NeurIPS 2026 Papers

Partial differential equations (PDEs) are foundational to modeling in science and engineering, but constructing reliable numerical solvers remains labor-intensive, demanding expert knowledge of discretization schemes, stability conditions, and boundary treatments. Recent work has begun to frame PDE solving as a code-generation task for large language models (LLMs), yet existing approaches operate primarily at inference time: relying on prompting, debugging, self-refinement, and test-time scaling rather than adapting the model itself. In parallel, reinforcement learning with verifiable rewards has emerged as a powerful post-training paradigm for code and math reasoning, but its verifiers are typically binary: a compiler runs, or a test passes. Such signals discard the graded structure of scientific correctness, where two solvers may both execute and yet differ in solution accuracy by orders of magnitude. In this work, we introduce RLVP: Reinforcement Learning with Verifiable Physics, an RL post-training framework for multi-PDE solver code generation.RLVP addresses this verifiability gap with a hybrid verifier: hard program-validity checks ensure executability, while continuous physics rewards score function-space accuracy and PDE-residual consistency. A single policy is post-trained across diverse PDE families spanning hyperbolic, parabolic, elliptic, and incompressible-flow systems. RLVP improves over both pre-trained and supervised-only baselines on PDE benchmarks, and shows zero-shot improvement transfer to held-out PDEs. We show that a smaller LLM post-trained with RLVP can outperform prompting a frontier model on in-distribution PDE solver generation. The trained policy shows evidence of compositionality in numerical motifs: it recombines stencils, time-stepping schemes, and boundary-handling primitives learned from the PDEs used in training into generated solvers for unseen PDE problems.

The human vocal tract, the cavity consisting of one's throat, mouth, lips opening, etc., filters one's voice to create the sounds we know as speech. In this paper, we present a differentiable and GPU parallelizable acoustic simulator that synthesizes speech by propagating sound along an acoustic tube representation of the vocal tract, and via its gradients, solves the inverse problem: reconstructing the geometry of their vocal tract solely from the sound it produces. Although learning the inverse mapping between geometry and sound is notoriously non-convex, we discover that gradient descent succeeds with three technical contributions: (1) we design a frequency domain formulation of the vocal tract's fluid dynamics that is 70x more GPU parallelizable than finite differences in time, (2) we integrate a differentiable model for turbulence to synthesize consonants, and (3) similar to prior work in implicit neural representations (INRs) and neural radiance fields (NeRFs), we find that parameterizing the geometry with a neural network accelerates convergence and escapes local minima that trap discrete representations. Because the simulator is differentiable, it is readily integrated with other deep learning pipelines to enable novel speech and medical imaging applications. Our model can be used for instance, in singing instruction or language learning, where a visualization of one's vocal tract can help people understand how their vocal tract maps to different speech sounds. To enable these tasks, (1) we demonstrate self-supervised autoencoding of vocal tract shapes across 11 languages, and (2) we couple our simulator with a generative model of MRI images to reconstruct one's moving vocal tract from only their speech, no paired data required.

Masked diffusion language models (MDLMs) generate text by unmasking tokens in parallel and have recently emerged as alternatives to autoregressive language models. They can be viewed as parallel decoders trained with a position-wise cross-entropy (CE) loss, the same setup as non-autoregressive translation (NAT). In NAT, CE-trained parallel decoders have been argued to be sensitive to small positional shifts, since CE penalizes them harshly. We ask whether CE-trained MDLMs are similarly sensitive to such shifts under iterative decoding. To probe this, we apply a controlled intervention that introduces them during decoding. On LLaDA-8B-Instruct with Arena-Hard, displacing as little as 1\% of generated tokens by one position substantially reduces win rates against the unintervened model, showing that MDLMs are sensitive to such small shifts under iterative parallel decoding. Motivated by this, we adapt connectionist temporal classification (CTC), an alignment-flexible objective known to mitigate it there, to MDLM supervised fine-tuning. By relaxing the strict position-wise match that CE imposes, CTC gives the loss room to absorb small positional shifts; concretely, we modified CTC objective to use a special \ token that absorbs positional uncertainty between target tokens and output positions, and a updated collapse map that preserves target surface forms. Across four open-ended generation benchmarks, the resulting model consistently improves over both the original model and a matched cross-entropy-trained baseline, with statistically significant gains on all four. These results identify training-side alignment flexibility as a useful design dimension for MDLM SFT, complementary to the inference-time approaches explored in prior work.

Causal modeling of physical temporal phenomena must handle interventions that act along trajectories, nonstationary induced laws, path-dependent effects, and feedback mediated by dynamics, all challenging in standard causal models. We introduce \emph{Hamiltonian Causal Models} (HCMs), a trajectory-level framework in which observed variables interact with local environments and interventions act as controls of Hamiltonian mechanisms. HCMs separate immutable equations of motion from intervenable mechanisms and define causal effects as discrepancies between interventional path laws. A key motivation for HCMs is their natural interface with non-equilibrium thermodynamics. Entropy production quantifies the irreversibility of a process and is a central causal observable: it is estimable from data and witnesses causal effects along the system's evolution that are invisible to endpoint and cumulative versions of the standard average treatment effect. As in physics, cause-and effect are not primitives of the relation between two random variables but arise from the non-invertibility of the thermodynamic arrow. With this, our paper reconciles the language of statistical causal models and non-stationary thermodynamics, offering new tools to describe causality in a wide range of physical systems.

Neural networks with randomly generated hidden weights (RaNNs) have been extensively studied, both as a standalone learning method and as an initialization for fully trainable deep learning methods. In this work, we study RaNN expressivity for learning solutions to non-linear partial differential equations (PDEs). Despite their widespread use in practical applications, a rigorous theoretical understanding of the approximation properties of RaNNs in this context remains limited. Here, we derive error bounds for RaNN approximations to time-dependent Sobolev functions and obtain a dimension-free approximation rate $\frac{1}{2}$ for sufficiently regular functions. We apply our results to two important classes of non-linear PDEs: Porous Medium Equations and Compressible Navier-Stokes Equations, showing that RaNNs are capable of efficiently approximating solutions to these complex, non-linear PDEs. Our theoretical analysis is supported by numerical experiments, validating the obtained convergence rates.

Parkinson's disease and other gait-related neurological disorders manifest in subtle but biomechanically meaningful changes in how patients move, making computational gait phenotyping a valuable tool for clinical assessment. Yet existing latent representations of motion are typically learned from data alone and miss the biomechanical structure that gives clinical motion its diagnostic meaning, leaving them vulnerable when training cohorts are small or acquisition sites differ. We propose REINS (Riemannian Embedding for INertia-induced motion Spaces), an Euler-Lagrange-induced motion representation that uses the generalized inertia matrix of articulated systems to define a biomechanically grounded geometry on configuration space. Because kinetic energy is itself a quadratic form in velocity weighted by the inertia matrix, this metric is a mechanically determined Riemannian structure, and we learn a latent embedding of the resulting structured space. By defining the representation space itself through rigid-body dynamics rather than enforcing physics as a post-hoc penalty, REINS aligns the geometry of motion representation with the same physical laws clinicians rely on to interpret gait. We apply the framework on Parkinson's disease gait analysis, where deviation from healthy motion is measured as a geodesic distance under the inertia-induced metric, and report results on healthy-control vs.\ PD discrimination and UPDRS-gait severity correlation.

Camera sensor RAW data offers intrinsic advantages for object detection, including richer bit depth, preserved physical information, and freedom from image signal processor (ISP) distortions. However, varying exposure conditions, spectral sensitivities, and bit depths across devices introduce substantially larger domain gaps than sRGB, making sensor-agnostic generalization a fundamental challenge. In this study, we present a physics-guided global-local tone mapping framework for sensor-agnostic RAW object detection. By factoring sensor-induced variations into a global tonal correction and a spatially adaptive local color adjustment, both driven by RAW distribution priors, our framework enables a single network to train jointly across heterogeneous sensors. To further support cross-sensor generalization, we construct a physics-based RAW simulation pipeline that synthesizes realistic sensor outputs spanning diverse spectral sensitivities, illuminants, and sensor non-idealities. Extensive experiments across multiple RAW benchmarks, covering bit depths from 12 to 24, demonstrate state-of-the-art (SOTA) performance in single-dataset, mixed-dataset, and challenging robustness settings.

Rheumatoid arthritis (RA) assessment from hand radiographs requires multi-level analysis and modeling of anatomical structures and fine-grained local pathological changes. However, existing public resources do not support such multi-level analysis, often lacking full-hand coverage, fine-grained annotations, and consistent integration with clinical scoring systems. In particular, annotations that enable quantitative analysis of bone erosion (BE) remain scarce. RAM-H1200 contains 1,200 hand radiographs from a regional, multi-institutional Japanese cohort collected across six medical centers in the Sapporo region, with multi-level annotations including (i) whole-hand bone structure instance segmentation, (ii) pixel-level BE masks, (iii) Sharp/van der Heijde (SvdH)-defined joint regions of interest, and (iv) joint-level SvdH scores for both BE and joint space narrowing (JSN). It provides a common evaluation resource covering anatomical structure modeling, localized erosive pathology analysis, and clinically standardized RA severity assessment from the same radiographs. The proposed BE masks enable, for the first time, quantitative BE analysis beyond coarse categorical grading by providing explicit spatial supervision for lesion extent and morphology. To our knowledge, RAM-H1200 is the first public benchmark resource providing whole-hand bone structure instance segmentation, pixel-level BE delineation, and clinically grounded joint-level SvdH scoring for both BE and JSN. Results across benchmark tasks show that anatomical modeling is substantially more mature than quantitative BE analysis: whole-hand bone segmentation achieves strong performance, whereas BE segmentation remains a major open challenge. By providing multi-level annotations spanning anatomical structure modeling, quantitative lesion analysis, and clinically grounded SvdH scoring, RAM-H1200 provides a comprehensive benchmark resource for RA analysis on hand radiographs. Benchmark \& Code: https://github.com/YSongxiao/RAM-H1200 Dataset Repository: https://huggingface.co/datasets/TokyoTechMagicYang/RAM-H1200-v1

We present a method of quantum composite hypothesis testing with small error, which enables us to establish quantum lower bounds in nonparametric statistics and high-dimensional functional estimation. This is achieved by introducing trigonometric polynomials into the quantum polynomial method, thereby generalizing the quantum phase-estimation lower bound of Mande and de Wolf (ESA 2023). As applications, we settle the quantum complexities of several problems in property testing and functional estimation with small error: - For $\ell_2$-closeness testing, we show that the approach of Luo et al. (*IEEE Trans. Inf. Theory* 2024) is optimal. - For Tsallis entropy estimation, where $q=2$ corresponds to the Gini impurity, we show that the approaches of Buhrman et al. (*Phys. Rev. Lett.* 2001) and Ekert et al. (*Phys. Rev. Lett.* 2002) are optimal for integer $q \geq 2$, and that the approach of Chen et al. (ICALP 2026) is near-optimal for real $q \geq 1.5$. - For pure-state Uhlmann fidelity and trace distance estimation, we show that the approach of Wang (*IEEE Trans. Inf. Theory* 2024) is optimal.

Best arm identification, also known as pure exploration, is a fundamental problem in the multi-armed bandit (MAB) model. Recent quantum algorithms achieve $O(n/\Delta_{[2]})$ query complexity for identifying the best arm with high constant probability, but all of them require $\Omega(\log(1/\Delta_{[2]}))$ or $\Omega(\log n)$ adaptive rounds. Motivated by the practical cost of adaptivity on near-term quantum hardware, where each round incurs substantial overhead from state re-preparation and classical--quantum communication, we study quantum best arm identification with no or very limited adaptivity. On the lower bound side, we prove that any non-adaptive quantum algorithm must make $\Omega(n\log n)$ queries when $\Delta_{[2]}=O(1)$, establishing a separation between adaptive and non-adaptive algorithms. To complement this lower bound, we present an algorithm that identifies the best arm with high constant probability using $O(n/\Delta_{[2]})$ queries and only $\log^*(n)+1$ adaptive rounds. En route to the lower bound, we show that the instances and techniques used to prove adaptivity lower bounds for classical bandits, such as those of Agarwal et al. [COLT'17], do not extend to the quantum setting. As such, we employ the polynomial method to prove our adaptivity lower bounds. To the best of our knowledge, this represents the first application of the polynomial method in this context, which may be of independent interest.

Atomic properties such as partial charges or multipoles encode chemically meaningful information that can inform downstream molecular property prediction, but their evaluation as machine learning targets has been complicated by the absence of a principled out-of-distribution evaluation protocol at the atomic level. In this work, we propose a held-out evaluation protocol that clusters atomic environments by SOAP descriptors and computes metrics accounting only for cluster labels unseen during training. Following this procedure, we use 5×5 cross-validation and Tukey's HSD to run a statistically rigorous comparison of E(3)-equivariant against non-equivariant, rotationally augmented models for predicting electron populations and multipoles of H, C, N, and O atoms. Building on our results, we introduce the Quantum Topological Neural Network (QT-Net), a rotationally augmented, non-equivariant graph neural network. We show that QT-Net can be used to infer properties of atoms in molecules from QM9 outside our training set, and that these inferred properties can yield improvement when used as input features for downstream molecular property prediction. To further validate the framework, molecular dipole moments computed from QT-Net's per-atom outputs recover the ground-truth values reported in QM9. We release all code and data, including a JAX implementation of QT-Net, to support the broader use of learned QTA properties as inductive biases for atomic-scale molecular machine learning.

Biomolecular generators are often adapted with reward feedback to improve task-specific utility, but pushing utility alone can concentrate generation on a narrow family of candidates. Maintaining diversity is difficult because sample diversity is a set-level property. We introduce Supergroup Relative Policy Optimization (SGRPO), a flexible GRPO-style framework that directly constructs rewards from set-level diversity. For each condition, SGRPO samples a supergroup of candidate sets, compares their diversity under the same condition, and redistributes the group diversity reward to individual rollouts through leave-one-out diversity contributions before combining it with rollout-level utility. This design decouples SGRPO from a particular generator, utility reward, or diversity metric, and allows instantiation with different GRPO-style approaches. We evaluate SGRPO on de novo small-molecule design, pocket-based small-molecule design, and de novo protein design, instantiating it with both GRPO and Coupled-GRPO across autoregressive and discrete diffusion generators. Across decoding sweeps, SGRPO expands the utility-diversity Pareto frontier and achieves the best frontier-level metrics relative to pretrained generators, GRPO, and memory-assisted GRPO when applicable. Our analyses further show that direct set-level diversity rewards remain effective with small groups and help preserve broader generation-distribution coverage during post-training. The code is available at https://anonymous.4open.science/r/SGRPO/README.md.

Shapley and Banzhaf interactions capture the complex dynamics inherent in modern machine learning applications. However, current estimators for these higher-order interactions trade off between speed and accuracy. To overcome this limitation, we introduce ProxySHAP. ProxySHAP reconciles the high sample efficiency of tree-based proxy models with a principled path to consistency via residual correction. On a theoretical level, we derive a polynomial-time generalization of interventional TreeSHAP to compute exact interaction indices for tree ensembles, successfully bypassing exponential tree-depth dependencies in prior methods. Furthermore, we formally analyze the residual adjustment strategy, characterizing the specific conditions under which Maximum Sample Reuse (MSR) corrects proxy bias without its variance scaling exponentially with interaction size. Extensive benchmarking demonstrates that ProxySHAP sets a new state-of-the-art standard for approximation quality, including in large-scale applications with thousands of features. By achieving the lowest error in both small- and large-budget regimes, ProxySHAP significantly outperforms the prior best estimators ProxySPEX and KernelSHAP-IQ, while also delivering superior performance on downstream explainability tasks.

Computing Shapley values exactly requires exponentially many evaluations of a cooperative value function, making approximation essential in high-dimensional applications. Recent leverage-score-based methods yield provably accurate estimators with sample complexity scaling linearly in the number of players, but this can still be too costly when only a small subset of players is truly influential. We study Shapley value estimation under a sparse influence model, where the Shapley vector is supported on an unknown set of size $s \ll n$. Using the regression characterization of Shapley values, we propose a sparse leverage sampling scheme and analyze it via sparse matrix concentration. Our main result shows that the resulting estimator achieves relative-error approximation with $\widetilde{O}(s^2)$ value function evaluations, up to logarithmic and accuracy factors, thereby replacing the ambient-dimension dependence of prior guarantees with dependence on the intrinsic support size. Empirically, our method matches or improves over dense Leverage SHAP on sparse synthetic and real-data benchmarks, while exhibiting better numerical stability at small sample sizes.

Vector quantization via random projection followed by scalar quantization is a fundamental primitive in machine learning, with applications ranging from similarity search to federated learning and KV compression. While dense random rotations yield clean theoretical guarantees, they require $\Theta(d^2)$ time. The randomized Hadamard transform $HD$ reduces this cost to $O(d \log d)$, but its discrete structure complicates analysis and leads to weaker or purely empirical compression guarantees. In this work, we study a variant of this approach: \emph{dithered quantization} with a single randomized Hadamard transform. Specifically, the quantizer applies $HD$ to the input vector and subtracts a random scalar offset before quantizing, injecting additional randomness at negligible cost. This approach is used by popular quantizers such as RaBitQ, and is a slight modification of others, including TurboQuant and EDEN. We prove that this approach provides mean squared error bounds that asymptotically match those achievable with truly random rotation matrices. For example, we prove a dithered version of TurboQuant achieves mean squared error $\bigl(\pi\sqrt{3}/2 + o(1)\bigr) \cdot 4^{-b}$ at $b$ bits per coordinate, where the $o(1)$ term vanishes uniformly over all unit vectors and all dimensions as the number of quantization levels grows.

Performing accurate protein-glycan docking is critical to understanding protein-glycan interactions which govern many biological processes. However, this important task is hampered by the lack of tailored datasets, benchmarks and models for protein-glycan docking. To fill this blank, we first curate a standard benchmark **ProtGlycanDock** for training and evaluating protein-glycan docking models under both protein and glycan generalization settings. On this benchmark, we evaluate ten models in three categories, physics-based docking methods, machine-learning-based docking models and molecular structure prediction models. To obtain more precise binding poses, we further propose two finetuning techniques to inject the knowledge of glycan conformations into a general-purpose molecular structure prediction model. These two techniques respectively augment model inputs with glycan topological features and refine glycan binding pose with a glycan energy loss. Benchmark results show the superiority of AlphaFold~3 among existing methods, and the two proposed finetuning techniques can be well combined to achieve a state-of-the-art model for protein-glycan docking. Also, we perform stratified benchmark analysis and case studies to better understand the strengths and weaknesses of current models, providing insights for future model improvements. We maintain datasets at https://huggingface.co/datasets/ProtGlycanDock/ProtGlycanDock and source codes at https://github.com/kasawa1234/ProtGlycanDock.

Astrophysical observations from Earth are subject to weather, environmental, and scientific constraints that lead to characteristically sparse and unevenly sampled multi-wavelength time series (light curves), severely limiting the performance on downstream tasks. Interpolating light curves help in mitigating this problem, where Gaussian Processes (GPs) are the current standard. However, GPs struggle with cross-wavelength correlations, require *a priori* kernel specification, and cannot handle ensembles of diverse transient and variable classes. Furthermore, GPs and other existing models (*e.g.*, Bayesian neural networks), scale poorly and must be fit to each light curve individually, transferring no information between light curves. We introduce Attentive Neural Processes (ANPs) for light curve reconstruction, combining the probabilistic framework of GPs with the scalability of deep learning, and we demonstrate their use as a fast, data-driven, universal light curve interpolator. Unlike prior processes tailored to single object classes, our approach is morphologically agnostic. By meta-learning on a diverse population of simulated transients, ANPs shift the computation cost to the training phase, enabling amortized inference - particularly important at the eve of the Vera C. Rubin Observatory Legacy Survey of Space and Time (LSST), which will produce massive datasets for transient astrophysics. We evaluate our models on realistic LSST cadences for its 6-filters across 14 transient classes. The ANP outperforms all our benchmark models, including a suite of GP and neural-network based models, on all regression quality, feature recovery, and rigorous probabilistic calibration metrics. Our ANP performs probabilistic interpolation on all filters in $\sim 10^{-6} $ seconds -- over four orders of magnitude faster than the next-best neural benchmark and over five orders of magnitude faster than GPs -- making them suitable for on-line use on the LSST alert stream estimated to carry $\sim10^{7}$ events/night. The ANP avoids neural networks' overconfidence and GPs' underconfidence, delivering sharp, highly calibrated uncertainties. This work establishes the neural process family as a scalable, probabilistic foundation for real-time transient science in the LSST era.

Evaluating physical reasoning in video models is difficult because absolute motion measurements depend on frame rate, object scale, and camera calibration, all of which are often ambiguous or unavailable in generated video. We propose a different approach. When two objects in the same scene obey the same physical law, their motions must satisfy predictable relationships, and these relationships hold independent of calibration. We introduce Principia, a benchmark that evaluates Newtonian physics through relational consistency between paired objects. Principia spans eight phenomena—gravity, restitution, friction, rotational inertia, projectile motion, momentum, pendulum, and mass-spring oscillation—across translational, rotational, collisional, and oscillatory dynamics, using real-world scenes recorded under controlled protocols. We also introduce a calibration-independent consistency score that quantifies physical violation directly in image space. Across thousands of generations from five state-of-the-art video generators, no model exceeds 0.42 on Principia despite all scoring above 0.7 on VBench. Vision-language models show inverse failure profiles, succeeding where generators fail and vice versa.

This paper presents a post-Bayesian approach to online filtering in nonlinear state-space models, capable of avoiding over-confident inferences in settings where either the dynamical model, the measurement model, or _both_, could be misspecified. This is addressed using _predictively oriented_ (PrO) posteriors, an emerging paradigm in which learning (i.e., posterior concentration) occurs if and only if the overall model is well-specified, without strict adherence to the Bayes' theorem. As the characterisation of PrO posteriors is challenging, our main technical contribution is a fast approximate linear-Gaussian update procedure, analogous to an (iterated) extended Kalman filter (EKF). The methodology, which we call PrO-EKF, has no tunable hyper-parameters and has a computational cost comparable to that of existing filtering methods. Performance is empirically assessed on a range of linear and non-linear applications, in which the state-space model is systematically misspecified.