Daily Papers

Hybrid attention architectures are becoming an increasingly important paradigm for improving LLM inference efficiency while preserving model quality, making hybrid architecture design a central problem. Existing designs often rely on manual empirical rules or proxy-based selector signals for layer-wise operator allocation. Recent NAS-style systems such as Jet-Nemotron demonstrate the promise of automated hybrid architecture search. However, Jet-Nemotron's PostNAS search stages alone use 200B tokens, making such search pipelines difficult to use as routine methods for hybrid architecture design. We introduce DASH, a fast differentiable search framework for hybrid attention architecture design, which relaxes discrete layer-wise attention operator placement into continuous architecture logits, prepares reusable teacher-aligned linear candidates, and performs architecture-only search with model and operator weights frozen to significantly enhance search efficiency. On Qwen2.5-3B-Instruct, DASH consistently outperforms a comprehensive suite of existing selector-style hybrid attention design baselines, showing that direct differentiable search can discover stronger hybrid architectures. Moreover, DASH achieves stronger RULER performance than released Jet-Nemotron models while remaining competitive on overlapping short-context and general benchmarks. Notably, each DASH search run uses only 12.3M tokens and takes about 20 minutes on a single RTX Pro 6000 GPU, corresponding to merely 0.006% of the PostNAS search tokens reported by Jet-Nemotron. These results suggest that high-quality hybrid attention architectures can be obtained through minutes-level differentiable search, providing a promising direction for hybrid architecture design.

We present an RL-driven compiler that jointly optimizes ASIC architecture, memory hierarchy, and workload partitioning for AI inference across 3nm to 28nm. The design space is formulated as a single Markov Decision Process with mixed discrete-continuous actions and a unified Power-Performance-Area (PPA) objective. Soft Actor-Critic (SAC) with Mixture-of-Experts gating explores the joint space of mesh topology, per-core microarchitecture, and operator placement. We validate on two workloads, Llama 3.1 8B FP16 (high-performance mode, 29809 tokens per second at 3nm) and SmolVLM (low-power mode, less than 13 mW at all nodes, 10 MHz). Across 7 process nodes, the RL automatically adapts mesh sizes and per-tile configurations, including heterogeneous FETCH, VLEN, and memory allocation without node-specific manual retuning.

Conventional multi-lead electrocardiogram (ECG) systems capture cardiac signals from a fixed set of anatomical viewpoints defined by lead placement. However, certain cardiac conditions (e.g., Brugada syndrome) require additional, non-standard viewpoints to reveal diagnostically critical patterns that may be absent in standard leads. To systematically overcome this limitation, Nef-Net was recently introduced to reconstruct a continuous electrocardiac field, enabling virtual observation of ECG signals from arbitrary views (termed Electrocardio Panorama). Despite its promise, Nef-Net operates under idealized assumptions and faces in-the-wild challenges, such as long-duration ECG modeling, robustness to device-specific signal artifacts, and suboptimal lead placement calibration. This paper presents NEF-NET+, an enhanced framework for realistic panoramic ECG synthesis that supports arbitrary-length signal synthesis from any desired view, generalizes across ECG devices, and compensates for operator-induced deviations in electrode placement. These capabilities are enabled by a newly designed model architecture that performs direct view transformation, incorporating a workflow comprising offline pretraining, device calibration tuning steps as well as an on-the-fly calibration step for patient-specific adaptation. To rigorously evaluate panoramic ECG synthesis, we construct a new Electrocardio Panorama benchmark, called Panobench, comprising 5367 recordings with 48-view per subject, capturing the full spatial variability of cardiac electrical activity. Experimental results show that NEF-NET+ delivers substantial improvements over Nef-Net, yielding an increase of around 6 dB in PSNR in real-world setting. The code and Panobench will be released in a subsequent publication.

A wide range of scientific problems, such as those described by continuous-time dynamical systems and partial differential equations (PDEs), are naturally formulated on function spaces. While function spaces are typically infinite-dimensional, deep learning has predominantly advanced through applications in computer vision and natural language processing that focus on mappings between finite-dimensional spaces. Such fundamental disparities in the nature of the data have limited neural networks from achieving a comparable level of success in scientific applications as seen in other fields. Neural operators are a principled way to generalize neural networks to mappings between function spaces, offering a pathway to replicate deep learning's transformative impact on scientific problems. For instance, neural operators can learn solution operators for entire classes of PDEs, e.g., physical systems with different boundary conditions, coefficient functions, and geometries. A key factor in deep learning's success has been the careful engineering of neural architectures through extensive empirical testing. Translating these neural architectures into neural operators allows operator learning to enjoy these same empirical optimizations. However, prior neural operator architectures have often been introduced as standalone models, not directly derived as extensions of existing neural network architectures. In this paper, we identify and distill the key principles for constructing practical implementations of mappings between infinite-dimensional function spaces. Using these principles, we propose a recipe for converting several popular neural architectures into neural operators with minimal modifications. This paper aims to guide practitioners through this process and details the steps to make neural operators work in practice. Our code can be found at https://github.com/neuraloperator/NNs-to-NOs

In this work, we propose a concise neural operator architecture for operator learning. Drawing an analogy with a conventional fully connected neural network, we define the neural operator as follows: the output of the i-th neuron in a nonlinear operator layer is defined by mathcal O_i(u) = sigmaleft( sum_j mathcal W_{ij} u + mathcal B_{ij}right). Here, mathcal W_{ij} denotes the bounded linear operator connecting j-th input neuron to i-th output neuron, and the bias mathcal B_{ij} takes the form of a function rather than a scalar. Given its new universal approximation property, the efficient parameterization of the bounded linear operators between two neurons (Banach spaces) plays a critical role. As a result, we introduce MgNO, utilizing multigrid structures to parameterize these linear operators between neurons. This approach offers both mathematical rigor and practical expressivity. Additionally, MgNO obviates the need for conventional lifting and projecting operators typically required in previous neural operators. Moreover, it seamlessly accommodates diverse boundary conditions. Our empirical observations reveal that MgNO exhibits superior ease of training compared to other CNN-based models, while also displaying a reduced susceptibility to overfitting when contrasted with spectral-type neural operators. We demonstrate the efficiency and accuracy of our method with consistently state-of-the-art performance on different types of partial differential equations (PDEs).

Macro placement is a fundamental step in modern chip physical design, playing a crucial role in determining the solution quality of high-dimensional combinatorial optimization problems. Despite recent advancements in machine learning for spatial coordinate determination, the temporal dimension of placement sequencing remains largely governed by static heuristics. In this work, we demonstrate that the placement sequence is not merely a preprocessing step but a decisive factor in optimization, where suboptimal early decisions trigger irreversible domino effects that constrain the solution space. To harness this unexplored dimension, we propose OrderPlace, a proxy-guided LLM evolution framework for automatically discovering macro placement order strategies. Instead of relying on manually crafted heuristics such as area- or connectivity-based ordering, OrderPlace explores a broader space of code-level policies, ranging from static scoring metrics to dynamic physics-inspired mechanisms. To mitigate the prohibitive cost of evaluating sequences, we introduce a lightweight proxy evaluation mechanism that efficiently filters candidates using a deterministic greedy probe. Experimental results on the standard ISPD 2005 benchmarks demonstrate that OrderPlace discovers novel ordering strategies. Compared with WireMask-EA and the state-of-the-art method EGPlace, OrderPlace reduces wirelength by 34.04\% and 14.08\%, respectively.

Geometric Attention (GA) specifies an attention layer by four independent inputs: a finite carrier (what indices are addressable), an evidence-kernel rule (how masked proto-scores and a link induce nonnegative weights), a probe family (which observables are treated as admissible), and an anchor/update rule (which representative kernel is selected and how it is applied). Probe families induce an operational equivalence relation on kernels and therefore a gauge; anchors select representatives relative to that probe. Under a scalar relational-work representation and a multiplicative compositionality law for evidence, the admissible link family is exponential, yielding Gibbs weights; with row anchoring this includes the softmax kernel family as a subregime. After quotienting unary row/column score fields, the remaining interaction component admits a canonical rank-r normal form (Eckart-Young/SVD); dot-product score charts implement the corresponding low-rank interaction regime. Fixing the carrier and extensionalizing the update yields the standard fixed-token Transformer attention operator; allowing carrier updates yields adaptive-carrier and staged-depth regimes. The operator language also supports multihead/mixed kernels, plan-based anchors (e.g., entropic OT/Sinkhorn), and unary operators (e.g., FFN-style fields) as explicit regime choices. This separates invariant structure from modeling choice, enabling principled comparison and extension of attention mechanisms, and attention-based architectures.

We analyze various consequences in relation to the extension of operators T:Xto Y that are p-compact, as well as the extension of operators T:Xto Y whose adjoints T^*:Y^*to X^* are p-compact. In most cases, we discuss these extension properties when the underlying spaces, either domain or codomain, are P_lambda spaces. We also answer if these extensions are almost norm-preserving in such circumstances where the extension T of a T exists. It is observed that an operator can often be extended to a larger domain when the codomain is appropriately extended as well. Specific assumptions might enable us to obtain an extension of an operator that maintains the same range. Necessary and sufficient conditions are derived for a Banach space to be L_1-predual.

Deep learning optimizers are often motivated through a mix of convex and approximate second-order theory. We select three such methods -- Adam, Shampoo and Prodigy -- and argue that each method can instead be understood as a squarely first-order method without convexity assumptions. In fact, after switching off exponential moving averages, each method is equivalent to steepest descent under a particular norm. By generalizing this observation, we chart a new design space for training algorithms. Different operator norms should be assigned to different tensors based on the role that the tensor plays within the network. For example, while linear and embedding layers may have the same weight space of R^{mtimes n}, these layers play different roles and should be assigned different norms. We hope that this idea of carefully metrizing the neural architecture might lead to more stable, scalable and indeed faster training.

Synthetic data generation has become a cornerstone for advancing large language models. However, the lack of the quantitative analysis for error tolerance became a critical bottleneck. Consequently, current filtering strategies fluctuate between two extremes: they are either overly aggressive, risking the exclusion of potentially valuable samples, or overly permissive, failing to eliminate erroneous samples effectively. To bridge this gap, this paper introduces Atomic Tree Operation Modeling (ATOM), a framework that decomposes data into functional units (f(x)rightarrow y). ATOM distinguishes benign Operand x perturbations from fatal Operator f perturbations. The former are needlessly discarded by aggressive filtering, while the latter slip through permissive filtering. Our experiments reveal a double dissociation: models are robust to operand perturbations but collapse under operator perturbations. By prioritizing operator over aggressive operand precision, our ATOM-synthesized data outperforms rigorous baselines (e.g., +3.1% gain over LIMA), suggesting that operator diversity matters more than operand precision. Our code is available at https://github.com/Lut-hub/ATOM.

In this work we study indoor scene object placement. Given a 3D indoor scene and an object, the task is to predict placement locations within the scene. Empirical observations of data-driven approaches to the problem show their tendency to miss placement modes. We introduce a system which helps to address this flaw. We design a Domain Specific Language (DSL) that specifies object relational constraints. Upon execution, programs from our language predict possible placements from a partial scene and object. We design a generative model which writes these programs automatically. Available 3D scene datasets do not contain programs to train on, and naively extracted programs only predict the original placement location of scene objects. Training on these programs results in subpar performance so we introduce a new program bootstrapping algorithm that improves our system's performance compared to the naive approach. To quantify our qualitative observations, we introduce a new evaluation procedure which captures how well a system models per-object location distributions. We ask human annotators to label all the possible places an object can go in a scene and compare this set against locations produced by the system in question. Our system produces per-object location distributions more consistent with human annotators than those produced by existing data-driven approaches and a zero-shot approach using an LLM. While other systems degrade in performance when training data is sparse, our system does not degrade to the same degree.

Efficient operator scheduling is a fundamental challenge in software compilation and hardware synthesis. While recent differentiable approaches have sought to replace traditional ones like exact solvers or heuristics with gradient-based search, they typically rely on categorical distributions that fail to capture the ordinal nature of time and suffer from a parameter space that scales poorly. In this paper, we propose a novel differentiable framework, GauS, that models operator scheduling as a stochastic relaxation using Gaussian distributions, which fully utilize modern parallel computing devices like GPUs. By representing schedules as continuous Gaussian variables, we successfully capture the ordinal nature of time and reduce the optimization space by orders of magnitude. Our method is highly flexible to represent various objectives and constraints, which provides the first differentiable formulation for the complex pipelined scheduling problem. We evaluate our method on a range of benchmarks, demonstrating that Gaus achieves Pareto-optimal results.

Object placement in robotic tasks is inherently challenging due to the diversity of object geometries and placement configurations. To address this, we propose AnyPlace, a two-stage method trained entirely on synthetic data, capable of predicting a wide range of feasible placement poses for real-world tasks. Our key insight is that by leveraging a Vision-Language Model (VLM) to identify rough placement locations, we focus only on the relevant regions for local placement, which enables us to train the low-level placement-pose-prediction model to capture diverse placements efficiently. For training, we generate a fully synthetic dataset of randomly generated objects in different placement configurations (insertion, stacking, hanging) and train local placement-prediction models. We conduct extensive evaluations in simulation, demonstrating that our method outperforms baselines in terms of success rate, coverage of possible placement modes, and precision. In real-world experiments, we show how our approach directly transfers models trained purely on synthetic data to the real world, where it successfully performs placements in scenarios where other models struggle -- such as with varying object geometries, diverse placement modes, and achieving high precision for fine placement. More at: https://any-place.github.io.

General-purpose object placement is a fundamental capability of an intelligent generalist robot: being capable of rearranging objects following precise human instructions even in novel environments. This work is dedicated to achieving general-purpose object placement with ``something something'' instructions. Specifically, we break the entire process down into three parts, including object localization, goal imagination and robot control, and propose a method named SPORT. SPORT leverages a pre-trained large vision model for broad semantic reasoning about objects, and learns a diffusion-based pose estimator to ensure physically-realistic results in 3D space. Only object types (movable or reference) are communicated between these two parts, which brings two benefits. One is that we can fully leverage the powerful ability of open-set object recognition and localization since no specific fine-tuning is needed for the robotic scenario. Moreover, the diffusion-based estimator only need to ``imagine" the object poses after the placement, while no necessity for their semantic information. Thus the training burden is greatly reduced and no massive training is required. The training data for the goal pose estimation is collected in simulation and annotated by using GPT-4. Experimental results demonstrate the effectiveness of our approach. SPORT can not only generate promising 3D goal poses for unseen simulated objects, but also be seamlessly applied to real-world settings.

We study the parameter placement problem: given a fixed budget of k trainable entries within the B matrix of a LoRA adapter (A frozen), does the choice of which k matter? Under supervised fine-tuning, random and informed subsets achieve comparable performance. Under GRPO on base models, random placement fails to improve over the base model, while gradient-informed placement recovers standard LoRA accuracy. This regime dependence traces to gradient structure: SFT gradients are low-rank and directionally stable, so any subset accumulates coherent updates; GRPO gradients are high-rank and near-orthogonal across steps, so only elements with consistently signed gradients retain the learning signal. Our scoring procedure identifies these critical parameters in under 10 seconds at less than 0.5% of training cost. Selected parameters concentrate on residual-stream-writing projections (V, O, Down), stable across model families and scales (1.5B - 8B).

In this paper, we investigate the behavior of gradient descent algorithms in physics-informed machine learning methods like PINNs, which minimize residuals connected to partial differential equations (PDEs). Our key result is that the difficulty in training these models is closely related to the conditioning of a specific differential operator. This operator, in turn, is associated to the Hermitian square of the differential operator of the underlying PDE. If this operator is ill-conditioned, it results in slow or infeasible training. Therefore, preconditioning this operator is crucial. We employ both rigorous mathematical analysis and empirical evaluations to investigate various strategies, explaining how they better condition this critical operator, and consequently improve training.

Training large language models requires distributing computation across many accelerators, yet practitioners select parallelism strategies (data, tensor, pipeline, ZeRO) through trial and error because no unified systematic framework predicts their behavior. We introduce placement semantics: each strategy is specified by how it places four training states (parameters, optimizer, gradients, activations) across devices using five modes (replicated, sharded, sharded-with-gather, materialized, offloaded). From placement alone, without implementation details, we derive memory consumption and communication volume. Our predictions match published results exactly: ZeRO-3 uses 8x less memory than data parallelism at 1.5x communication cost, as reported in the original paper. We prove two conditions (gradient integrity, state consistency) are necessary and sufficient for distributed training to match single-device results, and provide composition rules for combining strategies safely. The framework unifies ZeRO Stages 1-3, Fully Sharded Data Parallel (FSDP), tensor parallelism, and pipeline parallelism as instances with different placement choices.

This paper is an extended and reworked version of a short course given by the author at ''Uzbekistan-Ukrainian readings in stochastic processes'', Tashkent-Kyiv, 2022, and was prepared for a special issue of ''Theory of stochastic processes'', devoted to publishing lecture notes from the aforementioned workshop. The survey is devoted to operator splitting methods in the abstract formulation and their applications in probability. While the survey is focused on multiplicative methods, the BCH formula is used to discuss exponential splitting methods and a short informal introduction to additive splitting is presented. We introduce frameworks and available deterministic and probabilistic results and concentrate on constructing a wide picture of the field of operator splitting methods, providing a rigorous description in the setting of abstract Cauchy problems and an informal discussion for further and parallel advances. Some limitations and common difficulties are listed, as well as examples of works that provide solutions or hints. No new results are given. The bibliography contains illustrative deterministic examples and a selection of probability-related works.

We introduce Topological Neural Operators (TNOs), a principled framework for operator learning on cell complexes that lifts neural operators (NOs) from functions on points and/or edges to topological domains. TNOs represent data as features defined on cells of varying dimension and model their interactions through Discrete Exterior Calculus, enabling explicit cross-dimensional coupling via gradient-, curl-, and divergence-type operators. The key design principle is to decouple where information flows, as governed by fixed topological operators, from how it is transformed (which is learned), yielding models that respect the geometric support of physical quantities and expose conservation and compatibility structure. We further propose Hierarchical TNOs (HTNOs), which incorporate learned coarse complexes to propagate long-range and topology-dependent information. Our framework subsumes existing NOs as a special case, providing a unified perspective on operator learning across discretizations. Across a range of PDE benchmarks, including irregular-geometry flow problems, TNOs and HTNOs improve accuracy; controlled studies further isolate the benefits of native higher-rank and topological structure. Project page: https://circle-group.github.io/research/TNO

Most single-channel separators estimate a source by applying a real gain to the mixture in each time-frequency bin. The optimum of that format, which the oracle masks used as bounds do not attain, is the orthogonal projection of the source onto the line spanned by the mixture, its residual set by the angle between them. Locating an estimator reduces to the block structure of a real-linear operator on stacked spectra, giving a chain of four nested classes whose three larger terms match three assumptions on the prior: zero means, circularity and absence of inter-frequency coupling. Held fixed the chain is a cascade of four orthogonal projections; refitted per frame it collapses onto its first term, attributing the whole residual to one missing real parameter per bin, the phase. When the phase posterior is symmetric about the mixture direction, the minimum mean-square estimate falls back onto the line, with gain the posterior mean of the oracle gain and excess error its variance. On MUSDB18 a posterior mean under a non-circular Gaussian-mixture prior leaves the class yet stays 11.44 dB under the per-frame ceiling, which four times as many components and 7.5x the data do not close; a closed-form gate attributes some 70% of it, in decibels, to the predicted variance. The widest fixed class stays 6.70 dB under the same ceiling. Leaving the class and minimising squared error are conflicting requests: the barrier lies in the criterion rather than in the prior.

We investigate the construction of latent spaces through self-supervised learning to support semantically meaningful operations. Analogous to operational amplifiers, these "operational latent spaces" (OpLaS) not only demonstrate semantic structure such as clustering but also support common transformational operations with inherent semantic meaning. Some operational latent spaces are found to have arisen "unintentionally" in the progress toward some (other) self-supervised learning objective, in which unintended but still useful properties are discovered among the relationships of points in the space. Other spaces may be constructed "intentionally" by developers stipulating certain kinds of clustering or transformations intended to produce the desired structure. We focus on the intentional creation of operational latent spaces via self-supervised learning, including the introduction of rotation operators via a novel "FiLMR" layer, which can be used to enable ring-like symmetries found in some musical constructions.

Initial boundary value problems arise commonly in applications with engineering and natural systems governed by nonlinear partial differential equations (PDEs). Operator learning is an emerging field for solving these equations by using a neural network to learn a map between infinite dimensional input and output function spaces. These neural operators are trained using a combination of data (observations or simulations) and PDE-residuals (physics-loss). A major drawback of existing neural approaches is the requirement to retrain with new initial/boundary conditions, and the necessity for a large amount of simulation data for training. We develop a physics-informed transformer neural operator (named PINTO) that efficiently generalizes to unseen initial and boundary conditions, trained in a simulation-free setting using only physics loss. The main innovation lies in our new iterative kernel integral operator units, implemented using cross-attention, to transform the PDE solution's domain points into an initial/boundary condition-aware representation vector, enabling efficient learning of the solution function for new scenarios. The PINTO architecture is applied to simulate the solutions of important equations used in engineering applications: advection, Burgers, and steady and unsteady Navier-Stokes equations (three flow scenarios). For these five test cases, we show that the relative errors during testing under challenging conditions of unseen initial/boundary conditions are only one-fifth to one-third of other leading physics informed operator learning methods. Moreover, our PINTO model is able to accurately solve the advection and Burgers equations at time steps that are not included in the training collocation points. The code is available at https://github.com/quest-lab-iisc/PINTO

Placement is crucial in the physical design, as it greatly affects power, performance, and area metrics. Recent advancements in analytical methods, such as DREAMPlace, have demonstrated impressive performance in global placement. However, DREAMPlace has some limitations, e.g., may not guarantee legalizable placements under the same settings, leading to fragile and unpredictable results. This paper highlights the main issue as being stuck in local optima, and proposes a hybrid optimization framework to efficiently escape the local optima, by perturbing the placement result iteratively. The proposed framework achieves significant improvements compared to state-of-the-art methods on two popular benchmarks.

A popular way to approximate the Koopman operator's action on a finite-dimensional subspace of functions is via orthogonal projections. The quality of the projected model directly depends on the selected subspace, specifically on how close it is to being invariant under the Koopman operator. The notion of invariance proximity provides a tight upper bound on the worst-case relative prediction error of the finite-dimensional model. However, its direct calculation is computationally challenging. This paper leverages the geometric structure behind the definition of invariance proximity to provide a closed-form expression in terms of Jordan principal angles on general inner product spaces. Unveiling this connection allows us to exploit specific isomorphisms to circumvent the computational challenges associated with spaces of functions and enables the use of existing efficient numerical routines to compute invariance proximity.

In this work, we propose marginalized operators, a new class of off-policy evaluation operators for reinforcement learning. Marginalized operators strictly generalize generic multi-step operators, such as Retrace, as special cases. Marginalized operators also suggest a form of sample-based estimates with potential variance reduction, compared to sample-based estimates of the original multi-step operators. We show that the estimates for marginalized operators can be computed in a scalable way, which also generalizes prior results on marginalized importance sampling as special cases. Finally, we empirically demonstrate that marginalized operators provide performance gains to off-policy evaluation and downstream policy optimization algorithms.

Global placement is a critical step with high computational complexity in VLSI physical design. Modern analytical placers formulate the placement problem as a nonlinear optimization, where initialization strongly affects both convergence behavior and final placement quality. However, existing initialization methods exhibit a trade-off: area-aware initializers account for cell areas but are computationally expensive and can dominate total runtime, while fast point-based initializers ignore cell area, leading to a modeling gap that impairs convergence and solution quality. We propose a lightweight co-optimization framework that bridges this initialization gap through two strategies. First, an area-hint refinement initializer incorporates heuristic cell area information into a signed graph signal by augmenting the netlist graph with virtual nodes and negative-weight edges, yielding an area-aware and spectrally smooth placement initialization. Second, a macro-schedule placement procedure progressively restores area constraints, enabling a smooth transition from the refined initializer to the full area-aware objective and producing high-quality placement results. We evaluate the framework on macro-heavy ISPD2005 academic benchmarks and two real-world industrial designs across two technology nodes (12 cases in total). Experimental results show that our method consistently improves half-perimeter wirelength (HPWL) over point-based initializers in 11 out of 12 cases, achieving up to 2.2% HPWL reduction, while running approximately 100 times faster than the state-of-the-art area-aware initializer.

A single two-input gate suffices for all of Boolean logic in digital hardware. No comparable primitive has been known for continuous mathematics: computing elementary functions such as sin, cos, sqrt, and log has always required multiple distinct operations. Here I show that a single binary operator, eml(x,y)=exp(x)-ln(y), together with the constant 1, generates the standard repertoire of a scientific calculator. This includes constants such as e, pi, and i; arithmetic operations including addition, subtraction, multiplication, division, and exponentiation as well as the usual transcendental and algebraic functions. For example, exp(x)=eml(x,1), ln(x)=eml(1,eml(eml(1,x),1)), and likewise for all other operations. That such an operator exists was not anticipated; I found it by systematic exhaustive search and established constructively that it suffices for the concrete scientific-calculator basis. In EML (Exp-Minus-Log) form, every such expression becomes a binary tree of identical nodes, yielding a grammar as simple as S -> 1 | eml(S,S). This uniform structure also enables gradient-based symbolic regression: using EML trees as trainable circuits with standard optimizers (Adam), I demonstrate the feasibility of exact recovery of closed-form elementary functions from numerical data at shallow tree depths up to 4. The same architecture can fit arbitrary data, but when the generating law is elementary, it may recover the exact formula.

Solving linear systems of equations is a common problem that arises both on its own and as a subroutine in more complex problems: given a matrix A and a vector b, find a vector x such that Ax=b. We consider the case where one doesn't need to know the solution x itself, but rather an approximation of the expectation value of some operator associated with x, e.g., x'Mx for some matrix M. In this case, when A is sparse, N by N and has condition number kappa, classical algorithms can find x and estimate x'Mx in O(N sqrt(kappa)) time. Here, we exhibit a quantum algorithm for this task that runs in poly(log N, kappa) time, an exponential improvement over the best classical algorithm.

We present a deployed system that automates end-to-end customer support workflows inside an enterprise Business Process Management (BPM) platform. The approach is scalable in production and reaches selective automation within two weeks for a new process, leveraging supervision already generated at scale: structured per-case UI interaction traces and low-overhead copilot feedback, where operators either accept a suggestion or provide a correction. A staged deployment pipeline trains a next UI action policy, learns a critic from copilot feedback to calibrate abstention, and executes only high-confidence steps in the background while deferring uncertain decisions to operators and resuming from the updated UI state. This setup lets one operator supervise multiple concurrent sessions and be interrupted only when the system is uncertain. The system operates on a schema-driven view of the BPM interface and includes monitoring and safe fallbacks for production. In production, it automated 45% of sessions and reduced average handling time by 39% without degrading support quality level.

We extend JAX with the capability to automatically differentiate higher-order functions (functionals and operators). By representing functions as a generalization of arrays, we seamlessly use JAX's existing primitive system to implement higher-order functions. We present a set of primitive operators that serve as foundational building blocks for constructing several key types of functionals. For every introduced primitive operator, we derive and implement both linearization and transposition rules, aligning with JAX's internal protocols for forward and reverse mode automatic differentiation. This enhancement allows for functional differentiation in the same syntax traditionally use for functions. The resulting functional gradients are themselves functions ready to be invoked in python. We showcase this tool's efficacy and simplicity through applications where functional derivatives are indispensable. The source code of this work is released at https://github.com/sail-sg/autofd .

Neural operators, as an efficient surrogate model for learning the solutions of PDEs, have received extensive attention in the field of scientific machine learning. Among them, attention-based neural operators have become one of the mainstreams in related research. However, existing approaches overfit the limited training data due to the considerable number of parameters in the attention mechanism. To address this, we develop an orthogonal attention based on the eigendecomposition of the kernel integral operator and the neural approximation of eigenfunctions. The orthogonalization naturally poses a proper regularization effect on the resulting neural operator, which aids in resisting overfitting and boosting generalization. Experiments on six standard neural operator benchmark datasets comprising both regular and irregular geometries show that our method can outperform competing baselines with decent margins.

Transformers, and the attention mechanism in particular, have become ubiquitous in machine learning. Their success in modeling nonlocal, long-range correlations has led to their widespread adoption in natural language processing, computer vision, and time series problems. Neural operators, which map spaces of functions into spaces of functions, are necessarily both nonlinear and nonlocal if they are universal; it is thus natural to ask whether the attention mechanism can be used in the design of neural operators. Motivated by this, we study transformers in the function space setting. We formulate attention as a map between infinite dimensional function spaces and prove that the attention mechanism as implemented in practice is a Monte Carlo or finite difference approximation of this operator. The function space formulation allows for the design of transformer neural operators, a class of architectures designed to learn mappings between function spaces. In this paper, we state and prove the first universal approximation result for transformer neural operators, using only a slight modification of the architecture implemented in practice. The prohibitive cost of applying the attention operator to functions defined on multi-dimensional domains leads to the need for more efficient attention-based architectures. For this reason we also introduce a function space generalization of the patching strategy from computer vision, and introduce a class of associated neural operators. Numerical results, on an array of operator learning problems, demonstrate the promise of our approaches to function space formulations of attention and their use in neural operators.

In this paper, we apply the self-attention from the state-of-the-art Transformer in Attention Is All You Need for the first time to a data-driven operator learning problem related to partial differential equations. An effort is put together to explain the heuristics of, and to improve the efficacy of the attention mechanism. By employing the operator approximation theory in Hilbert spaces, it is demonstrated for the first time that the softmax normalization in the scaled dot-product attention is sufficient but not necessary. Without softmax, the approximation capacity of a linearized Transformer variant can be proved to be comparable to a Petrov-Galerkin projection layer-wise, and the estimate is independent with respect to the sequence length. A new layer normalization scheme mimicking the Petrov-Galerkin projection is proposed to allow a scaling to propagate through attention layers, which helps the model achieve remarkable accuracy in operator learning tasks with unnormalized data. Finally, we present three operator learning experiments, including the viscid Burgers' equation, an interface Darcy flow, and an inverse interface coefficient identification problem. The newly proposed simple attention-based operator learner, Galerkin Transformer, shows significant improvements in both training cost and evaluation accuracy over its softmax-normalized counterparts.

Neural operators learn to map initial conditions to the terminal solution of partial differential equations (PDEs), providing a surrogate for the full operator mapping. This enables rapid prediction across different input configurations. While recent neural operator architectures have demonstrated strong performance on diverse PDE tasks, their behavior under structured distribution shifts remains insufficiently understood. To investigate this, we study operator learning in a wave propagation setting governed by a one-dimensional variable-coefficient wave equation, using two representative architectures, the Fourier Neural Operator (FNO) and the Deep Operator Network (DeepONet). To examine their generalization under distribution shifts, we consider structured out-of-distribution (OOD) settings that independently vary input frequency and coefficient smoothness. The results show that under smoothness shifts, both models maintain stable performance, with FNO achieving lower error. In contrast, under frequency shifts, FNO exhibits a sharp increase in error under unseen high-frequency inputs, whereas DeepONet shows milder degradation despite higher overall error. Our analysis reveals that these differences arise from how each architecture represents and responds to variations in frequency structure. Together, these findings highlight a fundamental gap between strong in-distribution performance and generalization under distribution shifts in operator learning, underscoring the role of architectural representation bias in developing more reliable neural operators for physics-based PDE simulations beyond the training distribution.

Human assistance in robotics spans around several tasks such as navigation, object manipulation, and placement, where a key challenge is selecting target destinations that align with human intentions or preferences. We focus on this challenge in the context of Virtual Placement (VP), the task of identifying all plausible target locations given scene context and human-centric constraints. This differs from traditional placement tasks that typically focus on a single, predefined target location. The VP problem is complex, as it requires both global and local reasoning about the scene's geometry, semantics, and plausibility. To address this gap, we introduce {\bf Assistant Placement Aria}, the first benchmark to explore diverse aspects of VP, including global, local, and human-centric constraints. It contains both synthetic and real indoor scenes annotated for three tasks: (i)~2D Panel Placement, (ii)~Sitting Suggestion, and (iii)~TV Placement. Each scene includes 2D images, a 3D point cloud, and a textual description of the objects within the scene. By contributing this benchmark, we aim to encourage further research in this underexplored and challenging field that is critically dependent on relevant data. We also evaluate several foundation models for object detection and segmentation on our benchmark.

We present a feasibility-seeking approach to neural network training. This mathematical optimization framework is distinct from conventional gradient-based loss minimization and uses projection operators and iterative projection algorithms. We reformulate training as a large-scale feasibility problem: finding network parameters and states that satisfy local constraints derived from its elementary operations. Training then involves projecting onto these constraints, a local operation that can be parallelized across the network. We introduce PJAX, a JAX-based software framework that enables this paradigm. PJAX composes projection operators for elementary operations, automatically deriving the solution operators for the feasibility problems (akin to autodiff for derivatives). It inherently supports GPU/TPU acceleration, provides a familiar NumPy-like API, and is extensible. We train diverse architectures (MLPs, CNNs, RNNs) on standard benchmarks using PJAX, demonstrating its functionality and generality. Our results show that this approach is as a compelling alternative to gradient-based training, with clear advantages in parallelism and the ability to handle non-differentiable operations.

We introduce the novel task of Language-Guided Object Placement in Real 3D Scenes. Our model is given a 3D scene's point cloud, a 3D asset, and a textual prompt broadly describing where the 3D asset should be placed. The task here is to find a valid placement for the 3D asset that respects the prompt. Compared with other language-guided localization tasks in 3D scenes such as grounding, this task has specific challenges: it is ambiguous because it has multiple valid solutions, and it requires reasoning about 3D geometric relationships and free space. We inaugurate this task by proposing a new benchmark and evaluation protocol. We also introduce a new dataset for training 3D LLMs on this task, as well as the first method to serve as a non-trivial baseline. We believe that this challenging task and our new benchmark could become part of the suite of benchmarks used to evaluate and compare generalist 3D LLM models.

The semantic capabilities of large language models (LLMs) have the potential to enable rich analytics and reasoning over vast knowledge corpora. Unfortunately, existing systems either empirically optimize expensive LLM-powered operations with no performance guarantees, or serve a limited set of row-wise LLM operations, providing limited robustness, expressiveness and usability. We introduce semantic operators, the first formalism for declarative and general-purpose AI-based transformations based on natural language specifications (e.g., filtering, sorting, joining or aggregating records using natural language criteria). Each operator opens a rich space for execution plans, similar to relational operators. Our model specifies the expected behavior of each operator with a high-quality gold algorithm, and we develop an optimization framework that reduces cost, while providing accuracy guarantees with respect to a gold algorithm. Using this approach, we propose several novel optimizations to accelerate semantic filtering, joining, group-by and top-k operations by up to 1,000times. We implement semantic operators in the LOTUS system and demonstrate LOTUS' effectiveness on real, bulk-semantic processing applications, including fact-checking, biomedical multi-label classification, search, and topic analysis. We show that the semantic operator model is expressive, capturing state-of-the-art AI pipelines in a few operator calls, and making it easy to express new pipelines that match or exceed quality of recent LLM-based analytic systems by up to 170%, while offering accuracy guarantees. Overall, LOTUS programs match or exceed the accuracy of state-of-the-art AI pipelines for each task while running up to 3.6times faster than the highest-quality baselines. LOTUS is publicly available at https://github.com/lotus-data/lotus.

Although very successfully used in conventional machine learning, convolution based neural network architectures -- believed to be inconsistent in function space -- have been largely ignored in the context of learning solution operators of PDEs. Here, we present novel adaptations for convolutional neural networks to demonstrate that they are indeed able to process functions as inputs and outputs. The resulting architecture, termed as convolutional neural operators (CNOs), is designed specifically to preserve its underlying continuous nature, even when implemented in a discretized form on a computer. We prove a universality theorem to show that CNOs can approximate operators arising in PDEs to desired accuracy. CNOs are tested on a novel suite of benchmarks, encompassing a diverse set of PDEs with possibly multi-scale solutions and are observed to significantly outperform baselines, paving the way for an alternative framework for robust and accurate operator learning. Our code is publicly available at https://github.com/bogdanraonic3/ConvolutionalNeuralOperator

We propose a new approach for generative modeling based on training a neural network to be idempotent. An idempotent operator is one that can be applied sequentially without changing the result beyond the initial application, namely f(f(z))=f(z). The proposed model f is trained to map a source distribution (e.g, Gaussian noise) to a target distribution (e.g. realistic images) using the following objectives: (1) Instances from the target distribution should map to themselves, namely f(x)=x. We define the target manifold as the set of all instances that f maps to themselves. (2) Instances that form the source distribution should map onto the defined target manifold. This is achieved by optimizing the idempotence term, f(f(z))=f(z) which encourages the range of f(z) to be on the target manifold. Under ideal assumptions such a process provably converges to the target distribution. This strategy results in a model capable of generating an output in one step, maintaining a consistent latent space, while also allowing sequential applications for refinement. Additionally, we find that by processing inputs from both target and source distributions, the model adeptly projects corrupted or modified data back to the target manifold. This work is a first step towards a ``global projector'' that enables projecting any input into a target data distribution.

This paper presents the Relational Machine Calculus (RMC): a simple, foundational model of first-order relational programming. The RMC originates from the Functional Machine Calculus (FMC), which generalizes the lambda-calculus and its standard call-by-name stack machine in two directions. One, "locations", introduces multiple stacks, which enable effect operators to be encoded into the abstraction and application constructs. The second, "sequencing", introduces the imperative notions of "skip" and "sequence", similar to kappa-calculus and concatenative programming languages. The key observation of the RMC is that the first-order fragment of the FMC exhibits a latent duality which, given a simple decomposition of the relevant constructors, can be concretely expressed as an involution on syntax. Semantically, this gives rise to a sound and complete calculus for string diagrams of Frobenius monoids. We consider unification as the corresponding symmetric generalization of beta-reduction. By further including standard operators of Kleene algebra, the RMC embeds a range of computational models: the kappa-calculus, logic programming, automata, Interaction Nets, and Petri Nets, among others. These embeddings preserve operational semantics, which for the RMC is again given by a generalization of the standard stack machine for the lambda-calculus. The equational theory of the RMC (which supports reasoning about its operational semantics) is conservative over both the first-order lambda-calculus and Kleene algebra, and can be oriented to give a confluent reduction relation.

Neural operators have become increasingly popular in solving partial differential equations (PDEs) due to their superior capability to capture intricate mappings between function spaces over complex domains. However, the data-hungry nature of operator learning inevitably poses a bottleneck for their widespread applications. At the core of the challenge lies the absence of transferability of neural operators to new geometries. To tackle this issue, we propose operator learning with domain decomposition, a local-to-global framework to solve PDEs on arbitrary geometries. Under this framework, we devise an iterative scheme Schwarz Neural Inference (SNI). This scheme allows for partitioning of the problem domain into smaller subdomains, on which local problems can be solved with neural operators, and stitching local solutions to construct a global solution. Additionally, we provide a theoretical analysis of the convergence rate and error bound. We conduct extensive experiments on several representative PDEs with diverse boundary conditions and achieve remarkable geometry generalization compared to alternative methods. These analysis and experiments demonstrate the proposed framework's potential in addressing challenges related to geometry generalization and data efficiency.

Neural operators have proven to be a promising approach for modeling spatiotemporal systems in the physical sciences. However, training these models for large systems can be quite challenging as they incur significant computational and memory expense -- these systems are often forced to rely on autoregressive time-stepping of the neural network to predict future temporal states. While this is effective in managing costs, it can lead to uncontrolled error growth over time and eventual instability. We analyze the sources of this autoregressive error growth using prototypical neural operator models for physical systems and explore ways to mitigate it. We introduce architectural and application-specific improvements that allow for careful control of instability-inducing operations within these models without inflating the compute/memory expense. We present results on several scientific systems that include Navier-Stokes fluid flow, rotating shallow water, and a high-resolution global weather forecasting system. We demonstrate that applying our design principles to neural operators leads to significantly lower errors for long-term forecasts as well as longer time horizons without qualitative signs of divergence compared to the original models for these systems. We open-source our https://github.com/mikemccabe210/stabilizing_neural_operators{code} for reproducibility.

A variety of infinitely wide neural architectures (e.g., dense NNs, CNNs, and transformers) induce Gaussian process (GP) priors over their outputs. These relationships provide both an accurate characterization of the prior predictive distribution and enable the use of GP machinery to improve the uncertainty quantification of deep neural networks. In this work, we extend this connection to neural operators (NOs), a class of models designed to learn mappings between function spaces. Specifically, we show conditions for when arbitrary-depth NOs with Gaussian-distributed convolution kernels converge to function-valued GPs. Based on this result, we show how to compute the covariance functions of these NO-GPs for two NO parametrizations, including the popular Fourier neural operator (FNO). With this, we compute the posteriors of these GPs in regression scenarios, including PDE solution operators. This work is an important step towards uncovering the inductive biases of current FNO architectures and opens a path to incorporate novel inductive biases for use in kernel-based operator learning methods.

A sequence of operators T_n from a Hilbert space {mathfrak H} to Hilbert spaces {mathfrak K}_n which is nondecreasing in the sense of contractive domination is shown to have a limit which is still a linear operator T from {mathfrak H} to a Hilbert space {mathfrak K}. Moreover, the closability or closedness of T_n is preserved in the limit. The closures converge likewise and the connection between the limits is investigated. There is no similar way of dealing directly with linear relations. However, the sequence of closures is still nondecreasing and then the convergence is governed by the monotonicity principle. There are some related results for nonincreasing sequences.

We propose an abstract framework to establish the convergence of the iterates of stochastic versions of a broad range of monotone operator splitting methods in Hilbert spaces. This framework allows for the introduction of stochasticity at several levels: approximation of operators, selection of coordinates and operators in block-iterative implementations, and relaxation parameters. The proposed analysis involves a reduced inclusion model with two operators. At each iteration, stochastic approximations to points in the graphs of these two operators are used to form the update. The results are applied to derive the almost sure and L^2 convergence of stochastic versions of the proximal point algorithm, as well as of randomized block-iterative projective splitting methods for solving systems of coupled inclusions involving a mix of set-valued, cocoercive, and Lipschitzian monotone operators combined via various monotonicity-preserving operations.

Every Cartesian three-dimensional PDE solver hides a structural secret that production CFD codes have used for half a century and that graduate-level textbooks rarely state plainly. The derivative matrices, the compact Padé line solves, the Galerkin mass inversions, the alternating-direction-implicit substeps, and even the fast Poisson and Helmholtz diagonalization transforms all factor along the coordinate axes and collapse into repeated one-dimensional banded kernels executed along the grid lines. The three-dimensional operator exists only on paper; it is never assembled, factored, or stored. This paper is the manual for that collapse. We derive the Kronecker-product algebra that makes it exact, carry it cleanly through central differences, compact schemes, tensor-product Galerkin, B-spline and isogeometric methods, collocation, ADI time stepping, and direct Poisson and Helmholtz solves, and bring into the open the three production tricks that turn the reduction into hardware-conscious floating-point throughput on real machines: the multi-right-hand-side reshape that exposes a sweep as one batched line kernel (a dense BLAS-3 GEMM when the line factor is dense or element-local, a banded or stencil kernel when it is not), the sum factorization that rescues high-order Galerkin from the O(p^{2d}) quadrature trap, and the pencil decomposition that keeps every direction contiguous across an MPI cluster. For fixed stencil width or fixed polynomial degree, the compute cost stays O(N) in the total number of unknowns N = N_x N_y N_z; the operator storage drops to O(N_x + N_y + N_z) up to bandwidth constants; direct separable Poisson and Helmholtz solvers add the expected transform cost; the line kernels are embarrassingly parallel. These facts are familiar to practitioners but rarely assembled in one place; this paper collects them and shows how to use them.

A large class of inverse problems for PDEs are only well-defined as mappings from operators to functions. Existing operator learning frameworks map functions to functions and need to be modified to learn inverse maps from data. We propose a novel architecture termed Neural Inverse Operators (NIOs) to solve these PDE inverse problems. Motivated by the underlying mathematical structure, NIO is based on a suitable composition of DeepONets and FNOs to approximate mappings from operators to functions. A variety of experiments are presented to demonstrate that NIOs significantly outperform baselines and solve PDE inverse problems robustly, accurately and are several orders of magnitude faster than existing direct and PDE-constrained optimization methods.

Single-operator learning involves training a deep neural network to learn a specific operator, whereas recent work in multi-operator learning uses an operator embedding structure to train a single neural network on data from multiple operators. Thus, multi-operator learning is capable of predicting a range of operators within one model. In this work, we propose pretraining and fine-tuning strategies for solving PDEs using multi-operator learning. One key aspect is that by increasing the number of families of operators used in pretraining, a PDE foundation model can be fine-tuned to downstream tasks involving new PDEs with a limited number of samples, thus outperforming single operator neural networks. Specifically, a multi-operator learning model pre-trained with data from diverse PDE families can predict unseen operators after fine-tuning with only a limited number of operators from the new family, enabling them to serve as a data-free PDE solver. We also show that the proposed training and fine-tuning method is able to predict new operators in zero-shot prediction without samples. Additionally, we introduce a PDE-agnostic meta-learning algorithm to improve the adaptability of the model to various PDEs by providing a better parameter initialization process. To address the needs of applications with limited computing resources, we explore low-rank adaptation methods that reduce computational costs while enhancing solver accuracy. Lastly, by examining the scaling law with respect to the number of operator families, we establish and highlight its potential for broad adaptation in PDE-solving tasks.

We introduce Recursive Operator Localization (ROL), a framework for constructing hierarchies of local energy problems linked by linear maps between Hilbert spaces. To address spectral-gap problems, including those arising in quantum systems, we study a realization of ROL by nested subspaces indexed by a weighted simplicial complex. Using spectral gaps of auxiliary classical random walks on links, we propagate geometric estimates upward and use them to turn local spectral gaps at the bottom of the hierarchy into a spectral-gap bound for the global operator. For weighted scalar-valued functions on facets, the ROL decomposition becomes the law of total variance, recovering classical spectral-gap bounds from local link expansion (Alev and Lau, STOC 2020). Moreover, for positive binary spin distributions, conditional influence describes the overlap of the corresponding subspaces, yielding known heat-bath gap bounds from spectral independence (Anari, Liu, and Oveis Gharan, FOCS 2020). The framework also offers a route toward quantum Gibbs sampling, with applications requiring model-specific geometric estimates and energy comparisons.

We propose a new bound for generalization of neural networks using Koopman operators. Whereas most of existing works focus on low-rank weight matrices, we focus on full-rank weight matrices. Our bound is tighter than existing norm-based bounds when the condition numbers of weight matrices are small. Especially, it is completely independent of the width of the network if the weight matrices are orthogonal. Our bound does not contradict to the existing bounds but is a complement to the existing bounds. As supported by several existing empirical results, low-rankness is not the only reason for generalization. Furthermore, our bound can be combined with the existing bounds to obtain a tighter bound. Our result sheds new light on understanding generalization of neural networks with full-rank weight matrices, and it provides a connection between operator-theoretic analysis and generalization of neural networks.

Learning mappings between infinite-dimensional function spaces, or operator learning, is essential for many machine learning applications. Although transformer-based operators are popular, they often rely on token-wise attention. These methods treat continuous fields as discrete tokens and usually ignore the global functional structure. We introduce Functional Attention, which reinterprets attention as a functional correspondence between adaptive bases. Inspired by geometric functional maps, our method replaces softmax affinities with structured linear operators. This yields a compact, generalizable, resolution-invariant representation that explicitly captures global dependencies. Experiments demonstrate that Functional Attention can match state-of-the-art performance in many operator learning tasks, including solving PDEs, 3D segmentation, and regression, while remaining robust to varying discretizations. Project page is available at https://github.com/xjffff/FUNCATTN.

Safety alignment in large language models relies on behavioral training that can be overridden when sufficiently strong in-context patterns compete with learned refusal behaviors. We introduce Involuntary In-Context Learning (IICL), an attack class that uses abstract operator framing with few-shot examples to force pattern completion that overrides safety training. Through 3479 probes across 10 OpenAI models, we identify the attack's effective components through a seven-experiment ablation study. Key findings: (1)~semantic operator naming achieves 100% bypass rate (50/50, p < 0.001); (2)~the attack requires abstract framing, since identical examples in direct question-and-answer format yield 0%; (3)~example ordering matters strongly (interleaved: 76%, harmful-first: 6%); (4)~temperature has no meaningful effect (46-56% across 0.0--1.0). On the HarmBench benchmark, IICL achieves 24.0% bypass [18.6%, 30.4%] against GPT-5.4 with detailed 619-word responses, compared to 0.0% for direct queries.

Let Msubset B(H) be a semifinite von Neumann algebra, where B(H) denotes the algebra of all bounded linear operators on a Hilbert space H, and let τ be a fixed faithful normal semifinite trace on M.Let E_τ be the fully symmetric space associated with a fully symmetric Banach function space E on [0,infty).Using a complex interpolation argument based on the three-lines theorem on a strip, we show that for positive operators a,bin E_τ and tin[0,1], $ |a^t b^{1-t}+b^t a^{1-t}|_{E_τ}le 2^{max{2|t-1/2|-1/2,;0}};|a+b|_{E_τ}. In particular, we obtain the sharp constant 1 for t\in[1/4,3/4]: |a^t b^{1-t}+b^t a^{1-t}|_{E_τ}le |a+b|_{E_τ}. $ This extends the work of Kittaneh--Ricard in Linear Algebra Appl. 710 (2025), 356--362 and covers the results of Liu--He--Zhao in Acta Math. Sci. Ser. B (Engl. Ed.) 46 (2026), 62--68

We introduce Geometric Neural Operators (GNPs) for accounting for geometric contributions in data-driven deep learning of operators. We show how GNPs can be used (i) to estimate geometric properties, such as the metric and curvatures, (ii) to approximate Partial Differential Equations (PDEs) on manifolds, (iii) learn solution maps for Laplace-Beltrami (LB) operators, and (iv) to solve Bayesian inverse problems for identifying manifold shapes. The methods allow for handling geometries of general shape including point-cloud representations. The developed GNPs provide approaches for incorporating the roles of geometry in data-driven learning of operators.

Bayesian optimisation is a popular, surrogate model-based approach for optimising expensive black-box functions. Given a surrogate model, the next location to expensively evaluate is chosen via maximisation of a cheap-to-query acquisition function. We present an epsilon-greedy procedure for Bayesian optimisation in batch settings in which the black-box function can be evaluated multiple times in parallel. Our epsilon-shotgun algorithm leverages the model's prediction, uncertainty, and the approximated rate of change of the landscape to determine the spread of batch solutions to be distributed around a putative location. The initial target location is selected either in an exploitative fashion on the mean prediction, or -- with probability epsilon -- from elsewhere in the design space. This results in locations that are more densely sampled in regions where the function is changing rapidly and in locations predicted to be good (i.e close to predicted optima), with more scattered samples in regions where the function is flatter and/or of poorer quality. We empirically evaluate the epsilon-shotgun methods on a range of synthetic functions and two real-world problems, finding that they perform at least as well as state-of-the-art batch methods and in many cases exceed their performance.

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 a simple algorithm that efficiently computes tensor products of Pauli matrices. This is done by tailoring the calculations to this specific case, which allows to avoid unnecessary calculations. The strength of this strategy is benchmarked against state-of-the-art techniques, showing a remarkable acceleration. As a side product, we provide an optimized method for one key calculus in quantum simulations: the Pauli basis decomposition of Hamiltonians.

We integrate neural operators with diffusion models to address the spectral limitations of neural operators in surrogate modeling of turbulent flows. While neural operators offer computational efficiency, they exhibit deficiencies in capturing high-frequency flow dynamics, resulting in overly smooth approximations. To overcome this, we condition diffusion models on neural operators to enhance the resolution of turbulent structures. Our approach is validated for different neural operators on diverse datasets, including a high Reynolds number jet flow simulation and experimental Schlieren velocimetry. The proposed method significantly improves the alignment of predicted energy spectra with true distributions compared to neural operators alone. This enables the diffusion models to stabilize longer forecasts through diffusion-corrected autoregressive rollouts, as we demonstrate in this work. Additionally, proper orthogonal decomposition analysis demonstrates enhanced spectral fidelity in space-time. This work establishes a new paradigm for combining generative models with neural operators to advance surrogate modeling of turbulent systems, and it can be used in other scientific applications that involve microstructure and high-frequency content. See our project page: vivekoommen.github.io/NO_DM

We propose a hierarchical attention mechanism based on two-level overlapping Schwarz domain decomposition. The method is motivated by the observation that two-level Schwarz domain decomposition methods combine local subdomain corrections with a coarse level that communicates global, long-range information. We test its usefulness in the context of finite-dimensional operator learning using a simple, one-dimensional diffusion problem with homogeneous Dirichlet boundary conditions. Although elementary, this problem provides a controlled sequence-to-sequence setting in which the exact nonlocal solution operator is known. After discretization, learning the solution operator amounts to approximating the inverse of a symmetric positive definite matrix. As a baseline, we use a global softmax-free low-rank attention operator of the form QK^T. The proposed construction replaces this dense global factorization by a two-level additive structure: local low-rank attention blocks on overlapping subdomains are combined with a coarse attention block. The resulting operator has the form $M_θ^{-1} = ΦQ_0 K_0^T Φ^T + sum_{i=1}^{N} R_i^T D_i^{1/2} Q_i K_i^T D_i^{1/2} R_i. Here R_i restricts to an overlapping subdomain, D_i is a partition-of-unity weight, and Φ$ is a coarse interpolation (or prolongation) matrix. Numerical experiments for synthetic Fourier right-hand sides indicate that the domain-decomposition attention operator is able to train faster and can give more accurate approximations than a global low-rank attention baseline while using significantly fewer parameters.

Neural operator models trained on simulation data often lose accuracy when applied to experimental measurements due to the sim-to-real gap. Standard fine-tuning with limited real data can reduce this gap, but it may also damage the core physics-relevant representations learned during pretraining. Although knowledge-preserving adaptation has been widely investigated in vision or language tasks, it remains unclear whether these methods are suitable for neural operators whose architectures and protected knowledge are fundamentally different. Neural operators need to preserve core-scale physical structures rather than semantic or visual features. We propose PhysGuard, a physics-preserving framework for accurate sim-to-real adaptation of neural operators. Specifically, PhysGuard uses the empirical Fisher Information Matrix computed on simulation data to identify physics-critical parameter directions, then restricts fine-tuning updates to directions that do not interfere with them. A layer-wise Gram-matrix formulation makes this efficient for models with millions of parameters, while an adaptive threshold automatically determines the protected subspace size. A spectral probe experiment shows that the dominant Fisher directions are strongly associated with low-frequency output structures. Experiments on benchmark across four neural operator architectures and different physical systems show that PhysGuard performs strongly on most evaluation metrics compared to baselines. The benefits are most evident under severe domain shift, where it reduces low-frequency error by up to 32\% compared to standard fine-tuning while maintaining adaptability. Our code is available at https://github.com/ZhouChaunge/PhysGuard.

We show how imaginary numbers in quantum physics can be eliminated by enlarging the Hilbert Space followed by an imposition of - what effectively amounts to - a superselection rule. We illustrate this procedure with a qubit and apply it to the Mach-Zehnder interferometer. The procedure is somewhat reminiscent of the constrained quantization of the electromagnetic field, where, in order to manifestly comply with relativity, one enlargers the Hilbert Space by quantizing the longitudinal and scalar modes, only to subsequently introduce a constraint to make sure that they are actually not directly observable.

Surgical spatio-temporal grounding (STG) requires locating, at each video time specified by a procedural question, the object that the question asks about. Existing approaches face a trade-off: vision language models understand the question context but produce imprecise coordinates, whereas open-set detectors provide localized candidate boxes whose confidence does not reflect which box answers the question. We introduce RefineRank, which closes this gap at the candidate-box level. A compact trainable module, RefineNet, combines the language and regional features of a frozen medical vision language model with the proposals of a frozen open-set detector: it predicts a bounded coordinate correction and a quality score for every candidate box, and a fixed decoding rule returns the original or refined box with the highest score. On the MedVidBench Official Rankings (Verified), RefineRank records 0.421 STG mIoU, the highest displayed STG score, while its global multi-metric rank is 11. In a controlled evaluation on separate training and evaluation videos, coordinate correction raises the candidate oracle upper bound from 0.6772 to 0.7302, and ranking the joint pool of original and refined candidates by their RefineNet scores improves STG mIoU from 0.2719 to 0.4534, whereas separately trained selectors over the same pool reach at most 0.4186. These results show that a small box-level module can reconcile question understanding with precise localization without retraining either backbone. Code is available at [https://github.com/linzhe001/RefineRank](https://github.com/linzhe001/RefineRank).

Operator-learning benchmarks ship as full-precision arrays, and they have grown to terabyte scale. A curator who wants to distribute one has to decide how coarsely to store it. That decision is usually made by fixing a tolerance on the reconstruction error of the stored field. We show that this quantity is measured in the wrong place. It compares the stored field with the original, before any model is trained. What the curator is buying is the accuracy of a model trained on the compressed copy, and the two can disagree: two PDEBench families stored to identical field error differ threefold downstream. A solution operator smooths, so only part of the codec error ever reaches the model's output. That part can be measured. Push a compressed field through a model already trained at full precision, and compare its output against the output on the clean field. The measurement costs forward passes and no training on compressed data. It spans more than two orders of magnitude across six PDE families, and it orders downstream cost where field error does not, inverting 12 of 104 cross-dataset comparisons against field error's 36. Read as a budget, the same curve nominates a storage rate for each dataset. A back-check against trained models finds those rates wrong in both directions by up to a factor of two in stored bits.

Machine learning demand forecasts optimize statistical accuracy yet leave excess operational volatility that inflates safety stock and amplifies the Bullwhip effect. We introduce Contextual Deconvolution (CD), a two-stage estimator that reframes demand sensing as a convex decomposition: a kernel-modulated banded operator separates transient promotion-driven shocks from a smooth structural baseline, and hierarchical partial pooling enables catalog-scale deployment without per-SKU training. The operator is data-derived, not imposed---it reduces to the identity wherever the promotional response is impulsive (most of M5, all of Favorita) and contributes only where genuine multi-day carryover exists, so the gains rest on the structural decomposition itself. Evaluating strictly out-of-sample on 30,490 M5 SKUs and 2,845 Favorita items, with calendar-aware baselines given CD's identical future calendar, we anchor the contribution on a full inventory-cost accounting: CD lowers safety stock, holding cost, and order variance but under-provisions event spikes, reducing total cost only when holding costs exceed sim20\% of stockout costs (95\% CI [17%,25%]); otherwise it is an operational-stability and inventory-capital layer, not an expected-cost minimizer. Its accuracy contribution is reliability rather than central tendency: across eleven baselines, CD attains the lowest cross-sectional dispersion of per-SKU error and mis-forecasts by more than 200\% on 0.8\% of SKUs versus 9.9--20.6\% for every baseline, ranking first on both in all four M5 draws. Because the Variance Ratio and std-based safety stock are minimized by any sufficiently smooth forecast, we treat them as diagnostics, not objectives. A supporting analysis shows the learned demand operators are non-normal, yet CD's compact parametric kernel matches their operational performance interpretably.

Operator splitting algorithms are a cornerstone of modern first-order optimization, decomposing complex problems into simpler subproblems solved via proximal operators. However, most functions lack closed-form proximal operators, which has long restricted these methods to a narrow set of problems. Hamilton-Jacobi-based proximal operator (HJ-Prox) is a recent derivative-free Monte Carlo technique based on Hamilton-Jacobi PDE theory, that approximates proximal operators numerically. In this work, we introduce a unified framework for operator splitting via HJ-Prox, which allows for deployment of operator splitting even when functions are not proximable. We prove that replacing exact proximal steps with HJ-Prox in algorithms such as proximal point, proximal gradient descent, Douglas-Rachford splitting, Davis-Yin splitting, and primal-dual hybrid gradient preserves convergence guarantees under mild assumptions. Numerical experiments demonstrate HJ-Prox is competitive and effective on a wide variety of statistical learning tasks.

On-policy distillation (OPD) reduces train-test state mismatch by training a student on its own generated trajectories, but weak students may visit teacher-misaligned prefixes where supervision is less representative. We introduce SAKI (Supervision Allocation with KL-constrained Interpolation), which combines a KL-constrained teacher-guided rollout with maximal coupling and reuses realized accept/correction events to route token-level supervision. Accepted positions retain sampled-token reverse-KL supervision, while correction positions receive direct supervision on the teacher's highest-probability token. Under maximal coupling, the correction probability is exactly TV(p_t, q_t), so the same trust-region radius controls rollout deviation and upper-bounds intervention and specialized-supervision frequency. We further implement an engine-resident speculative verifier that preserves the exact-q trajectory distribution and coupling semantics while improving matched-workload rollout throughput by 4.22x. Across seven mathematical reasoning benchmarks, SAKI improves the matched teacher-guided baseline in Mean@8 and Pass@8 for both 1.7B and 0.6B students. Placement controls and fixed-prefix analysis further support correction-triggered routing as a conflict-adaptive supervision signal.

Recent studies have drawn attention to the untapped potential of the "star operation" (element-wise multiplication) in network design. While intuitive explanations abound, the foundational rationale behind its application remains largely unexplored. Our study attempts to reveal the star operation's ability to map inputs into high-dimensional, non-linear feature spaces -- akin to kernel tricks -- without widening the network. We further introduce StarNet, a simple yet powerful prototype, demonstrating impressive performance and low latency under compact network structure and efficient budget. Like stars in the sky, the star operation appears unremarkable but holds a vast universe of potential. Our work encourages further exploration across tasks, with codes available at https://github.com/ma-xu/Rewrite-the-Stars.

Spectral positional encodings (PEs) for directed graphs face two obstacles: magnetic Laplacians require an O(n^3) Hermitian eigendecomposition per potential, and their complex eigenvectors are defined only up to unitary gauge, which prior work handles with basis-invariant architectures. We propose learnable spectral PEs of the form h_θ(A_q),R, where A_q is a normalized magnetic operator, h_θ a learnable scalar spectral response, and R a block of random probes. Because the PE is a matrix function of the operator, it is gauge-invariant by construction. We compute it in a Hermitian block Krylov subspace from sparse matrix--vector products only, prove that k = O(log(1/varepsilon)) block steps suffice uniformly over heat--resolvent response families, and give a covering-number argument for why low-dimensional structured families generalize where free per-eigenvalue weights overfit. On a directed SBM whose symmetrization is uninformative by construction, direction-blind PEs stay at chance while magnetic Krylov PEs converge to the exact-eigendecomposition oracle as the depth grows. The same probes yield gauge-invariant pairwise features with 1/s Monte-Carlo error, and the undirected q{=}0 case improves heterophilous benchmarks over no-PE and polynomial baselines.

In this paper, we construct a mixture of neural operators (MoNOs) between function spaces whose complexity is distributed over a network of expert neural operators (NOs), with each NO satisfying parameter scaling restrictions. Our main result is a distributed universal approximation theorem guaranteeing that any Lipschitz non-linear operator between L^2([0,1]^d) spaces can be approximated uniformly over the Sobolev unit ball therein, to any given varepsilon>0 accuracy, by an MoNO while satisfying the constraint that: each expert NO has a depth, width, and rank of O(varepsilon^{-1}). Naturally, our result implies that the required number of experts must be large, however, each NO is guaranteed to be small enough to be loadable into the active memory of most computers for reasonable accuracies varepsilon. During our analysis, we also obtain new quantitative expression rates for classical NOs approximating uniformly continuous non-linear operators uniformly on compact subsets of L^2([0,1]^d).

In a class of quantum circuits known as peaked circuits, the goal is to predict the most probable bit string at the output of the circuit. Since these circuits are designed to have a sharp peak in their output distribution, in principle it should be possible to simulate them using a truncated state vector with a limited number of terms, or a fraction of the total probability mass. This approximate simulation can be carried out on a classical computer with a sparse representation that stores only the nonzero amplitudes of the state vector, in contrast to the dense representations that are common in most quantum simulators. For efficiency, all operations on the state vector should be vectorized to the furthest possible extent and, if available, hardware acceleration can also be used. This work describes how these requirements were met in an open-source implementation, and discusses its performance and limitations.

We release Super Apriel, a 15B-parameter supernet in which every decoder layer provides four trained mixer choices -- Full Attention (FA), Sliding Window Attention (SWA), Kimi Delta Attention (KDA), and Gated DeltaNet (GDN). A placement selects one mixer per layer; placements can be switched between requests at serving time without reloading weights, enabling multiple speed presets from a single checkpoint. The shared checkpoint also enables speculative decoding without a separate draft model. The all-FA preset matches the Apriel 1.6 teacher on all reported benchmarks; recommended hybrid presets span 2.9times to 10.7times decode throughput at 96% to 77% quality retention, with throughput advantages that compound at longer context lengths. With four mixer types across 48 layers, the configuration space is vast. A surrogate that predicts placement quality from the per-layer mixer assignment makes the speed-quality landscape tractable and identifies the best tradeoffs at each speed level. We investigate whether the best configurations at each speed level can be identified early in training or only after convergence. Rankings stabilize quickly at 0.5B scale, but the most efficient configurations exhibit higher instability at 15B, cautioning against extrapolation from smaller models. Super Apriel is trained by stochastic distillation from a frozen Apriel 1.6 teacher, followed by supervised fine-tuning. We release the supernet weights, Fast-LLM training code, vLLM serving code, and a placement optimization toolkit.

Object rearrangement is a significant task for collaborative robots, where they are directed to manipulate objects into a specified goal state. Determining the placement of objects is a major challenge that influences the efficiency of the rearrangement process. Most current methods heavily rely on pre-collected datasets to train the model for predicting the goal position and are restricted to specific instructions, which limits their broader applicability and effectiveness.In this paper, we propose a framework of language-conditioned object rearrangement based on the Large Language Model (LLM). Particularly, our approach mimics human reasoning by using past successful experiences as a reference to infer the desired goal position. Based on LLM's strong natural language comprehension and inference ability, our method can generalise to handle various everyday objects and free-form language instructions in a zero-shot manner. Experimental results demonstrate that our methods can effectively execute the robotic rearrangement tasks, even those involving long sequential orders.

While it is widely known that neural networks are universal approximators of continuous functions, a less known and perhaps more powerful result is that a neural network with a single hidden layer can approximate accurately any nonlinear continuous operator. This universal approximation theorem is suggestive of the potential application of neural networks in learning nonlinear operators from data. However, the theorem guarantees only a small approximation error for a sufficient large network, and does not consider the important optimization and generalization errors. To realize this theorem in practice, we propose deep operator networks (DeepONets) to learn operators accurately and efficiently from a relatively small dataset. A DeepONet consists of two sub-networks, one for encoding the input function at a fixed number of sensors x_i, i=1,dots,m (branch net), and another for encoding the locations for the output functions (trunk net). We perform systematic simulations for identifying two types of operators, i.e., dynamic systems and partial differential equations, and demonstrate that DeepONet significantly reduces the generalization error compared to the fully-connected networks. We also derive theoretically the dependence of the approximation error in terms of the number of sensors (where the input function is defined) as well as the input function type, and we verify the theorem with computational results. More importantly, we observe high-order error convergence in our computational tests, namely polynomial rates (from half order to fourth order) and even exponential convergence with respect to the training dataset size.

It has been conjectured that the operator norm of the Lyapunov operator induced by the Frobenius norm is always attained at a symmetric matrix. The conjecture is known to hold for all matrices of order at most five. We give an integer matrix of order seven for which the skew-symmetric restricted norm is strictly larger than the symmetric restricted norm. A rational separator and exact-arithmetic certificates establish the strict inequality without relying on floating-point computations. A direct-sum construction yields counterexamples in every order n geq 7; the case n = 6 remains open.

Data-driven surrogate models improve the efficiency of simulating continuous dynamical systems, yet their autoregressive rollouts are often limited by instability and spectral blow-up. While global regularization techniques can enforce contractive dynamics, they uniformly damp high-frequency features, introducing a contraction-dissipation dilemma. Furthermore, long-horizon trajectory optimization methods that explicitly correct drift are bottlenecked by memory constraints. In this work, we propose Jacobian-Adaptive Weighting for Stability (JAWS), a probabilistic regularization strategy designed to mitigate these limitations. By framing operator learning as Maximum A Posteriori (MAP) estimation with spatially heteroscedastic uncertainty, JAWS dynamically modulates the regularization strength based on local physical complexity. This allows the model to enforce contraction in smooth regions to suppress noise, while relaxing constraints near singular features to preserve gradients, effectively realizing a behavior similar to numerical shock-capturing schemes. Experiments demonstrate that this spatially-adaptive prior serves as an effective spectral pre-conditioner, which reduces the base operator's burden of handling high-frequency instabilities. This reduction enables memory-efficient, short-horizon trajectory optimization to match or exceed the long-term accuracy of long-horizon baselines. Evaluated on the 1D viscous Burgers' equation, our hybrid approach improves long-term stability, shock fidelity, and out-of-distribution generalization while reducing training computational costs.

The neural operator has emerged as a powerful tool in learning mappings between function spaces in PDEs. However, when faced with real-world physical data, which are often highly non-uniformly distributed, it is challenging to use mesh-based techniques such as the FFT. To address this, we introduce the Non-Uniform Neural Operator (NUNO), a comprehensive framework designed for efficient operator learning with non-uniform data. Leveraging a K-D tree-based domain decomposition, we transform non-uniform data into uniform grids while effectively controlling interpolation error, thereby paralleling the speed and accuracy of learning from non-uniform data. We conduct extensive experiments on 2D elasticity, (2+1)D channel flow, and a 3D multi-physics heatsink, which, to our knowledge, marks a novel exploration into 3D PDE problems with complex geometries. Our framework has reduced error rates by up to 60% and enhanced training speeds by 2x to 30x. The code is now available at https://github.com/thu-ml/NUNO.

Graphical user interface (GUI) grounding requires vision-language models (VLMs) to identify small target elements in high-resolution screenshots and predict precise screen coordinates. On-policy self-distillation (OPSD) is a promising post-training approach for this coordinate-sensitive task, since it provides dense token-level teacher signals beyond hard coordinate labels. However, naive OPSD is not well suited to GUI grounding: OPSD evaluates the teacher on student-generated prefixes, the quality of coordinate-token teacher signals can degrade when the prefix has already deviated from the target coordinate, leading to unreliable teacher signal. To mitigate this, We propose quality-aware self-distillation for VLM-based GUI grounding, which improves coordinate-token teacher-signal quality through soft correctness-aware gating and teacher-probability scaling. The soft correctness-aware gate checks whether the teacher's current coordinate-token prediction can still be completed into the ground-truth box under the student-generated prefix. If not, the corresponding teacher signal is down-weighted. Teacher-probability scaling then uses the teacher's confidence as a lightweight factor to further calibrate the strength of the gated supervision. A key empirical finding is that neither component alone improves overall performance, whereas combining them consistently improves performance. This suggests that the two mechanisms play complementary roles: correctness-aware gating suppresses unreliable coordinate-token supervision, while teacher-probability scaling calibrates the strength of the remaining signals. Experiments across six GUI grounding benchmarks show that our method consistently improves the base model and outperforms strong baselines.

We frame the problem of unifying representations in neural models as one of sparse model recovery and introduce a framework that extends sparse autoencoders (SAEs) to lifted spaces and infinite-dimensional function spaces, enabling mechanistic interpretability of large neural operators (NO). While the Platonic Representation Hypothesis suggests that neural networks converge to similar representations across architectures, the representational properties of neural operators remain underexplored despite their growing importance in scientific computing. We compare the inference and training dynamics of SAEs, lifted-SAE, and SAE neural operators. We highlight how lifting and operator modules introduce beneficial inductive biases, enabling faster recovery, improved recovery of smooth concepts, and robust inference across varying resolutions, a property unique to neural operators.

The very challenging task of learning solution operators of PDEs on arbitrary domains accurately and efficiently is of vital importance to engineering and industrial simulations. Despite the existence of many operator learning algorithms to approximate such PDEs, we find that accurate models are not necessarily computationally efficient and vice versa. We address this issue by proposing a geometry aware operator transformer (GAOT) for learning PDEs on arbitrary domains. GAOT combines novel multiscale attentional graph neural operator encoders and decoders, together with geometry embeddings and (vision) transformer processors to accurately map information about the domain and the inputs into a robust approximation of the PDE solution. Multiple innovations in the implementation of GAOT also ensure computational efficiency and scalability. We demonstrate this significant gain in both accuracy and efficiency of GAOT over several baselines on a large number of learning tasks from a diverse set of PDEs, including achieving state of the art performance on a large scale three-dimensional industrial CFD dataset.

Lattice surgery with two-dimensional quantum error correcting codes is among the leading schemes for fault-tolerant quantum computation, motivated by superconducting hardware architectures. In conventional lattice surgery compilation schemes, logical circuits are compiled following a place-and-route paradigm, where logical qubits remain statically fixed in space throughout the computation. In this work, we introduce a paradigm shift by exploiting movable logical qubits via teleportation during the logical lattice surgery CNOT gate. Focusing on lattice surgery with the color code, we propose a proof-of-concept compilation scheme that leverages this capability. Numerical simulations show that the proposed approach can substantially reduce the routed circuit depth compared to standard place-and-route compilation techniques. Our results demonstrate that optimizations based on movable logical qubits are not limited to architectures with physically movable qubits, such as neutral atoms or trapped ions - they are also readily applicable to superconducting quantum hardware. An open-source implementation of our method is available on GitHub https://github.com/munich-quantum-toolkit/qecc.

Recent work has made clear that the residual pathway is not mere optimization plumbing; it is part of the model's representational machinery. We agree, but argue that the cleanest way to organize this design space is through a two-axis view of the Transformer. A decoder evolves information along two ordered dimensions: sequence position and layer depth. Self-attention already provides adaptive mixing along the sequence axis, whereas the residual stream usually performs fixed addition along the depth axis. If we fix a token position and treat layer index as the ordered variable, then a causal depth-wise residual attention read is exactly the same local operator as causal short sliding-window attention (ShortSWA), except written over depth rather than over sequence. This is the core residual stream duality behind Transformer^2. This perspective also clarifies the recent literature. ELC-BERT and DenseFormer already show that learned aggregation over depth can outperform uniform residual accumulation, while Vertical Attention, DeepCrossAttention (DCA), MUDDFormer, and Attention Residuals move further toward explicit attention-based routing over earlier layers. The key point, however, is that operator-level duality does not imply systems-level symmetry. For large-scale autoregressive models, sequence-axis ShortSWA is usually the more hardware-friendly placement because it reuses token-side sliding-window kernels, KV-cache layouts, and chunked execution. If the goal is instead to change the shortcut itself, Deep Delta Learning (DDL) is the cleaner intervention because it modifies the residual operator directly rather than adding a separate cross-layer retrieval path. Our recommendation is therefore simple: use DDL when the shortcut is the object of interest, and use sequence-axis ShortSWA when the goal is local adaptive mixing.

Macro placement significantly affects a chip's post-route performance, power, and area (PPA). Most placement methods optimize half-perimeter wirelength (HPWL) as the primary objective. However, recent benchmarking shows a near-zero correlation between HPWL and post-route timing metrics such as the worst negative slack (WNS) and total negative slack (TNS). As a result, all six evaluated artificial intelligence (AI) placers degraded PPA relative to the hierarchical baseline. Recent efforts have tried to train cross-stage predictors to close this gap. However, existing methods focus on macro-only representations and use pre-route metrics as training labels. A label fidelity study of ten circuits at four design flow stages reveals that HPWL and pre-route timing poorly reflect final post-route timing rankings. In contrast, post-global-routing achieves the best balance between final timing fidelity and label generation cost-effectiveness. Based on this finding, PPAPlace is a timing-driven differentiable surrogate predicting post-route PPA from macro and standard-cell placements. The surrogate is a dual-stream predictor that combines graph attention over the chip netlist with spatial convolution over the placement grid. It is trained on post-global-routing labels. The predicted WNS and TNS gradients flow end-to-end back to cell coordinates. PPAPlace exploits these gradients in two ways: as a co-objective injected into an analytical placer's optimization loop (PPAPlace-CoOpt), and as a post-placement refinement step that adjusts macro positions via projected gradient descent (PPAPlace-Refine). On five ChiPBench test circuits excluded from training, PPAPlace improves average WNS and TNS by 22\% and 51\% over the hierarchical baseline while preserving power and routability, using the same predictor without test-circuit retraining. Code is available at https://github.com/ValleyC/PPAPlace.

It is common in graphic design humans visually arrange various elements according to their design intent and semantics. For example, a title text almost always appears on top of other elements in a document. In this work, we generate graphic layouts that can flexibly incorporate such design semantics, either specified implicitly or explicitly by a user. We optimize using the latent space of an off-the-shelf layout generation model, allowing our approach to be complementary to and used with existing layout generation models. Our approach builds on a generative layout model based on a Transformer architecture, and formulates the layout generation as a constrained optimization problem where design constraints are used for element alignment, overlap avoidance, or any other user-specified relationship. We show in the experiments that our approach is capable of generating realistic layouts in both constrained and unconstrained generation tasks with a single model. The code is available at https://github.com/ktrk115/const_layout .

Initialization of parameters in deep neural networks has been shown to have a big impact on the performance of the networks (Mishkin & Matas, 2015). The initialization scheme devised by He et al, allowed convolution activations to carry a constrained mean which allowed deep networks to be trained effectively (He et al., 2015a). Orthogonal initializations and more generally orthogonal matrices in standard recurrent networks have been proved to eradicate the vanishing and exploding gradient problem (Pascanu et al., 2012). Majority of current initialization schemes do not take fully into account the intrinsic structure of the convolution operator. Using the duality of the Fourier transform and the convolution operator, Convolution Aware Initialization builds orthogonal filters in the Fourier space, and using the inverse Fourier transform represents them in the standard space. With Convolution Aware Initialization we noticed not only higher accuracy and lower loss, but faster convergence. We achieve new state of the art on the CIFAR10 dataset, and achieve close to state of the art on various other tasks.

Packaging has become a crucial process due to the paradigm shift of More than Moore. Addressing manufacturing and yield issues is a significant challenge for modern layout algorithms. We propose to use printed circuit board (PCB) placement as a benchmark for the packaging problem. A maximum-margin formulation is devised to improve the separation between nets. Our framework includes seed layout proposals, a coordinate descent-based procedure to optimize routability, and a mixed-integer linear programming method to legalize the layout. We perform an extensive study with 14 PCB designs and an open-source router. We show that the placements produced by NS-place improve routed wirelength by up to 25\%, reduce the number of vias by up to 50\%, and reduce the number of DRVs by 79\% compared to manual and wirelength-minimal placements.

Practitioners accept a compressed language model once it clears a stack of data-cheap quality guards: perplexity within a small factor of the original, downstream accuracy (for example MMLU) inside a confidence interval, and data-free output-fidelity signals that compare the compressed and original network's internal representations under random probe inputs. This stack has a blind spot. Across three model families, gently-compressed models clear every guard and then invent procedure steps that were never in the instructions when they run a standard operating procedure (SOP) as an agent. The effect is operator-specific: coherent low-rank (SVD) truncation induces it, and magnitude pruning matched to the same perplexity does not. One dissociation isolates the cause. The same compressed weights that CI-win a paired output-fidelity test CI-fail the invented-step canary. The governing axis is the coherence of the compression error times its rate; the magnitude of the damage does not predict it. The data-free fidelity probe is a fidelity oracle by construction, so it cannot see this axis. We characterize the blindspot and dissociation with paired confidence intervals on a pre-registered, powered canary across three architectures. Operator-specificity replicates on all three, and the perplexity-guard evasion appears where the model admits in-guard low-rank headroom. We then give a data-free screen: a two-axis statistic of the compression error (coherent-fraction and error-rate) that flags the failing builds with fixed thresholds across architectures and matches the coherence-times-rate mechanism. Perplexity, MMLU, and fidelity acceptance do not certify agent safety. Screen gently-compressed low-rank builds before agentic deployment

Computer-use agents ground natural-language instructions in screenshots to locate interface elements, yet existing benchmarks do not isolate whether models bind relational language to the correct element. We introduce GUI-Primitives, a 994-item benchmark of contrastive instruction pairs over seven spatial relations in graphical user interfaces (left/right, above/below, containment, alignment, proximity, list ordinal, occlusion). Each pair holds the screenshot and anchor fixed while changing the relation expression, so the correct target moves between two designated candidates. Five annotators validate a 196-item subset (κ= 0.94 well-formedness; κ= 0.79 target selection). Nineteen vision-language models reach at most 32% strict point-in-box accuracy. Because models emit unconstrained coordinates, we classify each prediction by the candidate region it falls within. Predictions fall outside both candidates on 60-92% of items. Conditional on falling within a candidate region, target selection reaches 0.82-0.90 for horizontal position, vertical position, proximity, and list ordinal, but does not differ significantly from 0.50 for containment and occlusion: most failures reflect candidate localization rather than relation understanding. Across ten models, benchmark accuracy correlates with ScreenSpot-Pro accuracy (Spearman ρ= +0.74), an exploratory association at this sample size. Marking the two designated candidates raises selection accuracy by 35--57 percentage points, an oracle diagnostic that supplies the candidate set rather than a deployable method. We release the benchmark, predictions, and code.

On-policy self-distillation (OPSD) trains mathematical reasoning models using a privileged teacher that sees a reference solution and supervises student-sampled prefixes. Standard OPSD uses one fixed parameter setting at every state, but nearby settings may offer additional supervision. We find that local parameter perturbations reveal complementary reference-aligned corrections under the same reference context. Different experts supply these corrections at different reference positions. Their pool covers more such positions than the unperturbed privileged teacher. We introduce Neighborhood OPSD (N-OPSD) to turn these corrections into supervision at student-visited states. Offline, greedy selection builds a compact pool of frozen experts by rewarding filtered reference-token gains beyond the pool's current best at each position. The highest-peak expert need not provide the best training target. Online routing therefore separates the anchor direction from its level of support. MaxPeak selects the anchor token, and quantile selection chooses among experts whose top token matches it. The student learns from the chosen expert's full next-token distribution through the clipped forward-KL objective inherited from OPSD. We evaluate on AIME 2024, AIME 2025, and HMMT February 2025. Across three independent runs per method, Neighborhood OPSD improves the three-benchmark Average@12 over OPSD by 2.75, 1.67, and 1.94 points on Qwen3-1.7B, 4B, and 8B, respectively. Student-prefix continuations support using the pool beyond the reference trajectories used for selection. Matched ablations support filtered reference-token gains as a selection criterion. Accounting for overlap within the pool and routing by state further improve student accuracy. Inference uses only the distilled student.

Attention operators have been applied on both 1-D data like texts and higher-order data such as images and videos. Use of attention operators on high-order data requires flattening of the spatial or spatial-temporal dimensions into a vector, which is assumed to follow a multivariate normal distribution. This not only incurs excessive requirements on computational resources, but also fails to preserve structures in data. In this work, we propose to avoid flattening by assuming the data follow matrix-variate normal distributions. Based on this new view, we develop Kronecker attention operators (KAOs) that operate on high-order tensor data directly. More importantly, the proposed KAOs lead to dramatic reductions in computational resources. Experimental results show that our methods reduce the amount of required computational resources by a factor of hundreds, with larger factors for higher-dimensional and higher-order data. Results also show that networks with KAOs outperform models without attention, while achieving competitive performance as those with original attention operators.

Context: Just-in-Time (JIT) compilers are able to specialize the code they generate according to a continuous profiling of the running programs. This gives them an advantage when compared to Ahead-of-Time (AoT) compilers that must choose the code to generate once for all. Inquiry: Is it possible to improve the performance of AoT compilers by adding Dynamic Binary Modification (DBM) to the executions? Approach: We added to the Hopc AoT JavaScript compiler a new optimization based on DBM to the inline cache (IC), a classical optimization dynamic languages use to implement object property accesses efficiently. Knowledge: Reducing the number of memory accesses as the new optimization does, does not shorten execution times on contemporary architectures. Grounding: The DBM optimization we have implemented is fully operational on x86_64 architectures. We have conducted several experiments to evaluate its impact on performance and to study the reasons of the lack of acceleration. Importance: The (negative) result we present in this paper sheds new light on the best strategy to be used to implement dynamic languages. It tells that the old days were removing instructions or removing memory reads always yielded to speed up is over. Nowadays, implementing sophisticated compiler optimizations is only worth the effort if the processor is not able by itself to accelerate the code. This result applies to AoT compilers as well as JIT compilers.

We propose an extension of Thompson sampling to optimization problems over function spaces where the objective is a known functional of an unknown operator's output. We assume that queries to the operator (such as running a high-fidelity simulator or physical experiment) are costly, while functional evaluations on the operator's output are inexpensive. Our algorithm employs a sample-then-optimize approach using neural operator surrogates. This strategy avoids explicit uncertainty quantification by treating trained neural operators as approximate samples from a Gaussian process (GP) posterior. We derive regret bounds and theoretical results connecting neural operators with GPs in infinite-dimensional settings. Experiments benchmark our method against other Bayesian optimization baselines on functional optimization tasks involving partial differential equations of physical systems, demonstrating better sample efficiency and significant performance gains.

Given a remarkable representation of the generalized Pauli operators of two-qubits in terms of the points of the generalized quadrangle of order two, W(2), it is shown that specific subsets of these operators can also be associated with the points and lines of the four-dimensional projective space over the Galois field with two elements - the so-called Veldkamp space of W(2). An intriguing novelty is the recognition of (uni- and tri-centric) triads and specific pentads of the Pauli operators in addition to the "classical" subsets answering to geometric hyperplanes of W(2).

S. L. Woronowicz's theory of introducing C*-algebras generated by unbounded elements is applied to q-normal operators satisfying the defining relation of the quantum complex plane. The unique non-degenerate C*-algebra of bounded operators generated by a q-normal operator is computed and an abstract description is given by using crossed product algebras. If the spectrum of the modulus of the q-normal operator is the positive half line, this C*-algebra will be considered as the algebra of continuous functions on the quantum complex plane vanishing at infinity, and its unitization will be viewed as the algebra of continuous functions on a quantum 2-sphere.

The commitment to single-precision floating-point arithmetic is widespread in the deep learning community. To evaluate whether this commitment is justified, the influence of computing precision (single and double precision) on the optimization performance of the Conjugate Gradient (CG) method (a second-order optimization algorithm) and RMSprop (a first-order algorithm) has been investigated. Tests of neural networks with one to five fully connected hidden layers and moderate or strong nonlinearity with up to 4 million network parameters have been optimized for Mean Square Error (MSE). The training tasks have been set up so that their MSE minimum was known to be zero. Computing experiments have disclosed that single-precision can keep up (with superlinear convergence) with double-precision as long as line search finds an improvement. First-order methods such as RMSprop do not benefit from double precision. However, for moderately nonlinear tasks, CG is clearly superior. For strongly nonlinear tasks, both algorithm classes find only solutions fairly poor in terms of mean square error as related to the output variance. CG with double floating-point precision is superior whenever the solutions have the potential to be useful for the application goal.

Neural operators learn mappings between function spaces, which is practical for learning solution operators of PDEs and other scientific modeling applications. Among them, the Fourier neural operator (FNO) is a popular architecture that performs global convolutions in the Fourier space. However, such global operations are often prone to over-smoothing and may fail to capture local details. In contrast, convolutional neural networks (CNN) can capture local features but are limited to training and inference at a single resolution. In this work, we present a principled approach to operator learning that can capture local features under two frameworks by learning differential operators and integral operators with locally supported kernels. Specifically, inspired by stencil methods, we prove that we obtain differential operators under an appropriate scaling of the kernel values of CNNs. To obtain local integral operators, we utilize suitable basis representations for the kernels based on discrete-continuous convolutions. Both these approaches preserve the properties of operator learning and, hence, the ability to predict at any resolution. Adding our layers to FNOs significantly improves their performance, reducing the relative L2-error by 34-72% in our experiments, which include a turbulent 2D Navier-Stokes and the spherical shallow water equations.

Demonstrating the practical utility of Noisy Intermediate-Scale Quantum (NISQ) hardware for recurrent tasks in Computer-Aided Drug Discovery is of paramount importance. We tackle this challenge by performing three-dimensional protein pockets hydration-site prediction on a quantum computer. Formulating the water placement problem as a Quadratic Unconstrained Binary Optimization (QUBO), we use a hybrid approach coupling a classical three-dimensional reference-interaction site model (3D-RISM) to an efficient quantum optimization solver, to run various hardware experiments up to 123 qubits. Matching the precision of classical approaches, our results reproduced experimental predictions on real-life protein-ligand complexes. Furthermore, through a detailed resource estimation analysis, we show that accuracy can be systematically improved with increasing number of qubits, indicating that full quantum utility is in reach. Finally, we provide evidence that advantageous situations could be found for systems where classical optimization struggles to provide optimal solutions. The method has potential for assisting simulations of protein-ligand complexes for drug lead optimization and setup of docking calculations.

Transformer-based neural operators have achieved substantial progress in solving Partial Differential Equations (PDEs) by projecting spatial observations into compact latent tokens and learning physical interactions in latent spaces. However, we reveal that existing learnable projection mechanisms cannot ensure stable and balanced assignments from observation points to latent tokens, causing some latent tokens to be over-assigned while others remain underutilized. This limitation further restricts the design of hierarchical architectures, as assignment imbalance is continuously inherited and amplified across latent spaces, eventually causing severe token collapse in deeper spaces. To address these issues, we propose MoNo (Multiscale Optimal Transport Neural Operator), a progressive multiscale neural operator that efficiently solves PDEs on general geometries through stable latent-space construction. At its core is CoTAP (Cross-scale Optimal Transport Assignment and Projection), a novel latent-space construction method that formulates cross-space assignment between adjacent spaces as an entropy-regularized optimal transport problem, thereby constructing balanced bidirectional projections and stable latent spaces. CoTAP also ensures stable information transfer across multiple latent spaces, further enabling multiscale architectures on general geometries, which in turn support more efficient learning of long-range physical interactions. Extensive experiments demonstrate that MoNo outperforms existing state-of-the-art neural operators in both prediction performance and computational efficiency. Code is available at https://github.com/ZijiangY1116/MoNo.

Deep neural network (DNN) deployment has been confined to larger hardware devices due to their expensive computational requirements. This challenge has recently reached another scale with the emergence of large language models (LLMs). In order to reduce both their memory footprint and latency, a promising technique is quantization. It consists in converting floating point representations to low bit-width fixed point representations, usually by assuming a uniform mapping onto a regular grid. This process, referred to in the literature as uniform quantization, may however be ill-suited as most DNN weights and activations follow a bell-shaped distribution. This is even worse on LLMs whose weight distributions are known to exhibit large, high impact, outlier values. In this work, we propose an improvement over the most commonly adopted way to tackle this limitation in deep learning models quantization, namely, non-uniform quantization. NUPES leverages automorphisms to preserve the scalar multiplications. Such transformations are derived from power functions. However, the optimization of the exponent parameter and weight values remains a challenging and novel problem which could not be solved with previous post training optimization techniques which only learn to round up or down weight values in order to preserve the predictive function. We circumvent this limitation with a new paradigm: learning new quantized weights over the entire quantized space. Similarly, we enable the optimization of the power exponent, i.e. the optimization of the quantization operator itself during training by alleviating all the numerical instabilities. The resulting predictive function is compatible with integer-only low-bit inference. We show the ability of the method to achieve state-of-the-art compression rates in both, data-free and data-driven configurations.

Generative models in function spaces, situated at the intersection of generative modeling and operator learning, are attracting increasing attention due to their immense potential in diverse scientific and engineering applications. While functional generative models are theoretically domain- and discretization-agnostic, current implementations heavily rely on the Fourier Neural Operator (FNO), limiting their applicability to regular grids and rectangular domains. To overcome these critical limitations, we introduce the Mesh-Informed Neural Operator (MINO). By leveraging graph neural operators and cross-attention mechanisms, MINO offers a principled, domain- and discretization-agnostic backbone for generative modeling in function spaces. This advancement significantly expands the scope of such models to more diverse applications in generative, inverse, and regression tasks. Furthermore, MINO provides a unified perspective on integrating neural operators with general advanced deep learning architectures. Finally, we introduce a suite of standardized evaluation metrics that enable objective comparison of functional generative models, addressing another critical gap in the field.

Radio-frequency portion of the electromagnetic spectrum is a scarce resource. Cognitive radio technology has emerged as a promising solution to overcome the spectrum scarcity bottleneck. Through this technology, secondary users (SUs) sense the spectrum opportunities free from primary users (PUs), and opportunistically take advantage of these (temporarily) idle portions, known as spectrum holes. In this correspondence, we consider a variant of the cognitive radio resource allocation problem posed by Martinovic et al. in 2017. The distinguishing feature of this version of the problem is that each SU, due to its hardware limitations, imposes the requirement that the to-be-aggregated spectrum holes cannot be arbitrarily far from each other. We call this restriction as the Maximal Aggregation Range (MAR) constraint, and refer to this variant of the problem as the MAR-constrained hole assignment problem. The problem can be formalized as an NP-hard combinatorial optimization problem. We propose a novel binary integer linear programming (ILP) formulation to the problem. The number of constraints in this formulation is the number of spectrum holes plus the number of SUs. On the other hand, the number of binary decision variables in the formulation can be prohibitively large, as for each legitimate spectrum allocation to each SU, one variable is needed. We propose a branch-and-price (B&P) framework to tackle this challenge. This framework is in fact a branch-and-bound procedure in which at each node of the search tree, we utilize the so-called (delayed) column generation technique for solving the LP relaxation of the corresponding subproblem. As evidenced by the numerical results, the LP relaxation bounds are very tight. This allows for a very effective pruning of the search space. Compared to the previously suggested formulations, the proposed technique can require much less computational effort.

Language models hold latent quantities in a form they can report on, and more of a quantity is present in that form when the task requires reusing it flexibly. What causes a representation to enter that form is open, and the word workspace invites an admission story: a gate that decides what gets in. Testing it on open-weight models with Jacobian lenses, over a benchmark whose five arms share an identical context, we find no gate where it predicts one. Demand raises a concept's lens visibility beyond what applying an operator to a supplied value produces: +0.050 [+0.045, +0.057] in percentile rank on our primary checkpoint, positive on all four we measure, though that arm answers at ceiling and the accuracymatched contrast is stronger under that readout. At the same time one shared linear map decodes the variable from every arm, the control included, at 6.4-9.0x its selection-corrected floor. What produces the later readable form at the queried position is attention-mediated gathering inside a mid-depth window: separating patch depth from readout depth puts transport there at least 17x above anywhere shallower under non-saturating readouts, with no tested MLP output contributing positively inside it. Under the saturating percentile rank the same grid does not localise the window, which is a fact about that measure. An arm that needs the variable for nothing concentrates sevenfold less, so the window is demand-specific. That window has two measured edges, a survival failure below and destruction above, and it falls at the same fractional depth in a 64-layer hybrid and a 62-layer dense model from another family. We localise where the variable is installed and read, not the route from the passage, which transports nothing. But the readout is not a calibrated measure of use: three components move it to within 12% of one another and differ 7.4x in what they do to the answer.

We introduce a novel framework that directly learns a spectral basis for shape and manifold analysis from unstructured data, eliminating the need for traditional operator selection, discretization, and eigensolvers. Grounded in optimal-approximation theory, we train a network to decompose an implicit approximation operator by minimizing the reconstruction error in the learned basis over a chosen distribution of probe functions. For suitable distributions, they can be seen as an approximation of the Laplacian operator and its eigendecomposition, which are fundamental in geometry processing. Furthermore, our method recovers in a unified manner not only the spectral basis, but also the implicit metric's sampling density and the eigenvalues of the underlying operator. Notably, our unsupervised method makes no assumption on the data manifold, such as meshing or manifold dimensionality, allowing it to scale to arbitrary datasets of any dimension. On point clouds lying on surfaces in 3D and high-dimensional image manifolds, our approach yields meaningful spectral bases, that can resemble those of the Laplacian, without explicit construction of an operator. By replacing the traditional operator selection, construction, and eigendecomposition with a learning-based approach, our framework offers a principled, data-driven alternative to conventional pipelines. This opens new possibilities in geometry processing for unstructured data, particularly in high-dimensional spaces.

Image composition aims to generate realistic composite image by inserting an object from one image into another background image, where the placement (e.g., location, size, occlusion) of inserted object may be unreasonable, which would significantly degrade the quality of the composite image. Although some works attempted to learn object placement to create realistic composite images, they did not focus on assessing the plausibility of object placement. In this paper, we focus on object placement assessment task, which verifies whether a composite image is plausible in terms of the object placement. To accomplish this task, we construct the first Object Placement Assessment (OPA) dataset consisting of composite images and their rationality labels. We also propose a simple yet effective baseline for this task. Dataset is available at https://github.com/bcmi/Object-Placement-Assessment-Dataset-OPA.

We introduce a technique that uses gauge fixing to significantly improve the quantum error correcting performance of subsystem codes. By changing the order in which check operators are measured, valuable additional information can be gained, and we introduce a new method for decoding which uses this information to improve performance. Applied to the subsystem toric code with three-qubit check operators, we increase the threshold under circuit-level depolarising noise from 0.67% to 0.81%. The threshold increases further under a circuit-level noise model with small finite bias, up to 2.22% for infinite bias. Furthermore, we construct families of finite-rate subsystem LDPC codes with three-qubit check operators and optimal-depth parity-check measurement schedules. To the best of our knowledge, these finite-rate subsystem codes outperform all known codes at circuit-level depolarising error rates as high as 0.2%, where they have a qubit overhead that is 4.3times lower than the most efficient version of the surface code and 5.1times lower than the subsystem toric code. Their threshold and pseudo-threshold exceeds 0.42% for circuit-level depolarising noise, increasing to 2.4% under infinite bias using gauge fixing.

We introduce ECHO, a transformer-operator framework for generating million-point PDE trajectories. While existing neural operators (NOs) have shown promise for solving partial differential equations, they remain limited in practice due to poor scalability on dense grids, error accumulation during dynamic unrolling, and task-specific design. ECHO addresses these challenges through three key innovations. (i) It employs a hierarchical convolutional encode-decode architecture that achieves a 100 times spatio-temporal compression while preserving fidelity on mesh points. (ii) It incorporates a training and adaptation strategy that enables high-resolution PDE solution generation from sparse input grids. (iii) It adopts a generative modeling paradigm that learns complete trajectory segments, mitigating long-horizon error drift. The training strategy decouples representation learning from downstream task supervision, allowing the model to tackle multiple tasks such as trajectory generation, forward and inverse problems, and interpolation. The generative model further supports both conditional and unconditional generation. We demonstrate state-of-the-art performance on million-point simulations across diverse PDE systems featuring complex geometries, high-frequency dynamics, and long-term horizons.

Block-diffusion language models are served at hand-picked operating points, such as acceptance thresholds, buffer depth, schedule, checkpoint and precision, and each point is chosen by its mean benchmark accuracy. However, a mean does not tell an operator how often a faster configuration fails on prompts that the slower one answers correctly. On the serving engine and its decode traces, the default commit rule already commits every fully resolved block, a static skip rule captures nearly all of the compute that allocation can save, and self-distillation on engine-decoded targets adds speed at unchanged accuracy. Larger speedups come from lower thresholds, which commit tokens that are still uncertain. We therefore present Redline, a finite-sample procedure that selects operating points, hand-picked or learned, from the correctness of their answers on calibration prompts. Redline keeps the reference-relative risk, the joint probability that the reference answers correctly and a candidate configuration does not, within a user-chosen budget with high probability, and deploys the fastest configuration that passes. It speeds up math at a smaller risk budget than code in both model families, and at a budget of ten percent it deploys a LLaDA2 math configuration that commits over a third more tokens in each forward. It also applies without modification to the acceptance rule of speculative decoding and to weight quantization. On the same calibration data, Redline stays within its stated failure probability, whereas each tolerance of a mean-accuracy rule either gains less speed for some model and task or exceeds the risk budget far more often for another. Code is available at https://github.com/js-lee-AI/Redline.

Low Earth orbit (LEO) satellites increasingly carry compute hardware capable of on-board processing, yet each satellite generates roughly two orders of magnitude more data than it can downlink per orbit. This mismatch forces operators to decide, for every workload, which computation runs on-board and which runs on the ground, how intermediate data crosses the space-ground boundary through narrow contact windows, and how to maintain delivery guarantees over noisy channels. We present Constraint-Aware Execution (CAE), a planning system that takes a satellite identifier, a workload expressed as a directed acyclic graph of processing steps, and a set of orbital and resource constraints, and produces a deterministic, physically grounded execution plan. CAE operates in four phases: (1) orbital environment construction via SGP4 propagation with eclipse detection and ground station pass prediction, (2) compute placement using a cost model that compares on-board resource consumption against transfer overhead, (3) transfer insertion with adaptive forward error correction and security overhead modeling, and (4) greedy first-fit scheduling into orbital windows under power, thermal, compute, and communication constraints. We evaluate CAE against five representative workload patterns across satellites in distinct orbital regimes and demonstrate that the system produces feasible plans in under two seconds, correctly exploits onboard data reduction to minimize transfer volume, and adapts FEC and multi-pass allocation to varying channel conditions. CAE is deployed as a production API computing plans for any cataloged satellite using live two-line element data.

We present a methodology to price options and portfolios of options on a gate-based quantum computer using amplitude estimation, an algorithm which provides a quadratic speedup compared to classical Monte Carlo methods. The options that we cover include vanilla options, multi-asset options and path-dependent options such as barrier options. We put an emphasis on the implementation of the quantum circuits required to build the input states and operators needed by amplitude estimation to price the different option types. Additionally, we show simulation results to highlight how the circuits that we implement price the different option contracts. Finally, we examine the performance of option pricing circuits on quantum hardware using the IBM Q Tokyo quantum device. We employ a simple, yet effective, error mitigation scheme that allows us to significantly reduce the errors arising from noisy two-qubit gates.

Expanding on neural operators, we propose a novel framework for stochastic process learning across arbitrary domains. In particular, we develop operator flow matching (OFM) for learning stochastic process priors on function spaces. OFM provides the probability density of the values of any collection of points and enables mathematically tractable functional regression at new points with mean and density estimation. Our method outperforms state-of-the-art models in stochastic process learning, functional regression, and prior learning.

kooplearn is a machine-learning library that implements linear, kernel, and deep-learning estimators of dynamical operators and their spectral decompositions. kooplearn can model both discrete-time evolution operators (Koopman/Transfer) and continuous-time infinitesimal generators. By learning these operators, users can analyze dynamical systems via spectral methods, derive data-driven reduced-order models, and forecast future states and observables. kooplearn's interface is compliant with the scikit-learn API, facilitating its integration into existing machine learning and data science workflows. Additionally, kooplearn includes curated benchmark datasets to support experimentation, reproducibility, and the fair comparison of learning algorithms. The software is available at https://github.com/Machine-Learning-Dynamical-Systems/kooplearn.

The poor performance of transformers on arithmetic tasks seems to stem in large part from their inability to keep track of the exact position of each digit inside of a large span of digits. We mend this problem by adding an embedding to each digit that encodes its position relative to the start of the number. In addition to the boost these embeddings provide on their own, we show that this fix enables architectural modifications such as input injection and recurrent layers to improve performance even further. With positions resolved, we can study the logical extrapolation ability of transformers. Can they solve arithmetic problems that are larger and more complex than those in their training data? We find that training on only 20 digit numbers with a single GPU for one day, we can reach state-of-the-art performance, achieving up to 99% accuracy on 100 digit addition problems. Finally, we show that these gains in numeracy also unlock improvements on other multi-step reasoning tasks including sorting and multiplication.

A multiplicative dual-encoder network computes a real-valued output for a pair of inputs as the inner product of their separate encodings. This architecture has been developed independently in operator learning, bipartite matching, contrastive vision-language models, retrieval, and other areas, yet no unified theory guides the basic design decisions: how many interaction modes to represent, how to normalize the encoders, and when the architecture should be avoided. We provide such a foundation by introducing the class of functions of low interaction rank, a class whose intrinsic complexity is measured by its interaction spectrum. Within this framework, approximation error decomposes into a spectral truncation term and an encoder-realization term; sample complexity is governed by the sum of the two encoder complexities rather than their product; and a usability criterion based on spectral decay determines when the architecture can succeed. The same framework exposes a central identifiability problem: the encoders are defined only up to a linear gauge symmetry that leaves the learned coordinates arbitrary. We show that normalization is gauge fixing and that whitening pins the interaction modes up to permutation and sign, thereby explaining the uninterpretability of contrastive dimensions and providing a constructive remedy. Experiments on synthetic kernels, operator learning, and CLIP models validate the theoretical predictions: spectral decay rates match the predicted scaling, whitening recovers the true modes, and independently trained CLIP models are related by a single rotation which, after removal by whitening, exposes interpretable concept axes. The code of this paper is provided at https://github.com/RS2002/Mul-Net .

In dynamic programming (DP) and reinforcement learning (RL), an agent learns to act optimally in terms of expected long-term return by sequentially interacting with its environment modeled by a Markov decision process (MDP). More generally in distributional reinforcement learning (DRL), the focus is on the whole distribution of the return, not just its expectation. Although DRL-based methods produced state-of-the-art performance in RL with function approximation, they involve additional quantities (compared to the non-distributional setting) that are still not well understood. As a first contribution, we introduce a new class of distributional operators, together with a practical DP algorithm for policy evaluation, that come with a robust MDP interpretation. Indeed, our approach reformulates through an augmented state space where each state is split into a worst-case substate and a best-case substate, whose values are maximized by safe and risky policies respectively. Finally, we derive distributional operators and DP algorithms solving a new control task: How to distinguish safe from risky optimal actions in order to break ties in the space of optimal policies?

In this work, we introduce MO-SeGMan, a Multi-Objective Sequential and Guided Manipulation planner for highly constrained rearrangement problems. MO-SeGMan generates object placement sequences that minimize both replanning per object and robot travel distance while preserving critical dependency structures with a lazy evaluation method. To address highly cluttered, non-monotone scenarios, we propose a Selective Guided Forward Search (SGFS) that efficiently relocates only critical obstacles and to feasible relocation points. Furthermore, we adopt a refinement method for adaptive subgoal selection to eliminate unnecessary pick-and-place actions, thereby improving overall solution quality. Extensive evaluations on nine benchmark rearrangement tasks demonstrate that MO-SeGMan generates feasible motion plans in all cases, consistently achieving faster solution times and superior solution quality compared to the baselines. These results highlight the robustness and scalability of the proposed framework for complex rearrangement planning problems.

Generating 2-by-2 unitary matrices in floating-precision arithmetic is a delicate task. One way to reduce the accumulation error is to use less floating-point operations to compute each of the entries in the 2-by-2 unitary matrix. This paper shows an algorithm that reduces the number of operations to compute the entries of a Givens rotation. Overall, the new algorithm has more operations in total when compared to algorithms in different releases of LAPACK, but less operations per entry. Numerical tests show that the new algorithm is more accurate on average.

Supply chain operations traditionally involve a variety of complex decision making problems. Over the last few decades, supply chains greatly benefited from advances in computation, which allowed the transition from manual processing to automation and cost-effective optimization. Nonetheless, business operators still need to spend substantial efforts in explaining and interpreting the optimization outcomes to stakeholders. Motivated by the recent advances in Large Language Models (LLMs), we study how this disruptive technology can help bridge the gap between supply chain automation and human comprehension and trust thereof. We design -- a framework that accepts as input queries in plain text, and outputs insights about the underlying optimization outcomes. Our framework does not forgo the state-of-the-art combinatorial optimization technology, but rather leverages it to quantitatively answer what-if scenarios (e.g., how would the cost change if we used supplier B instead of supplier A for a given demand?). Importantly, our design does not require sending proprietary data over to LLMs, which can be a privacy concern in some circumstances. We demonstrate the effectiveness of our framework on a real server placement scenario within Microsoft's cloud supply chain. Along the way, we develop a general evaluation benchmark, which can be used to evaluate the accuracy of the LLM output in other scenarios.

Unsupervised learning with functional data is an emerging paradigm of machine learning research with applications to computer vision, climate modeling and physical systems. A natural way of modeling functional data is by learning operators between infinite dimensional spaces, leading to discretization invariant representations that scale independently of the sample grid resolution. Here we present Variational Autoencoding Neural Operators (VANO), a general strategy for making a large class of operator learning architectures act as variational autoencoders. For this purpose, we provide a novel rigorous mathematical formulation of the variational objective in function spaces for training. VANO first maps an input function to a distribution over a latent space using a parametric encoder and then decodes a sample from the latent distribution to reconstruct the input, as in classic variational autoencoders. We test VANO with different model set-ups and architecture choices for a variety of benchmarks. We start from a simple Gaussian random field where we can analytically track what the model learns and progressively transition to more challenging benchmarks including modeling phase separation in Cahn-Hilliard systems and real world satellite data for measuring Earth surface deformation.

Existing neural operator architectures face challenges when solving multiphysics problems with coupled partial differential equations (PDEs) due to complex geometries, interactions between physical variables, and the limited amounts of high-resolution training data. To address these issues, we propose Codomain Attention Neural Operator (CoDA-NO), which tokenizes functions along the codomain or channel space, enabling self-supervised learning or pretraining of multiple PDE systems. Specifically, we extend positional encoding, self-attention, and normalization layers to function spaces. CoDA-NO can learn representations of different PDE systems with a single model. We evaluate CoDA-NO's potential as a backbone for learning multiphysics PDEs over multiple systems by considering few-shot learning settings. On complex downstream tasks with limited data, such as fluid flow simulations, fluid-structure interactions, and Rayleigh-B\'enard convection, we found CoDA-NO to outperform existing methods by over 36%.

Neural operators have shown promise in learning solution maps of partial differential equations (PDEs), but they often struggle to generalize when test inputs lie outside the training distribution, such as novel initial conditions, unseen PDE coefficients or unseen physics. Prior works address this limitation with large-scale multiple physics pretraining followed by fine-tuning, but this still requires examples from the new dynamics, falling short of true zero-shot generalization. In this work, we propose a method to enhance generalization at test time, i.e., without modifying pretrained weights. Building on DISCO, which provides a dictionary of neural operators trained across different dynamics, we introduce a neural operator splitting strategy that, at test time, searches over compositions of training operators to approximate unseen dynamics. On challenging out-of-distribution tasks including parameter extrapolation and novel combinations of physics phenomena, our approach achieves state-of-the-art zero-shot generalization results, while being able to recover the underlying PDE parameters. These results underscore test-time computation as a key avenue for building flexible, compositional, and generalizable neural operators.