NeurIPS 2026 论文列表

Diffusion Maps (DM) is a well studied and popular non-linear dimension reduction algorithm. Surprisingly, while the convergence properties of DM are well understood, the {\em geometric} properties of the embedding, such as reach and smoothness have not yet been studied. Under a set of standard assumptions on a family of submanifolds $\subset \mathbb{R}^D$, we derive a series of geometric properties that are preserved by DM, including almost uniform density, finite polynomial approximation and reach. Leveraging these properties, we establish rigorous bounds on the embedding errors introduced by the DM algorithm, of the order $(\frac{\log n}{n})^{\frac{1}{8d+16}}$. These results offer a solid theoretical foundation for understanding the performance and reliability of DM in practical applications.

Understanding the behavior of stochastic gradient methods is a central problem in modern machine learning. Recent work has highlighted diagonal linear networks as a simplified yet expressive setting for analyzing the optimization and generalization properties of neural models. In this work, we show that in the high-dimensional regime, stochastic gradient descent on diagonal linear networks is well-approximated by continuous dynamics governed by a stochastic differential equation (SDE), which explicitly decouples the drift from the gradient noise. We further derive a deterministic partial differential equation whose solution propagates the relevant state of the iterates and characterizes the time evolution of a broad class of observable statistics, including the risk, curvature, and other metrics for optimality. To the best of our knowledge, this is the first deterministic high-dimensional description of SGD for a genuinely nonlinearly parametrized model. Finally, we show that, under a suitable parametrization, the stochastic dynamics are globally well posed and converge exponentially fast to zero risk with high probability, yielding a fully explicit non-asymptotic description of their long-time behavior. Numerical simulations corroborate our theoretical findings.

Video generation models have achieved remarkable visual fidelity and have strong potential to become general-purpose world simulators. Despite this progress, they still fail to generate videos which adhere to laws of physics. The problem becomes even more apparent in realistic settings where multiple physical principles must work together within the same video; for example, "a balloon floating upward while steam rises from a pot" requires buoyancy and fluid dynamics to unfold coherently and simultaneously. Yet existing methods largely ignore multi-principle interactions, focusing on a single principle per video. We propose HiPhy (Hierarchical Physical Alignment), a reinforcement learning framework that grounds video generation in physical laws through a dual-level objective: locally enforcing the temporal dynamics of individual physical principles, and globally ensuring the physical and semantic coherence of the entire scene. To support multi-principle generation, we construct a 50K-prompt dataset and introduce a prompt benchmark MultiPhyBench, spanning a diverse range of co-occurring physical events. Our experiments show that HiPhy significantly outperforms prior methods and baselines, improving physical commonsense and semantic alignment significantly across various benchmarks, with the largest gains on scenes involving multiple concurrent physical principles where competing methods degrade most sharply.

Parametric human models capture global pose but overlook fine-grained surface dynamics. Generic scene flow estimates dense motion but struggles with articulated humans and lacks 4D ground truth. To bridge this gap, we introduce H-Flow, a dense 4D motion representation that captures non-rigid deformations beyond skeletal kinematics. We estimate H-Flow from monocular video via a unified multi-head architecture that jointly perceives pose, depth, and flow. To overcome the scarcity of dense motion annotations, we propose a self-supervised learning paradigm that embeds geometric, structural, and biomechanical priors into optimization. These cross-modal constraints tightly couple pose, depth, and flow, so that improving any one modality simultaneously drives the others toward consistency. For evaluation, we present DynAct-4D, a high-fidelity synthetic benchmark providing dense 4D ground truth for complex human movements. Extensive experiments show that our method outperforms state-of-the-art scene flow baselines. Furthermore, H-Flow serves as an effective motion primitive, bringing substantial improvements to downstream tasks including action recognition and video generation. Models, code, and resources will be released upon publication.

Spatial and temporal resource constraints are critical for both biological and artificial intelligent systems. Here we define differentiable cost terms for breadth, depth, and time within a recurrent convolutional neural network conceived as a finite subset of an infinite lattice architecture. We optimize these costs jointly with task errors via backpropagation, and efficient computational graphs emerge organically through training. We find that breadth, depth, and time can be traded off against each other to achieve a given level of performance. Networks grow in all three dimensions with task complexity and spontaneously take more recurrent steps when inputs are occluded. Surprisingly, time used by the model correlates with human reaction times in an object recognition task. Our framework provides a normative account of how resource constraints shape neural architectures, connecting to questions about brain design in neuroscience, and may help illuminate the diversity of neural solutions found in nature.

The *Game of Twenty Questions* is a classical problem where a questioner must identify a hidden item by asking binary questions. Although simple in form, the game is strategically challenging, especially when the questions are expressed in natural language. It is also a useful abstraction for realistic information-seeking tasks, such as medical diagnosis and troubleshooting. Existing approaches often rely on simplifying assumptions that degrade *worst-case* performance. This is an issue with serious implications in high-stakes applications. In this work, we introduce *Strategic Language Search* (SLS), an adversarial counterpart of Twenty Questions, and formalize it and its variants as two-player zero-sum extensive-form games. We then propose *Game of Thought* (GoT), a framework that applies game-theoretic techniques to approximate a Nash equilibrium (NE) strategy for the restricted variant of the game. Empirical results demonstrate that our approach consistently improves worst-case performance compared to (1) direct prompting-based methods and (2) heuristic-guided search methods across all tested settings.

Neural operators provide fast surrogates for PDEs but their deterministic predictions limit their use in tasks requiring uncertainty quantification (UQ), especially under geometric variability. Existing approaches primarily model uncertainty in network parameters, largely overlooking the geometry-aware representations learned by the operator itself. We propose REEF-GP (Residual on Embedded Features Gaussian Process), a post-hoc UQ framework that fits a GP to the residuals of a frozen neural operator whose internal embeddings define the kernel feature space. Rather than learning a separate feature map, REEF-GP adapts the operator’s intrinsic coordinate-feature representations to construct geometry-aware uncertainties. To ensure stability and scalability on unstructured domains, REEF-GP incorporates spectral-normalized projections, heteroscedastic geometry-aware noise, and efficient subset-based training that avoids restrictive low-rank approximations. Across five PDE benchmarks with varying geometries, REEF-GP preserves predictive accuracy while achieving calibrated uncertainty estimates competitive with deep ensembles but at a fraction of their cost. Our approach remains robust under geometric distribution shift, with uncertainty concentrating in physically meaningful regions (e.g., shock fronts). Our results demonstrate that accurate and scalable post-hoc UQ for neural operators can be achieved directly in their learned feature space, offering a practical alternative to parameter-centric approaches.

Neural surrogate models for Partial Differential Equations (PDEs) on unstructured 3D geometries are often limited by poor generalization and the high cost of generating large-scale training datasets. Consequently, pre-training has emerged as a critical alternative to enhance the robustness and scalability of these models. In this work, we introduce a pre-training framework tailored to both steady-state and transient regimes. For steady-state problems, we propose a geometry-driven strategy that leverages intrinsic shape descriptors to learn representations of complex 3D domains. For transient problems, we introduce a physics-driven approach based on online generation of synthetic PDE data, enabling scalable pre-training without reliance on expensive datasets. This framework learns transferable features for challenging downstream tasks. Across multiple experiments, our approach achieves up to 3$\times$ faster convergence, 2$\times$ greater data efficiency, and up to 25\% higher accuracy during fine-tuning, particularly under realistic low-data regimes. This methodology provides a practical pathway toward data-efficient neural emulators for large-scale simulations.

In urban low-altitude flight, buildings reshape ambient wind into spatially varying 3D flow, making unmanned aerial vehicle (UAV) energy depend on local wind exposure as well as path length. However, building-resolved wind information is rarely available when a mission must be planned. Computational fluid dynamics (CFD) can produce high-fidelity urban flow fields, but each simulation is tied to a fixed inflow boundary condition and can take hours to days, which is incompatible with urban UAV missions that typically last minutes to tens of minutes. We present GeoWind2Plan, a geometry-to-wind-to-planning framework for mission-time 3D urban wind prediction and energy-efficient UAV planning. Given only a background wind vector, 3D building geometry, and a start-goal pair, GeoWind2Plan transforms the building geometry into a reference-wind frame, predicts mission-relevant 3D wind patches with a localized geometry-conditioned neural operator, stitches them into a queryable local wind field, and optimizes a feasible 3D path and speed profile using a physically grounded UAV energy model. Rather than pursuing CFD-perfect reconstruction, GeoWind2Plan targets decision-useful wind prediction: trajectories are planned with predicted wind and evaluated under high-fidelity CFD wind. Across held-out urban domains, wind speeds, and mission wind-angle regimes, GeoWind2Plan performs corridor-localized wind inference in about 3 seconds, compared with roughly 8 hours for CFD. Under CFD evaluation, trajectories planned with GeoWind2Plan reduce energy by 6.9%, 12.7%, and 4.5% in tailwind, headwind, and crosswind missions relative to wind-agnostic planning, recovering 87.9%, 85.7%, and 75.0% of CFD-reference savings. These results show that fast, corridor-localized 3D urban wind prediction can make wind-aware UAV energy planning practical at mission time.

We present GeoTransolver, a multiscale geometry-aware physics attention transformer for Computer Aided Engineering (CAE). GeoTransolver extends the Transolver backbone with GALE (Geometry-Aware Latent Embeddings) attention, which pairs physics-aware self-attention on learned state slices with cross-attention to a shared geometry and global context computed via multi-scale ball queries (inspired by Domino) and reused in every block. Implemented and released in NVIDIA PhysicsNeMo, GeoTransolver persistently projects geometry and global parameters, into physical state spaces to anchor computations to domain structure and operating regimes. We benchmark on DrivAerML, SHIFT-SUV, and SHIFT-Wing against Domino, Transolver (PhysicsNeMo implementation), and literature-reported AB-UPT, evaluating drag/lift $R^2$ and relative $L_1$ errors on field variables. As an additional nonlinear structural mechanics application, we also report Transolver and GeoTransolver results on bumper-beam and full-vehicle Body-in-White (BIW) crash-dynamics benchmarks, evaluating relative $L_2$ trajectory error and probe-level kinematic MSE. GeoTransolver delivers improved accuracy, robustness to geometry and regime shifts, and favorable data efficiency; we include DrivAerML ablations and qualitative contour and design-trend results, advancing operator learning for high-fidelity surrogates on complex, irregular, non-linear domains.

Worldwide image geolocalization aims to predict precise GPS coordinates for images captured anywhere on Earth, which is challenging due to the large visual and geographic diversity. Recent methods mainly follow two paradigms: retrieval-based approaches that match queries against a reference database, and generation-based approaches that directly predict coordinates using Large Vision-Language Models (LVLMs). However, we observe distinct error profiles between them: retrieval excels at fine-grained instance matching, while generation offers robust semantic reasoning. This complementary heterogeneity suggests that no single paradigm is universally superior. To harness this potential, we propose GeoRouter, a dynamic routing framework that adaptively assigns each query to the optimal paradigm. GeoRouter leverages an LVLM backbone to analyze visual content and provide routing decisions. To optimize GeoRouter, we introduce a distance-aware preference objective that converts the distance gap between paradigms into a continuous supervision signal, explicitly reflecting relative performance differences. Furthermore, we construct GeoRouting, the first large-scale dataset tailored for training routing policies with independent paradigm predictions. Extensive experiments on IM2GPS3K and YFCC4K demonstrate that GeoRouter significantly outperforms state-of-the-art baselines. Our dataset, code, and checkpoints are publicly available to facilitate future research.

Panoramic 3D reconstruction enables the holistic recovery of complete 360° scene geometry, offering significant benefits for robotics and AR/VR applications; however, direct estimation is persistently challenged by inherent spherical distortions. While decomposing panoramas into perspective views effectively circumvents these distortions and leverages established geometric priors, this strategy often fails to preserve global spatial consistency across the decomposed views. To address this fundamental limitation, we introduce an enhanced feed-forward 3D reconstruction framework tailored for the view decomposition paradigm, driven by two key innovations. First, we propose a geometry-aware attention aggregator, featuring a streamlined attention architecture equipped with a novel Geo-Attention mechanism, which explicitly exploits the intrinsic global information to guide multi-view interactions, thereby suppressing geometrically inconsistent matches. Furthermore, to jointly enforce fine-grained local details and global structural coherence, we formulate a hybrid supervision strategy. Specifically, we complement the standard local point cloud loss in perspective space with our newly proposed spatially uniform global point cloud loss in panoramic space, explicitly enforcing holistic constraints within a unified coordinate system. Experiments demonstrate that our method consistently outperforms prior single-view panoramic approaches in reconstruction accuracy, geometric consistency, and visual fidelity.

Molecules in equilibrium follow a Boltzmann distribution, making the underlying energy landscape a physically grounded modeling objective. However, such landscapes are difficult to learn from data and, once learned, hard to sample from. Diffusion and flow-matching models sidestep these difficulties by learning a time-conditional score or transport field between noise and data, losing the energy inductive bias in exchange for a more tractable training objective. We introduce EBMol, an energy-based model (EBM) that restores this inductive bias by learning an atom-additive scalar potential without explicit simulation during training. Our method employs a flow-inspired Restoring Field Matching objective to approximate the energy landscape. We adopt the Mirror-Langevin algorithm for sampling, enabling unified updates of atomic positions and types, and incorporate parallel tempering for inference-time compute scaling. EBMol is the first EBM for 3D molecular generation to achieve state-of-the-art performance on QM9 and GEOM-Drugs. Moreover, we show that the learned energy landscape serves as a principled quality metric for ranking and filtering configurations, and demonstrate controllable generation without retraining through shape-steered sampling via potential composition and zero-shot linker design.

Operator learning for partial differential equations (PDEs) aims to learn solution operators on infinite-dimensional function spaces from finite-resolution data. In this setting, it is important for the learned model to be discretization-invariant, or resolution-robust, and to reflect PDE-specific structure. It is therefore natural to ask how such structure should be encoded in the model architecture, hypothesis class, or learning procedure. In this paper, we study operator learning for solution operators of nonlinear parabolic PDEs based on Duhamel--Picard iteration. We formulate Picard iteration as an abstract state-transition model and present a theoretical framework for Picard-type operator learning. We derive implementation-agnostic generalization error bounds that separate the implementation error from the estimation error associated with the abstract state-transition model induced by Picard iteration. A key consequence is that increasing the Picard depth reduces the Picard truncation error without causing an unbounded growth of the entropy-based estimation error. We also extend the analysis to long-time prediction by rolling out the same learned local model over successive time blocks. Finally, we illustrate the theory for nonlinear heat equations on the torus using a Picard-type Fourier neural operator as a concrete implementation.

Learning the dynamics of interacting physical systems is a central challenge in machine learning for the physical sciences. Prevailing architectures such as Graph Neural Networks (GNNs) and feedforward models often fail to explicitly capture the compositional nature of physical laws. Instead of modeling the forces that govern a system, these networks learn to approximate aggregate state transitions. As a result, they lack a causal understanding of the underlying physics interactions, leading to poor generalization in out-of-distribution systems with varying numbers of entities or novel configurations. In this work, we introduce Compositional Potential Minimization (CPM), a framework that casts simulation as an energy minimization procedure over a composition of learned force potentials. Each force component is represented as a separate, interpretable energy landscape, and interaction dynamics emerge from their addition. Inference in CPM is framed as a global energy minimization problem over an arbitrary number of composed functions. Because CPM learns to disentangle separate forces during training, it enables seamless generalization during test time to larger-scale systems and unseen configurations through straightforward compositions of the learned energy components. Our experiments show that CPM significantly outperforms previous state-of-the-art GNN-based solutions in 2D, 3D, and particle-based simulations.

Electromagnetic inverse scattering (EIS), which aims to recover object permittivity from measured scattered fields, is central to a wide range of applications, from medical diagnostics to security screening. However, it is a highly nonlinear and ill-posed problem. Despite substantial progress, existing approaches face a fundamental trade-off between physical fidelity, efficiency, and generalization: data-driven methods are fast but tied to training geometries and sensor setups, while physics-based optimization is accurate but computationally expensive. We introduce a Gaussian-Galerkin (GauGal) method that reformulates EIS from a dense point-wise field reconstruction to a compact, primitive-level physical solving. Rather than using Gaussian primitive merely as a material representation, GauGal take it as the computational units for wave scattering. By projecting the continuous EIS scattering equation into a Gaussian primitive space, the material modulation, Green propagation, source excitation, and receiver observation are all realized as primitive-level operators. This yields a compact differentiable forward solver for physics-consistent reconstruction. Our method achieves state-of-the-art accuracy on synthetic and real benchmarks while reducing runtime from over 30 minutes by leading physics-driven baselines to under 10 seconds. It also generalizes significantly better to unseen geometries and sensor configurations than leading data-driven baselines. Overall, this framework establishes an accurate, efficient, and generalizable paradigm for electromagnetic inverse imaging, enabling fast, physics-consistent imaging in practical settings.

Adaptive volumetric finite element meshing is a critical step in computer-aided engineering and analysis that dictates the computational budget of a given problem. It traditionally requires iterative PDE solvers or heavily supervised, data-driven surrogates trained on large-scale simulation data. While Multimodal Large Language Models (MLLMs) excel in 2D visual tasks, their zero-shot capability to semantically ground regions based on geometric understanding and physics remains an open question. Overall, this study explores a significant question: can the high-level semantic understanding of off-the-shelf MLLMs serve as a viable, zero-shot geometric proxy for finite element mesh refinement? To investigate this, we introduce **GReFEM** (Geometric Reasoning Enhanced Multimodal LLMs for Finite Element Meshing), a framework that utilizes MLLMs to visually localize stress-critical regions based on physics-guided textual prompts. To bridge the gap between 2D MLLM pre-training and 3D geometries, we introduce *orthoViews*, a view-selection module that maximizes the observability of key geometric features. We conduct an in-depth empirical evaluation across diverse CAD geometries, loading cases, and SOTA MLLMs, comparing them against a tuned geometric heuristic under a strict, matched refinement budget. Our findings reveal that MLLMs demonstrate robust zero-shot capacity to accurately follow complex spatial-physical instructions, isolating stress-relevant features with higher precision than blind heuristics. By mapping both the successes and current limitations of MLLMs in physical grounding, this study defines the frontier of foundation models as semantic assistants in automated simulation workflows.

Accurate characterization of deep subsurface structures is a fundamental problem in geoscience, but geophysical inversion is inherently non-unique and therefore requires joint interpretation of gravity, magnetic, electromagnetic, and seismic observations. Despite recent advances in deep learning for geophysical inversion and PDE-based scientific modeling, existing large-scale open datasets are largely limited to a single modality and rarely provide physically consistent multi-source responses generated from the same 3D geological target. This limitation hinders the development and evaluation of multimodal joint inversion methods for deep subsurface characterization. We introduce GEMS-3D, a large-scale synthetic benchmark for 3D geological multiphysics learning. GEMS-3D couples four forward modeling engines—$\textbf{g}$ravity, $\textbf{e}$lectromagnetic, $\textbf{m}$agnetic, and 3D acoustic $\textbf{s}$eismic—to generate sample-aligned observations from shared geological realizations. Guided by facies-aware rock-physics relationships, the dataset provides spatially aligned volumes of P-wave velocity ($v_p$), density ($\rho$), resistivity ($\phi$), and magnetic susceptibility ($\chi$) across nine representative classes of complex geological anomalies, including dike swarms, salt domes, and gas reservoirs. Each sample includes gravity responses, magnetic responses, electromagnetic responses, and 3D acoustic seismic data, together with acquisition metadata and anomaly labels. By packaging co-registered property volumes, anomaly support, acquisition metadata, and aligned multiphysics responses as sample-level bundles, GEMS-3D provides a unified benchmark for operator learning, multimodal inverse interpretation, and missing-modality robustness in deep subsurface settings. The anonymized forward-modeling framework and corresponding dataset link is available at https://anonymous.4open.science/r/GEMS-3D-E4CA.

LLM-based coding agents are usually evaluated in familiar software settings: mainstream languages, common libraries, and public repositories. These benchmarks remain important, but they can hide how agents behave when the language itself is unfamiliar. We evaluate six contemporary coding agents on four esoteric programming languages using a sequential setup with file editing, local execution, and hidden-test grading. Our protocol exposes capability differences between these agents that mainstream coding and agentic benchmarks such as SWE-Bench Verified and Terminal-Bench 2.0 compress into much narrower bands. We observe that the strongest agents, Claude Opus 4.6 and GPT-5.4 xhigh, often avoid writing the target language directly. On Brainfuck and Befunge-98, they write Python programs that generate target-language code and debug those generators locally. Forbidding this metaprogramming strategy causes large performance drops. Text guidance distilled from this strategy does not materially improve weaker agents. In contrast, Opus-derived Python helper code for building generators, with no solved benchmark programs or hidden-test answers, sharply improves Sonnet 4.6 and GPT-5.4 mini on the same problems, while Haiku 4.5 remains low. More interpreter calls and output tokens improve stronger agents but leave weaker agents near their original performance, indicating that these resources amplify useful strategies rather than create them. Together, these results show that strong coding agents adapt to unfamiliar languages by using tools, feedback, and workspace state to build a working model of the target language. Metaprogramming is the clearest case, but the broader gap is constructing and debugging a strategy that works under the target language's rules.