Daily Papers

Safety-critical applications such as autonomous vehicles and social robots require fast computation and accurate probability density estimation on trajectory prediction. To address both requirements, this paper presents a new normalizing flow-based trajectory prediction model named FlowChain. FlowChain is a stack of conditional continuously-indexed flows (CIFs) that are expressive and allow analytical probability density computation. This analytical computation is faster than the generative models that need additional approximations such as kernel density estimation. Moreover, FlowChain is more accurate than the Gaussian mixture-based models due to fewer assumptions on the estimated density. FlowChain also allows a rapid update of estimated probability densities. This update is achieved by adopting the newest observed position and reusing the flow transformations and its log-det-jacobians that represent the motion trend. This update is completed in less than one millisecond because this reuse greatly omits the computational cost. Experimental results showed our FlowChain achieved state-of-the-art trajectory prediction accuracy compared to previous methods. Furthermore, our FlowChain demonstrated superiority in the accuracy and speed of density estimation. Our code is available at https://github.com/meaten/FlowChain-ICCV2023

Bayesian neural networks (BNNs) have been an important framework in the study of uncertainty quantification. Deterministic variational inference, one of the inference methods, utilizes moment propagation to compute the predictive distributions and objective functions. Unfortunately, deriving the moments requires computationally expensive Taylor expansion in nonlinear functions, such as a rectified linear unit (ReLU) or a sigmoid function. Therefore, a new nonlinear function that realizes faster moment propagation than conventional functions is required. In this paper, we propose a novel nonlinear function named moment propagating-Gaussian error linear unit (MP-GELU) that enables the fast derivation of first and second moments in BNNs. MP-GELU enables the analytical computation of moments by applying nonlinearity to the input statistics, thereby reducing the computationally expensive calculations required for nonlinear functions. In empirical experiments on regression tasks, we observed that the proposed MP-GELU provides higher prediction accuracy and better quality of uncertainty with faster execution than those of ReLU-based BNNs.

The cosmology of the F(R) gravity rebuilding by the Cartan formalism is investigated. This is called Cartan F(R) gravity. The well-known F(R) gravity has been introduced to extend the standard cosmology, e.g. to explain the cosmological accelerated expansion as the inflation. Cartan F(R) gravity is based on the Riemann-Cartan geometry. The curvature R can separate to two parts, one is derived from the Levi-Civita connection and the other from the torsion. Assuming the matter-independent spin connection, we have successfully rewritten the action of Cartan F(R) gravity into the Einstein-Hilbert action and a scalar field with canonical kinetic and potential terms without any conformal transformations. This feature simplifies building and analysis of new model of inflation. In this paper, we study two models, the power-law model and logarithmic model, and evaluate fluctuations in the cosmological microwave background (CMB) radiation. We found the robustness of CMB fluctuation by the analytical computation and confirm this feature by the numerical calculation.

Training deep generative models like Variational Autoencoders (VAEs) is often hindered by the need to backpropagate gradients through the stochastic sampling of their latent variables, a process that inherently introduces estimation variance, which can slow convergence and degrade performance. In this paper, we propose a new perspective that sidesteps this problem, which we call Silent Gradients. Instead of improving stochastic estimators, we leverage specific decoder architectures to analytically compute the expected ELBO, yielding a gradient with zero variance. We first provide a theoretical foundation for this method and demonstrate its superiority over existing estimators in a controlled setting with a linear decoder. To generalize our approach for practical use with complex, expressive decoders, we introduce a novel training dynamic that uses the exact, zero-variance gradient to guide the early stages of encoder training before annealing to a standard stochastic estimator. Our experiments show that this technique consistently improves the performance of established baselines, including reparameterization, Gumbel-Softmax, and REINFORCE, across multiple datasets. This work opens a new direction for training generative models by combining the stability of analytical computation with the expressiveness of deep, nonlinear architecture.

We propose a new dataset distillation algorithm using reparameterization and convexification of implicit gradients (RCIG), that substantially improves the state-of-the-art. To this end, we first formulate dataset distillation as a bi-level optimization problem. Then, we show how implicit gradients can be effectively used to compute meta-gradient updates. We further equip the algorithm with a convexified approximation that corresponds to learning on top of a frozen finite-width neural tangent kernel. Finally, we improve bias in implicit gradients by parameterizing the neural network to enable analytical computation of final-layer parameters given the body parameters. RCIG establishes the new state-of-the-art on a diverse series of dataset distillation tasks. Notably, with one image per class, on resized ImageNet, RCIG sees on average a 108% improvement over the previous state-of-the-art distillation algorithm. Similarly, we observed a 66% gain over SOTA on Tiny-ImageNet and 37% on CIFAR-100.

Monocular-to-stereo conversion synthesizes stereoscopic content from 2D videos for immersive 3D experiences. In modern Depth-Image-Based Rendering (DIBR) approaches, stereo inpainting of disocclusions is the critical bottleneck. Training-based methods achieve superior quality but rely on scarce stereo pairs or synthetic data with domain gaps. We address this through the first self-supervised framework learning from monocular videos via cycle consistency. Our key contribution is the Geometric Reciprocity Theorem (GRT): under the nearest-neighbor DIBR formulation, the disocclusion mask when synthesizing a target view equals the mask of pixels lost when warping back from target to source, enabling analytical computation of test-time disocclusion masks directly from monocular images. This yields train-test consistency for the stated warping formulation, supporting self-supervised learning from unlimited monocular videos and substantial improvements over training-free and supervised state-of-the-art methods. Project page: https://visual-ai.github.io/grt/

Evaluating enterprise agents on domain-specific benchmarks is critical, yet public benchmarks rarely evaluate whether agents can integrate business knowledge with analytical computation, and constructing such benchmarks manually is costly. We present DI-Bench, a pipeline for generating realistic benchmarks for data intelligence (DI), the practice of extracting insights from large volumes of enterprise data. To emulate realistic DI tasks that require both computation and knowledge retrieval, DI-Bench builds an artifact linkage graph over data tables, dimensions, metrics, and documents to form questions involving structured data and associated knowledge. Ground truth answers are derived via query execution, followed by LLM question generation and validation. Applied to two public datasets, the pipeline produces a 731-task benchmark covering knowledge retrieval, analytical computation, and rule-grounded reasoning. To show the discriminatory capability and difficulty of the benchmark, we evaluate four models, revealing a substantial finding: models achieve only 32% accuracy when doing computational tasks where retrieved business rules modify the computation.

We consider the matrix composite materials (CM) of either random (statistically homogeneous or inhomogeneous), periodic, or deterministic (neither random nor periodic) structures. CMs exhibit linear or nonlinear behavior, coupled or uncoupled multi-physical phenomena, locally elastic, weakly nonlocal (strain gradient and stress gradient), or strongly nonlocal (strain-type and displacement-type, peridynamics) phase properties. A modified Computational Analytical Micromechanics (CAM) approach introduces an exact Additive General Integral Equation (AGIE) for CMs of any structure and phase properties mentioned above. The unified iteration solution of static AGIEs is adapted to the body force with compact support serving as a fundamentally new universal training parameter. The approach also establishes a critical threshold for filtering out unsuitable sub-datasets of effective parameters through a novel Representative Volume Element (RVE) concept, which extends Hill's classical framework. This RVE concept eliminates sample size, boundary layer, and edge effects, making it applicable to CMs of any structure and phase properties, regardless of local or nonlocal, linear or nonlinear. Incorporating this new RVE concept into machine learning and neural network techniques enables the construction of any unpredefined surrogate nonlocal operators. The methodology is structured as a modular, block-based framework, allowing independent development and refinement of software components. This flexible, robust AGIE-CAM framework integrates data-driven, multi-scale, and multi-physics modeling, accelerating research in CM of any microtopology and phase properties considered. The AGIE-CAM framework represents a groundbreaking paradigm shift in the micromechanics of composites, redefining the very philosophy that underpins our understanding of their behavior at the microscopic level.

Large language models (LLMs) have shown remarkable progress in coding and math problem-solving, but evaluation on advanced research-level problems in hard sciences remains scarce. To fill this gap, we present CMT-Benchmark, a dataset of 50 problems covering condensed matter theory (CMT) at the level of an expert researcher. Topics span analytical and computational approaches in quantum many-body, and classical statistical mechanics. The dataset was designed and verified by a panel of expert researchers from around the world. We built the dataset through a collaborative environment that challenges the panel to write and refine problems they would want a research assistant to solve, including Hartree-Fock, exact diagonalization, quantum/variational Monte Carlo, density matrix renormalization group (DMRG), quantum/classical statistical mechanics, and model building. We evaluate LLMs by programmatically checking solutions against expert-supplied ground truth. We developed machine-grading, including symbolic handling of non-commuting operators via normal ordering. They generalize across tasks too. Our evaluations show that frontier models struggle with all of the problems in the dataset, highlighting a gap in the physical reasoning skills of current LLMs. Notably, experts identified strategies for creating increasingly difficult problems by interacting with the LLMs and exploiting common failure modes. The best model, GPT5, solves 30\% of the problems; average across 17 models (GPT, Gemini, Claude, DeepSeek, Llama) is 11.4pm2.1\%. Moreover, 18 problems are solved by none of the 17 models, and 26 by at most one. These unsolved problems span Quantum Monte Carlo, Variational Monte Carlo, and DMRG. Answers sometimes violate fundamental symmetries or have unphysical scaling dimensions. We believe this benchmark will guide development toward capable AI research assistants and tutors.

This work presents an enhanced Computational Analytical Micromechanics (CAM) framework for the analysis of linear thermoelastic composite materials (CMs) with random microstructure. The proposed approach is grounded in an exact Additive General Integral Equation (AGIE), specifically formulated for compactly supported loading, including both body forces and localized thermal changes (such as those from laser heating). New general integral equations (GIEs) for arbitrary mechanical and thermal loading are proposed. A unified iterative solution strategy is developed for the static AGIE, applicable to CMs with both perfectly and imperfectly bonded interfaces, where the compact support of loading is introduced as a new fundamental training parameter. Central to this methodology is a generalized Representative Volume Element (RVE) concept, which extends Hill classical definition. The resulting RVE is not predefined geometrically, but rather emerges from the characteristic scale of the localized loading, effectively reducing the analysis of an infinite, randomly heterogeneous medium to a finite, data-driven domain. This generalized RVE approach enables automatic exclusion of unrepresentative subsets of effective parameters, while inherently eliminating boundary effects, edge artifacts, and finite size limitations. Moreover, the AGIE-based CAM framework is naturally compatible with machine learning (ML) and neural network (NN) architectures, facilitating the construction of accurate and physically informed surrogate nonlocal operators.

This paper introduces an advanced Computational Analytical Micromechanics (CAM) framework for linear thermoelastic composites (CMs) with periodic microstructures. The approach is based on an exact new Additive General Integral Equation (AGIE), formulated for compactly supported loading conditions, such as body forces and localized thermal effects (for example laser heating). In addition, new general integral equations (GIEs) are established for arbitrary mechanical and thermal loading. A unified iterative scheme is developed for solving the static AGIEs, where the compact support of loading serves as a new fundamental training parameter. At the core of the methodology lies a generalized Representative Volume Element (RVE) concept that extends Hill classical definition of the RVE. Unlike conventional RVEs, this generalized RVE is not fixed geometrically but emerges naturally from the characteristic scale of localized loading, thereby reducing the analysis of an infinite periodic medium to a finite, data-driven domain. This formulation automatically filters out nonrepresentative subsets of effective parameters while eliminating boundary effects, edge artifacts, and finite-size sample dependencies. Furthermore, the AGIE-based CAM framework integrates seamlessly with machine learning (ML) and neural network (NN) architectures, supporting the development of accurate, physics-informed surrogate nonlocal operators.

Encoding input coordinates with sinusoidal functions into multi-layer perceptrons (MLPs) has proven effective for implicit neural representations (INRs) of surfaces defined as zero-level sets. However, existing methods often struggle to balance training efficiency, rendering speed, and noise robustness: single-MLP approaches are expensive at inference, grid-based representations are fast but can limit surface smoothness and overfit input noise, and previous multiscale approaches frequently capture noise and produce artifacts due to hard spectral truncation. To address these limitations, we propose M-plicits, a multiscale framework that models surfaces as a residual sum of MLPs trained via a sequence of nested neighborhoods. Unlike existing residual approaches that rely on standard domain-wide sampling and require costly mesh extraction for visualization, our method strictly localizes supervision to narrow bands around the previous zero-level sets. This nested design naturally provides robustness against noisy input data: the coarse network acts as a low-pass filter that establishes a clean geometric prior, while subsequent residuals progressively refine the geometry without fitting to high-frequency artifacts. We further introduce a multiscale sphere-tracing algorithm and a GEMM-based analytical normal computation that bypasses auto-differentiation entirely, yielding high-fidelity real-time rendering. On Stanford and Thingi32, M-plicits achieves the best mean Chamfer distance in the coarse configuration and the best median Chamfer distance and IoU in the fine configuration, with substantially better noise robustness than iNGP, BACON, and IDF, while using an order of magnitude fewer parameters than grid-based baselines. Code, models, and data are available at https://github.com/dsilvavinicius/m-plicits.

Fluorescence LiDAR (FLiDAR), a Light Detection and Ranging (LiDAR) technology employed for distance and depth estimation across medical, automotive, and other fields, encounters significant computational challenges in scattering media. The complex nature of the acquired FLiDAR signal, particularly in such environments, makes isolating photon time-of-flight (related to target depth) and intrinsic fluorescence lifetime exceptionally difficult, thus limiting the effectiveness of current analytical and computational methodologies. To overcome this limitation, we present a Physics-Guided Mixture-of-Experts (MoE) framework tailored for specialized modeling of diverse temporal components. In contrast to the conventional MoE approaches our expert models are informed by underlying physics, such as the radiative transport equation governing photon propagation in scattering media. Central to our approach is EvidenceMoE, which integrates Evidence-Based Dirichlet Critics (EDCs). These critic models assess the reliability of each expert's output by providing per-expert quality scores and corrective feedback. A Decider Network then leverages this information to fuse expert predictions into a robust final estimate adaptively. We validate our method using realistically simulated Fluorescence LiDAR (FLiDAR) data for non-invasive cancer cell depth detection generated from photon transport models in tissue. Our framework demonstrates strong performance, achieving a normalized root mean squared error (NRMSE) of 0.030 for depth estimation and 0.074 for fluorescence lifetime.

Breast Cancer is a major public health problem and the most common diagnosed malignancy in woman. There have been significant developments in clinical approaches and theoretical experimental to understand the interactions of cancer cells dynamics with the immune system, also developments on analytical and computational models to help provide insights into clinical observations for a better understanding of cancer cells, but more are needed, especially at the genetic and molecular levels mathematically. Treatments such as immunotherapy, chemotherapy, hormone therapy, radiotherapy, and gene therapy are the main strategies in the fight against breast cancer. The present study aims at investigating the effects of estrogen derived from recent models, but this time combined with immunotherapy as a way to treat or inhibit the cancer growth by a mathematical model of breast cancer in situ, governed by a simplified model of nonlinear-coupled ordinary differential equations, that combines important interactions between natural cells, tumor cells, immune cells, ketogenic diet in the presence of an anticancer drug. Another contribution was to introduce the inhibition effect epsilon for new results and conclusions, A qualitative study was performed and biological interpretations were included to understand the conditions of stability in a realistic way.

As the demand for emotional intelligence in large language models (LLMs) grows, a key challenge lies in understanding the internal mechanisms that give rise to emotional expression and in controlling emotions in generated text. This study addresses three core questions: (1) Do LLMs contain context-agnostic mechanisms shaping emotional expression? (2) What form do these mechanisms take? (3) Can they be harnessed for universal emotion control? We first construct a controlled dataset, SEV (Scenario-Event with Valence), to elicit comparable internal states across emotions. Subsequently, we extract context-agnostic emotion directions that reveal consistent, cross-context encoding of emotion (Q1). We identify neurons and attention heads that locally implement emotional computation through analytical decomposition and causal analysis, and validate their causal roles via ablation and enhancement interventions. Next, we quantify each sublayer's causal influence on the model's final emotion representation and integrate the identified local components into coherent global emotion circuits that drive emotional expression (Q2). Directly modulating these circuits achieves 99.65% emotion-expression accuracy on the test set, surpassing prompting- and steering-based methods (Q3). To our knowledge, this is the first systematic study to uncover and validate emotion circuits in LLMs, offering new insights into interpretability and controllable emotional intelligence.

We introduce ToRL (Tool-Integrated Reinforcement Learning), a framework for training large language models (LLMs) to autonomously use computational tools via reinforcement learning. Unlike supervised fine-tuning, ToRL allows models to explore and discover optimal strategies for tool use. Experiments with Qwen2.5-Math models show significant improvements: ToRL-7B reaches 43.3\% accuracy on AIME~24, surpassing reinforcement learning without tool integration by 14\% and the best existing Tool-Integrated Reasoning (TIR) model by 17\%. Further analysis reveals emergent behaviors such as strategic tool invocation, self-regulation of ineffective code, and dynamic adaptation between computational and analytical reasoning, all arising purely through reward-driven learning.

We introduce a novel Longitudinal Focused Attention Meditation Electroencephalography (L-FAME) dataset and an accompanying benchmark, designed to foster research into the neural effects of various meditation practices and the evolution of these effects over a six-week training period. The dataset contains EEG recordings and psychological assessments from 74 healthy college participants, collected at two distinct time points: pre-intervention and post-intervention. Participants were randomly assigned to one of three distinct meditation groups: two mantra-based techniques (SA-TA-NA-MA and Hare Krishna) and one Breath Focus practice. Leveraging this unique longitudinal and comparative dataset, we propose a benchmark suite comprising three distinct classification tasks: (1) cognitive state decoding to distinguish between resting and meditation states, (2) fine-grained classification of the specific meditation techniques, and (3) cross-session adaptation to evaluate model generalization across the longitudinal time gap. We provide comprehensive baseline results for these tasks utilizing a range of classical machine learning algorithms and deep learning architectures. The complete dataset, preprocessing pipelines, and benchmark evaluation code will be publicly released, offering a valuable resource and a standardized framework for the development and comparison of new analytical methods in computational meditation research and EEG-based machine learning. The dataset is available at https://huggingface.co/datasets/L-FAME-Dataset-Benchmark/L-FAME

Physics simulation for contact-rich robotics is often bottlenecked by contact resolution: mainstream engines enforce non-penetration and Coulomb friction via complementarity constraints or constrained optimization, requiring per-step iterative solves whose cost grows superlinearly with contact density. We present ComFree-Sim, a GPU-parallelized analytical contact physics engine built on complementarity-free contact modeling. ComFree-Sim computes contact impulses in closed form via an impedance-style prediction--correction update in the dual cone of Coulomb friction. Contact computation decouples across contact pairs and becomes separable across cone facets, mapping naturally to GPU kernels and yielding near-linear runtime scaling with the number of contacts. We further extend the formulation to a unified 6D contact model capturing tangential, torsional, and rolling friction, and introduce a practical dual-cone impedance heuristic. ComFree-Sim is implemented in Warp and exposed through a MuJoCo-compatible interface as a drop-in backend alternative to MuJoCo Warp (MJWarp). Experiments benchmark penetration, friction behaviors, stability, and simulation runtime scaling against MJWarp, demonstrating near-linear scaling and 2--3 times higher throughput in dense contact scenes with comparable physical fidelity. We deploy ComFree-Sim in real-time MPC for in-hand dexterous manipulation on a real-world multi-fingered LEAP hand and in dynamics-aware motion retargeting, demonstrating that low-latency simulation yields higher closed-loop success rates and enables practical high-frequency control in contact-rich tasks.

The growing number of parameters and computational demands of large language models (LLMs) present significant challenges for their efficient deployment. Recently, there is an increasing interest in quantizing weights to extremely low precision while offsetting the resulting error with low-rank, high-precision error reconstruction terms. The combination of quantization and low-rank approximation is now popular in both adapter-based, parameter-efficient fine-tuning methods such as LoftQ and low-precision inference techniques including ZeroQuant-V2. Usually, the low-rank terms are calculated via the singular value decomposition (SVD) of the weight quantization error, minimizing the Frobenius and spectral norms of the weight approximation error. Recent methods like LQ-LoRA and LQER introduced hand-crafted heuristics to minimize errors in layer outputs (activations) rather than weights, resulting improved quantization results. However, these heuristic methods lack an analytical solution to guide the design of quantization error reconstruction terms. In this paper, we revisit this problem and formulate an analytical framework, named Quantization Error Reconstruction Analysis (QERA), and offer a closed-form solution to the problem. We show QERA benefits both existing low-precision fine-tuning and inference methods -- QERA achieves a fine-tuned accuracy gain of Δ_{acc} = 6.05% of 2-bit RoBERTa-base on GLUE compared to LoftQ; and obtains Δ_{acc} = 2.97% higher post-training quantization accuracy of 4-bit Llama-3.1-70B on average than ZeroQuant-V2 and Δ_{ppl} = - 0.28 lower perplexity on WikiText2 than LQER.

The finetuning of Large Language Models (LLMs) has significantly advanced their instruction-following capabilities, yet the underlying computational mechanisms driving these improvements remain poorly understood. This study systematically examines how fine-tuning reconfigures LLM computations by isolating and analyzing instruction-specific sparse components, i.e., neurons in dense models and both neurons and experts in Mixture-of-Experts (MoE) architectures. In particular, we introduce HexaInst, a carefully curated and balanced instructional dataset spanning six distinct categories, and propose SPARCOM, a novel analytical framework comprising three key contributions: (1) a method for identifying these sparse components, (2) an evaluation of their functional generality and uniqueness, and (3) a systematic comparison of their alterations. Through experiments, we demonstrate functional generality, uniqueness, and the critical role of these components in instruction execution. By elucidating the relationship between fine-tuning-induced adaptations and sparse computational substrates, this work provides deeper insights into how LLMs internalize instruction-following behavior for the trustworthy LLM community.

The independent atom ansatz of density functional theory yields an accurate analytical expression for dynamic correlation energy in the H_{2} molecule: E_{c} = 0.5(1 - 2)(ab|ba) for the atom-additive self-consistent density rho = |a|^{2} + |b|^{2}. Combined with exact atomic self-exchange, it recovers more than 99.5 % of nearly exact SCAN exchange-correlation energy at R > 0.5 A, differing by less than 0.12 eV. The total energy functional correctly dissociates the H-H bond and yields absolute errors of 0.002 A, 0.19 eV, and 13 cm^{-1} relative to experiment at the tight binding computational cost. The chemical bond formation is attributed to the asymptotic Heitler-London resonance of quasi-orthogonal atomic states (- (ab|ba)) with no contributions from kinetic energy or charge accumulation in the bond.

Understanding and analyzing markets is crucial, yet analytical equilibrium solutions remain largely infeasible. Recent breakthroughs in equilibrium computation rely on zeroth-order policy gradient estimation. These approaches commonly suffer from high variance and are computationally expensive. The use of fully differentiable simulators would enable more efficient gradient estimation. However, the discrete allocation of goods in economic simulations is a non-differentiable operation. This renders the first-order Monte Carlo gradient estimator inapplicable and the learning feedback systematically misleading. We propose a novel smoothing technique that creates a surrogate market game, in which first-order methods can be applied. We provide theoretical bounds on the resulting bias which justifies solving the smoothed game instead. These bounds also allow choosing the smoothing strength a priori such that the resulting estimate has low variance. Furthermore, we validate our approach via numerous empirical experiments. Our method theoretically and empirically outperforms zeroth-order methods in approximation quality and computational efficiency.

Drug-side effect research is vital for understanding adverse reactions arising in complex multi-drug therapies. However, the scarcity of higher-order datasets that capture the combinatorial effects of multiple drugs severely limits progress in this field. Existing resources such as TWOSIDES primarily focus on pairwise interactions. To fill this critical gap, we introduce HODDI, the first Higher-Order Drug-Drug Interaction Dataset, constructed from U.S. Food and Drug Administration (FDA) Adverse Event Reporting System (FAERS) records spanning the past decade, to advance computational pharmacovigilance. HODDI contains 109,744 records involving 2,506 unique drugs and 4,569 unique side effects, specifically curated to capture multi-drug interactions and their collective impact on adverse effects. Comprehensive statistical analyses demonstrate HODDI's extensive coverage and robust analytical metrics, making it a valuable resource for studying higher-order drug relationships. Evaluating HODDI with multiple models, we found that simple Multi-Layer Perceptron (MLP) can outperform graph models, while hypergraph models demonstrate superior performance in capturing complex multi-drug interactions, further validating HODDI's effectiveness. Our findings highlight the inherent value of higher-order information in drug-side effect prediction and position HODDI as a benchmark dataset for advancing research in pharmacovigilance, drug safety, and personalized medicine. The dataset and codes are available at https://github.com/TIML-Group/HODDI.

The rapid integration of renewable energy sources is transforming modern electric power systems into converter-dominated networks characterized by complex multi-timescale dynamics and new stability interactions over a wide frequency range. Accurate analysis of these phenomena requires high-fidelity models of transmission networks and power electronic converters, which often result in large-scale differential-algebraic systems with significant computational burden. This paper provides an analytical review of model order reduction techniques for converter-rich power systems, with particular emphasis on the key equations employed in the reduction process. Advantages and limitations of commonly used strategies, including timescale-based singular perturbation, projection-based, operator-based, data-driven, and machine-learning based methods, are discussed. The synthesis highlights key challenges related to harmonic interactions and multi-frequency dynamics, and outlines directions for the development of computationally efficient yet dynamically accurate reduced-order models for future power systems.

This letter presents an analytical linear parameter-varying (LPV) representation of quadrotor dynamics utilizing Koopman theory, facilitating computationally efficient linear model predictive control (LMPC) for real-time trajectory tracking. By leveraging carefully designed Koopman observables, the proposed approach enables a compact lifted-space evolution that mitigates the curse of dimensionality while preserving the nonlinear characteristics of the system. Although model predictive control (MPC) is a powerful strategy for quadrotor control, it faces a trade-off between the high computational cost of nonlinear MPC (NMPC) and the reduced accuracy of LMPC. To address this gap, we introduce KQ-LMPC (Koopman Quasilinear LPV MPC), which leverages the Koopman-lifted LPV formulation to enforce constraints, ensure lower computational burden and real-time feasibility, and deliver tracking performance comparable to NMPC. Experimental validation confirms the effectiveness of the framework in reasonably agile flight. To the best of our knowledge, this is the first experimentally validated LMPC for quadrotors that employs analytically derived Koopman observables without requiring training data.

Advances in LLMs have produced agents with knowledge and operational capabilities comparable to human scientists, suggesting potential to assist, accelerate, and automate research. However, existing studies mainly evaluate such systems on well-defined benchmarks or general tasks like literature retrieval, limiting their end-to-end problem-solving ability in open scientific scenarios. This is particularly true in physics, which is abstract, mathematically intensive, and requires integrating analytical reasoning with code-based computation. To address this, we propose PhysMaster, an LLM-based agent functioning as an autonomous theoretical and computational physicist. PhysMaster couples absract reasoning with numerical computation and leverages LANDAU, the Layered Academic Data Universe, which preserves retrieved literature, curated prior knowledge, and validated methodological traces, enhancing decision reliability and stability. It also employs an adaptive exploration strategy balancing efficiency and open-ended exploration, enabling robust performance in ultra-long-horizon tasks. We evaluate PhysMaster on problems from high-energy theory, condensed matter theory to astrophysics, including: (i) acceleration, compressing labor-intensive research from months to hours; (ii) automation, autonomously executing hypothesis-driven loops ; and (iii) autonomous discovery, independently exploring open problems.

Large Language Models (LLMs) and LLM-based agents show great promise in accelerating scientific research. Existing benchmarks for measuring this potential and guiding future development continue to evolve from pure recall and rote knowledge tasks, towards more practical work such as literature review and experimental planning. Bioinformatics is a domain where fully autonomous AI-driven discovery may be near, but no extensive benchmarks for measuring progress have been introduced to date. We therefore present the Bioinformatics Benchmark (BixBench), a dataset comprising over 50 real-world scenarios of practical biological data analysis with nearly 300 associated open-answer questions designed to measure the ability of LLM-based agents to explore biological datasets, perform long, multi-step analytical trajectories, and interpret the nuanced results of those analyses. We evaluate the performance of two frontier LLMs (GPT-4o and Claude 3.5 Sonnet) using a custom agent framework we open source. We find that even the latest frontier models only achieve 17% accuracy in the open-answer regime, and no better than random in a multiple-choice setting. By exposing the current limitations of frontier models, we hope BixBench can spur the development of agents capable of conducting rigorous bioinformatic analysis and accelerate scientific discovery.

How can we develop visual analytics (VA) tools that can be easily adopted? Visualization researchers have developed a large number of web-based VA tools to help data scientists in a wide range of tasks. However, adopting these standalone systems can be challenging, as they require data scientists to create new workflows to streamline the VA processes. Recent surveys suggest computational notebooks have been dominating data scientists' analytical workflows, as these notebooks seamlessly combine text, code, and visualization, allowing users to rapidly iterate code experiments. To help visualization researchers develop VA tools that can be easily integrated into existing data science workflows, we present NOVA, a simple and flexible method to adapt web-based VA systems for notebooks. We provide detailed examples of using this method with diverse web development technologies and different types of computational notebooks. Deployed application examples highlight that NOVA is easy to adopt, and data scientists appreciate in-notebook VA. NOVA is available at https://github.com/poloclub/nova.

We present EE-LLM, a framework for large-scale training and inference of early-exit large language models (LLMs). While recent works have shown preliminary evidence for the efficacy of early exiting in accelerating LLM inference, EE-LLM makes a foundational step towards scaling up early-exit LLMs by supporting their training and inference with massive 3D parallelism. Built upon Megatron-LM, EE-LLM implements a variety of algorithmic innovations and performance optimizations tailored to early exiting, including a lightweight method that facilitates backpropagation for the early-exit training objective with pipeline parallelism, techniques of leveraging idle resources in the original pipeline schedule for computation related to early-exit layers, and two approaches of early-exit inference that are compatible with KV caching for autoregressive generation. Our analytical and empirical study shows that EE-LLM achieves great training efficiency with negligible computational overhead compared to standard LLM training, as well as outstanding inference speedup without compromising output quality. To facilitate further research and adoption, we release EE-LLM at https://github.com/pan-x-c/EE-LLM.

Estimating the symbolic or analytical form of probability density functions (PDFs) from observed samples is a fundamental challenge in statistical and computational modelling. This process is critical for deriving interpretable and generalizable relationships characterizing the underlying phenomenon. Traditionally, this estimation depends strongly on domain expertise and prior field-specific knowledge, with experts selecting appropriate functional forms or parametric families based on empirical evidence and theoretical understanding. The coefficients of these forms are then typically determined through parameter estimation. In this paper, we develop a framework for estimating symbolic expressions of unnormalized distributions from observed samples using domain-specific prior knowledge, such as the range of interactions and a predefined set of primitive functions. We integrate deep generative models with symbolic regression (SR), incorporating inductive biases, such as factorizing large distributions, to keep the problem tractable. The deep generative models we examine include likelihood-based models, viz., flow models, and score-based models. Experiments show the effectiveness of the proposed framework for estimating density functions for multivariate toy distributions as well as lattices from computational physics, namely, XY model and ϕ^4 theory. When applied to the renormalization problem in ϕ^4 theory, the proposed framework estimates compact symbolic approximations of the hamiltonian function at different scales directly from samples, yielding expressions that may be challenging to derive using traditional perturbative or analytic approaches in nonperturbative settings.

Radiance fields have emerged as a predominant representation for modeling 3D scene appearance. Neural formulations such as Neural Radiance Fields provide high expressivity but require costly ray marching for rendering, whereas primitive-based methods such as 3D Gaussian Splatting offer real-time efficiency through splatting, yet at the expense of representational power. Inspired by advances in both these directions, we introduce splattable neural primitives, a new volumetric representation that reconciles the expressivity of neural models with the efficiency of primitive-based splatting. Each primitive encodes a bounded neural density field parameterized by a shallow neural network. Our formulation admits an exact analytical solution for line integrals, enabling efficient computation of perspectively accurate splatting kernels. As a result, our representation supports integration along view rays without the need for costly ray marching. The primitives flexibly adapt to scene geometry and, being larger than prior analytic primitives, reduce the number required per scene. On novel-view synthesis benchmarks, our approach matches the quality and speed of 3D Gaussian Splatting while using 10times fewer primitives and 6times fewer parameters. These advantages arise directly from the representation itself, without reliance on complex control or adaptation frameworks. The project page is https://vcai.mpi-inf.mpg.de/projects/SplatNet/.

Fine-tuning large language models (LLMs) on devices is attracting increasing interest. Recent works have fused low-rank adaptation (LoRA) techniques with federated fine-tuning to mitigate challenges associated with device model sizes and data scarcity. Still, the heterogeneity of computational resources remains a critical bottleneck: while higher-rank modules generally enhance performance, varying device capabilities constrain LoRA's feasible rank range. Existing approaches attempting to resolve this issue either lack analytical justification or impose additional computational overhead, leaving a wide gap for an efficient and theoretically-grounded solution. To address these challenges, we propose federated sketching LoRA (FSLoRA), which leverages a sketching mechanism to enable devices to selectively update submatrices of global LoRA modules maintained by the server. By adjusting the sketching ratios, which determine the ranks of the submatrices on the devices, FSLoRA flexibly adapts to device-specific communication and computational constraints. We provide a rigorous convergence analysis of FSLoRA that characterizes how the sketching ratios affect the convergence rate. Through comprehensive experiments on multiple datasets and LLM models, we demonstrate FSLoRA's superior performance compared to various baselines.

Recent years have seen a boom in computational approaches to music analysis, yet each one is typically tailored to a specific analytical domain. In this work, we introduce AnalysisGNN, a novel graph neural network framework that leverages a data-shuffling strategy with a custom weighted multi-task loss and logit fusion between task-specific classifiers to integrate heterogeneously annotated symbolic datasets for comprehensive score analysis. We further integrate a Non-Chord-Tone prediction module, which identifies and excludes passing and non-functional notes from all tasks, thereby improving the consistency of label signals. Experimental evaluations demonstrate that AnalysisGNN achieves performance comparable to traditional static-dataset approaches, while showing increased resilience to domain shifts and annotation inconsistencies across multiple heterogeneous corpora.

We present shadow spectroscopy as a simulator-agnostic quantum algorithm for estimating energy gaps using very few circuit repetitions (shots) and no extra resources (ancilla qubits) beyond performing time evolution and measurements. The approach builds on the fundamental feature that every observable property of a quantum system must evolve according to the same harmonic components: we can reveal them by post-processing classical shadows of time-evolved quantum states to extract a large number of time-periodic signals N_opropto 10^8, whose frequencies correspond to Hamiltonian energy differences with Heisenberg-limited precision. We provide strong analytical guarantees that (a) quantum resources scale as O(log N_o), while the classical computational complexity is linear O(N_o), (b) the signal-to-noise ratio increases with the number of processed signals as propto N_o, and (c) spectral peak positions are immune to reasonable levels of noise. We demonstrate our approach on model spin systems and the excited state conical intersection of molecular CH_2 and verify that our method is indeed intuitively easy to use in practice, robust against gate noise, amiable to a new type of algorithmic-error mitigation technique, and uses orders of magnitude fewer number of shots than typical near-term quantum algorithms -- as low as 10 shots per timestep is sufficient. Finally, we measured a high-quality, experimental shadow spectrum of a spin chain on readily-available IBM quantum computers, achieving the same precision as in noise-free simulations without using any advanced error mitigation, and verified scalability in tensor-network simulations of up to 100-qubit systems.

The analytical description of charts is an exciting and important research area with many applications in academia and industry. Yet, this challenging task has received limited attention from the computational linguistics research community. This paper proposes AutoChart, a large dataset for the analytical description of charts, which aims to encourage more research into this important area. Specifically, we offer a novel framework that generates the charts and their analytical description automatically. We conducted extensive human and machine evaluations on the generated charts and descriptions and demonstrate that the generated texts are informative, coherent, and relevant to the corresponding charts.

Machine learning methods can be a valuable aid in the scientific process, but they need to face challenging settings where data come from inhomogeneous experimental conditions. Recent meta-learning methods have made significant progress in multi-task learning, but they rely on black-box neural networks, resulting in high computational costs and limited interpretability. Leveraging the structure of the learning problem, we argue that multi-environment generalization can be achieved using a simpler learning model, with an affine structure with respect to the learning task. Crucially, we prove that this architecture can identify the physical parameters of the system, enabling interpreable learning. We demonstrate the competitive generalization performance and the low computational cost of our method by comparing it to state-of-the-art algorithms on physical systems, ranging from toy models to complex, non-analytical systems. The interpretability of our method is illustrated with original applications to physical-parameter-induced adaptation and to adaptive control.

Attractor dynamics are a fundamental computational motif in neural circuits, supporting diverse cognitive functions through stable, self-sustaining patterns of neural activity. In these lecture notes, we review four key examples that demonstrate how autoassociative neural network models can elucidate the computational mechanisms underlying attractor-based information processing in biological neural systems performing cognitive functions. Drawing on empirical evidence, we explore hippocampal spatial representations, visual classification in the inferotemporal cortex, perceptual adaptation and priming, and working-memory biases shaped by sensory history. Across these domains, attractor network models reveal common computational principles and provide analytical insights into how experience shapes neural activity and behavior. Our synthesis underscores the value of attractor models as powerful tools for probing the neural basis of cognition and behavior.

Spatial proteomics enables single-cell-resolution characterization of protein expression within tissue architecture, playing a critical role in understanding tumor microenvironments and guiding precision medicine. However, current analysis workflows remain fragmented, requiring expert manual orchestration of heterogeneous tools and limiting research scalability and reproducibility. We present SP-Mind, the first autonomous AI agent designed to unify the spatial proteomics analysis pipeline, from raw multiplexed tissue imaging to downstream phenotype discovery. Equipped with expert-curated biological analysis skills and specialized computational tools, SP-Mind converts natural-language queries into end-to-end analytical workflows without task-specific fine-tuning. To rigorously evaluate its capabilities, we introduce SP-Bench, a comprehensive benchmark spanning diverse tissue types, comprising 102 tasks across 18 distinct categories. Through extensive evaluation on SP-Bench and established downstream tasks, SP-Mind achieves state-of-the-art performance compared to existing open-source biomedical agent baselines.

Time series analysis is crucial in real-world applications, yet traditional methods focus on isolated tasks only, and recent studies on time series reasoning remain limited to either single-step inference or are constrained to natural language answers. In this work, we introduce TS-Reasoner, a domain-specialized agent designed for multi-step time series inference. By integrating large language model (LLM) reasoning with domain-specific computational tools and an error feedback loop, TS-Reasoner enables domain-informed, constraint-aware analytical workflows that combine symbolic reasoning with precise numerical analysis. We assess the system's capabilities along two axes: (1) fundamental time series understanding assessed by TimeSeriesExam and (2) complex, multi-step inference evaluated by a newly proposed dataset designed to test both compositional reasoning and computational precision in time series analysis. Experiments show that our approach outperforms standalone general-purpose LLMs in both basic time series concept understanding as well as the multi-step time series inference task, highlighting the promise of domain-specialized agents for automating real-world time series reasoning and analysis.

Variational Quantum Algorithms (VQAs) rely heavily on classical optimization routines to navigate high-dimensional, noisy parameter landscapes. However, the evaluation of analytical gradients via the parameter-shift rule requires O(p) circuit executions, rendering this approach computationally prohibitive for deep circuits. While simultaneous perturbation methods provide an O(1) alternative, their gradient estimates suffer from severe O(p) spatial variance, which impedes convergence. To resolve this tension, we introduce the Cached Recycled Variance-Reduced Gradient (CRVG) by leveraging variance reduction techniques. We formulate two variants: a 3-Circuit (1-Sided) Non-Recursive CRVG and a Recursive CRVG. By introducing a center-point caching mechanism, the 3-Circuit CRVG mathematically recycles quantum measurements to reduce the inner-loop circuit complexity by 25\% across both variants. We establish a theoretical separation between the two routing paths to achieve an ε^2-stationary point: the non-recursive variant reaches a sample complexity of O(max{p/ε^2, p^{2/3}/ε^{10/3}}), while the recursive variant achieves an improved complexity of O(max{p/ε^2, p/ε^3}). Empirically, extensive evaluations on MaxCut and Maximum Independent Set (MIS) benchmarks validate these findings. While standard simultaneous perturbation remains a robust baseline across varying depths, CRVG demonstrates targeted advantages in specific amortizable regimes, such as the non-recursive variant on EfficientSU2 circuits and both variants on depth-three QAOA, yielding superior energy minimums and tighter output variances. Ultimately, CRVG establishes a principled bias-throughput trade-off for scaling near-term quantum optimization.

External potentials break the translational invariance of nonlinear Schrödinger (NLS) equations and generally convert a continuous family of standing wave solutions into isolated pinned states. We study this mechanism for the prototypical case of a kink in the NLS equation with competing focusing cubic and defocusing quintic nonlinearities. The Lyapunov--Schmidt reduction method is used to prove that the location s_0 of the pinned kinks is selected near roots of an effective potential V_{rm eff}(s). We then determine how the translational zero eigenvalue of the Jacobian operator splits under the perturbation and prove that the sign of V_{rm eff}'(s_0) distinguishes eigendirections of positive and negative energy. We further show that if the Jacobian operator admits a negative eigenvalue, then the stability problem contains a real unstable eigenvalue. The derivation of the asymptotic expansion for the unstable eigenvalue is nonstandard because the zero eigenvalue is embedded in the continuous spectrum of the linearized operator. Finally, we show that the persistence of kinks lacks parity symmetry even if the external potential is odd. Numerical continuation, spectral computations with domains adapted to the slowly decaying eigenfunctions, and direct dynamical simulations of the time evolution corroborate the analytical predictions for different representative examples.

Inferring cellular dynamics from unpaired single-cell snapshots requires modeling both state transitions and population growth or death. Unbalanced dynamic optimal transport (UDOT) addresses this by penalizing growth along transport paths, making the choice of growth penalty a key way to encode biological priors on proliferation and apoptosis. However, existing UDOT solvers either rely on computationally expensive NeuralODE simulations or depend on analytical solutions of conditional paths, restricting their efficiency solely to quadratic penalties, i.e. Wasserstein-Fisher-Rao (WFR) geodesics. To enable an efficient UDOT solver for general growth penalties, we first show that concave growth penalties lead to degenerate solutions where growth and transport are separated. We then introduce Simulation-free Unbalanced Dynamic Optimal transport (SUDO), a simulation-free framework for UDOT with general non-quadratic convex growth penalties. SUDO learns the conditional paths and transport costs, solves the induced semi-coupling problem, and subsequently leverages unbalanced flow matching to achieve a simulation-free solution. On WFR benchmarks, SUDO matches the accuracy of efficient, analytical solution-driven algorithms while outperforming simulation-based methods in computational speed. Beyond WFR, SUDO supports asymmetric penalties that encode proliferation-dominant priors and produce more plausible trajectories and growth estimates on synthetic and single-cell datasets.

Continuous latent variables (LVs) are a key ingredient of many generative models, as they allow modelling expressive mixtures with an uncountable number of components. In contrast, probabilistic circuits (PCs) are hierarchical discrete mixtures represented as computational graphs composed of input, sum and product units. Unlike continuous LV models, PCs provide tractable inference but are limited to discrete LVs with categorical (i.e. unordered) states. We bridge these model classes by introducing probabilistic integral circuits (PICs), a new language of computational graphs that extends PCs with integral units representing continuous LVs. In the first place, PICs are symbolic computational graphs and are fully tractable in simple cases where analytical integration is possible. In practice, we parameterise PICs with light-weight neural nets delivering an intractable hierarchical continuous mixture that can be approximated arbitrarily well with large PCs using numerical quadrature. On several distribution estimation benchmarks, we show that such PIC-approximating PCs systematically outperform PCs commonly learned via expectation-maximization or SGD.

Single-cell sequencing technologies reveal cellular heterogeneity at high resolution, advancing our understanding of biological complexity. As datasets start to scale to tens of millions of cells, computational workflows face substantial bottlenecks, with CPU-based analytical pipelines requiring hours or days for routine processing steps like filtering, normalization, and clustering. These scalability limitations fundamentally restrict common interactive data exploration and iterative hypothesis testing. Here we introduce rapids-singlecell, a GPU-accelerated framework that integrates natively with the scverse ecosystem and operates directly on the AnnData data structure, which delivers orders-of-magnitude speedups for single-cell workflows. Built on CuPy arrays and the NVIDIA CUDA-X Data Science (RAPIDS) ecosystem, rapids-singlecell provides near drop-in GPU replacements for core scanpy-based analysis steps. Across standard single-cell workflows such as preprocessing, dimensionality reduction, neighborhood graph construction, clustering, and batch correction, rapids-singlecell achieves speedups of up to several hundred-fold compared to optimized CPU baselines. This reduces analysis time from hours to minutes on standard hardware, while maintaining consistent biological interpretations. These performance improvements make it possible to analyze large data sets in close to real time, without the need for data splitting. Together with real-time parameter tuning and iterative workflows, rapids-singlecell makes interactive large-scale single-cell analysis possible.

Large language models have made significant progress in various language tasks, yet they still struggle with complex mathematics. In this paper, we propose ToRA a series of Tool-integrated Reasoning Agents designed to solve challenging mathematical problems by seamlessly integrating natural language reasoning with the utilization of external tools (e.g., computation libraries and symbolic solvers), thereby amalgamating the analytical prowess of language and the computational efficiency of tools. To train ToRA, we curate interactive tool-use trajectories on mathematical datasets, apply imitation learning on the annotations, and propose output space shaping to further refine models' reasoning behavior. As a result, ToRA models significantly outperform open-source models on 10 mathematical reasoning datasets across all scales with 13%-19% absolute improvements on average. Notably, ToRA-7B reaches 44.6% on the competition-level dataset MATH, surpassing the best open-source model WizardMath-70B by 22% absolute. ToRA-34B is also the first open-source model that achieves an accuracy exceeding 50% on MATH, which significantly outperforms GPT-4's CoT result, and is competitive with GPT-4 solving problems with programs. Additionally, we conduct a comprehensive analysis of the benefits and remaining challenges of tool interaction for mathematical reasoning, providing valuable insights for future research.

The complexity of modern bioinformatics analysis has created a critical gap between data generation and developing scientific insights. While large language models (LLMs) have shown promise in scientific reasoning, they remain fundamentally limited when dealing with real-world analytical workflows that demand iterative computation, tool integration and rigorous validation. We introduce K-Dense Analyst, a hierarchical multi-agent system that achieves autonomous bioinformatics analysis through a dual-loop architecture. K-Dense Analyst, part of the broader K-Dense platform, couples planning with validated execution using specialized agents to decompose complex objectives into executable, verifiable tasks within secure computational environments. On BixBench, a comprehensive benchmark for open-ended biological analysis, K-Dense Analyst achieves 29.2% accuracy, surpassing the best-performing language model (GPT-5) by 6.3 percentage points, representing nearly 27% improvement over what is widely considered the most powerful LLM available. Remarkably, K-Dense Analyst achieves this performance using Gemini 2.5 Pro, which attains only 18.3% accuracy when used directly, demonstrating that our architectural innovations unlock capabilities far beyond the underlying model's baseline performance. Our insights demonstrate that autonomous scientific reasoning requires more than enhanced language models, it demands purpose-built systems that can bridge the gap between high-level scientific objectives and low-level computational execution. These results represent a significant advance toward fully autonomous computational biologists capable of accelerating discovery across the life sciences.

Quantitative chemistry is central to modern chemical research, yet the ability of large language models (LLMs) to perform its rigorous, step-by-step calculations remains underexplored. To fill this blank, we propose QCBench, a Quantitative Chemistry oriented benchmark comprising 350 computational chemistry problems across 7 chemistry subfields, which contains analytical chemistry, bio/organic chemistry, general chemistry, inorganic chemistry, physical chemistry, polymer chemistry and quantum chemistry. To systematically evaluate the mathematical reasoning abilities of large language models (LLMs), they are categorized into three tiers: easy, medium, and difficult. Each problem, rooted in realistic chemical scenarios, is structured to prevent heuristic shortcuts and demand explicit numerical reasoning. QCBench enables fine-grained diagnosis of computational weaknesses, reveals model-specific limitations across difficulty levels, and lays the groundwork for future improvements such as domain-adaptive fine-tuning or multi-modal integration. Evaluations on 24 LLMs demonstrate a consistent performance degradation with increasing task complexity, highlighting the current gap between language fluency and scientific computation accuracy. Code for QCBench is available at https://github.com/jiaqingxie/QCBench.

Structural survival of offshore structures is crucial for the growing marine economy. Calculating the added mass, radiation damping, and excitation coefficients to quantify wave loads with the traditional boundary element method (BEM) presents a computational bottleneck. The matched eigenfunction expansion method (MEEM), a long-known but rarely-used alternative, offers computational benefits due to its semi-analytical nature. However, previous work fails to directly compare its accuracy and computational performance with BEM, leaving the extent of its utility unknown. Furthermore, the geometry-dependent convergence for cylindrical and slanted geometries has not yet been documented, making the method's practicality for general geometries unclear. This paper presents a unifying MEEM framework for modeling an arbitrary number of fixed or heaving surface-piercing annular cylinders with continuous and radially-monotonic body profiles, and explores the method's block matrix structure, convergence behavior, ability to accurately approximate slanted geometries, and computational advantages over the BEM solver Capytaine. The numerical experiments show that MEEM can compute hydrodynamic coefficients of slanted geometries within 5% of Capytaine, even for angles as steep as 15 degrees from vertical. Finally, MEEM can achieve 2% convergence of its hydrodynamic coefficients an order of magnitude faster than Capytaine with a matrix size two orders of magnitude smaller, making it a computationally effective alternative to traditional BEM solvers. These contributions enable hydrodynamic analysis of a broad range of shapes with increased speed and confidence, paving the way for future optimization studies to yield improved designs.

Accurately modeling chemical reactions at the atomistic level requires high-level electronic structure theory due to the presence of unpaired electrons and the need to properly describe bond breaking and making energetics. Commonly used approaches such as Density Functional Theory (DFT) frequently fail for this task due to deficiencies that are well recognized. However, for high-fidelity approaches, creating large datasets of energies and forces for reactive processes to train machine learning interatomic potentials or force fields is daunting. For example, the use of the unrestricted coupled cluster level of theory has previously been seen as unfeasible due to high computational costs, the lack of analytical gradients in many computational codes, and additional challenges such as constructing suitable basis set corrections for forces. In this work, we develop new methods and workflows to overcome the challenges inherent to automating unrestricted coupled cluster calculations. Using these advancements, we create a dataset of gas-phase reactions containing energies and forces for 3119 different organic molecules configurations calculated at the gold-standard level of unrestricted CCSD(T) (coupled cluster singles doubles and perturbative triples). With this dataset, we provide an analysis of the differences between the density functional and unrestricted CCSD(T) descriptions. We develop a transferable machine learning interatomic potential for gas-phase reactions, trained on unrestricted CCSD(T) data, and demonstrate the advantages of transitioning away from DFT data. Transitioning from training to DFT to training to UCCSD(T) datasets yields an improvement of more than 0.1 eV/Å in force accuracy and over 0.1 eV in activation energy reproduction.

Machine unlearning has become a critical capability for safely removing specific, sensitive knowledge from large language models (LLMs). Current state-of-the-art approaches primarily rely on iterative, training-time unlearning via fine-tuning. However, even when utilizing parameter-efficient dimensionality reduction techniques like LoRA, gradient-based optimization remains computationally expensive and lacks explicit analytical formulations. It can also leave the targeted knowledge merely hidden rather than removed, to the point that simply quantizing the unlearned model restores much of what it was supposed to have erased. To resolve this, we propose a novel one-shot unlearning approach, abandoning iterative optimization in favor of a direct, exact analytical solution. We frame the unlearning process as a ridge-regularized least-squares optimization problem, deriving a closed-form additive update for targeted weight matrices. This update forces the selected layer to suppress unwanted content while strictly preserving its behavior on retained data. Computed from gradient-free forward passes alone, with no backpropagation and no iteration to convergence, GROM applies the weight edit in mere seconds, which makes it orders of magnitude faster than traditional fine-tuning. Extensive evaluations demonstrate that GROM achieves state-of-the-art forgetting-utility trade-offs on TOFU-5%, TOFU-10%, MUSE-Books, MUSE-News and WMDP, significantly reducing computational overhead without sacrificing overall model performance. Because the update removes the targeted content from the weights instead of masking it, GROM also withstands the low-bit quantization attack that recovers much of the content a gradient-based baseline had appeared to forget. Our code is publicly available at https://github.com/Batorskq/GROM.

Analytical reasoning is an essential and challenging task that requires a system to analyze a scenario involving a set of particular circumstances and perform reasoning over it to make conclusions. In this paper, we study the challenge of analytical reasoning of text and introduce a new dataset consisting of questions from the Law School Admission Test from 1991 to 2016. We analyze what knowledge understanding and reasoning abilities are required to do well on this task. Furthermore, to address this reasoning challenge, we design two different baselines: (1) a Transformer-based method which leverages the state-of-the-art pre-trained language models and (2) Analytical Reasoning Machine (ARM), a logical-level reasoning framework extracting symbolic knowledge (e.g, participants, facts, logical functions) to deduce legitimate solutions. In our experiments, we find that the Transformer-based models struggle to solve this task as their performance is close to random guess and ARM achieves better performance by leveraging symbolic knowledge and interpretable reasoning steps. Results show that both methods still lag far behind human performance, which leave further space for future research.

Computers calculate transcendental functions by approximating them through the composition of a few limited-precision instructions. For example, an exponential can be calculated with a Taylor series. These approximation methods were developed over the centuries by mathematicians, who emphasized the attainability of arbitrary precision. Computers, however, operate on few limited precision types, such as the popular float32. In this study, we show that when aiming for limited precision, existing approximation methods can be outperformed by programs automatically discovered from scratch by a simple evolutionary algorithm. In particular, over real numbers, our method can approximate the exponential function reaching orders of magnitude more precision for a given number of operations when compared to previous approaches. More practically, over float32 numbers and constrained to less than 1 ULP of error, the same method attains a speedup over baselines by generating code that triggers better XLA/LLVM compilation paths. In other words, in both cases, evolution searched a vast space of possible programs, without knowledge of mathematics, to discover previously unknown optimized approximations to high precision, for the first time. We also give evidence that these results extend beyond the exponential. The ubiquity of transcendental functions suggests that our method has the potential to reduce the cost of scientific computing applications.

In recent decades, a growing number of discoveries in fields of mathematics have been assisted by computer algorithms, primarily for exploring large parameter spaces that humans would take too long to investigate. As computers and algorithms become more powerful, an intriguing possibility arises - the interplay between human intuition and computer algorithms can lead to discoveries of novel mathematical concepts that would otherwise remain elusive. To realize this perspective, we have developed a massively parallel computer algorithm that discovers an unprecedented number of continued fraction formulas for fundamental mathematical constants. The sheer number of formulas discovered by the algorithm unveils a novel mathematical structure that we call the conservative matrix field. Such matrix fields (1) unify thousands of existing formulas, (2) generate infinitely many new formulas, and most importantly, (3) lead to unexpected relations between different mathematical constants, including multiple integer values of the Riemann zeta function. Conservative matrix fields also enable new mathematical proofs of irrationality. In particular, we can use them to generalize the celebrated proof by Ap\'ery for the irrationality of zeta(3). Utilizing thousands of personal computers worldwide, our computer-supported research strategy demonstrates the power of experimental mathematics, highlighting the prospects of large-scale computational approaches to tackle longstanding open problems and discover unexpected connections across diverse fields of science.

How can we perform computations over natural language representations to solve tasks that require symbolic and numeric reasoning? We propose natural language embedded programs (NLEP) as a unifying framework for addressing math/symbolic reasoning, natural language understanding, and instruction following tasks. Our approach prompts a language model to generate full Python programs that define functions over data structures which contain natural language representations of structured knowledge. A Python interpreter then executes the generated code and prints the output. Despite using a task-general prompt, we find that this approach can improve upon strong baselines across a range of different tasks including math and symbolic reasoning, text classification, question answering, and instruction following. We further find the generated programs are often interpretable and enable post-hoc verification of the intermediate reasoning steps.

Lara is a key-value algebra that aims at unifying linear and relational algebra with three types of operation abstraction. The study of Lara's expressive ability reports that it can represent relational algebra and most linear algebra operations. However, several essential computations, such as matrix inversion and determinant, cannot be expressed in Lara. Lara cannot represent global and iterative computation, either. This article proposes IterLara, extending Lara with iterative operators, to provide an algebraic model that unifies operations in general-purpose computing, like big data, AI, scientific computing, and database. We study the expressive ability of Lara and IterLara and prove that IterLara with aggregation functions can represent matrix inversion, determinant. Besides, we demonstrate that IterLara with no limitation of function utility is Turing complete. We also propose the Operation Count (OP) as a metric of computation amount for IterLara and ensure that the OP metric is in accordance with the existing computation metrics.

Mathematical problem-solving is a key field in artificial intelligence (AI) and a critical benchmark for evaluating the capabilities of large language models (LLMs). While extensive research has focused on mathematical problem-solving, most existing work and datasets concentrate on computational tasks, leaving gaps in areas like mathematical analysis, which demands rigorous proofs and formal reasoning. We developed the DEMI-MathAnalysis dataset, comprising proof-based problems from mathematical analysis topics such as Sequences and Limits, Infinite Series, and Convex Functions. We also designed a guiding framework to rigorously enhance LLMs' ability to solve these problems. Through fine-tuning LLMs on this dataset and employing our framework, we observed significant improvements in their capability to generate logical, complete, and elegant proofs. This work addresses critical gaps in mathematical reasoning and contributes to advancing trustworthy AI capable of handling formalized mathematical language. The code is publicly accessible at LLMs for Mathematical Analysis.

AI for Mathematics (AI4Math) has emerged as a distinct field that leverages machine learning to navigate mathematical landscapes historically intractable for early symbolic systems. While mid-20th-century symbolic approaches successfully automated formal logic, they faced severe scalability limitations due to the combinatorial explosion of the search space. The recent integration of data-driven approaches has revitalized this pursuit. In this review, we provide a systematic overview of AI4Math, highlighting its primary focus on developing AI models to support mathematical research. Crucially, we emphasize that this is not merely the application of AI to mathematical activities; it also encompasses the development of stronger AI systems where the rigorous nature of mathematics serves as a premier testbed for advancing general reasoning capabilities. We categorize existing research into two complementary directions: problem-specific modeling, involving the design of specialized architectures for distinct mathematical tasks, and general-purpose modeling, focusing on foundation models capable of broader reasoning, retrieval, and exploratory workflows. We conclude by discussing key challenges and prospects, advocating for AI systems that go beyond facilitating formal correctness to enabling the discovery of meaningful results and unified theories, recognizing that the true value of a proof lies in the insights and tools it offers to the broader mathematical landscape.

While machine learning has advanced through massive parallelization, we identify a critical blind spot: some problems are fundamentally sequential. These "inherently serial" problems-from mathematical reasoning to physical simulations to sequential decision-making-require dependent computational steps that cannot be parallelized. Drawing from complexity theory, we formalize this distinction and demonstrate that current parallel-centric architectures face fundamental limitations on such tasks. We argue that recognizing the serial nature of computation holds profound implications on machine learning, model design, hardware development. As AI tackles increasingly complex reasoning, deliberately scaling serial computation-not just parallel computation-is essential for continued progress.

The Functional Machine Calculus (Heijltjes 2022) is a new approach to unifying the imperative and functional programming paradigms. It extends the lambda-calculus, preserving the key features of confluent reduction and typed termination, to embed computational effects, evaluation strategies, and control flow operations. The first instalment modelled sequential higher-order computation with global store, input/output, probabilities, and non-determinism, and embedded both the call-by-name and call-by-value lambda-calculus, as well as Moggi's computational metalanguage and Levy's call-by-push-value. The present paper extends the calculus from sequential to branching and looping control flow. This allows the faithful embedding of a minimal but complete imperative language, including conditionals, exception handling, and iteration, as well as constants and algebraic data types. The calculus is defined through a simple operational semantics, extending the (simplified) Krivine machine for the lambda-calculus with multiple operand stacks to model effects and a continuation stack to model sequential, branching, and looping computation. It features a confluent reduction relation and a system of simple types that guarantees termination of the machine and strong normalization of reduction (in the absence of iteration). These properties carry over to the embedded imperative language, providing a unified functional-imperative model of computation that supports simple types, a direct and intuitive operational semantics, and a confluent reduction semantics.

The research in AI-based formal mathematical reasoning has shown an unstoppable growth trend. These studies have excelled in mathematical competitions like IMO, showing significant progress. However, these studies intertwined multiple skills simultaneously, i.e., problem-solving, reasoning, and writing formal specifications, making it hard to precisely identify the LLMs' strengths and weaknesses in each task. This paper focuses on formal verification, an immediate application scenario of formal reasoning, and decomposes it into six sub-tasks. We constructed 18k high-quality instruction-response pairs across five mainstream formal specification languages (Coq, Lean4, Dafny, ACSL, and TLA+) in six formal-verification-related tasks by distilling GPT-4o. They are split into a 14k+ fine-tuning dataset FM-alpaca and a 4k benchmark FM-Bench. We found that LLMs are good at writing proof segments when given either the code, or the detailed description of proof steps. Also, the fine-tuning brought about a nearly threefold improvement at most. Interestingly, we observed that fine-tuning with formal data also enhances mathematics, reasoning, and coding abilities. We hope our findings inspire further research. Fine-tuned models are released to facilitate subsequent studies

Current evaluations of mathematical reasoning in large language models (LLMs) are dominated by static benchmarks, either derived from competition-style problems or curated through costly expert effort, resulting in limited coverage of research-level mathematics and rapid performance saturation. We propose a fully automated, theorem-grounded pipeline for evaluating frontier mathematical reasoning, which directly transforms recent peer-reviewed mathematical literature into executable and verifiable reasoning tasks. The pipeline identifies constructive or quantitative results, instantiates them into parameterized problem templates, and generates deterministic solutions through execution-based verification, enabling scalable, reproducible, and continuously updatable evaluation without reliance on large-scale expert authoring. By design, this approach supports temporal extensibility, intrinsic correctness checking, and domain-specific customization across mathematical subfields. Applying this pipeline yields EternalMath, an evolving evaluation suite derived from contemporary research papers. Experiments with state-of-the-art LLMs reveal substantial performance gaps, indicating that mathematical reasoning at the research frontier remains far from saturated and underscoring the need for evaluation methodologies that evolve in step with human mathematical discovery.

In recent years, large language models (LLMs) have made significant progress in code intelligence, yet systematically evaluating their code understanding and reasoning abilities remains challenging. Mainstream benchmarks such as HumanEval and MBPP primarily assess functional correctness, while reasoning benchmarks like CRUXEVAL are limited to single-function, low-complexity scenarios. As a result, advanced models achieve nearly saturated scores, limiting their discriminative power. To address this, we present STEPWISE-CODEX-Bench (SX-Bench), a novel benchmark designed for complex multi-function understanding and fine-grained execution reasoning. SX-Bench features tasks involving collaboration among multiple sub-functions (e.g., chained calls, nested loops), shifting evaluation towards overall control and data flow modeling. It defines "computation steps" as the minimal execution unit and requires models to predict the total number of steps in reasoning tasks, thereby assessing a model's in-depth understanding of dynamic execution beyond simple I/O matching. Evaluation on over 20 mainstream models (including 14 reasoning-enhanced models) demonstrates that SX-Bench is highly discriminative: even the state-of-the-art OpenAI-O3 achieves only 78.37 percent accuracy on Hard-Reasoning tasks, much lower than its saturated scores on previous benchmarks, thereby revealing bottlenecks in complex and fine-grained reasoning. We also release an automated pipeline combining program synthesis, symbolic execution, and LLM-aided validation for efficient benchmark generation and quality assurance. SX-Bench advances code evaluation from "single-function verification" to "multi-function dynamic reasoning," providing a key tool for the in-depth assessment of advanced code intelligence models.

Sophisticated computation methods were developed 4000 years ago in Mesopotamia in the context of scribal schools. The basics of the computation can be detected in clay tablets written by young students educated in these scribal schools. At first glance, the mathematical exercises contained in school tablets seem to be very simple and quite familiar, and therefore, they have little attracted the attention of historians of mathematics. Yet if we look more closely at these modest writings, their simplicity proves deceptive. The feeling of familiarity primarily results from a projection on to the past of thought patterns ingrained from our own early learning. Careful observation of school tablets reveals subtle differences between ancient and modern notions of number, quantity, measurement unit, order, divisibility, algorithm, etc. In fact, we find a completely original mathematical world with rules of its own. This article explores the computation practices developed in scholarly milieus in the light of the mathematical basics that was taught in elementary education.

Ongoing efforts that span over decades show a rise of AI methods for accelerating scientific discovery, yet accelerating discovery in mathematics remains a persistent challenge for AI. Specifically, AI methods were not effective in creation of formulas for mathematical constants because each such formula must be correct for infinite digits of precision, with "near-true" formulas providing no insight toward the correct ones. Consequently, formula discovery lacks a clear distance metric needed to guide automated discovery in this realm. In this work, we propose a systematic methodology for categorization, characterization, and pattern identification of such formulas. The key to our methodology is introducing metrics based on the convergence dynamics of the formulas, rather than on the numerical value of the formula. These metrics enable the first automated clustering of mathematical formulas. We demonstrate this methodology on Polynomial Continued Fraction formulas, which are ubiquitous in their intrinsic connections to mathematical constants, and generalize many mathematical functions and structures. We test our methodology on a set of 1,768,900 such formulas, identifying many known formulas for mathematical constants, and discover previously unknown formulas for pi, ln(2), Gauss', and Lemniscate's constants. The uncovered patterns enable a direct generalization of individual formulas to infinite families, unveiling rich mathematical structures. This success paves the way towards a generative model that creates formulas fulfilling specified mathematical properties, accelerating the rate of discovery of useful formulas.

Reviewing the progress in artificial intelligence over the past decade, various significant advances (e.g. object detection, image generation, large language models) have enabled AI systems to produce more semantically meaningful outputs and achieve widespread adoption in internet scenarios. Nevertheless, AI systems still struggle when it comes to understanding and interacting with the physical world. This reveals an important issue: relying solely on semantic-level concepts learned from internet data (e.g. texts, images) to understand the physical world is far from sufficient -- machine intelligence currently lacks an effective way to learn about the physical world. This research introduces the idea of analytic concept -- representing the concepts related to the physical world through programs of mathematical procedures, providing machine intelligence a portal to perceive, reason about, and interact with the physical world. Except for detailing the design philosophy and providing guidelines for the application of analytic concepts, this research also introduce about the infrastructure that has been built around analytic concepts. I aim for my research to contribute to addressing these questions: What is a proper abstraction of general concepts in the physical world for machine intelligence? How to systematically integrate structured priors with neural networks to constrain AI systems to comply with physical laws?

Recent progress in large language models (LLMs) like GPT-4 and PaLM-2 has brought significant advancements in addressing math reasoning problems. In particular, OpenAI's latest version of GPT-4, known as GPT-4 Code Interpreter, shows remarkable performance on challenging math datasets. In this paper, we explore the effect of code on enhancing LLMs' reasoning capability by introducing different constraints on the Code Usage Frequency of GPT-4 Code Interpreter. We found that its success can be largely attributed to its powerful skills in generating and executing code, evaluating the output of code execution, and rectifying its solution when receiving unreasonable outputs. Based on this insight, we propose a novel and effective prompting method, explicit code-based self-verification~(CSV), to further boost the mathematical reasoning potential of GPT-4 Code Interpreter. This method employs a zero-shot prompt on GPT-4 Code Interpreter to encourage it to use code to self-verify its answers. In instances where the verification state registers as ``False'', the model shall automatically amend its solution, analogous to our approach of rectifying errors during a mathematics examination. Furthermore, we recognize that the states of the verification result indicate the confidence of a solution, which can improve the effectiveness of majority voting. With GPT-4 Code Interpreter and CSV, we achieve an impressive zero-shot accuracy on MATH dataset (53.9\% to 84.3\%).

Formulas involving fundamental mathematical constants had a great impact on various fields of science and mathematics, for example aiding in proofs of irrationality of constants. However, the discovery of such formulas has historically remained scarce, often perceived as an act of mathematical genius by great mathematicians such as Ramanujan, Euler, and Gauss. Recent efforts to automate the discovery of formulas for mathematical constants, such as the Ramanujan Machine project, relied on exhaustive search. Despite several successful discoveries, exhaustive search remains limited by the space of options that can be covered and by the need for vast amounts of computational resources. Here we propose a fundamentally different method to search for conjectures on mathematical constants: through analysis of integer sequences. We introduce the Enumerated Signed-continued-fraction Massey Approve (ESMA) algorithm, which builds on the Berlekamp-Massey algorithm to identify patterns in integer sequences that represent mathematical constants. The ESMA algorithm found various known formulas for e, e^2, tan(1), and ratios of values of Bessel functions. The algorithm further discovered a large number of new conjectures for these constants, some providing simpler representations and some providing faster numerical convergence than the corresponding simple continued fractions. Along with the algorithm, we present mathematical tools for manipulating continued fractions. These connections enable us to characterize what space of constants can be found by ESMA and quantify its algorithmic advantage in certain scenarios. Altogether, this work continues in the development of augmenting mathematical intuition by computer algorithms, to help reveal mathematical structures and accelerate mathematical research.

Can AI make progress on important, unsolved mathematical problems? Large language models are now capable of sophisticated mathematical and scientific reasoning, but whether they can perform novel research is still widely debated and underexplored. We introduce HorizonMath, a benchmark of over 100 predominantly unsolved problems spanning 8 domains in computational and applied mathematics, paired with an open-source evaluation framework for automated verification. Our benchmark targets a class of problems where discovery is hard, requiring meaningful mathematical insight, but verification is computationally efficient and simple. Because these solutions are unknown, HorizonMath is immune to data contamination, and most state-of-the-art models score near 0%. Existing research-level benchmarks instead rely on formal proof verification or manual review, both of which are expensive to scale. Using this platform, we find two problems for which GPT 5.4 Pro proposes solutions that improve on the best-known published results, representing potential novel contributions (pending expert review). We release HorizonMath as an open challenge and a growing community resource, where correct solutions to problems in the unsolved problem classes could constitute novel results in the mathematical literature.

Over the last decades, deep neural networks based-models became the dominant paradigm in machine learning. Further, the use of artificial neural networks in symbolic learning has been seen as increasingly relevant recently. To study the capabilities of neural networks in the symbolic AI domain, researchers have explored the ability of deep neural networks to learn mathematical constructions, such as addition and multiplication, logic inference, such as theorem provers, and even the execution of computer programs. The latter is known to be too complex a task for neural networks. Therefore, the results were not always successful, and often required the introduction of biased elements in the learning process, in addition to restricting the scope of possible programs to be executed. In this work, we will analyze the ability of neural networks to learn how to execute programs as a whole. To do so, we propose a different approach. Instead of using an imperative programming language, with complex structures, we use the Lambda Calculus (λ-Calculus), a simple, but Turing-Complete mathematical formalism, which serves as the basis for modern functional programming languages and is at the heart of computability theory. We will introduce the use of integrated neural learning and lambda calculi formalization. Finally, we explore execution of a program in λ-Calculus is based on reductions, we will show that it is enough to learn how to perform these reductions so that we can execute any program. Keywords: Machine Learning, Lambda Calculus, Neurosymbolic AI, Neural Networks, Transformer Model, Sequence-to-Sequence Models, Computational Models

This report overviews our ongoing work in enriching chain-of-thoughts datasets requiring arithmetical reasoning with the integration of non-parametric components, such as a calculator. We conduct an analysis of prominent relevant datasets such as GSM8K, Ape210K, AQuA-RAT, and MathQA and propose a machine-processable HTML-like format specifically tailored for working with semi-structured chains. By converting the datasets into this unified format, we enable the effective integration of large language models and symbolic systems, empowering them to tackle arithmetical reasoning tasks more efficiently.

Large language models (LLMs) have demonstrated remarkable capabilities in code-related tasks, such as code understanding and code generation. However, an equally important yet underexplored question is whether LLMs can serve as general-purpose surrogate code executors, to predict the output and behavior of a program without actually running it. To systematically investigate this capability, we introduce SURGE, a comprehensive benchmark covering eight key aspects: multi-language programming tasks, competition-level programming problems, repository-level code analysis, high-cost scientific computing, time-complexity-intensive algorithms, buggy code analysis, programs dependent on specific compilers or execution environments, and formal mathematical proof verification. We evaluate multiple open-source and proprietary LLMs on SURGE and conduct a scaling study to analyze the impact of model size and training data scale on surrogate execution accuracy. Additionally, we categorize model prediction errors and explore potential areas for improvement. Our findings indicate that while LLMs can predict code execution results in certain cases, they exhibit limitations in general-purpose surrogate execution. This study provides empirical insights into the feasibility of using LLMs as surrogate code executors. Code and dataset are released at https://github.com/Imbernoulli/SURGE.

AI systems are increasingly capable of contributing to mathematical research. In research practice, frontier-model reasoning is a limited resource, and expert mathematical review is even more sharply constrained. Allocating these scarce resources well is therefore central to making AI-assisted mathematical discovery efficient. In most current AI-for-math workflows, human effort is concentrated at the beginning and end, in selecting suitable research problems and later reviewing the resulting artifacts. These two stages are becoming bottlenecks for research-level mathematics. We address them by proposing a new human-AI discovery paradigm. The human input is no longer a single problem selected in advance, but a research direction in which the experts have interest and expertise. The system then searches a broad literature corpus for candidate problems in that direction. Inspired by search and recommender systems, we build Find, Attempt, and Recommend (FAR), a literature-to-review cascade that automates the search for suitable problems and focuses human attention on artifacts that have passed several stages of filtering. In a combinatorics pilot, the pipeline starts from 5,245 combinatorics papers, recovers 6,453 candidate conjectures or open problems, and filters them to 4,717 apparently well-posed and still-open conjectures. Subsequent reasoning and automated triage stages surface 598 potential resolutions and select 77 items for author-team review. Among them, we identify many interesting discoveries, including results on conjectures and questions of Davies--Jenssen--Perkins--Roberts, Erdős--Straus, Ikenmeyer--Pak--Panova, and Lund--Saraf--Wolf. These results demonstrate the effectiveness of this new mode of human-AI collaboration for mathematical discovery.

The application of physics formulas is a fundamental human capability in numerical reasoning. While existing datasets often rely on implicit mathematical knowledge, they rarely explicitate the underlying formulas. To address this, we introduce FormulaReasoning, a new benchmark for formula-based numerical reasoning comprising 5,324 questions requiring calculations grounded in external physics principles. We provide high-quality, fine-grained annotations in English and Chinese--including formula structures, parameter names, symbols, values, and units--curated through manual effort and LLM-assisted validation. Additionally, we provide a consolidated formula database as an external knowledge source. To further challenge model performance, we develop an extended version of the dataset by coupling multiple questions. We evaluate various architectural and methodological frameworks, including retrieval-augmented methods, modular reasoning (formula generation, parameter extraction, and calculation), and preference-based optimization. Our analysis identifies critical challenges in formula-based reasoning, highlighting significant opportunities for future methodological advancement.

We present a reproducible, parameter-driven software workflow for optimizing approximate mutually unbiased basis (AMUB) configurations in arbitrary dimensions d using a Lie-algebra unitary parameterization. The workflow is designed for portable execution across CPU, Apple MPS, CUDA-capable GPU, and HPC backends, using a Taylor-series matrix exponential layer as an accelerator compatibility pathway. As a dimension-six case study, we optimize unanchored configurations across 100 random seeds for basis counts n = 3, 4, 5, 6 in complex128 and complex64 arithmetic. The workflow recovers exact three-basis configurations, identifies a recurrent four-basis partial-exact hub-and-triangle structure, and finds no near-exact pairs for n = 5 or n = 6 in the reported campaigns under the primary tolerance. As a hardware-execution check, we embed the representative d = 6, n = 4 transition unitaries into three-qubit 8x8 unitaries and execute the resulting circuits on the 156-qubit Heron processor ibm-marrakesh using subspace post-selection. The measured QPU pairwise losses are dominated by a hardware and compilation noise floor of approximately 0.02-0.08, associated with compiled circuits averaging 37 native CZ gates, which obscures the distinction between classically near-exact and defective pairs. The results provide a reproducible computational framework for exploring AMUB landscapes, together with an initial assessment of the challenges involved in executing optimized dimension-six unitaries on current quantum hardware.

As mathematical computing becomes more democratized in high-level languages, high-performance symbolic-numeric systems are necessary for domain scientists and engineers to get the best performance out of their machine without deep knowledge of code optimization. Naturally, users need different term types either to have different algebraic properties for them, or to use efficient data structures. To this end, we developed Symbolics.jl, an extendable symbolic system which uses dynamic multiple dispatch to change behavior depending on the domain needs. In this work we detail an underlying abstract term interface which allows for speed without sacrificing generality. We show that by formalizing a generic API on actions independent of implementation, we can retroactively add optimized data structures to our system without changing the pre-existing term rewriters. We showcase how this can be used to optimize term construction and give a 113x acceleration on general symbolic transformations. Further, we show that such a generic API allows for complementary term-rewriting implementations. We demonstrate the ability to swap between classical term-rewriting simplifiers and e-graph-based term-rewriting simplifiers. We showcase an e-graph ruleset which minimizes the number of CPU cycles during expression evaluation, and demonstrate how it simplifies a real-world reaction-network simulation to halve the runtime. Additionally, we show a reaction-diffusion partial differential equation solver which is able to be automatically converted into symbolic expressions via multiple dispatch tracing, which is subsequently accelerated and parallelized to give a 157x simulation speedup. Together, this presents Symbolics.jl as a next-generation symbolic-numeric computing environment geared towards modeling and simulation.

Large Language Models (LLMs) have demonstrated remarkable capabilities across a wide range of natural language processing and reasoning tasks. However, their performance in the foundational domain of arithmetic remains unsatisfactory. When dealing with arithmetic tasks, LLMs often memorize specific examples rather than learning the underlying computational logic, limiting their ability to generalize to new problems. In this paper, we propose a Composable Arithmetic Execution Framework (CAEF) that enables LLMs to learn to execute step-by-step computations by emulating Turing Machines, thereby gaining a genuine understanding of computational logic. Moreover, the proposed framework is highly scalable, allowing composing learned operators to significantly reduce the difficulty of learning complex operators. In our evaluation, CAEF achieves nearly 100% accuracy across seven common mathematical operations on the LLaMA 3.1-8B model, effectively supporting computations involving operands with up to 100 digits, a level where GPT-4o falls short noticeably in some settings.

The rapidly growing demand for high-quality data in Large Language Models (LLMs) has intensified the need for scalable, reliable, and semantically rich data preparation pipelines. However, current practices remain dominated by ad-hoc scripts and loosely specified workflows, which lack principled abstractions, hinder reproducibility, and offer limited support for model-in-the-loop data generation. To address these challenges, we present DataFlow, a unified and extensible LLM-driven data preparation framework. DataFlow is designed with system-level abstractions that enable modular, reusable, and composable data transformations, and provides a PyTorch-style pipeline construction API for building debuggable and optimizable dataflows. The framework consists of nearly 200 reusable operators and six domain-general pipelines spanning text, mathematical reasoning, code, Text-to-SQL, agentic RAG, and large-scale knowledge extraction. To further improve usability, we introduce DataFlow-Agent, which automatically translates natural-language specifications into executable pipelines via operator synthesis, pipeline planning, and iterative verification. Across six representative use cases, DataFlow consistently improves downstream LLM performance. Our math, code, and text pipelines outperform curated human datasets and specialized synthetic baselines, achieving up to +3\% execution accuracy in Text-to-SQL over SynSQL, +7\% average improvements on code benchmarks, and 1--3 point gains on MATH, GSM8K, and AIME. Moreover, a unified 10K-sample dataset produced by DataFlow enables base models to surpass counterparts trained on 1M Infinity-Instruct data. These results demonstrate that DataFlow provides a practical and high-performance substrate for reliable, reproducible, and scalable LLM data preparation, and establishes a system-level foundation for future data-centric AI development.

We present a case study where an automatic AI system formalizes a textbook with more than 500 pages of graduate-level algebraic combinatorics to Lean. The resulting formalization represents a new milestone in textbook formalization scale and proficiency, moving from early results in undergraduate topology and restructuring of existing library content to a full standalone formalization of a graduate textbook. The formalization comprises 130K lines of code and 5900 Lean declarations and was conducted within one week by a total of 30K Claude 4.5 Opus agents collaborating in parallel on a shared code base via version control, simultaneously setting a record in multi-agent software engineering with usable results. The inference cost matches or undercuts what we estimate as the salaries required for a team of human experts, and we expect there is still the potential for large efficiencies to be made without the need for better models. We make our code, the resulting Lean code base and a side-by-side blueprint website available open-source.

Large language models (LLMs) have achieved impressive results on multi-step mathematical reasoning, yet at the cost of high computational overhead. This challenge is particularly acute for test-time scaling methods such as parallel decoding, which increase answer diversity but scale poorly in efficiency. To address this efficiency-accuracy trade-off, we propose SSR (Speculative Parallel Scaling Reasoning), a training-free framework that leverages a key insight: by introducing speculative decoding at the step level, we can accelerate reasoning without sacrificing correctness. SSR integrates two components: a Selective Parallel Module (SPM) that identifies a small set of promising reasoning strategies via model-internal scoring, and Step-level Speculative Decoding (SSD), which enables efficient draft-target collaboration for fine-grained reasoning acceleration. Experiments on three mathematical benchmarks-AIME 2024, MATH-500, and LiveMathBench - demonstrate that SSR achieves strong gains over baselines. For instance, on LiveMathBench, SSR improves pass@1 accuracy by 13.84% while reducing computation to 80.5% of the baseline FLOPs. On MATH-500, SSR reduces compute to only 30% with no loss in accuracy.

There is a substantial body of literature examining the mathematical reasoning capabilities of large language models (LLMs), particularly their performance on precise arithmetic operations in autoregressive architectures. However, their ability to perform approximate reasoning in informal, fast-paced mathematical operations has received far less attention, especially among non-autoregressive decoder models. Our work addresses this gap by introducing StreetMath, a benchmark designed to evaluate models' approximation abilities under real-world approximation scenarios. We conduct extensive evaluations across different LLM architectures: Qwen3-4B-Instruct-2507, Qwen3-4B-Thinking-2507, Dream-v0-Instruct-7B, Falcon-Mamba-7B-Instruct, and Mamba-GPT-3B. Furthermore, we apply mechanistic interpretability techniques to probe their internal computational states. Our analysis reveals that LLMs generally attempt to compute exact values or invoke external tools even in tasks that call for approximation. Moreover, while models sometimes reach the correct answer in early layers or steps, they still consume more tokens when solving approximation tasks. Additional experiments indicate that exact and approximate arithmetic operations rely on largely separate neural components. Drawing upon research on cognitive psychology, we argue that LLMs do not exhibit cognitive miserliness in the same way humans do in street math settings. We open source our work https://github.com/ctseng777/StreetMath

Recent advancements in Chain-of-Thoughts (CoT) and Program-of-Thoughts (PoT) methods have greatly enhanced language models' mathematical reasoning capabilities, facilitating their integration into instruction tuning datasets with LLMs. However, existing methods for large-scale dataset creation require substantial seed data and high computational costs for data synthesis, posing significant challenges for scalability. We introduce InfinityMATH, a scalable instruction tuning dataset for programmatic mathematical reasoning. The construction pipeline emphasizes decoupling numbers from mathematical problems to synthesize number-independent programs, enabling efficient and flexible scaling while minimizing dependency on specific numerical values. Fine-tuning experiments with open-source language and code models, such as Llama2 and CodeLlama, demonstrate the practical benefits of InfinityMATH. These fine-tuned models, showed significant relative improvements on both in-domain and out-of-domain benchmarks, ranging from 184.7% to 514.3% on average. Additionally, these models exhibited high robustness on the GSM8K+ and MATH+ benchmarks, which are enhanced version of test sets with simply the number variations. InfinityMATH ensures that models are more versatile and effective across a broader range of mathematical problems. The data is available at https://huggingface.co/datasets/flagopen/InfinityMATH.

This definitive research memoria presents a comprehensive, mathematically verified paradigm for neural communication with Bitcoin mining Application-Specific Integrated Circuits (ASICs), integrating five complementary frameworks: thermodynamic reservoir computing, hierarchical number system theory, algorithmic analysis, network latency optimization, and machine-checked mathematical formalization. We establish that obsolete cryptocurrency mining hardware exhibits emergent computational properties enabling bidirectional information exchange between AI systems and silicon substrates. The research program demonstrates: (1) reservoir computing with NARMA-10 Normalized Root Mean Square Error (NRMSE) of 0.8661; (2) the Thermodynamic Probability Filter (TPF) achieving 92.19% theoretical energy reduction; (3) the Virtual Block Manager achieving +25% effective hashrate; and (4) hardware universality across multiple ASIC families including Antminer S9, Lucky Miner LV06, and Goldshell LB-Box. A significant contribution is the machine-checked mathematical formalization using Lean 4 and Mathlib, providing unambiguous definitions, machine-verified theorems, and reviewer-proof claims. Key theorems proven include: independence implies zero leakage, predictor beats baseline implies non-independence (the logical core of TPF), energy savings theoretical maximum, and Physical Unclonable Function (PUF) distinguishability witnesses. Vladimir Veselov's hierarchical number system theory explains why early-round information contains predictive power. This work establishes a new paradigm: treating ASICs not as passive computational substrates but as active conversational partners whose thermodynamic state encodes exploitable computational information.

The widespread adoption of machine learning algorithms necessitates hardware acceleration to ensure efficient performance. This acceleration relies on custom matrix engines that operate on full or reduced-precision floating-point arithmetic. However, conventional floating-point implementations can be power hungry. This paper proposes a method to improve the energy efficiency of the matrix engines used in machine learning algorithm acceleration. Our approach leverages approximate normalization within the floating-point multiply-add units as a means to reduce their hardware complexity, without sacrificing overall machine-learning model accuracy. Hardware synthesis results show that this technique reduces area and power consumption roughly by 16% and 13% on average for Bfloat16 format. Also, the error introduced in transformer model accuracy is 1% on average, for the most efficient configuration of the proposed approach.

Mathematical reasoning models are widely deployed in education, automated tutoring, and decision support systems despite exhibiting fundamental computational instabilities. We demonstrate that state-of-the-art models (Qwen2.5-Math-7B) achieve 61% accuracy through a mixture of reliable and unreliable reasoning pathways: 18.4% of correct predictions employ stable, faithful reasoning while 81.6% emerge through computationally inconsistent pathways. Additionally, 8.8% of all predictions are silent failures -- confident yet incorrect outputs. Through comprehensive analysis using novel faithfulness metrics, we reveal: (1) reasoning quality shows weak negative correlation with correctness (r=-0.21, p=0.002), reflecting a binary classification threshold artifact rather than a monotonic inverse relationship; (2) scaling from 1.5B to 7B parameters (4.7x increase) provides zero accuracy benefit on our evaluated subset (6% of GSM8K), requiring validation on the complete benchmark; and (3) latent reasoning employs diverse computational strategies, with ~20% sharing CoT-like patterns. These findings highlight that benchmark accuracy can mask computational unreliability, demanding evaluation reforms measuring stability beyond single-sample metrics.

When verifying computer systems we sometimes want to study their asymptotic behaviors, i.e., how they behave in the long run. In such cases, we need real analysis, the area of mathematics that deals with limits and the foundations of calculus. In a prior work, we used real analysis in ACL2s to study the asymptotic behavior of the RTO computation, commonly used in congestion control algorithms across the Internet. One key component in our RTO computation analysis was proving in ACL2s that for all alpha in [0, 1), the limit as n approaches infinity of alpha raised to n is zero. Whereas the most obvious proof strategy involves the logarithm, whose codomain includes irrationals, by default ACL2 only supports rationals, which forced us to take a non-standard approach. In this paper, we explore different approaches to proving the above result in ACL2(r) and ACL2s, from the perspective of a relatively new user to each. We also contextualize the theorem by showing how it allowed us to prove important asymptotic properties of the RTO computation. Finally, we discuss tradeoffs between the various proof strategies and directions for future research.

Bessel functions are critical in scientific computing for applications such as machine learning, protein structure modeling, and robotics. However, currently, available routines lack precision or fail for certain input ranges, such as when the order v is large, and GPU-specific implementations are limited. We address the precision limitations of current numerical implementations while dramatically improving the runtime. We propose two novel algorithms for computing the logarithm of modified Bessel functions of the first and second kinds by computing intermediate values on a logarithmic scale. Our algorithms are robust and never have issues with underflows or overflows while having relative errors on the order of machine precision, even for inputs where existing libraries fail. In C++/CUDA, our algorithms have median and maximum speedups of 45x and 6150x for GPU and 17x and 3403x for CPU, respectively, over the ranges of inputs and third-party libraries tested. Compared to SciPy, the algorithms have median and maximum speedups of 77x and 300x for GPU and 35x and 98x for CPU, respectively, over the tested inputs. The ability to robustly compute a solution and the low relative errors allow us to fit von Mises-Fisher, vMF, distributions to high-dimensional neural network features. This is, e.g., relevant for uncertainty quantification in metric learning. We obtain image feature data by processing CIFAR10 training images with the convolutional layers of a pre-trained ResNet50. We successfully fit vMF distributions to 2048-, 8192-, and 32768-dimensional image feature data using our algorithms. Our approach provides fast and accurate results while existing implementations in SciPy and mpmath fail to fit successfully. Our approach is readily implementable on GPUs, and we provide a fast open-source implementation alongside this paper.

We present Hermes 4, a family of hybrid reasoning models that combine structured, multi-turn reasoning with broad instruction-following ability. We describe the challenges encountered during data curation, synthesis, training, and evaluation, and outline the solutions employed to address these challenges at scale. We comprehensively evaluate across mathematical reasoning, coding, knowledge, comprehension, and alignment benchmarks, and we report both quantitative performance and qualitative behavioral analysis. To support open research, all model weights are published publicly at https://huggingface.co/collections/NousResearch/hermes-4-collection-68a731bfd452e20816725728

We study a governed approach to enterprise analytics: a language model interprets the question, while deterministic policy selects and runs a pre-approved analytical program that returns both results and evidence. We show that this restriction can remain expressive within a defined analytical class, using relational operations plus aggregation, comparison, windows, ranking, and similarity. Fixed meaning, policy, data, and execution rules also make results replayable. Across 440 runs, three 8B models generated SQL and selected tools at runtime, while Qwen3-8B interpreted intent only and policy executed the approved program. None of 330 runtime-planning episodes matched the full answer-and-evidence contract across all test datasets; the policy-executed analyzer matched 110 of 110. This is a configuration-specific result, not evidence that runtime agents cannot succeed under other designs.

Large-scale formalization of advanced mathematics requires more than translating individual statements: it must reconstruct a coherent theory distributed across heterogeneous sources. This process raises four challenges: discovering implicit dependencies, correcting source defects, preserving semantic fidelity, and reconciling cross-source misalignments. We present FormaTheoria, an end-to-end, AI-assisted workflow that coordinates source acquisition, formalization, proof construction, recursive dependency discovery, independent review, and reconciliation, while preserving provenance and protecting approved declarations. A shared agent framework supports long-horizon execution through tool use, context compaction, review-gated termination, section-level source context, and dependency-aware batch parallelization. Applying FormaTheoria to major components of the Classification of Finite Simple Groups (CFSG), we construct a machine-checked Lean development extending through the Bender--Suzuki theorem and encompassing the Feit--Thompson Odd Order Theorem, Glauberman's Z^* theorem, and the Brauer--Suzuki theorem. This development verifies an extensive body of deeply interdependent finite-group theory while providing a foundation for continuing the CFSG formalization. An empirical analysis of the code and recorded construction process supports the practical relevance of the identified challenges and illustrates the roles of the corresponding workflow components. Together, these results demonstrate how AI-assisted workflows can reconstruct mathematically significant formal theories from distributed literature by combining language-model agents with formal verification, structured review, and explicit dependency management.

Text-to-SQL is a key natural language processing task that maps natural language questions to SQL queries, enabling intuitive interaction with web-based databases. Although current methods perform well on benchmarks like BIRD and Spider, they struggle with complex reasoning, domain knowledge, and hypothetical queries, and remain costly in enterprise deployment. To address these issues, we propose a framework named IESR(Information Enhanced Structured Reasoning) for lightweight large language models: (i) leverages LLMs for key information understanding and schema linking, and decoupling mathematical computation and SQL generation, (ii) integrates a multi-path reasoning mechanism based on Monte Carlo Tree Search (MCTS) with majority voting, and (iii) introduces a trajectory consistency verification module with a discriminator model to ensure accuracy and consistency. Experimental results demonstrate that IESR achieves state-of-the-art performance on the complex reasoning benchmark LogicCat (24.28 EX) and the Archer dataset (37.28 EX) using only compact lightweight models without fine-tuning. Furthermore, our analysis reveals that current coder models exhibit notable biases and deficiencies in physical knowledge, mathematical computation, and common-sense reasoning, highlighting important directions for future research. We released code at https://github.com/Ffunkytao/IESR-SLM.

The key limitation of the verification performance lies in the ability of error detection. With this intuition we designed several variants of pessimistic verification, which are simple workflows that could significantly improve the verification of open-ended math questions. In pessimistic verification we construct multiple parallel verifications for the same proof, and the proof is deemed incorrect if any one of them reports an error. This simple technique significantly improves the performance across many math verification benchmarks without incurring substantial computational resources. Its token efficiency even surpassed extended long-CoT in test-time scaling. Our case studies further indicate that the majority of false negatives in stronger models are actually caused by annotation errors in the original dataset, so our method's performance is in fact underestimated. Self-verification for mathematical problems can effectively improve the reliability and performance of language model outputs, and it also plays a critical role in enabling long-horizon mathematical tasks. We believe that research on pessimistic verification will help enhance the mathematical capabilities of language models across a wide range of tasks.

Existing math datasets evaluate the reasoning abilities of large language models (LLMs) by either using the final answer or the intermediate reasoning steps derived from static examples. However, the former approach fails to surface model's uses of shortcuts and wrong reasoning while the later poses challenges in accommodating alternative solutions. In this work, we seek to use symbolic programs as a means for automated evaluation if a model can consistently produce correct final answers across various inputs to the program. We begin by extracting programs for popular math datasets (GSM8K and MATH) using GPT4-o. For those executable programs verified using the original input-output pairs, they are found to encapsulate the proper reasoning required to solve the original text questions. We then prompt GPT4-o to generate new questions using alternative input-output pairs based the extracted program. We apply the resulting datasets to evaluate a collection of LLMs. In our experiments, we observe significant accuracy drops using our proposed evaluation compared with original static examples, suggesting the fragility of math reasoning in state-of-the-art LLMs.

During early optimization passes, compilers must make predictions for machine-dependent characteristics such as execution unit utilization, number of register spills, latency, throughput etc. to generate better code. Often a hand-written static/analytical hardware cost model is built into the compiler. However, the need for more sophisticated and varied predictions has become more pronounced with the development of deep learning compilers which need to optimize dataflow graphs. Such compilers usually employ a much higher level MLIR form as an IR representation before lowering to traditional LLVM-IR. A static/analytical cost model in such a scenario is cumbersome and error prone as the opcodes represent very high level algebraic/arithmetic operations. Hence, we develop a machine learning-based cost model for high-level MLIR which can predict different target variables of interest such as CPU/GPU/xPU utilization, instructions executed, register usage etc. By considering the incoming MLIR as a text input a la NLP models we can apply well-known techniques from modern NLP research to help predict hardware characteristics more accurately. We expect such precise ML-driven hardware cost models to guide our deep learning compiler in graph level optimizations around operator fusion, local memory allocation, kernel scheduling etc. as well as in many kernel-level optimizations such as loop interchange, LICM and unroll. We report early work-in -progress results of developing such models on high-level MLIR representing dataflow graphs emitted by Pytorch/Tensorflow-like frameworks as well as lower-level dialects like affine. We show that these models can provide reasonably good estimates with low error bounds for various hardware characteristics of interest and can be a go-to mechanism for hardware cost modelling in the future.

Language models are not accurate in numerical problems. Their architecture does not allow for anything less than a probabilistic next word. This paper introduces ComputeGPT: an approach of creating a chat model able to answer computational problems through running on-demand code. ComputeGPT converts each question to relevant code, runs the code, and returns the computed answer as part of the chat. We combine this approach with a local browser-based Python interpretation and fine-tuned prompts in order to achieve state-of-the-art efficiency on numerical problems and provide a suitable front-end and safe environment for the code to be executed in.

We prove that any Turing machine running on inputs of arbitrary length can be simulated by a constant bit-size transformer, as long as the context window is sufficiently long. This improves previous works, which require scaling up either the model's precision or the number of parameters on longer inputs. Furthermore, we prove that the complexity class SPACE[s(n)] exactly characterizes the expressive power of a constant bit-size transformer with a context window of length s(n). Our approach relies on simulating Post machines, a Turing-complete computational model. Post machines can be modeled as automata equipped with a queue, exhibiting computational behaviors naturally aligned with those of transformers. The behavioral similarity between transformers and Post machines may offer new insights into the mechanisms underlying the reasoning abilities of transformers.

As automated reasoning systems advance rapidly, there is a growing need for research-level formal mathematical problems to accurately evaluate their capabilities. To address this, we present Formal Conjectures, an evolving benchmark of currently 2615 mathematical problem statements formalized in Lean 4. Sourced from areas of active mathematical research, the dataset features 1029 open research conjectures providing a zero-contamination benchmark for mathematical proof discovery, and 836 solved problems for proof autoformalization. Notably, the repository provides a structured interface connecting mathematicians who formalize and clarify problems with the AI systems and humans attempting to solve them. Demonstrating its immediate utility, the benchmark has already been leveraged to make new mathematical discoveries, including the resolution of open research conjectures. We describe our approach to ensuring the correctness of these formalizations in a collaborative open-source project where contributions stem from an active community. In this framework, AI-generated proofs and disproofs serve as a valuable auditing mechanism to iteratively improve the fidelity of the benchmark. Finally, we provide a standardized evaluation setup and report baseline results on frozen evaluation subsets, demonstrating a climbable signal that measures the current frontier of automated reasoning on research-level mathematics.

Recent advancements in software engineering agents have demonstrated promising capabilities in automating program improvements. However, their reliance on closed-source or resource-intensive models introduces significant deployment challenges in private environments, prompting a critical question: How can personally deployable open-source LLMs achieve comparable code reasoning performance? To this end, we propose a unified Test-Time Compute scaling framework that leverages increased inference-time computation instead of larger models. Our framework incorporates two complementary strategies: internal TTC and external TTC. Internally, we introduce a development-contextualized trajectory synthesis method leveraging real-world software repositories to bootstrap multi-stage reasoning processes, such as fault localization and patch generation. We further enhance trajectory quality through rejection sampling, rigorously evaluating trajectories along accuracy and complexity. Externally, we propose a novel development-process-based search strategy guided by reward models and execution verification. This approach enables targeted computational allocation at critical development decision points, overcoming limitations of existing "end-point only" verification methods. Evaluations on SWE-bench Verified demonstrate our 32B model achieves a 46\% issue resolution rate, surpassing significantly larger models such as DeepSeek R1 671B and OpenAI o1. Additionally, we provide the empirical validation of the test-time scaling phenomenon within SWE agents, revealing that models dynamically allocate more tokens to increasingly challenging problems, effectively enhancing reasoning capabilities. We publicly release all training data, models, and code to facilitate future research. https://github.com/yingweima2022/SWE-Reasoner

This paper presents our winning submission to the AI Mathematical Olympiad - Progress Prize 2 (AIMO-2) competition. Our recipe for building state-of-the-art mathematical reasoning models relies on three key pillars. First, we create a large-scale dataset comprising 540K unique high-quality math problems, including olympiad-level problems, and their 3.2M long-reasoning solutions. Second, we develop a novel method to integrate code execution with long reasoning models through iterative training, generation, and quality filtering, resulting in 1.7M high-quality Tool-Integrated Reasoning solutions. Third, we create a pipeline to train models to select the most promising solution from many candidates. We show that such generative solution selection (GenSelect) can significantly improve upon majority voting baseline. Combining these ideas, we train a series of models that achieve state-of-the-art results on mathematical reasoning benchmarks. To facilitate further research, we release our code, models, and the complete OpenMathReasoning dataset under a commercially permissive license.

Large Language Models (LLMs) have recently achieved remarkable progress in mathematical reasoning. To enable such capabilities, many existing works distill strong reasoning models into long chains of thought or design algorithms to construct high-quality math QA data for training. However, these efforts primarily focus on generating correct reasoning paths and answers, while largely overlooking the validity of the questions themselves. In this work, we propose Math Question Verification (MathQ-Verify), a novel five-stage pipeline designed to rigorously filter ill-posed or under-specified math problems. MathQ-Verify first performs format-level validation to remove redundant instructions and ensure that each question is syntactically well-formed. It then formalizes each question, decomposes it into atomic conditions, and verifies them against mathematical definitions. Next, it detects logical contradictions among these conditions, followed by a goal-oriented completeness check to ensure the question provides sufficient information for solving. To evaluate this task, we use existing benchmarks along with an additional dataset we construct, containing 2,147 math questions with diverse error types, each manually double-validated. Experiments show that MathQ-Verify achieves state-of-the-art performance across multiple benchmarks, improving the F1 score by up to 25 percentage points over the direct verification baseline. It further attains approximately 90% precision and 63% recall through a lightweight model voting scheme. MathQ-Verify offers a scalable and accurate solution for curating reliable mathematical datasets, reducing label noise and avoiding unnecessary computation on invalid questions. Our code and data are available at https://github.com/scuuy/MathQ-Verify.

Code has been shown to be effective in enhancing the mathematical reasoning abilities of large language models due to its precision and accuracy. Previous works involving continued mathematical pretraining often include code that utilizes math-related packages, which are primarily designed for fields such as engineering, machine learning, signal processing, or module testing, rather than being directly focused on mathematical reasoning. In this paper, we introduce a novel method for generating mathematical code accompanied with corresponding reasoning steps for continued pretraining. Our approach begins with the construction of a high-quality mathematical continued pretraining dataset by incorporating math-related web data, code using mathematical packages, math textbooks, and synthetic data. Next, we construct reasoning steps by extracting LaTeX expressions, the conditions needed for the expressions, and the results of the expressions from the previously collected dataset. Based on this extracted information, we generate corresponding code to accurately capture the mathematical reasoning process. Appending the generated code to each reasoning step results in data consisting of paired natural language reasoning steps and their corresponding code. Combining this data with the original dataset results in a 19.2B-token high-performing mathematical pretraining corpus, which we name MathCode-Pile. Training several popular base models with this corpus significantly improves their mathematical abilities, leading to the creation of the MathCoder2 family of models. All of our data processing and training code is open-sourced, ensuring full transparency and easy reproducibility of the entire data collection and training pipeline. The code is released at https://github.com/mathllm/MathCoder2 .

The study demonstrates the capabilities of a vector-based approach for calculating stoichiometric coefficients in chemical equations, using black powder as an illustrative example. A method is proposed for selecting and constraining intermediate interactions between reactants, as well as for identifying final products. It is shown that even a small number of components can lead to a large number of final and intermediate products. Through concrete calculations, a correlation is established between the number of possible chemical equations and the number of reactants. A methodology is proposed for computing all possible chemical equations within a reaction system for arbitrary component ratios, enabling the derivation of all feasible chemical reactions. Additionally, a method is developed for calculating the chemical composition for a fixed set of reactants, allowing for the evaluation of the set of products resulting from all possible chemical interactions given a specified initial composition.

This book presents a code-first introduction to computer vision, spanning classical 2D image processing, classical 3D vision, and deep learning. Organized as 44 short chapters across three parts, the book builds each topic from first principles: image arithmetic and morphology; convolution, pyramids, and frequency-domain filtering; feature detection, optical flow, and stereo; projective geometry, camera calibration, and structure from motion; and the full arc of modern deep learning, from a single neuron through convolutional networks, backpropagation, classic architectures, transfer learning, object detection, and semantic and instance segmentation, concluding with engineering considerations like mixed-precision and parallel training. Every technique is implemented directly in Python and NumPy or PyTorch and checked numerically against the corresponding OpenCV or PyTorch library function, so readers see not just the mathematics but its concrete behavior on real and synthetic data. The material was distilled with AI assistance from freely available online course notes, condensing extensive working code into concise mathematical exposition while preserving verified, reproducible results throughout. It is intended as a self-contained reference for students and practitioners who want to understand computer vision algorithms and their Python implementations.

Linear attention has emerged as a cornerstone for efficient long-context architectures, as evidenced by its integration into state-of-the-art open-source models including Qwen3.5/3.6, Kimi Linear, and RWKV-7. Models that incorporate linear attention layers with the so-called Delta-Rule involve the inversion of triangular matrices as a core sub-routine. This operation often forms a performance bottleneck, and, due to its high-sensitivity to numerical errors, it can significantly deteriorate end-to-end model accuracy if it is not carefully implemented. This work provides a systematic analysis of both direct and iterative triangular inversion algorithms, targeting methods that are rich in matrix products, and, therefore, have the potential to efficiently utilize modern hardware. To that end, our analysis covers a broad spectrum of mathematical and practical aspects, with a heavy focus on numerical stability, computational complexity, and, ultimately, hardware efficiency and practical considerations. We provide a rigorous experimental evaluation to verify these properties in practical scenarios, and in low-precision floating-point representations, highlighting the strengths and limitations of each method. Performance benchmarks on NPUs reveal up to 4.3times speed-up against the state-of-the-art implementations of SGLang for triangular matrix inversion, leading to significant performance improvements on the entire layer level, while maintaining full end-to-end model accuracy.

Feedforward computation, such as evaluating a neural network or sampling from an autoregressive model, is ubiquitous in machine learning. The sequential nature of feedforward computation, however, requires a strict order of execution and cannot be easily accelerated with parallel computing. To enable parallelization, we frame the task of feedforward computation as solving a system of nonlinear equations. We then propose to find the solution using a Jacobi or Gauss-Seidel fixed-point iteration method, as well as hybrid methods of both. Crucially, Jacobi updates operate independently on each equation and can be executed in parallel. Our method is guaranteed to give exactly the same values as the original feedforward computation with a reduced (or equal) number of parallelizable iterations, and hence reduced time given sufficient parallel computing power. Experimentally, we demonstrate the effectiveness of our approach in accelerating (i) backpropagation of RNNs, (ii) evaluation of DenseNets, and (iii) autoregressive sampling of MADE and PixelCNN++, with speedup factors between 2.1 and 26 under various settings.

This paper presents the Functional Machine Calculus (FMC) as a simple model of higher-order computation with "reader/writer" effects: higher-order mutable store, input/output, and probabilistic and non-deterministic computation. The FMC derives from the lambda-calculus by taking the standard operational perspective of a call-by-name stack machine as primary, and introducing two natural generalizations. One, "locations", introduces multiple stacks, which each may represent an effect and so enable effect operators to be encoded into the abstraction and application constructs of the calculus. The second, "sequencing", is known from kappa-calculus and concatenative programming languages, and introduces the imperative notions of "skip" and "sequence". This enables the encoding of reduction strategies, including call-by-value lambda-calculus and monadic constructs. The encoding of effects into generalized abstraction and application means that standard results from the lambda-calculus may carry over to effects. The main result is confluence, which is possible because encoded effects reduce algebraically rather than operationally. Reduction generates the familiar algebraic laws for state, and unlike in the monadic setting, reader/writer effects combine seamlessly. A system of simple types confers termination of the machine.

We introduce the AI co-mathematician, a workbench for mathematicians to interactively leverage AI agents to pursue open-ended research. The AI co-mathematician is optimized to provide holistic support for the exploratory and iterative reality of mathematical workflows, including ideation, literature search, computational exploration, theorem proving and theory building. By providing an asynchronous, stateful workspace that manages uncertainty, refines user intent, tracks failed hypotheses, and outputs native mathematical artifacts, the system mirrors human collaborative workflows. In early tests, the AI co-mathematician helped researchers solve open problems, identify new research directions, and uncover overlooked literature references. Besides demonstrating a highly interactive paradigm for AI-assisted mathematical discovery, the AI co-mathematician also achieves state of the art results on hard problem-solving benchmarks, including scoring 48% on FrontierMath Tier 4, a new high score among all AI systems evaluated.

Large language models now solve many benchmark math problems at near-expert levels, yet this progress has not fully translated into reliable performance in real-world applications. We study this gap through contextual mathematical reasoning, where the mathematical core must be formulated from descriptive scenarios. We introduce ContextMATH, a benchmark that repurposes AIME and MATH-500 problems into two contextual settings: Scenario Grounding (SG), which embeds abstract problems into realistic narratives without increasing reasoning complexity, and Complexity Scaling (CS), which transforms explicit conditions into sub-problems to capture how constraints often appear in practice. Evaluating 61 proprietary and open-source models, we observe sharp drops: on average, open-source models decline by 13 and 34 points on SG and CS, while proprietary models drop by 13 and 20. Error analysis shows that errors are dominated by incorrect problem formulation, with formulation accuracy declining as original problem difficulty increases. Correct formulation emerges as a prerequisite for success, and its sufficiency improves with model scale, indicating that larger models advance in both understanding and reasoning. Nevertheless, formulation and reasoning remain two complementary bottlenecks that limit contextual mathematical problem solving. Finally, we find that fine-tuning with scenario data improves performance, whereas formulation-only training is ineffective. However, performance gaps are only partially alleviated, highlighting contextual mathematical reasoning as a central unsolved challenge for LLMs.

We present and evaluate LeanFlow, an LLM agent system specialized for translating mathematical papers into buildable Lean projects. Recent verifier-in-the-loop systems show that large formal artifacts can be produced, but it remains unclear which runtime mechanisms affect completion, auditability, or efficiency in document-to-project formalization. We study this question through case studies on two previously unformalized mathematical papers in number theory and measure theory, using model, proof-workflow, and toolset ablations with Kimi2.6 and GPT5.5; we report task outcome, API calls, input tokens, and output tokens. With Kimi2.6, the full workflow completes both document-level projects within the 2000-call budget, while no-queue variants reach the budget limit; with GPT5.5, all document-level variants complete, and the full workflow has the lowest or tied-lowest input-token cost on both sources. As complementary calibration, LeanFlow reaches 75.7% BEq+ on the PFR slice of RLM25 and solves all five ICML 2026 AI for Math TCS challenge projects in our GPT5.5 runs.

In this paper, we present techniques used to implement our portable vectorized library of C standard mathematical functions written entirely in C language. In order to make the library portable while maintaining good performance, intrinsic functions of vector extensions are abstracted by inline functions or preprocessor macros. We implemented the functions so that they can use sub-features of vector extensions such as fused multiply-add, mask registers and extraction of mantissa. In order to make computation with SIMD instructions efficient, the library only uses a small number of conditional branches, and all the computation paths are vectorized. We devised a variation of the Payne-Hanek argument reduction for trigonometric functions and a floating point remainder, both of which are suitable for vector computation. We compare the performance of our library to Intel SVML.

Tool-integrated reasoning (TIR) has become a key approach for improving large reasoning models (LRMs) on complex problems. Prior work has mainly studied when to invoke tools, while overlooking how tools are applied. We identify two common patterns: a calculator pattern that uses code for direct computation, and an algorithmic pattern that encodes problems as programs. Misaligned choices often cause failures even when reasoning is sound. We propose a two-stage framework that first builds code competence from both patterns and then aligns pattern selection with teacher preferences. Across challenging math datasets, our pattern-aware method substantially improves both code usage and accuracy, for instance raising Code@1 on MATH500 from 64.0% to 70.5% and on AIME24 from 26.7% to 50.0%. These gains highlight the effectiveness of a pattern-aware approach for tool-integrated reasoning.

Large language models are unreliable at arithmetic, which is a problem for clinical calculators where a single numerical error changes the recommendation. The standard response is to hardcode each calculator as a validated function, one at a time. We test an alternative: the model does not calculate. Instead, it writes case-specific Python that a restricted local executor runs as a deterministic solver, and the model's task reduces to deciding how to use it. We evaluate this Program-Solve interface on MedCalc-Bench Verified (1,100 cases, 55 calculators) against direct model arithmetic and a hand-written 22-calculator library, using Qwen2.5-7B and Qwen2.5-32B-AWQ, after auditing the benchmark's formulas against current clinical guidelines and flagging 16 of 55 with version, use or coefficient concerns. With formulas and gold variables supplied and both routes reading the whole note, handing off to the solver is not a reliable advantage at 7B (75.31% against 72.02%, a paired +3.29 points with a 95% calculator-cluster interval of [-3.49, 10.38]) but is one at 32B (90.53% against 83.47%, +7.05 [0.47, 14.60], clear of zero). The hand-written library is exact on its 440 supported cases but abstains elsewhere (40.0% overall). Adding an executor thus helps some open-weight models more than others even under matched formula, variable and note access, and is not a substitute for verified formulas or reliable variable extraction either way.

Mathematical reasoning has long served as a stringent test of machine intelligence; over the past decade, it has moved from a niche problem within NLP to one of the most consequential AI frontiers. This survey provides a unified account of the field's evolution, from early rule-based math word problem (MWP) solvers and template-driven geometry systems, through neural expression generation and LLM prompting, to contemporary reasoning models, multi-agent systems, neuro-symbolic theorem provers, and verified discovery workflows. We organize the landscape along four axes: (i) informal reasoning over text and diagrams, spanning MWP solving, multimodal geometry, and VLMs; (ii) formal reasoning in proof assistants, including autoformalization, tactic prediction, compiler-guided repair, and proof search; (iii) mathematical discovery, where systems propose constructions, improve bounds, or assist attacks on open problems; and (iv) the inference and training-time techniques, including CoT prompting, tool use, process reward models, and RLVR, that increasingly connect generation with verification. We catalog major benchmarks across grade-school arithmetic, competition mathematics, geometry, formal proving, multimodal and multilingual reasoning, and expert evaluation, and we examine benchmark saturation, contamination, reporting mismatches, and the distinction between pass@1, majority voting, and verifier-assisted pass@k. We critically assess failure modes: brittleness under perturbation, reward hacking, multimodal grounding failures, fragile formalization, and the energy cost of reasoning-scale inference. Drawing on recent perspectives from working mathematicians, we identify future directions centered on verified-discovery workflows, reasoning efficiency, and infrastructure to make AI-assisted formalization broadly usable. Companion materials: https://github.com/Starscream-11813/awesome-AI4Math.

Large Language Models (LLMs) have shown impressive progress in mathematical reasoning. While data augmentation is promising to enhance mathematical problem-solving ability, current approaches are predominantly limited to instance-level modifications-such as rephrasing or generating syntactic variations-which fail to capture and leverage the intrinsic relational structures inherent in mathematical knowledge. Inspired by human learning processes, where mathematical proficiency develops through systematic exposure to interconnected concepts, we introduce MathFusion, a novel framework that enhances mathematical reasoning through cross-problem instruction synthesis. MathFusion implements this through three fusion strategies: (1) sequential fusion, which chains related problems to model solution dependencies; (2) parallel fusion, which combines analogous problems to reinforce conceptual understanding; and (3) conditional fusion, which creates context-aware selective problems to enhance reasoning flexibility. By applying these strategies, we generate a new dataset, MathFusionQA, followed by fine-tuning models (DeepSeekMath-7B, Mistral-7B, Llama3-8B) on it. Experimental results demonstrate that MathFusion achieves substantial improvements in mathematical reasoning while maintaining high data efficiency, boosting performance by 18.0 points in accuracy across diverse benchmarks while requiring only 45K additional synthetic instructions, representing a substantial improvement over traditional single-instruction approaches. Our datasets, models, and code are publicly available at https://github.com/QizhiPei/mathfusion.

Test-time compute can improve mathematical reasoning, but can short-budget runs predict how mathematical reasoning scales with additional compute? We introduce a Discovery--Execution (DE) framework that predicts the aggregate held-out scaling curves through a convolution of strategy discovery and conditional execution. From independent short-budget attempts and oracle-sketch-conditioned runs, the framework estimates cumulative success along held-out reasoning trajectories under alternate compute allocations. We evaluate four models on 35 fresh Olympiad problems and non-geometry problems from IMO-ProofBench Advanced. Under the DE framework, near-saturated execution predicts geometric scaling, as observed for the GPT models. For Claude Opus 4.8, incorporating measured execution substantially improves held-out forecasts over geometric extrapolation across one- and two-arm allocations. As a secondary application, regularized DE (R-DE) decisions to continue or restart yield lower average regret than the best model-specific retrospective policy. Together, these results show that measuring conditional execution provides information about longer reasoning that short-budget success rates do not always capture.

The correctness of computations remains a significant challenge in computer science, with traditional approaches relying on automated testing or formal verification. Self-testing/correcting programs introduce an alternative paradigm, allowing a program to verify and correct its own outputs via randomized reductions, a concept that previously required manual derivation. In this paper, we present Bitween, a method and tool for automated learning of randomized (self)-reductions and program properties in numerical programs. Bitween combines symbolic analysis and machine learning, with a surprising finding: polynomial-time linear regression, a basic optimization method, is not only sufficient but also highly effective for deriving complex randomized self-reductions and program invariants, often outperforming sophisticated mixed-integer linear programming solvers. We establish a theoretical framework for learning these reductions and introduce RSR-Bench, a benchmark suite for evaluating Bitween's capabilities on scientific and machine learning functions. Our empirical results show that Bitween surpasses state-of-the-art tools in scalability, stability, and sample efficiency when evaluated on nonlinear invariant benchmarks like NLA-DigBench. Bitween is open-source as a Python package and accessible via a web interface that supports C language programs.

Large language models (LLMs) have exhibited great potential in mathematical reasoning. However, there remains a performance gap in this area between existing open-source models and closed-source models such as GPT-4. In this paper, we introduce MathGenie, a novel method for generating diverse and reliable math problems from a small-scale problem-solution dataset (denoted as seed data). We augment the ground-truth solutions of our seed data and train a back-translation model to translate the augmented solutions back into new questions. Subsequently, we generate code-integrated solutions for the new questions. To ensure the correctness of the code-integrated solutions, we employ rationale-based strategy for solution verification. Various pretrained models, ranging from 7B to 70B, are trained on the newly curated data to test the effectiveness of the proposed augmentation technique, resulting in a family of models known as MathGenieLM. These models consistently outperform previous open-source models across five representative mathematical reasoning datasets, achieving state-of-the-art performance. In particular, MathGenieLM-InternLM2 achieves an accuracy of 87.7% on GSM8K and 55.7% on MATH, securing the best overall score among open-source language models.

Modern AI systems rely on vector embeddings stored and searched using floating-point arithmetic. While effective for approximate similarity search, this design introduces fundamental non-determinism: identical models, inputs, and code can produce different memory states and retrieval results across hardware architectures (e.g., x86 vs. ARM). This prevents replayability and safe deployment, leading to silent data divergence that prevents post-hoc verification and compromises audit trails in regulated sectors. We present Valori, a deterministic AI memory substrate that replaces floating-point memory operations with fixed-point arithmetic (Q16.16) and models memory as a replayable state machine. Valori guarantees bit-identical memory states, snapshots, and search results across platforms. We demonstrate that non-determinism arises before indexing or retrieval and show how Valori enforces determinism at the memory boundary. Our results suggest that deterministic memory is a necessary primitive for trustworthy AI systems. The reference implementation is open-source and available at https://github.com/varshith-Git/Valori-Kernel (archived at https://zenodo.org/records/18022660).

We study a class of optimization problems motivated by automating the design and update of AI systems like coding assistants, robots, and copilots. We propose an end-to-end optimization framework, Trace, which treats the computational workflow of an AI system as a graph akin to neural networks, based on a generalization of back-propagation. Optimization of computational workflows often involves rich feedback (e.g. console output or user's responses), heterogeneous parameters (e.g. prompts, hyper-parameters, codes), and intricate objectives (beyond maximizing a score). Moreover, its computation graph can change dynamically with the inputs and parameters. We frame a new mathematical setup of iterative optimization, Optimization with Trace Oracle (OPTO), to capture and abstract these properties so as to design optimizers that work across many domains. In OPTO, an optimizer receives an execution trace along with feedback on the computed output and updates parameters iteratively. Trace is the tool to implement OPTO in practice. Trace has a Python interface that efficiently converts a computational workflow into an OPTO instance using a PyTorch-like interface. Using Trace, we develop a general-purpose LLM-based optimizer called OptoPrime that can effectively solve OPTO problems. In empirical studies, we find that OptoPrime is capable of first-order numerical optimization, prompt optimization, hyper-parameter tuning, robot controller design, code debugging, etc., and is often competitive with specialized optimizers for each domain. We believe that Trace, OptoPrime and the OPTO framework will enable the next generation of interactive agents that automatically adapt using various kinds of feedback. Website: https://microsoft.github.io/Trace

We develop tools for explicitly constructing categories enriched over generating data and that compose via ordinary scalar and matrix arithmetic arithmetic operations. We characterize meaningful size maps, weightings, and magnitude that reveal features analogous to outliers that these same notions have previously been shown to reveal in the context of metric spaces. Throughout, we provide examples of such "outlier detection" relevant to the analysis of computer programs, neural networks, cyber-physical systems, and networks of communications channels.

We introduce Vibe Reasoning, a human-AI collaborative paradigm for solving complex mathematical problems. Our key insight is that frontier AI models already possess the knowledge required to solve challenging problems -- they simply do not know how, what, or when to apply it. Vibe Reasoning transforms AI's latent potential into manifested capability through generic meta-prompts, agentic grounding, and model orchestration. We demonstrate this paradigm through IMO 2025 Problem 6, a combinatorial optimization problem where autonomous AI systems publicly reported failures. Our solution combined GPT-5's exploratory capabilities with Gemini 3 Pro's proof strengths, leveraging agentic workflows with Python code execution and file-based memory, to derive both the correct answer (2112) and a rigorous mathematical proof. Through iterative refinement across multiple attempts, we discovered the necessity of agentic grounding and model orchestration, while human prompts evolved from problem-specific hints to generic, transferable meta-prompts. We analyze why capable AI fails autonomously, how each component addresses specific failure modes, and extract principles for effective vibe reasoning. Our findings suggest that lightweight human guidance can unlock frontier models' mathematical reasoning potential. This is ongoing work; we are developing automated frameworks and conducting broader evaluations to further validate Vibe Reasoning's generality and effectiveness.