k$ as 150 steps from the end of training and the Bernoulli parameter $p$ controlling the step size as 0.2. At each step, we choose a \"minibatch\" with a size equal to half the target set and a quarter of the target set for BBH and MMLU, respectively (that is, we only select to optimize performance on a fraction of the target set at a time). We model select over iterates and hyperparameters by (a) choosing the top three steps in terms of validation loss for each run (b) selecting the best one in terms of full train set accuracy (including the part that we trained on). We perform this procedure—akin to Pareto optimization [JS08]—because the validation set is so small (as the overall set of samples is very small) that it is difficult to select models without overfitting otherwise.",
+ "bbox": [
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+ "page_idx": 33
+ },
+ {
+ "type": "text",
+ "text": "We compare with two baselines: training on the full dataset (i.e., training on the entirety of all the data for a single epoch), and LESS (we use the data selected according to \"LESS-T\" [XMG+24], following the recommendation of 4 epochs).",
+ "bbox": [
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+ "page_idx": 33
+ },
+ {
+ "type": "text",
+ "text": "For model training, we train with ADAM ( $\\beta_{1} = 0.95$ , $\\beta_{2} = 0.975$ , decoupled weight decay as $10^{-5}$ ) and a one-cycle linear schedule starting at $10^{-6}$ of the maximum learning rate, reaching the peak over $25\\%$ of training, then ending at 0.1 of the maximum learning rate. We insert a positive $\\varepsilon_{\\mathrm{root}}$ into the inverse square root term in the ADAM update to prevent metagradient (and to a lesser extent update) blowup (see Eq. 12). The model training is the same across selected data, except that we use $\\varepsilon_{\\mathrm{root}} = 10^{-7}$ for MGD-selected data and $\\varepsilon_{\\mathrm{root}} = 10^{-9}$ for the other runs (we select the optimal parameter for each class of method). We additionally hyperparameter select for the best learning rate across each baseline by minimizing validation set loss; LESS performs best with a smaller learning rate (0.00024 for BBH and 0.00012 for MMLU) than training on the full dataset or with MGD (0.0006 for both). We normalize the loss of each training sample by taking the mean across predicted tokens during training, and do not divide by the batch size (important for scaling the $\\varepsilon_{\\mathrm{root}}$ term, but otherwise ADAM is invariant to the scale).",
+ "bbox": [
+ 109,
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+ "page_idx": 33
+ },
+ {
+ "type": "text",
+ "text": "Selecting smooth model training for MGD. For MGD runs, we jointly select learning rate and $\\varepsilon_{\\mathrm{root}}$ using the smoothness metric of Section 3. We find that the choice of $\\varepsilon_{\\mathrm{root}}$ term is important (just as the choice of $\\varepsilon$ is important in standard ADAM training); choosing a much larger term results in non-smooth training. We also find that metagradients are sensitive to learning rate schedule; choosing a much larger or smaller maximum learning rate results in non-smooth training.",
+ "bbox": [
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+ "page_idx": 33
+ },
+ {
+ "type": "table",
+ "img_path": "images/0ee01eb4fa01b3734087808afd61cf8ee91b915d35fec2c214b7e2bfd450099d.jpg",
+ "table_caption": [
+ "Table 3: Overview of datasets used in IFT dataset selection (from Xia et al. [XMG+24])."
+ ],
+ "table_footnote": [],
+ "table_body": "| Parameter | Values |
| Peak learning rate | [7.0, 7.5, 8.0, 8.5, 9.0, 9.5, 10.0, 10.5, 11.0, 11.5, 12.0] |
| Initial LR multiplier | [0.05, 0.15, 0.25, 0.35, 0.45, 0.55] |
| Final LR multiplier | [0.05, 0.15, 0.25, 0.35, 0.45, 0.55] |
| LR peak time | [0.25, 0.5, 0.75] |
",
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+ "image_caption": [
+ "MGD step",
+ "Figure 14: Graphs of our learned LR schedules."
+ ],
+ "image_footnote": [],
+ "bbox": [
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diff --git a/data/2025/2503_13xxx/2503.13751/2a994aa1-2c31-48af-8c61-ad007e40c304_model.json b/data/2025/2503_13xxx/2503.13751/2a994aa1-2c31-48af-8c61-ad007e40c304_model.json
new file mode 100644
index 0000000000000000000000000000000000000000..98d1f78ab5a75a016e0bbb2c1400fc8c1c0cb56a
--- /dev/null
+++ b/data/2025/2503_13xxx/2503.13751/2a994aa1-2c31-48af-8c61-ad007e40c304_model.json
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+[
+ [
+ {
+ "type": "aside_text",
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+ "angle": 270,
+ "content": "arXiv:2503.13751v1 [stat.ML] 17 Mar 2025"
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.17,
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+ "angle": 0,
+ "content": "Optimizing ML Training with Metagradient Descent"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.257,
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+ "angle": 0,
+ "content": "Logan Engstrom\\*1, Andrew Ilyas\\*2†, Benjamin Chen\\*1, Axel Feldmann\\*1, William Moses\\*3, Aleksander Madry\\*1"
+ },
+ {
+ "type": "text",
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+ ],
+ "angle": 0,
+ "content": "*Equal contribution \\( {}^{1} \\) MIT, \\( {}^{2} \\) Stanford, \\( {}^{3} \\) UIUC"
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.468,
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+ "angle": 0,
+ "content": "Abstract"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.152,
+ 0.299,
+ 0.846,
+ 0.399
+ ],
+ "angle": 0,
+ "content": "A major challenge in training large-scale machine learning models is configuring the training process to maximize model performance, i.e., finding the best training setup from a vast design space. In this work, we unlock a gradient-based approach to this problem. We first introduce an algorithm for efficiently calculating metagradient gradients through model training at scale. We then introduce a \"smooth model training\" framework that enables effective optimization using metagradient. With metagradient descent (MGD), we greatly improve on existing dataset selection methods, outperform accuracy-degrading data poisoning attacks by an order of magnitude, and automatically find competitive learning rate schedules."
+ },
+ {
+ "type": "title",
+ "bbox": [
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+ "angle": 0,
+ "content": "1 Introduction"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.453,
+ 0.885,
+ 0.559
+ ],
+ "angle": 0,
+ "content": "How should I clean my data? What architecture should I use? Training large-scale (i.e., deep) machine learning models entails making many design decisions. When making such decisions, typical practice is to exhaustively search over a small set of standard options. For example, we might try a few well-known data cleaning heuristics, construct a grid over a hyperparameters, and choose the options that yield the best models. However, given that this process explores only a small part of the overall design space (e.g., one can construct \\(2^{n}\\) possible training datasets from a pool of \\(n\\) candidate datapoints), it is unlikely that this approach really yields the optimal training configuration."
+ },
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+ "type": "text",
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+ "angle": 0,
+ "content": "How can we find optimal (or at least, better) training configurations? To do so, we take the optimization perspective on designing model training. From this well-studied perspective, deciding on a training configuration—or as we will call it, a set of metaparameters—is just a high-dimensional optimization problem. The input space of this problem comprises all possible metaparameter choices, including which datapoints to train on, what model architecture to use, and how to initialize model weights. The objective function takes in a set of metaparameters, trains a machine learning model according to those metaparameters, and then returns a target metric evaluated on that model (e.g., test accuracy). From this perspective, any procedure for selecting metaparameters—including the typical practice of grid-searching over standard options—is just an optimization algorithm, whose goal is to maximize the objective function with respect to the (high-dimensional) input."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.71,
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+ ],
+ "angle": 0,
+ "content": "Given that selecting metaparameters is \"just\" a high-dimensional optimization problem, a natural tool to consider is the gradient. After all, in many contexts, gradients offer a more effective approach to maximizing high-dimensional functions than grid search. Indeed, for a sufficiently \"well-behaved\" function \\( f(x) \\) with gradient \\( \\nabla f(x) \\), we can optimize \\( f \\) by iteratively updating \\( x \\) in the direction of \\( \\nabla f(x) \\). This insight suggests a generic recipe for selecting metaparameters: first, make the objective differentiable with respect to the metaparameters; second, update via gradient steps."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.801,
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+ 0.862
+ ],
+ "angle": 0,
+ "content": "Now, the idea of using gradients to search for metaparameters is not new. Indeed, there is a substantial line of work that aims to optimize metaparameters (e.g., architectures, regularizers, or data augmentation schemes) with gradient-based methods [MDA15; LSY18; LVD20]. However, such methods have not managed to scale beyond relatively small settings. This state of affairs prompts our main question:"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ ],
+ "angle": 0,
+ "content": "Can we scalably configure model training using gradient-based methods?"
+ },
+ {
+ "type": "page_footnote",
+ "bbox": [
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+ ],
+ "angle": 0,
+ "content": "\\(^{\\dagger}\\)Work done at MIT EECS. Correspondence to {engstrom,ailyas,benchen}@mit.edu."
+ },
+ {
+ "type": "page_number",
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+ "angle": 0,
+ "content": "1"
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+ [
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+ "type": "image",
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+ "type": "image_caption",
+ "bbox": [
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+ "angle": 0,
+ "content": "Figure 1: Our proto-algorithm, metagradient descent (MGD), uses gradients to achieve state-of-the-art performance across a variety of applications, including data selection and data poisoning."
+ },
+ {
+ "type": "title",
+ "bbox": [
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+ "content": "1.1 Contributions"
+ },
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+ "type": "text",
+ "bbox": [
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+ ],
+ "angle": 0,
+ "content": "In this work, we answer this question in the affirmative, adding \"gradient descent on metaparameters\" to the large-scale machine learning toolkit. Along the way, we will face—and address—two main challenges."
+ },
+ {
+ "type": "text",
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+ "content": "First, existing methods for computing metagradients do not scale. In response, we devise an algorithm, REPLAY, that can take metagradients in large-scale settings. By combining reverse-mode autodifferentiation (AD) with an efficient data structure, REPLAY can calculate exact metagradients for models with billions of parameters and thousands of training steps."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "content": "Second, we find that metagradients of standard training routines are not necessarily helpful for optimization, which we connect to non-smoothness of the metaparameter optimization landscape. Borrowing tools from convex optimization, we devise a framework for designing \"metasmooth\" training routines that do admit helpful metagradients."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "content": "Addressing the challenges above unlocks a simple recipe for solving a broad range of machine learning tasks: (a) frame the task as a continuous optimization problem over metaparameters; (b) design a metasmooth training routine; (c) perform metagradient descent (MGD). Applying this recipe:"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "- In the DataComp-small11 competition [GIF+24], we achieve state-of-the-art pre-training data selection for CLIP (2x larger performance improvement than the previous DataComp-small1 leader [Eco24]);"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.137,
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+ "angle": 0,
+ "content": "- In the context of data selection for instruction tuning (as introduced by Xia et al. [XMG+24]), we substantially improve on data selection for Gemma-2B (outperforming existing selection methods as well as full-data training);"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.137,
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+ "angle": 0,
+ "content": "- In the accuracy-degrading data poisoning setting (defined by Huber [Hub64] and pioneered by Lu et al. [LKY22] for deep neural networks), we improve attacks on DNNs by an order of magnitude, dropping CIFAR-10 accuracy from \\(92\\% \\rightarrow 78\\%\\) (the best previous attack [LKY23] only reduces accuracy to \\(91\\%\\));"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "- For the task of hyperparameter optimization, we efficiently find a competitive CIFAR-10 learning rate schedule (matching the performance of a schedule found by grid search)."
+ },
+ {
+ "type": "list",
+ "bbox": [
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+ {
+ "type": "title",
+ "bbox": [
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+ "angle": 0,
+ "content": "2 Scalably computing metagradients"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "In this section we present REPLAY, an algorithm for computing metagradients of large-scale iterative ML algorithms. We first detail the setting, then discuss existing approaches to computing metagradients, and conclude by describing REPLAY."
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+ "content": "2"
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+ ],
+ [
+ {
+ "type": "image",
+ "bbox": [
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+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
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+ "angle": 0,
+ "content": "Training setup"
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
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+ "content": "Trained model"
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+ {
+ "type": "image_caption",
+ "bbox": [
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+ "angle": 0,
+ "content": "Observed behavior"
+ },
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+ "type": "image_caption",
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+ "content": "Figure 2: An illustration of the metagradient. We embed a given aspect of the training setup (e.g., the training dataset, or optimizer hyperparameters) into a continuous metaparameter vector \\( z \\in \\mathbb{R}^d \\). This metaparameter defines a model \\( \\mathcal{A}(z) \\) by way of the learning algorithm \\( \\mathcal{A} \\), which in turn defines an output \\( \\phi(z) \\). The metagradient \\( \\nabla_z \\phi(\\mathcal{A}(z)) \\in \\mathbb{R}^d \\) is the gradient of this model output with respect to the metaparameter."
+ },
+ {
+ "type": "title",
+ "bbox": [
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+ "angle": 0,
+ "content": "2.1 What is a metagradient?"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Training a machine learning model is a two-step process. First, we decide on a training setup—we must pick, for example, a neural network architecture, a training dataset, and an optimizer for training. Second, we apply the algorithm defined by this training setup to train a model."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Our overall goal in this paper is to optimize model behavior as a function of the training setup (or, as we call it, the metaparameters) using gradient-based methods. To this end, we define the following notation:"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "- Let \\(\\mathbf{z} \\in \\mathbb{R}^n\\) be a vector of continuous metaparameters representing the aspects of the training setup we aim to optimize. For example, if we only want to adjust the learning rate and weight decay of SGD then \\(n = 2\\). We handle discrete metaparameters (e.g., choice of training data) by finding a continuous relaxation (e.g., importance weights)."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "- Let \\(\\mathcal{A}\\) be an algorithm mapping \\(\\mathbf{z}\\) to a trained machine learning model; we assume all other aspects of the training setup outside \\(\\mathbf{z}\\) are fixed and thus part of the algorithm \\(\\mathcal{A}\\)."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.137,
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+ "angle": 0,
+ "content": "- Finally, let \\(\\phi\\) be an output function mapping a model \\(\\theta\\) to a vector \\(\\phi(\\theta) \\in \\mathbb{R}\\). For example, \\(\\phi(\\theta)\\) might represent the validation loss of the model \\(\\theta\\). We require that \\(\\phi\\) be differentiable with respect to \\(\\theta\\), but otherwise make no assumptions on \\(\\phi\\)."
+ },
+ {
+ "type": "list",
+ "bbox": [
+ 0.137,
+ 0.405,
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+ ],
+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "With this notation in place, we define the training function \\( f \\coloneqq \\phi \\circ \\mathcal{A} \\) mapping the training setup \\( \\mathbf{z} \\) directly to the output function \\( \\phi \\) evaluated on the corresponding model."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Finally, the metagradient is the gradient of the training function with respect to the metaparameters, \\(\\nabla_{\\mathbf{z}}f(\\mathbf{z})\\). Intuitively, the metagradient defines the \"direction of steepest ascent\" in metaparameter space."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Our focus: iterative algorithms. To efficiently compute the metagradient, we restrict our focus to cases where the algorithm \\(\\mathcal{A}\\) is iterative, i.e., when it can be written in the form"
+ },
+ {
+ "type": "equation",
+ "bbox": [
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+ "angle": 0,
+ "content": "\\[\n\\underbrace {\\mathcal {A} (z) := \\mathbf {s} _ {T}} _ {\\text {m o d e l s t a t e a f t e r T s t e p s}}, \\quad \\text {w h e r e} \\quad \\underbrace {\\mathbf {s} _ {t + 1} : = h _ {t} (\\mathbf {s} _ {t} , \\mathbf {z})} _ {\\text {o p t i m i z e r s t e p t}}. \\tag {1}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ 0.741,
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+ "angle": 0,
+ "content": "Here, \\(\\mathbf{s}_t\\) is the optimizer state at step \\(t\\) (with \\(\\mathbf{s}_0\\) being the initial state) and \\(h_t\\) is the update mapping from state \\(t\\) to state \\(t + 1\\). The form of (1) captures most large-scale training algorithms. For example, if the setup \\(\\mathbf{z} \\in \\mathbb{R}^T\\) is a per-step learning rate, and the algorithm \\(\\mathcal{A}\\) is full batch gradient descent, then each update \\(h_t\\) is"
+ },
+ {
+ "type": "equation",
+ "bbox": [
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+ 0.799,
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+ "angle": 0,
+ "content": "\\[\nh _ {t} (\\mathbf {s} _ {t}, \\mathbf {z}) := \\mathbf {s} _ {t} - z _ {t} \\nabla \\ell (\\mathbf {s} _ {t}),\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.826,
+ 0.884,
+ 0.859
+ ],
+ "angle": 0,
+ "content": "where \\( z_{t} \\) is the learning rate at step \\( t \\), \\( \\ell \\) is the training loss, and the state \\( \\mathbf{s}_t \\) comprises the parameters at step \\( t \\). For more complex algorithms like Adam [KB15], the state \\( \\mathbf{s}_t \\) includes terms like gradient moments."
+ },
+ {
+ "type": "page_number",
+ "bbox": [
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+ "angle": 0,
+ "content": "3"
+ }
+ ],
+ [
+ {
+ "type": "title",
+ "bbox": [
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+ "angle": 0,
+ "content": "2.2 Warmup: Metagradients via autodifferentiation"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "A key primitive we leverage to calculate metagradients is automatic differentiation (AD)—a standard tool for taking gradients through computer-defined functions. AD takes gradients by decomposing functions into elementary operations with known derivatives, then combining these derivatives using the chain rule. Concretely, AD operates in two passes: a \"forward pass,\" which executes the function of interest and stores intermediate products for each elementary operation; and a \"backward pass,\" which calculates the gradient by propagating chains of partial derivatives using these stored products. For the purposes of this paper, we will view AD as a black box that calculates the gradient of a many-to-one function (i.e., any \\( f: \\mathbb{R}^d \\to \\mathbb{R} \\)) at a given point using only a small constant factor more time than calculating the function itself (along with the space cost of storing the necessary forward-pass products)."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.252,
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+ ],
+ "angle": 0,
+ "content": "What does this have to do with metagradients? Well, seeing as how training itself is a computer-defined function, AD is a natural tool for calculating the metagradient. The main challenge, as we discuss in the sequel, is that AD-based approaches to calculating the metagradient tend to be too resource-intensive for the large-scale machine learning algorithms we consider. In the remainder of this section we build up background before finally describing REPLAY, our algorithm for scalably computing (exact) metagradients."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.346,
+ 0.884,
+ 0.391
+ ],
+ "angle": 0,
+ "content": "Approach #1: Direct AD. The direct approach to calculating metagradients exploits the fact that nearly any learning algorithm is itself a sequence of differentiable computer-defined operations—meaning the training function \\( f \\) is also differentiable."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.392,
+ 0.884,
+ 0.468
+ ],
+ "angle": 0,
+ "content": "However, operationalizing this observation to compute metagradients turns out to be challenging. The reason is that AD stores intermediate products for each operation. The amount of data stored thus scales with the number of operations in the function of interest. In the case of our training function \\( f \\), this number encompasses all the operations used to train a machine learning model. As a result, even in a toy scenario like MNIST training, computing metagradients with naive AD would require storing terabytes of data."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.485,
+ 0.884,
+ 0.532
+ ],
+ "angle": 0,
+ "content": "Approach #2: Exploiting structure with step-wise AD. A more efficient method for calculating the metagradient, step-wise AD, leverages the structure of iterative learning algorithms [Wer90; MDA15; FDF+17]. Recall from (1) that such algorithms take the form"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.356,
+ 0.542,
+ 0.64,
+ 0.559
+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathcal {A} (\\mathbf {z}) := \\mathbf {s} _ {T}, \\quad \\text {w h e r e} \\quad \\mathbf {s} _ {t + 1} := h _ {t} (\\mathbf {s} _ {t}, \\mathbf {z}).\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.568,
+ 0.884,
+ 0.6
+ ],
+ "angle": 0,
+ "content": "Algebraic manipulation (in particular, using the chain rule, the law of the total derivative, and the identity \\(\\mathbf{s}_t = h_{t-1}(\\mathbf{s}_{t-1}, \\mathbf{z})\\)) allows us to write the metagradient over an iterative algorithm as"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.318,
+ 0.61,
+ 0.884,
+ 0.679
+ ],
+ "angle": 0,
+ "content": "\\[\n\\frac {\\partial f (\\mathbf {z})}{\\partial \\mathbf {z}} = \\frac {\\partial \\phi (\\mathcal {A} (\\mathbf {z}))}{\\partial \\mathbf {z}} = \\sum_ {t = 1} ^ {T} \\underbrace {\\overbrace {\\partial \\phi (\\mathbf {s} _ {T})} ^ {A _ {t}} . \\overbrace {\\partial \\mathbf {s} _ {t}} ^ {\\partial \\phi (\\mathbf {s} _ {T})}} _ {B _ {t}}, \\tag {2}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.688,
+ 0.884,
+ 0.75
+ ],
+ "angle": 0,
+ "content": "where we have introduced the notation \\( A_{t} \\) and \\( B_{t} \\) for notational convenience. Step-wise AD computes the metagradient by calculating each term in the sum of (2) one at a time. For each term, the main challenge lies in computing \\( A_{t} \\), since given \\( A_{t} \\) we can straightforwardly compute \\( B_{t} \\) (the entire term) by differentiating through a single model update, i.e.,"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.323,
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+ ],
+ "angle": 0,
+ "content": "\\[\nB _ {t} := A _ {t} \\cdot \\frac {\\partial h _ {t - 1} (\\mathbf {s} _ {t - 1} , \\mathbf {z})}{\\partial \\mathbf {z}} = \\frac {\\partial (A _ {t} \\cdot h _ {t - 1} (\\mathbf {s} _ {t - 1} , \\mathbf {z}))}{\\partial \\mathbf {z}},\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.799,
+ 0.884,
+ 0.831
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+ "angle": 0,
+ "content": "which is just a single call to our assumed \"AD oracle\" on the function \\(\\mathbf{z} \\mapsto A_t \\cdot h_{t-1}(\\mathbf{s}_{t-1}, \\mathbf{z})\\). Computing the \\(A_t\\) terms is less straightforward as we need to relate \\(s_t\\) and \\(s_T\\); to do so, we exploit the recurrence"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.292,
+ 0.839,
+ 0.884,
+ 0.873
+ ],
+ "angle": 0,
+ "content": "\\[\nA _ {t} := \\frac {\\partial \\phi (\\mathbf {s} _ {T})}{\\partial \\mathbf {s} _ {t}} = \\frac {\\partial \\phi (\\mathbf {s} _ {T})}{\\partial \\mathbf {s} _ {t + 1}} \\cdot \\frac {\\partial h _ {t} (\\mathbf {s} _ {t} , \\mathbf {z})}{\\partial \\mathbf {s} _ {t}} = \\frac {\\partial \\left(A _ {t + 1} \\cdot h _ {t} (\\mathbf {s} _ {t} , \\mathbf {z})\\right)}{\\partial \\mathbf {s} _ {t}}, \\tag {3}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.882,
+ 0.884,
+ 0.914
+ ],
+ "angle": 0,
+ "content": "making \\( A_{t} \\) straightforward to compute (again, a single \"AD oracle\" call) given \\( A_{t+1} \\). Step-wise AD exploits this fact to successively calculate the gradient with respect to each state, from state \\( T \\) down to state 0."
+ },
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.494,
+ 0.936,
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+ "angle": 0,
+ "content": "4"
+ }
+ ],
+ [
+ {
+ "type": "image",
+ "bbox": [
+ 0.148,
+ 0.111,
+ 0.852,
+ 0.375
+ ],
+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 0.111,
+ 0.388,
+ 0.884,
+ 0.482
+ ],
+ "angle": 0,
+ "content": "Figure 3: The lazy \\( k \\)-ary tree structure for traversing optimizer states in reverse order, with \\( k = 2 \\). Recall that \\( n \\) is the number of states (parameterized such that \\( n = T + 1 \\)). Each node represents the correspondingly numbered state. We give an example of the traversal using the blue arrows in the figure, which denote the traversal path up to state \\( s_{\\frac{3n}{4} + 1} \\). The gray cylinders indicate the states that are stored when the traversal is at state \\( s_{\\frac{3n}{4} + 1} \\); the other states are not stored at this point in the traversal. Traversing this structure requires storing \\( \\mathcal{O}(\\log(n)) \\) state and computing \\( \\mathcal{O}(n \\log(n)) \\) optimizer steps—compared to \\( n \\) for simply training."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.506,
+ 0.884,
+ 0.612
+ ],
+ "angle": 0,
+ "content": "Bringing these ingredients together, the algorithm executes as follows. As a preprocessing step, it trains the model and stores all intermediate states \\(\\mathbf{s}_0,\\dots ,\\mathbf{s}_T\\). Then, the algorithm calculates and sums the terms in (2). It first computes \\(A_{T}\\coloneqq \\partial \\phi (\\mathbf{s}_{T}) / \\mathbf{s}_{T}\\), the gradient of the output function \\(\\phi\\) with respect to the final state. Then, the algorithm steps through \\(\\mathbf{s}_{T - 1},\\ldots ,\\mathbf{s}_0\\) in reverse order, calculating (a) the gradient with respect to each state \\(A_{t}\\) (via (3)) and (b) the gradient with respect to \\(\\mathbf{z}\\) at that step \\(B_{t}\\) (via (2), using the previously calculated gradient with respect to that state). AD calculates both quantities--each requires differentiating over only one train step. Finally, the algorithm returns the final metagradient as the sum of the terms."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ 0.612,
+ 0.884,
+ 0.645
+ ],
+ "angle": 0,
+ "content": "Despite improving storage overhead compared to \"direct AD\", step-wise AD is still too space-intensive at scale. After all, this algorithm saves every optimizer state."
+ },
+ {
+ "type": "title",
+ "bbox": [
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+ "angle": 0,
+ "content": "2.3 REPLAY"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ 0.687,
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+ ],
+ "angle": 0,
+ "content": "REPLAY is our algorithm for efficiently and exactly computing metagradients. It uses \\(\\mathcal{O}(k\\log_k(T))\\) space and requires running the learning algorithm \\(\\mathcal{A}\\) a total of \\(1 + \\log_{k}(T)\\) times, with \\(k\\) a user-chosen constant. The main idea is to make the space-intensive subroutine of step-wise AD—a reverse-order traversal of the optimizer states at each step—much more efficient. After all, step-wise AD stores all the states to reverse traverse them. REPLAY modifies step-wise AD to traverse states in less space by exploiting a simple observation: when training is deterministic, one can reinstantiate an optimizer state \\(\\mathbf{s}_t\\) by \"replaying\" training from a fixed point \\(t' < t\\) at the compute cost of \\(t - t'\\) training steps. For example, one simple scheme saves every other state, then \"replays\" the remaining states when (reverse) traversing; this routine stores \\(T/2\\) states but computes an extra \\(T/2\\) model updates compared to storing all the states."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.824,
+ 0.884,
+ 0.903
+ ],
+ "angle": 0,
+ "content": "REPLAY performs a reverse-order traversal the optimizer states while balancing the compute cost of \"replaying\" training with the storage cost of saving states. We use a combination of deterministic training (fixing data ordering, data augmentation, and any other randomness in the training process) and an efficient data structure (similar to a segment tree; see Figure 3) to reverse-order traverse the optimizer states with \\(\\mathcal{O}(k\\log_k(T))\\) space and an additional \\(T\\log_k(T)\\) model steps."
+ },
+ {
+ "type": "page_number",
+ "bbox": [
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+ "angle": 0,
+ "content": "5"
+ }
+ ],
+ [
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.092,
+ 0.888,
+ 0.214
+ ],
+ "angle": 0,
+ "content": "Specifically, REPLAY recursively saves and replays training states. The algorithm splits the training trajectory into \\( k \\) segments, performs the full training routine while saving only the start of each segment, then recurses into each segment (in reverse) to retrieve the states in reverse-order. The recursion depth bottoms out at \\( \\log_k(T) \\), at which point the algorithm has \\( k \\) consecutive optimizer states in memory; the algorithm then backpropagates along this segment, before deleting all these states from memory and then reinstantating the next \\( k \\)-length segment of optimizer states. We provide additional details on the algorithm in Appendix A.2. REPLAY unlocks computing large-scale metagradients by requiring only logarithmic storage and additional compute time."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.222,
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+ ],
+ "angle": 0,
+ "content": "Remark 1 (Connection to rematerialization). In a broad sense, both REPLAY and step-wise AD above can be viewed as special cases of a classical approach in AD (and computing broadly) known as rematerialization [CAC+81; BCT92; ZP00; GW08; CXZ+16]. To our knowledge, however, REPLAY is the first application of this particular rematerialization technique to the problem of computing metagradients through model training."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Remark 2 (Reversible learning). An alternative approach to calculating metagradients that does not save any state is reversible learning [MDA15], for which one can \"invert\" previous training states from future ones. We focus here on general (non-reversible) learning algorithms for two reasons: first, even simple algorithms such as SGD without momentum are non-reversible; second, reversibility in practice introduces numerical precision issues."
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.112,
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+ "angle": 0,
+ "content": "3 Designing metasmooth training routines"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.41,
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+ ],
+ "angle": 0,
+ "content": "Given a training function \\( f \\), REPLAY enables us to compute metagradients \\( \\nabla f(\\mathbf{z}) \\) for any setup \\( \\mathbf{z} \\). Can we immediately use these metagradients to optimize model training setups? The answer is (generally) no: we find that applying REPLAY to a function \\( f \\) representing a standard model training and evaluation routine yields metagradients that are often \\( \\pm \\infty \\)-valued and generally unhelpful for optimization. Indeed, previous work has observed similar issues optimizing over even (very) small-scale training [BSF94; Pea96; MDA15]."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.486,
+ 0.884,
+ 0.562
+ ],
+ "angle": 0,
+ "content": "In this section, we show that an underlying source of the issue is the landscape of the metaparameter optimization problem. We then present a framework for modifying standard learning algorithms to admit useful metagradient, i.e., to be metasmooth. To use a familiar analogy: just as residual connections and improved initialization schemes can improve optimization in standard deep learning algorithms, our framework introduces an analogous set of modifications to enable optimization with metagradient."
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.112,
+ 0.581,
+ 0.55,
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+ ],
+ "angle": 0,
+ "content": "3.1 The metaparameter optimization landscape"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.607,
+ 0.886,
+ 0.655
+ ],
+ "angle": 0,
+ "content": "We first review the notion of smoothness from optimization theory, and then adapt it to the setting of metagradients. The resulting metasmoothness metric allows us to quantify (and later, improve) the amenability of the metaparameter optimization problem to gradient-based methods."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.671,
+ 0.884,
+ 0.718
+ ],
+ "angle": 0,
+ "content": "Smoothness. In optimization theory, the basic property of a function that controls how effectively it can be optimized with first-order methods is smoothness. Specifically, a function \\( f(\\mathbf{z}) \\) is \\( \\beta \\)-smooth at a point \\( \\mathbf{z} \\) if its gradient \\( \\nabla f \\) satisfies the property that"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.327,
+ 0.727,
+ 0.884,
+ 0.746
+ ],
+ "angle": 0,
+ "content": "\\[\n\\left\\| \\nabla f (\\mathbf {z}) - \\nabla f \\left(\\mathbf {z} ^ {\\prime}\\right) \\right\\| \\leq \\beta \\cdot \\left\\| \\mathbf {z} - \\mathbf {z} ^ {\\prime} \\right\\| \\quad \\text {f o r a l l} \\mathbf {z} ^ {\\prime}, \\tag {4}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.757,
+ 0.883,
+ 0.802
+ ],
+ "angle": 0,
+ "content": "or in other words, if its gradient does not change too quickly around \\(\\mathbf{z}\\). To motivate this definition: if a function \\(f\\) is \\(\\beta\\)-smooth at \\(\\mathbf{z}\\), then a step of gradient descent with step size \\(1 / \\beta\\) will successfully decrease the value of the function:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.347,
+ 0.81,
+ 0.65,
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+ ],
+ "angle": 0,
+ "content": "\\[\nf \\left(\\mathbf {z} - \\frac {1}{\\beta} \\nabla f (\\mathbf {z})\\right) \\leq f (\\mathbf {z}) - \\frac {1}{2 \\beta} \\| \\nabla f (\\mathbf {z}) \\| ^ {2}.\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.113,
+ 0.854,
+ 0.633,
+ 0.872
+ ],
+ "angle": 0,
+ "content": "This guarantee makes \\(\\beta\\)-smoothness a good measure of gradient utility."
+ },
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.494,
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+ ],
+ "angle": 0,
+ "content": "6"
+ }
+ ],
+ [
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
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+ ],
+ "angle": 0,
+ "content": "Metasmoothness. There are two main challenges in adapting the smoothness property to the metagradient setting. First, evaluating (4) requires a search over all possible \\(\\mathbf{z}'\\), which is infeasible. Second, even if we could exactly evaluate the left-hand side of (4), it would be difficult to disentangle non-smoothness of the training function \\(f\\) from potential error in metagradient computation (e.g., a numerically unstable operation in REPLAY)."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.168,
+ 0.881,
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+ ],
+ "angle": 0,
+ "content": "To sidestep these issues, we propose a metric called metasmoothness, given in Definition 1. Metasmoothness is cheap to compute—requiring only three evaluations of the training function—and does not rely on metagradient computation. For the remainder of this section, we fix a small constant \\( h > 0 \\), and define the corresponding finite-differences estimator of the directional derivative \\( \\Delta_f \\) as"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.389,
+ 0.239,
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+ ],
+ "angle": 0,
+ "content": "\\[\n\\Delta_ {f} (\\mathbf {z}; \\mathbf {v}) := \\frac {f (\\mathbf {z} + h \\mathbf {v}) - f (\\mathbf {z})}{h}.\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
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+ 0.881,
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+ ],
+ "angle": 0,
+ "content": "Definition 1 (Metasmoothness of \\( f \\) at \\( \\mathbf{z} \\) towards \\( \\mathbf{v} \\)). Consider a training function \\( f \\) mapping metaparameters \\( \\mathbf{z} \\in \\mathbb{R}^n \\) to model output \\( f(\\mathbf{z}) \\in \\mathbb{R} \\). Given a metaparameter \\( \\mathbf{z} \\) and a vector \\( \\mathbf{v} \\in \\mathbb{R}^n \\), the metasmoothness of \\( f \\) at \\( \\mathbf{z} \\) towards \\( \\mathbf{v} \\) is given by"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.369,
+ 0.333,
+ 0.882,
+ 0.368
+ ],
+ "angle": 0,
+ "content": "\\[\nS _ {h, \\mathbf {v}} (f; \\mathbf {z}) := \\left| \\frac {\\Delta_ {f} (\\mathbf {z} + h \\mathbf {v}) - \\Delta_ {f} (\\mathbf {z})}{h} \\right|. \\tag {5}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.377,
+ 0.881,
+ 0.409
+ ],
+ "angle": 0,
+ "content": "Definition 1 measures the rate of change of the derivative of \\( f(\\mathbf{z}) \\) in the direction of a given vector \\( \\mathbf{v} \\), and is therefore related to \\( \\beta \\)-smoothness in that:"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.125,
+ 0.415,
+ 0.86,
+ 0.432
+ ],
+ "angle": 0,
+ "content": "(a) If \\( f \\) is \\( \\beta \\)-smooth at \\( \\mathbf{z} \\), then \\( S_{h,\\mathbf{v}}(f;\\mathbf{z}) \\leq \\beta \\) for any \\( (h,\\mathbf{v}) \\) (so Definition 1 is necessary for smoothness)."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.125,
+ 0.441,
+ 0.88,
+ 0.472
+ ],
+ "angle": 0,
+ "content": "(b) If \\(\\lim_{h\\to 0}S_{h,\\mathbf{v}}(f;\\mathbf{z})\\leq \\beta\\) for all \\(\\mathbf{z}\\in \\mathbb{R}^n\\) and \\(\\mathbf{v}\\in \\mathbb{S}^{n - 1}\\), then \\(f\\) is \\(\\beta\\)-smooth everywhere (so a global version of Definition 1 is sufficient for smoothness)."
+ },
+ {
+ "type": "list",
+ "bbox": [
+ 0.125,
+ 0.415,
+ 0.88,
+ 0.472
+ ],
+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.491,
+ 0.881,
+ 0.628
+ ],
+ "angle": 0,
+ "content": "Empirical metasmoothness. Definition 1 lets us measure the meta-smoothness of a training function \\( f \\) at a particular metaparameter \\( \\mathbf{z} \\) (towards a direction \\( \\mathbf{v} \\)). This definition, however, has two shortcomings. First, recall that the training function \\( f \\) is a composition of a learning algorithm \\( \\mathcal{A} \\) and an output function \\( \\phi \\), so the smoothness of \\( f \\) depends on that of both \\( \\mathcal{A} \\) and \\( \\phi \\) (in particular, \\( \\frac{\\partial f}{\\partial \\mathbf{z}} = \\frac{\\partial \\phi}{\\partial \\mathcal{A}} \\cdot \\frac{\\partial \\mathcal{A}}{\\partial \\mathbf{z}} \\)). Since the output function \\( \\phi \\) might be unknown ahead of time, we are most interested in measuring the overall metasmoothness of a learning algorithm \\( \\mathcal{A} \\). Second, while the result of (5) does have a concrete basis in optimization theory, it may not be easy to interpret in practice (e.g., what does \\( S = 200 \\) mean?). We address both issues simultaneously by (a) proposing an interpretable \"binarized\" version of Definition 1, and (b) studying metasmoothness in the space of model parameters \\( \\theta \\), instead of the output space."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.635,
+ 0.881,
+ 0.68
+ ],
+ "angle": 0,
+ "content": "Definition 2 (Empirical metasmoothness of \\(\\mathcal{A}\\)). Let \\(\\mathcal{A}\\) be a learning algorithm which maps metaparameters \\(\\mathbf{z} \\in \\mathbb{R}^n\\) to model parameters \\(\\theta \\in \\mathbb{R}^d\\), let \\(\\mathbf{z}\\) be a metaparameter vector, and let \\(\\mathbf{v}\\) be a given direction. Let \\(\\mathbf{d} \\in \\mathbb{R}^d\\) be the per-coordinate variation in \\(\\theta\\), i.e.,"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.404,
+ 0.69,
+ 0.589,
+ 0.707
+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathbf {d} = \\left| \\mathcal {A} (\\mathbf {z} + 2 h \\mathbf {v}) - \\mathcal {A} (\\mathbf {z}) \\right|\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.113,
+ 0.716,
+ 0.502,
+ 0.733
+ ],
+ "angle": 0,
+ "content": "The empirical \\((h,\\mathbf{v})\\)-metasmoothness of \\(\\mathcal{A}\\) at \\(\\mathbf{z}\\) is given by"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.256,
+ 0.742,
+ 0.884,
+ 0.775
+ ],
+ "angle": 0,
+ "content": "\\[\n\\widehat {S} _ {h, \\mathbf {v}} (\\mathcal {A}; \\mathbf {z}) = \\operatorname {s i g n} \\left(\\Delta_ {\\mathcal {A}} (\\mathbf {z}; \\mathbf {v})\\right) ^ {\\top} \\cdot \\operatorname {d i a g} \\left(\\frac {\\mathbf {d}}{\\| \\mathbf {d} \\| _ {1}}\\right) \\cdot \\operatorname {s i g n} \\left(\\Delta_ {\\mathcal {A}} (\\mathbf {z} + h \\mathbf {v}; \\mathbf {v})\\right), \\tag {6}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.113,
+ 0.783,
+ 0.355,
+ 0.8
+ ],
+ "angle": 0,
+ "content": "weights each parameter by its range."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.807,
+ 0.881,
+ 0.913
+ ],
+ "angle": 0,
+ "content": "Intuitively, (6) measures the agreement in sign between the (finite-difference approximation of the) metagradient in the direction of \\(\\mathbf{v}\\) at \\(\\mathbf{z}\\) and at \\(\\mathbf{z} + h\\mathbf{v}\\), averaged across parameter coordinates and weighted by the variation in each coordinate. Taking a weighted average of sign agreements ensures that \\(\\widehat{S} \\in [-1,1]\\) (making it easier to interpret than Definition 1). The \\(\\mathrm{diag}(\\mathsf{d} / \\| \\mathsf{d}\\|_1)\\) term weights each agreement proportionally to the scale of the corresponding parameter change (downweighting, e.g., coordinates \\(i\\) that are essentially constant). Finally, observe that Definition 2 is efficient to compute in practice: it requires only three calls to the learning algorithm \\(\\mathcal{A}\\)."
+ },
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.494,
+ 0.936,
+ 0.504,
+ 0.947
+ ],
+ "angle": 0,
+ "content": "7"
+ }
+ ],
+ [
+ {
+ "type": "image",
+ "bbox": [
+ 0.125,
+ 0.087,
+ 0.388,
+ 0.314
+ ],
+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 0.258,
+ 0.324,
+ 0.278,
+ 0.337
+ ],
+ "angle": 0,
+ "content": "(a)"
+ },
+ {
+ "type": "image",
+ "bbox": [
+ 0.399,
+ 0.088,
+ 0.589,
+ 0.282
+ ],
+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "image",
+ "bbox": [
+ 0.602,
+ 0.087,
+ 0.876,
+ 0.313
+ ],
+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 0.702,
+ 0.324,
+ 0.724,
+ 0.337
+ ],
+ "angle": 0,
+ "content": "(b)"
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 0.111,
+ 0.349,
+ 0.884,
+ 0.427
+ ],
+ "angle": 0,
+ "content": "Figure 4: (a) For a variety of training configurations of a ResNet-9 model, we plot metasmoothness (Def. 2) against test accuracy. Strategies such as increasing width, placing batch normalization before activations, and scaling down network outputs consistently improve metasmoothness, at a minor cost to accuracy. (b) Smoother training configurations can be optimized via metagradients more effectively. Here, as in Section 4.3, we use metagradients to gradient ascend on validation loss."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.447,
+ 0.884,
+ 0.495
+ ],
+ "angle": 0,
+ "content": "Remark 3. Ideally, recalling the smoothness definition (4), we would evaluate metasmoothness in all possible directions \\(\\mathbf{v}\\) and all points \\(\\mathbf{z}\\). Empirically, we find in the sequel (Section 3.2) that this single-direction approximation at a single point \\(\\mathbf{z}\\) still yields a useful estimate of metasmoothness (e.g., one that correlates with metagradients utility)."
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.112,
+ 0.518,
+ 0.55,
+ 0.536
+ ],
+ "angle": 0,
+ "content": "3.2 Estimating and improving metasmoothness"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.543,
+ 0.884,
+ 0.633
+ ],
+ "angle": 0,
+ "content": "Having established a method for quantifying metasmoothness, we turn to the practical question: how can we design learning algorithms that are amenable to metagradient optimization? To answer this question, we introduce a straightforward framework: given a learning algorithm, explore a fixed menu of possible modifications to the training setup, and choose the combination that maximizes empirical metasmoothness. In practice, we find that this framework allows us to slightly modify learning algorithms in a way that makes them amenable to first-order methods."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.634,
+ 0.884,
+ 0.695
+ ],
+ "angle": 0,
+ "content": "As a case study, we study the task of training ResNet-9 on the CIFAR-10 dataset [Kri09]. We let the metaparameters \\(\\mathbf{z}\\) be a perturbation to the pixels of 1000 random training images (so \\(\\mathbf{z} \\in \\mathbb{R}^{1000 \\times 32 \\times 32 \\times 3}\\)). We estimate the empirical metasmoothness of different learning algorithms \\(\\mathcal{A}\\) at \\(\\mathbf{z} = \\mathbf{0}\\) using Definition 2. Concretely, we proceed as follows for each learning algorithm \\(\\mathcal{A}\\):"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.132,
+ 0.699,
+ 0.667,
+ 0.716
+ ],
+ "angle": 0,
+ "content": "1. Let \\( \\mathbf{z}_0 = \\mathbf{0} \\) be the metaparameter corresponding to the original dataset."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.132,
+ 0.722,
+ 0.524,
+ 0.739
+ ],
+ "angle": 0,
+ "content": "2. Sample a random perturbation vector \\(\\mathbf{v} \\sim \\mathcal{N}(0,1)\\)."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.132,
+ 0.745,
+ 0.507,
+ 0.762
+ ],
+ "angle": 0,
+ "content": "3. Compute the empirical metasmoothness (6), i.e.,"
+ },
+ {
+ "type": "list",
+ "bbox": [
+ 0.132,
+ 0.699,
+ 0.667,
+ 0.762
+ ],
+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.161,
+ 0.768,
+ 0.884,
+ 0.8
+ ],
+ "angle": 0,
+ "content": "(a) Let \\(\\theta_0\\coloneqq \\mathcal{A}(\\mathbf{z}_0)\\), \\(\\theta_h\\coloneqq \\mathcal{A}(\\mathbf{z}_0 + h\\cdot \\mathbf{v})\\), and \\(\\theta_{2h}\\coloneqq \\mathcal{A}(\\mathbf{z}_0 + 2h\\cdot \\mathbf{v})\\) be the model parameters that result from training with training dataset perturbations \\(\\mathbf{z}_0,\\mathbf{z}_0 + h\\mathbf{v}\\), and \\(\\mathbf{z}_0 + 2h\\mathbf{v}\\), respectively."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.161,
+ 0.801,
+ 0.471,
+ 0.817
+ ],
+ "angle": 0,
+ "content": "(b) Compute the approximate derivatives"
+ },
+ {
+ "type": "list",
+ "bbox": [
+ 0.161,
+ 0.768,
+ 0.884,
+ 0.817
+ ],
+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.321,
+ 0.823,
+ 0.75,
+ 0.841
+ ],
+ "angle": 0,
+ "content": "\\[\n\\Delta_ {\\mathcal {A}} (\\mathbf {z} _ {0}; \\mathbf {v}) = \\left(\\theta_ {h} - \\theta_ {0}\\right) / h, \\quad \\Delta_ {\\mathcal {A}} (\\mathbf {z} _ {0} + h \\mathbf {v}; \\mathbf {v}) = \\left(\\theta_ {2 h} - \\theta_ {h}\\right) / h.\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.162,
+ 0.847,
+ 0.884,
+ 0.864
+ ],
+ "angle": 0,
+ "content": "(c) Compute the weighting vector \\(\\mathbf{d} = |\\theta_{2h} - \\theta_0|\\), and compute the average metasmoothness (6), i.e.,"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.284,
+ 0.868,
+ 0.787,
+ 0.902
+ ],
+ "angle": 0,
+ "content": "\\[\n\\widehat {S} _ {h, \\mathbf {v}} (\\mathcal {A}; z _ {0}) = \\operatorname {s i g n} \\left(\\Delta_ {\\mathcal {A}} \\left(\\mathbf {z} _ {0} + h \\mathbf {v}; \\mathbf {v}\\right)\\right) ^ {\\top} \\cdot \\operatorname {d i a g} \\left(\\frac {\\mathbf {d}}{\\| \\mathbf {d} \\| _ {1}}\\right) \\cdot \\operatorname {s i g n} \\left(\\Delta_ {\\mathcal {A}} \\left(\\mathbf {z} _ {0}; \\mathbf {v}\\right)\\right).\n\\]"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.494,
+ 0.936,
+ 0.504,
+ 0.948
+ ],
+ "angle": 0,
+ "content": "8"
+ }
+ ],
+ [
+ {
+ "type": "image",
+ "bbox": [
+ 0.152,
+ 0.09,
+ 0.49,
+ 0.284
+ ],
+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "image",
+ "bbox": [
+ 0.502,
+ 0.091,
+ 0.837,
+ 0.282
+ ],
+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "image",
+ "bbox": [
+ 0.152,
+ 0.291,
+ 0.49,
+ 0.481
+ ],
+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "image",
+ "bbox": [
+ 0.501,
+ 0.291,
+ 0.839,
+ 0.481
+ ],
+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 0.111,
+ 0.494,
+ 0.884,
+ 0.586
+ ],
+ "angle": 0,
+ "content": "Figure 5: The effect of metasmoothness on the optimization landscape. Each plot above visualizes the loss landscape of a (deterministic) learning algorithm \\(\\mathcal{A}\\), with the \\(x\\)- and \\(y\\)-axes representing additive perturbations to 1000 examples in the training set and the \\(z\\)-axis representing the resulting model's loss on the test example given in the title. In each row, the left plot is a non-smooth algorithm, and the right plot is a smooth algorithm (as per Definition 2) evaluated on the same example. Overall, empirical metasmoothness seems to strongly correlate with qualitative landscape smoothness. See Figure 12 for more examples."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.612,
+ 0.884,
+ 0.732
+ ],
+ "angle": 0,
+ "content": "Metasmooth learning algorithms. We apply the procedure above to estimate the metasmoothness of learning algorithms induced by different design choices (batch size, network width, BatchNorm placement, gradient scaling), and report the results in Figure 4 (left). On one hand, \"standard\" learning algorithms (i.e., those designed without metasmoothness in mind) are not metasmooth. On the other hand, our investigation reveals central factors driving metasmoothness. In addition to \"standard\" hyperparameters such as batch size and network width playing a role, we find that placing Batch Normalization layers prior to nonlinearities (instead of after) and scaling the final layer output are both crucial to metasmoothness. Note that the modifications we consider above are not exhaustive—see Appendix E for the full training setup."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.112,
+ 0.733,
+ 0.884,
+ 0.779
+ ],
+ "angle": 0,
+ "content": "Finally, in Figure 5, we plot the optimization landscape of both metasmooth (right) and non-metasmooth (left) models. We find that the landscapes of metasmooth models are much smoother and—qualitatively—more straightforward to optimize."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.797,
+ 0.885,
+ 0.904
+ ],
+ "angle": 0,
+ "content": "Metasmoothness/performance tradeoffs? Figure 4 (left) relates metasmoothness to model accuracy for the considered learning algorithms. While there is no clear trend, the top-performing learning algorithms are not always metasmooth. However, the trade-off is not too severe: the most metasmooth algorithms still achieve near-optimal accuracy. Furthermore, it is possible that with additional searching we could identify even more accurate metasmooth models. Taken together with our previous experiment, our results suggest that jointly searching over metasmoothness and model accuracy is a general recipe for designing learning algorithms that are both performant and metasmooth. Finally, as we discuss in Section 5, a fruitful avenue"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
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+ ],
+ "angle": 0,
+ "content": "9"
+ }
+ ],
+ [
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
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+ "angle": 0,
+ "content": "for future work may be to design metasmooth learning algorithms directly, i.e., without relying on stability heuristics or grid search."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.141,
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+ "angle": 0,
+ "content": "Does metasmoothness aid downstream optimization? Recall that our motivation for studying metasmoothness is to develop learning algorithms that we can optimize the metaparameters of via metagradients (using first-order methods). We started with the notion of \\(\\beta\\)-smoothness from optimization theory, and we adapted it to the setting of metagradients by making a series of approximations and modifications. The final question we address is: does our final notion of metasmoothness actually predict the utility of metagradients for optimization? Figure 4 (right) demonstrates that metasmoothness strongly predicts our ability to optimize the metaparameters of a given learning algorithm. We use metagradients (computed by REPLAY) to gradient ascend on validation loss with respect to the metaparameters \\(\\mathbf{z}\\), and measure the change in model loss."
+ },
+ {
+ "type": "title",
+ "bbox": [
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+ ],
+ "angle": 0,
+ "content": "4 Applications"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.333,
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+ ],
+ "angle": 0,
+ "content": "In this section, apply metagradients to three problems in machine learning: selecting training data, poisoning training data, and searching for hyperparameters. In each setting we follow the same recipe: we frame the task as an optimization problem, modify the learning algorithm of interest to be smooth, then solve by first-order optimizing with meta-gradients—which we refer to, in a catch-all manner across algorithms, as metagradient descent (MGD). In particular: we substantially improve on existing dataset selection methods (Section 4.1, Section 4.2), perform the first effective accuracy-degrading data poisoning attack (Section 4.3), and discover one-cycle learning rate schedules with MGD (Section 4.4)."
+ },
+ {
+ "type": "title",
+ "bbox": [
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+ "angle": 0,
+ "content": "4.1 Selecting multimodal training data"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ ],
+ "angle": 0,
+ "content": "Curating a training dataset from a mass of unfiltered data is a necessary and influential step in any large-scale machine learning pipeline. Deciding how to curate such a dataset is a challenging problem that has attracted substantial recent interest [FiW+22; ATS+23; EFM24; GIF+24]. In this section, we frame pre-training data selection as an optimization problem, and then solve this problem by first-order optimizing with metagradient. Applying our method to the DataComp-small benchmark [GIF+24], we greatly improve on the state-of-the-art (our improvement over state-of-the-art is roughly the same as the improvement of state-ofthe-art over training on random data)."
+ },
+ {
+ "type": "title",
+ "bbox": [
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+ "angle": 0,
+ "content": "4.1.1 Setup"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "The goal of dataset selection is to choose a training data subset (out of a broad pool of data) that maximizes trained machine learning model performance. Given this goal, dataset selection has a natural interpretation as a combinatorial metaparameter optimization problem. In particular, in the language of Section 2.1, for a training set of size \\( n \\), let"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.126,
+ 0.703,
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+ "angle": 0,
+ "content": "(a) the metaparameters \\(\\mathbf{c} \\in \\mathcal{C} \\coloneqq \\mathbb{Z}_{\\geq 0}^{n}\\) be non-negative data counts representing the number of times each training sample repeats in the training data;"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ 0.744,
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+ ],
+ "angle": 0,
+ "content": "(b) the algorithm \\(\\mathcal{A}\\) be a standard large-scale learning procedure, which runs on a training set comprising \\(c_{i}\\) copies of each sample \\(i\\) for \\(i\\in [n]\\);"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.126,
+ 0.785,
+ 0.735,
+ 0.8
+ ],
+ "angle": 0,
+ "content": "(c) the output function \\(\\phi\\) be the loss of the trained model on a target distribution \\(D\\)."
+ },
+ {
+ "type": "list",
+ "bbox": [
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+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Then, defining \\( f(\\mathbf{c}) \\coloneqq \\phi(\\mathcal{A}(\\mathbf{c})) \\) (as in Section 2.1), our goal is to find the data counts \\( \\mathbf{c}^* \\) that solve"
+ },
+ {
+ "type": "equation",
+ "bbox": [
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+ 0.837,
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+ "angle": 0,
+ "content": "\\[\n\\mathbf {c} ^ {*} := \\underset {\\mathbf {c} \\in \\mathcal {C}} {\\arg \\min } f (\\mathbf {c}). \\tag {7}\n\\]"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
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+ "angle": 0,
+ "content": "10"
+ }
+ ],
+ [
+ {
+ "type": "title",
+ "bbox": [
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+ "angle": 0,
+ "content": "4.1.2 Gradient descent on training data"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Metagradients let us directly minimize the target task loss (7) with respect to the choice of training data. At a high level, our algorithm operates as follows: we start with a randomly chosen set of training data, then iteratively update the dataset selection using metagradients with respect to importance weights placed on each training datapoint. The specifics of our method are in Algorithm 1; we describe its core ideas below."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Idea 1: A surrogate algorithm. We cannot use metagradients to optimize (7) directly, because the metaparameters of interest \\( \\mathbf{c} \\) are discrete counts (and so the algorithm \\( \\mathcal{A} \\) is non-differentiable with respect to \\( \\mathbf{c} \\)). To circumvent this problem, we relax \\( \\mathcal{A} \\): we define a surrogate algorithm \\( \\mathcal{A}_{\\mathbf{c}}^{\\prime} \\) that takes in a continuous metaparameter \\( \\mathbf{z} \\in \\mathbb{R}^{n} \\), whose metagradient we can compute, then optimize using the metagradient on \\( \\mathcal{A}_{\\mathbf{c}}^{\\prime} \\)."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "This surrogate learning algorithm \\(\\mathcal{A}_{\\mathrm{c}}^{\\prime}\\) maps a metaparameter \\(\\mathbf{z} \\in \\mathbb{R}^{n}\\) (representing a perturbation to training data weights) to a machine learning model. The surrogate is defined by a set of counts \\(\\mathbf{c} \\in \\mathbb{Z}_{+}^{n}\\), and a hyperparameter \\(k\\) denoting a specific training iteration, both of which we bake into the surrogate algorithm itself. Given a metaparameter \\(\\mathbf{z} \\in \\mathbb{R}^{n}\\), the algorithm \\(\\mathcal{A}_{\\mathrm{c}}^{\\prime}\\) trains a model \"as usual\" using the fixed counts \\(\\mathbf{c}\\). That is, it makes \\(c_{i}\\) copies of each training sample \\(i\\), shuffles and partitions the data into batches, and then at each iteration minimizes the batch loss with a step—just as the original learning algorithm \\(\\mathcal{A}\\). At iteration \\(k\\), however, in addition to the original loss on the \\(k\\)-th batch, the algorithm upweights each training sample \\(i\\) according to the metaparameter \\(z_{i}\\). In other words, the objective at iteration \\(t\\) of the surrogate algorithm \\(\\mathcal{A}_{\\mathrm{c}}^{\\prime}\\) is"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.304,
+ 0.392,
+ 0.693,
+ 0.432
+ ],
+ "angle": 0,
+ "content": "\\[\n\\ell_ {t} ^ {\\prime} (\\theta) := \\left\\{ \\begin{array}{l l} \\sum_ {x \\in t ^ {\\text {t h}} \\text {b a t c h}} \\ell (x; \\theta) & \\text {i f} t \\neq k \\\\ \\sum_ {x \\in t ^ {\\text {t h}} \\text {b a t c h}} \\ell (x; \\theta) + \\sum_ {i = 1} ^ {n} z _ {i} \\ell (x _ {i}; \\theta) & \\text {i f} t = k \\end{array} \\right.\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.113,
+ 0.437,
+ 0.453,
+ 0.453
+ ],
+ "angle": 0,
+ "content": "where \\(\\ell (x;\\theta)\\) is the training loss on example \\(x\\)"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.112,
+ 0.453,
+ 0.884,
+ 0.499
+ ],
+ "angle": 0,
+ "content": "Observe that when \\( \\mathbf{z} = \\mathbf{0}_n \\), the algorithm \\( \\mathcal{A}_{\\mathbf{c}}' \\) is identical to the standard learning algorithm \\( \\mathcal{A} \\). And while \\( \\mathcal{A} \\) was a function of (nondifferentiable) discrete data counts \\( \\mathbf{c} \\), \\( \\mathcal{A}_{\\mathbf{c}}' \\) is differentiable with respect to its input \\( \\mathbf{z} \\), and so we can compute the metagradient"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.411,
+ 0.509,
+ 0.585,
+ 0.531
+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathbf {g} := \\nabla_ {\\mathbf {z}} \\phi \\big (\\mathcal {A} _ {\\mathbf {c}} ^ {\\prime} (\\mathbf {z}) \\big) \\big | _ {\\mathbf {z} = \\mathbf {0} _ {n}}.\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.112,
+ 0.54,
+ 0.884,
+ 0.618
+ ],
+ "angle": 0,
+ "content": "Intuitively, the entries of the metagradient \\(\\mathbf{g}\\) capture the effect of adding an infinitesimal amount of each training sample \\(i\\) to the training data at iteration \\(k\\). A positive entry \\(g_{i}\\) indicates that adding an infinitesimal amount of sample \\(i\\) to the training data would increase the loss, and a negative entry indicates that adding an infinitesimal amount of sample \\(i\\) to the training data would decrease the loss; the slot at \\(i\\) represents the (estimated) effect of adding a copy of sample \\(i\\) to the training data at every batch containing the sample."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ ],
+ "angle": 0,
+ "content": "Idea 2: Block coordinate descent. We then use the metagradient \\(\\mathbf{g}\\) to iteratively update our selected dataset. We update data counts as"
+ },
+ {
+ "type": "equation",
+ "bbox": [
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+ 0.678,
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+ 0.695
+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathbf {c} \\leftarrow \\mathbf {c} - \\operatorname {s i g n} (\\mathbf {g}) \\odot \\mathbf {m}, \\quad \\mathbf {m} \\sim \\text {B e r n o u l l i} (p), \\tag {8}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ 0.705,
+ 0.884,
+ 0.766
+ ],
+ "angle": 0,
+ "content": "where \\( p \\) is a hyperparameter controlling the fraction of sample counts to update. This algorithm resembles a block coordinate descent algorithm [OR00], with the main difference being that we take signed gradient steps with step size 1 (projected onto non-negative integers) to ensure that the counts remain well-defined. As a result, \\( p \\) implicitly controls the algorithm's step size."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ 0.766,
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+ "angle": 0,
+ "content": "Applying (8) concludes a single optimization step. By repeating this process of estimating the metagradient, updating our counts vector, then constructing a new training dataset, we iteratively improve the selected data. Pseudocode for our algorithm can be found in Algorithm 1."
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.113,
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+ "angle": 0,
+ "content": "4.1.3 Results"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.112,
+ 0.854,
+ 0.884,
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+ ],
+ "angle": 0,
+ "content": "We evaluate our data selection algorithm using DataComp [GIF+24], a standardized framework for evaluating data selection methods for multimodal models. Algorithm 1 greatly improves on the state-of-the-art for the benchmark. Below, we describe the setting, outline our method, and conclude with our results."
+ },
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+ "type": "page_number",
+ "bbox": [
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+ "angle": 0,
+ "content": "11"
+ }
+ ],
+ [
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+ "type": "code_caption",
+ "bbox": [
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+ "angle": 0,
+ "content": "Algorithm 1: Dataset selection using using metagradient descent (MGD)."
+ },
+ {
+ "type": "algorithm",
+ "bbox": [
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+ "angle": 0,
+ "content": "Input: initial data counts \\(\\mathbf{c}\\in \\mathbb{Z}_{\\geq 0}^{n}\\) , learning algorithm \\(\\mathcal{A}\\) output function Hyperparameters: step size \\(p\\) # opt steps \\(T\\) iteration number \\(k\\) for \\(t\\gets 1\\) to \\(T\\) do \n2 \\(\\mathbf{z}\\gets \\mathbf{0}_n / /\\) Build input to surrogate \n3 \\(\\mathbf{g}\\leftarrow \\frac{\\partial\\phi(\\mathcal{A}_c'(\\mathbf{z}))}{\\partial\\mathbf{z}} / /\\) Calculate metagradient using REPLAY \n4 m<- sample from Bernoulli(p) // Sample indices to step on \n5 c<- c-sign(g) \\(\\odot\\) m// Take optimization step \n6 Return c// Return final data counts"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.282,
+ 0.885,
+ 0.404
+ ],
+ "angle": 0,
+ "content": "Setting. DataComp [GIF+24] is a multimodal model training competition and benchmark for evaluating dataset selection methods. DataComp provides a fixed learning algorithm chosen in advance by the organizers and a large fixed candidate pool of internet data. The goal is to choose a subset of the candidate pool—possibly with repeated datapoints—that yields the best-performing model after training with the given learning algorithm, as measured by a predetermined set of 38 benchmarks. Given a submission subset, the mean score on the evaluation datasets for a model trained with that subset is taken as the final \"score.\" DataComp offers four separate \"scales\" requiring different amounts of compute; we focus on the small scale in this paper due to compute limitations."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.422,
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+ 0.53
+ ],
+ "angle": 0,
+ "content": "Method. We select data with MGD (Algorithm 1) to minimize loss on data on a \"target set\" that is distributionally similar to the DataComp benchmark tasks, and select hyperparameters with a held-out \"validation set.\" In particular, we construct target and validation sets by taking samples from the DataComp evaluation tasks with extra samples available beyond those used in the DataComp test set (e.g., ImageNet, one of the tasks in DataComp, has a training set in addition to the test set evaluated in DataComp). See Appendix C for the exact details of the target and validation sets, the precise hyperparameters used with Algorithm 1, and a discussion on scalability (including further engineering details on executing our algorithm efficiently)."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.547,
+ 0.884,
+ 0.609
+ ],
+ "angle": 0,
+ "content": "Results. MGD greatly outperforms the current state-of-the-art: the difference in accuracy between MGD and the current best method is roughly as large as the difference between the previous state-of-the-art (EcoDatum [Eco24]) and training on randomly chosen data (cf. Figure 6). Inspecting scores over the course of the optimization in Figure 6, we find that only a few steps are necessary to outperform previous methods."
+ },
+ {
+ "type": "image",
+ "bbox": [
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+ ],
+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "table",
+ "bbox": [
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+ 0.622,
+ 0.857,
+ 0.715
+ ],
+ "angle": 0,
+ "content": "| Dataset | Task | Test size | Train size | Val size | Main metric | Target set |
| Caltech-101 [FFP04] | Object recognition | 6085 | 2754 | 306 | mean per class | ✓ |
| CIFAR-10 [Kri09] | Visual recognition | 10000 | 45000 | 5000 | accuracy | ✓ |
| CIFAR-100 [Kri09] | Visual recognition | 10000 | 45000 | 5000 | accuracy | ✓ |
| CLEVR Counts [JHV+17; ZPK+19] | Counting | 15000 | 65000 | 5000 | accuracy | ✓ |
| CLEVR Distance [JHV+17; ZPK+19] | Distance prediction | 15000 | 65000 | 5000 | accuracy | ✓ |
| Country211 [RKH+21; TSF+16] | Geolocation | 21100 | 37980 | 4220 | accuracy | ✓ |
| DTD [CMK+14] | Texture classification | 1880 | 3384 | 376 | accuracy | ✓ |
| EuroSAT [HBD+19; ZPK+19] | Satellite imagery recognition | 5400 | 19440 | 2160 | accuracy | ✓ |
| FGVC Aircraft [MRK+13] | Aircraft recognition | 3333 | 6001 | 666 | mean per class | ✓ |
| Food-101 [BGV14] | Food recognition | 25250 | 70750 | 5000 | accuracy | ✓ |
| GTSRB [SSS+11] | Traffic sign recognition | 12630 | 35289 | 3920 | accuracy | ✓ |
| ImageNet 1k [DDS+09] | Visual recognition | 50000 | 1276167 | 5000 | accuracy | ✓ |
| ImageNet Sketch [WGX+19] | Visual recognition | 50889 | N/A | N/A | accuracy | * |
| ImageNet V2 [RKS+19] | Visual recognition | 10000 | N/A | N/A | accuracy | * |
| ImageNet-A [HZB+19] | Visual recognition | 7500 | N/A | N/A | accuracy | * |
| ImageNet-O [HZB+19] | Visual recognition | 2000 | N/A | N/A | accuracy | * |
| ImageNet-R [HBM+20] | Visual recognition | 30000 | N/A | N/A | accuracy | * |
| KITTI distance [GLU12; ZPK+19] | Distance prediction | 711 | N/A | N/A | accuracy | † |
| MNIST [LeC98] | Digit recognition | 10000 | 55000 | 5000 | accuracy | ✓ |
| ObjectNet [BMA+19] | Visual recognition | 18574 | N/A | N/A | accuracy | * |
| Oxford Flowers-102 [NZ08] | Flower recognition | 6149 | 1836 | 204 | mean per class | ✓ |
| Oxford-IIIT Pet [PVZ+12; ZPK+19] | Pet classification | 3669 | 3312 | 368 | mean per class | ✓ |
| Pascal VOC 2007 [EVW+10] | Object recognition | 14976 | 14096 | 1566 | accuracy | ✓ |
| PatchCamelyon [VLW+18; ZPK+19] | Metastatic tissue cls. | 32768 | 289912 | 5000 | accuracy | ✓ |
| Rendered SST2 [ZPK+19] | Sentiment classification | 1821 | 7013 | 779 | accuracy | ✓ |
| RESISC45 [CHL17; ZPK+19] | Satellite imagery recognition | 6300 | 22680 | 2520 | accuracy | ✓ |
| Stanford Cars [KSD+13] | Vehicle recognition | 8041 | 7329 | 814 | accuracy | ✓ |
| STL-10 [CNL11] | Visual recognition | 8000 | 4500 | 500 | accuracy | ✓ |
| SUN-397 [XHE+10] | Scene recognition | 108753 | N/A | N/A | accuracy | ‡ |
| SVHN [NWC+11; ZPK+19] | Digit recognition | 26032 | 68257 | 5000 | accuracy | ✓ |
| iWildCam [BAC+21; KSM+20] | Animal recognition | 42791 | 147084 | 5000 | macro F1 score | ✓ |
| Camelyon17 [BGM+18; KSM+20] | Metastatic tissue cls. | 85054 | 365900 | 5000 | accuracy | ✓ |
| FMoW [CFW+18; KSM+20] | Satellite imagery recognition | 22108 | 103261 | 5000 | worst-region acc. | ✓ |
| Dollar Street [RDK+22] | Object recognition | 3503 | 13842 | 1537 | worst-income top-5 acc. | ✓ |
| GeoDE [RLZ+24] | Object recognition | 12438 | 44488 | 4943 | worst-region acc. | ✓ |
| Flickr30k [YLH+14] | Image and text retrieval | 31014 | N/A | N/A | R@1 | § |
| MSCOCO [LMB+14] | Image and text retrieval | 5000 | N/A | N/A | R@1 | § |
| WinoGAViL [BBY+22] | Commonsense association | 3563 | N/A | N/A | Jaccard score | § |
"
+ },
+ {
+ "type": "page_footnote",
+ "bbox": [
+ 0.131,
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+ 0.875
+ ],
+ "angle": 0,
+ "content": "*No train or val set exists for this dataset, so we were unable to create disjoint target and val sets."
+ },
+ {
+ "type": "page_footnote",
+ "bbox": [
+ 0.133,
+ 0.875,
+ 0.503,
+ 0.885
+ ],
+ "angle": 0,
+ "content": "We were unable to use this dataset due to technical difficulties."
+ },
+ {
+ "type": "page_footnote",
+ "bbox": [
+ 0.133,
+ 0.887,
+ 0.882,
+ 0.9
+ ],
+ "angle": 0,
+ "content": "Both the train and val sets were used by DataComp to make their test set, so we were unable to create disjoint target and val sets."
+ },
+ {
+ "type": "page_footnote",
+ "bbox": [
+ 0.133,
+ 0.9,
+ 0.42,
+ 0.912
+ ],
+ "angle": 0,
+ "content": "\\( {}^{S} \\) Retrieval tasks were not used for metagradients."
+ },
+ {
+ "type": "list",
+ "bbox": [
+ 0.131,
+ 0.862,
+ 0.882,
+ 0.912
+ ],
+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "page_number",
+ "bbox": [
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+ "angle": 0,
+ "content": "33"
+ }
+ ],
+ [
+ {
+ "type": "title",
+ "bbox": [
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+ ],
+ "angle": 0,
+ "content": "D Selecting IFT data"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.12,
+ 0.884,
+ 0.152
+ ],
+ "angle": 0,
+ "content": "In this section, we describe the details of the IFT setting of Xia et al. [XMG+24], as well as the details of our method."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
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+ 0.885,
+ 0.248
+ ],
+ "angle": 0,
+ "content": "Setting. The setting contains a fixed data pool: instruction fine-tuning data from a data pool consisting of four combined IFT datasets (cf. Table 4 and Xia et al. [XMG+24] for more information). The goal is to select the data that yields the best possible task performance for a LoRA fine-tuning run. We adapt a LoRA to a Gemma-2B model (the pretraining-only Gemma-2B model) using the LoRA configuration from Xia et al. [XMG+24]."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ ],
+ "angle": 0,
+ "content": "Data splits. See Table 3 for a description of the available data for each task, along with the task setup details. Xia et al. [XMG+24] constructed these extra samples by drawing from the ICL samples given in the tasks originally. Note that we drop TydiQA from the original work of Xia et al. [XMG+24] as there are not enough samples to select with (there is only one from each category, for a total of 7)."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ ],
+ "angle": 0,
+ "content": "Method. We execute Algorithm 1 with \\( k \\) as 150 steps from the end of training and the Bernoulli parameter \\( p \\) controlling the step size as 0.2. At each step, we choose a \"minibatch\" with a size equal to half the target set and a quarter of the target set for BBH and MMLU, respectively (that is, we only select to optimize performance on a fraction of the target set at a time). We model select over iterates and hyperparameters by (a) choosing the top three steps in terms of validation loss for each run (b) selecting the best one in terms of full train set accuracy (including the part that we trained on). We perform this procedure—akin to Pareto optimization [JS08]—because the validation set is so small (as the overall set of samples is very small) that it is difficult to select models without overfitting otherwise."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.465,
+ 0.884,
+ 0.51
+ ],
+ "angle": 0,
+ "content": "We compare with two baselines: training on the full dataset (i.e., training on the entirety of all the data for a single epoch), and LESS (we use the data selected according to \"LESS-T\" [XMG+24], following the recommendation of 4 epochs)."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.51,
+ 0.885,
+ 0.678
+ ],
+ "angle": 0,
+ "content": "For model training, we train with ADAM (\\(\\beta_{1} = 0.95\\), \\(\\beta_{2} = 0.975\\), decoupled weight decay as \\(10^{-5}\\)) and a one-cycle linear schedule starting at \\(10^{-6}\\) of the maximum learning rate, reaching the peak over \\(25\\%\\) of training, then ending at 0.1 of the maximum learning rate. We insert a positive \\(\\varepsilon_{\\mathrm{root}}\\) into the inverse square root term in the ADAM update to prevent metagradient (and to a lesser extent update) blowup (see Eq. 12). The model training is the same across selected data, except that we use \\(\\varepsilon_{\\mathrm{root}} = 10^{-7}\\) for MGD-selected data and \\(\\varepsilon_{\\mathrm{root}} = 10^{-9}\\) for the other runs (we select the optimal parameter for each class of method). We additionally hyperparameter select for the best learning rate across each baseline by minimizing validation set loss; LESS performs best with a smaller learning rate (0.00024 for BBH and 0.00012 for MMLU) than training on the full dataset or with MGD (0.0006 for both). We normalize the loss of each training sample by taking the mean across predicted tokens during training, and do not divide by the batch size (important for scaling the \\(\\varepsilon_{\\mathrm{root}}\\) term, but otherwise ADAM is invariant to the scale)."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.695,
+ 0.884,
+ 0.773
+ ],
+ "angle": 0,
+ "content": "Selecting smooth model training for MGD. For MGD runs, we jointly select learning rate and \\(\\varepsilon_{\\mathrm{root}}\\) using the smoothness metric of Section 3. We find that the choice of \\(\\varepsilon_{\\mathrm{root}}\\) term is important (just as the choice of \\(\\varepsilon\\) is important in standard ADAM training); choosing a much larger term results in non-smooth training. We also find that metagradients are sensitive to learning rate schedule; choosing a much larger or smaller maximum learning rate results in non-smooth training."
+ },
+ {
+ "type": "table_caption",
+ "bbox": [
+ 0.185,
+ 0.785,
+ 0.812,
+ 0.802
+ ],
+ "angle": 0,
+ "content": "Table 3: Overview of datasets used in IFT dataset selection (from Xia et al. [XMG+24])."
+ },
+ {
+ "type": "table",
+ "bbox": [
+ 0.155,
+ 0.812,
+ 0.846,
+ 0.878
+ ],
+ "angle": 0,
+ "content": "| Parameter | Values |
| Peak learning rate | [7.0, 7.5, 8.0, 8.5, 9.0, 9.5, 10.0, 10.5, 11.0, 11.5, 12.0] |
| Initial LR multiplier | [0.05, 0.15, 0.25, 0.35, 0.45, 0.55] |
| Final LR multiplier | [0.05, 0.15, 0.25, 0.35, 0.45, 0.55] |
| LR peak time | [0.25, 0.5, 0.75] |
"
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 0.449,
+ 0.286,
+ 0.529,
+ 0.303
+ ],
+ "angle": 0,
+ "content": "MGD step"
+ },
+ {
+ "type": "image",
+ "bbox": [
+ 0.241,
+ 0.31,
+ 0.761,
+ 0.543
+ ],
+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 0.326,
+ 0.557,
+ 0.672,
+ 0.574
+ ],
+ "angle": 0,
+ "content": "Figure 14: Graphs of our learned LR schedules."
+ },
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.49,
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+ "content": "37"
+ }
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+# Optimizing ML Training with Metagradient Descent
+
+Logan Engstrom\*1, Andrew Ilyas\*2†, Benjamin Chen\*1, Axel Feldmann\*1, William Moses\*3, Aleksander Madry\*1
+
+*Equal contribution ${}^{1}$ MIT, ${}^{2}$ Stanford, ${}^{3}$ UIUC
+
+# Abstract
+
+A major challenge in training large-scale machine learning models is configuring the training process to maximize model performance, i.e., finding the best training setup from a vast design space. In this work, we unlock a gradient-based approach to this problem. We first introduce an algorithm for efficiently calculating metagradient gradients through model training at scale. We then introduce a "smooth model training" framework that enables effective optimization using metagradient. With metagradient descent (MGD), we greatly improve on existing dataset selection methods, outperform accuracy-degrading data poisoning attacks by an order of magnitude, and automatically find competitive learning rate schedules.
+
+# 1 Introduction
+
+How should I clean my data? What architecture should I use? Training large-scale (i.e., deep) machine learning models entails making many design decisions. When making such decisions, typical practice is to exhaustively search over a small set of standard options. For example, we might try a few well-known data cleaning heuristics, construct a grid over a hyperparameters, and choose the options that yield the best models. However, given that this process explores only a small part of the overall design space (e.g., one can construct $2^{n}$ possible training datasets from a pool of $n$ candidate datapoints), it is unlikely that this approach really yields the optimal training configuration.
+
+How can we find optimal (or at least, better) training configurations? To do so, we take the optimization perspective on designing model training. From this well-studied perspective, deciding on a training configuration—or as we will call it, a set of metaparameters—is just a high-dimensional optimization problem. The input space of this problem comprises all possible metaparameter choices, including which datapoints to train on, what model architecture to use, and how to initialize model weights. The objective function takes in a set of metaparameters, trains a machine learning model according to those metaparameters, and then returns a target metric evaluated on that model (e.g., test accuracy). From this perspective, any procedure for selecting metaparameters—including the typical practice of grid-searching over standard options—is just an optimization algorithm, whose goal is to maximize the objective function with respect to the (high-dimensional) input.
+
+Given that selecting metaparameters is "just" a high-dimensional optimization problem, a natural tool to consider is the gradient. After all, in many contexts, gradients offer a more effective approach to maximizing high-dimensional functions than grid search. Indeed, for a sufficiently "well-behaved" function $f(x)$ with gradient $\nabla f(x)$ , we can optimize $f$ by iteratively updating $x$ in the direction of $\nabla f(x)$ . This insight suggests a generic recipe for selecting metaparameters: first, make the objective differentiable with respect to the metaparameters; second, update via gradient steps.
+
+Now, the idea of using gradients to search for metaparameters is not new. Indeed, there is a substantial line of work that aims to optimize metaparameters (e.g., architectures, regularizers, or data augmentation schemes) with gradient-based methods [MDA15; LSY18; LVD20]. However, such methods have not managed to scale beyond relatively small settings. This state of affairs prompts our main question:
+
+Can we scalably configure model training using gradient-based methods?
+
+
+Figure 1: Our proto-algorithm, metagradient descent (MGD), uses gradients to achieve state-of-the-art performance across a variety of applications, including data selection and data poisoning.
+
+
+
+
+
+# 1.1 Contributions
+
+In this work, we answer this question in the affirmative, adding "gradient descent on metaparameters" to the large-scale machine learning toolkit. Along the way, we will face—and address—two main challenges.
+
+First, existing methods for computing metagradients do not scale. In response, we devise an algorithm, REPLAY, that can take metagradients in large-scale settings. By combining reverse-mode autodifferentiation (AD) with an efficient data structure, REPLAY can calculate exact metagradients for models with billions of parameters and thousands of training steps.
+
+Second, we find that metagradients of standard training routines are not necessarily helpful for optimization, which we connect to non-smoothness of the metaparameter optimization landscape. Borrowing tools from convex optimization, we devise a framework for designing "metasmooth" training routines that do admit helpful metagradients.
+
+Addressing the challenges above unlocks a simple recipe for solving a broad range of machine learning tasks: (a) frame the task as a continuous optimization problem over metaparameters; (b) design a metasmooth training routine; (c) perform metagradient descent (MGD). Applying this recipe:
+
+- In the DataComp-small11 competition [GIF+24], we achieve state-of-the-art pre-training data selection for CLIP (2x larger performance improvement than the previous DataComp-small1 leader [Eco24]);
+- In the context of data selection for instruction tuning (as introduced by Xia et al. [XMG+24]), we substantially improve on data selection for Gemma-2B (outperforming existing selection methods as well as full-data training);
+- In the accuracy-degrading data poisoning setting (defined by Huber [Hub64] and pioneered by Lu et al. [LKY22] for deep neural networks), we improve attacks on DNNs by an order of magnitude, dropping CIFAR-10 accuracy from $92\% \rightarrow 78\%$ (the best previous attack [LKY23] only reduces accuracy to $91\%$ );
+- For the task of hyperparameter optimization, we efficiently find a competitive CIFAR-10 learning rate schedule (matching the performance of a schedule found by grid search).
+
+# 2 Scalably computing metagradients
+
+In this section we present REPLAY, an algorithm for computing metagradients of large-scale iterative ML algorithms. We first detail the setting, then discuss existing approaches to computing metagradients, and conclude by describing REPLAY.
+
+
+Training setup
+Trained model
+Observed behavior
+Figure 2: An illustration of the metagradient. We embed a given aspect of the training setup (e.g., the training dataset, or optimizer hyperparameters) into a continuous metaparameter vector $z \in \mathbb{R}^d$ . This metaparameter defines a model $\mathcal{A}(z)$ by way of the learning algorithm $\mathcal{A}$ , which in turn defines an output $\phi(z)$ . The metagradient $\nabla_z \phi(\mathcal{A}(z)) \in \mathbb{R}^d$ is the gradient of this model output with respect to the metaparameter.
+
+# 2.1 What is a metagradient?
+
+Training a machine learning model is a two-step process. First, we decide on a training setup—we must pick, for example, a neural network architecture, a training dataset, and an optimizer for training. Second, we apply the algorithm defined by this training setup to train a model.
+
+Our overall goal in this paper is to optimize model behavior as a function of the training setup (or, as we call it, the metaparameters) using gradient-based methods. To this end, we define the following notation:
+
+- Let $\mathbf{z} \in \mathbb{R}^n$ be a vector of continuous metaparameters representing the aspects of the training setup we aim to optimize. For example, if we only want to adjust the learning rate and weight decay of SGD then $n = 2$ . We handle discrete metaparameters (e.g., choice of training data) by finding a continuous relaxation (e.g., importance weights).
+- Let $\mathcal{A}$ be an algorithm mapping $\mathbf{z}$ to a trained machine learning model; we assume all other aspects of the training setup outside $\mathbf{z}$ are fixed and thus part of the algorithm $\mathcal{A}$ .
+- Finally, let $\phi$ be an output function mapping a model $\theta$ to a vector $\phi(\theta) \in \mathbb{R}$ . For example, $\phi(\theta)$ might represent the validation loss of the model $\theta$ . We require that $\phi$ be differentiable with respect to $\theta$ , but otherwise make no assumptions on $\phi$ .
+
+With this notation in place, we define the training function $f \coloneqq \phi \circ \mathcal{A}$ mapping the training setup $\mathbf{z}$ directly to the output function $\phi$ evaluated on the corresponding model.
+
+Finally, the metagradient is the gradient of the training function with respect to the metaparameters, $\nabla_{\mathbf{z}}f(\mathbf{z})$ . Intuitively, the metagradient defines the "direction of steepest ascent" in metaparameter space.
+
+Our focus: iterative algorithms. To efficiently compute the metagradient, we restrict our focus to cases where the algorithm $\mathcal{A}$ is iterative, i.e., when it can be written in the form
+
+$$
+\underbrace {\mathcal {A} (z) := \mathbf {s} _ {T}} _ {\text {m o d e l s t a t e a f t e r T s t e p s}}, \quad \text {w h e r e} \quad \underbrace {\mathbf {s} _ {t + 1} : = h _ {t} (\mathbf {s} _ {t} , \mathbf {z})} _ {\text {o p t i m i z e r s t e p t}}. \tag {1}
+$$
+
+Here, $\mathbf{s}_t$ is the optimizer state at step $t$ (with $\mathbf{s}_0$ being the initial state) and $h_t$ is the update mapping from state $t$ to state $t + 1$ . The form of (1) captures most large-scale training algorithms. For example, if the setup $\mathbf{z} \in \mathbb{R}^T$ is a per-step learning rate, and the algorithm $\mathcal{A}$ is full batch gradient descent, then each update $h_t$ is
+
+$$
+h _ {t} (\mathbf {s} _ {t}, \mathbf {z}) := \mathbf {s} _ {t} - z _ {t} \nabla \ell (\mathbf {s} _ {t}),
+$$
+
+where $z_{t}$ is the learning rate at step $t$ , $\ell$ is the training loss, and the state $\mathbf{s}_t$ comprises the parameters at step $t$ . For more complex algorithms like Adam [KB15], the state $\mathbf{s}_t$ includes terms like gradient moments.
+
+# 2.2 Warmup: Metagradients via autodifferentiation
+
+A key primitive we leverage to calculate metagradients is automatic differentiation (AD)—a standard tool for taking gradients through computer-defined functions. AD takes gradients by decomposing functions into elementary operations with known derivatives, then combining these derivatives using the chain rule. Concretely, AD operates in two passes: a "forward pass," which executes the function of interest and stores intermediate products for each elementary operation; and a "backward pass," which calculates the gradient by propagating chains of partial derivatives using these stored products. For the purposes of this paper, we will view AD as a black box that calculates the gradient of a many-to-one function (i.e., any $f: \mathbb{R}^d \to \mathbb{R}$ ) at a given point using only a small constant factor more time than calculating the function itself (along with the space cost of storing the necessary forward-pass products).
+
+What does this have to do with metagradients? Well, seeing as how training itself is a computer-defined function, AD is a natural tool for calculating the metagradient. The main challenge, as we discuss in the sequel, is that AD-based approaches to calculating the metagradient tend to be too resource-intensive for the large-scale machine learning algorithms we consider. In the remainder of this section we build up background before finally describing REPLAY, our algorithm for scalably computing (exact) metagradients.
+
+Approach #1: Direct AD. The direct approach to calculating metagradients exploits the fact that nearly any learning algorithm is itself a sequence of differentiable computer-defined operations—meaning the training function $f$ is also differentiable.
+
+However, operationalizing this observation to compute metagradients turns out to be challenging. The reason is that AD stores intermediate products for each operation. The amount of data stored thus scales with the number of operations in the function of interest. In the case of our training function $f$ , this number encompasses all the operations used to train a machine learning model. As a result, even in a toy scenario like MNIST training, computing metagradients with naive AD would require storing terabytes of data.
+
+Approach #2: Exploiting structure with step-wise AD. A more efficient method for calculating the metagradient, step-wise AD, leverages the structure of iterative learning algorithms [Wer90; MDA15; FDF+17]. Recall from (1) that such algorithms take the form
+
+$$
+\mathcal {A} (\mathbf {z}) := \mathbf {s} _ {T}, \quad \text {w h e r e} \quad \mathbf {s} _ {t + 1} := h _ {t} (\mathbf {s} _ {t}, \mathbf {z}).
+$$
+
+Algebraic manipulation (in particular, using the chain rule, the law of the total derivative, and the identity $\mathbf{s}_t = h_{t-1}(\mathbf{s}_{t-1}, \mathbf{z})$ ) allows us to write the metagradient over an iterative algorithm as
+
+$$
+\frac {\partial f (\mathbf {z})}{\partial \mathbf {z}} = \frac {\partial \phi (\mathcal {A} (\mathbf {z}))}{\partial \mathbf {z}} = \sum_ {t = 1} ^ {T} \underbrace {\overbrace {\partial \phi (\mathbf {s} _ {T})} ^ {A _ {t}} . \overbrace {\partial \mathbf {s} _ {t}} ^ {\partial \phi (\mathbf {s} _ {T})}} _ {B _ {t}}, \tag {2}
+$$
+
+where we have introduced the notation $A_{t}$ and $B_{t}$ for notational convenience. Step-wise AD computes the metagradient by calculating each term in the sum of (2) one at a time. For each term, the main challenge lies in computing $A_{t}$ , since given $A_{t}$ we can straightforwardly compute $B_{t}$ (the entire term) by differentiating through a single model update, i.e.,
+
+$$
+B _ {t} := A _ {t} \cdot \frac {\partial h _ {t - 1} (\mathbf {s} _ {t - 1} , \mathbf {z})}{\partial \mathbf {z}} = \frac {\partial (A _ {t} \cdot h _ {t - 1} (\mathbf {s} _ {t - 1} , \mathbf {z}))}{\partial \mathbf {z}},
+$$
+
+which is just a single call to our assumed "AD oracle" on the function $\mathbf{z} \mapsto A_t \cdot h_{t-1}(\mathbf{s}_{t-1}, \mathbf{z})$ . Computing the $A_t$ terms is less straightforward as we need to relate $s_t$ and $s_T$ ; to do so, we exploit the recurrence
+
+$$
+A _ {t} := \frac {\partial \phi (\mathbf {s} _ {T})}{\partial \mathbf {s} _ {t}} = \frac {\partial \phi (\mathbf {s} _ {T})}{\partial \mathbf {s} _ {t + 1}} \cdot \frac {\partial h _ {t} (\mathbf {s} _ {t} , \mathbf {z})}{\partial \mathbf {s} _ {t}} = \frac {\partial \left(A _ {t + 1} \cdot h _ {t} (\mathbf {s} _ {t} , \mathbf {z})\right)}{\partial \mathbf {s} _ {t}}, \tag {3}
+$$
+
+making $A_{t}$ straightforward to compute (again, a single "AD oracle" call) given $A_{t+1}$ . Step-wise AD exploits this fact to successively calculate the gradient with respect to each state, from state $T$ down to state 0.
+
+
+Figure 3: The lazy $k$ -ary tree structure for traversing optimizer states in reverse order, with $k = 2$ . Recall that $n$ is the number of states (parameterized such that $n = T + 1$ ). Each node represents the correspondingly numbered state. We give an example of the traversal using the blue arrows in the figure, which denote the traversal path up to state $s_{\frac{3n}{4} + 1}$ . The gray cylinders indicate the states that are stored when the traversal is at state $s_{\frac{3n}{4} + 1}$ ; the other states are not stored at this point in the traversal. Traversing this structure requires storing $\mathcal{O}(\log(n))$ state and computing $\mathcal{O}(n \log(n))$ optimizer steps—compared to $n$ for simply training.
+
+Bringing these ingredients together, the algorithm executes as follows. As a preprocessing step, it trains the model and stores all intermediate states $\mathbf{s}_0,\dots ,\mathbf{s}_T$ . Then, the algorithm calculates and sums the terms in (2). It first computes $A_{T}\coloneqq \partial \phi (\mathbf{s}_{T}) / \mathbf{s}_{T}$ , the gradient of the output function $\phi$ with respect to the final state. Then, the algorithm steps through $\mathbf{s}_{T - 1},\ldots ,\mathbf{s}_0$ in reverse order, calculating (a) the gradient with respect to each state $A_{t}$ (via (3)) and (b) the gradient with respect to $\mathbf{z}$ at that step $B_{t}$ (via (2), using the previously calculated gradient with respect to that state). AD calculates both quantities--each requires differentiating over only one train step. Finally, the algorithm returns the final metagradient as the sum of the terms.
+
+Despite improving storage overhead compared to "direct AD", step-wise AD is still too space-intensive at scale. After all, this algorithm saves every optimizer state.
+
+# 2.3 REPLAY
+
+REPLAY is our algorithm for efficiently and exactly computing metagradients. It uses $\mathcal{O}(k\log_k(T))$ space and requires running the learning algorithm $\mathcal{A}$ a total of $1 + \log_{k}(T)$ times, with $k$ a user-chosen constant. The main idea is to make the space-intensive subroutine of step-wise AD—a reverse-order traversal of the optimizer states at each step—much more efficient. After all, step-wise AD stores all the states to reverse traverse them. REPLAY modifies step-wise AD to traverse states in less space by exploiting a simple observation: when training is deterministic, one can reinstantiate an optimizer state $\mathbf{s}_t$ by "replaying" training from a fixed point $t' < t$ at the compute cost of $t - t'$ training steps. For example, one simple scheme saves every other state, then "replays" the remaining states when (reverse) traversing; this routine stores $T/2$ states but computes an extra $T/2$ model updates compared to storing all the states.
+
+REPLAY performs a reverse-order traversal the optimizer states while balancing the compute cost of "replaying" training with the storage cost of saving states. We use a combination of deterministic training (fixing data ordering, data augmentation, and any other randomness in the training process) and an efficient data structure (similar to a segment tree; see Figure 3) to reverse-order traverse the optimizer states with $\mathcal{O}(k\log_k(T))$ space and an additional $T\log_k(T)$ model steps.
+
+Specifically, REPLAY recursively saves and replays training states. The algorithm splits the training trajectory into $k$ segments, performs the full training routine while saving only the start of each segment, then recurses into each segment (in reverse) to retrieve the states in reverse-order. The recursion depth bottoms out at $\log_k(T)$ , at which point the algorithm has $k$ consecutive optimizer states in memory; the algorithm then backpropagates along this segment, before deleting all these states from memory and then reinstantating the next $k$ -length segment of optimizer states. We provide additional details on the algorithm in Appendix A.2. REPLAY unlocks computing large-scale metagradients by requiring only logarithmic storage and additional compute time.
+
+Remark 1 (Connection to rematerialization). In a broad sense, both REPLAY and step-wise AD above can be viewed as special cases of a classical approach in AD (and computing broadly) known as rematerialization [CAC+81; BCT92; ZP00; GW08; CXZ+16]. To our knowledge, however, REPLAY is the first application of this particular rematerialization technique to the problem of computing metagradients through model training.
+
+Remark 2 (Reversible learning). An alternative approach to calculating metagradients that does not save any state is reversible learning [MDA15], for which one can "invert" previous training states from future ones. We focus here on general (non-reversible) learning algorithms for two reasons: first, even simple algorithms such as SGD without momentum are non-reversible; second, reversibility in practice introduces numerical precision issues.
+
+# 3 Designing metasmooth training routines
+
+Given a training function $f$ , REPLAY enables us to compute metagradients $\nabla f(\mathbf{z})$ for any setup $\mathbf{z}$ . Can we immediately use these metagradients to optimize model training setups? The answer is (generally) no: we find that applying REPLAY to a function $f$ representing a standard model training and evaluation routine yields metagradients that are often $\pm \infty$ -valued and generally unhelpful for optimization. Indeed, previous work has observed similar issues optimizing over even (very) small-scale training [BSF94; Pea96; MDA15].
+
+In this section, we show that an underlying source of the issue is the landscape of the metaparameter optimization problem. We then present a framework for modifying standard learning algorithms to admit useful metagradient, i.e., to be metasmooth. To use a familiar analogy: just as residual connections and improved initialization schemes can improve optimization in standard deep learning algorithms, our framework introduces an analogous set of modifications to enable optimization with metagradient.
+
+# 3.1 The metaparameter optimization landscape
+
+We first review the notion of smoothness from optimization theory, and then adapt it to the setting of metagradients. The resulting metasmoothness metric allows us to quantify (and later, improve) the amenability of the metaparameter optimization problem to gradient-based methods.
+
+Smoothness. In optimization theory, the basic property of a function that controls how effectively it can be optimized with first-order methods is smoothness. Specifically, a function $f(\mathbf{z})$ is $\beta$ -smooth at a point $\mathbf{z}$ if its gradient $\nabla f$ satisfies the property that
+
+$$
+\left\| \nabla f (\mathbf {z}) - \nabla f \left(\mathbf {z} ^ {\prime}\right) \right\| \leq \beta \cdot \left\| \mathbf {z} - \mathbf {z} ^ {\prime} \right\| \quad \text {f o r a l l} \mathbf {z} ^ {\prime}, \tag {4}
+$$
+
+or in other words, if its gradient does not change too quickly around $\mathbf{z}$ . To motivate this definition: if a function $f$ is $\beta$ -smooth at $\mathbf{z}$ , then a step of gradient descent with step size $1 / \beta$ will successfully decrease the value of the function:
+
+$$
+f \left(\mathbf {z} - \frac {1}{\beta} \nabla f (\mathbf {z})\right) \leq f (\mathbf {z}) - \frac {1}{2 \beta} \| \nabla f (\mathbf {z}) \| ^ {2}.
+$$
+
+This guarantee makes $\beta$ -smoothness a good measure of gradient utility.
+
+Metasmoothness. There are two main challenges in adapting the smoothness property to the metagradient setting. First, evaluating (4) requires a search over all possible $\mathbf{z}'$ , which is infeasible. Second, even if we could exactly evaluate the left-hand side of (4), it would be difficult to disentangle non-smoothness of the training function $f$ from potential error in metagradient computation (e.g., a numerically unstable operation in REPLAY).
+
+To sidestep these issues, we propose a metric called metasmoothness, given in Definition 1. Metasmoothness is cheap to compute—requiring only three evaluations of the training function—and does not rely on metagradient computation. For the remainder of this section, we fix a small constant $h > 0$ , and define the corresponding finite-differences estimator of the directional derivative $\Delta_f$ as
+
+$$
+\Delta_ {f} (\mathbf {z}; \mathbf {v}) := \frac {f (\mathbf {z} + h \mathbf {v}) - f (\mathbf {z})}{h}.
+$$
+
+Definition 1 (Metasmoothness of $f$ at $\mathbf{z}$ towards $\mathbf{v}$ ). Consider a training function $f$ mapping metaparameters $\mathbf{z} \in \mathbb{R}^n$ to model output $f(\mathbf{z}) \in \mathbb{R}$ . Given a metaparameter $\mathbf{z}$ and a vector $\mathbf{v} \in \mathbb{R}^n$ , the metasmoothness of $f$ at $\mathbf{z}$ towards $\mathbf{v}$ is given by
+
+$$
+S _ {h, \mathbf {v}} (f; \mathbf {z}) := \left| \frac {\Delta_ {f} (\mathbf {z} + h \mathbf {v}) - \Delta_ {f} (\mathbf {z})}{h} \right|. \tag {5}
+$$
+
+Definition 1 measures the rate of change of the derivative of $f(\mathbf{z})$ in the direction of a given vector $\mathbf{v}$ , and is therefore related to $\beta$ -smoothness in that:
+
+(a) If $f$ is $\beta$ -smooth at $\mathbf{z}$ , then $S_{h,\mathbf{v}}(f;\mathbf{z}) \leq \beta$ for any $(h,\mathbf{v})$ (so Definition 1 is necessary for smoothness).
+(b) If $\lim_{h\to 0}S_{h,\mathbf{v}}(f;\mathbf{z})\leq \beta$ for all $\mathbf{z}\in \mathbb{R}^n$ and $\mathbf{v}\in \mathbb{S}^{n - 1}$ , then $f$ is $\beta$ -smooth everywhere (so a global version of Definition 1 is sufficient for smoothness).
+
+Empirical metasmoothness. Definition 1 lets us measure the meta-smoothness of a training function $f$ at a particular metaparameter $\mathbf{z}$ (towards a direction $\mathbf{v}$ ). This definition, however, has two shortcomings. First, recall that the training function $f$ is a composition of a learning algorithm $\mathcal{A}$ and an output function $\phi$ , so the smoothness of $f$ depends on that of both $\mathcal{A}$ and $\phi$ (in particular, $\frac{\partial f}{\partial \mathbf{z}} = \frac{\partial \phi}{\partial \mathcal{A}} \cdot \frac{\partial \mathcal{A}}{\partial \mathbf{z}}$ ). Since the output function $\phi$ might be unknown ahead of time, we are most interested in measuring the overall metasmoothness of a learning algorithm $\mathcal{A}$ . Second, while the result of (5) does have a concrete basis in optimization theory, it may not be easy to interpret in practice (e.g., what does $S = 200$ mean?). We address both issues simultaneously by (a) proposing an interpretable "binarized" version of Definition 1, and (b) studying metasmoothness in the space of model parameters $\theta$ , instead of the output space.
+
+Definition 2 (Empirical metasmoothness of $\mathcal{A}$ ). Let $\mathcal{A}$ be a learning algorithm which maps metaparameters $\mathbf{z} \in \mathbb{R}^n$ to model parameters $\theta \in \mathbb{R}^d$ , let $\mathbf{z}$ be a metaparameter vector, and let $\mathbf{v}$ be a given direction. Let $\mathbf{d} \in \mathbb{R}^d$ be the per-coordinate variation in $\theta$ , i.e.,
+
+$$
+\mathbf {d} = \left| \mathcal {A} (\mathbf {z} + 2 h \mathbf {v}) - \mathcal {A} (\mathbf {z}) \right|
+$$
+
+The empirical $(h,\mathbf{v})$ -metasmoothness of $\mathcal{A}$ at $\mathbf{z}$ is given by
+
+$$
+\widehat {S} _ {h, \mathbf {v}} (\mathcal {A}; \mathbf {z}) = \operatorname {s i g n} \left(\Delta_ {\mathcal {A}} (\mathbf {z}; \mathbf {v})\right) ^ {\top} \cdot \operatorname {d i a g} \left(\frac {\mathbf {d}}{\| \mathbf {d} \| _ {1}}\right) \cdot \operatorname {s i g n} \left(\Delta_ {\mathcal {A}} (\mathbf {z} + h \mathbf {v}; \mathbf {v})\right), \tag {6}
+$$
+
+weights each parameter by its range.
+
+Intuitively, (6) measures the agreement in sign between the (finite-difference approximation of the) metagradient in the direction of $\mathbf{v}$ at $\mathbf{z}$ and at $\mathbf{z} + h\mathbf{v}$ , averaged across parameter coordinates and weighted by the variation in each coordinate. Taking a weighted average of sign agreements ensures that $\widehat{S} \in [-1,1]$ (making it easier to interpret than Definition 1). The $\mathrm{diag}(\mathsf{d} / \| \mathsf{d}\|_1)$ term weights each agreement proportionally to the scale of the corresponding parameter change (downweighting, e.g., coordinates $i$ that are essentially constant). Finally, observe that Definition 2 is efficient to compute in practice: it requires only three calls to the learning algorithm $\mathcal{A}$ .
+
+
+(a)
+
+
+Figure 4: (a) For a variety of training configurations of a ResNet-9 model, we plot metasmoothness (Def. 2) against test accuracy. Strategies such as increasing width, placing batch normalization before activations, and scaling down network outputs consistently improve metasmoothness, at a minor cost to accuracy. (b) Smoother training configurations can be optimized via metagradients more effectively. Here, as in Section 4.3, we use metagradients to gradient ascend on validation loss.
+
+
+(b)
+
+Remark 3. Ideally, recalling the smoothness definition (4), we would evaluate metasmoothness in all possible directions $\mathbf{v}$ and all points $\mathbf{z}$ . Empirically, we find in the sequel (Section 3.2) that this single-direction approximation at a single point $\mathbf{z}$ still yields a useful estimate of metasmoothness (e.g., one that correlates with metagradients utility).
+
+# 3.2 Estimating and improving metasmoothness
+
+Having established a method for quantifying metasmoothness, we turn to the practical question: how can we design learning algorithms that are amenable to metagradient optimization? To answer this question, we introduce a straightforward framework: given a learning algorithm, explore a fixed menu of possible modifications to the training setup, and choose the combination that maximizes empirical metasmoothness. In practice, we find that this framework allows us to slightly modify learning algorithms in a way that makes them amenable to first-order methods.
+
+As a case study, we study the task of training ResNet-9 on the CIFAR-10 dataset [Kri09]. We let the metaparameters $\mathbf{z}$ be a perturbation to the pixels of 1000 random training images (so $\mathbf{z} \in \mathbb{R}^{1000 \times 32 \times 32 \times 3}$ ). We estimate the empirical metasmoothness of different learning algorithms $\mathcal{A}$ at $\mathbf{z} = \mathbf{0}$ using Definition 2. Concretely, we proceed as follows for each learning algorithm $\mathcal{A}$ :
+
+1. Let $\mathbf{z}_0 = \mathbf{0}$ be the metaparameter corresponding to the original dataset.
+2. Sample a random perturbation vector $\mathbf{v} \sim \mathcal{N}(0,1)$ .
+3. Compute the empirical metasmoothness (6), i.e.,
+
+(a) Let $\theta_0\coloneqq \mathcal{A}(\mathbf{z}_0)$ , $\theta_h\coloneqq \mathcal{A}(\mathbf{z}_0 + h\cdot \mathbf{v})$ , and $\theta_{2h}\coloneqq \mathcal{A}(\mathbf{z}_0 + 2h\cdot \mathbf{v})$ be the model parameters that result from training with training dataset perturbations $\mathbf{z}_0,\mathbf{z}_0 + h\mathbf{v}$ , and $\mathbf{z}_0 + 2h\mathbf{v}$ , respectively.
+(b) Compute the approximate derivatives
+
+$$
+\Delta_ {\mathcal {A}} (\mathbf {z} _ {0}; \mathbf {v}) = \left(\theta_ {h} - \theta_ {0}\right) / h, \quad \Delta_ {\mathcal {A}} (\mathbf {z} _ {0} + h \mathbf {v}; \mathbf {v}) = \left(\theta_ {2 h} - \theta_ {h}\right) / h.
+$$
+
+(c) Compute the weighting vector $\mathbf{d} = |\theta_{2h} - \theta_0|$ , and compute the average metasmoothness (6), i.e.,
+
+$$
+\widehat {S} _ {h, \mathbf {v}} (\mathcal {A}; z _ {0}) = \operatorname {s i g n} \left(\Delta_ {\mathcal {A}} \left(\mathbf {z} _ {0} + h \mathbf {v}; \mathbf {v}\right)\right) ^ {\top} \cdot \operatorname {d i a g} \left(\frac {\mathbf {d}}{\| \mathbf {d} \| _ {1}}\right) \cdot \operatorname {s i g n} \left(\Delta_ {\mathcal {A}} \left(\mathbf {z} _ {0}; \mathbf {v}\right)\right).
+$$
+
+
+
+
+
+
+Figure 5: The effect of metasmoothness on the optimization landscape. Each plot above visualizes the loss landscape of a (deterministic) learning algorithm $\mathcal{A}$ , with the $x$ - and $y$ -axes representing additive perturbations to 1000 examples in the training set and the $z$ -axis representing the resulting model's loss on the test example given in the title. In each row, the left plot is a non-smooth algorithm, and the right plot is a smooth algorithm (as per Definition 2) evaluated on the same example. Overall, empirical metasmoothness seems to strongly correlate with qualitative landscape smoothness. See Figure 12 for more examples.
+
+
+
+Metasmooth learning algorithms. We apply the procedure above to estimate the metasmoothness of learning algorithms induced by different design choices (batch size, network width, BatchNorm placement, gradient scaling), and report the results in Figure 4 (left). On one hand, "standard" learning algorithms (i.e., those designed without metasmoothness in mind) are not metasmooth. On the other hand, our investigation reveals central factors driving metasmoothness. In addition to "standard" hyperparameters such as batch size and network width playing a role, we find that placing Batch Normalization layers prior to nonlinearities (instead of after) and scaling the final layer output are both crucial to metasmoothness. Note that the modifications we consider above are not exhaustive—see Appendix E for the full training setup.
+
+Finally, in Figure 5, we plot the optimization landscape of both metasmooth (right) and non-metasmooth (left) models. We find that the landscapes of metasmooth models are much smoother and—qualitatively—more straightforward to optimize.
+
+Metasmoothness/performance tradeoffs? Figure 4 (left) relates metasmoothness to model accuracy for the considered learning algorithms. While there is no clear trend, the top-performing learning algorithms are not always metasmooth. However, the trade-off is not too severe: the most metasmooth algorithms still achieve near-optimal accuracy. Furthermore, it is possible that with additional searching we could identify even more accurate metasmooth models. Taken together with our previous experiment, our results suggest that jointly searching over metasmoothness and model accuracy is a general recipe for designing learning algorithms that are both performant and metasmooth. Finally, as we discuss in Section 5, a fruitful avenue
+
+for future work may be to design metasmooth learning algorithms directly, i.e., without relying on stability heuristics or grid search.
+
+Does metasmoothness aid downstream optimization? Recall that our motivation for studying metasmoothness is to develop learning algorithms that we can optimize the metaparameters of via metagradients (using first-order methods). We started with the notion of $\beta$ -smoothness from optimization theory, and we adapted it to the setting of metagradients by making a series of approximations and modifications. The final question we address is: does our final notion of metasmoothness actually predict the utility of metagradients for optimization? Figure 4 (right) demonstrates that metasmoothness strongly predicts our ability to optimize the metaparameters of a given learning algorithm. We use metagradients (computed by REPLAY) to gradient ascend on validation loss with respect to the metaparameters $\mathbf{z}$ , and measure the change in model loss.
+
+# 4 Applications
+
+In this section, apply metagradients to three problems in machine learning: selecting training data, poisoning training data, and searching for hyperparameters. In each setting we follow the same recipe: we frame the task as an optimization problem, modify the learning algorithm of interest to be smooth, then solve by first-order optimizing with meta-gradients—which we refer to, in a catch-all manner across algorithms, as metagradient descent (MGD). In particular: we substantially improve on existing dataset selection methods (Section 4.1, Section 4.2), perform the first effective accuracy-degrading data poisoning attack (Section 4.3), and discover one-cycle learning rate schedules with MGD (Section 4.4).
+
+# 4.1 Selecting multimodal training data
+
+Curating a training dataset from a mass of unfiltered data is a necessary and influential step in any large-scale machine learning pipeline. Deciding how to curate such a dataset is a challenging problem that has attracted substantial recent interest [FiW+22; ATS+23; EFM24; GIF+24]. In this section, we frame pre-training data selection as an optimization problem, and then solve this problem by first-order optimizing with metagradient. Applying our method to the DataComp-small benchmark [GIF+24], we greatly improve on the state-of-the-art (our improvement over state-of-the-art is roughly the same as the improvement of state-ofthe-art over training on random data).
+
+# 4.1.1 Setup
+
+The goal of dataset selection is to choose a training data subset (out of a broad pool of data) that maximizes trained machine learning model performance. Given this goal, dataset selection has a natural interpretation as a combinatorial metaparameter optimization problem. In particular, in the language of Section 2.1, for a training set of size $n$ , let
+
+(a) the metaparameters $\mathbf{c} \in \mathcal{C} \coloneqq \mathbb{Z}_{\geq 0}^{n}$ be non-negative data counts representing the number of times each training sample repeats in the training data;
+(b) the algorithm $\mathcal{A}$ be a standard large-scale learning procedure, which runs on a training set comprising $c_{i}$ copies of each sample $i$ for $i\in [n]$ ;
+(c) the output function $\phi$ be the loss of the trained model on a target distribution $D$ .
+
+Then, defining $f(\mathbf{c}) \coloneqq \phi(\mathcal{A}(\mathbf{c}))$ (as in Section 2.1), our goal is to find the data counts $\mathbf{c}^*$ that solve
+
+$$
+\mathbf {c} ^ {*} := \underset {\mathbf {c} \in \mathcal {C}} {\arg \min } f (\mathbf {c}). \tag {7}
+$$
+
+# 4.1.2 Gradient descent on training data
+
+Metagradients let us directly minimize the target task loss (7) with respect to the choice of training data. At a high level, our algorithm operates as follows: we start with a randomly chosen set of training data, then iteratively update the dataset selection using metagradients with respect to importance weights placed on each training datapoint. The specifics of our method are in Algorithm 1; we describe its core ideas below.
+
+Idea 1: A surrogate algorithm. We cannot use metagradients to optimize (7) directly, because the metaparameters of interest $\mathbf{c}$ are discrete counts (and so the algorithm $\mathcal{A}$ is non-differentiable with respect to $\mathbf{c}$ ). To circumvent this problem, we relax $\mathcal{A}$ : we define a surrogate algorithm $\mathcal{A}_{\mathbf{c}}^{\prime}$ that takes in a continuous metaparameter $\mathbf{z} \in \mathbb{R}^{n}$ , whose metagradient we can compute, then optimize using the metagradient on $\mathcal{A}_{\mathbf{c}}^{\prime}$ .
+
+This surrogate learning algorithm $\mathcal{A}_{\mathrm{c}}^{\prime}$ maps a metaparameter $\mathbf{z} \in \mathbb{R}^{n}$ (representing a perturbation to training data weights) to a machine learning model. The surrogate is defined by a set of counts $\mathbf{c} \in \mathbb{Z}_{+}^{n}$ , and a hyperparameter $k$ denoting a specific training iteration, both of which we bake into the surrogate algorithm itself. Given a metaparameter $\mathbf{z} \in \mathbb{R}^{n}$ , the algorithm $\mathcal{A}_{\mathrm{c}}^{\prime}$ trains a model "as usual" using the fixed counts $\mathbf{c}$ . That is, it makes $c_{i}$ copies of each training sample $i$ , shuffles and partitions the data into batches, and then at each iteration minimizes the batch loss with a step—just as the original learning algorithm $\mathcal{A}$ . At iteration $k$ , however, in addition to the original loss on the $k$ -th batch, the algorithm upweights each training sample $i$ according to the metaparameter $z_{i}$ . In other words, the objective at iteration $t$ of the surrogate algorithm $\mathcal{A}_{\mathrm{c}}^{\prime}$ is
+
+$$
+\ell_ {t} ^ {\prime} (\theta) := \left\{ \begin{array}{l l} \sum_ {x \in t ^ {\text {t h}} \text {b a t c h}} \ell (x; \theta) & \text {i f} t \neq k \\ \sum_ {x \in t ^ {\text {t h}} \text {b a t c h}} \ell (x; \theta) + \sum_ {i = 1} ^ {n} z _ {i} \ell (x _ {i}; \theta) & \text {i f} t = k \end{array} \right.
+$$
+
+where $\ell (x;\theta)$ is the training loss on example $x$
+
+Observe that when $\mathbf{z} = \mathbf{0}_n$ , the algorithm $\mathcal{A}_{\mathbf{c}}'$ is identical to the standard learning algorithm $\mathcal{A}$ . And while $\mathcal{A}$ was a function of (nondifferentiable) discrete data counts $\mathbf{c}$ , $\mathcal{A}_{\mathbf{c}}'$ is differentiable with respect to its input $\mathbf{z}$ , and so we can compute the metagradient
+
+$$
+\mathbf {g} := \nabla_ {\mathbf {z}} \phi \big (\mathcal {A} _ {\mathbf {c}} ^ {\prime} (\mathbf {z}) \big) \big | _ {\mathbf {z} = \mathbf {0} _ {n}}.
+$$
+
+Intuitively, the entries of the metagradient $\mathbf{g}$ capture the effect of adding an infinitesimal amount of each training sample $i$ to the training data at iteration $k$ . A positive entry $g_{i}$ indicates that adding an infinitesimal amount of sample $i$ to the training data would increase the loss, and a negative entry indicates that adding an infinitesimal amount of sample $i$ to the training data would decrease the loss; the slot at $i$ represents the (estimated) effect of adding a copy of sample $i$ to the training data at every batch containing the sample.
+
+Idea 2: Block coordinate descent. We then use the metagradient $\mathbf{g}$ to iteratively update our selected dataset. We update data counts as
+
+$$
+\mathbf {c} \leftarrow \mathbf {c} - \operatorname {s i g n} (\mathbf {g}) \odot \mathbf {m}, \quad \mathbf {m} \sim \text {B e r n o u l l i} (p), \tag {8}
+$$
+
+where $p$ is a hyperparameter controlling the fraction of sample counts to update. This algorithm resembles a block coordinate descent algorithm [OR00], with the main difference being that we take signed gradient steps with step size 1 (projected onto non-negative integers) to ensure that the counts remain well-defined. As a result, $p$ implicitly controls the algorithm's step size.
+
+Applying (8) concludes a single optimization step. By repeating this process of estimating the metagradient, updating our counts vector, then constructing a new training dataset, we iteratively improve the selected data. Pseudocode for our algorithm can be found in Algorithm 1.
+
+# 4.1.3 Results
+
+We evaluate our data selection algorithm using DataComp [GIF+24], a standardized framework for evaluating data selection methods for multimodal models. Algorithm 1 greatly improves on the state-of-the-art for the benchmark. Below, we describe the setting, outline our method, and conclude with our results.
+
+Algorithm 1: Dataset selection using using metagradient descent (MGD).
+Input: initial data counts $\mathbf{c}\in \mathbb{Z}_{\geq 0}^{n}$ , learning algorithm $\mathcal{A}$ output function Hyperparameters: step size $p$ # opt steps $T$ iteration number $k$ for $t\gets 1$ to $T$ do
+2 $\mathbf{z}\gets \mathbf{0}_n / /$ Build input to surrogate
+3 $\mathbf{g}\leftarrow \frac{\partial\phi(\mathcal{A}_c'(\mathbf{z}))}{\partial\mathbf{z}} / /$ Calculate metagradient using REPLAY
+4 m<- sample from Bernoulli(p) // Sample indices to step on
+5 c<- c-sign(g) $\odot$ m// Take optimization step
+6 Return c// Return final data counts
+
+Setting. DataComp [GIF+24] is a multimodal model training competition and benchmark for evaluating dataset selection methods. DataComp provides a fixed learning algorithm chosen in advance by the organizers and a large fixed candidate pool of internet data. The goal is to choose a subset of the candidate pool—possibly with repeated datapoints—that yields the best-performing model after training with the given learning algorithm, as measured by a predetermined set of 38 benchmarks. Given a submission subset, the mean score on the evaluation datasets for a model trained with that subset is taken as the final "score." DataComp offers four separate "scales" requiring different amounts of compute; we focus on the small scale in this paper due to compute limitations.
+
+Method. We select data with MGD (Algorithm 1) to minimize loss on data on a "target set" that is distributionally similar to the DataComp benchmark tasks, and select hyperparameters with a held-out "validation set." In particular, we construct target and validation sets by taking samples from the DataComp evaluation tasks with extra samples available beyond those used in the DataComp test set (e.g., ImageNet, one of the tasks in DataComp, has a training set in addition to the test set evaluated in DataComp). See Appendix C for the exact details of the target and validation sets, the precise hyperparameters used with Algorithm 1, and a discussion on scalability (including further engineering details on executing our algorithm efficiently).
+
+Results. MGD greatly outperforms the current state-of-the-art: the difference in accuracy between MGD and the current best method is roughly as large as the difference between the previous state-of-the-art (EcoDatum [Eco24]) and training on randomly chosen data (cf. Figure 6). Inspecting scores over the course of the optimization in Figure 6, we find that only a few steps are necessary to outperform previous methods.
+
+
+Figure 6: MGD dataset selection greatly outperforms existing methods (improving over the previous SOTA by as much as the previous SOTA improves over no filtering at all). We compare DataComp scores for MGD (over optimization steps), training on the entire candidate pool, the best baseline originally proposed by DataComp, and the previous SOTA [Eco24].
+
+| Dataset | Task | Test size | Train size | Val size | Main metric | Target set |
| Caltech-101 [FFP04] | Object recognition | 6085 | 2754 | 306 | mean per class | ✓ |
| CIFAR-10 [Kri09] | Visual recognition | 10000 | 45000 | 5000 | accuracy | ✓ |
| CIFAR-100 [Kri09] | Visual recognition | 10000 | 45000 | 5000 | accuracy | ✓ |
| CLEVR Counts [JHV+17; ZPK+19] | Counting | 15000 | 65000 | 5000 | accuracy | ✓ |
| CLEVR Distance [JHV+17; ZPK+19] | Distance prediction | 15000 | 65000 | 5000 | accuracy | ✓ |
| Country211 [RKH+21; TSF+16] | Geolocation | 21100 | 37980 | 4220 | accuracy | ✓ |
| DTD [CMK+14] | Texture classification | 1880 | 3384 | 376 | accuracy | ✓ |
| EuroSAT [HBD+19; ZPK+19] | Satellite imagery recognition | 5400 | 19440 | 2160 | accuracy | ✓ |
| FGVC Aircraft [MRK+13] | Aircraft recognition | 3333 | 6001 | 666 | mean per class | ✓ |
| Food-101 [BGV14] | Food recognition | 25250 | 70750 | 5000 | accuracy | ✓ |
| GTSRB [SSS+11] | Traffic sign recognition | 12630 | 35289 | 3920 | accuracy | ✓ |
| ImageNet 1k [DDS+09] | Visual recognition | 50000 | 1276167 | 5000 | accuracy | ✓ |
| ImageNet Sketch [WGX+19] | Visual recognition | 50889 | N/A | N/A | accuracy | * |
| ImageNet V2 [RKS+19] | Visual recognition | 10000 | N/A | N/A | accuracy | * |
| ImageNet-A [HZB+19] | Visual recognition | 7500 | N/A | N/A | accuracy | * |
| ImageNet-O [HZB+19] | Visual recognition | 2000 | N/A | N/A | accuracy | * |
| ImageNet-R [HBM+20] | Visual recognition | 30000 | N/A | N/A | accuracy | * |
| KITTI distance [GLU12; ZPK+19] | Distance prediction | 711 | N/A | N/A | accuracy | † |
| MNIST [LeC98] | Digit recognition | 10000 | 55000 | 5000 | accuracy | ✓ |
| ObjectNet [BMA+19] | Visual recognition | 18574 | N/A | N/A | accuracy | * |
| Oxford Flowers-102 [NZ08] | Flower recognition | 6149 | 1836 | 204 | mean per class | ✓ |
| Oxford-IIIT Pet [PVZ+12; ZPK+19] | Pet classification | 3669 | 3312 | 368 | mean per class | ✓ |
| Pascal VOC 2007 [EVW+10] | Object recognition | 14976 | 14096 | 1566 | accuracy | ✓ |
| PatchCamelyon [VLW+18; ZPK+19] | Metastatic tissue cls. | 32768 | 289912 | 5000 | accuracy | ✓ |
| Rendered SST2 [ZPK+19] | Sentiment classification | 1821 | 7013 | 779 | accuracy | ✓ |
| RESISC45 [CHL17; ZPK+19] | Satellite imagery recognition | 6300 | 22680 | 2520 | accuracy | ✓ |
| Stanford Cars [KSD+13] | Vehicle recognition | 8041 | 7329 | 814 | accuracy | ✓ |
| STL-10 [CNL11] | Visual recognition | 8000 | 4500 | 500 | accuracy | ✓ |
| SUN-397 [XHE+10] | Scene recognition | 108753 | N/A | N/A | accuracy | ‡ |
| SVHN [NWC+11; ZPK+19] | Digit recognition | 26032 | 68257 | 5000 | accuracy | ✓ |
| iWildCam [BAC+21; KSM+20] | Animal recognition | 42791 | 147084 | 5000 | macro F1 score | ✓ |
| Camelyon17 [BGM+18; KSM+20] | Metastatic tissue cls. | 85054 | 365900 | 5000 | accuracy | ✓ |
| FMoW [CFW+18; KSM+20] | Satellite imagery recognition | 22108 | 103261 | 5000 | worst-region acc. | ✓ |
| Dollar Street [RDK+22] | Object recognition | 3503 | 13842 | 1537 | worst-income top-5 acc. | ✓ |
| GeoDE [RLZ+24] | Object recognition | 12438 | 44488 | 4943 | worst-region acc. | ✓ |
| Flickr30k [YLH+14] | Image and text retrieval | 31014 | N/A | N/A | R@1 | § |
| MSCOCO [LMB+14] | Image and text retrieval | 5000 | N/A | N/A | R@1 | § |
| WinoGAViL [BBY+22] | Commonsense association | 3563 | N/A | N/A | Jaccard score | § |
+
+# D Selecting IFT data
+
+In this section, we describe the details of the IFT setting of Xia et al. [XMG+24], as well as the details of our method.
+
+Setting. The setting contains a fixed data pool: instruction fine-tuning data from a data pool consisting of four combined IFT datasets (cf. Table 4 and Xia et al. [XMG+24] for more information). The goal is to select the data that yields the best possible task performance for a LoRA fine-tuning run. We adapt a LoRA to a Gemma-2B model (the pretraining-only Gemma-2B model) using the LoRA configuration from Xia et al. [XMG+24].
+
+Data splits. See Table 3 for a description of the available data for each task, along with the task setup details. Xia et al. [XMG+24] constructed these extra samples by drawing from the ICL samples given in the tasks originally. Note that we drop TydiQA from the original work of Xia et al. [XMG+24] as there are not enough samples to select with (there is only one from each category, for a total of 7).
+
+Method. We execute Algorithm 1 with $k$ as 150 steps from the end of training and the Bernoulli parameter $p$ controlling the step size as 0.2. At each step, we choose a "minibatch" with a size equal to half the target set and a quarter of the target set for BBH and MMLU, respectively (that is, we only select to optimize performance on a fraction of the target set at a time). We model select over iterates and hyperparameters by (a) choosing the top three steps in terms of validation loss for each run (b) selecting the best one in terms of full train set accuracy (including the part that we trained on). We perform this procedure—akin to Pareto optimization [JS08]—because the validation set is so small (as the overall set of samples is very small) that it is difficult to select models without overfitting otherwise.
+
+We compare with two baselines: training on the full dataset (i.e., training on the entirety of all the data for a single epoch), and LESS (we use the data selected according to "LESS-T" [XMG+24], following the recommendation of 4 epochs).
+
+For model training, we train with ADAM ( $\beta_{1} = 0.95$ , $\beta_{2} = 0.975$ , decoupled weight decay as $10^{-5}$ ) and a one-cycle linear schedule starting at $10^{-6}$ of the maximum learning rate, reaching the peak over $25\%$ of training, then ending at 0.1 of the maximum learning rate. We insert a positive $\varepsilon_{\mathrm{root}}$ into the inverse square root term in the ADAM update to prevent metagradient (and to a lesser extent update) blowup (see Eq. 12). The model training is the same across selected data, except that we use $\varepsilon_{\mathrm{root}} = 10^{-7}$ for MGD-selected data and $\varepsilon_{\mathrm{root}} = 10^{-9}$ for the other runs (we select the optimal parameter for each class of method). We additionally hyperparameter select for the best learning rate across each baseline by minimizing validation set loss; LESS performs best with a smaller learning rate (0.00024 for BBH and 0.00012 for MMLU) than training on the full dataset or with MGD (0.0006 for both). We normalize the loss of each training sample by taking the mean across predicted tokens during training, and do not divide by the batch size (important for scaling the $\varepsilon_{\mathrm{root}}$ term, but otherwise ADAM is invariant to the scale).
+
+Selecting smooth model training for MGD. For MGD runs, we jointly select learning rate and $\varepsilon_{\mathrm{root}}$ using the smoothness metric of Section 3. We find that the choice of $\varepsilon_{\mathrm{root}}$ term is important (just as the choice of $\varepsilon$ is important in standard ADAM training); choosing a much larger term results in non-smooth training. We also find that metagradients are sensitive to learning rate schedule; choosing a much larger or smaller maximum learning rate results in non-smooth training.
+
+Table 3: Overview of datasets used in IFT dataset selection (from Xia et al. [XMG+24]).
+
+| Parameter | Values |
| Peak learning rate | [7.0, 7.5, 8.0, 8.5, 9.0, 9.5, 10.0, 10.5, 11.0, 11.5, 12.0] |
| Initial LR multiplier | [0.05, 0.15, 0.25, 0.35, 0.45, 0.55] |
| Final LR multiplier | [0.05, 0.15, 0.25, 0.35, 0.45, 0.55] |
| LR peak time | [0.25, 0.5, 0.75] |
+
+
+MGD step
+Figure 14: Graphs of our learned LR schedules.
\ No newline at end of file
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+ "content": "How should I clean my data? What architecture should I use? Training large-scale (i.e., deep) machine learning models entails making many design decisions. When making such decisions, typical practice is to exhaustively search over a small set of standard options. For example, we might try a few well-known data cleaning heuristics, construct a grid over a hyperparameters, and choose the options that yield the best models. However, given that this process explores only a small part of the overall design space (e.g., one can construct "
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+ "content": " candidate datapoints), it is unlikely that this approach really yields the optimal training configuration."
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+ "content": "How can we find optimal (or at least, better) training configurations? To do so, we take the optimization perspective on designing model training. From this well-studied perspective, deciding on a training configuration—or as we will call it, a set of metaparameters—is just a high-dimensional optimization problem. The input space of this problem comprises all possible metaparameter choices, including which datapoints to train on, what model architecture to use, and how to initialize model weights. The objective function takes in a set of metaparameters, trains a machine learning model according to those metaparameters, and then returns a target metric evaluated on that model (e.g., test accuracy). From this perspective, any procedure for selecting metaparameters—including the typical practice of grid-searching over standard options—is just an optimization algorithm, whose goal is to maximize the objective function with respect to the (high-dimensional) input."
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+ "content": "Given that selecting metaparameters is \"just\" a high-dimensional optimization problem, a natural tool to consider is the gradient. After all, in many contexts, gradients offer a more effective approach to maximizing high-dimensional functions than grid search. Indeed, for a sufficiently \"well-behaved\" function "
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+ "content": ". This insight suggests a generic recipe for selecting metaparameters: first, make the objective differentiable with respect to the metaparameters; second, update via gradient steps."
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+ "content": "- In the DataComp-small11 competition [GIF+24], we achieve state-of-the-art pre-training data selection for CLIP (2x larger performance improvement than the previous DataComp-small1 leader [Eco24]);"
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+ "content": "Figure 2: An illustration of the metagradient. We embed a given aspect of the training setup (e.g., the training dataset, or optimizer hyperparameters) into a continuous metaparameter vector "
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+ "content": "2.1 What is a metagradient?"
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+ "content": "Training a machine learning model is a two-step process. First, we decide on a training setup—we must pick, for example, a neural network architecture, a training dataset, and an optimizer for training. Second, we apply the algorithm defined by this training setup to train a model."
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+ "content": "Our overall goal in this paper is to optimize model behavior as a function of the training setup (or, as we call it, the metaparameters) using gradient-based methods. To this end, we define the following notation:"
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+ "content": ". We handle discrete metaparameters (e.g., choice of training data) by finding a continuous relaxation (e.g., importance weights)."
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+ "content": "A key primitive we leverage to calculate metagradients is automatic differentiation (AD)—a standard tool for taking gradients through computer-defined functions. AD takes gradients by decomposing functions into elementary operations with known derivatives, then combining these derivatives using the chain rule. Concretely, AD operates in two passes: a \"forward pass,\" which executes the function of interest and stores intermediate products for each elementary operation; and a \"backward pass,\" which calculates the gradient by propagating chains of partial derivatives using these stored products. For the purposes of this paper, we will view AD as a black box that calculates the gradient of a many-to-one function (i.e., any "
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+ "content": "What does this have to do with metagradients? Well, seeing as how training itself is a computer-defined function, AD is a natural tool for calculating the metagradient. The main challenge, as we discuss in the sequel, is that AD-based approaches to calculating the metagradient tend to be too resource-intensive for the large-scale machine learning algorithms we consider. In the remainder of this section we build up background before finally describing REPLAY, our algorithm for scalably computing (exact) metagradients."
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+ "content": " (via (2), using the previously calculated gradient with respect to that state). AD calculates both quantities--each requires differentiating over only one train step. Finally, the algorithm returns the final metagradient as the sum of the terms."
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+ "content": " a user-chosen constant. The main idea is to make the space-intensive subroutine of step-wise AD—a reverse-order traversal of the optimizer states at each step—much more efficient. After all, step-wise AD stores all the states to reverse traverse them. REPLAY modifies step-wise AD to traverse states in less space by exploiting a simple observation: when training is deterministic, one can reinstantiate an optimizer state "
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+ "content": "-length segment of optimizer states. We provide additional details on the algorithm in Appendix A.2. REPLAY unlocks computing large-scale metagradients by requiring only logarithmic storage and additional compute time."
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+ "content": " is a hyperparameter controlling the fraction of sample counts to update. This algorithm resembles a block coordinate descent algorithm [OR00], with the main difference being that we take signed gradient steps with step size 1 (projected onto non-negative integers) to ensure that the counts remain well-defined. As a result, "
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+ "content": "We evaluate our data selection algorithm using DataComp [GIF+24], a standardized framework for evaluating data selection methods for multimodal models. Algorithm 1 greatly improves on the state-of-the-art for the benchmark. Below, we describe the setting, outline our method, and conclude with our results."
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+ "content": "Setting. DataComp [GIF+24] is a multimodal model training competition and benchmark for evaluating dataset selection methods. DataComp provides a fixed learning algorithm chosen in advance by the organizers and a large fixed candidate pool of internet data. The goal is to choose a subset of the candidate pool—possibly with repeated datapoints—that yields the best-performing model after training with the given learning algorithm, as measured by a predetermined set of 38 benchmarks. Given a submission subset, the mean score on the evaluation datasets for a model trained with that subset is taken as the final \"score.\" DataComp offers four separate \"scales\" requiring different amounts of compute; we focus on the small scale in this paper due to compute limitations."
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+ "content": "Method. We select data with MGD (Algorithm 1) to minimize loss on data on a \"target set\" that is distributionally similar to the DataComp benchmark tasks, and select hyperparameters with a held-out \"validation set.\" In particular, we construct target and validation sets by taking samples from the DataComp evaluation tasks with extra samples available beyond those used in the DataComp test set (e.g., ImageNet, one of the tasks in DataComp, has a training set in addition to the test set evaluated in DataComp). See Appendix C for the exact details of the target and validation sets, the precise hyperparameters used with Algorithm 1, and a discussion on scalability (including further engineering details on executing our algorithm efficiently)."
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+ "content": "Results. MGD greatly outperforms the current state-of-the-art: the difference in accuracy between MGD and the current best method is roughly as large as the difference between the previous state-of-the-art (EcoDatum [Eco24]) and training on randomly chosen data (cf. Figure 6). Inspecting scores over the course of the optimization in Figure 6, we find that only a few steps are necessary to outperform previous methods."
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+ "content": "Figure 6: MGD dataset selection greatly outperforms existing methods (improving over the previous SOTA by as much as the previous SOTA improves over no filtering at all). We compare DataComp scores for MGD (over optimization steps), training on the entire candidate pool, the best baseline originally proposed by DataComp, and the previous SOTA [Eco24]."
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+ "html": "| Dataset | Task | Test size | Train size | Val size | Main metric | Target set |
| Caltech-101 [FFP04] | Object recognition | 6085 | 2754 | 306 | mean per class | ✓ |
| CIFAR-10 [Kri09] | Visual recognition | 10000 | 45000 | 5000 | accuracy | ✓ |
| CIFAR-100 [Kri09] | Visual recognition | 10000 | 45000 | 5000 | accuracy | ✓ |
| CLEVR Counts [JHV+17; ZPK+19] | Counting | 15000 | 65000 | 5000 | accuracy | ✓ |
| CLEVR Distance [JHV+17; ZPK+19] | Distance prediction | 15000 | 65000 | 5000 | accuracy | ✓ |
| Country211 [RKH+21; TSF+16] | Geolocation | 21100 | 37980 | 4220 | accuracy | ✓ |
| DTD [CMK+14] | Texture classification | 1880 | 3384 | 376 | accuracy | ✓ |
| EuroSAT [HBD+19; ZPK+19] | Satellite imagery recognition | 5400 | 19440 | 2160 | accuracy | ✓ |
| FGVC Aircraft [MRK+13] | Aircraft recognition | 3333 | 6001 | 666 | mean per class | ✓ |
| Food-101 [BGV14] | Food recognition | 25250 | 70750 | 5000 | accuracy | ✓ |
| GTSRB [SSS+11] | Traffic sign recognition | 12630 | 35289 | 3920 | accuracy | ✓ |
| ImageNet 1k [DDS+09] | Visual recognition | 50000 | 1276167 | 5000 | accuracy | ✓ |
| ImageNet Sketch [WGX+19] | Visual recognition | 50889 | N/A | N/A | accuracy | * |
| ImageNet V2 [RKS+19] | Visual recognition | 10000 | N/A | N/A | accuracy | * |
| ImageNet-A [HZB+19] | Visual recognition | 7500 | N/A | N/A | accuracy | * |
| ImageNet-O [HZB+19] | Visual recognition | 2000 | N/A | N/A | accuracy | * |
| ImageNet-R [HBM+20] | Visual recognition | 30000 | N/A | N/A | accuracy | * |
| KITTI distance [GLU12; ZPK+19] | Distance prediction | 711 | N/A | N/A | accuracy | † |
| MNIST [LeC98] | Digit recognition | 10000 | 55000 | 5000 | accuracy | ✓ |
| ObjectNet [BMA+19] | Visual recognition | 18574 | N/A | N/A | accuracy | * |
| Oxford Flowers-102 [NZ08] | Flower recognition | 6149 | 1836 | 204 | mean per class | ✓ |
| Oxford-IIIT Pet [PVZ+12; ZPK+19] | Pet classification | 3669 | 3312 | 368 | mean per class | ✓ |
| Pascal VOC 2007 [EVW+10] | Object recognition | 14976 | 14096 | 1566 | accuracy | ✓ |
| PatchCamelyon [VLW+18; ZPK+19] | Metastatic tissue cls. | 32768 | 289912 | 5000 | accuracy | ✓ |
| Rendered SST2 [ZPK+19] | Sentiment classification | 1821 | 7013 | 779 | accuracy | ✓ |
| RESISC45 [CHL17; ZPK+19] | Satellite imagery recognition | 6300 | 22680 | 2520 | accuracy | ✓ |
| Stanford Cars [KSD+13] | Vehicle recognition | 8041 | 7329 | 814 | accuracy | ✓ |
| STL-10 [CNL11] | Visual recognition | 8000 | 4500 | 500 | accuracy | ✓ |
| SUN-397 [XHE+10] | Scene recognition | 108753 | N/A | N/A | accuracy | ‡ |
| SVHN [NWC+11; ZPK+19] | Digit recognition | 26032 | 68257 | 5000 | accuracy | ✓ |
| iWildCam [BAC+21; KSM+20] | Animal recognition | 42791 | 147084 | 5000 | macro F1 score | ✓ |
| Camelyon17 [BGM+18; KSM+20] | Metastatic tissue cls. | 85054 | 365900 | 5000 | accuracy | ✓ |
| FMoW [CFW+18; KSM+20] | Satellite imagery recognition | 22108 | 103261 | 5000 | worst-region acc. | ✓ |
| Dollar Street [RDK+22] | Object recognition | 3503 | 13842 | 1537 | worst-income top-5 acc. | ✓ |
| GeoDE [RLZ+24] | Object recognition | 12438 | 44488 | 4943 | worst-region acc. | ✓ |
| Flickr30k [YLH+14] | Image and text retrieval | 31014 | N/A | N/A | R@1 | § |
| MSCOCO [LMB+14] | Image and text retrieval | 5000 | N/A | N/A | R@1 | § |
| WinoGAViL [BBY+22] | Commonsense association | 3563 | N/A | N/A | Jaccard score | § |
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+ "content": " as 150 steps from the end of training and the Bernoulli parameter "
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+ "content": " controlling the step size as 0.2. At each step, we choose a \"minibatch\" with a size equal to half the target set and a quarter of the target set for BBH and MMLU, respectively (that is, we only select to optimize performance on a fraction of the target set at a time). We model select over iterates and hyperparameters by (a) choosing the top three steps in terms of validation loss for each run (b) selecting the best one in terms of full train set accuracy (including the part that we trained on). We perform this procedure—akin to Pareto optimization [JS08]—because the validation set is so small (as the overall set of samples is very small) that it is difficult to select models without overfitting otherwise."
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+ "type": "inline_equation",
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+ "content": " into the inverse square root term in the ADAM update to prevent metagradient (and to a lesser extent update) blowup (see Eq. 12). The model training is the same across selected data, except that we use "
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+ "type": "text",
+ "content": " for the other runs (we select the optimal parameter for each class of method). We additionally hyperparameter select for the best learning rate across each baseline by minimizing validation set loss; LESS performs best with a smaller learning rate (0.00024 for BBH and 0.00012 for MMLU) than training on the full dataset or with MGD (0.0006 for both). We normalize the loss of each training sample by taking the mean across predicted tokens during training, and do not divide by the batch size (important for scaling the "
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+ "content": " using the smoothness metric of Section 3. We find that the choice of "
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+ "content": " is important in standard ADAM training); choosing a much larger term results in non-smooth training. We also find that metagradients are sensitive to learning rate schedule; choosing a much larger or smaller maximum learning rate results in non-smooth training."
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+ "html": "| Parameter | Values |
| Peak learning rate | [7.0, 7.5, 8.0, 8.5, 9.0, 9.5, 10.0, 10.5, 11.0, 11.5, 12.0] |
| Initial LR multiplier | [0.05, 0.15, 0.25, 0.35, 0.45, 0.55] |
| Final LR multiplier | [0.05, 0.15, 0.25, 0.35, 0.45, 0.55] |
| LR peak time | [0.25, 0.5, 0.75] |
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diff --git a/data/2025/2503_13xxx/2503.13804/53558f89-5ff8-41c8-b6ca-fe406c0656ac_content_list.json b/data/2025/2503_13xxx/2503.13804/53558f89-5ff8-41c8-b6ca-fe406c0656ac_content_list.json
new file mode 100644
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+ {
+ "type": "text",
+ "text": "Empowering GraphRAG with Knowledge Filtering and Integration",
+ "text_level": 1,
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+ "page_idx": 0
+ },
+ {
+ "type": "text",
+ "text": "Kai Guo $^{1}$ , Harry Shomer $^{1}$ , Shenglai Zeng $^{1}$ , Haoyu Han $^{1}$ , Yu Wang $^{2}$ , Jiliang Tang $^{1}$",
+ "bbox": [
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+ "type": "text",
+ "text": "$^{1}$ Michigan State University $^{2}$ University of Oregon",
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+ },
+ {
+ "type": "text",
+ "text": "{guokai1, shomerha, zengshe1, hanhaoy1, tangjili} @msu.edu,",
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+ "type": "text",
+ "text": "{yuwang} @uoregon.edu",
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+ {
+ "type": "text",
+ "text": "Abstract",
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+ "text": "In recent years, large language models (LLMs) have revolutionized the field of natural language processing. However, they often suffer from knowledge gaps and hallucinations. Graph retrieval-augmented generation (GraphRAG) enhances LLM reasoning by integrating structured knowledge from external graphs. However, we identify two key challenges that plague GraphRAG: (1) Retrieving noisy and irrelevant information can degrade performance and (2) Excessive reliance on external knowledge suppresses the model's intrinsic reasoning. To address these issues, we propose GraphRAG-FI (Filtering & Integration), consisting of GraphRAG-Filtering and GraphRAG-Integration. GraphRAG-Filtering employs a two-stage filtering mechanism to refine retrieved information. GraphRAG-Integration employs a logits-based selection strategy to balance external knowledge from GraphRAG with the LLM's intrinsic reasoning, reducing over-reliance on retrievals. Experiments on knowledge graph QA tasks demonstrate that GraphRAG-FI significantly improves reasoning performance across multiple backbone models, establishing a more reliable and effective GraphRAG framework.",
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+ "type": "text",
+ "text": "1 Introduction",
+ "text_level": 1,
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+ "text": "Large language models (LLMs) have achieved remarkable success in NLP tasks, particularly in tasks that require complex reasoning (Havrilla et al.; Wu et al., 2023; Hao et al., 2023). However, despite their strengths, LLMs are prone to hallucinations, resulting in incorrect or poor reasoning (Ji et al., 2023; Huang et al., 2024; Sriramanan et al., 2025). GraphRAG techniques have emerged as a promising solution to this problem (Han et al., 2024; Zhang et al., 2025; He et al., 2025; Mavromatis and Karypis, 2024), by integrating relevant information from external graphs. Knowledge graphs, which store facts in the form of a graph, are commonly used for this problem. Specifically, relevant",
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+ "text": "facts (i.e., triples) or paths are extracted from the knowledge graph and used to enrich the context of the LLMs with structured and reliable information (Luo et al., 2024; Li et al., 2025; Ma et al., 2024). This approach has shown ability to improve the reasoning capabilities and reduce the presence of hallucinations in LLMs (Sun et al.; Li et al., 2025; Dong et al., 2024).",
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+ "text": "To better assess the efficacy of GraphRAG, in Section 3 we conduct a preliminary study comparing its performance with an LLM-only model (i.e., LLM without GraphRAG). This comparison reveals both the advantages and limitations of GraphRAG. While GraphRAG improved reasoning accuracy by correcting some LLM errors, it also introduces some notable weaknesses. For example, incorporating external knowledge will sometimes cause questions that were originally answered correctly by the LLM to be misclassified. This highlights the dangers of retrieving irrelevant information. Furthermore, excessive retrieval compounds this issue by introducing both noise and redundant information, thus further hindering the reasoning process.",
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+ "text": "Meanwhile, we find that LLM-only and GraphRAG can complement one another. Specifically, GraphRAG can enhance reasoning for those questions LLMs lack knowledge of; while excessive reliance on external information may cause the model to overlook internally known correct answers. These findings highlight two key limitations of existing GraphRAG methods. First, GraphRAG is highly susceptible to retrieving irrelevant or misleading information. Second, GraphRAG struggles to balance external retrieval with the LLM's internal knowledge, often missing parts of the answer that the LLM-only model can provide using its own knowledge.",
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+ "text": "Inspired by these findings, we propose a novel design that aims to address these issues. First, we aim to enhance the retrieval quality to better avoid",
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+ "text": "arXiv:2503.13804v1 [cs.AI] 18 Mar 2025",
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+ "type": "text",
+ "text": "retrieving irrelevant information. Second, we integrate GraphRAG with an LLM's intrinsic reasoning ability, thus only using GraphRAG when external knowledge is necessary. In particular, to mitigate the issue of retrieving irrelevant information, we introduce a two-stage filtering process. Furthermore, to mitigate GraphRAG from overrelying on retrieved information while underutilizing the LLM's inherent reasoning ability, we introduce a logits-based selection mechanism that dynamically integrates LLMs' standalone answers with GraphRAG's outputs. This approach ensures that the final response effectively balances external knowledge with the model's internal reasoning. The main contributions of our work are summarized as follows:",
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+ "- We identify two key challenges in GraphRAG: (1) It is susceptible to errors by retrieving irrelevant or misleading information. (2) It overemphasizes the externally retrieved knowledge, at the expense of the intrinsic reasoning capabilities of LLMs.",
+ "- We introduce a novel approach that enhances GraphRAG by incorporating a two-stage filtering mechanism to refine the retrieved knowledge and dynamically integrate this knowledge with a LLMs' standalone reasoning capabilities.",
+ "- Extensive experiments on knowledge graph QA demonstrate the effectiveness of our method across multiple backbone models."
+ ],
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+ "type": "text",
+ "text": "2 Related work",
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+ "text": "GraphRAG. GraphRAG aims to address hallucinations and outdated knowledge in LLMs by incorporating additional information retrieved from external knowledge bases (Sun et al.; Li et al., 2025; Dong et al., 2024). G-R retriever (He et al., 2025) identifies relevant nodes and edges for a given query based on cosine similarity, and then constructs a subgraph to aid in the generation process. Similarly, RoG (Luo et al., 2024) introduces a planning-retrieval-reasoning framework, where it retrieves reasoning paths guided by a planning module and performs reasoning using these paths. On the other hand, GNN-RAG (Mavromatis and Karypis, 2024) leverages Graph Neural Networks (GNNs) (Kipf and Welling, 2016) to process the intricate graph structures within knowledge graphs, enabling effective retrieval. They also use",
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+ "type": "text",
+ "text": "retrieval augmentation techniques to enhance diversity. However, the effectiveness of these methods is heavily dependent on the quality of the retrieved information, and their performance significantly declines when the retrieved graph data is either noisy or unrelated to the query (He et al., 2025).",
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+ "text": "Filter Methods. Filtering attempts to only keep those pieces of retrieved information that are relevant to the given query (Gao et al., 2025). ChunkRAG (Singh et al., 2024) tries to improve RAG systems by assessing and filtering retrieved data at the chunk level, with each \"chunk\" representing a concise and coherent segment of a document. This method first applies semantic chunking to partition documents into meaningful sections. It then leverages LLM-based relevance scoring to evaluate how well each chunk aligns with the user query. Zeng et al. (2024b) thoroughly investigate LLM representation behaviors in relation to RAG, uncovering distinct patterns between positive and negative samples in the representation space. This distinction enables representation-based methods to achieve significantly better performance for certain tasks. Building on these insights, they introduce Rep-PCA, which employs representation classifiers for knowledge filtering. RoK (Wang et al., 2024) refines the reasoning paths within the subgraph by computing the average PageRank score for each path. Similarly, He et al. (2024) use PageRank to identify the most relevant entities.",
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+ "text": "3 Preliminary studies",
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+ "text": "To evaluate the effectiveness of GraphRAG, we compare the performance with and without retrieved external knowledge. Furthermore, we analyze the attention scores of the LLM to assess its ability to discern both the relevance and importance of the retrieved information. Lastly, we evaluate the performance of internal knowledge filtering.",
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+ "text": "3.1 Experimental settings",
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+ "text": "In this section, we aim to study the importance of retrieving external information when using GraphRAG for knowledge graph QA. To do so, we report the QA performance when using: LLM with GraphRAG and LLM w/o GraphRAG (i.e., LLM-only). For GraphRAG, we use RoG (Luo et al., 2024) and GNN-RAG (Mavromatis and Karypis, 2024). For the LLM-only experiments, we use the fine-tuned LLaMA 2-7B model, which is the same LLM used by RoG. The experiments are con",
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+ "Figure 1: Category A includes cases where both GraphRAG and the LLM-only model are correct. Category B covers instances where GraphRAG outperforms the LLM-only model, while Category C includes cases where the LLM-only model performs better than GraphRAG. Category D represents cases where both models fail."
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+ "type": "text",
+ "text": "duced on two common datasets the WebQSP (Yih et al., 2016) and CWQ (Talmor and Berant, 2018) datasets. In this study, we mainly use the F1 score to evaluate the performance.",
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+ "Figure 2: The relationship between path number and average F1"
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+ "text": "3.2 The Impact of GraphRAG",
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+ "text": "To understanding the effectiveness of GraphRAG, we compare prediction outcomes between LLM with GraphRAG and LLM w/o GraphRAG (i.e., LLM-only). We categorize the results into four groups based on F1 scores, as shown in the Figure 1. Category A includes cases where both GraphRAG and the LLM-only model provide correct answers. Category B consists of instances where GraphRAG produces a more accurate answer than the LLM-only model. Category C includes cases where the LLM-only model outperforms GraphRAG. Finally, Category D represents instances where both GraphRAG and the LLM-only model fail to generate the correct answer. Figure 1 illustrates the key observations from our experiments. While GraphRAG enhances certain predictions, it also",
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+ "type": "text",
+ "text": "introduces notable challenges that require further investigation.",
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+ "text": "Positive Impact of GraphRAG GraphRAG can enhance the LLM's reasoning capabilities by correcting errors that the standalone model would typically commit. Notably, in the category B, $45.64\\%$ of previously incorrect responses were successfully rectified with the integration of GraphRAG. This highlights the advantage of leveraging structured knowledge graphs to boost LLM performance.",
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+ "text": "Limited Impact of GraphRAG Category A contains those answers where both GraphRAG and LLM-only are correct. This show that GraphRAG can sometimes preserve the performance of a LLM when the LLM already possesses the correct knowledge. Conversely, category D, representing $9.03\\%$ of cases, corresponds to those cases where GraphRAG fails to enhance the model's accuracy. For this category, neither the standalone LLM nor GraphRAG are able to provide the correct answer. This pattern implies that GraphRAG does not always access or incorporate sufficiently informative or relevant knowledge.",
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+ "text": "Negative Impact of GraphRAG A notable drawback of GraphRAG is that will occasionally degrade the performance of a standalone LLM. That is, it will sometimes lead to wrong predictions for queries that the standalone LLM originally got right. These instances are represented by category C and accounts for $16.89\\%$ of samples when evaluating via the F1 score. In these cases, GraphRAG misleads the model rather than improving it. This suggests that some of the retrieved information may be incorrect, noisy, or irrelevant, ultimately leading to poorer predictions. Therefore, in some cases, LLMs without GraphRAG outperform those with GraphRAG, because existing works have shown that LLMs tend to over-rely on external information (Ren et al., 2023; Tan et al., 2024; Wang et al., 2023; Ni et al., 2024; Zeng et al., 2024a). When retrieval is insufficient or the quality of retrieved knowledge is low, this reliance can degrade generation quality.",
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+ "type": "text",
+ "text": "3.3 The Impact of the Number of Retrieved Paths",
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+ "text": "Due to the structure of knowledge graphs, nodes with high degrees and numerous relational edges have a greater likelihood of yielding a large number of retrieved paths. In this subsection, we study",
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+ "text": "the impact of the number of retrieved paths on performance. Figure 2 illustrates the relationship between the number of retrieved paths and the model's performance. Interestingly, as indicated by the smoothed line (blue), incorporating a moderate amount of retrieved information enhances performance. However, increasing the number of retrieved paths ultimately leads to a decline in performance. This trend (green line) suggests that retrieving too much information will introduce noise, making it harder for the model to use the correct and relevant knowledge for the task. This phenomenon thus highlights an important insight - more information does not necessarily indicate better performance. Instead, an overabundance of retrieved data can overwhelm the model with irrelevant details. This observation underscores the necessity for effective filtering mechanisms that can prioritize high-quality, relevant knowledge while discarding extraneous or misleading information.",
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+ "img_path": "images/5394ce95b061ee2998994297ff15b21d1f49085458e088ff35602ddff5e6c4a4.jpg",
+ "image_caption": [
+ "3.4 Attention Reflects the Importance of Retrieved Information",
+ "Figure 3: Attention Scores for Retrieved Information With/Without Ground Truth"
+ ],
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+ "text": "In this subsection, we analyze the ability of the LLM to distinguish the importance of retrieved external knowledge. The attention scores of a LLM can provide a natural indicator of the relevance and significance of the retrieved knowledge (Yang et al., 2024; Ben-Artzy and Schwartz, 2024). The attention scores, derived from the model's internal mechanisms, effectively capture which pieces of information are most influential in reaching the final decision. Inspired by recent work (Chuang et al., 2023; Halawi et al., 2023), which suggests that attention scores in the middle layers are more effective. We examine the attention scores of the (middle + 2)-th layer in the LLM for each retrieved path. We obtain the attention scores for all retrieved paths and categorize them into two groups: (1) paths that",
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+ "text": "contain the ground truth and (2) paths that do not. We then compute the average attention score for each group and present the results in Figure 3. As demonstrated in Figure 3, there is a clear alignment between the attention scores and the ground truth labels, suggesting that these scores can be used to assess the relevance of retrieved information.",
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+ "text": "This observation inspires a key insight: The attention scores highlight the most significant retrieved information, suggesting their potential use in filtering out noisy or irrelevant knowledge. Since retrieved information with lower attention scores contribute minimally to the final output, they can be pruned to streamline retrieval and enhance overall performance.",
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+ "text": "3.5 Internal Knowledge Filtering",
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+ "text": "Large language models (LLMs) generate responses that may contain both correct and incorrect information. To assess the reliability of these responses, we analyze the associated logits, which represent the model's confidence in its predictions. Typically, higher confidence correlates with correctness (Ma et al., 2025; Virk et al., 2024). Leveraging this property, we implement \"Internal Knowledge Filtering\", which uses the logits to help refine the answer selection. The logits of answer can be directly obtained from the LLM's output. Formally, let $A_{L}$ denote the sets of answer candidates from the LLM model. Furthermore, let it's corresponding logits after softmax function be given by $\\ell_{L}(a)$ . The filtering step is given by the following:",
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+ "type": "equation",
+ "text": "\n$$\nA _ {L} ^ {\\text {f i l t e r e d}} = \\left\\{a \\in A _ {L} \\mid \\ell_ {L} (a) \\geq \\tau_ {L} \\right\\}, \\tag {1}\n$$\n",
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+ "text": "where $\\tau_{L} = 1$ . This allows us to filter out the responses that the LLM has low-confidence in. The experimental results are shown in Table 1. We can clearly see that that leveraging logits to filter out low-confidence responses has a large positive effect on performance. In this way, we can reconsider intrinsic knowledge and apply this approach to GraphRAG to better balance internal and external knowledge base on logits.",
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+ "Table 1: Impact of logits on LLM performance"
+ ],
+ "table_footnote": [],
+ "table_body": "| Variables | Description |
| uh(x) | Interpolated scalar field defined inside of an element |
| Ae | Nodes within element e |
| As | Nodes within patch domain of element e |
| WiS,a,p,j(x) | Convolution patch function at node j for i-th nodal patch with hyperparameters s, a, and p |
| M | Total number of modes in tensor decomposition (TD) |
| m | Index for mode |
| D | Total number of dimensions |
| d | Index for dimension |
| x | Independent variable which includes spatial variable xs, parametric variable xp and temporal variable xt |
| Nd(xd;ad,sd,pd) | Global C-HiDeNN shape function for dimension d with dilation parameter ad, patch size sd and reproducing polynomial order pd |
| b | Source function in laser powder bed fusion process |
| uTD | Approximation of the solution field expressed via TD |
| T | Time slab index for space-parameter-time problem |
| k | Thermal conductivity |
| ρ | Material density |
| cp | Heat capacity |
| η | Material absorptivity |
| P | Laser power |
| r | Standard deviation that characterizes the width of the heat source |
| q | Heat flux |
| qconv | Heat flux from convection |
| qrad | Heat flux from radiation |
| qevap | Heat flux from evaporative cooling |
| hconv | Convection coefficient |
| σSB | Stefan-Boltzman constant |
| mevap | Mass evaporation flux |
| Levap | Heat of evaporation |
| Bnd | Shape function derivative |
| Ud | Solution matrix (Rnd×M) for dimension d that contains all the modes |
",
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+ "text": "To fulfill the ever-growing challenges in predictive scientific models, data-driven surrogates, especially artificial intelligence (AI)-based models, present an alternative to conventional numerical models by significantly reducing the forward prediction time. These models can be treated as a reasonably accurate, reduced representation of real physics. Once trained properly, they can be used for fast prediction on unseen parameters [1, 2]. However, it is still uncertain whether a data-driven surrogate model can be trained to achieve the level of accuracy required in engineering design. Recently, it has been pointed out by Wolfram Research that standard AI models cannot easily fulfill the high accuracy requirement of predictive scientific tasks [3]. Furthermore, as suggested by Google Deepmind, the real potential of AI models lies in enhancing, rather than thoroughly replacing, well-established classical numerical algorithms [4]. In addition, the current standard data-driven approaches follow an offline-online scheme, where the offline stage involves a huge amount of training data, which can again be prohibitive. For problems with known physics, this data can be obtained by running multiple expensive simulations relying on standard numerical algorithms. In scenarios involving high-dimensional design spaces governed by parameterized partial differential equations (PDEs), such as in additive manufacturing (AM), conducting repetitive simulations with varying parameters in this offline stage becomes exceedingly expensive both in terms of computation time and data storage.",
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+ "type": "text",
+ "text": "To avoid the prohibitive offline stage, one can try to obtain a surrogate model directly from governing equations",
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+ "type": "page_number",
+ "text": "2",
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+ "img_path": "images/ba32edcab434e8ad5e0767139009d16908e9de00ed4f1dfddd99493d69460436.jpg",
+ "image_caption": [
+ "Figure 1: The parameterized PDE is a PDE that includes parameters $\\mathbf{x}_p$ that can vary and influence the solution $\\mathbf{u}(\\mathbf{x}_s, \\mathbf{x}_p, x_t)$ , where $\\mathbf{x}_s$ and $x_t$ are the spatial and time variables, respectively. The a priori approach directly finds a surrogate model from the governing parameterized PDE, whereas the data-driven approach has to solve the parameter-fixed PDE on sampled parameters to generate simulation data, followed by training tasks. FEM: Finite Element Method [5], C-HiDeNN: Convolution Hierarchical Deep-learning Neural Network [6], TAPS: Tensor-decomposition-based A Priori Surrogate, PINN: Physics Informed Neural Network [7]."
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+ "page_idx": 2
+ },
+ {
+ "type": "text",
+ "text": "without generating any data. As shown in Fig. 1 denoted by the words \"A Priori\", this approach aims to find the surrogate model before actually \"seeing\" any data. For example, multilayer perceptron (MLP) architectures have been vastly used in physics-informed neural networks (PINNs) and their variations to approximate solutions to PDEs without requiring data [7, 8]. However, the results of these efforts are underwhelming, as it has been shown that PINN results have often been compared to weak baselines [9], and it is unclear if they guarantee convergence. Moreover, this method is still susceptible to high computational costs for both large-scale and high-dimensional problems [10].",
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+ "image_caption": [
+ "Figure 2: Development history of INN [11]. Figures are borrowed from references: HiDeNN [12], HiDeNN-TD [13], C-HiDeNN [14], C-HiDeNN-TD [6]."
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+ "text": "Instead of developing solvers solely based on machine learning concepts, a new class of Hierarchical Deep-learning Neural Networks (HiDeNN) has been developed recently. This network architecture incorporates principles from the finite element method (FEM) to construct their architecture [15, 12]. Originally designed to advance FEM as opposed to solve parameterized PDEs, this approach significantly enhances computational accuracy and efficiency for both linear and nonlinear problems compared to standard FEM [16]. HiDeNN was then enhanced by adding an additional hidden layer in the form of a nonlinear convolutional filter, formulating a new neural network architecture named Convolutional HiDeNN (C-HiDeNN) [6, 14]. C-HiDeNN mimics the structure of the generalized finite element method but leverages machine learning to optimize its hyperparameters to further improve accuracy and efficiency. Arbitrary orders of convergence have been observed for C-HiDeNN despite utilizing a linear finite element mesh [14]. Although these methods offers greater accuracy with fewer DoFs, like FEM, they still encounter computational challenges such as balancing memory usage against mesh resolution, which limits their efficiency in modeling ultra large-scale and high-dimensional problems. Therefore, it becomes necessary to employ model order reduction",
+ "bbox": [
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+ "page_idx": 2
+ },
+ {
+ "type": "page_number",
+ "text": "3",
+ "bbox": [
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+ "page_idx": 2
+ },
+ {
+ "type": "text",
+ "text": "techniques to address these limitations.",
+ "bbox": [
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+ ],
+ "page_idx": 3
+ },
+ {
+ "type": "text",
+ "text": "Model order reduction techniques have been widely used to tackle the ever-growing challenges from high-dimensional and large-scale problems. For example, proper generalized decomposition (PGD) [17, 18, 19] has been proposed to efficiently solve high-dimensional PDEs. Recently, tensor decomposition (TD) has been successfully leveraged within the HiDeNN framework. For example, Zhang showed that HiDeNN combined with TD (HiDeNN-TD) significantly improved the speed of HiDeNN while maintaining higher accuracy [13]. Li proposed C-HiDeNN combined with TD (C-HiDeNN-TD) for extremely large-scale nested topology optimization problems [20]. Recently, Park generalized the HiDeNN-family networks under the umbrella of Interpolating Neural Networks (INNs) and demonstrated that the network can be used for both data-driven learning and data-free (i.e., a priori) solving [11]. The development history of HiDeNN family networks and INN is summarized in Fig. 2. While INN clearly explains how to construct the network architecture, an efficient optimization scheme for solving ultra large-scale and high-dimensional problems remains underdeveloped. In this paper, ultra large-scale problems refer to problems on the zetta-scale $(10^{21})$ in terms of DoFs.",
+ "bbox": [
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+ "page_idx": 3
+ },
+ {
+ "type": "text",
+ "text": "The demand for high-precision engineering simulations and efficient solution schemes highlights the need for innovative modeling approaches that swiftly solve large-scale problems while optimizing the design space. This research aims to fulfill this need by developing tensor-decomposition-based A Priori Surrogate (TAPS), a data-free predictive AI model, which aims to enhance high-resolution capabilities while simultaneously optimizing computational efficiency with a minimal memory footprint, low data storage needs, and fast prediction. The proposed comprehensive framework sets a foundation for scalable, adaptable, and future-proof solutions to counter the ever-growing complexity in simulation-driven advanced industries. TAPS is particularly well-suited for engineering challenges where: 1) the finite element method and other conventional methods are unsuitable due to excessively long simulation times or high RAM and storage demands needed to achieve high accuracy, 2) the model must accommodate design parameters as inputs, or 3) fast prediction is required once the model is obtained.",
+ "bbox": [
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+ "page_idx": 3
+ },
+ {
+ "type": "text",
+ "text": "This paper is structured as follows. We first introduce the formulation of TAPS in section 2. In section 3, we examine the numerical convergence of TAPS for both space-time (S-T) and space-parameter-time (S-P-T) problems (i.e., problems that are dependent on spatial, parametric, and temporal inputs). In section 4, TAPS is applied to large-scale additive manufacturing problems that are considered intractable with standard numerical algorithms. This application effectively demonstrates TAPS's capability to address all of the three identified challenges.",
+ "bbox": [
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+ "page_idx": 3
+ },
+ {
+ "type": "text",
+ "text": "2. Theory",
+ "text_level": 1,
+ "bbox": [
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+ "page_idx": 3
+ },
+ {
+ "type": "text",
+ "text": "2.1. Review of C-HiDeNN interpolation theory",
+ "text_level": 1,
+ "bbox": [
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+ "page_idx": 3
+ },
+ {
+ "type": "text",
+ "text": "Leveraging the universal approximation theorem, multilayer perceptrons (MLPs) have been successfully applied as global basis functions in deep learning-based solvers [7]. However, as shown in Table 2, MLPs have a few potential caveats when approximating PDE solutions. To overcome these limitations, we leverage the Convolutional HiDeNN (C-HiDeNN) interpolation function, which leverages the merits of both locally supported finite element shape functions and the flexibility of machine learning. Note that C-HiDeNN also belongs to the INN category as shown in Fig. 2. C-HiDeNN maintains all the essential finite element approximation properties such as Kronecker delta and partition of unity [14].",
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+ {
+ "type": "table",
+ "img_path": "images/043303db4e5c82962bde585e2e5978667fa6324d753cd893b2e7c4471a16091e.jpg",
+ "table_caption": [
+ "Table 2: Comparison of MLP and C-HiDeNN as approximation functions of PDE solutions."
+ ],
+ "table_footnote": [],
+ "table_body": "| PDE | Differential operator L | Dependent variable | xs | xp | xt |
| ∂2u/∂x12 + ∂2u/∂x22 + ... + ∂2u/∂x2D = f(x) | ∂2/∂x12 + ∂2/∂x22 + ... + ∂2/∂x2D | u | (x1, x2, ...,xD) | - | - |
| μui, jj + (μ + λ)uj,ij + Fi = e(x12+x22+x32) | μ(·)i,jj + (μ + λ)(·)j,ij | ui, i = 1, 2, 3 | (x1, x2, x3) | (λ,μ) | - |
| ρcp,du/dt + k(∂2u/∂x12 + ∂2u/∂x22 + ∂2u/∂x32) = Pe(x12+x22+x32) | ρcp,du/dt + k(∂2/∂x12 + ∂2/∂x22 + ∂2/∂x32) | u | (x1, x2, x3) | (ρ, cp, k, P) | t |
",
+ "bbox": [
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+ "page_idx": 7
+ },
+ {
+ "type": "text",
+ "text": "The weighted-sum residual form of the PDE with TD interpolation can be written as:",
+ "bbox": [
+ 127,
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+ 702,
+ 560
+ ],
+ "page_idx": 7
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\int_ {\\Omega} \\delta u ^ {T D} (\\boldsymbol {x}) \\left[ \\mathcal {L} \\left(u ^ {T D} (\\boldsymbol {x})\\right) - f (\\boldsymbol {x}) \\right] d \\Omega = 0 \\tag {15}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 361,
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+ 892,
+ 601
+ ],
+ "page_idx": 7
+ },
+ {
+ "type": "text",
+ "text": "where $u^{TD}$ is the approximation of the solution (i.e., trial function), $\\delta u^{TD}$ is the test function, and $d\\Omega = d\\Omega_{x_s}d\\Omega_{x_p}d\\Omega_{x_t}$ .",
+ "bbox": [
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+ "page_idx": 7
+ },
+ {
+ "type": "text",
+ "text": "Depending on how $\\delta u^{TD}$ is adopted, different mathematical formulations can be obtained. If the test function resides in the same function space as the trial function, it becomes the Galerkin formulation. When the test function space differs from the trial function space, it becomes the Petrov-Galerkin formulation [22]. If the Dirac delta function is used for the test function, then Eq. 15 corresponds to the collocation method [24]. In this paper, we employ the Galerkin formulation. However, the proposed framework is versatile and can be extended to accommodate other formulations as well.",
+ "bbox": [
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+ ],
+ "page_idx": 7
+ },
+ {
+ "type": "text",
+ "text": "In Eq. 12, the entire S-P-T domain is approximated using TD interpolation. However, this approach may result in a large system of equations due to the rapid increase in the number of TD modes for certain cases. For example, if the forcing function represents a moving source function in Eq. 14), this complexity may arise. To maintain computational efficiency, we can partition the temporal domain into a series of time slabs. As illustrated in Fig. 6(a), the S-P-T continuum is divided into S-P-T slabs $\\mathcal{T}_1,\\mathcal{T}_2,\\dots ,\\mathcal{T}_T$ . The solution within each time slab is then approximated individually using the TD interpolation.",
+ "bbox": [
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+ "page_idx": 7
+ },
+ {
+ "type": "text",
+ "text": "Between consecutive S-P-T slabs, either a continuous or discontinuous formulation can be employed. As shown in Fig. 6(b) for the continuous Galerkin scheme, the continuity of the solution in time is enforced by imposing the solution at the end of slab $\\mathcal{T}_{i-1}$ as the initial condition of $\\mathcal{T}_i$ :",
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+ "page_idx": 7
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n{ } ^ { [ \\mathcal { T } + 1 ] } u ( \\boldsymbol { x } _ { s } , \\boldsymbol { x } _ { p } , 0 ) = { } ^ { [ \\mathcal { T } ] } u ( \\boldsymbol { x } _ { s } , \\boldsymbol { x } _ { p } , x _ { t } ^ { m a x } ) \\tag {16}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 7
+ },
+ {
+ "type": "page_number",
+ "text": "8",
+ "bbox": [
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+ "page_idx": 7
+ },
+ {
+ "type": "text",
+ "text": "Discontinuous Galerkin method can be used when a discontinuity is allowed between S-P-T slabs, as illustrated in Fig. 6(c). Discontinuity in time can be modeled using the jump operator $\\llbracket \\dots \\rrbracket$ [25].",
+ "bbox": [
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+ ],
+ "page_idx": 8
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\llbracket u \\left(\\boldsymbol {x} _ {s}, \\boldsymbol {x} _ {p}, t\\right) \\rrbracket = \\lim _ {\\epsilon \\rightarrow 0 ^ {+}} \\left(^ {\\mathcal {T} + 1} u \\left(\\boldsymbol {x} _ {s}, \\boldsymbol {x} _ {p}, \\epsilon\\right) - ^ {\\mathcal {T}} u \\left(\\boldsymbol {x} _ {s}, \\boldsymbol {x} _ {p}, x _ {t} ^ {\\max } - \\epsilon\\right)\\right) \\tag {17}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 8
+ },
+ {
+ "type": "image",
+ "img_path": "images/6c2d52bb0f68ce1b2b23d130b8706969e5dd55fc54afb40e93be9e7a63dfc8a8.jpg",
+ "image_caption": [
+ "(a)"
+ ],
+ "image_footnote": [],
+ "bbox": [
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+ ],
+ "page_idx": 8
+ },
+ {
+ "type": "image",
+ "img_path": "images/2211d1927532087afa8c9ab6bf9f0e46ebf0f9fcc68e95f91179e9e07f54486f.jpg",
+ "image_caption": [
+ "(b)",
+ "Figure 6: (a) Multiple S-P-T slabs along the temporal dimension. (b) Continuous Galerkin: the solution is continuous across different S-P-T slabs. (c) Discontinuous Galerkin: jumps are allowed across the slab boundaries"
+ ],
+ "image_footnote": [],
+ "bbox": [
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+ ],
+ "page_idx": 8
+ },
+ {
+ "type": "image",
+ "img_path": "images/f9df96ab1e94c3f197c125cc26bdae791e3769bfcd4aa7fddc2a762fde3a337b.jpg",
+ "image_caption": [
+ "(c)"
+ ],
+ "image_footnote": [],
+ "bbox": [
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+ "page_idx": 8
+ },
+ {
+ "type": "text",
+ "text": "Keeping in mind that this approach can be applied generally to a range of engineering problems, we will demonstrate an example of the Galerkin formulation using a single space-parameter-time partition (S-P-T) slab in the remainder of this section. For illustrative purposes, the transient heat transfer equation will be utilized:",
+ "bbox": [
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+ "page_idx": 8
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\rho c _ {p} \\nabla_ {x _ {t}} u - \\nabla_ {\\boldsymbol {x} _ {s}} \\cdot k \\nabla_ {\\boldsymbol {x} _ {s}} u = f (\\boldsymbol {x} _ {s}, \\boldsymbol {x} _ {p}, x _ {t}) \\tag {18}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 8
+ },
+ {
+ "type": "text",
+ "text": "as we focus on the example of modeling the laser powder bed fusion (LPBF) process in additive manufacturing (AM). In an LPBF simulation, we adopt the following time-dependent moving heat source function:",
+ "bbox": [
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+ ],
+ "page_idx": 8
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\nf \\left(\\boldsymbol {x} _ {s}, \\boldsymbol {x} _ {p}, x _ {t}\\right) = \\frac {2 \\eta P}{\\pi r ^ {2} d _ {\\nu}} \\exp \\left(- \\frac {2 \\left((x - x _ {0} (t)) ^ {2} + (y - y _ {0} (t)) ^ {2}\\right)}{r ^ {2}}\\right) \\cdot \\mathbf {1} _ {\\left(x _ {3} \\geq d _ {\\nu}\\right)} \\tag {19}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 8
+ },
+ {
+ "type": "text",
+ "text": "Summarizing the independent variables in Eq. 18, there are spatial variables $\\mathbf{x}_s = (x_1, x_2, x_3)$ ; parametric variables $\\mathbf{x}_p = (k, \\rho, c_p, \\eta, P, r, d_v)$ ; and a temporal variable $x_t = t$ . Among the parametric variables, $k$ is conductivity; $\\rho$ is the material density; $c_p$ is heat capacity; $\\eta$ is the material absorptivity; $P$ represents laser power; $r$ is the standard deviation that characterizes the width of the heat source; $d_v$ is the penetration depth of the heat source. In Eq. 19, $[x_0(t), y_0(t)]$ represents the center of the moving heat source; $\\mathbf{1}_{(x_3 \\geq d_v)}$ is the indicator function where $\\mathbf{1}_{(x_3 \\geq d_v)} = 1$ if $x_3 \\geq d_v$ or $\\mathbf{1}_{(x_3 \\geq d_v)} = 0$ if $x_3 < d_v$ . Note that the discretization of the material parameters, in particular, in a random field setting, has been previously proposed by Liu et al. [26, 27].",
+ "bbox": [
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+ "page_idx": 8
+ },
+ {
+ "type": "text",
+ "text": "As shown in the schematic below, we classify the boundary surfaces into 2 categories: the Dirichlet boundary surface $\\Gamma_{D}$ and the Neumann boundary surface $\\Gamma_{N}$ .",
+ "bbox": [
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+ "page_idx": 8
+ },
+ {
+ "type": "text",
+ "text": "A uniform ambient temperature is used as the initial condition. The bottom of the powder bed is subject to the Dirichlet boundary condition and the Neumann boundary conditions are prescribed on the other surfaces. The initial and boundary conditions are:",
+ "bbox": [
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+ "page_idx": 8
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\nu (\\boldsymbol {x} _ {s}, \\boldsymbol {x} _ {p}, 0) | _ {\\Omega} = u _ {0},\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 396,
+ 728,
+ 529,
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+ ],
+ "page_idx": 8
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\left. u \\left(\\boldsymbol {x} _ {s}, \\boldsymbol {x} _ {p}, x _ {t}\\right) \\right| _ {\\Gamma_ {D}} = u _ {0}, \\tag {20}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 8
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\boldsymbol {n} \\cdot \\boldsymbol {q} | _ {\\Gamma_ {N}} = q _ {\\text {c o n v}} + q _ {\\text {r a d}} + q _ {\\text {e v a p}}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 8
+ },
+ {
+ "type": "text",
+ "text": "where $u_{0}$ is the ambient temperature, $q_{conv}$ accounts for free convection, $q_{rad}$ accounts for radiation, and $q_{evap}$ imposes evaporative cooling when any material surface reaches the evaporation temperature [28]. Each flux is defined as:",
+ "bbox": [
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+ "page_idx": 8
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\nq _ {c o n v} = h _ {c o n v} [ u (x, t) - u _ {0} ],\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 393,
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+ ],
+ "page_idx": 8
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\nq _ {r a d} = - \\sigma_ {S B} \\epsilon \\left(u ^ {4} \\left(\\boldsymbol {x} _ {s}, x _ {t}\\right) - u _ {0} ^ {4}\\right), \\tag {21}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 8
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\nq _ {e v a p} = - m _ {e v a p} L _ {e v a p}.\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 394,
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+ ],
+ "page_idx": 8
+ },
+ {
+ "type": "page_number",
+ "text": "9",
+ "bbox": [
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+ ],
+ "page_idx": 8
+ },
+ {
+ "type": "image",
+ "img_path": "images/a36972000cd774cc5ab556e0900c90f4c6eea0a3a42c11d6c7322779ba30c687.jpg",
+ "image_caption": [
+ "Figure 7: Transient heat transfer with initial condition and boundary conditions."
+ ],
+ "image_footnote": [],
+ "bbox": [
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+ 261
+ ],
+ "page_idx": 9
+ },
+ {
+ "type": "text",
+ "text": "where $\\sigma_{SB}$ is the Stefan-Boltzmann constant; $\\epsilon$ is the material emissivity; $u_0$ is the ambient temperature; $h_{conv}$ is the convection coefficient of the surrounding gas, $m_{evap}$ is the mass evaporation flux and $L_{evap}$ is the heat of evaporation. In the following numerical examples, we only consider the free convection term in the Neumann boundary condition. The solution to Eq. 18 is approximated using TD interpolation function:",
+ "bbox": [
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+ "page_idx": 9
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\nu ^ {T D} \\left(\\boldsymbol {x} _ {s}, \\boldsymbol {x} _ {p}, x _ {t}\\right) = \\sum_ {m = 1} ^ {M} u _ {\\boldsymbol {x} _ {s}} ^ {(m)} \\left(\\boldsymbol {x} _ {s}\\right) u _ {\\boldsymbol {x} _ {p}} ^ {(m)} \\left(\\boldsymbol {x} _ {p}\\right) u _ {x _ {t}} ^ {(m)} \\left(x _ {t}\\right) \\tag {22}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 9
+ },
+ {
+ "type": "text",
+ "text": "Here a general notation is employed to represent different types of components in Eq. 22. For example, the spatial component $u_{\\boldsymbol{x}_s}^{(m)}(\\boldsymbol{x}_s)$ is equivalent to $u_{x_1}^{(m)}(x_1)u_{x_2}^{(m)}(x_2)u_{x_3}^{(m)}(x_3)$ . The corresponding test function can be obtained using the variational principle:",
+ "bbox": [
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+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\delta u ^ {T D} \\left(\\boldsymbol {x} _ {s}, \\boldsymbol {x} _ {p}, x _ {t}\\right) = \\sum_ {m = 1} ^ {M} \\left[ \\delta u _ {\\boldsymbol {x} _ {s}} ^ {(m)} \\left(\\boldsymbol {x} _ {s}\\right) u _ {\\boldsymbol {x} _ {p}} ^ {(m)} \\left(\\boldsymbol {x} _ {p}\\right) u _ {x _ {t}} ^ {(m)} \\left(x _ {t}\\right) + u _ {\\boldsymbol {x} _ {s}} ^ {(m)} \\left(\\boldsymbol {x} _ {s}\\right) \\delta u _ {\\boldsymbol {x} _ {p}} ^ {(m)} \\left(\\boldsymbol {x} _ {p}\\right) u _ {x _ {t}} ^ {(m)} \\left(x _ {t}\\right) + u _ {\\boldsymbol {x} _ {s}} ^ {(m)} \\left(\\boldsymbol {x} _ {s}\\right) u _ {\\boldsymbol {x} _ {p}} ^ {(m)} \\left(\\boldsymbol {x} _ {p}\\right) \\delta u _ {x _ {t}} ^ {(m)} \\left(x _ {t}\\right) \\right] \\tag {23}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 9
+ },
+ {
+ "type": "text",
+ "text": "Plugging the trial and test functions, the S-P-T Galerkin form of Eq. 18 can be obtained by following Eq. 15:",
+ "bbox": [
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+ 550
+ ],
+ "page_idx": 9
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\int_ {\\Omega} \\delta u ^ {T D} \\left[ \\rho c _ {p} \\nabla_ {x _ {t}} u ^ {T D} - \\nabla_ {x _ {s}} \\cdot k \\nabla_ {x _ {s}} u ^ {T D} - f \\right] d \\Omega = 0 \\tag {24}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 9
+ },
+ {
+ "type": "text",
+ "text": "Using integration by parts on the diffusion term, we get the corresponding general S-P-T Galerkin weak form in TAPS formulation:",
+ "bbox": [
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+ ],
+ "page_idx": 9
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\int_ {\\Omega} \\delta u ^ {T D} \\rho c _ {p} \\nabla_ {x _ {t}} u ^ {T D} d \\Omega + \\int_ {\\Omega} \\nabla_ {\\boldsymbol {x} _ {s}} \\delta u ^ {T D} \\cdot k \\nabla_ {\\boldsymbol {x} _ {s}} u ^ {T D} d \\Omega - \\int_ {\\partial \\boldsymbol {x} _ {s}, \\boldsymbol {x} _ {p}, t} \\delta u ^ {T D} \\boldsymbol {n} \\cdot \\boldsymbol {q} | _ {\\Gamma_ {N}} d s d \\Omega_ {\\boldsymbol {x} _ {p}} d \\Omega_ {t} - \\int_ {\\Omega} \\delta u ^ {T D} f (\\boldsymbol {x} _ {s}, \\boldsymbol {x} _ {p}, x _ {t}) d \\Omega = 0 \\tag {25}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 9
+ },
+ {
+ "type": "text",
+ "text": "where $q$ is the heat flux on the Neumann boundary.",
+ "bbox": [
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+ "page_idx": 9
+ },
+ {
+ "type": "text",
+ "text": "2.5. Discretized matrix form",
+ "text_level": 1,
+ "bbox": [
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+ "page_idx": 9
+ },
+ {
+ "type": "text",
+ "text": "The S-P-T Galerkin weak form shown in Eq. 25 is nonlinear in nature due to the tensor product structure in TD interpolation, necessitating efficient solution schemes. To illustrate the detailed solution approach for the general S-P-T weak form, we simplify the governing equation Eq. 18 by considering a one-dimensional spatial problem where $x_{s} = x$ . We assume that the product of density and specific heat capacity $\\rho c_{p}$ is equal to 1. Additionally, the forcing term is solely dependent on $x$ . Therefore, the simplified governing equation for this example is given by:",
+ "bbox": [
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+ ],
+ "page_idx": 9
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\frac {\\partial u}{\\partial t} - \\frac {\\partial u}{\\partial x} \\cdot k \\frac {\\partial x}{\\partial x} = f (x) \\tag {26}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 9
+ },
+ {
+ "type": "text",
+ "text": "subject to homogeneous boundary conditions and initial conditions. This equation has 3 independent variables ( $D = 3$ ), i.e., spatial variable $x_{s} = x_{1} = x$ , parametric variable $x_{p} = x_{2} = k$ and temporal variable $x_{t} = x_{3} = t$ . The S-P-T",
+ "bbox": [
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+ ],
+ "page_idx": 9
+ },
+ {
+ "type": "page_number",
+ "text": "10",
+ "bbox": [
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+ "page_idx": 9
+ },
+ {
+ "type": "text",
+ "text": "Galerkin weak form of this problem can be written as follows according to Eq. 25 (the superscripts \"TD\" for both trial and test functions are omitted for brevity).",
+ "bbox": [
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+ ],
+ "page_idx": 10
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\int_ {\\Omega} \\delta u \\nabla_ {t} u d \\Omega + \\int_ {\\Omega} \\nabla_ {x} \\delta u \\cdot k \\nabla_ {x} u d \\Omega - \\int_ {\\Omega} \\delta u f d \\Omega = 0 \\tag {27}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 10
+ },
+ {
+ "type": "text",
+ "text": "The corresponding trial and test functions can be obtained using Eqs. 22-23:",
+ "bbox": [
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+ ],
+ "page_idx": 10
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\nu (x, k, t) = \\sum_ {m = 1} ^ {M} u _ {x} ^ {(m)} (x) u _ {k} ^ {(m)} (k) u _ {t} ^ {(m)} (t) \\tag {28}\n$$\n",
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+ "bbox": [
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+ ],
+ "page_idx": 10
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\delta u (x, k, t) = \\underbrace {\\sum_ {m = 1} ^ {M} \\delta u _ {x} ^ {(m)} (x) u _ {k} ^ {(m)} (k) u _ {t} ^ {(m)} (t)} _ {\\text {s p a t i a l v a r i a t i o n}} + \\underbrace {\\sum_ {m = 1} ^ {M} u _ {x} ^ {(m)} (x) \\delta u _ {k} ^ {(m)} (k) u _ {t} ^ {(m)} (t)} _ {\\text {p a r a m e t r i c v a r i a t i o n}} + \\underbrace {\\sum_ {m = 1} ^ {M} u _ {x} ^ {(m)} (x) u _ {k} ^ {(m)} (k) \\delta u _ {t} ^ {(m)} (t)} _ {\\text {t e m p o r a l v a r i a t i o n}} \\tag {29}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 10
+ },
+ {
+ "type": "text",
+ "text": "As shown in Eq. 29, the test function is further split into $D$ variational terms for a general $D$ dimensional problem (in the current example, $D = 3$ ). As an example, we first plug Eq. 28 and the spatial variation term of Eq. 29 into the Galerkin weak form in Eq. 27 to obtain the S-P-T weak form terms corresponding to spatial variation:",
+ "bbox": [
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+ "page_idx": 10
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\underbrace {\\int_ {\\Omega} \\sum_ {m = 1} ^ {M} \\sum_ {n = 1} ^ {M} \\left[ \\nabla \\delta u _ {x} ^ {(m)} (x) \\nabla u _ {x} ^ {(n)} (x) d x \\right] \\cdot \\left[ u _ {k} ^ {(m)} (k) k u _ {k} ^ {(n)} (k) d k \\right] \\cdot \\left[ u _ {t} ^ {(m)} (t) u _ {t} ^ {(n)} (t) d t \\right]} _ {\\text {d i f f u s i o n t e r m}} +\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 10
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\underbrace {\\int_ {\\Omega} \\sum_ {m = 1} ^ {M} \\sum_ {n = 1} ^ {M} \\left[ \\delta u _ {x} ^ {(m)} (x) u _ {x} ^ {(n)} (x) d x \\right] \\cdot \\left[ u _ {k} ^ {(m)} (k) u _ {k} ^ {(n)} (k) d k \\right] \\cdot \\left[ u _ {t} ^ {(m)} (t) \\nabla_ {t} u _ {t} ^ {(n)} (t) d t \\right]} _ {\\text {t i m e d e r i v a t i v e t e r m}} - \\tag {30}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 10
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\underbrace {\\int_ {\\Omega} \\sum_ {m = 1} ^ {M} \\left[ \\delta u _ {x} ^ {(m)} (x) f (x) d x \\right] \\cdot \\left[ u _ {k} ^ {(m)} (k) d k \\right] \\cdot \\left[ u _ {t} ^ {(m)} (t) d t \\right]} _ {\\text {f o r c i n g t e r m}}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 10
+ },
+ {
+ "type": "text",
+ "text": "We use 1D C-HiDeNN shape functions to approximate each univariate function:",
+ "bbox": [
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+ ],
+ "page_idx": 10
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\nu _ {d} ^ {(n)} (x _ {d}) = \\widetilde {N} _ {n _ {d} ^ {\\prime}} ^ {[ d ]} (x _ {d}) u _ {n _ {d} ^ {\\prime} n} ^ {[ d ]} \\quad (\\text {n o s u m o n} d)\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 359,
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+ ],
+ "page_idx": 10
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\delta u _ {d} ^ {(m)} \\left(x _ {d}\\right) = \\widetilde {N} _ {n _ {d}} ^ {[ d ]} \\left(x _ {d}\\right) \\delta u _ {n _ {d} m} ^ {[ d ]} \\quad (\\text {n o s u m o n} d) \\tag {31}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 10
+ },
+ {
+ "type": "text",
+ "text": "where Einstein summation is used. The free index $d$ refers to dimension and $d = x,k$ or $t$ . The gradient of the interpolated variable can be computed using the shape function derivative $\\widetilde{B}_{n_d}^{[d]}(x_d) = \\frac{d\\widetilde{N}_{n_d}^{[d]}(x_d)}{dx_d}$ .",
+ "bbox": [
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+ ],
+ "page_idx": 10
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\nabla_ {x _ {d}} u _ {d} ^ {(n)} (x _ {d}) = \\widetilde {B} _ {n _ {d} ^ {\\prime}} ^ {[ d ]} (x _ {d}) u _ {n _ {d} ^ {\\prime} n} ^ {[ d ]} \\quad (\\text {n o s u m o n} d)\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 10
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\nabla_ {x _ {d}} \\delta u _ {d} ^ {(m)} (x _ {d}) = \\widetilde {B} _ {n _ {d}} ^ {[ d ]} (x _ {d}) \\delta u _ {n _ {d} m} ^ {[ d ]} \\quad (\\text {n o s u m o n} d) \\tag {32}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 10
+ },
+ {
+ "type": "text",
+ "text": "Plugging Eq. 31 - 32 into Eq. 30, the diffusion term can be rewritten as:",
+ "bbox": [
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+ 589,
+ 774
+ ],
+ "page_idx": 10
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\sum_ {m = 1} ^ {M} \\sum_ {n = 1} ^ {M} \\underbrace {\\int_ {\\Omega_ {x}} \\widetilde {B} _ {n _ {x}} (x) \\delta u _ {n _ {x} m} ^ {[ x ]} \\widetilde {B} _ {n _ {x} ^ {\\prime}} (x) u _ {n _ {x} ^ {\\prime} n} ^ {[ x ]} d x} _ {\\text {s p a t i a l t e r m}} \\cdot \\underbrace {\\int_ {\\Omega_ {k}} \\widetilde {N} _ {n _ {k}} (k) u _ {n _ {k} m} ^ {[ k ]} k \\widetilde {N} _ {n _ {k} ^ {\\prime}} (k) u _ {n _ {k} ^ {\\prime} n} ^ {[ k ]} d k} _ {\\text {p a r a m e t r i c t e r m}} \\cdot \\underbrace {\\int_ {\\Omega_ {t}} \\widetilde {N} _ {n _ {t}} (t) u _ {n _ {t} m} ^ {[ t ]} \\widetilde {N} _ {n _ {t} ^ {\\prime}} (t) u _ {n _ {t} ^ {\\prime} n} ^ {[ t ]} d t} _ {\\text {t e m p o r a l t e r m}} \\tag {33}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 10
+ },
+ {
+ "type": "text",
+ "text": "As can be readily seen from Eq. 33, after doing 1D integration of each term, the parametric and temporal terms can be treated as coefficient matrices:",
+ "bbox": [
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+ ],
+ "page_idx": 10
+ },
+ {
+ "type": "page_number",
+ "text": "11",
+ "bbox": [
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+ ],
+ "page_idx": 10
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\nC _ {m n} ^ {[ k ]} = \\underbrace {\\int_ {\\Omega_ {k}} \\widetilde {N} _ {n _ {k}} (k) u _ {n _ {k} m} ^ {[ k ]} k \\widetilde {N} _ {n _ {k} ^ {\\prime}} (k) u _ {n _ {k} ^ {\\prime} n} ^ {[ k ]} d k} _ {\\text {p a r a m e t r i c t e r m}}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ 601,
+ 200
+ ],
+ "page_idx": 11
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\nC _ {m n} ^ {[ t ]} = \\underbrace {\\int_ {\\Omega_ {t}} \\widetilde {N} _ {n _ {t}} (t) u _ {n _ {t} m} ^ {[ t ]} \\widetilde {N} _ {n _ {t} ^ {\\prime}} (t) u _ {n _ {t} ^ {\\prime} n} ^ {[ t ]} d t} _ {\\text {t e m p o r a l t e r m}} \\tag {34}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 383,
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+ ],
+ "page_idx": 11
+ },
+ {
+ "type": "text",
+ "text": "as the only free indices are $m$ and $n$ . Substituting the coefficient matrices and rearranging different terms in Eq. 33, we have:",
+ "bbox": [
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+ 288
+ ],
+ "page_idx": 11
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\sum_ {m = 1} ^ {M} \\delta u _ {n _ {x} m} ^ {[ x ]} \\sum_ {n = 1} ^ {M} \\left[ \\int_ {\\Omega_ {x}} \\widetilde {B} _ {n _ {x}} (x) \\widetilde {B} _ {n _ {x} ^ {\\prime}} (x) d x \\right] \\cdot C _ {m n} ^ {[ k ]} C _ {m n} ^ {[ t ]} \\cdot u _ {n _ {x} ^ {\\prime} n} ^ {[ x ]} \\tag {35}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ 892,
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+ ],
+ "page_idx": 11
+ },
+ {
+ "type": "text",
+ "text": "Like standard FEM, we can define $\\int_{x}\\widetilde{B}_{n_x}(x)\\widetilde{B}_{n_x'}(x)dx$ as the 1D stiffness matrix $K_{n_x n_x'}^{[x]}$ of $x$ dimension in Eq. 35. We let $C_{mn}^{[x]} = C_{mn}^{[k]}C_{mn}^{[t]}$ with no summation on $(m,n)$ . Furthermore, let us define the following 4-th order tensor:",
+ "bbox": [
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+ ],
+ "page_idx": 11
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\nA _ {n _ {x} n _ {x} ^ {\\prime} m n} ^ {[ x ]} = K _ {n _ {x} n _ {x} ^ {\\prime}} ^ {[ x ] ^ {\\prime}} C _ {m n} ^ {[ x ]} \\tag {36}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 430,
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+ 409
+ ],
+ "page_idx": 11
+ },
+ {
+ "type": "text",
+ "text": "where $A_{n_s n_s'mn}^{[x]}$ is a function of solution vectors $u_{n_k m}^{[k]}$ and $u_{n_m m}^{[t]}$ since the coefficient matrix $C_{mn}^{[x]}$ depends on these solution vectors as shown in Eq. 34. This dependency reflects the interconnected nature of the variables across different dimensions in the S-P-T framework, highlighting how the spatial, parameter, and temporal components influence each other through the coefficients. As a result, Eq. 33 can be further simplified as follows:",
+ "bbox": [
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+ ],
+ "page_idx": 11
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\delta u _ {n _ {x} m} ^ {[ x ]} A _ {n _ {x} n _ {x} ^ {\\prime} m n} ^ {[ x ]} u _ {n _ {x} ^ {\\prime} n} ^ {[ x ]} \\tag {37}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 438,
+ 481,
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+ 502
+ ],
+ "page_idx": 11
+ },
+ {
+ "type": "text",
+ "text": "where the summation signs are neglected since $m$ and $n$ become dummy variables. The 4-th order tensor $A_{n_x n_x' mn}^{[x]}$ can be reshaped as a 2nd order tensor $\\mathbb{A}_{IJ}^{[x]}$ : the indices $n_x$ and $m$ are combined into a single composite index $I$ , and the indices $n_x'$ and $n$ are combined into a single composite index $J$ .",
+ "bbox": [
+ 102,
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+ ],
+ "page_idx": 11
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\nA _ {n _ {x} n _ {x} ^ {\\prime} m n} ^ {[ x ]} = \\mathbb {A} _ {I J} ^ {[ x ]} \\tag {38}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ 590
+ ],
+ "page_idx": 11
+ },
+ {
+ "type": "text",
+ "text": "Define the following vectorization:",
+ "bbox": [
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+ 608
+ ],
+ "page_idx": 11
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\delta \\mathbb {U} _ {I} ^ {[ x ]} = \\left[ \\operatorname {v e c} \\left(\\delta u _ {n _ {x} m} ^ {[ x ]}\\right) \\right] _ {I}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 403,
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+ 652
+ ],
+ "page_idx": 11
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\mathbb {U} _ {J} ^ {[ x ]} = \\left[ \\operatorname {v e c} \\left(u _ {n _ {x} ^ {\\prime} n} ^ {[ x ]}\\right) \\right] _ {J} \\tag {39}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 425,
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+ ],
+ "page_idx": 11
+ },
+ {
+ "type": "text",
+ "text": "As a result, Eq. 37 is equivalent to:",
+ "bbox": [
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+ ],
+ "page_idx": 11
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\delta \\mathbb {U} ^ {[ x ] T} \\mathbb {A} ^ {[ x ]} \\mathbb {U} ^ {[ x ]} \\tag {40}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 448,
+ 709,
+ 890,
+ 726
+ ],
+ "page_idx": 11
+ },
+ {
+ "type": "text",
+ "text": "Following the same procedure, we can obtain matrix forms corresponding to the time derivative term $\\delta \\mathbb{U}^{[x]^T}\\mathbb{B}^{[x]}\\mathbb{U}^{[x]}$ , and the forcing term $\\delta \\mathbb{U}^{[x]^T}\\mathbb{Q}^{[x]}$ for the spatial variational part of Eq. 30. Similar structures can also be obtained for the parametric and temporal variational parts of the test function in Eq. 29. Denoting $\\mathbb{K}^{[d]} = \\mathbb{A}^{[d]} + \\mathbb{B}^{[d]}$ , the matrix form of the generalized S-P-T Galerkin form in Eq. 27 can be written as:",
+ "bbox": [
+ 100,
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+ ],
+ "page_idx": 11
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\underbrace {\\delta \\mathbb {U} ^ {[ x ] ^ {T}} \\mathbb {K} ^ {[ x ]} \\mathbb {U} ^ {[ x ]} - \\delta \\mathbb {U} ^ {[ x ] ^ {T}} \\mathbb {Q} ^ {[ x ]}} _ {\\text {s p a t i a l v a r i a t i o n a l p a r t}} + \\underbrace {\\delta \\mathbb {U} ^ {[ k ] ^ {T}} \\mathbb {K} ^ {[ k ]} \\mathbb {U} ^ {[ k ]} - \\delta \\mathbb {U} ^ {[ k ] ^ {T}} \\mathbb {Q} ^ {[ k ]}} _ {\\text {p a r a m e t r i c v a r i a t i o n a l p a r t}} + \\underbrace {\\delta \\mathbb {U} ^ {[ t ] ^ {T}} \\mathbb {K} ^ {[ t ]} \\mathbb {U} ^ {[ t ]} - \\delta \\mathbb {U} ^ {[ t ] ^ {T}} \\mathbb {Q} ^ {[ t ]}} _ {\\text {t e m p o r a l v a r i a t i o n a l p a r t}} = 0 \\tag {41}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 198,
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+ 834
+ ],
+ "page_idx": 11
+ },
+ {
+ "type": "text",
+ "text": "Eq. 41 is equivalent to the following nonlinear system of equations. Note that the nonlinearity comes from the fact that $\\mathbb{K}^{[d]}$ is solution dependent:",
+ "bbox": [
+ 100,
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+ 892,
+ 870
+ ],
+ "page_idx": 11
+ },
+ {
+ "type": "page_number",
+ "text": "12",
+ "bbox": [
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+ ],
+ "page_idx": 11
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\left[ \\delta \\mathbb {U} ^ {[ x ] ^ {T}}, \\delta \\mathbb {U} ^ {[ k ] ^ {T}}, \\delta \\mathbb {U} ^ {[ t ] ^ {T}} \\right] \\left\\{\\left[ \\begin{array}{c c c} \\mathbb {K} ^ {[ x ]} (\\mathbb {U} ^ {[ k ]}, \\mathbb {U} ^ {[ t ]}) & 0 & 0 \\\\ 0 & \\mathbb {K} ^ {[ k ]} (\\mathbb {U} ^ {[ x ]}, \\mathbb {U} ^ {[ t ]}) & 0 \\\\ 0 & 0 & \\mathbb {K} ^ {[ t ]} (\\mathbb {U} ^ {[ x ]}, \\mathbb {U} ^ {[ k ]}) \\end{array} \\right] \\left[ \\begin{array}{l} \\mathbb {U} ^ {[ x ]} \\\\ \\mathbb {U} ^ {[ k ]} \\\\ \\mathbb {U} ^ {[ t ]} \\end{array} \\right] - \\left[ \\begin{array}{l} \\mathbb {Q} ^ {[ x ]} (\\mathbb {U} ^ {[ k ]}, \\mathbb {U} ^ {[ t ]}) \\\\ \\mathbb {Q} ^ {[ k ]} (\\mathbb {U} ^ {[ x ]}, \\mathbb {U} ^ {[ t ]}) \\\\ \\mathbb {Q} ^ {[ t ]} (\\mathbb {U} ^ {[ x ]}, \\mathbb {U} ^ {[ k ]}) \\end{array} \\right] \\right\\} = 0 \\tag {42}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 131,
+ 153,
+ 892,
+ 200
+ ],
+ "page_idx": 12
+ },
+ {
+ "type": "text",
+ "text": "where we can treat the solution vector $\\left[\\mathbb{U}^{[x]^T},\\mathbb{U}^{[k]^T},\\mathbb{U}^{[t]^T}\\right]$ as generalized DoFs like standard FEM. There are many ways to solve Eq. 42. For example, standard linearization schemes such as Newton's method have been used [29]. However, this method may suffer from ill-conditioning since the mismatch of scales for different dimensions can be significant. In this paper, we use the concept of subspace iteration to efficiently approximate the solution by iterating in the subspace of the test function space until a convergence criteria is met [19]. A similar counterpart has been widely adopted as the gold standard in discrete tensor decomposition [23].",
+ "bbox": [
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+ ],
+ "page_idx": 12
+ },
+ {
+ "type": "text",
+ "text": "2.6. Solution scheme of TAPS: subspace iteration",
+ "text_level": 1,
+ "bbox": [
+ 102,
+ 312,
+ 442,
+ 326
+ ],
+ "page_idx": 12
+ },
+ {
+ "type": "text",
+ "text": "For subspace iteration in $d$ -th dimension, only the solution matrix $\\mathbb{U}^{[d]}$ is treated as unknown while all other functions are considered as known constants. Consequently, the variations of the univariate functions other than $d$ -th dimension will vanish. From Eq. 42, it can be seen that this will lead to a linear system of equations for the unknowns in the $d$ -th dimension. The updated solution matrix $\\mathbb{U}^{[d]}$ from this process is then used in the next subspace iteration for dimension $d + 1$ (when $d = D$ , we come back to the first dimension $d = 1$ ). The complete solution scheme for subspace iteration is shown in Algorithm 1.",
+ "bbox": [
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+ "page_idx": 12
+ },
+ {
+ "type": "text",
+ "text": "Algorithm 1 TAPS solution scheme (subspace iteration)",
+ "bbox": [
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+ 442
+ ],
+ "page_idx": 12
+ },
+ {
+ "type": "text",
+ "text": "1: Initialize solution vector $\\mathbb{U}^{[x_1][0]}$ ..., $\\mathbb{U}^{[x_D][0]}$ with random values and compute $\\mathbb{K}^{[x_1][0]}$ , and $\\mathbb{Q}^{[x_1][0]}$",
+ "bbox": [
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+ "page_idx": 12
+ },
+ {
+ "type": "list",
+ "sub_type": "text",
+ "list_items": [
+ "2: for iter = 0 to iter_max do",
+ "3: for $d = 1$ to D do",
+ "4: Update iteration number $\\mathcal{K} = iter\\times D + d$",
+ "5: Solve TD linear system $\\mathbb{K}^{[x_d][\\mathcal{K} - 1]}\\mathbb{U}^{[x_d][\\mathcal{K}]} = \\mathbb{Q}^{[x_d][\\mathcal{K} - 1]}$",
+ "6: Update matrices $\\mathbb{K}^{[x_{d + 1}][\\mathcal{K}]}$ and force vector $\\mathbb{Q}^{[x_{d + 1}][\\mathcal{K}]}$",
+ "7: end for",
+ "8: Check convergence",
+ "9: end for"
+ ],
+ "bbox": [
+ 112,
+ 444,
+ 546,
+ 571
+ ],
+ "page_idx": 12
+ },
+ {
+ "type": "text",
+ "text": "To illustrate the details of the subspace iteration algorithm, we consider the $\\mathcal{K}$ -th subspace iteration (which is on spatial variable $x$ ). Here, we assume that the parametric and temporal solutions have been updated from the previous $(\\mathcal{K} - 1)$ -th iteration, leaving the spatial solution as unknown to be solved in $\\mathcal{K}$ -th iteration. Moreover, instead of the full variation form of the test function as in Eq. 42, we only consider the subspace $x$ of the test function by setting the parametric and temporal variational parts as 0. As a result, we have:",
+ "bbox": [
+ 100,
+ 590,
+ 892,
+ 662
+ ],
+ "page_idx": 12
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\mathbb {K} ^ {[ x ] [ \\mathcal {K} - 1 ]} \\left(\\mathbb {U} ^ {[ k ] [ \\mathcal {K} - 1 ]}, \\mathbb {U} ^ {[ t ] [ \\mathcal {K} - 1 ]}\\right) \\mathbb {U} ^ {[ x ] [ \\mathcal {K} ]} = \\mathbb {Q} ^ {[ x ] [ \\mathcal {K} - 1 ]} \\left(\\mathbb {U} ^ {[ k ] [ \\mathcal {K} - 1 ]}, \\mathbb {U} ^ {[ t ] [ \\mathcal {K} - 1 ]}\\right) \\tag {43}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 265,
+ 673,
+ 890,
+ 694
+ ],
+ "page_idx": 12
+ },
+ {
+ "type": "text",
+ "text": "which is a linear system of equations with unknown $\\mathbb{U}^{[x][\\mathcal{K}]}$ . This is a general Sylvester equation which can be solved using many efficient solution schemes [30, 31]. In this paper, sparse direct solvers based on fast diagonalization/complex Schur decomposition methods are adopted [32]. The computational complexity of the sparse direct solver is $O(M^3 + M^2 n_d + C_c(n_d))$ for the $d$ -th dimension subspace iteration, where $M$ is the total number of modes; $n_d$ is the number of grid points for $d$ -th dimension; $C_c(n_d)$ refers to the computational cost of the banded sparse mass/stiffness matrix for $d$ -th dimension with a shape of $(n_d \\times n_d)$ .",
+ "bbox": [
+ 100,
+ 700,
+ 892,
+ 785
+ ],
+ "page_idx": 12
+ },
+ {
+ "type": "text",
+ "text": "Once $\\mathbb{U}^{[x][\\mathcal{K}]}$ is obtained, we then update matrix $\\mathbb{K}^{[k][\\mathcal{K}]}(\\mathbb{U}^{[x][\\mathcal{K}]},\\mathbb{U}^{[t][\\mathcal{K}]})$ and forcing vector $\\mathbb{Q}^{[k][\\mathcal{K}]}(\\mathbb{U}^{[x][\\mathcal{K}]},\\mathbb{U}^{[t][\\mathcal{K}]})$ . In the next iteration (for dimension $k$ ), we treat $\\mathbb{U}^{[k][\\mathcal{K} + 1]}$ as the only unknown. Subspace iteration will continue unless the relative change of all solution matrices (for example, $L_{2}$ norm) is within the tolerance.",
+ "bbox": [
+ 100,
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+ ],
+ "page_idx": 12
+ },
+ {
+ "type": "page_number",
+ "text": "13",
+ "bbox": [
+ 489,
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+ ],
+ "page_idx": 12
+ },
+ {
+ "type": "text",
+ "text": "2.7. Error estimates of TAPS",
+ "text_level": 1,
+ "bbox": [
+ 102,
+ 131,
+ 307,
+ 145
+ ],
+ "page_idx": 13
+ },
+ {
+ "type": "text",
+ "text": "Since the TAPS solution is based on the C-HiDeNN-TD approximation and the generalized Galerkin formulation, we can have the following theoretical results on the error bounds, as demonstrated in our previous work on C-HiDeNN [6]:",
+ "bbox": [
+ 102,
+ 149,
+ 892,
+ 191
+ ],
+ "page_idx": 13
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\left\\| u ^ {\\mathrm {C} - \\mathrm {H i D e N N}} - u ^ {\\mathrm {e x}} \\right\\| _ {E} \\leq \\left\\| u ^ {\\mathrm {T A P S}} - u ^ {\\mathrm {e x}} \\right\\| _ {E} \\leq \\left\\| u ^ {\\mathrm {F E M}} - u ^ {\\mathrm {e x}} \\right\\| _ {E} \\tag {44}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 321,
+ 191,
+ 890,
+ 206
+ ],
+ "page_idx": 13
+ },
+ {
+ "type": "text",
+ "text": "where $\\| \\cdot \\|_E$ denotes the energy norm, $u^{\\mathrm{ex}}$ denotes the exact solution, $u^{\\mathrm{C - HiDeNN}}$ denotes the solution obtained by the full C-HiDeNN method without tensor decomposition, $u^{\\mathrm{TAPS}}$ denotes the TAPS solution with a sufficient number of modes, $u^{\\mathrm{FEM}}$ denotes the FEM solution. The proof of the above results is based on the fact that the full C-HiDeNN approximation can provide a larger function space and therefore more accurate solutions than conventional FEM [6]. The subspace iteration can be considered as a local (directional) version of the Galerkin formulation and is expected to enable an optimized solution for the tensor decomposition that will converge to the Galerkin-based full C-HiDeNN method.",
+ "bbox": [
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+ ],
+ "page_idx": 13
+ },
+ {
+ "type": "text",
+ "text": "3. Results",
+ "text_level": 1,
+ "bbox": [
+ 102,
+ 335,
+ 181,
+ 348
+ ],
+ "page_idx": 13
+ },
+ {
+ "type": "text",
+ "text": "3.1. Convergence study for moving heat source",
+ "text_level": 1,
+ "bbox": [
+ 102,
+ 360,
+ 426,
+ 375
+ ],
+ "page_idx": 13
+ },
+ {
+ "type": "text",
+ "text": "In this section, we first analyze the convergence of the TAPS solver for a space-time (S-T) transient heat transfer problem. A single NVIDIA RTX A6000 GPU is used for all the following analyses. In Eq. 18, we let $\\rho c_{p} = 1$ , $k = 1$ , and replace the heat source as shown in Eq. 45. In this example, we have the spatial variable $x_{s} = (x,y,z)$ and the temporal variable $x_{t} = t$ .",
+ "bbox": [
+ 102,
+ 378,
+ 892,
+ 434
+ ],
+ "page_idx": 13
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\begin{array}{l} f \\left(\\boldsymbol {x} _ {s}, x _ {t}\\right) = 2 (1 - 2 y ^ {2}) \\left(1 - e ^ {- 1 5 t}\\right) e ^ {- y ^ {2} - (1 0 0 t - x - 5) ^ {2}} \\\\ + 2 (1 - 2 (1 0 0 t - x - 5) ^ {2}) \\left(1 - e ^ {- 1 5 t}\\right) e ^ {- y ^ {2} - (1 0 0 t - x - 5) ^ {2}} + (1 \\tag {45} \\\\ - e ^ {- 1 5 t}) (2 0 0 x + 1 0 0 0 - 2 0 0 0 0 t) e ^ {- y ^ {2} - (1 0 0 t - x - 5) ^ {2}} \\\\ - 1 5 e ^ {- 1 5 t} e ^ {- y ^ {2} - (1 0 0 t - x - 5) ^ {2}} \\\\ \\end{array}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 265,
+ 445,
+ 890,
+ 521
+ ],
+ "page_idx": 13
+ },
+ {
+ "type": "text",
+ "text": "The analytical solution to the PDE is inherently non-separable.",
+ "bbox": [
+ 102,
+ 531,
+ 529,
+ 545
+ ],
+ "page_idx": 13
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\nu ^ {\\mathrm {e x}} \\left(\\boldsymbol {x} _ {s}, x _ {t}\\right) = (1 - e ^ {- 1 5 t}) e ^ {- y ^ {2} - (x - 1 0 0 t - 5) ^ {2}} \\tag {46}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 373,
+ 556,
+ 890,
+ 573
+ ],
+ "page_idx": 13
+ },
+ {
+ "type": "text",
+ "text": "The initial and boundary conditions are:",
+ "bbox": [
+ 102,
+ 585,
+ 376,
+ 598
+ ],
+ "page_idx": 13
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\nu \\left(x _ {s}, 0\\right) = 0,\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 406,
+ 612,
+ 490,
+ 624
+ ],
+ "page_idx": 13
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\left. u \\left(\\boldsymbol {x} _ {s}, x _ {t}\\right) \\right| _ {\\partial \\Omega} = \\left. u ^ {\\mathrm {e x}} \\left(\\boldsymbol {x} _ {s}, x _ {t}\\right) \\right| _ {\\partial \\Omega}. \\tag {47}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 406,
+ 623,
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+ 646
+ ],
+ "page_idx": 13
+ },
+ {
+ "type": "text",
+ "text": "The relative $L_{2}$ norm error is defined as:",
+ "bbox": [
+ 102,
+ 651,
+ 376,
+ 664
+ ],
+ "page_idx": 13
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\epsilon_ {L _ {2}} = \\frac {\\| u ^ {T D} \\left(\\boldsymbol {x} _ {s} , x _ {t}\\right) - u ^ {\\mathrm {e x}} \\left(\\boldsymbol {x} _ {s} , x _ {t}\\right) \\| _ {L _ {2} \\left(\\Omega_ {\\boldsymbol {x} _ {s}} \\otimes \\Omega_ {x _ {t}}\\right)}}{\\| u ^ {\\mathrm {e x}} \\left(\\boldsymbol {x} _ {s} , x _ {t}\\right) \\| _ {L _ {2} \\left(\\Omega_ {\\boldsymbol {x} _ {s}} \\otimes \\Omega_ {x _ {t}}\\right)}} \\tag {48}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 359,
+ 675,
+ 890,
+ 709
+ ],
+ "page_idx": 13
+ },
+ {
+ "type": "text",
+ "text": "First, we investigate the influence of the number of subspace iterations. As shown in Fig. 8(a), 3 iterations are enough to obtain an accurate result. Next, we investigate the convergence in terms of the number of modes. Here we compare the relative $L_{2}$ norm error for both TAPS and proper generalized decomposition (PGD) methods [17, 18]. To this aim, we use the same discretization for the space-time domain with each dimension discretized by 100 grid points, the same reproducing polynomial order $p = 1$ and convolution patch size $s = 1$ . As can be seen from Fig. 8(b), TAPS requires a much smaller number of modes than PGD. For TAPS, when the number of modes equals 25, the relative $L_{2}$ norm error decreases to $2.5 \\times 10^{-3}$ . The total solution time is 15.2 s. However, PGD requires 1,000 modes which takes 60.6 s solution time to reach the same level of accuracy. This is because the test function space in PGD is a subspace of TAPS [29]. Furthermore, the modal decomposition obtained from PGD is not optimal and thus requires a larger storage requirement due to the increased number of modes.",
+ "bbox": [
+ 102,
+ 718,
+ 892,
+ 860
+ ],
+ "page_idx": 13
+ },
+ {
+ "type": "page_number",
+ "text": "14",
+ "bbox": [
+ 489,
+ 879,
+ 507,
+ 889
+ ],
+ "page_idx": 13
+ },
+ {
+ "type": "image",
+ "img_path": "images/71b70cc86e81f29f17717ebc5e990a233cea4fd792c202ed7d02aa919c3be394.jpg",
+ "image_caption": [
+ "(a)"
+ ],
+ "image_footnote": [],
+ "bbox": [
+ 183,
+ 133,
+ 505,
+ 281
+ ],
+ "page_idx": 14
+ },
+ {
+ "type": "image",
+ "img_path": "images/fc1a326bac6804be45fa1c9b3cf27c427a62e03aa83dfcf816fe3cdb17e70be1.jpg",
+ "image_caption": [
+ "(b)",
+ "Figure 8: Relative L2 norm error with respect to (a) the number of subspace iterations (b) the number of modes"
+ ],
+ "image_footnote": [],
+ "bbox": [
+ 512,
+ 131,
+ 810,
+ 284
+ ],
+ "page_idx": 14
+ },
+ {
+ "type": "text",
+ "text": "The spatial and temporal convergence are also studied. In Fig. 9(a), the number of temporal nodes is fixed as 500, and the spatial mesh is refined. It shows the relative $L_{2}$ norm error with respect to the number of nodes along each spatial dimension. As can be readily seen from the figure, larger patch size $s$ leads to smaller error given the same reproducing polynomial orders $p$ . Moreover, we can adjust $p$ to control the spatial convergence rate. Similarly, Fig. 9(b) demonstrates the convergence rate in the temporal domain where we fix the spatial discretization as 500 along each spatial dimension. By adjusting $s$ and $p$ , we can obtain different temporal convergence rates.",
+ "bbox": [
+ 100,
+ 347,
+ 892,
+ 432
+ ],
+ "page_idx": 14
+ },
+ {
+ "type": "text",
+ "text": "Finally, we refine the spatial and temporal mesh simultaneously and study the spatio-temporal convergence rate in Fig. 9(c). As can be observed from the figure, higher reproducing polynomial order $p$ will lead to a higher-order convergence rate.",
+ "bbox": [
+ 100,
+ 432,
+ 892,
+ 475
+ ],
+ "page_idx": 14
+ },
+ {
+ "type": "image",
+ "img_path": "images/2345a49f326f69ec67725fc081645d1b37d8869821b2514f92f47f0a0b9cc7e8.jpg",
+ "image_caption": [
+ "(a)"
+ ],
+ "image_footnote": [],
+ "bbox": [
+ 110,
+ 489,
+ 381,
+ 645
+ ],
+ "page_idx": 14
+ },
+ {
+ "type": "image",
+ "img_path": "images/aa5c83a28c6329ef2d94f536ced88460656363be508afd476967c069343f0768.jpg",
+ "image_caption": [
+ "(b)"
+ ],
+ "image_footnote": [],
+ "bbox": [
+ 386,
+ 489,
+ 620,
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+ ],
+ "page_idx": 14
+ },
+ {
+ "type": "image",
+ "img_path": "images/d1d0eb95867b86e555e3a95df3d5b5a14cd34d608385d9234dcdbecea9446986.jpg",
+ "image_caption": [
+ "(c)",
+ "Figure 9: Relative $L_{2}$ norm error with respect to the number of grid points (a) spatial convergence (b) temporal convergence (c) spatio-temporal convergence"
+ ],
+ "image_footnote": [],
+ "bbox": [
+ 623,
+ 489,
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+ ],
+ "page_idx": 14
+ },
+ {
+ "type": "text",
+ "text": "3.2. Convergence study of $S-P-T$ problems up to equivalent zetta-scale (10 $^{21}$ ) full models",
+ "text_level": 1,
+ "bbox": [
+ 102,
+ 720,
+ 705,
+ 734
+ ],
+ "page_idx": 14
+ },
+ {
+ "type": "text",
+ "text": "In this example, we study the convergence of the TAPS solver for the time-dependent parametric heat transfer problem in a S-P-T setting. In Eq. 18, we adopt the heat source as shown in Eq. 49. In this example, we have spatial variable $\\boldsymbol{x}_s = (x,y,z)$ , parametric variable $\\boldsymbol{x}_p = (k,P,\\rho ,c_p)$ and temporal variable $x_{t} = t$ .",
+ "bbox": [
+ 100,
+ 736,
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+ ],
+ "page_idx": 14
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\begin{array}{l} f \\left(\\boldsymbol {x} _ {s}, \\boldsymbol {x} _ {p}, x _ {t}\\right) = 1 5 \\rho^ {2} c _ {p} ^ {2} k p e ^ {- 1 5 k t} e ^ {- 2 5. 0 x ^ {2} - 2 5. 0 y ^ {2}} \\\\ + 5 0 \\rho c _ {p} k p \\left(1 - e ^ {- 1 5 k t}\\right) e ^ {- 2 5. 0 x ^ {2} - 2 5. 0 y ^ {2}} \\left[ \\left(1 - 5 0 x ^ {2}\\right) + \\left(1 - 5 0 y ^ {2}\\right) \\right] \\tag {49} \\\\ \\end{array}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 240,
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+ ],
+ "page_idx": 14
+ },
+ {
+ "type": "text",
+ "text": "The analytical solution to the PDE is inherently non-separable.",
+ "bbox": [
+ 102,
+ 832,
+ 529,
+ 847
+ ],
+ "page_idx": 14
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\nu ^ {\\mathrm {e x}} \\left(\\boldsymbol {x} _ {s}, \\boldsymbol {x} _ {p}, x _ {t}\\right) = \\rho c _ {p} P \\left(1 - e ^ {- 1 5 k t}\\right) e ^ {- 2 5. 0 x ^ {2} - 2 5. 0 y ^ {2}} \\tag {50}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 342,
+ 853,
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+ 872
+ ],
+ "page_idx": 14
+ },
+ {
+ "type": "page_number",
+ "text": "15",
+ "bbox": [
+ 490,
+ 878,
+ 507,
+ 889
+ ],
+ "page_idx": 14
+ },
+ {
+ "type": "text",
+ "text": "The initial and boundary conditions are:",
+ "bbox": [
+ 102,
+ 131,
+ 376,
+ 146
+ ],
+ "page_idx": 15
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\nu \\left(\\boldsymbol {x} _ {s}, \\boldsymbol {x} _ {p}, 0\\right) = 0\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 376,
+ 156,
+ 489,
+ 172
+ ],
+ "page_idx": 15
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\nu \\left(\\boldsymbol {x} _ {s}, \\boldsymbol {x} _ {p}, x _ {t}\\right) | _ {\\partial \\Omega} = u ^ {\\mathrm {e x}} \\left(\\boldsymbol {x} _ {s}, \\boldsymbol {x} _ {p}, x _ {t}\\right) | _ {\\partial \\Omega}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 379,
+ 175,
+ 616,
+ 195
+ ],
+ "page_idx": 15
+ },
+ {
+ "type": "text",
+ "text": "The relative $L_{2}$ norm error is defined as:",
+ "bbox": [
+ 102,
+ 200,
+ 379,
+ 212
+ ],
+ "page_idx": 15
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\epsilon_ {L _ {2}} = \\frac {\\left\\| u ^ {T D} \\left(\\boldsymbol {x} _ {s} , \\boldsymbol {x} _ {p} , x _ {t}\\right) - u ^ {\\mathrm {e x}} \\left(\\boldsymbol {x} _ {s} , \\boldsymbol {x} _ {p} , x _ {t}\\right) \\right\\| _ {L _ {2} \\left(\\Omega_ {x _ {s}} \\otimes \\Omega_ {x _ {p}} \\otimes \\Omega_ {x _ {t}}\\right)}}{\\left\\| u ^ {\\mathrm {e x}} \\left(\\boldsymbol {x} _ {s} , \\boldsymbol {x} _ {p} , x _ {t}\\right) \\right\\| _ {L _ {2} \\left(\\Omega_ {x _ {s}} \\otimes \\Omega_ {x _ {p}} \\otimes \\Omega_ {x _ {t}}\\right)}} \\tag {51}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 317,
+ 222,
+ 892,
+ 260
+ ],
+ "page_idx": 15
+ },
+ {
+ "type": "text",
+ "text": "To study the convergence of TAPS for S-P-T problems, the number of grid points is refined simultaneously in each dimension and corresponding relative $L_{2}$ norm errors are computed as shown in Fig. 10. When the number of grid points in each dimension is 450, the equivalent DoFs of a full model achieves $450^{8} = 1.68 \\times 10^{21}$ . Consequently, it is equivalent to a zetta-scale $(10^{21})$ full problem. As can be seen from the figure, a larger patch size $s$ leads to a smaller error and faster convergence. A higher reproducing polynomial order $p$ also leads to a higher convergence rate. It can be noticed that the convergence rate for $p = 3$ case is smaller than expected $p + 1 = 4$ . This is attributed to the fact that the S-P-T mesh is not fine enough. However, due to the rounding error in computing the relative $L_{2}$ norm error, we can only accurately compute the error up to 450 grid points per dimension.",
+ "bbox": [
+ 100,
+ 268,
+ 892,
+ 382
+ ],
+ "page_idx": 15
+ },
+ {
+ "type": "image",
+ "img_path": "images/7fb0c377642dbc2df6a083a06da8df232e8bdb8c9587d121c766dfaf94864ad3.jpg",
+ "image_caption": [
+ "Figure 10: Relative $L_{2}$ norm error with respect to the number of grid points in each dimension"
+ ],
+ "image_footnote": [],
+ "bbox": [
+ 344,
+ 395,
+ 655,
+ 596
+ ],
+ "page_idx": 15
+ },
+ {
+ "type": "text",
+ "text": "In summary, we have the flexibility to choose different $s$ and $p$ to control the accuracy of TAPS by directly solving the S-P-T PDE. This is different from other data-driven modeling approaches (for instance, neural networks-based data-driven methods) in two notable ways. First, unlike a black-box neural network interpolator where the accuracy of the model is not guaranteed, our method is built upon the AI-enhanced finite element method, and we can control the convergence rate by choosing suitable hyperparameters $s$ and $p$ . Second, unlike most data-driven reduced-order models for physical problems, our method directly solves the governing PDE by plugging in the TD interpolation without seeing any training data. As a result, we can avoid the most expensive offline data generation stage as opposed to data-driven methods.",
+ "bbox": [
+ 100,
+ 636,
+ 894,
+ 749
+ ],
+ "page_idx": 15
+ },
+ {
+ "type": "text",
+ "text": "3.3. Moving source with solution dependent material parameters",
+ "text_level": 1,
+ "bbox": [
+ 102,
+ 763,
+ 547,
+ 778
+ ],
+ "page_idx": 15
+ },
+ {
+ "type": "text",
+ "text": "In this section, we model moving heat sources using temperature-dependent material parameters. The solution scheme of this problem is provided in detail in Appendix A. Figure 11(a) illustrates a typical representation of temperature-dependent heat conductivity and capacity for Inconel 718 [33].",
+ "bbox": [
+ 100,
+ 781,
+ 894,
+ 824
+ ],
+ "page_idx": 15
+ },
+ {
+ "type": "text",
+ "text": "Since the temperature dependency of $k(u)$ and $\\rho c_{p}(u)$ can be approximated using a linear relationship. As a result, we can directly rewrite $k(u(\\boldsymbol{x}_s,x_t))$ and $\\rho c_{p}(u(\\boldsymbol{x}_{s},x_{t}))$ in the TD format.",
+ "bbox": [
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+ ],
+ "page_idx": 15
+ },
+ {
+ "type": "page_number",
+ "text": "16",
+ "bbox": [
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+ ],
+ "page_idx": 15
+ },
+ {
+ "type": "image",
+ "img_path": "images/5e704b162ebdd448550d9186205e95bdb2ff6e27cf2d3fe5c63a122256540197.jpg",
+ "image_caption": [
+ "(a)"
+ ],
+ "image_footnote": [],
+ "bbox": [
+ 186,
+ 143,
+ 500,
+ 286
+ ],
+ "page_idx": 16
+ },
+ {
+ "type": "image",
+ "img_path": "images/716c83297ce005647d99e5f23cc46e6e3024853b336292259b2f47b9f8a009d8.jpg",
+ "image_caption": [
+ "(b)",
+ "Figure 11: (a) Temperature dependent material properties for Inconel 718 [33] (b) Schematic of numerical simulation, where the solution along the center line is compared for FEM and TAPS."
+ ],
+ "image_footnote": [],
+ "bbox": [
+ 526,
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+ 280
+ ],
+ "page_idx": 16
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\nk \\left(\\boldsymbol {x} _ {s}, x _ {t}\\right) \\approx \\sum_ {m = 1} ^ {M} m _ {k} u _ {x _ {1}} ^ {(m)} \\left(x _ {1}\\right) u _ {x _ {2}} ^ {(m)} \\left(x _ {2}\\right) u _ {x _ {3}} ^ {(m)} \\left(x _ {3}\\right) u _ {x _ {t}} ^ {(m)} \\left(x _ {t}\\right) + n _ {k}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 315,
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+ ],
+ "page_idx": 16
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\rho c _ {p} \\left(\\boldsymbol {x} _ {s}, x _ {t}\\right) \\approx \\sum_ {m = 1} ^ {M} m _ {c _ {p}} u _ {x _ {1}} ^ {(m)} \\left(x _ {1}\\right) u _ {x _ {2}} ^ {(m)} \\left(x _ {2}\\right) u _ {x _ {3}} ^ {(m)} \\left(x _ {3}\\right) u _ {x _ {t}} ^ {(m)} \\left(x _ {t}\\right) + n _ {c _ {p}} \\tag {52}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ 428,
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+ 464
+ ],
+ "page_idx": 16
+ },
+ {
+ "type": "text",
+ "text": "where $M$ is the decomposition modes of the TAPS solution; $m_{k} = 1.52 \\times 10^{-5} \\mathrm{~W} / (\\mathrm{mmK}^{2})$ ; $n_{k} = 5.29 \\times 10^{-3} \\mathrm{~W} / (\\mathrm{mmK})$ ; $m_{c_{p}} = 6.11 \\times 10^{-7} \\mathrm{~mm}^{-3} \\mathrm{~K}^{-2}$ ; $n_{c p} = 3.25 \\times 10^{-3} \\mathrm{~mm}^{-3} \\mathrm{~K}^{-1}$",
+ "bbox": [
+ 102,
+ 476,
+ 892,
+ 506
+ ],
+ "page_idx": 16
+ },
+ {
+ "type": "text",
+ "text": "The problem setup is shown in Fig. 11 (b). The spatial domain size is $10\\mathrm{mm} \\times 10\\mathrm{mm} \\times 1\\mathrm{mm}$ where homogeneous Dirichlet boundary conditions are assumed for the left and right surfaces; homogeneous Neumann boundary conditions are applied to all other surfaces. As shown in Eq. 19, a moving Gaussian source term $f(\\boldsymbol{x}_s, t)$ is applied as a volumetric source term with a radius $r = 0.5\\mathrm{mm}$ and moving velocity $500\\mathrm{mm/s}$ . The diameter is discretized using 10 spatial elements.",
+ "bbox": [
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+ ],
+ "page_idx": 16
+ },
+ {
+ "type": "text",
+ "text": "Since there is no analytical solution available to this problem, we use implicit finite element analysis as the baseline for validation. JAX-FEM [34] is used to generate the nonlinear FEM solution. For ease of comparison, we use the same time increment as $1.60 \\times 10^{-4}$ sec for both TAPS and FEM. The solution along the center line, as shown in Fig. 11 (b) is compared. As can be seen from Fig. 12, the result of the nonlinear TAPS solver agrees well with FEM.",
+ "bbox": [
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+ "page_idx": 16
+ },
+ {
+ "type": "text",
+ "text": "3.4. Simulation of LPBF process",
+ "text_level": 1,
+ "bbox": [
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+ "page_idx": 16
+ },
+ {
+ "type": "text",
+ "text": "In this section, we use TAPS to efficiently model the laser powder bed fusion process (LPBF) in additive manufacturing. Here we only consider the free convection term in the Neumann boundary condition. The initial condition can be considered by splitting the total solution as a summation of the homogeneous part and the inhomogeneous part.",
+ "bbox": [
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+ "page_idx": 16
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\nu \\left(\\boldsymbol {x} _ {s}, x _ {t}\\right) = u _ {0} \\left(\\boldsymbol {x} _ {s}, x _ {t}\\right) + u _ {\\text {i n i t}} \\left(\\boldsymbol {x} _ {s}\\right) \\tag {53}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 16
+ },
+ {
+ "type": "text",
+ "text": "As a result, $u_{0}(\\pmb{x}_{s}, x_{t})$ is subject to homogeneous initial conditions. In this section, we assume Ti-6Al-4V is used as the powder bed materials. The detailed material parameters can be found in Table 4.",
+ "bbox": [
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+ "page_idx": 16
+ },
+ {
+ "type": "text",
+ "text": "3.4.1. Single-track simulation",
+ "text_level": 1,
+ "bbox": [
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+ "page_idx": 16
+ },
+ {
+ "type": "text",
+ "text": "In this example, we investigate the computational complexity numerically for single-track LPBF simulation with a single S-T slab, as shown in Fig. 13 (a). A single NVIDIA RTX A6000 GPU is used for all the following analyses. To ensure accuracy, the number of modes is adopted as 5 times larger than the number of time steps in the following examples. In the first case, within the S-T slab, the spatial mesh is refined uniformly along each spatial dimension",
+ "bbox": [
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+ "page_idx": 16
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+ {
+ "type": "page_number",
+ "text": "17",
+ "bbox": [
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+ "page_idx": 16
+ },
+ {
+ "type": "image",
+ "img_path": "images/3d5930d344ad41e50bbdf9aa4fa400ae67373da8806b2d086f22076597db5e79.jpg",
+ "image_caption": [
+ "Figure 12: Comparison of Nonlinear TAPS solution versus finite element solution at different times."
+ ],
+ "image_footnote": [],
+ "bbox": [
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+ "page_idx": 17
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+ {
+ "type": "table",
+ "img_path": "images/e26a208925086b40e365433211679472b0588b4faef6f759f0fae6134d0dc341.jpg",
+ "table_caption": [
+ "Table 4: Parameters used in the simulation"
+ ],
+ "table_footnote": [],
+ "table_body": "| Variables | Description |
| uh(x) | Interpolated scalar field defined inside of an element |
| Ae | Nodes within element e |
| As | Nodes within patch domain of element e |
| WiS,a,p,j(x) | Convolution patch function at node j for i-th nodal patch with hyperparameters s, a, and p |
| M | Total number of modes in tensor decomposition (TD) |
| m | Index for mode |
| D | Total number of dimensions |
| d | Index for dimension |
| x | Independent variable which includes spatial variable xs, parametric variable xp and temporal variable xt |
| Nd(xd;ad,sd,pd) | Global C-HiDeNN shape function for dimension d with dilation parameter ad, patch size sd and reproducing polynomial order pd |
| b | Source function in laser powder bed fusion process |
| uTD | Approximation of the solution field expressed via TD |
| T | Time slab index for space-parameter-time problem |
| k | Thermal conductivity |
| ρ | Material density |
| cp | Heat capacity |
| η | Material absorptivity |
| P | Laser power |
| r | Standard deviation that characterizes the width of the heat source |
| q | Heat flux |
| qconv | Heat flux from convection |
| qrad | Heat flux from radiation |
| qevap | Heat flux from evaporative cooling |
| hconv | Convection coefficient |
| σSB | Stefan-Boltzman constant |
| mevap | Mass evaporation flux |
| Levap | Heat of evaporation |
| Bnd | Shape function derivative |
| Ud | Solution matrix (Rnd×M) for dimension d that contains all the modes |
"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.102,
+ 0.647,
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+ ],
+ "angle": 0,
+ "content": "To fulfill the ever-growing challenges in predictive scientific models, data-driven surrogates, especially artificial intelligence (AI)-based models, present an alternative to conventional numerical models by significantly reducing the forward prediction time. These models can be treated as a reasonably accurate, reduced representation of real physics. Once trained properly, they can be used for fast prediction on unseen parameters [1, 2]. However, it is still uncertain whether a data-driven surrogate model can be trained to achieve the level of accuracy required in engineering design. Recently, it has been pointed out by Wolfram Research that standard AI models cannot easily fulfill the high accuracy requirement of predictive scientific tasks [3]. Furthermore, as suggested by Google Deepmind, the real potential of AI models lies in enhancing, rather than thoroughly replacing, well-established classical numerical algorithms [4]. In addition, the current standard data-driven approaches follow an offline-online scheme, where the offline stage involves a huge amount of training data, which can again be prohibitive. For problems with known physics, this data can be obtained by running multiple expensive simulations relying on standard numerical algorithms. In scenarios involving high-dimensional design spaces governed by parameterized partial differential equations (PDEs), such as in additive manufacturing (AM), conducting repetitive simulations with varying parameters in this offline stage becomes exceedingly expensive both in terms of computation time and data storage."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.129,
+ 0.847,
+ 0.893,
+ 0.861
+ ],
+ "angle": 0,
+ "content": "To avoid the prohibitive offline stage, one can try to obtain a surrogate model directly from governing equations"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.494,
+ 0.88,
+ 0.505,
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+ ],
+ "angle": 0,
+ "content": "2"
+ }
+ ],
+ [
+ {
+ "type": "image",
+ "bbox": [
+ 0.189,
+ 0.131,
+ 0.818,
+ 0.245
+ ],
+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 0.102,
+ 0.261,
+ 0.894,
+ 0.319
+ ],
+ "angle": 0,
+ "content": "Figure 1: The parameterized PDE is a PDE that includes parameters \\( \\mathbf{x}_p \\) that can vary and influence the solution \\( \\mathbf{u}(\\mathbf{x}_s, \\mathbf{x}_p, x_t) \\), where \\( \\mathbf{x}_s \\) and \\( x_t \\) are the spatial and time variables, respectively. The a priori approach directly finds a surrogate model from the governing parameterized PDE, whereas the data-driven approach has to solve the parameter-fixed PDE on sampled parameters to generate simulation data, followed by training tasks. FEM: Finite Element Method [5], C-HiDeNN: Convolution Hierarchical Deep-learning Neural Network [6], TAPS: Tensor-decomposition-based A Priori Surrogate, PINN: Physics Informed Neural Network [7]."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.102,
+ 0.342,
+ 0.893,
+ 0.429
+ ],
+ "angle": 0,
+ "content": "without generating any data. As shown in Fig. 1 denoted by the words \"A Priori\", this approach aims to find the surrogate model before actually \"seeing\" any data. For example, multilayer perceptron (MLP) architectures have been vastly used in physics-informed neural networks (PINNs) and their variations to approximate solutions to PDEs without requiring data [7, 8]. However, the results of these efforts are underwhelming, as it has been shown that PINN results have often been compared to weak baselines [9], and it is unclear if they guarantee convergence. Moreover, this method is still susceptible to high computational costs for both large-scale and high-dimensional problems [10]."
+ },
+ {
+ "type": "image",
+ "bbox": [
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+ ],
+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 0.103,
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+ ],
+ "angle": 0,
+ "content": "Figure 2: Development history of INN [11]. Figures are borrowed from references: HiDeNN [12], HiDeNN-TD [13], C-HiDeNN [14], C-HiDeNN-TD [6]."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.102,
+ 0.691,
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+ ],
+ "angle": 0,
+ "content": "Instead of developing solvers solely based on machine learning concepts, a new class of Hierarchical Deep-learning Neural Networks (HiDeNN) has been developed recently. This network architecture incorporates principles from the finite element method (FEM) to construct their architecture [15, 12]. Originally designed to advance FEM as opposed to solve parameterized PDEs, this approach significantly enhances computational accuracy and efficiency for both linear and nonlinear problems compared to standard FEM [16]. HiDeNN was then enhanced by adding an additional hidden layer in the form of a nonlinear convolutional filter, formulating a new neural network architecture named Convolutional HiDeNN (C-HiDeNN) [6, 14]. C-HiDeNN mimics the structure of the generalized finite element method but leverages machine learning to optimize its hyperparameters to further improve accuracy and efficiency. Arbitrary orders of convergence have been observed for C-HiDeNN despite utilizing a linear finite element mesh [14]. Although these methods offers greater accuracy with fewer DoFs, like FEM, they still encounter computational challenges such as balancing memory usage against mesh resolution, which limits their efficiency in modeling ultra large-scale and high-dimensional problems. Therefore, it becomes necessary to employ model order reduction"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
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+ "angle": 0,
+ "content": "3"
+ }
+ ],
+ [
+ {
+ "type": "text",
+ "bbox": [
+ 0.103,
+ 0.133,
+ 0.373,
+ 0.146
+ ],
+ "angle": 0,
+ "content": "techniques to address these limitations."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.102,
+ 0.147,
+ 0.916,
+ 0.315
+ ],
+ "angle": 0,
+ "content": "Model order reduction techniques have been widely used to tackle the ever-growing challenges from high-dimensional and large-scale problems. For example, proper generalized decomposition (PGD) [17, 18, 19] has been proposed to efficiently solve high-dimensional PDEs. Recently, tensor decomposition (TD) has been successfully leveraged within the HiDeNN framework. For example, Zhang showed that HiDeNN combined with TD (HiDeNN-TD) significantly improved the speed of HiDeNN while maintaining higher accuracy [13]. Li proposed C-HiDeNN combined with TD (C-HiDeNN-TD) for extremely large-scale nested topology optimization problems [20]. Recently, Park generalized the HiDeNN-family networks under the umbrella of Interpolating Neural Networks (INNs) and demonstrated that the network can be used for both data-driven learning and data-free (i.e., a priori) solving [11]. The development history of HiDeNN family networks and INN is summarized in Fig. 2. While INN clearly explains how to construct the network architecture, an efficient optimization scheme for solving ultra large-scale and high-dimensional problems remains underdeveloped. In this paper, ultra large-scale problems refer to problems on the zetta-scale \\((10^{21})\\) in terms of DoFs."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.102,
+ 0.317,
+ 0.894,
+ 0.459
+ ],
+ "angle": 0,
+ "content": "The demand for high-precision engineering simulations and efficient solution schemes highlights the need for innovative modeling approaches that swiftly solve large-scale problems while optimizing the design space. This research aims to fulfill this need by developing tensor-decomposition-based A Priori Surrogate (TAPS), a data-free predictive AI model, which aims to enhance high-resolution capabilities while simultaneously optimizing computational efficiency with a minimal memory footprint, low data storage needs, and fast prediction. The proposed comprehensive framework sets a foundation for scalable, adaptable, and future-proof solutions to counter the ever-growing complexity in simulation-driven advanced industries. TAPS is particularly well-suited for engineering challenges where: 1) the finite element method and other conventional methods are unsuitable due to excessively long simulation times or high RAM and storage demands needed to achieve high accuracy, 2) the model must accommodate design parameters as inputs, or 3) fast prediction is required once the model is obtained."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.102,
+ 0.459,
+ 0.894,
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+ ],
+ "angle": 0,
+ "content": "This paper is structured as follows. We first introduce the formulation of TAPS in section 2. In section 3, we examine the numerical convergence of TAPS for both space-time (S-T) and space-parameter-time (S-P-T) problems (i.e., problems that are dependent on spatial, parametric, and temporal inputs). In section 4, TAPS is applied to large-scale additive manufacturing problems that are considered intractable with standard numerical algorithms. This application effectively demonstrates TAPS's capability to address all of the three identified challenges."
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.104,
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+ ],
+ "angle": 0,
+ "content": "2. Theory"
+ },
+ {
+ "type": "title",
+ "bbox": [
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+ ],
+ "angle": 0,
+ "content": "2.1. Review of C-HiDeNN interpolation theory"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.102,
+ 0.591,
+ 0.894,
+ 0.69
+ ],
+ "angle": 0,
+ "content": "Leveraging the universal approximation theorem, multilayer perceptrons (MLPs) have been successfully applied as global basis functions in deep learning-based solvers [7]. However, as shown in Table 2, MLPs have a few potential caveats when approximating PDE solutions. To overcome these limitations, we leverage the Convolutional HiDeNN (C-HiDeNN) interpolation function, which leverages the merits of both locally supported finite element shape functions and the flexibility of machine learning. Note that C-HiDeNN also belongs to the INN category as shown in Fig. 2. C-HiDeNN maintains all the essential finite element approximation properties such as Kronecker delta and partition of unity [14]."
+ },
+ {
+ "type": "table_caption",
+ "bbox": [
+ 0.249,
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+ "angle": 0,
+ "content": "Table 2: Comparison of MLP and C-HiDeNN as approximation functions of PDE solutions."
+ },
+ {
+ "type": "table",
+ "bbox": [
+ 0.174,
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+ "angle": 0,
+ "content": "| PDE | Differential operator L | Dependent variable | xs | xp | xt |
| ∂2u/∂x12 + ∂2u/∂x22 + ... + ∂2u/∂x2D = f(x) | ∂2/∂x12 + ∂2/∂x22 + ... + ∂2/∂x2D | u | (x1, x2, ...,xD) | - | - |
| μui, jj + (μ + λ)uj,ij + Fi = e(x12+x22+x32) | μ(·)i,jj + (μ + λ)(·)j,ij | ui, i = 1, 2, 3 | (x1, x2, x3) | (λ,μ) | - |
| ρcp,du/dt + k(∂2u/∂x12 + ∂2u/∂x22 + ∂2u/∂x32) = Pe(x12+x22+x32) | ρcp,du/dt + k(∂2/∂x12 + ∂2/∂x22 + ∂2/∂x32) | u | (x1, x2, x3) | (ρ, cp, k, P) | t |
"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.128,
+ 0.547,
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+ "angle": 0,
+ "content": "The weighted-sum residual form of the PDE with TD interpolation can be written as:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.362,
+ 0.571,
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+ "angle": 0,
+ "content": "\\[\n\\int_ {\\Omega} \\delta u ^ {T D} (\\boldsymbol {x}) \\left[ \\mathcal {L} \\left(u ^ {T D} (\\boldsymbol {x})\\right) - f (\\boldsymbol {x}) \\right] d \\Omega = 0 \\tag {15}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.103,
+ 0.606,
+ 0.891,
+ 0.621
+ ],
+ "angle": 0,
+ "content": "where \\( u^{TD} \\) is the approximation of the solution (i.e., trial function), \\( \\delta u^{TD} \\) is the test function, and \\( d\\Omega = d\\Omega_{x_s}d\\Omega_{x_p}d\\Omega_{x_t} \\)."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.102,
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+ ],
+ "angle": 0,
+ "content": "Depending on how \\(\\delta u^{TD}\\) is adopted, different mathematical formulations can be obtained. If the test function resides in the same function space as the trial function, it becomes the Galerkin formulation. When the test function space differs from the trial function space, it becomes the Petrov-Galerkin formulation [22]. If the Dirac delta function is used for the test function, then Eq. 15 corresponds to the collocation method [24]. In this paper, we employ the Galerkin formulation. However, the proposed framework is versatile and can be extended to accommodate other formulations as well."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.102,
+ 0.707,
+ 0.894,
+ 0.791
+ ],
+ "angle": 0,
+ "content": "In Eq. 12, the entire S-P-T domain is approximated using TD interpolation. However, this approach may result in a large system of equations due to the rapid increase in the number of TD modes for certain cases. For example, if the forcing function represents a moving source function in Eq. 14), this complexity may arise. To maintain computational efficiency, we can partition the temporal domain into a series of time slabs. As illustrated in Fig. 6(a), the S-P-T continuum is divided into S-P-T slabs \\(\\mathcal{T}_1,\\mathcal{T}_2,\\dots ,\\mathcal{T}_T\\) . The solution within each time slab is then approximated individually using the TD interpolation."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.102,
+ 0.792,
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+ ],
+ "angle": 0,
+ "content": "Between consecutive S-P-T slabs, either a continuous or discontinuous formulation can be employed. As shown in Fig. 6(b) for the continuous Galerkin scheme, the continuity of the solution in time is enforced by imposing the solution at the end of slab \\(\\mathcal{T}_{i-1}\\) as the initial condition of \\(\\mathcal{T}_i\\):"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.375,
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+ "angle": 0,
+ "content": "\\[\n{ } ^ { [ \\mathcal { T } + 1 ] } u ( \\boldsymbol { x } _ { s } , \\boldsymbol { x } _ { p } , 0 ) = { } ^ { [ \\mathcal { T } ] } u ( \\boldsymbol { x } _ { s } , \\boldsymbol { x } _ { p } , x _ { t } ^ { m a x } ) \\tag {16}\n\\]"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
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+ "angle": 0,
+ "content": "8"
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+ ],
+ [
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+ "type": "text",
+ "bbox": [
+ 0.103,
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+ ],
+ "angle": 0,
+ "content": "Discontinuous Galerkin method can be used when a discontinuity is allowed between S-P-T slabs, as illustrated in Fig. 6(c). Discontinuity in time can be modeled using the jump operator \\(\\llbracket \\dots \\rrbracket\\) [25]."
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.291,
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+ "angle": 0,
+ "content": "\\[\n\\llbracket u \\left(\\boldsymbol {x} _ {s}, \\boldsymbol {x} _ {p}, t\\right) \\rrbracket = \\lim _ {\\epsilon \\rightarrow 0 ^ {+}} \\left(^ {\\mathcal {T} + 1} u \\left(\\boldsymbol {x} _ {s}, \\boldsymbol {x} _ {p}, \\epsilon\\right) - ^ {\\mathcal {T}} u \\left(\\boldsymbol {x} _ {s}, \\boldsymbol {x} _ {p}, x _ {t} ^ {\\max } - \\epsilon\\right)\\right) \\tag {17}\n\\]"
+ },
+ {
+ "type": "image",
+ "bbox": [
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+ "type": "image_caption",
+ "bbox": [
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+ "angle": 0,
+ "content": "(a)"
+ },
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+ "type": "image",
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+ "type": "image_caption",
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+ "angle": 0,
+ "content": "(b)"
+ },
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+ "type": "image",
+ "bbox": [
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+ {
+ "type": "image_caption",
+ "bbox": [
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+ "angle": 0,
+ "content": "(c)"
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+ {
+ "type": "image_caption",
+ "bbox": [
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+ "angle": 0,
+ "content": "Figure 6: (a) Multiple S-P-T slabs along the temporal dimension. (b) Continuous Galerkin: the solution is continuous across different S-P-T slabs. (c) Discontinuous Galerkin: jumps are allowed across the slab boundaries"
+ },
+ {
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+ ],
+ "angle": 0,
+ "content": "Keeping in mind that this approach can be applied generally to a range of engineering problems, we will demonstrate an example of the Galerkin formulation using a single space-parameter-time partition (S-P-T) slab in the remainder of this section. For illustrative purposes, the transient heat transfer equation will be utilized:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.371,
+ 0.452,
+ 0.893,
+ 0.468
+ ],
+ "angle": 0,
+ "content": "\\[\n\\rho c _ {p} \\nabla_ {x _ {t}} u - \\nabla_ {\\boldsymbol {x} _ {s}} \\cdot k \\nabla_ {\\boldsymbol {x} _ {s}} u = f (\\boldsymbol {x} _ {s}, \\boldsymbol {x} _ {p}, x _ {t}) \\tag {18}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.103,
+ 0.476,
+ 0.892,
+ 0.504
+ ],
+ "angle": 0,
+ "content": "as we focus on the example of modeling the laser powder bed fusion (LPBF) process in additive manufacturing (AM). In an LPBF simulation, we adopt the following time-dependent moving heat source function:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.271,
+ 0.511,
+ 0.893,
+ 0.551
+ ],
+ "angle": 0,
+ "content": "\\[\nf \\left(\\boldsymbol {x} _ {s}, \\boldsymbol {x} _ {p}, x _ {t}\\right) = \\frac {2 \\eta P}{\\pi r ^ {2} d _ {\\nu}} \\exp \\left(- \\frac {2 \\left((x - x _ {0} (t)) ^ {2} + (y - y _ {0} (t)) ^ {2}\\right)}{r ^ {2}}\\right) \\cdot \\mathbf {1} _ {\\left(x _ {3} \\geq d _ {\\nu}\\right)} \\tag {19}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.103,
+ 0.558,
+ 0.894,
+ 0.656
+ ],
+ "angle": 0,
+ "content": "Summarizing the independent variables in Eq. 18, there are spatial variables \\( \\mathbf{x}_s = (x_1, x_2, x_3) \\); parametric variables \\( \\mathbf{x}_p = (k, \\rho, c_p, \\eta, P, r, d_v) \\); and a temporal variable \\( x_t = t \\). Among the parametric variables, \\( k \\) is conductivity; \\( \\rho \\) is the material density; \\( c_p \\) is heat capacity; \\( \\eta \\) is the material absorptivity; \\( P \\) represents laser power; \\( r \\) is the standard deviation that characterizes the width of the heat source; \\( d_v \\) is the penetration depth of the heat source. In Eq. 19, \\( [x_0(t), y_0(t)] \\) represents the center of the moving heat source; \\( \\mathbf{1}_{(x_3 \\geq d_v)} \\) is the indicator function where \\( \\mathbf{1}_{(x_3 \\geq d_v)} = 1 \\) if \\( x_3 \\geq d_v \\) or \\( \\mathbf{1}_{(x_3 \\geq d_v)} = 0 \\) if \\( x_3 < d_v \\). Note that the discretization of the material parameters, in particular, in a random field setting, has been previously proposed by Liu et al. [26, 27]."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.103,
+ 0.658,
+ 0.892,
+ 0.685
+ ],
+ "angle": 0,
+ "content": "As shown in the schematic below, we classify the boundary surfaces into 2 categories: the Dirichlet boundary surface \\(\\Gamma_{D}\\) and the Neumann boundary surface \\(\\Gamma_{N}\\)."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.103,
+ 0.686,
+ 0.894,
+ 0.727
+ ],
+ "angle": 0,
+ "content": "A uniform ambient temperature is used as the initial condition. The bottom of the powder bed is subject to the Dirichlet boundary condition and the Neumann boundary conditions are prescribed on the other surfaces. The initial and boundary conditions are:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.398,
+ 0.729,
+ 0.531,
+ 0.743
+ ],
+ "angle": 0,
+ "content": "\\[\nu (\\boldsymbol {x} _ {s}, \\boldsymbol {x} _ {p}, 0) | _ {\\Omega} = u _ {0},\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.398,
+ 0.745,
+ 0.892,
+ 0.761
+ ],
+ "angle": 0,
+ "content": "\\[\n\\left. u \\left(\\boldsymbol {x} _ {s}, \\boldsymbol {x} _ {p}, x _ {t}\\right) \\right| _ {\\Gamma_ {D}} = u _ {0}, \\tag {20}\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.398,
+ 0.764,
+ 0.597,
+ 0.779
+ ],
+ "angle": 0,
+ "content": "\\[\n\\boldsymbol {n} \\cdot \\boldsymbol {q} | _ {\\Gamma_ {N}} = q _ {\\text {c o n v}} + q _ {\\text {r a d}} + q _ {\\text {e v a p}}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.103,
+ 0.783,
+ 0.893,
+ 0.811
+ ],
+ "angle": 0,
+ "content": "where \\( u_{0} \\) is the ambient temperature, \\( q_{conv} \\) accounts for free convection, \\( q_{rad} \\) accounts for radiation, and \\( q_{evap} \\) imposes evaporative cooling when any material surface reaches the evaporation temperature [28]. Each flux is defined as:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.394,
+ 0.822,
+ 0.573,
+ 0.836
+ ],
+ "angle": 0,
+ "content": "\\[\nq _ {c o n v} = h _ {c o n v} [ u (x, t) - u _ {0} ],\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.395,
+ 0.839,
+ 0.892,
+ 0.856
+ ],
+ "angle": 0,
+ "content": "\\[\nq _ {r a d} = - \\sigma_ {S B} \\epsilon \\left(u ^ {4} \\left(\\boldsymbol {x} _ {s}, x _ {t}\\right) - u _ {0} ^ {4}\\right), \\tag {21}\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.395,
+ 0.859,
+ 0.536,
+ 0.874
+ ],
+ "angle": 0,
+ "content": "\\[\nq _ {e v a p} = - m _ {e v a p} L _ {e v a p}.\n\\]"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.493,
+ 0.88,
+ 0.505,
+ 0.89
+ ],
+ "angle": 0,
+ "content": "9"
+ }
+ ],
+ [
+ {
+ "type": "image",
+ "bbox": [
+ 0.349,
+ 0.136,
+ 0.648,
+ 0.262
+ ],
+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 0.281,
+ 0.275,
+ 0.715,
+ 0.287
+ ],
+ "angle": 0,
+ "content": "Figure 7: Transient heat transfer with initial condition and boundary conditions."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.103,
+ 0.311,
+ 0.894,
+ 0.368
+ ],
+ "angle": 0,
+ "content": "where \\(\\sigma_{SB}\\) is the Stefan-Boltzmann constant; \\(\\epsilon\\) is the material emissivity; \\(u_0\\) is the ambient temperature; \\(h_{conv}\\) is the convection coefficient of the surrounding gas, \\(m_{evap}\\) is the mass evaporation flux and \\(L_{evap}\\) is the heat of evaporation. In the following numerical examples, we only consider the free convection term in the Neumann boundary condition. The solution to Eq. 18 is approximated using TD interpolation function:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.347,
+ 0.378,
+ 0.893,
+ 0.415
+ ],
+ "angle": 0,
+ "content": "\\[\nu ^ {T D} \\left(\\boldsymbol {x} _ {s}, \\boldsymbol {x} _ {p}, x _ {t}\\right) = \\sum_ {m = 1} ^ {M} u _ {\\boldsymbol {x} _ {s}} ^ {(m)} \\left(\\boldsymbol {x} _ {s}\\right) u _ {\\boldsymbol {x} _ {p}} ^ {(m)} \\left(\\boldsymbol {x} _ {p}\\right) u _ {x _ {t}} ^ {(m)} \\left(x _ {t}\\right) \\tag {22}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.103,
+ 0.425,
+ 0.893,
+ 0.468
+ ],
+ "angle": 0,
+ "content": "Here a general notation is employed to represent different types of components in Eq. 22. For example, the spatial component \\( u_{\\boldsymbol{x}_s}^{(m)}(\\boldsymbol{x}_s) \\) is equivalent to \\( u_{x_1}^{(m)}(x_1)u_{x_2}^{(m)}(x_2)u_{x_3}^{(m)}(x_3) \\). The corresponding test function can be obtained using the variational principle:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.14,
+ 0.488,
+ 0.893,
+ 0.527
+ ],
+ "angle": 0,
+ "content": "\\[\n\\delta u ^ {T D} \\left(\\boldsymbol {x} _ {s}, \\boldsymbol {x} _ {p}, x _ {t}\\right) = \\sum_ {m = 1} ^ {M} \\left[ \\delta u _ {\\boldsymbol {x} _ {s}} ^ {(m)} \\left(\\boldsymbol {x} _ {s}\\right) u _ {\\boldsymbol {x} _ {p}} ^ {(m)} \\left(\\boldsymbol {x} _ {p}\\right) u _ {x _ {t}} ^ {(m)} \\left(x _ {t}\\right) + u _ {\\boldsymbol {x} _ {s}} ^ {(m)} \\left(\\boldsymbol {x} _ {s}\\right) \\delta u _ {\\boldsymbol {x} _ {p}} ^ {(m)} \\left(\\boldsymbol {x} _ {p}\\right) u _ {x _ {t}} ^ {(m)} \\left(x _ {t}\\right) + u _ {\\boldsymbol {x} _ {s}} ^ {(m)} \\left(\\boldsymbol {x} _ {s}\\right) u _ {\\boldsymbol {x} _ {p}} ^ {(m)} \\left(\\boldsymbol {x} _ {p}\\right) \\delta u _ {x _ {t}} ^ {(m)} \\left(x _ {t}\\right) \\right] \\tag {23}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.104,
+ 0.536,
+ 0.84,
+ 0.551
+ ],
+ "angle": 0,
+ "content": "Plugging the trial and test functions, the S-P-T Galerkin form of Eq. 18 can be obtained by following Eq. 15:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.33,
+ 0.56,
+ 0.893,
+ 0.59
+ ],
+ "angle": 0,
+ "content": "\\[\n\\int_ {\\Omega} \\delta u ^ {T D} \\left[ \\rho c _ {p} \\nabla_ {x _ {t}} u ^ {T D} - \\nabla_ {x _ {s}} \\cdot k \\nabla_ {x _ {s}} u ^ {T D} - f \\right] d \\Omega = 0 \\tag {24}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.103,
+ 0.594,
+ 0.892,
+ 0.621
+ ],
+ "angle": 0,
+ "content": "Using integration by parts on the diffusion term, we get the corresponding general S-P-T Galerkin weak form in TAPS formulation:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.112,
+ 0.643,
+ 0.893,
+ 0.677
+ ],
+ "angle": 0,
+ "content": "\\[\n\\int_ {\\Omega} \\delta u ^ {T D} \\rho c _ {p} \\nabla_ {x _ {t}} u ^ {T D} d \\Omega + \\int_ {\\Omega} \\nabla_ {\\boldsymbol {x} _ {s}} \\delta u ^ {T D} \\cdot k \\nabla_ {\\boldsymbol {x} _ {s}} u ^ {T D} d \\Omega - \\int_ {\\partial \\boldsymbol {x} _ {s}, \\boldsymbol {x} _ {p}, t} \\delta u ^ {T D} \\boldsymbol {n} \\cdot \\boldsymbol {q} | _ {\\Gamma_ {N}} d s d \\Omega_ {\\boldsymbol {x} _ {p}} d \\Omega_ {t} - \\int_ {\\Omega} \\delta u ^ {T D} f (\\boldsymbol {x} _ {s}, \\boldsymbol {x} _ {p}, x _ {t}) d \\Omega = 0 \\tag {25}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.104,
+ 0.685,
+ 0.452,
+ 0.699
+ ],
+ "angle": 0,
+ "content": "where \\(q\\) is the heat flux on the Neumann boundary."
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.104,
+ 0.713,
+ 0.304,
+ 0.727
+ ],
+ "angle": 0,
+ "content": "2.5. Discretized matrix form"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.103,
+ 0.73,
+ 0.894,
+ 0.801
+ ],
+ "angle": 0,
+ "content": "The S-P-T Galerkin weak form shown in Eq. 25 is nonlinear in nature due to the tensor product structure in TD interpolation, necessitating efficient solution schemes. To illustrate the detailed solution approach for the general S-P-T weak form, we simplify the governing equation Eq. 18 by considering a one-dimensional spatial problem where \\( x_{s} = x \\). We assume that the product of density and specific heat capacity \\( \\rho c_{p} \\) is equal to 1. Additionally, the forcing term is solely dependent on \\( x \\). Therefore, the simplified governing equation for this example is given by:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.421,
+ 0.81,
+ 0.893,
+ 0.839
+ ],
+ "angle": 0,
+ "content": "\\[\n\\frac {\\partial u}{\\partial t} - \\frac {\\partial u}{\\partial x} \\cdot k \\frac {\\partial x}{\\partial x} = f (x) \\tag {26}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.103,
+ 0.843,
+ 0.893,
+ 0.872
+ ],
+ "angle": 0,
+ "content": "subject to homogeneous boundary conditions and initial conditions. This equation has 3 independent variables (\\(D = 3\\)), i.e., spatial variable \\(x_{s} = x_{1} = x\\), parametric variable \\(x_{p} = x_{2} = k\\) and temporal variable \\(x_{t} = x_{3} = t\\). The S-P-T"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.49,
+ 0.879,
+ 0.51,
+ 0.89
+ ],
+ "angle": 0,
+ "content": "10"
+ }
+ ],
+ [
+ {
+ "type": "text",
+ "bbox": [
+ 0.102,
+ 0.132,
+ 0.894,
+ 0.162
+ ],
+ "angle": 0,
+ "content": "Galerkin weak form of this problem can be written as follows according to Eq. 25 (the superscripts \"TD\" for both trial and test functions are omitted for brevity)."
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.324,
+ 0.171,
+ 0.894,
+ 0.202
+ ],
+ "angle": 0,
+ "content": "\\[\n\\int_ {\\Omega} \\delta u \\nabla_ {t} u d \\Omega + \\int_ {\\Omega} \\nabla_ {x} \\delta u \\cdot k \\nabla_ {x} u d \\Omega - \\int_ {\\Omega} \\delta u f d \\Omega = 0 \\tag {27}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.104,
+ 0.205,
+ 0.623,
+ 0.22
+ ],
+ "angle": 0,
+ "content": "The corresponding trial and test functions can be obtained using Eqs. 22-23:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.38,
+ 0.229,
+ 0.893,
+ 0.267
+ ],
+ "angle": 0,
+ "content": "\\[\nu (x, k, t) = \\sum_ {m = 1} ^ {M} u _ {x} ^ {(m)} (x) u _ {k} ^ {(m)} (k) u _ {t} ^ {(m)} (t) \\tag {28}\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.192,
+ 0.277,
+ 0.893,
+ 0.335
+ ],
+ "angle": 0,
+ "content": "\\[\n\\delta u (x, k, t) = \\underbrace {\\sum_ {m = 1} ^ {M} \\delta u _ {x} ^ {(m)} (x) u _ {k} ^ {(m)} (k) u _ {t} ^ {(m)} (t)} _ {\\text {s p a t i a l v a r i a t i o n}} + \\underbrace {\\sum_ {m = 1} ^ {M} u _ {x} ^ {(m)} (x) \\delta u _ {k} ^ {(m)} (k) u _ {t} ^ {(m)} (t)} _ {\\text {p a r a m e t r i c v a r i a t i o n}} + \\underbrace {\\sum_ {m = 1} ^ {M} u _ {x} ^ {(m)} (x) u _ {k} ^ {(m)} (k) \\delta u _ {t} ^ {(m)} (t)} _ {\\text {t e m p o r a l v a r i a t i o n}} \\tag {29}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.102,
+ 0.339,
+ 0.894,
+ 0.382
+ ],
+ "angle": 0,
+ "content": "As shown in Eq. 29, the test function is further split into \\( D \\) variational terms for a general \\( D \\) dimensional problem (in the current example, \\( D = 3 \\)). As an example, we first plug Eq. 28 and the spatial variation term of Eq. 29 into the Galerkin weak form in Eq. 27 to obtain the S-P-T weak form terms corresponding to spatial variation:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.253,
+ 0.392,
+ 0.744,
+ 0.446
+ ],
+ "angle": 0,
+ "content": "\\[\n\\underbrace {\\int_ {\\Omega} \\sum_ {m = 1} ^ {M} \\sum_ {n = 1} ^ {M} \\left[ \\nabla \\delta u _ {x} ^ {(m)} (x) \\nabla u _ {x} ^ {(n)} (x) d x \\right] \\cdot \\left[ u _ {k} ^ {(m)} (k) k u _ {k} ^ {(n)} (k) d k \\right] \\cdot \\left[ u _ {t} ^ {(m)} (t) u _ {t} ^ {(n)} (t) d t \\right]} _ {\\text {d i f f u s i o n t e r m}} +\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.261,
+ 0.449,
+ 0.893,
+ 0.505
+ ],
+ "angle": 0,
+ "content": "\\[\n\\underbrace {\\int_ {\\Omega} \\sum_ {m = 1} ^ {M} \\sum_ {n = 1} ^ {M} \\left[ \\delta u _ {x} ^ {(m)} (x) u _ {x} ^ {(n)} (x) d x \\right] \\cdot \\left[ u _ {k} ^ {(m)} (k) u _ {k} ^ {(n)} (k) d k \\right] \\cdot \\left[ u _ {t} ^ {(m)} (t) \\nabla_ {t} u _ {t} ^ {(n)} (t) d t \\right]} _ {\\text {t i m e d e r i v a t i v e t e r m}} - \\tag {30}\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.336,
+ 0.506,
+ 0.662,
+ 0.561
+ ],
+ "angle": 0,
+ "content": "\\[\n\\underbrace {\\int_ {\\Omega} \\sum_ {m = 1} ^ {M} \\left[ \\delta u _ {x} ^ {(m)} (x) f (x) d x \\right] \\cdot \\left[ u _ {k} ^ {(m)} (k) d k \\right] \\cdot \\left[ u _ {t} ^ {(m)} (t) d t \\right]} _ {\\text {f o r c i n g t e r m}}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.104,
+ 0.566,
+ 0.647,
+ 0.58
+ ],
+ "angle": 0,
+ "content": "We use 1D C-HiDeNN shape functions to approximate each univariate function:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.36,
+ 0.603,
+ 0.624,
+ 0.624
+ ],
+ "angle": 0,
+ "content": "\\[\nu _ {d} ^ {(n)} (x _ {d}) = \\widetilde {N} _ {n _ {d} ^ {\\prime}} ^ {[ d ]} (x _ {d}) u _ {n _ {d} ^ {\\prime} n} ^ {[ d ]} \\quad (\\text {n o s u m o n} d)\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.34,
+ 0.626,
+ 0.893,
+ 0.647
+ ],
+ "angle": 0,
+ "content": "\\[\n\\delta u _ {d} ^ {(m)} \\left(x _ {d}\\right) = \\widetilde {N} _ {n _ {d}} ^ {[ d ]} \\left(x _ {d}\\right) \\delta u _ {n _ {d} m} ^ {[ d ]} \\quad (\\text {n o s u m o n} d) \\tag {31}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.103,
+ 0.654,
+ 0.893,
+ 0.69
+ ],
+ "angle": 0,
+ "content": "where Einstein summation is used. The free index \\(d\\) refers to dimension and \\(d = x,k\\) or \\(t\\). The gradient of the interpolated variable can be computed using the shape function derivative \\(\\widetilde{B}_{n_d}^{[d]}(x_d) = \\frac{d\\widetilde{N}_{n_d}^{[d]}(x_d)}{dx_d}\\)."
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.349,
+ 0.709,
+ 0.636,
+ 0.731
+ ],
+ "angle": 0,
+ "content": "\\[\n\\nabla_ {x _ {d}} u _ {d} ^ {(n)} (x _ {d}) = \\widetilde {B} _ {n _ {d} ^ {\\prime}} ^ {[ d ]} (x _ {d}) u _ {n _ {d} ^ {\\prime} n} ^ {[ d ]} \\quad (\\text {n o s u m o n} d)\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.329,
+ 0.733,
+ 0.892,
+ 0.753
+ ],
+ "angle": 0,
+ "content": "\\[\n\\nabla_ {x _ {d}} \\delta u _ {d} ^ {(m)} (x _ {d}) = \\widetilde {B} _ {n _ {d}} ^ {[ d ]} (x _ {d}) \\delta u _ {n _ {d} m} ^ {[ d ]} \\quad (\\text {n o s u m o n} d) \\tag {32}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.104,
+ 0.761,
+ 0.591,
+ 0.775
+ ],
+ "angle": 0,
+ "content": "Plugging Eq. 31 - 32 into Eq. 30, the diffusion term can be rewritten as:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.185,
+ 0.786,
+ 0.893,
+ 0.839
+ ],
+ "angle": 0,
+ "content": "\\[\n\\sum_ {m = 1} ^ {M} \\sum_ {n = 1} ^ {M} \\underbrace {\\int_ {\\Omega_ {x}} \\widetilde {B} _ {n _ {x}} (x) \\delta u _ {n _ {x} m} ^ {[ x ]} \\widetilde {B} _ {n _ {x} ^ {\\prime}} (x) u _ {n _ {x} ^ {\\prime} n} ^ {[ x ]} d x} _ {\\text {s p a t i a l t e r m}} \\cdot \\underbrace {\\int_ {\\Omega_ {k}} \\widetilde {N} _ {n _ {k}} (k) u _ {n _ {k} m} ^ {[ k ]} k \\widetilde {N} _ {n _ {k} ^ {\\prime}} (k) u _ {n _ {k} ^ {\\prime} n} ^ {[ k ]} d k} _ {\\text {p a r a m e t r i c t e r m}} \\cdot \\underbrace {\\int_ {\\Omega_ {t}} \\widetilde {N} _ {n _ {t}} (t) u _ {n _ {t} m} ^ {[ t ]} \\widetilde {N} _ {n _ {t} ^ {\\prime}} (t) u _ {n _ {t} ^ {\\prime} n} ^ {[ t ]} d t} _ {\\text {t e m p o r a l t e r m}} \\tag {33}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.102,
+ 0.843,
+ 0.893,
+ 0.87
+ ],
+ "angle": 0,
+ "content": "As can be readily seen from Eq. 33, after doing 1D integration of each term, the parametric and temporal terms can be treated as coefficient matrices:"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.49,
+ 0.879,
+ 0.508,
+ 0.89
+ ],
+ "angle": 0,
+ "content": "11"
+ }
+ ],
+ [
+ {
+ "type": "equation",
+ "bbox": [
+ 0.36,
+ 0.153,
+ 0.603,
+ 0.202
+ ],
+ "angle": 0,
+ "content": "\\[\nC _ {m n} ^ {[ k ]} = \\underbrace {\\int_ {\\Omega_ {k}} \\widetilde {N} _ {n _ {k}} (k) u _ {n _ {k} m} ^ {[ k ]} k \\widetilde {N} _ {n _ {k} ^ {\\prime}} (k) u _ {n _ {k} ^ {\\prime} n} ^ {[ k ]} d k} _ {\\text {p a r a m e t r i c t e r m}}\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.384,
+ 0.204,
+ 0.893,
+ 0.252
+ ],
+ "angle": 0,
+ "content": "\\[\nC _ {m n} ^ {[ t ]} = \\underbrace {\\int_ {\\Omega_ {t}} \\widetilde {N} _ {n _ {t}} (t) u _ {n _ {t} m} ^ {[ t ]} \\widetilde {N} _ {n _ {t} ^ {\\prime}} (t) u _ {n _ {t} ^ {\\prime} n} ^ {[ t ]} d t} _ {\\text {t e m p o r a l t e r m}} \\tag {34}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.103,
+ 0.262,
+ 0.893,
+ 0.289
+ ],
+ "angle": 0,
+ "content": "as the only free indices are \\( m \\) and \\( n \\). Substituting the coefficient matrices and rearranging different terms in Eq. 33, we have:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.329,
+ 0.3,
+ 0.893,
+ 0.339
+ ],
+ "angle": 0,
+ "content": "\\[\n\\sum_ {m = 1} ^ {M} \\delta u _ {n _ {x} m} ^ {[ x ]} \\sum_ {n = 1} ^ {M} \\left[ \\int_ {\\Omega_ {x}} \\widetilde {B} _ {n _ {x}} (x) \\widetilde {B} _ {n _ {x} ^ {\\prime}} (x) d x \\right] \\cdot C _ {m n} ^ {[ k ]} C _ {m n} ^ {[ t ]} \\cdot u _ {n _ {x} ^ {\\prime} n} ^ {[ x ]} \\tag {35}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.103,
+ 0.344,
+ 0.894,
+ 0.379
+ ],
+ "angle": 0,
+ "content": "Like standard FEM, we can define \\(\\int_{x}\\widetilde{B}_{n_x}(x)\\widetilde{B}_{n_x'}(x)dx\\) as the 1D stiffness matrix \\(K_{n_x n_x'}^{[x]}\\) of \\(x\\) dimension in Eq. 35. We let \\(C_{mn}^{[x]} = C_{mn}^{[k]}C_{mn}^{[t]}\\) with no summation on \\((m,n)\\). Furthermore, let us define the following 4-th order tensor:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.431,
+ 0.39,
+ 0.893,
+ 0.41
+ ],
+ "angle": 0,
+ "content": "\\[\nA _ {n _ {x} n _ {x} ^ {\\prime} m n} ^ {[ x ]} = K _ {n _ {x} n _ {x} ^ {\\prime}} ^ {[ x ] ^ {\\prime}} C _ {m n} ^ {[ x ]} \\tag {36}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.103,
+ 0.415,
+ 0.894,
+ 0.474
+ ],
+ "angle": 0,
+ "content": "where \\( A_{n_s n_s'mn}^{[x]} \\) is a function of solution vectors \\( u_{n_k m}^{[k]} \\) and \\( u_{n_m m}^{[t]} \\) since the coefficient matrix \\( C_{mn}^{[x]} \\) depends on these solution vectors as shown in Eq. 34. This dependency reflects the interconnected nature of the variables across different dimensions in the S-P-T framework, highlighting how the spatial, parameter, and temporal components influence each other through the coefficients. As a result, Eq. 33 can be further simplified as follows:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.44,
+ 0.482,
+ 0.893,
+ 0.503
+ ],
+ "angle": 0,
+ "content": "\\[\n\\delta u _ {n _ {x} m} ^ {[ x ]} A _ {n _ {x} n _ {x} ^ {\\prime} m n} ^ {[ x ]} u _ {n _ {x} ^ {\\prime} n} ^ {[ x ]} \\tag {37}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.103,
+ 0.513,
+ 0.893,
+ 0.561
+ ],
+ "angle": 0,
+ "content": "where the summation signs are neglected since \\(m\\) and \\(n\\) become dummy variables. The 4-th order tensor \\(A_{n_x n_x' mn}^{[x]}\\) can be reshaped as a 2nd order tensor \\(\\mathbb{A}_{IJ}^{[x]}\\): the indices \\(n_x\\) and \\(m\\) are combined into a single composite index \\(I\\), and the indices \\(n_x'\\) and \\(n\\) are combined into a single composite index \\(J\\)."
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.448,
+ 0.572,
+ 0.892,
+ 0.592
+ ],
+ "angle": 0,
+ "content": "\\[\nA _ {n _ {x} n _ {x} ^ {\\prime} m n} ^ {[ x ]} = \\mathbb {A} _ {I J} ^ {[ x ]} \\tag {38}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.104,
+ 0.596,
+ 0.343,
+ 0.609
+ ],
+ "angle": 0,
+ "content": "Define the following vectorization:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.405,
+ 0.632,
+ 0.555,
+ 0.653
+ ],
+ "angle": 0,
+ "content": "\\[\n\\delta \\mathbb {U} _ {I} ^ {[ x ]} = \\left[ \\operatorname {v e c} \\left(\\delta u _ {n _ {x} m} ^ {[ x ]}\\right) \\right] _ {I}\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.426,
+ 0.655,
+ 0.892,
+ 0.675
+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathbb {U} _ {J} ^ {[ x ]} = \\left[ \\operatorname {v e c} \\left(u _ {n _ {x} ^ {\\prime} n} ^ {[ x ]}\\right) \\right] _ {J} \\tag {39}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.104,
+ 0.684,
+ 0.344,
+ 0.698
+ ],
+ "angle": 0,
+ "content": "As a result, Eq. 37 is equivalent to:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.449,
+ 0.71,
+ 0.892,
+ 0.727
+ ],
+ "angle": 0,
+ "content": "\\[\n\\delta \\mathbb {U} ^ {[ x ] T} \\mathbb {A} ^ {[ x ]} \\mathbb {U} ^ {[ x ]} \\tag {40}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.102,
+ 0.732,
+ 0.894,
+ 0.791
+ ],
+ "angle": 0,
+ "content": "Following the same procedure, we can obtain matrix forms corresponding to the time derivative term \\(\\delta \\mathbb{U}^{[x]^T}\\mathbb{B}^{[x]}\\mathbb{U}^{[x]}\\), and the forcing term \\(\\delta \\mathbb{U}^{[x]^T}\\mathbb{Q}^{[x]}\\) for the spatial variational part of Eq. 30. Similar structures can also be obtained for the parametric and temporal variational parts of the test function in Eq. 29. Denoting \\(\\mathbb{K}^{[d]} = \\mathbb{A}^{[d]} + \\mathbb{B}^{[d]}\\), the matrix form of the generalized S-P-T Galerkin form in Eq. 27 can be written as:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.199,
+ 0.798,
+ 0.892,
+ 0.835
+ ],
+ "angle": 0,
+ "content": "\\[\n\\underbrace {\\delta \\mathbb {U} ^ {[ x ] ^ {T}} \\mathbb {K} ^ {[ x ]} \\mathbb {U} ^ {[ x ]} - \\delta \\mathbb {U} ^ {[ x ] ^ {T}} \\mathbb {Q} ^ {[ x ]}} _ {\\text {s p a t i a l v a r i a t i o n a l p a r t}} + \\underbrace {\\delta \\mathbb {U} ^ {[ k ] ^ {T}} \\mathbb {K} ^ {[ k ]} \\mathbb {U} ^ {[ k ]} - \\delta \\mathbb {U} ^ {[ k ] ^ {T}} \\mathbb {Q} ^ {[ k ]}} _ {\\text {p a r a m e t r i c v a r i a t i o n a l p a r t}} + \\underbrace {\\delta \\mathbb {U} ^ {[ t ] ^ {T}} \\mathbb {K} ^ {[ t ]} \\mathbb {U} ^ {[ t ]} - \\delta \\mathbb {U} ^ {[ t ] ^ {T}} \\mathbb {Q} ^ {[ t ]}} _ {\\text {t e m p o r a l v a r i a t i o n a l p a r t}} = 0 \\tag {41}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.102,
+ 0.843,
+ 0.893,
+ 0.871
+ ],
+ "angle": 0,
+ "content": "Eq. 41 is equivalent to the following nonlinear system of equations. Note that the nonlinearity comes from the fact that \\(\\mathbb{K}^{[d]}\\) is solution dependent:"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.49,
+ 0.879,
+ 0.509,
+ 0.89
+ ],
+ "angle": 0,
+ "content": "12"
+ }
+ ],
+ [
+ {
+ "type": "equation",
+ "bbox": [
+ 0.132,
+ 0.154,
+ 0.894,
+ 0.201
+ ],
+ "angle": 0,
+ "content": "\\[\n\\left[ \\delta \\mathbb {U} ^ {[ x ] ^ {T}}, \\delta \\mathbb {U} ^ {[ k ] ^ {T}}, \\delta \\mathbb {U} ^ {[ t ] ^ {T}} \\right] \\left\\{\\left[ \\begin{array}{c c c} \\mathbb {K} ^ {[ x ]} (\\mathbb {U} ^ {[ k ]}, \\mathbb {U} ^ {[ t ]}) & 0 & 0 \\\\ 0 & \\mathbb {K} ^ {[ k ]} (\\mathbb {U} ^ {[ x ]}, \\mathbb {U} ^ {[ t ]}) & 0 \\\\ 0 & 0 & \\mathbb {K} ^ {[ t ]} (\\mathbb {U} ^ {[ x ]}, \\mathbb {U} ^ {[ k ]}) \\end{array} \\right] \\left[ \\begin{array}{l} \\mathbb {U} ^ {[ x ]} \\\\ \\mathbb {U} ^ {[ k ]} \\\\ \\mathbb {U} ^ {[ t ]} \\end{array} \\right] - \\left[ \\begin{array}{l} \\mathbb {Q} ^ {[ x ]} (\\mathbb {U} ^ {[ k ]}, \\mathbb {U} ^ {[ t ]}) \\\\ \\mathbb {Q} ^ {[ k ]} (\\mathbb {U} ^ {[ x ]}, \\mathbb {U} ^ {[ t ]}) \\\\ \\mathbb {Q} ^ {[ t ]} (\\mathbb {U} ^ {[ x ]}, \\mathbb {U} ^ {[ k ]}) \\end{array} \\right] \\right\\} = 0 \\tag {42}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.102,
+ 0.211,
+ 0.894,
+ 0.299
+ ],
+ "angle": 0,
+ "content": "where we can treat the solution vector \\(\\left[\\mathbb{U}^{[x]^T},\\mathbb{U}^{[k]^T},\\mathbb{U}^{[t]^T}\\right]\\) as generalized DoFs like standard FEM. There are many ways to solve Eq. 42. For example, standard linearization schemes such as Newton's method have been used [29]. However, this method may suffer from ill-conditioning since the mismatch of scales for different dimensions can be significant. In this paper, we use the concept of subspace iteration to efficiently approximate the solution by iterating in the subspace of the test function space until a convergence criteria is met [19]. A similar counterpart has been widely adopted as the gold standard in discrete tensor decomposition [23]."
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.103,
+ 0.313,
+ 0.443,
+ 0.327
+ ],
+ "angle": 0,
+ "content": "2.6. Solution scheme of TAPS: subspace iteration"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.102,
+ 0.33,
+ 0.894,
+ 0.416
+ ],
+ "angle": 0,
+ "content": "For subspace iteration in \\(d\\)-th dimension, only the solution matrix \\(\\mathbb{U}^{[d]}\\) is treated as unknown while all other functions are considered as known constants. Consequently, the variations of the univariate functions other than \\(d\\)-th dimension will vanish. From Eq. 42, it can be seen that this will lead to a linear system of equations for the unknowns in the \\(d\\)-th dimension. The updated solution matrix \\(\\mathbb{U}^{[d]}\\) from this process is then used in the next subspace iteration for dimension \\(d + 1\\) (when \\(d = D\\), we come back to the first dimension \\(d = 1\\)). The complete solution scheme for subspace iteration is shown in Algorithm 1."
+ },
+ {
+ "type": "code_caption",
+ "bbox": [
+ 0.107,
+ 0.429,
+ 0.488,
+ 0.443
+ ],
+ "angle": 0,
+ "content": "Algorithm 1 TAPS solution scheme (subspace iteration)"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.114,
+ 0.445,
+ 0.783,
+ 0.461
+ ],
+ "angle": 0,
+ "content": "1: Initialize solution vector \\(\\mathbb{U}^{[x_1][0]}\\) ..., \\(\\mathbb{U}^{[x_D][0]}\\) with random values and compute \\(\\mathbb{K}^{[x_1][0]}\\), and \\(\\mathbb{Q}^{[x_1][0]}\\)"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.114,
+ 0.462,
+ 0.303,
+ 0.476
+ ],
+ "angle": 0,
+ "content": "2: for iter = 0 to iter_max do"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.114,
+ 0.476,
+ 0.281,
+ 0.488
+ ],
+ "angle": 0,
+ "content": "3: for \\(d = 1\\) to D do"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.114,
+ 0.489,
+ 0.476,
+ 0.502
+ ],
+ "angle": 0,
+ "content": "4: Update iteration number \\(\\mathcal{K} = iter\\times D + d\\)"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.114,
+ 0.502,
+ 0.547,
+ 0.517
+ ],
+ "angle": 0,
+ "content": "5: Solve TD linear system \\(\\mathbb{K}^{[x_d][\\mathcal{K} - 1]}\\mathbb{U}^{[x_d][\\mathcal{K}]} = \\mathbb{Q}^{[x_d][\\mathcal{K} - 1]}\\)"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.114,
+ 0.517,
+ 0.539,
+ 0.532
+ ],
+ "angle": 0,
+ "content": "6: Update matrices \\(\\mathbb{K}^{[x_{d + 1}][\\mathcal{K}]}\\) and force vector \\(\\mathbb{Q}^{[x_{d + 1}][\\mathcal{K}]}\\)"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.114,
+ 0.533,
+ 0.215,
+ 0.545
+ ],
+ "angle": 0,
+ "content": "7: end for"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.114,
+ 0.546,
+ 0.293,
+ 0.561
+ ],
+ "angle": 0,
+ "content": "8: Check convergence"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.114,
+ 0.561,
+ 0.189,
+ 0.573
+ ],
+ "angle": 0,
+ "content": "9: end for"
+ },
+ {
+ "type": "list",
+ "bbox": [
+ 0.114,
+ 0.445,
+ 0.547,
+ 0.573
+ ],
+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.102,
+ 0.592,
+ 0.894,
+ 0.663
+ ],
+ "angle": 0,
+ "content": "To illustrate the details of the subspace iteration algorithm, we consider the \\(\\mathcal{K}\\)-th subspace iteration (which is on spatial variable \\(x\\)). Here, we assume that the parametric and temporal solutions have been updated from the previous \\((\\mathcal{K} - 1)\\)-th iteration, leaving the spatial solution as unknown to be solved in \\(\\mathcal{K}\\)-th iteration. Moreover, instead of the full variation form of the test function as in Eq. 42, we only consider the subspace \\(x\\) of the test function by setting the parametric and temporal variational parts as 0. As a result, we have:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.267,
+ 0.674,
+ 0.892,
+ 0.695
+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathbb {K} ^ {[ x ] [ \\mathcal {K} - 1 ]} \\left(\\mathbb {U} ^ {[ k ] [ \\mathcal {K} - 1 ]}, \\mathbb {U} ^ {[ t ] [ \\mathcal {K} - 1 ]}\\right) \\mathbb {U} ^ {[ x ] [ \\mathcal {K} ]} = \\mathbb {Q} ^ {[ x ] [ \\mathcal {K} - 1 ]} \\left(\\mathbb {U} ^ {[ k ] [ \\mathcal {K} - 1 ]}, \\mathbb {U} ^ {[ t ] [ \\mathcal {K} - 1 ]}\\right) \\tag {43}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.102,
+ 0.701,
+ 0.894,
+ 0.786
+ ],
+ "angle": 0,
+ "content": "which is a linear system of equations with unknown \\(\\mathbb{U}^{[x][\\mathcal{K}]}\\). This is a general Sylvester equation which can be solved using many efficient solution schemes [30, 31]. In this paper, sparse direct solvers based on fast diagonalization/complex Schur decomposition methods are adopted [32]. The computational complexity of the sparse direct solver is \\(O(M^3 + M^2 n_d + C_c(n_d))\\) for the \\(d\\)-th dimension subspace iteration, where \\(M\\) is the total number of modes; \\(n_d\\) is the number of grid points for \\(d\\)-th dimension; \\(C_c(n_d)\\) refers to the computational cost of the banded sparse mass/stiffness matrix for \\(d\\)-th dimension with a shape of \\((n_d \\times n_d)\\)."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.102,
+ 0.785,
+ 0.894,
+ 0.829
+ ],
+ "angle": 0,
+ "content": "Once \\(\\mathbb{U}^{[x][\\mathcal{K}]}\\) is obtained, we then update matrix \\(\\mathbb{K}^{[k][\\mathcal{K}]}(\\mathbb{U}^{[x][\\mathcal{K}]},\\mathbb{U}^{[t][\\mathcal{K}]})\\) and forcing vector \\(\\mathbb{Q}^{[k][\\mathcal{K}]}(\\mathbb{U}^{[x][\\mathcal{K}]},\\mathbb{U}^{[t][\\mathcal{K}]})\\). In the next iteration (for dimension \\(k\\)), we treat \\(\\mathbb{U}^{[k][\\mathcal{K} + 1]}\\) as the only unknown. Subspace iteration will continue unless the relative change of all solution matrices (for example, \\(L_{2}\\) norm) is within the tolerance."
+ },
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.49,
+ 0.879,
+ 0.508,
+ 0.89
+ ],
+ "angle": 0,
+ "content": "13"
+ }
+ ],
+ [
+ {
+ "type": "title",
+ "bbox": [
+ 0.104,
+ 0.132,
+ 0.308,
+ 0.146
+ ],
+ "angle": 0,
+ "content": "2.7. Error estimates of TAPS"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.103,
+ 0.15,
+ 0.894,
+ 0.192
+ ],
+ "angle": 0,
+ "content": "Since the TAPS solution is based on the C-HiDeNN-TD approximation and the generalized Galerkin formulation, we can have the following theoretical results on the error bounds, as demonstrated in our previous work on C-HiDeNN [6]:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.322,
+ 0.192,
+ 0.892,
+ 0.208
+ ],
+ "angle": 0,
+ "content": "\\[\n\\left\\| u ^ {\\mathrm {C} - \\mathrm {H i D e N N}} - u ^ {\\mathrm {e x}} \\right\\| _ {E} \\leq \\left\\| u ^ {\\mathrm {T A P S}} - u ^ {\\mathrm {e x}} \\right\\| _ {E} \\leq \\left\\| u ^ {\\mathrm {F E M}} - u ^ {\\mathrm {e x}} \\right\\| _ {E} \\tag {44}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.103,
+ 0.215,
+ 0.894,
+ 0.314
+ ],
+ "angle": 0,
+ "content": "where \\(\\| \\cdot \\|_E\\) denotes the energy norm, \\(u^{\\mathrm{ex}}\\) denotes the exact solution, \\(u^{\\mathrm{C - HiDeNN}}\\) denotes the solution obtained by the full C-HiDeNN method without tensor decomposition, \\(u^{\\mathrm{TAPS}}\\) denotes the TAPS solution with a sufficient number of modes, \\(u^{\\mathrm{FEM}}\\) denotes the FEM solution. The proof of the above results is based on the fact that the full C-HiDeNN approximation can provide a larger function space and therefore more accurate solutions than conventional FEM [6]. The subspace iteration can be considered as a local (directional) version of the Galerkin formulation and is expected to enable an optimized solution for the tensor decomposition that will converge to the Galerkin-based full C-HiDeNN method."
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.104,
+ 0.336,
+ 0.183,
+ 0.349
+ ],
+ "angle": 0,
+ "content": "3. Results"
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.104,
+ 0.361,
+ 0.428,
+ 0.376
+ ],
+ "angle": 0,
+ "content": "3.1. Convergence study for moving heat source"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.103,
+ 0.379,
+ 0.894,
+ 0.435
+ ],
+ "angle": 0,
+ "content": "In this section, we first analyze the convergence of the TAPS solver for a space-time (S-T) transient heat transfer problem. A single NVIDIA RTX A6000 GPU is used for all the following analyses. In Eq. 18, we let \\(\\rho c_{p} = 1\\), \\(k = 1\\), and replace the heat source as shown in Eq. 45. In this example, we have the spatial variable \\(x_{s} = (x,y,z)\\) and the temporal variable \\(x_{t} = t\\)."
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.267,
+ 0.447,
+ 0.891,
+ 0.523
+ ],
+ "angle": 0,
+ "content": "\\[\n\\begin{array}{l} f \\left(\\boldsymbol {x} _ {s}, x _ {t}\\right) = 2 (1 - 2 y ^ {2}) \\left(1 - e ^ {- 1 5 t}\\right) e ^ {- y ^ {2} - (1 0 0 t - x - 5) ^ {2}} \\\\ + 2 (1 - 2 (1 0 0 t - x - 5) ^ {2}) \\left(1 - e ^ {- 1 5 t}\\right) e ^ {- y ^ {2} - (1 0 0 t - x - 5) ^ {2}} + (1 \\tag {45} \\\\ - e ^ {- 1 5 t}) (2 0 0 x + 1 0 0 0 - 2 0 0 0 0 t) e ^ {- y ^ {2} - (1 0 0 t - x - 5) ^ {2}} \\\\ - 1 5 e ^ {- 1 5 t} e ^ {- y ^ {2} - (1 0 0 t - x - 5) ^ {2}} \\\\ \\end{array}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.104,
+ 0.532,
+ 0.53,
+ 0.546
+ ],
+ "angle": 0,
+ "content": "The analytical solution to the PDE is inherently non-separable."
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.374,
+ 0.557,
+ 0.892,
+ 0.574
+ ],
+ "angle": 0,
+ "content": "\\[\nu ^ {\\mathrm {e x}} \\left(\\boldsymbol {x} _ {s}, x _ {t}\\right) = (1 - e ^ {- 1 5 t}) e ^ {- y ^ {2} - (x - 1 0 0 t - 5) ^ {2}} \\tag {46}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.104,
+ 0.586,
+ 0.377,
+ 0.599
+ ],
+ "angle": 0,
+ "content": "The initial and boundary conditions are:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.407,
+ 0.613,
+ 0.491,
+ 0.625
+ ],
+ "angle": 0,
+ "content": "\\[\nu \\left(x _ {s}, 0\\right) = 0,\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.407,
+ 0.624,
+ 0.892,
+ 0.647
+ ],
+ "angle": 0,
+ "content": "\\[\n\\left. u \\left(\\boldsymbol {x} _ {s}, x _ {t}\\right) \\right| _ {\\partial \\Omega} = \\left. u ^ {\\mathrm {e x}} \\left(\\boldsymbol {x} _ {s}, x _ {t}\\right) \\right| _ {\\partial \\Omega}. \\tag {47}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.104,
+ 0.652,
+ 0.377,
+ 0.665
+ ],
+ "angle": 0,
+ "content": "The relative \\(L_{2}\\) norm error is defined as:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.36,
+ 0.676,
+ 0.892,
+ 0.71
+ ],
+ "angle": 0,
+ "content": "\\[\n\\epsilon_ {L _ {2}} = \\frac {\\| u ^ {T D} \\left(\\boldsymbol {x} _ {s} , x _ {t}\\right) - u ^ {\\mathrm {e x}} \\left(\\boldsymbol {x} _ {s} , x _ {t}\\right) \\| _ {L _ {2} \\left(\\Omega_ {\\boldsymbol {x} _ {s}} \\otimes \\Omega_ {x _ {t}}\\right)}}{\\| u ^ {\\mathrm {e x}} \\left(\\boldsymbol {x} _ {s} , x _ {t}\\right) \\| _ {L _ {2} \\left(\\Omega_ {\\boldsymbol {x} _ {s}} \\otimes \\Omega_ {x _ {t}}\\right)}} \\tag {48}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.103,
+ 0.719,
+ 0.894,
+ 0.861
+ ],
+ "angle": 0,
+ "content": "First, we investigate the influence of the number of subspace iterations. As shown in Fig. 8(a), 3 iterations are enough to obtain an accurate result. Next, we investigate the convergence in terms of the number of modes. Here we compare the relative \\( L_{2} \\) norm error for both TAPS and proper generalized decomposition (PGD) methods [17, 18]. To this aim, we use the same discretization for the space-time domain with each dimension discretized by 100 grid points, the same reproducing polynomial order \\( p = 1 \\) and convolution patch size \\( s = 1 \\). As can be seen from Fig. 8(b), TAPS requires a much smaller number of modes than PGD. For TAPS, when the number of modes equals 25, the relative \\( L_{2} \\) norm error decreases to \\( 2.5 \\times 10^{-3} \\). The total solution time is 15.2 s. However, PGD requires 1,000 modes which takes 60.6 s solution time to reach the same level of accuracy. This is because the test function space in PGD is a subspace of TAPS [29]. Furthermore, the modal decomposition obtained from PGD is not optimal and thus requires a larger storage requirement due to the increased number of modes."
+ },
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.49,
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+ ],
+ "angle": 0,
+ "content": "14"
+ }
+ ],
+ [
+ {
+ "type": "image",
+ "bbox": [
+ 0.184,
+ 0.134,
+ 0.506,
+ 0.282
+ ],
+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 0.375,
+ 0.286,
+ 0.395,
+ 0.298
+ ],
+ "angle": 0,
+ "content": "(a)"
+ },
+ {
+ "type": "image",
+ "bbox": [
+ 0.514,
+ 0.133,
+ 0.811,
+ 0.285
+ ],
+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 0.679,
+ 0.287,
+ 0.699,
+ 0.298
+ ],
+ "angle": 0,
+ "content": "(b)"
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 0.197,
+ 0.315,
+ 0.799,
+ 0.327
+ ],
+ "angle": 0,
+ "content": "Figure 8: Relative L2 norm error with respect to (a) the number of subspace iterations (b) the number of modes"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.102,
+ 0.348,
+ 0.894,
+ 0.433
+ ],
+ "angle": 0,
+ "content": "The spatial and temporal convergence are also studied. In Fig. 9(a), the number of temporal nodes is fixed as 500, and the spatial mesh is refined. It shows the relative \\( L_{2} \\) norm error with respect to the number of nodes along each spatial dimension. As can be readily seen from the figure, larger patch size \\( s \\) leads to smaller error given the same reproducing polynomial orders \\( p \\). Moreover, we can adjust \\( p \\) to control the spatial convergence rate. Similarly, Fig. 9(b) demonstrates the convergence rate in the temporal domain where we fix the spatial discretization as 500 along each spatial dimension. By adjusting \\( s \\) and \\( p \\), we can obtain different temporal convergence rates."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.102,
+ 0.434,
+ 0.894,
+ 0.476
+ ],
+ "angle": 0,
+ "content": "Finally, we refine the spatial and temporal mesh simultaneously and study the spatio-temporal convergence rate in Fig. 9(c). As can be observed from the figure, higher reproducing polynomial order \\( p \\) will lead to a higher-order convergence rate."
+ },
+ {
+ "type": "image",
+ "bbox": [
+ 0.111,
+ 0.49,
+ 0.382,
+ 0.646
+ ],
+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 0.259,
+ 0.647,
+ 0.277,
+ 0.657
+ ],
+ "angle": 0,
+ "content": "(a)"
+ },
+ {
+ "type": "image",
+ "bbox": [
+ 0.387,
+ 0.49,
+ 0.621,
+ 0.647
+ ],
+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 0.511,
+ 0.647,
+ 0.529,
+ 0.657
+ ],
+ "angle": 0,
+ "content": "(b)"
+ },
+ {
+ "type": "image",
+ "bbox": [
+ 0.624,
+ 0.491,
+ 0.885,
+ 0.646
+ ],
+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 0.764,
+ 0.646,
+ 0.78,
+ 0.656
+ ],
+ "angle": 0,
+ "content": "(c)"
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 0.103,
+ 0.673,
+ 0.893,
+ 0.697
+ ],
+ "angle": 0,
+ "content": "Figure 9: Relative \\( L_{2} \\) norm error with respect to the number of grid points (a) spatial convergence (b) temporal convergence (c) spatio-temporal convergence"
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.103,
+ 0.721,
+ 0.707,
+ 0.735
+ ],
+ "angle": 0,
+ "content": "3.2. Convergence study of \\( S-P-T \\) problems up to equivalent zetta-scale (10\\(^{21}\\)) full models"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.102,
+ 0.737,
+ 0.894,
+ 0.78
+ ],
+ "angle": 0,
+ "content": "In this example, we study the convergence of the TAPS solver for the time-dependent parametric heat transfer problem in a S-P-T setting. In Eq. 18, we adopt the heat source as shown in Eq. 49. In this example, we have spatial variable \\( \\boldsymbol{x}_s = (x,y,z) \\), parametric variable \\( \\boldsymbol{x}_p = (k,P,\\rho ,c_p) \\) and temporal variable \\( x_{t} = t \\)."
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.241,
+ 0.789,
+ 0.892,
+ 0.832
+ ],
+ "angle": 0,
+ "content": "\\[\n\\begin{array}{l} f \\left(\\boldsymbol {x} _ {s}, \\boldsymbol {x} _ {p}, x _ {t}\\right) = 1 5 \\rho^ {2} c _ {p} ^ {2} k p e ^ {- 1 5 k t} e ^ {- 2 5. 0 x ^ {2} - 2 5. 0 y ^ {2}} \\\\ + 5 0 \\rho c _ {p} k p \\left(1 - e ^ {- 1 5 k t}\\right) e ^ {- 2 5. 0 x ^ {2} - 2 5. 0 y ^ {2}} \\left[ \\left(1 - 5 0 x ^ {2}\\right) + \\left(1 - 5 0 y ^ {2}\\right) \\right] \\tag {49} \\\\ \\end{array}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.103,
+ 0.833,
+ 0.53,
+ 0.848
+ ],
+ "angle": 0,
+ "content": "The analytical solution to the PDE is inherently non-separable."
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.343,
+ 0.854,
+ 0.892,
+ 0.873
+ ],
+ "angle": 0,
+ "content": "\\[\nu ^ {\\mathrm {e x}} \\left(\\boldsymbol {x} _ {s}, \\boldsymbol {x} _ {p}, x _ {t}\\right) = \\rho c _ {p} P \\left(1 - e ^ {- 1 5 k t}\\right) e ^ {- 2 5. 0 x ^ {2} - 2 5. 0 y ^ {2}} \\tag {50}\n\\]"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.491,
+ 0.879,
+ 0.508,
+ 0.89
+ ],
+ "angle": 0,
+ "content": "15"
+ }
+ ],
+ [
+ {
+ "type": "text",
+ "bbox": [
+ 0.104,
+ 0.133,
+ 0.378,
+ 0.147
+ ],
+ "angle": 0,
+ "content": "The initial and boundary conditions are:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.378,
+ 0.158,
+ 0.49,
+ 0.173
+ ],
+ "angle": 0,
+ "content": "\\[\nu \\left(\\boldsymbol {x} _ {s}, \\boldsymbol {x} _ {p}, 0\\right) = 0\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.38,
+ 0.177,
+ 0.617,
+ 0.196
+ ],
+ "angle": 0,
+ "content": "\\[\nu \\left(\\boldsymbol {x} _ {s}, \\boldsymbol {x} _ {p}, x _ {t}\\right) | _ {\\partial \\Omega} = u ^ {\\mathrm {e x}} \\left(\\boldsymbol {x} _ {s}, \\boldsymbol {x} _ {p}, x _ {t}\\right) | _ {\\partial \\Omega}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.104,
+ 0.201,
+ 0.38,
+ 0.214
+ ],
+ "angle": 0,
+ "content": "The relative \\(L_{2}\\) norm error is defined as:"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.319,
+ 0.223,
+ 0.894,
+ 0.261
+ ],
+ "angle": 0,
+ "content": "\\[\n\\epsilon_ {L _ {2}} = \\frac {\\left\\| u ^ {T D} \\left(\\boldsymbol {x} _ {s} , \\boldsymbol {x} _ {p} , x _ {t}\\right) - u ^ {\\mathrm {e x}} \\left(\\boldsymbol {x} _ {s} , \\boldsymbol {x} _ {p} , x _ {t}\\right) \\right\\| _ {L _ {2} \\left(\\Omega_ {x _ {s}} \\otimes \\Omega_ {x _ {p}} \\otimes \\Omega_ {x _ {t}}\\right)}}{\\left\\| u ^ {\\mathrm {e x}} \\left(\\boldsymbol {x} _ {s} , \\boldsymbol {x} _ {p} , x _ {t}\\right) \\right\\| _ {L _ {2} \\left(\\Omega_ {x _ {s}} \\otimes \\Omega_ {x _ {p}} \\otimes \\Omega_ {x _ {t}}\\right)}} \\tag {51}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.102,
+ 0.269,
+ 0.894,
+ 0.384
+ ],
+ "angle": 0,
+ "content": "To study the convergence of TAPS for S-P-T problems, the number of grid points is refined simultaneously in each dimension and corresponding relative \\( L_{2} \\) norm errors are computed as shown in Fig. 10. When the number of grid points in each dimension is 450, the equivalent DoFs of a full model achieves \\( 450^{8} = 1.68 \\times 10^{21} \\). Consequently, it is equivalent to a zetta-scale \\( (10^{21}) \\) full problem. As can be seen from the figure, a larger patch size \\( s \\) leads to a smaller error and faster convergence. A higher reproducing polynomial order \\( p \\) also leads to a higher convergence rate. It can be noticed that the convergence rate for \\( p = 3 \\) case is smaller than expected \\( p + 1 = 4 \\). This is attributed to the fact that the S-P-T mesh is not fine enough. However, due to the rounding error in computing the relative \\( L_{2} \\) norm error, we can only accurately compute the error up to 450 grid points per dimension."
+ },
+ {
+ "type": "image",
+ "bbox": [
+ 0.345,
+ 0.397,
+ 0.657,
+ 0.598
+ ],
+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 0.243,
+ 0.61,
+ 0.754,
+ 0.624
+ ],
+ "angle": 0,
+ "content": "Figure 10: Relative \\( L_{2} \\) norm error with respect to the number of grid points in each dimension"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.102,
+ 0.637,
+ 0.895,
+ 0.75
+ ],
+ "angle": 0,
+ "content": "In summary, we have the flexibility to choose different \\( s \\) and \\( p \\) to control the accuracy of TAPS by directly solving the S-P-T PDE. This is different from other data-driven modeling approaches (for instance, neural networks-based data-driven methods) in two notable ways. First, unlike a black-box neural network interpolator where the accuracy of the model is not guaranteed, our method is built upon the AI-enhanced finite element method, and we can control the convergence rate by choosing suitable hyperparameters \\( s \\) and \\( p \\). Second, unlike most data-driven reduced-order models for physical problems, our method directly solves the governing PDE by plugging in the TD interpolation without seeing any training data. As a result, we can avoid the most expensive offline data generation stage as opposed to data-driven methods."
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.104,
+ 0.764,
+ 0.548,
+ 0.779
+ ],
+ "angle": 0,
+ "content": "3.3. Moving source with solution dependent material parameters"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.102,
+ 0.782,
+ 0.895,
+ 0.825
+ ],
+ "angle": 0,
+ "content": "In this section, we model moving heat sources using temperature-dependent material parameters. The solution scheme of this problem is provided in detail in Appendix A. Figure 11(a) illustrates a typical representation of temperature-dependent heat conductivity and capacity for Inconel 718 [33]."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.103,
+ 0.825,
+ 0.894,
+ 0.854
+ ],
+ "angle": 0,
+ "content": "Since the temperature dependency of \\( k(u) \\) and \\( \\rho c_{p}(u) \\) can be approximated using a linear relationship. As a result, we can directly rewrite \\( k(u(\\boldsymbol{x}_s,x_t)) \\) and \\( \\rho c_{p}(u(\\boldsymbol{x}_{s},x_{t})) \\) in the TD format."
+ },
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.49,
+ 0.879,
+ 0.51,
+ 0.89
+ ],
+ "angle": 0,
+ "content": "16"
+ }
+ ],
+ [
+ {
+ "type": "image",
+ "bbox": [
+ 0.187,
+ 0.144,
+ 0.501,
+ 0.287
+ ],
+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 0.324,
+ 0.289,
+ 0.349,
+ 0.302
+ ],
+ "angle": 0,
+ "content": "(a)"
+ },
+ {
+ "type": "image",
+ "bbox": [
+ 0.527,
+ 0.178,
+ 0.812,
+ 0.281
+ ],
+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 0.626,
+ 0.285,
+ 0.65,
+ 0.301
+ ],
+ "angle": 0,
+ "content": "(b)"
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 0.102,
+ 0.318,
+ 0.894,
+ 0.342
+ ],
+ "angle": 0,
+ "content": "Figure 11: (a) Temperature dependent material properties for Inconel 718 [33] (b) Schematic of numerical simulation, where the solution along the center line is compared for FEM and TAPS."
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.317,
+ 0.388,
+ 0.674,
+ 0.426
+ ],
+ "angle": 0,
+ "content": "\\[\nk \\left(\\boldsymbol {x} _ {s}, x _ {t}\\right) \\approx \\sum_ {m = 1} ^ {M} m _ {k} u _ {x _ {1}} ^ {(m)} \\left(x _ {1}\\right) u _ {x _ {2}} ^ {(m)} \\left(x _ {2}\\right) u _ {x _ {3}} ^ {(m)} \\left(x _ {3}\\right) u _ {x _ {t}} ^ {(m)} \\left(x _ {t}\\right) + n _ {k}\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.288,
+ 0.429,
+ 0.893,
+ 0.466
+ ],
+ "angle": 0,
+ "content": "\\[\n\\rho c _ {p} \\left(\\boldsymbol {x} _ {s}, x _ {t}\\right) \\approx \\sum_ {m = 1} ^ {M} m _ {c _ {p}} u _ {x _ {1}} ^ {(m)} \\left(x _ {1}\\right) u _ {x _ {2}} ^ {(m)} \\left(x _ {2}\\right) u _ {x _ {3}} ^ {(m)} \\left(x _ {3}\\right) u _ {x _ {t}} ^ {(m)} \\left(x _ {t}\\right) + n _ {c _ {p}} \\tag {52}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.103,
+ 0.477,
+ 0.893,
+ 0.507
+ ],
+ "angle": 0,
+ "content": "where \\(M\\) is the decomposition modes of the TAPS solution; \\(m_{k} = 1.52 \\times 10^{-5} \\mathrm{~W} / (\\mathrm{mmK}^{2})\\); \\(n_{k} = 5.29 \\times 10^{-3} \\mathrm{~W} / (\\mathrm{mmK})\\); \\(m_{c_{p}} = 6.11 \\times 10^{-7} \\mathrm{~mm}^{-3} \\mathrm{~K}^{-2}\\); \\(n_{c p} = 3.25 \\times 10^{-3} \\mathrm{~mm}^{-3} \\mathrm{~K}^{-1}\\)"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.103,
+ 0.507,
+ 0.894,
+ 0.576
+ ],
+ "angle": 0,
+ "content": "The problem setup is shown in Fig. 11 (b). The spatial domain size is \\(10\\mathrm{mm} \\times 10\\mathrm{mm} \\times 1\\mathrm{mm}\\) where homogeneous Dirichlet boundary conditions are assumed for the left and right surfaces; homogeneous Neumann boundary conditions are applied to all other surfaces. As shown in Eq. 19, a moving Gaussian source term \\(f(\\boldsymbol{x}_s, t)\\) is applied as a volumetric source term with a radius \\(r = 0.5\\mathrm{mm}\\) and moving velocity \\(500\\mathrm{mm/s}\\). The diameter is discretized using 10 spatial elements."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.103,
+ 0.577,
+ 0.894,
+ 0.635
+ ],
+ "angle": 0,
+ "content": "Since there is no analytical solution available to this problem, we use implicit finite element analysis as the baseline for validation. JAX-FEM [34] is used to generate the nonlinear FEM solution. For ease of comparison, we use the same time increment as \\(1.60 \\times 10^{-4}\\) sec for both TAPS and FEM. The solution along the center line, as shown in Fig. 11 (b) is compared. As can be seen from Fig. 12, the result of the nonlinear TAPS solver agrees well with FEM."
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.104,
+ 0.648,
+ 0.332,
+ 0.662
+ ],
+ "angle": 0,
+ "content": "3.4. Simulation of LPBF process"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.102,
+ 0.666,
+ 0.894,
+ 0.71
+ ],
+ "angle": 0,
+ "content": "In this section, we use TAPS to efficiently model the laser powder bed fusion process (LPBF) in additive manufacturing. Here we only consider the free convection term in the Neumann boundary condition. The initial condition can be considered by splitting the total solution as a summation of the homogeneous part and the inhomogeneous part."
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.392,
+ 0.723,
+ 0.892,
+ 0.739
+ ],
+ "angle": 0,
+ "content": "\\[\nu \\left(\\boldsymbol {x} _ {s}, x _ {t}\\right) = u _ {0} \\left(\\boldsymbol {x} _ {s}, x _ {t}\\right) + u _ {\\text {i n i t}} \\left(\\boldsymbol {x} _ {s}\\right) \\tag {53}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.103,
+ 0.745,
+ 0.893,
+ 0.773
+ ],
+ "angle": 0,
+ "content": "As a result, \\( u_{0}(\\pmb{x}_{s}, x_{t}) \\) is subject to homogeneous initial conditions. In this section, we assume Ti-6Al-4V is used as the powder bed materials. The detailed material parameters can be found in Table 4."
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.104,
+ 0.787,
+ 0.312,
+ 0.801
+ ],
+ "angle": 0,
+ "content": "3.4.1. Single-track simulation"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.102,
+ 0.802,
+ 0.894,
+ 0.86
+ ],
+ "angle": 0,
+ "content": "In this example, we investigate the computational complexity numerically for single-track LPBF simulation with a single S-T slab, as shown in Fig. 13 (a). A single NVIDIA RTX A6000 GPU is used for all the following analyses. To ensure accuracy, the number of modes is adopted as 5 times larger than the number of time steps in the following examples. In the first case, within the S-T slab, the spatial mesh is refined uniformly along each spatial dimension"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.49,
+ 0.879,
+ 0.509,
+ 0.89
+ ],
+ "angle": 0,
+ "content": "17"
+ }
+ ],
+ [
+ {
+ "type": "image",
+ "bbox": [
+ 0.34,
+ 0.131,
+ 0.65,
+ 0.337
+ ],
+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 0.227,
+ 0.353,
+ 0.768,
+ 0.366
+ ],
+ "angle": 0,
+ "content": "Figure 12: Comparison of Nonlinear TAPS solution versus finite element solution at different times."
+ },
+ {
+ "type": "table_caption",
+ "bbox": [
+ 0.383,
+ 0.389,
+ 0.615,
+ 0.398
+ ],
+ "angle": 0,
+ "content": "Table 4: Parameters used in the simulation"
+ },
+ {
+ "type": "table",
+ "bbox": [
+ 0.106,
+ 0.398,
+ 0.894,
+ 0.487
+ ],
+ "angle": 0,
+ "content": "| Variables | Description |
| uh(x) | Interpolated scalar field defined inside of an element |
| Ae | Nodes within element e |
| As | Nodes within patch domain of element e |
| WiS,a,p,j(x) | Convolution patch function at node j for i-th nodal patch with hyperparameters s, a, and p |
| M | Total number of modes in tensor decomposition (TD) |
| m | Index for mode |
| D | Total number of dimensions |
| d | Index for dimension |
| x | Independent variable which includes spatial variable xs, parametric variable xp and temporal variable xt |
| Nd(xd;ad,sd,pd) | Global C-HiDeNN shape function for dimension d with dilation parameter ad, patch size sd and reproducing polynomial order pd |
| b | Source function in laser powder bed fusion process |
| uTD | Approximation of the solution field expressed via TD |
| T | Time slab index for space-parameter-time problem |
| k | Thermal conductivity |
| ρ | Material density |
| cp | Heat capacity |
| η | Material absorptivity |
| P | Laser power |
| r | Standard deviation that characterizes the width of the heat source |
| q | Heat flux |
| qconv | Heat flux from convection |
| qrad | Heat flux from radiation |
| qevap | Heat flux from evaporative cooling |
| hconv | Convection coefficient |
| σSB | Stefan-Boltzman constant |
| mevap | Mass evaporation flux |
| Levap | Heat of evaporation |
| Bnd | Shape function derivative |
| Ud | Solution matrix (Rnd×M) for dimension d that contains all the modes |
+
+To fulfill the ever-growing challenges in predictive scientific models, data-driven surrogates, especially artificial intelligence (AI)-based models, present an alternative to conventional numerical models by significantly reducing the forward prediction time. These models can be treated as a reasonably accurate, reduced representation of real physics. Once trained properly, they can be used for fast prediction on unseen parameters [1, 2]. However, it is still uncertain whether a data-driven surrogate model can be trained to achieve the level of accuracy required in engineering design. Recently, it has been pointed out by Wolfram Research that standard AI models cannot easily fulfill the high accuracy requirement of predictive scientific tasks [3]. Furthermore, as suggested by Google Deepmind, the real potential of AI models lies in enhancing, rather than thoroughly replacing, well-established classical numerical algorithms [4]. In addition, the current standard data-driven approaches follow an offline-online scheme, where the offline stage involves a huge amount of training data, which can again be prohibitive. For problems with known physics, this data can be obtained by running multiple expensive simulations relying on standard numerical algorithms. In scenarios involving high-dimensional design spaces governed by parameterized partial differential equations (PDEs), such as in additive manufacturing (AM), conducting repetitive simulations with varying parameters in this offline stage becomes exceedingly expensive both in terms of computation time and data storage.
+
+To avoid the prohibitive offline stage, one can try to obtain a surrogate model directly from governing equations
+
+
+Figure 1: The parameterized PDE is a PDE that includes parameters $\mathbf{x}_p$ that can vary and influence the solution $\mathbf{u}(\mathbf{x}_s, \mathbf{x}_p, x_t)$ , where $\mathbf{x}_s$ and $x_t$ are the spatial and time variables, respectively. The a priori approach directly finds a surrogate model from the governing parameterized PDE, whereas the data-driven approach has to solve the parameter-fixed PDE on sampled parameters to generate simulation data, followed by training tasks. FEM: Finite Element Method [5], C-HiDeNN: Convolution Hierarchical Deep-learning Neural Network [6], TAPS: Tensor-decomposition-based A Priori Surrogate, PINN: Physics Informed Neural Network [7].
+
+without generating any data. As shown in Fig. 1 denoted by the words "A Priori", this approach aims to find the surrogate model before actually "seeing" any data. For example, multilayer perceptron (MLP) architectures have been vastly used in physics-informed neural networks (PINNs) and their variations to approximate solutions to PDEs without requiring data [7, 8]. However, the results of these efforts are underwhelming, as it has been shown that PINN results have often been compared to weak baselines [9], and it is unclear if they guarantee convergence. Moreover, this method is still susceptible to high computational costs for both large-scale and high-dimensional problems [10].
+
+
+Figure 2: Development history of INN [11]. Figures are borrowed from references: HiDeNN [12], HiDeNN-TD [13], C-HiDeNN [14], C-HiDeNN-TD [6].
+
+Instead of developing solvers solely based on machine learning concepts, a new class of Hierarchical Deep-learning Neural Networks (HiDeNN) has been developed recently. This network architecture incorporates principles from the finite element method (FEM) to construct their architecture [15, 12]. Originally designed to advance FEM as opposed to solve parameterized PDEs, this approach significantly enhances computational accuracy and efficiency for both linear and nonlinear problems compared to standard FEM [16]. HiDeNN was then enhanced by adding an additional hidden layer in the form of a nonlinear convolutional filter, formulating a new neural network architecture named Convolutional HiDeNN (C-HiDeNN) [6, 14]. C-HiDeNN mimics the structure of the generalized finite element method but leverages machine learning to optimize its hyperparameters to further improve accuracy and efficiency. Arbitrary orders of convergence have been observed for C-HiDeNN despite utilizing a linear finite element mesh [14]. Although these methods offers greater accuracy with fewer DoFs, like FEM, they still encounter computational challenges such as balancing memory usage against mesh resolution, which limits their efficiency in modeling ultra large-scale and high-dimensional problems. Therefore, it becomes necessary to employ model order reduction
+
+techniques to address these limitations.
+
+Model order reduction techniques have been widely used to tackle the ever-growing challenges from high-dimensional and large-scale problems. For example, proper generalized decomposition (PGD) [17, 18, 19] has been proposed to efficiently solve high-dimensional PDEs. Recently, tensor decomposition (TD) has been successfully leveraged within the HiDeNN framework. For example, Zhang showed that HiDeNN combined with TD (HiDeNN-TD) significantly improved the speed of HiDeNN while maintaining higher accuracy [13]. Li proposed C-HiDeNN combined with TD (C-HiDeNN-TD) for extremely large-scale nested topology optimization problems [20]. Recently, Park generalized the HiDeNN-family networks under the umbrella of Interpolating Neural Networks (INNs) and demonstrated that the network can be used for both data-driven learning and data-free (i.e., a priori) solving [11]. The development history of HiDeNN family networks and INN is summarized in Fig. 2. While INN clearly explains how to construct the network architecture, an efficient optimization scheme for solving ultra large-scale and high-dimensional problems remains underdeveloped. In this paper, ultra large-scale problems refer to problems on the zetta-scale $(10^{21})$ in terms of DoFs.
+
+The demand for high-precision engineering simulations and efficient solution schemes highlights the need for innovative modeling approaches that swiftly solve large-scale problems while optimizing the design space. This research aims to fulfill this need by developing tensor-decomposition-based A Priori Surrogate (TAPS), a data-free predictive AI model, which aims to enhance high-resolution capabilities while simultaneously optimizing computational efficiency with a minimal memory footprint, low data storage needs, and fast prediction. The proposed comprehensive framework sets a foundation for scalable, adaptable, and future-proof solutions to counter the ever-growing complexity in simulation-driven advanced industries. TAPS is particularly well-suited for engineering challenges where: 1) the finite element method and other conventional methods are unsuitable due to excessively long simulation times or high RAM and storage demands needed to achieve high accuracy, 2) the model must accommodate design parameters as inputs, or 3) fast prediction is required once the model is obtained.
+
+This paper is structured as follows. We first introduce the formulation of TAPS in section 2. In section 3, we examine the numerical convergence of TAPS for both space-time (S-T) and space-parameter-time (S-P-T) problems (i.e., problems that are dependent on spatial, parametric, and temporal inputs). In section 4, TAPS is applied to large-scale additive manufacturing problems that are considered intractable with standard numerical algorithms. This application effectively demonstrates TAPS's capability to address all of the three identified challenges.
+
+# 2. Theory
+
+# 2.1. Review of C-HiDeNN interpolation theory
+
+Leveraging the universal approximation theorem, multilayer perceptrons (MLPs) have been successfully applied as global basis functions in deep learning-based solvers [7]. However, as shown in Table 2, MLPs have a few potential caveats when approximating PDE solutions. To overcome these limitations, we leverage the Convolutional HiDeNN (C-HiDeNN) interpolation function, which leverages the merits of both locally supported finite element shape functions and the flexibility of machine learning. Note that C-HiDeNN also belongs to the INN category as shown in Fig. 2. C-HiDeNN maintains all the essential finite element approximation properties such as Kronecker delta and partition of unity [14].
+
+Table 2: Comparison of MLP and C-HiDeNN as approximation functions of PDE solutions.
+
+| PDE | Differential operator L | Dependent variable | xs | xp | xt |
| ∂2u/∂x12 + ∂2u/∂x22 + ... + ∂2u/∂x2D = f(x) | ∂2/∂x12 + ∂2/∂x22 + ... + ∂2/∂x2D | u | (x1, x2, ...,xD) | - | - |
| μui, jj + (μ + λ)uj,ij + Fi = e(x12+x22+x32) | μ(·)i,jj + (μ + λ)(·)j,ij | ui, i = 1, 2, 3 | (x1, x2, x3) | (λ,μ) | - |
| ρcp,du/dt + k(∂2u/∂x12 + ∂2u/∂x22 + ∂2u/∂x32) = Pe(x12+x22+x32) | ρcp,du/dt + k(∂2/∂x12 + ∂2/∂x22 + ∂2/∂x32) | u | (x1, x2, x3) | (ρ, cp, k, P) | t |
+
+The weighted-sum residual form of the PDE with TD interpolation can be written as:
+
+$$
+\int_ {\Omega} \delta u ^ {T D} (\boldsymbol {x}) \left[ \mathcal {L} \left(u ^ {T D} (\boldsymbol {x})\right) - f (\boldsymbol {x}) \right] d \Omega = 0 \tag {15}
+$$
+
+where $u^{TD}$ is the approximation of the solution (i.e., trial function), $\delta u^{TD}$ is the test function, and $d\Omega = d\Omega_{x_s}d\Omega_{x_p}d\Omega_{x_t}$ .
+
+Depending on how $\delta u^{TD}$ is adopted, different mathematical formulations can be obtained. If the test function resides in the same function space as the trial function, it becomes the Galerkin formulation. When the test function space differs from the trial function space, it becomes the Petrov-Galerkin formulation [22]. If the Dirac delta function is used for the test function, then Eq. 15 corresponds to the collocation method [24]. In this paper, we employ the Galerkin formulation. However, the proposed framework is versatile and can be extended to accommodate other formulations as well.
+
+In Eq. 12, the entire S-P-T domain is approximated using TD interpolation. However, this approach may result in a large system of equations due to the rapid increase in the number of TD modes for certain cases. For example, if the forcing function represents a moving source function in Eq. 14), this complexity may arise. To maintain computational efficiency, we can partition the temporal domain into a series of time slabs. As illustrated in Fig. 6(a), the S-P-T continuum is divided into S-P-T slabs $\mathcal{T}_1,\mathcal{T}_2,\dots ,\mathcal{T}_T$ . The solution within each time slab is then approximated individually using the TD interpolation.
+
+Between consecutive S-P-T slabs, either a continuous or discontinuous formulation can be employed. As shown in Fig. 6(b) for the continuous Galerkin scheme, the continuity of the solution in time is enforced by imposing the solution at the end of slab $\mathcal{T}_{i-1}$ as the initial condition of $\mathcal{T}_i$ :
+
+$$
+{ } ^ { [ \mathcal { T } + 1 ] } u ( \boldsymbol { x } _ { s } , \boldsymbol { x } _ { p } , 0 ) = { } ^ { [ \mathcal { T } ] } u ( \boldsymbol { x } _ { s } , \boldsymbol { x } _ { p } , x _ { t } ^ { m a x } ) \tag {16}
+$$
+
+Discontinuous Galerkin method can be used when a discontinuity is allowed between S-P-T slabs, as illustrated in Fig. 6(c). Discontinuity in time can be modeled using the jump operator $\llbracket \dots \rrbracket$ [25].
+
+$$
+\llbracket u \left(\boldsymbol {x} _ {s}, \boldsymbol {x} _ {p}, t\right) \rrbracket = \lim _ {\epsilon \rightarrow 0 ^ {+}} \left(^ {\mathcal {T} + 1} u \left(\boldsymbol {x} _ {s}, \boldsymbol {x} _ {p}, \epsilon\right) - ^ {\mathcal {T}} u \left(\boldsymbol {x} _ {s}, \boldsymbol {x} _ {p}, x _ {t} ^ {\max } - \epsilon\right)\right) \tag {17}
+$$
+
+
+(a)
+
+
+(b)
+Figure 6: (a) Multiple S-P-T slabs along the temporal dimension. (b) Continuous Galerkin: the solution is continuous across different S-P-T slabs. (c) Discontinuous Galerkin: jumps are allowed across the slab boundaries
+
+
+(c)
+
+Keeping in mind that this approach can be applied generally to a range of engineering problems, we will demonstrate an example of the Galerkin formulation using a single space-parameter-time partition (S-P-T) slab in the remainder of this section. For illustrative purposes, the transient heat transfer equation will be utilized:
+
+$$
+\rho c _ {p} \nabla_ {x _ {t}} u - \nabla_ {\boldsymbol {x} _ {s}} \cdot k \nabla_ {\boldsymbol {x} _ {s}} u = f (\boldsymbol {x} _ {s}, \boldsymbol {x} _ {p}, x _ {t}) \tag {18}
+$$
+
+as we focus on the example of modeling the laser powder bed fusion (LPBF) process in additive manufacturing (AM). In an LPBF simulation, we adopt the following time-dependent moving heat source function:
+
+$$
+f \left(\boldsymbol {x} _ {s}, \boldsymbol {x} _ {p}, x _ {t}\right) = \frac {2 \eta P}{\pi r ^ {2} d _ {\nu}} \exp \left(- \frac {2 \left((x - x _ {0} (t)) ^ {2} + (y - y _ {0} (t)) ^ {2}\right)}{r ^ {2}}\right) \cdot \mathbf {1} _ {\left(x _ {3} \geq d _ {\nu}\right)} \tag {19}
+$$
+
+Summarizing the independent variables in Eq. 18, there are spatial variables $\mathbf{x}_s = (x_1, x_2, x_3)$ ; parametric variables $\mathbf{x}_p = (k, \rho, c_p, \eta, P, r, d_v)$ ; and a temporal variable $x_t = t$ . Among the parametric variables, $k$ is conductivity; $\rho$ is the material density; $c_p$ is heat capacity; $\eta$ is the material absorptivity; $P$ represents laser power; $r$ is the standard deviation that characterizes the width of the heat source; $d_v$ is the penetration depth of the heat source. In Eq. 19, $[x_0(t), y_0(t)]$ represents the center of the moving heat source; $\mathbf{1}_{(x_3 \geq d_v)}$ is the indicator function where $\mathbf{1}_{(x_3 \geq d_v)} = 1$ if $x_3 \geq d_v$ or $\mathbf{1}_{(x_3 \geq d_v)} = 0$ if $x_3 < d_v$ . Note that the discretization of the material parameters, in particular, in a random field setting, has been previously proposed by Liu et al. [26, 27].
+
+As shown in the schematic below, we classify the boundary surfaces into 2 categories: the Dirichlet boundary surface $\Gamma_{D}$ and the Neumann boundary surface $\Gamma_{N}$ .
+
+A uniform ambient temperature is used as the initial condition. The bottom of the powder bed is subject to the Dirichlet boundary condition and the Neumann boundary conditions are prescribed on the other surfaces. The initial and boundary conditions are:
+
+$$
+u (\boldsymbol {x} _ {s}, \boldsymbol {x} _ {p}, 0) | _ {\Omega} = u _ {0},
+$$
+
+$$
+\left. u \left(\boldsymbol {x} _ {s}, \boldsymbol {x} _ {p}, x _ {t}\right) \right| _ {\Gamma_ {D}} = u _ {0}, \tag {20}
+$$
+
+$$
+\boldsymbol {n} \cdot \boldsymbol {q} | _ {\Gamma_ {N}} = q _ {\text {c o n v}} + q _ {\text {r a d}} + q _ {\text {e v a p}}
+$$
+
+where $u_{0}$ is the ambient temperature, $q_{conv}$ accounts for free convection, $q_{rad}$ accounts for radiation, and $q_{evap}$ imposes evaporative cooling when any material surface reaches the evaporation temperature [28]. Each flux is defined as:
+
+$$
+q _ {c o n v} = h _ {c o n v} [ u (x, t) - u _ {0} ],
+$$
+
+$$
+q _ {r a d} = - \sigma_ {S B} \epsilon \left(u ^ {4} \left(\boldsymbol {x} _ {s}, x _ {t}\right) - u _ {0} ^ {4}\right), \tag {21}
+$$
+
+$$
+q _ {e v a p} = - m _ {e v a p} L _ {e v a p}.
+$$
+
+
+Figure 7: Transient heat transfer with initial condition and boundary conditions.
+
+where $\sigma_{SB}$ is the Stefan-Boltzmann constant; $\epsilon$ is the material emissivity; $u_0$ is the ambient temperature; $h_{conv}$ is the convection coefficient of the surrounding gas, $m_{evap}$ is the mass evaporation flux and $L_{evap}$ is the heat of evaporation. In the following numerical examples, we only consider the free convection term in the Neumann boundary condition. The solution to Eq. 18 is approximated using TD interpolation function:
+
+$$
+u ^ {T D} \left(\boldsymbol {x} _ {s}, \boldsymbol {x} _ {p}, x _ {t}\right) = \sum_ {m = 1} ^ {M} u _ {\boldsymbol {x} _ {s}} ^ {(m)} \left(\boldsymbol {x} _ {s}\right) u _ {\boldsymbol {x} _ {p}} ^ {(m)} \left(\boldsymbol {x} _ {p}\right) u _ {x _ {t}} ^ {(m)} \left(x _ {t}\right) \tag {22}
+$$
+
+Here a general notation is employed to represent different types of components in Eq. 22. For example, the spatial component $u_{\boldsymbol{x}_s}^{(m)}(\boldsymbol{x}_s)$ is equivalent to $u_{x_1}^{(m)}(x_1)u_{x_2}^{(m)}(x_2)u_{x_3}^{(m)}(x_3)$ . The corresponding test function can be obtained using the variational principle:
+
+$$
+\delta u ^ {T D} \left(\boldsymbol {x} _ {s}, \boldsymbol {x} _ {p}, x _ {t}\right) = \sum_ {m = 1} ^ {M} \left[ \delta u _ {\boldsymbol {x} _ {s}} ^ {(m)} \left(\boldsymbol {x} _ {s}\right) u _ {\boldsymbol {x} _ {p}} ^ {(m)} \left(\boldsymbol {x} _ {p}\right) u _ {x _ {t}} ^ {(m)} \left(x _ {t}\right) + u _ {\boldsymbol {x} _ {s}} ^ {(m)} \left(\boldsymbol {x} _ {s}\right) \delta u _ {\boldsymbol {x} _ {p}} ^ {(m)} \left(\boldsymbol {x} _ {p}\right) u _ {x _ {t}} ^ {(m)} \left(x _ {t}\right) + u _ {\boldsymbol {x} _ {s}} ^ {(m)} \left(\boldsymbol {x} _ {s}\right) u _ {\boldsymbol {x} _ {p}} ^ {(m)} \left(\boldsymbol {x} _ {p}\right) \delta u _ {x _ {t}} ^ {(m)} \left(x _ {t}\right) \right] \tag {23}
+$$
+
+Plugging the trial and test functions, the S-P-T Galerkin form of Eq. 18 can be obtained by following Eq. 15:
+
+$$
+\int_ {\Omega} \delta u ^ {T D} \left[ \rho c _ {p} \nabla_ {x _ {t}} u ^ {T D} - \nabla_ {x _ {s}} \cdot k \nabla_ {x _ {s}} u ^ {T D} - f \right] d \Omega = 0 \tag {24}
+$$
+
+Using integration by parts on the diffusion term, we get the corresponding general S-P-T Galerkin weak form in TAPS formulation:
+
+$$
+\int_ {\Omega} \delta u ^ {T D} \rho c _ {p} \nabla_ {x _ {t}} u ^ {T D} d \Omega + \int_ {\Omega} \nabla_ {\boldsymbol {x} _ {s}} \delta u ^ {T D} \cdot k \nabla_ {\boldsymbol {x} _ {s}} u ^ {T D} d \Omega - \int_ {\partial \boldsymbol {x} _ {s}, \boldsymbol {x} _ {p}, t} \delta u ^ {T D} \boldsymbol {n} \cdot \boldsymbol {q} | _ {\Gamma_ {N}} d s d \Omega_ {\boldsymbol {x} _ {p}} d \Omega_ {t} - \int_ {\Omega} \delta u ^ {T D} f (\boldsymbol {x} _ {s}, \boldsymbol {x} _ {p}, x _ {t}) d \Omega = 0 \tag {25}
+$$
+
+where $q$ is the heat flux on the Neumann boundary.
+
+# 2.5. Discretized matrix form
+
+The S-P-T Galerkin weak form shown in Eq. 25 is nonlinear in nature due to the tensor product structure in TD interpolation, necessitating efficient solution schemes. To illustrate the detailed solution approach for the general S-P-T weak form, we simplify the governing equation Eq. 18 by considering a one-dimensional spatial problem where $x_{s} = x$ . We assume that the product of density and specific heat capacity $\rho c_{p}$ is equal to 1. Additionally, the forcing term is solely dependent on $x$ . Therefore, the simplified governing equation for this example is given by:
+
+$$
+\frac {\partial u}{\partial t} - \frac {\partial u}{\partial x} \cdot k \frac {\partial x}{\partial x} = f (x) \tag {26}
+$$
+
+subject to homogeneous boundary conditions and initial conditions. This equation has 3 independent variables ( $D = 3$ ), i.e., spatial variable $x_{s} = x_{1} = x$ , parametric variable $x_{p} = x_{2} = k$ and temporal variable $x_{t} = x_{3} = t$ . The S-P-T
+
+Galerkin weak form of this problem can be written as follows according to Eq. 25 (the superscripts "TD" for both trial and test functions are omitted for brevity).
+
+$$
+\int_ {\Omega} \delta u \nabla_ {t} u d \Omega + \int_ {\Omega} \nabla_ {x} \delta u \cdot k \nabla_ {x} u d \Omega - \int_ {\Omega} \delta u f d \Omega = 0 \tag {27}
+$$
+
+The corresponding trial and test functions can be obtained using Eqs. 22-23:
+
+$$
+u (x, k, t) = \sum_ {m = 1} ^ {M} u _ {x} ^ {(m)} (x) u _ {k} ^ {(m)} (k) u _ {t} ^ {(m)} (t) \tag {28}
+$$
+
+$$
+\delta u (x, k, t) = \underbrace {\sum_ {m = 1} ^ {M} \delta u _ {x} ^ {(m)} (x) u _ {k} ^ {(m)} (k) u _ {t} ^ {(m)} (t)} _ {\text {s p a t i a l v a r i a t i o n}} + \underbrace {\sum_ {m = 1} ^ {M} u _ {x} ^ {(m)} (x) \delta u _ {k} ^ {(m)} (k) u _ {t} ^ {(m)} (t)} _ {\text {p a r a m e t r i c v a r i a t i o n}} + \underbrace {\sum_ {m = 1} ^ {M} u _ {x} ^ {(m)} (x) u _ {k} ^ {(m)} (k) \delta u _ {t} ^ {(m)} (t)} _ {\text {t e m p o r a l v a r i a t i o n}} \tag {29}
+$$
+
+As shown in Eq. 29, the test function is further split into $D$ variational terms for a general $D$ dimensional problem (in the current example, $D = 3$ ). As an example, we first plug Eq. 28 and the spatial variation term of Eq. 29 into the Galerkin weak form in Eq. 27 to obtain the S-P-T weak form terms corresponding to spatial variation:
+
+$$
+\underbrace {\int_ {\Omega} \sum_ {m = 1} ^ {M} \sum_ {n = 1} ^ {M} \left[ \nabla \delta u _ {x} ^ {(m)} (x) \nabla u _ {x} ^ {(n)} (x) d x \right] \cdot \left[ u _ {k} ^ {(m)} (k) k u _ {k} ^ {(n)} (k) d k \right] \cdot \left[ u _ {t} ^ {(m)} (t) u _ {t} ^ {(n)} (t) d t \right]} _ {\text {d i f f u s i o n t e r m}} +
+$$
+
+$$
+\underbrace {\int_ {\Omega} \sum_ {m = 1} ^ {M} \sum_ {n = 1} ^ {M} \left[ \delta u _ {x} ^ {(m)} (x) u _ {x} ^ {(n)} (x) d x \right] \cdot \left[ u _ {k} ^ {(m)} (k) u _ {k} ^ {(n)} (k) d k \right] \cdot \left[ u _ {t} ^ {(m)} (t) \nabla_ {t} u _ {t} ^ {(n)} (t) d t \right]} _ {\text {t i m e d e r i v a t i v e t e r m}} - \tag {30}
+$$
+
+$$
+\underbrace {\int_ {\Omega} \sum_ {m = 1} ^ {M} \left[ \delta u _ {x} ^ {(m)} (x) f (x) d x \right] \cdot \left[ u _ {k} ^ {(m)} (k) d k \right] \cdot \left[ u _ {t} ^ {(m)} (t) d t \right]} _ {\text {f o r c i n g t e r m}}
+$$
+
+We use 1D C-HiDeNN shape functions to approximate each univariate function:
+
+$$
+u _ {d} ^ {(n)} (x _ {d}) = \widetilde {N} _ {n _ {d} ^ {\prime}} ^ {[ d ]} (x _ {d}) u _ {n _ {d} ^ {\prime} n} ^ {[ d ]} \quad (\text {n o s u m o n} d)
+$$
+
+$$
+\delta u _ {d} ^ {(m)} \left(x _ {d}\right) = \widetilde {N} _ {n _ {d}} ^ {[ d ]} \left(x _ {d}\right) \delta u _ {n _ {d} m} ^ {[ d ]} \quad (\text {n o s u m o n} d) \tag {31}
+$$
+
+where Einstein summation is used. The free index $d$ refers to dimension and $d = x,k$ or $t$ . The gradient of the interpolated variable can be computed using the shape function derivative $\widetilde{B}_{n_d}^{[d]}(x_d) = \frac{d\widetilde{N}_{n_d}^{[d]}(x_d)}{dx_d}$ .
+
+$$
+\nabla_ {x _ {d}} u _ {d} ^ {(n)} (x _ {d}) = \widetilde {B} _ {n _ {d} ^ {\prime}} ^ {[ d ]} (x _ {d}) u _ {n _ {d} ^ {\prime} n} ^ {[ d ]} \quad (\text {n o s u m o n} d)
+$$
+
+$$
+\nabla_ {x _ {d}} \delta u _ {d} ^ {(m)} (x _ {d}) = \widetilde {B} _ {n _ {d}} ^ {[ d ]} (x _ {d}) \delta u _ {n _ {d} m} ^ {[ d ]} \quad (\text {n o s u m o n} d) \tag {32}
+$$
+
+Plugging Eq. 31 - 32 into Eq. 30, the diffusion term can be rewritten as:
+
+$$
+\sum_ {m = 1} ^ {M} \sum_ {n = 1} ^ {M} \underbrace {\int_ {\Omega_ {x}} \widetilde {B} _ {n _ {x}} (x) \delta u _ {n _ {x} m} ^ {[ x ]} \widetilde {B} _ {n _ {x} ^ {\prime}} (x) u _ {n _ {x} ^ {\prime} n} ^ {[ x ]} d x} _ {\text {s p a t i a l t e r m}} \cdot \underbrace {\int_ {\Omega_ {k}} \widetilde {N} _ {n _ {k}} (k) u _ {n _ {k} m} ^ {[ k ]} k \widetilde {N} _ {n _ {k} ^ {\prime}} (k) u _ {n _ {k} ^ {\prime} n} ^ {[ k ]} d k} _ {\text {p a r a m e t r i c t e r m}} \cdot \underbrace {\int_ {\Omega_ {t}} \widetilde {N} _ {n _ {t}} (t) u _ {n _ {t} m} ^ {[ t ]} \widetilde {N} _ {n _ {t} ^ {\prime}} (t) u _ {n _ {t} ^ {\prime} n} ^ {[ t ]} d t} _ {\text {t e m p o r a l t e r m}} \tag {33}
+$$
+
+As can be readily seen from Eq. 33, after doing 1D integration of each term, the parametric and temporal terms can be treated as coefficient matrices:
+
+$$
+C _ {m n} ^ {[ k ]} = \underbrace {\int_ {\Omega_ {k}} \widetilde {N} _ {n _ {k}} (k) u _ {n _ {k} m} ^ {[ k ]} k \widetilde {N} _ {n _ {k} ^ {\prime}} (k) u _ {n _ {k} ^ {\prime} n} ^ {[ k ]} d k} _ {\text {p a r a m e t r i c t e r m}}
+$$
+
+$$
+C _ {m n} ^ {[ t ]} = \underbrace {\int_ {\Omega_ {t}} \widetilde {N} _ {n _ {t}} (t) u _ {n _ {t} m} ^ {[ t ]} \widetilde {N} _ {n _ {t} ^ {\prime}} (t) u _ {n _ {t} ^ {\prime} n} ^ {[ t ]} d t} _ {\text {t e m p o r a l t e r m}} \tag {34}
+$$
+
+as the only free indices are $m$ and $n$ . Substituting the coefficient matrices and rearranging different terms in Eq. 33, we have:
+
+$$
+\sum_ {m = 1} ^ {M} \delta u _ {n _ {x} m} ^ {[ x ]} \sum_ {n = 1} ^ {M} \left[ \int_ {\Omega_ {x}} \widetilde {B} _ {n _ {x}} (x) \widetilde {B} _ {n _ {x} ^ {\prime}} (x) d x \right] \cdot C _ {m n} ^ {[ k ]} C _ {m n} ^ {[ t ]} \cdot u _ {n _ {x} ^ {\prime} n} ^ {[ x ]} \tag {35}
+$$
+
+Like standard FEM, we can define $\int_{x}\widetilde{B}_{n_x}(x)\widetilde{B}_{n_x'}(x)dx$ as the 1D stiffness matrix $K_{n_x n_x'}^{[x]}$ of $x$ dimension in Eq. 35. We let $C_{mn}^{[x]} = C_{mn}^{[k]}C_{mn}^{[t]}$ with no summation on $(m,n)$ . Furthermore, let us define the following 4-th order tensor:
+
+$$
+A _ {n _ {x} n _ {x} ^ {\prime} m n} ^ {[ x ]} = K _ {n _ {x} n _ {x} ^ {\prime}} ^ {[ x ] ^ {\prime}} C _ {m n} ^ {[ x ]} \tag {36}
+$$
+
+where $A_{n_s n_s'mn}^{[x]}$ is a function of solution vectors $u_{n_k m}^{[k]}$ and $u_{n_m m}^{[t]}$ since the coefficient matrix $C_{mn}^{[x]}$ depends on these solution vectors as shown in Eq. 34. This dependency reflects the interconnected nature of the variables across different dimensions in the S-P-T framework, highlighting how the spatial, parameter, and temporal components influence each other through the coefficients. As a result, Eq. 33 can be further simplified as follows:
+
+$$
+\delta u _ {n _ {x} m} ^ {[ x ]} A _ {n _ {x} n _ {x} ^ {\prime} m n} ^ {[ x ]} u _ {n _ {x} ^ {\prime} n} ^ {[ x ]} \tag {37}
+$$
+
+where the summation signs are neglected since $m$ and $n$ become dummy variables. The 4-th order tensor $A_{n_x n_x' mn}^{[x]}$ can be reshaped as a 2nd order tensor $\mathbb{A}_{IJ}^{[x]}$ : the indices $n_x$ and $m$ are combined into a single composite index $I$ , and the indices $n_x'$ and $n$ are combined into a single composite index $J$ .
+
+$$
+A _ {n _ {x} n _ {x} ^ {\prime} m n} ^ {[ x ]} = \mathbb {A} _ {I J} ^ {[ x ]} \tag {38}
+$$
+
+Define the following vectorization:
+
+$$
+\delta \mathbb {U} _ {I} ^ {[ x ]} = \left[ \operatorname {v e c} \left(\delta u _ {n _ {x} m} ^ {[ x ]}\right) \right] _ {I}
+$$
+
+$$
+\mathbb {U} _ {J} ^ {[ x ]} = \left[ \operatorname {v e c} \left(u _ {n _ {x} ^ {\prime} n} ^ {[ x ]}\right) \right] _ {J} \tag {39}
+$$
+
+As a result, Eq. 37 is equivalent to:
+
+$$
+\delta \mathbb {U} ^ {[ x ] T} \mathbb {A} ^ {[ x ]} \mathbb {U} ^ {[ x ]} \tag {40}
+$$
+
+Following the same procedure, we can obtain matrix forms corresponding to the time derivative term $\delta \mathbb{U}^{[x]^T}\mathbb{B}^{[x]}\mathbb{U}^{[x]}$ , and the forcing term $\delta \mathbb{U}^{[x]^T}\mathbb{Q}^{[x]}$ for the spatial variational part of Eq. 30. Similar structures can also be obtained for the parametric and temporal variational parts of the test function in Eq. 29. Denoting $\mathbb{K}^{[d]} = \mathbb{A}^{[d]} + \mathbb{B}^{[d]}$ , the matrix form of the generalized S-P-T Galerkin form in Eq. 27 can be written as:
+
+$$
+\underbrace {\delta \mathbb {U} ^ {[ x ] ^ {T}} \mathbb {K} ^ {[ x ]} \mathbb {U} ^ {[ x ]} - \delta \mathbb {U} ^ {[ x ] ^ {T}} \mathbb {Q} ^ {[ x ]}} _ {\text {s p a t i a l v a r i a t i o n a l p a r t}} + \underbrace {\delta \mathbb {U} ^ {[ k ] ^ {T}} \mathbb {K} ^ {[ k ]} \mathbb {U} ^ {[ k ]} - \delta \mathbb {U} ^ {[ k ] ^ {T}} \mathbb {Q} ^ {[ k ]}} _ {\text {p a r a m e t r i c v a r i a t i o n a l p a r t}} + \underbrace {\delta \mathbb {U} ^ {[ t ] ^ {T}} \mathbb {K} ^ {[ t ]} \mathbb {U} ^ {[ t ]} - \delta \mathbb {U} ^ {[ t ] ^ {T}} \mathbb {Q} ^ {[ t ]}} _ {\text {t e m p o r a l v a r i a t i o n a l p a r t}} = 0 \tag {41}
+$$
+
+Eq. 41 is equivalent to the following nonlinear system of equations. Note that the nonlinearity comes from the fact that $\mathbb{K}^{[d]}$ is solution dependent:
+
+$$
+\left[ \delta \mathbb {U} ^ {[ x ] ^ {T}}, \delta \mathbb {U} ^ {[ k ] ^ {T}}, \delta \mathbb {U} ^ {[ t ] ^ {T}} \right] \left\{\left[ \begin{array}{c c c} \mathbb {K} ^ {[ x ]} (\mathbb {U} ^ {[ k ]}, \mathbb {U} ^ {[ t ]}) & 0 & 0 \\ 0 & \mathbb {K} ^ {[ k ]} (\mathbb {U} ^ {[ x ]}, \mathbb {U} ^ {[ t ]}) & 0 \\ 0 & 0 & \mathbb {K} ^ {[ t ]} (\mathbb {U} ^ {[ x ]}, \mathbb {U} ^ {[ k ]}) \end{array} \right] \left[ \begin{array}{l} \mathbb {U} ^ {[ x ]} \\ \mathbb {U} ^ {[ k ]} \\ \mathbb {U} ^ {[ t ]} \end{array} \right] - \left[ \begin{array}{l} \mathbb {Q} ^ {[ x ]} (\mathbb {U} ^ {[ k ]}, \mathbb {U} ^ {[ t ]}) \\ \mathbb {Q} ^ {[ k ]} (\mathbb {U} ^ {[ x ]}, \mathbb {U} ^ {[ t ]}) \\ \mathbb {Q} ^ {[ t ]} (\mathbb {U} ^ {[ x ]}, \mathbb {U} ^ {[ k ]}) \end{array} \right] \right\} = 0 \tag {42}
+$$
+
+where we can treat the solution vector $\left[\mathbb{U}^{[x]^T},\mathbb{U}^{[k]^T},\mathbb{U}^{[t]^T}\right]$ as generalized DoFs like standard FEM. There are many ways to solve Eq. 42. For example, standard linearization schemes such as Newton's method have been used [29]. However, this method may suffer from ill-conditioning since the mismatch of scales for different dimensions can be significant. In this paper, we use the concept of subspace iteration to efficiently approximate the solution by iterating in the subspace of the test function space until a convergence criteria is met [19]. A similar counterpart has been widely adopted as the gold standard in discrete tensor decomposition [23].
+
+# 2.6. Solution scheme of TAPS: subspace iteration
+
+For subspace iteration in $d$ -th dimension, only the solution matrix $\mathbb{U}^{[d]}$ is treated as unknown while all other functions are considered as known constants. Consequently, the variations of the univariate functions other than $d$ -th dimension will vanish. From Eq. 42, it can be seen that this will lead to a linear system of equations for the unknowns in the $d$ -th dimension. The updated solution matrix $\mathbb{U}^{[d]}$ from this process is then used in the next subspace iteration for dimension $d + 1$ (when $d = D$ , we come back to the first dimension $d = 1$ ). The complete solution scheme for subspace iteration is shown in Algorithm 1.
+
+Algorithm 1 TAPS solution scheme (subspace iteration)
+
+1: Initialize solution vector $\mathbb{U}^{[x_1][0]}$ ..., $\mathbb{U}^{[x_D][0]}$ with random values and compute $\mathbb{K}^{[x_1][0]}$ , and $\mathbb{Q}^{[x_1][0]}$
+
+2: for iter = 0 to iter_max do
+3: for $d = 1$ to D do
+4: Update iteration number $\mathcal{K} = iter\times D + d$
+5: Solve TD linear system $\mathbb{K}^{[x_d][\mathcal{K} - 1]}\mathbb{U}^{[x_d][\mathcal{K}]} = \mathbb{Q}^{[x_d][\mathcal{K} - 1]}$
+6: Update matrices $\mathbb{K}^{[x_{d + 1}][\mathcal{K}]}$ and force vector $\mathbb{Q}^{[x_{d + 1}][\mathcal{K}]}$
+7: end for
+8: Check convergence
+9: end for
+
+To illustrate the details of the subspace iteration algorithm, we consider the $\mathcal{K}$ -th subspace iteration (which is on spatial variable $x$ ). Here, we assume that the parametric and temporal solutions have been updated from the previous $(\mathcal{K} - 1)$ -th iteration, leaving the spatial solution as unknown to be solved in $\mathcal{K}$ -th iteration. Moreover, instead of the full variation form of the test function as in Eq. 42, we only consider the subspace $x$ of the test function by setting the parametric and temporal variational parts as 0. As a result, we have:
+
+$$
+\mathbb {K} ^ {[ x ] [ \mathcal {K} - 1 ]} \left(\mathbb {U} ^ {[ k ] [ \mathcal {K} - 1 ]}, \mathbb {U} ^ {[ t ] [ \mathcal {K} - 1 ]}\right) \mathbb {U} ^ {[ x ] [ \mathcal {K} ]} = \mathbb {Q} ^ {[ x ] [ \mathcal {K} - 1 ]} \left(\mathbb {U} ^ {[ k ] [ \mathcal {K} - 1 ]}, \mathbb {U} ^ {[ t ] [ \mathcal {K} - 1 ]}\right) \tag {43}
+$$
+
+which is a linear system of equations with unknown $\mathbb{U}^{[x][\mathcal{K}]}$ . This is a general Sylvester equation which can be solved using many efficient solution schemes [30, 31]. In this paper, sparse direct solvers based on fast diagonalization/complex Schur decomposition methods are adopted [32]. The computational complexity of the sparse direct solver is $O(M^3 + M^2 n_d + C_c(n_d))$ for the $d$ -th dimension subspace iteration, where $M$ is the total number of modes; $n_d$ is the number of grid points for $d$ -th dimension; $C_c(n_d)$ refers to the computational cost of the banded sparse mass/stiffness matrix for $d$ -th dimension with a shape of $(n_d \times n_d)$ .
+
+Once $\mathbb{U}^{[x][\mathcal{K}]}$ is obtained, we then update matrix $\mathbb{K}^{[k][\mathcal{K}]}(\mathbb{U}^{[x][\mathcal{K}]},\mathbb{U}^{[t][\mathcal{K}]})$ and forcing vector $\mathbb{Q}^{[k][\mathcal{K}]}(\mathbb{U}^{[x][\mathcal{K}]},\mathbb{U}^{[t][\mathcal{K}]})$ . In the next iteration (for dimension $k$ ), we treat $\mathbb{U}^{[k][\mathcal{K} + 1]}$ as the only unknown. Subspace iteration will continue unless the relative change of all solution matrices (for example, $L_{2}$ norm) is within the tolerance.
+
+# 2.7. Error estimates of TAPS
+
+Since the TAPS solution is based on the C-HiDeNN-TD approximation and the generalized Galerkin formulation, we can have the following theoretical results on the error bounds, as demonstrated in our previous work on C-HiDeNN [6]:
+
+$$
+\left\| u ^ {\mathrm {C} - \mathrm {H i D e N N}} - u ^ {\mathrm {e x}} \right\| _ {E} \leq \left\| u ^ {\mathrm {T A P S}} - u ^ {\mathrm {e x}} \right\| _ {E} \leq \left\| u ^ {\mathrm {F E M}} - u ^ {\mathrm {e x}} \right\| _ {E} \tag {44}
+$$
+
+where $\| \cdot \|_E$ denotes the energy norm, $u^{\mathrm{ex}}$ denotes the exact solution, $u^{\mathrm{C - HiDeNN}}$ denotes the solution obtained by the full C-HiDeNN method without tensor decomposition, $u^{\mathrm{TAPS}}$ denotes the TAPS solution with a sufficient number of modes, $u^{\mathrm{FEM}}$ denotes the FEM solution. The proof of the above results is based on the fact that the full C-HiDeNN approximation can provide a larger function space and therefore more accurate solutions than conventional FEM [6]. The subspace iteration can be considered as a local (directional) version of the Galerkin formulation and is expected to enable an optimized solution for the tensor decomposition that will converge to the Galerkin-based full C-HiDeNN method.
+
+# 3. Results
+
+# 3.1. Convergence study for moving heat source
+
+In this section, we first analyze the convergence of the TAPS solver for a space-time (S-T) transient heat transfer problem. A single NVIDIA RTX A6000 GPU is used for all the following analyses. In Eq. 18, we let $\rho c_{p} = 1$ , $k = 1$ , and replace the heat source as shown in Eq. 45. In this example, we have the spatial variable $x_{s} = (x,y,z)$ and the temporal variable $x_{t} = t$ .
+
+$$
+\begin{array}{l} f \left(\boldsymbol {x} _ {s}, x _ {t}\right) = 2 (1 - 2 y ^ {2}) \left(1 - e ^ {- 1 5 t}\right) e ^ {- y ^ {2} - (1 0 0 t - x - 5) ^ {2}} \\ + 2 (1 - 2 (1 0 0 t - x - 5) ^ {2}) \left(1 - e ^ {- 1 5 t}\right) e ^ {- y ^ {2} - (1 0 0 t - x - 5) ^ {2}} + (1 \tag {45} \\ - e ^ {- 1 5 t}) (2 0 0 x + 1 0 0 0 - 2 0 0 0 0 t) e ^ {- y ^ {2} - (1 0 0 t - x - 5) ^ {2}} \\ - 1 5 e ^ {- 1 5 t} e ^ {- y ^ {2} - (1 0 0 t - x - 5) ^ {2}} \\ \end{array}
+$$
+
+The analytical solution to the PDE is inherently non-separable.
+
+$$
+u ^ {\mathrm {e x}} \left(\boldsymbol {x} _ {s}, x _ {t}\right) = (1 - e ^ {- 1 5 t}) e ^ {- y ^ {2} - (x - 1 0 0 t - 5) ^ {2}} \tag {46}
+$$
+
+The initial and boundary conditions are:
+
+$$
+u \left(x _ {s}, 0\right) = 0,
+$$
+
+$$
+\left. u \left(\boldsymbol {x} _ {s}, x _ {t}\right) \right| _ {\partial \Omega} = \left. u ^ {\mathrm {e x}} \left(\boldsymbol {x} _ {s}, x _ {t}\right) \right| _ {\partial \Omega}. \tag {47}
+$$
+
+The relative $L_{2}$ norm error is defined as:
+
+$$
+\epsilon_ {L _ {2}} = \frac {\| u ^ {T D} \left(\boldsymbol {x} _ {s} , x _ {t}\right) - u ^ {\mathrm {e x}} \left(\boldsymbol {x} _ {s} , x _ {t}\right) \| _ {L _ {2} \left(\Omega_ {\boldsymbol {x} _ {s}} \otimes \Omega_ {x _ {t}}\right)}}{\| u ^ {\mathrm {e x}} \left(\boldsymbol {x} _ {s} , x _ {t}\right) \| _ {L _ {2} \left(\Omega_ {\boldsymbol {x} _ {s}} \otimes \Omega_ {x _ {t}}\right)}} \tag {48}
+$$
+
+First, we investigate the influence of the number of subspace iterations. As shown in Fig. 8(a), 3 iterations are enough to obtain an accurate result. Next, we investigate the convergence in terms of the number of modes. Here we compare the relative $L_{2}$ norm error for both TAPS and proper generalized decomposition (PGD) methods [17, 18]. To this aim, we use the same discretization for the space-time domain with each dimension discretized by 100 grid points, the same reproducing polynomial order $p = 1$ and convolution patch size $s = 1$ . As can be seen from Fig. 8(b), TAPS requires a much smaller number of modes than PGD. For TAPS, when the number of modes equals 25, the relative $L_{2}$ norm error decreases to $2.5 \times 10^{-3}$ . The total solution time is 15.2 s. However, PGD requires 1,000 modes which takes 60.6 s solution time to reach the same level of accuracy. This is because the test function space in PGD is a subspace of TAPS [29]. Furthermore, the modal decomposition obtained from PGD is not optimal and thus requires a larger storage requirement due to the increased number of modes.
+
+
+(a)
+
+
+(b)
+Figure 8: Relative L2 norm error with respect to (a) the number of subspace iterations (b) the number of modes
+
+The spatial and temporal convergence are also studied. In Fig. 9(a), the number of temporal nodes is fixed as 500, and the spatial mesh is refined. It shows the relative $L_{2}$ norm error with respect to the number of nodes along each spatial dimension. As can be readily seen from the figure, larger patch size $s$ leads to smaller error given the same reproducing polynomial orders $p$ . Moreover, we can adjust $p$ to control the spatial convergence rate. Similarly, Fig. 9(b) demonstrates the convergence rate in the temporal domain where we fix the spatial discretization as 500 along each spatial dimension. By adjusting $s$ and $p$ , we can obtain different temporal convergence rates.
+
+Finally, we refine the spatial and temporal mesh simultaneously and study the spatio-temporal convergence rate in Fig. 9(c). As can be observed from the figure, higher reproducing polynomial order $p$ will lead to a higher-order convergence rate.
+
+
+(a)
+
+
+(b)
+
+
+(c)
+Figure 9: Relative $L_{2}$ norm error with respect to the number of grid points (a) spatial convergence (b) temporal convergence (c) spatio-temporal convergence
+
+# 3.2. Convergence study of $S-P-T$ problems up to equivalent zetta-scale (10 $^{21}$ ) full models
+
+In this example, we study the convergence of the TAPS solver for the time-dependent parametric heat transfer problem in a S-P-T setting. In Eq. 18, we adopt the heat source as shown in Eq. 49. In this example, we have spatial variable $\boldsymbol{x}_s = (x,y,z)$ , parametric variable $\boldsymbol{x}_p = (k,P,\rho ,c_p)$ and temporal variable $x_{t} = t$ .
+
+$$
+\begin{array}{l} f \left(\boldsymbol {x} _ {s}, \boldsymbol {x} _ {p}, x _ {t}\right) = 1 5 \rho^ {2} c _ {p} ^ {2} k p e ^ {- 1 5 k t} e ^ {- 2 5. 0 x ^ {2} - 2 5. 0 y ^ {2}} \\ + 5 0 \rho c _ {p} k p \left(1 - e ^ {- 1 5 k t}\right) e ^ {- 2 5. 0 x ^ {2} - 2 5. 0 y ^ {2}} \left[ \left(1 - 5 0 x ^ {2}\right) + \left(1 - 5 0 y ^ {2}\right) \right] \tag {49} \\ \end{array}
+$$
+
+The analytical solution to the PDE is inherently non-separable.
+
+$$
+u ^ {\mathrm {e x}} \left(\boldsymbol {x} _ {s}, \boldsymbol {x} _ {p}, x _ {t}\right) = \rho c _ {p} P \left(1 - e ^ {- 1 5 k t}\right) e ^ {- 2 5. 0 x ^ {2} - 2 5. 0 y ^ {2}} \tag {50}
+$$
+
+The initial and boundary conditions are:
+
+$$
+u \left(\boldsymbol {x} _ {s}, \boldsymbol {x} _ {p}, 0\right) = 0
+$$
+
+$$
+u \left(\boldsymbol {x} _ {s}, \boldsymbol {x} _ {p}, x _ {t}\right) | _ {\partial \Omega} = u ^ {\mathrm {e x}} \left(\boldsymbol {x} _ {s}, \boldsymbol {x} _ {p}, x _ {t}\right) | _ {\partial \Omega}
+$$
+
+The relative $L_{2}$ norm error is defined as:
+
+$$
+\epsilon_ {L _ {2}} = \frac {\left\| u ^ {T D} \left(\boldsymbol {x} _ {s} , \boldsymbol {x} _ {p} , x _ {t}\right) - u ^ {\mathrm {e x}} \left(\boldsymbol {x} _ {s} , \boldsymbol {x} _ {p} , x _ {t}\right) \right\| _ {L _ {2} \left(\Omega_ {x _ {s}} \otimes \Omega_ {x _ {p}} \otimes \Omega_ {x _ {t}}\right)}}{\left\| u ^ {\mathrm {e x}} \left(\boldsymbol {x} _ {s} , \boldsymbol {x} _ {p} , x _ {t}\right) \right\| _ {L _ {2} \left(\Omega_ {x _ {s}} \otimes \Omega_ {x _ {p}} \otimes \Omega_ {x _ {t}}\right)}} \tag {51}
+$$
+
+To study the convergence of TAPS for S-P-T problems, the number of grid points is refined simultaneously in each dimension and corresponding relative $L_{2}$ norm errors are computed as shown in Fig. 10. When the number of grid points in each dimension is 450, the equivalent DoFs of a full model achieves $450^{8} = 1.68 \times 10^{21}$ . Consequently, it is equivalent to a zetta-scale $(10^{21})$ full problem. As can be seen from the figure, a larger patch size $s$ leads to a smaller error and faster convergence. A higher reproducing polynomial order $p$ also leads to a higher convergence rate. It can be noticed that the convergence rate for $p = 3$ case is smaller than expected $p + 1 = 4$ . This is attributed to the fact that the S-P-T mesh is not fine enough. However, due to the rounding error in computing the relative $L_{2}$ norm error, we can only accurately compute the error up to 450 grid points per dimension.
+
+
+Figure 10: Relative $L_{2}$ norm error with respect to the number of grid points in each dimension
+
+In summary, we have the flexibility to choose different $s$ and $p$ to control the accuracy of TAPS by directly solving the S-P-T PDE. This is different from other data-driven modeling approaches (for instance, neural networks-based data-driven methods) in two notable ways. First, unlike a black-box neural network interpolator where the accuracy of the model is not guaranteed, our method is built upon the AI-enhanced finite element method, and we can control the convergence rate by choosing suitable hyperparameters $s$ and $p$ . Second, unlike most data-driven reduced-order models for physical problems, our method directly solves the governing PDE by plugging in the TD interpolation without seeing any training data. As a result, we can avoid the most expensive offline data generation stage as opposed to data-driven methods.
+
+# 3.3. Moving source with solution dependent material parameters
+
+In this section, we model moving heat sources using temperature-dependent material parameters. The solution scheme of this problem is provided in detail in Appendix A. Figure 11(a) illustrates a typical representation of temperature-dependent heat conductivity and capacity for Inconel 718 [33].
+
+Since the temperature dependency of $k(u)$ and $\rho c_{p}(u)$ can be approximated using a linear relationship. As a result, we can directly rewrite $k(u(\boldsymbol{x}_s,x_t))$ and $\rho c_{p}(u(\boldsymbol{x}_{s},x_{t}))$ in the TD format.
+
+
+(a)
+
+
+(b)
+Figure 11: (a) Temperature dependent material properties for Inconel 718 [33] (b) Schematic of numerical simulation, where the solution along the center line is compared for FEM and TAPS.
+
+$$
+k \left(\boldsymbol {x} _ {s}, x _ {t}\right) \approx \sum_ {m = 1} ^ {M} m _ {k} u _ {x _ {1}} ^ {(m)} \left(x _ {1}\right) u _ {x _ {2}} ^ {(m)} \left(x _ {2}\right) u _ {x _ {3}} ^ {(m)} \left(x _ {3}\right) u _ {x _ {t}} ^ {(m)} \left(x _ {t}\right) + n _ {k}
+$$
+
+$$
+\rho c _ {p} \left(\boldsymbol {x} _ {s}, x _ {t}\right) \approx \sum_ {m = 1} ^ {M} m _ {c _ {p}} u _ {x _ {1}} ^ {(m)} \left(x _ {1}\right) u _ {x _ {2}} ^ {(m)} \left(x _ {2}\right) u _ {x _ {3}} ^ {(m)} \left(x _ {3}\right) u _ {x _ {t}} ^ {(m)} \left(x _ {t}\right) + n _ {c _ {p}} \tag {52}
+$$
+
+where $M$ is the decomposition modes of the TAPS solution; $m_{k} = 1.52 \times 10^{-5} \mathrm{~W} / (\mathrm{mmK}^{2})$ ; $n_{k} = 5.29 \times 10^{-3} \mathrm{~W} / (\mathrm{mmK})$ ; $m_{c_{p}} = 6.11 \times 10^{-7} \mathrm{~mm}^{-3} \mathrm{~K}^{-2}$ ; $n_{c p} = 3.25 \times 10^{-3} \mathrm{~mm}^{-3} \mathrm{~K}^{-1}$
+
+The problem setup is shown in Fig. 11 (b). The spatial domain size is $10\mathrm{mm} \times 10\mathrm{mm} \times 1\mathrm{mm}$ where homogeneous Dirichlet boundary conditions are assumed for the left and right surfaces; homogeneous Neumann boundary conditions are applied to all other surfaces. As shown in Eq. 19, a moving Gaussian source term $f(\boldsymbol{x}_s, t)$ is applied as a volumetric source term with a radius $r = 0.5\mathrm{mm}$ and moving velocity $500\mathrm{mm/s}$ . The diameter is discretized using 10 spatial elements.
+
+Since there is no analytical solution available to this problem, we use implicit finite element analysis as the baseline for validation. JAX-FEM [34] is used to generate the nonlinear FEM solution. For ease of comparison, we use the same time increment as $1.60 \times 10^{-4}$ sec for both TAPS and FEM. The solution along the center line, as shown in Fig. 11 (b) is compared. As can be seen from Fig. 12, the result of the nonlinear TAPS solver agrees well with FEM.
+
+# 3.4. Simulation of LPBF process
+
+In this section, we use TAPS to efficiently model the laser powder bed fusion process (LPBF) in additive manufacturing. Here we only consider the free convection term in the Neumann boundary condition. The initial condition can be considered by splitting the total solution as a summation of the homogeneous part and the inhomogeneous part.
+
+$$
+u \left(\boldsymbol {x} _ {s}, x _ {t}\right) = u _ {0} \left(\boldsymbol {x} _ {s}, x _ {t}\right) + u _ {\text {i n i t}} \left(\boldsymbol {x} _ {s}\right) \tag {53}
+$$
+
+As a result, $u_{0}(\pmb{x}_{s}, x_{t})$ is subject to homogeneous initial conditions. In this section, we assume Ti-6Al-4V is used as the powder bed materials. The detailed material parameters can be found in Table 4.
+
+# 3.4.1. Single-track simulation
+
+In this example, we investigate the computational complexity numerically for single-track LPBF simulation with a single S-T slab, as shown in Fig. 13 (a). A single NVIDIA RTX A6000 GPU is used for all the following analyses. To ensure accuracy, the number of modes is adopted as 5 times larger than the number of time steps in the following examples. In the first case, within the S-T slab, the spatial mesh is refined uniformly along each spatial dimension
+
+
+Figure 12: Comparison of Nonlinear TAPS solution versus finite element solution at different times.
+
+Table 4: Parameters used in the simulation
+
+| Variables | Description |
| uh(x) | Interpolated scalar field defined inside of an element |
| Ae | Nodes within element e |
| As | Nodes within patch domain of element e |
| WiS,a,p,j(x) | Convolution patch function at node j for i-th nodal patch with hyperparameters s, a, and p |
| M | Total number of modes in tensor decomposition (TD) |
| m | Index for mode |
| D | Total number of dimensions |
| d | Index for dimension |
| x | Independent variable which includes spatial variable xs, parametric variable xp and temporal variable xt |
| Nd(xd;ad,sd,pd) | Global C-HiDeNN shape function for dimension d with dilation parameter ad, patch size sd and reproducing polynomial order pd |
| b | Source function in laser powder bed fusion process |
| uTD | Approximation of the solution field expressed via TD |
| T | Time slab index for space-parameter-time problem |
| k | Thermal conductivity |
| ρ | Material density |
| cp | Heat capacity |
| η | Material absorptivity |
| P | Laser power |
| r | Standard deviation that characterizes the width of the heat source |
| q | Heat flux |
| qconv | Heat flux from convection |
| qrad | Heat flux from radiation |
| qevap | Heat flux from evaporative cooling |
| hconv | Convection coefficient |
| σSB | Stefan-Boltzman constant |
| mevap | Mass evaporation flux |
| Levap | Heat of evaporation |
| Bnd | Shape function derivative |
| Ud | Solution matrix (Rnd×M) for dimension d that contains all the modes |
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+ "content": "{ } ^ { [ \\mathcal { T } + 1 ] } u ( \\boldsymbol { x } _ { s } , \\boldsymbol { x } _ { p } , 0 ) = { } ^ { [ \\mathcal { T } ] } u ( \\boldsymbol { x } _ { s } , \\boldsymbol { x } _ { p } , x _ { t } ^ { m a x } ) \\tag {16}",
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+ "content": "Keeping in mind that this approach can be applied generally to a range of engineering problems, we will demonstrate an example of the Galerkin formulation using a single space-parameter-time partition (S-P-T) slab in the remainder of this section. For illustrative purposes, the transient heat transfer equation will be utilized:"
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+ "content": "\\rho c _ {p} \\nabla_ {x _ {t}} u - \\nabla_ {\\boldsymbol {x} _ {s}} \\cdot k \\nabla_ {\\boldsymbol {x} _ {s}} u = f (\\boldsymbol {x} _ {s}, \\boldsymbol {x} _ {p}, x _ {t}) \\tag {18}",
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+ "type": "text",
+ "content": "as we focus on the example of modeling the laser powder bed fusion (LPBF) process in additive manufacturing (AM). In an LPBF simulation, we adopt the following time-dependent moving heat source function:"
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+ "content": "f \\left(\\boldsymbol {x} _ {s}, \\boldsymbol {x} _ {p}, x _ {t}\\right) = \\frac {2 \\eta P}{\\pi r ^ {2} d _ {\\nu}} \\exp \\left(- \\frac {2 \\left((x - x _ {0} (t)) ^ {2} + (y - y _ {0} (t)) ^ {2}\\right)}{r ^ {2}}\\right) \\cdot \\mathbf {1} _ {\\left(x _ {3} \\geq d _ {\\nu}\\right)} \\tag {19}",
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+ "type": "text",
+ "content": "Summarizing the independent variables in Eq. 18, there are spatial variables "
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+ "content": "\\mathbf{x}_s = (x_1, x_2, x_3)"
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+ "content": "\\mathbf{x}_p = (k, \\rho, c_p, \\eta, P, r, d_v)"
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+ "content": " is conductivity; "
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+ "content": " is the material density; "
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+ "content": " is heat capacity; "
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+ "content": " represents laser power; "
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+ "type": "text",
+ "content": " is the standard deviation that characterizes the width of the heat source; "
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+ "type": "text",
+ "content": " is the penetration depth of the heat source. In Eq. 19, "
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+ "content": "[x_0(t), y_0(t)]"
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+ "content": " represents the center of the moving heat source; "
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+ "type": "text",
+ "content": ". Note that the discretization of the material parameters, in particular, in a random field setting, has been previously proposed by Liu et al. [26, 27]."
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+ "content": "As shown in the schematic below, we classify the boundary surfaces into 2 categories: the Dirichlet boundary surface "
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+ "content": "A uniform ambient temperature is used as the initial condition. The bottom of the powder bed is subject to the Dirichlet boundary condition and the Neumann boundary conditions are prescribed on the other surfaces. The initial and boundary conditions are:"
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+ "content": "\\left. u \\left(\\boldsymbol {x} _ {s}, \\boldsymbol {x} _ {p}, x _ {t}\\right) \\right| _ {\\Gamma_ {D}} = u _ {0}, \\tag {20}",
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+ "content": " is the ambient temperature, "
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+ "content": " accounts for radiation, and "
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+ "type": "text",
+ "content": " imposes evaporative cooling when any material surface reaches the evaporation temperature [28]. Each flux is defined as:"
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+ "content": " is the Stefan-Boltzmann constant; "
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+ "content": " is the material emissivity; "
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+ "content": " is the convection coefficient of the surrounding gas, "
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+ "content": " is the mass evaporation flux and "
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+ "type": "text",
+ "content": " is the heat of evaporation. In the following numerical examples, we only consider the free convection term in the Neumann boundary condition. The solution to Eq. 18 is approximated using TD interpolation function:"
+ }
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+ "content": "u ^ {T D} \\left(\\boldsymbol {x} _ {s}, \\boldsymbol {x} _ {p}, x _ {t}\\right) = \\sum_ {m = 1} ^ {M} u _ {\\boldsymbol {x} _ {s}} ^ {(m)} \\left(\\boldsymbol {x} _ {s}\\right) u _ {\\boldsymbol {x} _ {p}} ^ {(m)} \\left(\\boldsymbol {x} _ {p}\\right) u _ {x _ {t}} ^ {(m)} \\left(x _ {t}\\right) \\tag {22}",
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+ "type": "text",
+ "content": "Here a general notation is employed to represent different types of components in Eq. 22. For example, the spatial component "
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+ "content": "u_{\\boldsymbol{x}_s}^{(m)}(\\boldsymbol{x}_s)"
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+ "content": " is equivalent to "
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+ "content": "u_{x_1}^{(m)}(x_1)u_{x_2}^{(m)}(x_2)u_{x_3}^{(m)}(x_3)"
+ },
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+ "type": "text",
+ "content": ". The corresponding test function can be obtained using the variational principle:"
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+ }
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+ "content": "\\delta u ^ {T D} \\left(\\boldsymbol {x} _ {s}, \\boldsymbol {x} _ {p}, x _ {t}\\right) = \\sum_ {m = 1} ^ {M} \\left[ \\delta u _ {\\boldsymbol {x} _ {s}} ^ {(m)} \\left(\\boldsymbol {x} _ {s}\\right) u _ {\\boldsymbol {x} _ {p}} ^ {(m)} \\left(\\boldsymbol {x} _ {p}\\right) u _ {x _ {t}} ^ {(m)} \\left(x _ {t}\\right) + u _ {\\boldsymbol {x} _ {s}} ^ {(m)} \\left(\\boldsymbol {x} _ {s}\\right) \\delta u _ {\\boldsymbol {x} _ {p}} ^ {(m)} \\left(\\boldsymbol {x} _ {p}\\right) u _ {x _ {t}} ^ {(m)} \\left(x _ {t}\\right) + u _ {\\boldsymbol {x} _ {s}} ^ {(m)} \\left(\\boldsymbol {x} _ {s}\\right) u _ {\\boldsymbol {x} _ {p}} ^ {(m)} \\left(\\boldsymbol {x} _ {p}\\right) \\delta u _ {x _ {t}} ^ {(m)} \\left(x _ {t}\\right) \\right] \\tag {23}",
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+ }
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+ "type": "text",
+ "content": "Plugging the trial and test functions, the S-P-T Galerkin form of Eq. 18 can be obtained by following Eq. 15:"
+ }
+ ]
+ }
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+ "content": "\\int_ {\\Omega} \\delta u ^ {T D} \\left[ \\rho c _ {p} \\nabla_ {x _ {t}} u ^ {T D} - \\nabla_ {x _ {s}} \\cdot k \\nabla_ {x _ {s}} u ^ {T D} - f \\right] d \\Omega = 0 \\tag {24}",
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+ "content": "Using integration by parts on the diffusion term, we get the corresponding general S-P-T Galerkin weak form in TAPS formulation:"
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+ "content": "\\int_ {\\Omega} \\delta u ^ {T D} \\rho c _ {p} \\nabla_ {x _ {t}} u ^ {T D} d \\Omega + \\int_ {\\Omega} \\nabla_ {\\boldsymbol {x} _ {s}} \\delta u ^ {T D} \\cdot k \\nabla_ {\\boldsymbol {x} _ {s}} u ^ {T D} d \\Omega - \\int_ {\\partial \\boldsymbol {x} _ {s}, \\boldsymbol {x} _ {p}, t} \\delta u ^ {T D} \\boldsymbol {n} \\cdot \\boldsymbol {q} | _ {\\Gamma_ {N}} d s d \\Omega_ {\\boldsymbol {x} _ {p}} d \\Omega_ {t} - \\int_ {\\Omega} \\delta u ^ {T D} f (\\boldsymbol {x} _ {s}, \\boldsymbol {x} _ {p}, x _ {t}) d \\Omega = 0 \\tag {25}",
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+ "content": "where "
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+ "content": " is the heat flux on the Neumann boundary."
+ }
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+ "content": "2.5. Discretized matrix form"
+ }
+ ]
+ }
+ ],
+ "index": 11
+ },
+ {
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+ "type": "text",
+ "content": "The S-P-T Galerkin weak form shown in Eq. 25 is nonlinear in nature due to the tensor product structure in TD interpolation, necessitating efficient solution schemes. To illustrate the detailed solution approach for the general S-P-T weak form, we simplify the governing equation Eq. 18 by considering a one-dimensional spatial problem where "
+ },
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+ "content": "x_{s} = x"
+ },
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+ "type": "text",
+ "content": ". We assume that the product of density and specific heat capacity "
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+ "type": "inline_equation",
+ "content": "\\rho c_{p}"
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+ "type": "text",
+ "content": " is equal to 1. Additionally, the forcing term is solely dependent on "
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+ "type": "inline_equation",
+ "content": "x"
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+ "type": "text",
+ "content": ". Therefore, the simplified governing equation for this example is given by:"
+ }
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+ }
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+ "type": "interline_equation",
+ "content": "\\frac {\\partial u}{\\partial t} - \\frac {\\partial u}{\\partial x} \\cdot k \\frac {\\partial x}{\\partial x} = f (x) \\tag {26}",
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+ "type": "text",
+ "content": "subject to homogeneous boundary conditions and initial conditions. This equation has 3 independent variables ("
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+ "content": "D = 3"
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+ "type": "text",
+ "content": "), i.e., spatial variable "
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+ "type": "inline_equation",
+ "content": "x_{s} = x_{1} = x"
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+ "type": "text",
+ "content": ", parametric variable "
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+ "content": "x_{p} = x_{2} = k"
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+ "content": " and temporal variable "
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+ "type": "inline_equation",
+ "content": "x_{t} = x_{3} = t"
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+ "type": "text",
+ "content": ". The S-P-T"
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+ "content": "Galerkin weak form of this problem can be written as follows according to Eq. 25 (the superscripts \"TD\" for both trial and test functions are omitted for brevity)."
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+ "content": "\\int_ {\\Omega} \\delta u \\nabla_ {t} u d \\Omega + \\int_ {\\Omega} \\nabla_ {x} \\delta u \\cdot k \\nabla_ {x} u d \\Omega - \\int_ {\\Omega} \\delta u f d \\Omega = 0 \\tag {27}",
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+ "content": "The corresponding trial and test functions can be obtained using Eqs. 22-23:"
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+ "content": "As shown in Eq. 29, the test function is further split into "
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+ "type": "text",
+ "content": " dimensional problem (in the current example, "
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+ "content": "D = 3"
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+ "type": "text",
+ "content": "). As an example, we first plug Eq. 28 and the spatial variation term of Eq. 29 into the Galerkin weak form in Eq. 27 to obtain the S-P-T weak form terms corresponding to spatial variation:"
+ }
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+ }
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+ "content": "\\underbrace {\\int_ {\\Omega} \\sum_ {m = 1} ^ {M} \\sum_ {n = 1} ^ {M} \\left[ \\delta u _ {x} ^ {(m)} (x) u _ {x} ^ {(n)} (x) d x \\right] \\cdot \\left[ u _ {k} ^ {(m)} (k) u _ {k} ^ {(n)} (k) d k \\right] \\cdot \\left[ u _ {t} ^ {(m)} (t) \\nabla_ {t} u _ {t} ^ {(n)} (t) d t \\right]} _ {\\text {t i m e d e r i v a t i v e t e r m}} - \\tag {30}",
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+ "content": "\\underbrace {\\int_ {\\Omega} \\sum_ {m = 1} ^ {M} \\left[ \\delta u _ {x} ^ {(m)} (x) f (x) d x \\right] \\cdot \\left[ u _ {k} ^ {(m)} (k) d k \\right] \\cdot \\left[ u _ {t} ^ {(m)} (t) d t \\right]} _ {\\text {f o r c i n g t e r m}}",
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+ "content": "We use 1D C-HiDeNN shape functions to approximate each univariate function:"
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+ "content": "u _ {d} ^ {(n)} (x _ {d}) = \\widetilde {N} _ {n _ {d} ^ {\\prime}} ^ {[ d ]} (x _ {d}) u _ {n _ {d} ^ {\\prime} n} ^ {[ d ]} \\quad (\\text {n o s u m o n} d)",
+ "image_path": "c2e33388cf4eff2cfecbd23aa72b77c80f3c2433581d947423a327c19d842950.jpg"
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+ "content": "\\delta u _ {d} ^ {(m)} \\left(x _ {d}\\right) = \\widetilde {N} _ {n _ {d}} ^ {[ d ]} \\left(x _ {d}\\right) \\delta u _ {n _ {d} m} ^ {[ d ]} \\quad (\\text {n o s u m o n} d) \\tag {31}",
+ "image_path": "7941905e0df1d4b2af4263592fe2ecf1aa7ed89fb9abe6f6c67bb8b0be136701.jpg"
+ }
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+ "type": "text",
+ "content": "where Einstein summation is used. The free index "
+ },
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+ "content": " refers to dimension and "
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+ "content": "d = x,k"
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+ "content": " or "
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+ "type": "text",
+ "content": ". The gradient of the interpolated variable can be computed using the shape function derivative "
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+ "type": "inline_equation",
+ "content": "\\widetilde{B}_{n_d}^{[d]}(x_d) = \\frac{d\\widetilde{N}_{n_d}^{[d]}(x_d)}{dx_d}"
+ },
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+ "type": "text",
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+ "content": "\\nabla_ {x _ {d}} u _ {d} ^ {(n)} (x _ {d}) = \\widetilde {B} _ {n _ {d} ^ {\\prime}} ^ {[ d ]} (x _ {d}) u _ {n _ {d} ^ {\\prime} n} ^ {[ d ]} \\quad (\\text {n o s u m o n} d)",
+ "image_path": "6228caa736b34f1a766b1c32d5292f854237e8521a4f50f7ab19126515009330.jpg"
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+ "content": "\\nabla_ {x _ {d}} \\delta u _ {d} ^ {(m)} (x _ {d}) = \\widetilde {B} _ {n _ {d}} ^ {[ d ]} (x _ {d}) \\delta u _ {n _ {d} m} ^ {[ d ]} \\quad (\\text {n o s u m o n} d) \\tag {32}",
+ "image_path": "43c0b7c65e4bba26c746c9a922247c0398fa273e4791579a9699590c69d1bac1.jpg"
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+ "type": "text",
+ "content": "Plugging Eq. 31 - 32 into Eq. 30, the diffusion term can be rewritten as:"
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+ }
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+ "type": "interline_equation",
+ "content": "\\sum_ {m = 1} ^ {M} \\sum_ {n = 1} ^ {M} \\underbrace {\\int_ {\\Omega_ {x}} \\widetilde {B} _ {n _ {x}} (x) \\delta u _ {n _ {x} m} ^ {[ x ]} \\widetilde {B} _ {n _ {x} ^ {\\prime}} (x) u _ {n _ {x} ^ {\\prime} n} ^ {[ x ]} d x} _ {\\text {s p a t i a l t e r m}} \\cdot \\underbrace {\\int_ {\\Omega_ {k}} \\widetilde {N} _ {n _ {k}} (k) u _ {n _ {k} m} ^ {[ k ]} k \\widetilde {N} _ {n _ {k} ^ {\\prime}} (k) u _ {n _ {k} ^ {\\prime} n} ^ {[ k ]} d k} _ {\\text {p a r a m e t r i c t e r m}} \\cdot \\underbrace {\\int_ {\\Omega_ {t}} \\widetilde {N} _ {n _ {t}} (t) u _ {n _ {t} m} ^ {[ t ]} \\widetilde {N} _ {n _ {t} ^ {\\prime}} (t) u _ {n _ {t} ^ {\\prime} n} ^ {[ t ]} d t} _ {\\text {t e m p o r a l t e r m}} \\tag {33}",
+ "image_path": "af36e8b78f8671a9d8dbb5b6d0a5894167bc9a3b14b2f2e6073c308d17cc95a0.jpg"
+ }
+ ]
+ }
+ ],
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+ {
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+ "type": "text",
+ "content": "As can be readily seen from Eq. 33, after doing 1D integration of each term, the parametric and temporal terms can be treated as coefficient matrices:"
+ }
+ ]
+ }
+ ],
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+ }
+ ],
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+ "content": "C _ {m n} ^ {[ k ]} = \\underbrace {\\int_ {\\Omega_ {k}} \\widetilde {N} _ {n _ {k}} (k) u _ {n _ {k} m} ^ {[ k ]} k \\widetilde {N} _ {n _ {k} ^ {\\prime}} (k) u _ {n _ {k} ^ {\\prime} n} ^ {[ k ]} d k} _ {\\text {p a r a m e t r i c t e r m}}",
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+ "content": "C _ {m n} ^ {[ t ]} = \\underbrace {\\int_ {\\Omega_ {t}} \\widetilde {N} _ {n _ {t}} (t) u _ {n _ {t} m} ^ {[ t ]} \\widetilde {N} _ {n _ {t} ^ {\\prime}} (t) u _ {n _ {t} ^ {\\prime} n} ^ {[ t ]} d t} _ {\\text {t e m p o r a l t e r m}} \\tag {34}",
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+ "type": "text",
+ "content": "as the only free indices are "
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+ "type": "text",
+ "content": ". Substituting the coefficient matrices and rearranging different terms in Eq. 33, we have:"
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+ "content": "\\sum_ {m = 1} ^ {M} \\delta u _ {n _ {x} m} ^ {[ x ]} \\sum_ {n = 1} ^ {M} \\left[ \\int_ {\\Omega_ {x}} \\widetilde {B} _ {n _ {x}} (x) \\widetilde {B} _ {n _ {x} ^ {\\prime}} (x) d x \\right] \\cdot C _ {m n} ^ {[ k ]} C _ {m n} ^ {[ t ]} \\cdot u _ {n _ {x} ^ {\\prime} n} ^ {[ x ]} \\tag {35}",
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+ }
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+ "type": "inline_equation",
+ "content": "\\int_{x}\\widetilde{B}_{n_x}(x)\\widetilde{B}_{n_x'}(x)dx"
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+ {
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+ "content": " as the 1D stiffness matrix "
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+ "type": "inline_equation",
+ "content": "K_{n_x n_x'}^{[x]}"
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+ "type": "text",
+ "content": " of "
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+ "type": "inline_equation",
+ "content": "x"
+ },
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+ "type": "text",
+ "content": " dimension in Eq. 35. We let "
+ },
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+ "type": "inline_equation",
+ "content": "C_{mn}^{[x]} = C_{mn}^{[k]}C_{mn}^{[t]}"
+ },
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+ "bbox": [
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+ "type": "text",
+ "content": " with no summation on "
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+ "type": "inline_equation",
+ "content": "(m,n)"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": ". Furthermore, let us define the following 4-th order tensor:"
+ }
+ ]
+ }
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+ },
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+ "content": "A _ {n _ {x} n _ {x} ^ {\\prime} m n} ^ {[ x ]} = K _ {n _ {x} n _ {x} ^ {\\prime}} ^ {[ x ] ^ {\\prime}} C _ {m n} ^ {[ x ]} \\tag {36}",
+ "image_path": "a018defcc9f90fd14ccd98acbd00e8f58264429bc1da13c7878a25904266e95c.jpg"
+ }
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+ }
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+ "type": "inline_equation",
+ "content": "A_{n_s n_s'mn}^{[x]}"
+ },
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+ "type": "text",
+ "content": " is a function of solution vectors "
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+ "type": "inline_equation",
+ "content": "u_{n_k m}^{[k]}"
+ },
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+ "type": "text",
+ "content": " and "
+ },
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+ "bbox": [
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+ "type": "inline_equation",
+ "content": "u_{n_m m}^{[t]}"
+ },
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+ "type": "text",
+ "content": " since the coefficient matrix "
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+ {
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+ "type": "inline_equation",
+ "content": "C_{mn}^{[x]}"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": " depends on these solution vectors as shown in Eq. 34. This dependency reflects the interconnected nature of the variables across different dimensions in the S-P-T framework, highlighting how the spatial, parameter, and temporal components influence each other through the coefficients. As a result, Eq. 33 can be further simplified as follows:"
+ }
+ ]
+ }
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+ "type": "interline_equation",
+ "content": "\\delta u _ {n _ {x} m} ^ {[ x ]} A _ {n _ {x} n _ {x} ^ {\\prime} m n} ^ {[ x ]} u _ {n _ {x} ^ {\\prime} n} ^ {[ x ]} \\tag {37}",
+ "image_path": "2850ea393a41fa50b8726a0787ed58a81109d8f74bcacd2cf3ae869fd55d92b6.jpg"
+ }
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+ }
+ ],
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+ "type": "text",
+ "content": "where the summation signs are neglected since "
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+ "content": "m"
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+ "type": "text",
+ "content": " and "
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+ "type": "text",
+ "content": " become dummy variables. The 4-th order tensor "
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+ "type": "inline_equation",
+ "content": "A_{n_x n_x' mn}^{[x]}"
+ },
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+ "type": "text",
+ "content": " can be reshaped as a 2nd order tensor "
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+ "type": "inline_equation",
+ "content": "\\mathbb{A}_{IJ}^{[x]}"
+ },
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+ "type": "text",
+ "content": ": the indices "
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+ "type": "inline_equation",
+ "content": "n_x"
+ },
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+ "type": "text",
+ "content": " and "
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+ {
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+ "type": "inline_equation",
+ "content": "m"
+ },
+ {
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+ ],
+ "type": "text",
+ "content": " are combined into a single composite index "
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+ {
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+ "type": "inline_equation",
+ "content": "I"
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+ "type": "text",
+ "content": ", and the indices "
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+ {
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+ "type": "inline_equation",
+ "content": "n_x'"
+ },
+ {
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+ "type": "text",
+ "content": " and "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "n"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": " are combined into a single composite index "
+ },
+ {
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+ ],
+ "type": "inline_equation",
+ "content": "J"
+ },
+ {
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+ ],
+ "type": "text",
+ "content": "."
+ }
+ ]
+ }
+ ],
+ "index": 8
+ },
+ {
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+ ],
+ "type": "interline_equation",
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+ "lines": [
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+ "type": "interline_equation",
+ "content": "A _ {n _ {x} n _ {x} ^ {\\prime} m n} ^ {[ x ]} = \\mathbb {A} _ {I J} ^ {[ x ]} \\tag {38}",
+ "image_path": "88a6dfc3df4ae1b1e46ae11bb2c9bb454322138eaa5dae837daeb486544ab148.jpg"
+ }
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+ }
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+ "type": "text",
+ "content": "Define the following vectorization:"
+ }
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+ }
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+ "type": "interline_equation",
+ "content": "\\delta \\mathbb {U} _ {I} ^ {[ x ]} = \\left[ \\operatorname {v e c} \\left(\\delta u _ {n _ {x} m} ^ {[ x ]}\\right) \\right] _ {I}",
+ "image_path": "1ff792e01f6a4782e6a7a960c755bc1261ea125e23cd82a65e43e7b723d1f98f.jpg"
+ }
+ ]
+ }
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+ "type": "interline_equation",
+ "content": "\\mathbb {U} _ {J} ^ {[ x ]} = \\left[ \\operatorname {v e c} \\left(u _ {n _ {x} ^ {\\prime} n} ^ {[ x ]}\\right) \\right] _ {J} \\tag {39}",
+ "image_path": "2070e61fd49ddc5fc7f11d25203116c3b06771ac17d03dde98df5bdaae2bcd76.jpg"
+ }
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+ }
+ ],
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+ "type": "text",
+ "content": "As a result, Eq. 37 is equivalent to:"
+ }
+ ]
+ }
+ ],
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+ "type": "interline_equation",
+ "content": "\\delta \\mathbb {U} ^ {[ x ] T} \\mathbb {A} ^ {[ x ]} \\mathbb {U} ^ {[ x ]} \\tag {40}",
+ "image_path": "455e449627117c21f40ecb4a6ab5bd0b4fee5a46f3b2c0e54b88fc6f4f3fa290.jpg"
+ }
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+ }
+ ],
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+ ],
+ "type": "text",
+ "content": "Following the same procedure, we can obtain matrix forms corresponding to the time derivative term "
+ },
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+ "type": "inline_equation",
+ "content": "\\delta \\mathbb{U}^{[x]^T}\\mathbb{B}^{[x]}\\mathbb{U}^{[x]}"
+ },
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+ "type": "text",
+ "content": ", and the forcing term "
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+ "type": "inline_equation",
+ "content": "\\delta \\mathbb{U}^{[x]^T}\\mathbb{Q}^{[x]}"
+ },
+ {
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+ ],
+ "type": "text",
+ "content": " for the spatial variational part of Eq. 30. Similar structures can also be obtained for the parametric and temporal variational parts of the test function in Eq. 29. Denoting "
+ },
+ {
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+ "type": "inline_equation",
+ "content": "\\mathbb{K}^{[d]} = \\mathbb{A}^{[d]} + \\mathbb{B}^{[d]}"
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+ "type": "text",
+ "content": ", the matrix form of the generalized S-P-T Galerkin form in Eq. 27 can be written as:"
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+ }
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+ "type": "interline_equation",
+ "content": "\\underbrace {\\delta \\mathbb {U} ^ {[ x ] ^ {T}} \\mathbb {K} ^ {[ x ]} \\mathbb {U} ^ {[ x ]} - \\delta \\mathbb {U} ^ {[ x ] ^ {T}} \\mathbb {Q} ^ {[ x ]}} _ {\\text {s p a t i a l v a r i a t i o n a l p a r t}} + \\underbrace {\\delta \\mathbb {U} ^ {[ k ] ^ {T}} \\mathbb {K} ^ {[ k ]} \\mathbb {U} ^ {[ k ]} - \\delta \\mathbb {U} ^ {[ k ] ^ {T}} \\mathbb {Q} ^ {[ k ]}} _ {\\text {p a r a m e t r i c v a r i a t i o n a l p a r t}} + \\underbrace {\\delta \\mathbb {U} ^ {[ t ] ^ {T}} \\mathbb {K} ^ {[ t ]} \\mathbb {U} ^ {[ t ]} - \\delta \\mathbb {U} ^ {[ t ] ^ {T}} \\mathbb {Q} ^ {[ t ]}} _ {\\text {t e m p o r a l v a r i a t i o n a l p a r t}} = 0 \\tag {41}",
+ "image_path": "1ffca6770318f49f4d362fee97c27f8a0a54ef98fbc6d78dd5736e0864301f88.jpg"
+ }
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+ }
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+ "type": "text",
+ "content": "Eq. 41 is equivalent to the following nonlinear system of equations. Note that the nonlinearity comes from the fact that "
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+ "type": "text",
+ "content": " is solution dependent:"
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+ "type": "interline_equation",
+ "content": "\\left[ \\delta \\mathbb {U} ^ {[ x ] ^ {T}}, \\delta \\mathbb {U} ^ {[ k ] ^ {T}}, \\delta \\mathbb {U} ^ {[ t ] ^ {T}} \\right] \\left\\{\\left[ \\begin{array}{c c c} \\mathbb {K} ^ {[ x ]} (\\mathbb {U} ^ {[ k ]}, \\mathbb {U} ^ {[ t ]}) & 0 & 0 \\\\ 0 & \\mathbb {K} ^ {[ k ]} (\\mathbb {U} ^ {[ x ]}, \\mathbb {U} ^ {[ t ]}) & 0 \\\\ 0 & 0 & \\mathbb {K} ^ {[ t ]} (\\mathbb {U} ^ {[ x ]}, \\mathbb {U} ^ {[ k ]}) \\end{array} \\right] \\left[ \\begin{array}{l} \\mathbb {U} ^ {[ x ]} \\\\ \\mathbb {U} ^ {[ k ]} \\\\ \\mathbb {U} ^ {[ t ]} \\end{array} \\right] - \\left[ \\begin{array}{l} \\mathbb {Q} ^ {[ x ]} (\\mathbb {U} ^ {[ k ]}, \\mathbb {U} ^ {[ t ]}) \\\\ \\mathbb {Q} ^ {[ k ]} (\\mathbb {U} ^ {[ x ]}, \\mathbb {U} ^ {[ t ]}) \\\\ \\mathbb {Q} ^ {[ t ]} (\\mathbb {U} ^ {[ x ]}, \\mathbb {U} ^ {[ k ]}) \\end{array} \\right] \\right\\} = 0 \\tag {42}",
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+ "type": "text",
+ "content": "where we can treat the solution vector "
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+ "type": "inline_equation",
+ "content": "\\left[\\mathbb{U}^{[x]^T},\\mathbb{U}^{[k]^T},\\mathbb{U}^{[t]^T}\\right]"
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+ "content": " as generalized DoFs like standard FEM. There are many ways to solve Eq. 42. For example, standard linearization schemes such as Newton's method have been used [29]. However, this method may suffer from ill-conditioning since the mismatch of scales for different dimensions can be significant. In this paper, we use the concept of subspace iteration to efficiently approximate the solution by iterating in the subspace of the test function space until a convergence criteria is met [19]. A similar counterpart has been widely adopted as the gold standard in discrete tensor decomposition [23]."
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+ "type": "inline_equation",
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+ "type": "text",
+ "content": " ..., "
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+ "type": "inline_equation",
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+ "type": "text",
+ "content": " with random values and compute "
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+ "type": "inline_equation",
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+ "type": "text",
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+ "content": "\\mathbb{K}^{[x_d][\\mathcal{K} - 1]}\\mathbb{U}^{[x_d][\\mathcal{K}]} = \\mathbb{Q}^{[x_d][\\mathcal{K} - 1]}"
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+ "type": "text",
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+ "type": "text",
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+ "type": "text",
+ "content": "-th iteration. Moreover, instead of the full variation form of the test function as in Eq. 42, we only consider the subspace "
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+ "type": "text",
+ "content": " of the test function by setting the parametric and temporal variational parts as 0. As a result, we have:"
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+ "content": ". This is a general Sylvester equation which can be solved using many efficient solution schemes [30, 31]. In this paper, sparse direct solvers based on fast diagonalization/complex Schur decomposition methods are adopted [32]. The computational complexity of the sparse direct solver is "
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+ "content": " is obtained, we then update matrix "
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+ "type": "inline_equation",
+ "content": "\\mathbb{K}^{[k][\\mathcal{K}]}(\\mathbb{U}^{[x][\\mathcal{K}]},\\mathbb{U}^{[t][\\mathcal{K}]})"
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+ "content": " and forcing vector "
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+ "type": "inline_equation",
+ "content": "\\mathbb{Q}^{[k][\\mathcal{K}]}(\\mathbb{U}^{[x][\\mathcal{K}]},\\mathbb{U}^{[t][\\mathcal{K}]})"
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+ "type": "text",
+ "content": ". In the next iteration (for dimension "
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+ "content": "k"
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+ "type": "text",
+ "content": "), we treat "
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+ "content": "\\mathbb{U}^{[k][\\mathcal{K} + 1]}"
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+ "type": "text",
+ "content": " as the only unknown. Subspace iteration will continue unless the relative change of all solution matrices (for example, "
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+ "type": "inline_equation",
+ "content": "L_{2}"
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+ "type": "text",
+ "content": " norm) is within the tolerance."
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+ "content": "Since the TAPS solution is based on the C-HiDeNN-TD approximation and the generalized Galerkin formulation, we can have the following theoretical results on the error bounds, as demonstrated in our previous work on C-HiDeNN [6]:"
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+ "content": "\\left\\| u ^ {\\mathrm {C} - \\mathrm {H i D e N N}} - u ^ {\\mathrm {e x}} \\right\\| _ {E} \\leq \\left\\| u ^ {\\mathrm {T A P S}} - u ^ {\\mathrm {e x}} \\right\\| _ {E} \\leq \\left\\| u ^ {\\mathrm {F E M}} - u ^ {\\mathrm {e x}} \\right\\| _ {E} \\tag {44}",
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+ "content": "First, we investigate the influence of the number of subspace iterations. As shown in Fig. 8(a), 3 iterations are enough to obtain an accurate result. Next, we investigate the convergence in terms of the number of modes. Here we compare the relative "
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+ "content": " norm error for both TAPS and proper generalized decomposition (PGD) methods [17, 18]. To this aim, we use the same discretization for the space-time domain with each dimension discretized by 100 grid points, the same reproducing polynomial order "
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+ "content": ". As can be seen from Fig. 8(b), TAPS requires a much smaller number of modes than PGD. For TAPS, when the number of modes equals 25, the relative "
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+ "type": "text",
+ "content": " where homogeneous Dirichlet boundary conditions are assumed for the left and right surfaces; homogeneous Neumann boundary conditions are applied to all other surfaces. As shown in Eq. 19, a moving Gaussian source term "
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+ "content": ". The diameter is discretized using 10 spatial elements."
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+ "content": "Since there is no analytical solution available to this problem, we use implicit finite element analysis as the baseline for validation. JAX-FEM [34] is used to generate the nonlinear FEM solution. For ease of comparison, we use the same time increment as "
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+ "content": " sec for both TAPS and FEM. The solution along the center line, as shown in Fig. 11 (b) is compared. As can be seen from Fig. 12, the result of the nonlinear TAPS solver agrees well with FEM."
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+ "content": "In this section, we use TAPS to efficiently model the laser powder bed fusion process (LPBF) in additive manufacturing. Here we only consider the free convection term in the Neumann boundary condition. The initial condition can be considered by splitting the total solution as a summation of the homogeneous part and the inhomogeneous part."
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+ "content": "u \\left(\\boldsymbol {x} _ {s}, x _ {t}\\right) = u _ {0} \\left(\\boldsymbol {x} _ {s}, x _ {t}\\right) + u _ {\\text {i n i t}} \\left(\\boldsymbol {x} _ {s}\\right) \\tag {53}",
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+ "type": "text",
+ "content": " is subject to homogeneous initial conditions. In this section, we assume Ti-6Al-4V is used as the powder bed materials. The detailed material parameters can be found in Table 4."
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+ "content": "In this example, we investigate the computational complexity numerically for single-track LPBF simulation with a single S-T slab, as shown in Fig. 13 (a). A single NVIDIA RTX A6000 GPU is used for all the following analyses. To ensure accuracy, the number of modes is adopted as 5 times larger than the number of time steps in the following examples. In the first case, within the S-T slab, the spatial mesh is refined uniformly along each spatial dimension"
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