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5 papers in the last four weeks, against 1 the four weeks before. 0.1% of all new papers.

Jul 6Week of Sep 21

Latest papers 22

Sep 29, 2026cs.LG

Geometry-physics confounding impairs PDE learning across varying domains

Learning partial differential equation (PDE) dynamics across varying domains is central to predictive modelling and data-driven discovery of governing equations. However, geometric variation alters both field representation and the governing differential operators, confounding geometric effects with intrinsic physical properties in the observed dynamics. This work identifies geometry-physics confounding as a unified failure mechanism for PDE learning across varying domains. In forward operator learning, this confounding increases the burden of inferring geometry-dependent operator changes from finite data, reducing data efficiency and generalisation. In equation discovery, omitting geometry-induced operators misspecifies the candidate library, leading to biased parameters, missed governing terms and spurious terms. We propose a de-confounding framework that makes the known geometry-to-operator transformation explicit. Geometry-induced coefficient fields improve prediction and data efficiency across five operator-learning benchmarks, while geometry-complete candidate libraries recover the generating equations and reduce held-out PDE residuals by more than two orders of magnitude in both evolving-domain systems. By separating known geometric action from intrinsic physics, the proposed framework supports more reliable and data-efficient PDE learning across scientific and engineering problems with varying geometries.

Yinghao Cheng, Gengxiang Chen, Xu Liu +6

College of Mechanical & Electrical Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, China · School of Electronics, Electrical Engineering and Computer Science, Queen’s University Belfast, Belfast, United Kingdom · School of Mechanical and Aerospace Engineering, Queen’s University Belfast, Belfast, United Kingdom +3

Partial Differential Equations Probabilistic Operator Learning Physics-Informed Learning Scalar Field Visual Geometry Grounded Transformer Latent Confounders+1

Sep 27, 2026cs.LG

Dynamic Kuramoto-Hodge Operators for PDEs on Complex Geometries and Topologies

Learning PDE operators on complex domains requires capturing interactions among fields on vertices, edges, and faces, alongside global responses shaped by topology. Existing neural operators accommodate irregular geometries but often overlook these distinct field supports or their condition-dependent coupling. We introduce the Dynamic Kuramoto--Hodge Operator (DKHO), which combines topology-constrained interactions with learned coordination. DKHO encodes conditions on their native cochain supports, evolves Kuramoto-inspired relation states through the boundary and coboundary operators that compose the Dirac operator, and decodes non-harmonic and harmonic responses in orthogonal Hodge subspaces. Topology thus determines where information can flow, while learned dynamics adapts how it is exchanged to each PDE instance. Across porous-medium Darcy flow, torus transport--diffusion, and cavity magnetostatics, DKHO-large reduces prediction error by approximately 61% on average over leading baselines, while DKHO-small remains competitive using only 11.5--24.3% as many parameters. These results suggest that coupling topological structure with adaptive dynamics provides an effective inductive bias for accurate and parameter-efficient PDE operator learning on complex geometries and topologies.

Xiang Li, Yue Song

Tsinghua University · Beijing University of Chemical Technology

Neural Operators Partial Differential Equations Scalar Field Topology

Sep 24, 2026cs.LG

Image Fidelity is Not Field Fidelity: Joint Thermodynamic Reconstruction and Error Localization in Neural Tomography

Neural fields for scientific tomography are optimized from 2D images, but the actual quantity of interest is often a latent 3D physical field. Because the forward map is many-to-one, low 2D image error need not certify a correct 3D field. Moreover, the latent field is not directly supervised during training, and its error cannot be evaluated against truth at deployment. We develop CoroNeRF to jointly optimize 3D electron density and temperature fields directly from multiview, multiline intensities through a differentiable atomic-emission renderer. Using solar coronal tomography as a controlled testbed, we evaluate physical-field recovery and test whether cross-seed instability provides a ground-truth-free-at-inference indicator of local physical-field error. We underscore the following two observations. (i) Image fidelity is not field fidelity: spectral ablations show that limited-channel reconstructions can fit their available observations well while recovering substantially worse fields, whereas evaluation on a common richer probe exposes the discrepancy. (ii) Cross-seed instability ranks local physical-field error across tested matched-model conditions, supported by sparsification and physical signal-strength controls. Seed-deviation projections provide complementary directional validation, but shared forward-model mismatch can still produce incorrect cross-seed consensus. These results characterize joint thermodynamic recovery and the usefulness and limits of seed-based error localization in a controlled, single-scene solar tomography testbed.

Alan Hsu, Jenna Samra, Alin Razvan Paraschiv +1

Harvard University Cambridge, MA, USA · Center for Astrophysics | Harvard & Smithsonian Cambridge, MA, USA · National Solar Observatory Boulder, CO, USA

Cone-Beam Computed Tomography Neural Fields Scalar Field Thermodynamics Localization

Sep 15, 2026cs.LG

Can Deep Learning Achieve Cross-Physics Mapping?

Can deep learning translate physical fields governed by fundamentally different equations? We address this question by introducing Cross-Physics Mapping (CPM), an operator-learning framework for mappings between heterogeneous physical domains. We formulate sufficient conditions for such mappings through compatible latent representations and propose a dimensionless scaling principle that aligns the characteristic evolution scales of the source and target systems without assuming their dynamical equivalence. As a representative test, paired diffusion and wave fields are generated independently from their respective parabolic and hyperbolic equations while sharing the same latent geometry, material heterogeneity, excitation, and dimensionless scale. Seven architectures-ResUNet, DeepONet, Fourier, latent, wavelet, U-shaped, and Galerkin neural operators-are evaluated for both diffusion-to-wave and wave-to-diffusion mappings. The results reveal a strong directional asymmetry. Diffusion-to-wave reconstruction is more challenging because it requires recovering wavefront, phase, and time-of-flight information attenuated by diffusion; U-NO performs best in this direction, achieving a relative

\ell_2

error of

0.307

and an

R^2

of

0.905

. Wave-to-diffusion mapping is considerably more stable, with GNO attaining a relative

\ell_2

error of

0.154

and an

R^2

of

0.935

. Neural operators generally outperform the conventional convolutional baseline, highlighting the nonlocal nature of cross-physics transformations. These findings demonstrate that deep learning can establish useful mappings between distinct physical modalities on a shared latent manifold, while the achievable accuracy remains fundamentally constrained by the direction-dependent information content of the governing physics.

Pengfei Zhu, Julien Lecompagnon, Mathias Ziegler

Bundesanstalt für Materialforschung and -prüfung (BAM), 12205 Berlin, Germany

Scalar Field Deep Learning Transform Concepts

Sep 7, 2026cs.LG

PCFlow: Physics-Conditioned Flow Matching for GPR B-Scan Image Synthesis

Ground-penetrating radar (GPR) B-scan image synthesis is important for data augmentation, algorithm validation, and simulation acceleration, yet generating radargrams with both visual realism and physical consistency remains challenging. Existing learning-based generative models often emphasize visual appearance but provide limited control over response geometry. In this paper, we propose PCFlow, a physics-conditioned flow matching framework for fast GPR B-scan image synthesis. The core of PCFlow is a Maxwell-informed dense physical condition field constructed from the parameterized physical model used for electromagnetic simulation, including material properties, target geometry, propagation cues, and response-domain priors. This condition field provides an interpretable interface between physical scene parameters and radar response geometry, and guides conditional flow matching in the VAE latent space toward physically feasible generation paths. We evaluate PCFlow on a gprMax-based buried-pipeline dataset with both in-distribution and out-of-distribution test cases. Experimental results show that PCFlow generates images with more accurate response geometry and high visual fidelity, demonstrating its effectiveness for controllable and physically faithful radar image synthesis.

Zhijie Shen, Chenchen Fu, Xuanhao Chang +2

College of Computer Science and Technology, Jilin University, Changchun 130012, China · Symbol Computation and Knowledge Engineering of the Ministry of Education, Jilin University, Changchun 130012, China

Physics-Guided Machine Learning Synthetic Aperture Radar Wireless Radiance Field Image Reconstruction Conditional Flow Matching Subsurface Scattering+2

Aug 31, 2026cond-mat.stat-mech

A Human-AI Theorem Connecting Spontaneous and Field-Induced Mechanisms of Collective Behavior in One Dimension

Can an artificial intelligence (AI) generate a scientific hypothesis outside a human collaborator's active hypothesis space (AHS), and can human-AI research be organized to make such breakthroughs more likely? We document such a case while proving a theorem that connects two basic organizing mechanisms of statistical physics: collective behavior arising in zero field from competing interactions and that induced or controlled by an external field. A zero-field

O(n)

-vector open chain with arbitrary inhomogeneous nearest- and next-nearest-neighbor interaction functions

U_i(S_i\cdot{S}_{i+1})

and

V_i(S_i\cdot{S}_{i+2})

is microscopically, via a temperature-independent mapping at the Hamiltonian level, equivalent to a simpler

O(n)

open chain with nearest-neighbor interaction

V_i(\boldsymbolσ_i\cdot\boldsymbolσ_{i+1})

and axial single-spin potential

U_i(σ_i^z)

for every integer

n\ge1

and every system size

L\ge1

. The homogeneous linear specialization maps the foundational frustrated

J_1

-

J_2

model onto the canonical

J

-

h

field model---with

n=1,2,3

being the Ising, XY, and Heisenberg classical spin models, respectively; the theorem resolved a longstanding challenge for

n=3

published in 1990. Its proof was done with an AI-synthesized recursive Householder moving frame and understood via a human-recognized hidden reciprocity. An analogous theorem holds when the continuous

O(n)

spins are replaced by the

q

-state Potts spins, implying a closed-form exact solution of the

J_1

-

J_2

standard Potts open chain for every

q\ge2

and every

L\ge1

. The emergence of these theorems from a human-AI co-development framework suggests that sustained AI involvement throughout a systematic research program may incubate autonomous scientific breakthroughs and make aspects of the discovery process experimentally testable.

Weiguo Yin

Condensed Matter Physics and Materials Science Division, Brookhaven National Laboratory, Upton, New York 11973, USA

Human-Ai Collaboration Spatial Photonic Ising Machines Collective Behaviors Scientific Agents Emergence Hypothesis+2

Aug 3, 2026cs.AI

The Field Knows: Cross-Dimensional Geometry from Navigation to Black Holes

We introduce a continuous metric field framework trained by a single causal contrastive loss. The framework encodes a scene into coefficients of a fixed symmetric matrix basis, assembles them into a Lie algebra element, and exponentiates the result to a Riemannian or Lorentzian metric. Across dimensions, this field discovers the full spectrum of geometric structures: from obstacle-avoiding geodesics in robot navigation across planar and manipulator configuration spaces, to event horizons of black holes in Lorentzian spacetime. Extensive zero-shot generalization studies demonstrate that the field captures transferable geometric structure rather than memorizing specific configurations. In the black hole setting, the causal loss spontaneously evolves genuine black-hole-like structures with the correct Lorentzian signature. The same loss, the same architecture, and the same training protocol produce the full range of geometric phenomena across dimensions. The field knows geometry, and geometry knows physics.

Chenghao Xu

School of Artificial Intelligence and Robotics and National Engineering Research Center of Robot Visual Perception and Control Technology, Hunan University, Changsha, Hunan, China

Geometric Metrics General Relativity Scalar Field Geometric Machine Learning Contrastive Loss Navigation

Jul 31, 2026cs.LG

SILVA Networks as Structured Implicit Layers and Vector Attractors via Dynamic Interaction Fields

Many learning problems require representations that reconcile direct input, nearby structure, and broader context. In implicit neural layers, these influences are usually absorbed into a single fixed-point update, making it hard to identify what enters from the stimulus, what propagates locally, what comes from global context, and what is produced by solver dynamics. Here we introduce SILVA Networks, Structured Implicit Layers and Vector Attractors via Dynamic Interaction Fields. SILVA separates stimulus, local interaction, global interaction, damping, and readout inside one fixed-point architecture. The same template is instantiated for images, molecules, citation networks, and long-range graph benchmarks through domain-specific definitions of nodes, neighborhoods, and global summaries. Experiments and ablations show task-dependent roles for these terms: local interactions are load-bearing in the graph tasks, MNIST gains little from recurrence at the tested capacity, and the clearest global benefit appears in a long-range node-classification benchmark. SILVA therefore provides an implicit representation whose internal interaction dynamics can be trained, ablated, visualized, and diagnosed.

Jose Luis Lima de Jesus Silva

Federal University of Bahia, Department of Geophysics, Salvador, BA 40170-115, Brazil · Grupo de Estudos e Aplicação de Inteligência Artificial em Geofísica (GAIA), Federal University of Bahia, Salvador, BA 40170-115, Brazil

Neural Representations Attractors Scalar Field

Jul 13, 2026cs.CV

Self-Consistent Flow: Unifying Velocity and Endpoint Prediction for Rectified Flow Models

In rectified-flow-based generative models, the neural network can be trained to predict two different targets, such as the instantaneous velocity or the data endpoint, to perform denoising. Although prior work shows that these parameterizations lead to different empirical behaviors, the mechanisms underlying their respective advantages remain to be underexplored, and how to combine them effectively is still unclear. In this work, we analyze how learning errors from different parameterizations affect the generation performance. We show that predicting the data endpoint has a clear training signal that stabilizes training, whereas predicting the velocity maintains stable sampling dynamics near the data manifold. Motivated by these insights, we propose Self-Consistent Flow (SC-Flow), a new method that unifies the benefits of both parameterizations. By employing a lightweight consistency loss, SC-Flow jointly trains a single network to predict both the local velocity and the data endpoint, and the consistency between the two predictions improves the model's performance. The method requires no major architectural changes and adds minimal computational overhead. Extensive experiments on image generation tasks demonstrate that SC-Flow substantially stabilizes optimization and improves the straightness of generation paths, leading to significant gains in generation quality over standard rectified-flow baselines.

Xu Han, Jiajing Hu, Li-Ping Liu

Department of Computer Science Tufts University

Rectified Flow Generative Flow Networks Scalar Field

Jun 29, 2026cs.LG

ScaleAware-JEPA: Latent Representation for Discovery in Multiscale Physical Fields

Continuous physical fields represent a large fraction of data under scientific investigation. Their multiscale structures are central to discovery, yet useful coordinates are not known in advance. Standard self-supervised methods define context and targets in fixed image coordinates, posing a predictive task misaligned with fields organized across a continuous scale hierarchy. We introduce ScaleAware-JEPA, a framework that constructs dense, label-free latent coordinates for continuous scalar fields. Constrained Diffusion Decomposition (CDD) separates each field into pixel-registered scale components and provides the scale coordinates that define the masking geometry. The resulting JEPA objective predicts hidden structure with a context footprint tied to the diffusion scale of each component rather than to an arbitrary patch size. Across MHD turbulence, interstellar molecular gas and urban nighttime-light structure, the learned geometry maps back to coherent morphology, forming dense structural atlases without labels or predefined segmentation rules. By tying latent prediction to the scale hierarchy of a field, ScaleAware-JEPA constructs latent coordinates through which complex physical patterns can be inspected before their relevant structures have been prescribed. Code is available at https://github.com/gxli/SA-JEPA.

Guang-Xing Li

Scalar Field Multiscale Dynamics Latent Space Atlas Discovery

Jun 22, 2026cs.LG

One-Step Flow Matching for Generative Modeling of Path-Dependent Physical Fields

Physical simulations for intricate geometries with path-dependent constitutive models face difficulties due to the enormous computational cost they require. Recently, the emergence of generative AI models, which succeed in image and video synthesis tasks, has provided a promise to further improve simulations. Although U-Net-based denoising diffusion probabilistic models (DDPMs) have been adopted for elastic stress field generation, they typically require hundreds of sampling steps, and applications of generative models to path-dependent, e.g. plastic, stress fields remain very limited. In this work, we propose a novel flow matching (FM) model based on a transformer backbone for high-resolution path-dependent stress field generation with stochastic loading-unloading paths and geometry. The proposed model operates within the latent space of a variational autoencoder (VAE) and formulates the simulation of plastic fields as a video synthesis task, directly generating the stress fields across all time steps. Meanwhile, we design a non-Gaussian source distribution for flow matching, such that crossings among conditional transport paths are reduced during training. This enables our model to generate satisfactory samples in one step without relying on distillation. In addition, we introduce token-level loading embeddings and two auxiliary networks to further enhance the model performance in path-dependent simulation. The results demonstrate that, even with a limited training dataset, our model can accurately generate high-resolution path-dependent fields. It is much more computationally efficient than finite element analysis, providing a speedup of 6 to 7 times over FEM on CPUs and approximately two orders of magnitude speedup on consumer-grade GPUs.

Yijing Zhou, Jasmin Jelovica

aDepartment of Mechanical Engineering, The University of British Columbia, Vancouver, Canada · bDepartment of Civil Engineering, The University of British Columbia, Vancouver, Canada

Conditional Flow Matching Generative Models Scalar Field Pathways

Jun 8, 2026cs.LG

Learning Dynamics Reveal a Hierarchy of Weight-Induced Layerwise Gram Metrics

We study feed-forward ReLU networks with fixed readout and quadratic loss, and rewrite gradient descent as a collective dynamics of activation fields and conjugate fields on the training set. Working to first order in the learning rate inside a fixed activation chamber, we derive explicitly the one-, two- and three-hidden-layer cases, and then give the arbitrary-depth recursion. For one hidden layer the activation dynamics closes directly and the residual update is governed by the product of an input Gram matrix and a co-activation/backpropagation Gram matrix. For two hidden layers a conjugate field is required, but no nontrivial pullback Gram metric has yet appeared. For three hidden layers the first weight-induced pullback Gram metric enters the conjugate-field dynamics. At arbitrary depth, activation variations propagate forward through a recursive response operator

\cU_\ell^{αβ}

, while conjugate-field variations propagate backward through an effective transport operator

\cM_\ell^{αβ}

. Their contractions reconstruct a layerwise residual kernel

K_{αβ}^{(L)}=\sum_{\ell=1}^{L}Q_{αβ}^{(\ell-1)}S_{αβ}^{(\ell)}.

The resulting description exposes a duality between push-forward and pullback transport across every layer cut, and identifies the first Gram metrics as the lowest nontrivial terms in a broader hierarchy of activation-conditioned transport operators. We deliberately stop at the level of collective fields, conjugate fields, residual kernels and cut-wise transport metrics, leaving the later tensorial geometric formulation outside the scope of this paper.

Claudio Nordio

Rectified Linear Unit Networks Learning Dynamics Layer-Wise Scalar Field Riemannian Manifolds Hierarchical

Jun 6, 2026cs.GR

MS-COOT: Comparing Morse-Smale Complexes with Co-Optimal Transport

Understanding and comparing structures in scalar fields is a central challenge in scientific visualization, with applications ranging from feature analysis to temporal and structural comparison. The Morse-Smale (MS) complex provides a natural representation by decomposing a scalar field into regions induced by gradient flow. However, existing approaches typically rely on graph-based representations, capturing relationships between critical points while discarding region-level structure. In this work, we represent the MS complex as a hypergraph, where critical points form nodes and regions define hyperedges. We introduce MS-COOT, a co-optimal transport distance that jointly computes correspondences between critical points and regions. This formulation enables explicit region-to-region matching within a distance-based framework, allowing identification of region-level events such as splitting and merging. We instantiate this framework with domain-specific components, including a hypernetwork function encoding critical point-region relationships, persistence-based probability measures that emphasize topologically significant features, and a sample cost term that incorporates critical point attributes. We evaluate MS-COOT on five datasets spanning 2D simulations, 3D surface meshes, and volumetric data. Our results show that MS-COOT captures region-level structural changes that are not reflected by graph-based distances, while achieving strong performance in downstream tasks such as classification and resolution discrimination.

Guangyu Meng, Mingzhe Li, Erin Wolf Chambers

Department of Computer Science and Engineering, University of Notre Dame, Notre Dame, Indiana, United States

Multiscale Differentiable Optimal Transport Scalar Field Stationary Point Visual Analytics Hypernetwork

Jun 3, 2026cs.CV

UniPixie: Unified and Probabilistic 3D Physics Learning via Flow Matching

Existing feed-forward networks excel at predicting a single set of physical properties from visual appearance, but this point-estimate paradigm fundamentally fails to capture the real world's inherent physical ambiguity. We address this by reframing physics prediction as a task of learning a controllable, continuous distribution of material properties. We introduce UNIPIXIE, a framework trained to predict a continuous and parameterized path of physically plausible material properties from a single visual input. By learning a direct mapping along an object's softest-to-stiffest spectrum on our PIXIEMULTIVERSE dataset, UNIPIXIE allows for controllable generation of diverse, physically valid material fields via a single intuitive parameter. Crucially, UNIPIXIE introduces a novel unified architecture to produce simulation-ready parameters for diverse physics solvers, including continuum-based Material Point Method (MPM), reduced-order deformation based on Linear Blend Skinning (LBS), and anchor-based Spring-Mass systems, addressing a key portability issue in prior work. Experiments show our approach not only generates a rich variety of plausible dynamics but also reduces Young's Modulus prediction error by over 50% against the strongest deterministic baseline, bridging the gap between static point estimates and the continuous nature of physical reality. Project page: https://unipixie.github.io/

Qilin Huang, Quynh Anh Huynh, Long Le +5

University of Pennsylvania · Southern University of Science and Technology · Work done during an internship at UPenn +1

Scalar Field Pixelloop 3D Generation

May 21, 2026cs.LG

Physics-Informed Generative Solver: Bridging Data-Driven Priors and Conservation Laws for Stable Spatiotemporal Field Reconstruction

Reconstructing continuous physical fields from sparse measurements is a central inverse problem, but data-driven generative models can produce states that violate governing dynamics. We introduce a physics-informed generative solver that separates stable prior learning from inference-time enforcement of conservation laws. Martingale-Regularized Score Matching regularizes score pretraining with a Score Fokker-Planck constraint, yielding a dynamically stable prior. Physics-Informed Implicit Score Sampling then guides denoising trajectories by gradients of physical residuals, projecting samples toward admissible manifolds without retraining. In acoustics, the method co-generates pressure and particle velocity from sparse sensors, enabling dense virtual arrays that suppress spatial aliasing. The same framework generalizes to real-world ERA5 meteorological fields under extreme sparsity. Together, this work establishes a rigorous and generalizable paradigm for solving high-dimensional inverse problems, bridging the gap between generative artificial intelligence and first-principles science.

Ziyuan Zhu, Keyu Hu, Zhifei Chen +10

School of Advanced Manufacturing Engineering, Nanjing University, Suzhou, 215163, China. · National Laboratory of Solid State Microstructures, Nanjing University, Nanjing, 210093, China. · Suzhou Acoustics Industry Technology Research Institute Co., Ltd., Suzhou, 215513, China. +8

Bayesian Inverse Problems Scalar Field Fokker--Planck Equation

May 13, 2026cs.LG

Topology-Preserving Neural Operator Learning via Hodge Decomposition

In this paper, we study solution operators of physical field equations on geometric meshes from a function-space perspective. We reveal that Hodge orthogonality fundamentally resolves spectral interference by isolating unlearnable topological degrees of freedom from learnable geometric dynamics, enabling an additive approximation confined to structure-preserving subspaces. Building on Hodge theory and operator splitting, we derive a principled operator-level decomposition. The result is a Hybrid Eulerian-Lagrangian architecture with an algebraic-level inductive bias we call Hodge Spectral Duality (HSD). In our framework, we use discrete differential forms to capture topology-dominated components and an orthogonal auxiliary ambient space to represent complex local dynamics. Our method achieves superior accuracy and efficiency on geometric graphs with enhanced fidelity to physical invariants. Our code is available at https://github.com/ContinuumCoder/Hodge-Spectral-Duality

Dongzhe Zheng, Tao Zhong, Christine Allen-Blanchette

Princeton University

Neural Operators Graph Laplacians Scalar Field Riemannian Manifolds

May 11, 2026cs.RO

JODA: Composable Joint Dynamics for Articulated Objects

Articulated objects used in simulation and embodied AI are typically specified by geometry and kinematic structure, but lack the fine-grained dynamical effects that govern realistic mechanical behavior, such as frictional holding, detents, soft closing, and snap latching. Existing approaches either ignore the detailed structure of dynamics entirely, or use simple models with limited expressiveness. We introduce JODA, a framework for generating joint-level dynamics as a structured three-channel field over the joint degree of freedom, capturing conservative forces, dry friction, and damping. Instantiated using shape-constrained piecewise cubic interpolation (PCHIP), this formulation defines a compact and expressive function space that is both interpretable and compatible with differentiable simulation. Building on this representation, we develop methods for inferring and refining joint dynamics from multimodal inputs. Given visual observations and joint context, a vision-language model proposes structured dynamical primitives, which are composed into a unified dynamics field. The resulting representation supports both direct manipulation and gradient-based refinement. We demonstrate that JODA enables plausible and controllable modeling of diverse joint behaviors, providing a unified interface for inference, editing, and optimization. Code and example assets with their generated profiles will be released upon publication.

Tianhong Gao, Cheng Yu, Yinghao Xu +1

Peking University · Ant Group, Robbyant

Articulated Objects Kinematics Mujoco Scalar Field

May 1, 2026cs.LG

Mesh Field Theory: Port-Hamiltonian Formulation of Mesh-Based Physics

We present Mesh Field Theory (MeshFT) and its neural realization, MeshFT-Net: a structure-preserving framework for mesh-based continuum physics that cleanly separates the physics' topological structure from its metric structure. Imposing minimal physical principles (locality, permutation equivariance, orientation covariance, and energy balance/dissipation inequality), we prove a reduction theorem for mesh-based physics. Under these conditions, the physical dynamics admit a local factorization into a port-Hamiltonian form: the conservative interconnection is fixed uniquely by mesh topology, whereas metric effects enter only through constitutive relations and dissipation. This reduction clarifies what must be fixed and what should be learned, directly informing MeshFT-Net's design. Across evaluations on analytic and realistic datasets, physics-consistency tests, and out-of-distribution validation, MeshFT-Net achieves near-zero energy drift and strong physical fidelity (correct dispersion and momentum conservation) along with robust extrapolation and high data efficiency. By eliminating non-physical degrees of freedom and learning only metric-dependent structure, MeshFT provides a principled inductive bias for stable, faithful, and data-efficient learning-based physical simulation.

Satoshi Noguchi, Yoshinobu Kawahara

Center for Mathematical Science and Advanced Technology, JAMSTEC · Center for Advanced Intelligence Project, RIKEN · Graduate School of Information Science and Technology, The University of Osaka.

Mesh Generation Scalar Field Hamiltonian

Apr 24, 2026stat.ML

Explanation of Dynamic Physical Field Predictions using WassersteinGrad: Application to Autoregressive Weather Forecasting

As the demand to integrate Artificial Intelligence into high-stakes environments continues to grow, explaining the reasoning behind neural-network predictions has shifted from a theoretical curiosity to a strict operational requirement. Our work is motivated by the explanations of autoregressive neural predictions on dynamic physical fields, as in weather forecasting. Gradient-based feature attribution methods are widely used to explain the predictions on such data, in particular due to their scalability to high-dimensional inputs. It is also interesting to remark that gradient-based techniques such as SmoothGrad are now standard on images to robustify the explanations using pointwise averages of the attribution maps obtained from several noised inputs. Our goal is to efficiently adapt this aggregation strategy to dynamic physical fields. To do so, our first contribution is to identify a fundamental failure mode when averaging perturbed attribution maps on dynamic physical fields: stochastic input perturbations do not induce stationary amplitude noise in attribution maps, but instead cause a geometric displacement of the attributions. Consequently, pointwise averaging blurs these spatially misaligned features. To tackle this issue, we introduce WassersteinGrad, which extracts a geometric consensus of perturbed attribution maps by computing their entropic Wasserstein barycenter. The results, obtained on regional weather data and a meteorologist-validated neural model, demonstrate promising explainability properties of WassersteinGrad over gradient-based baselines across both single-step and autoregressive forecasting settings.

Younes Essafouri, Laure Raynaud, Luciano Drozda +1

Univ. Toulouse, INSA Toulouse, CNRS UMR 5219, IMT, Toulouse, France · Météo-France, CNRS, Univ. Toulouse, CNRM, Toulouse, France · Cerfacs, CNRS/Cerfacs/IRD, CECI, Toulouse, France

Artificial Intelligence Weather Models Scalar Field Autoregressive Model Wasserstein Distance Explainable AI Methods

Apr 22, 2026cs.RO

A Hough transform approach to safety-aware scalar field mapping using Gaussian Processes

This paper presents a framework for mapping unknown scalar fields using a sensor-equipped autonomous robot operating in unsafe environments. The unsafe regions are defined as regions of high-intensity, where the field value exceeds a predefined safety threshold. For safe and efficient mapping of the scalar field, the sensor-equipped robot must avoid high-intensity regions during the measurement process. In this paper, the scalar field is modeled as a sample from a Gaussian process (GP), which enables Bayesian inference and provides closed-form expressions for both the predictive mean and the uncertainty. Concurrently, the spatial structure of the high-intensity regions is estimated in real-time using the Hough transform (HT), leveraging the evolving GP posterior. A safe sampling strategy is then employed to guide the robot towards safe measurement locations, using probabilistic safety guarantees on the evolving GP posterior. The estimated high-intensity regions also facilitate the design of safe motion plans for the robot. The effectiveness of the approach is verified through two numerical simulation studies and an indoor experiment for mapping a light-intensity field using a wheeled mobile robot.

Muzaffar Qureshi, Trivikram Satharasi, Tochukwu E. Ogri +2

Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, FL, USA · Torch Technologies, Shalimar, Florida, USA

Gaussian Process Scalar Field Dense

Mar 20, 2026math.NA

An Adaptive Machine Learning Framework for Fluid Flow in Dual-Network Porous Media

Porous materials -- natural or engineered -- often exhibit dual pore-network structures that govern processes such as mineral exploration and hydrocarbon recovery from tight shales. Double porosity/permeability (DPP) mathematical models describe incompressible fluid flow through two interacting pore networks with inter-network mass exchange. Despite significant advances in numerical methods, there remains a need for computational frameworks that enable rapid forecasting, data assimilation, and reliable inverse analysis. To address this, we present a physics-informed neural network (PINN) framework for forward and inverse modeling of DPP systems. The proposed approach encodes the governing equations in mixed form, along with boundary conditions, directly into the loss function, with adaptive weighting strategies to balance their contributions. Key features of the framework include adaptive weight tuning, dynamic collocation point selection, and the use of shared trunk neural architectures to efficiently capture the coupled behavior of the dual pore networks. It is inherently mesh-free, making it well-suited for complex geometries typical of porous media. It accurately captures discontinuities in solution fields across layered domains without introducing spurious oscillations commonly observed in classical finite element formulations. Importantly, the framework is well-suited for inverse analysis, enabling robust parameter identification in scenarios where key physical quantities -- such as the mass transfer coefficient in DPP models -- are difficult to measure directly. In addition, a systematic convergence analysis is provided to rigorously assess the stability, accuracy, and reliability of the method. The effectiveness and computational advantages of the approach are demonstrated through a series of representative numerical experiments.

V. S. Maduri, K. B. Nakshatrala

Department of Civil & Environmental Engineering University of Houston, Houston, Texas 77204, USA.

Parametric Physics-Informed Neural Network Fluid Dynamics Porous Media Inverse Problem Scalar Field

Jan 28, 2026stat.ML

Leveraging Differentiable PDE Solvers for Semi-Neural Spatial Reconstruction From Sparse Measurements

Generating dense physical fields from sparse measurements is a fundamental question in sampling, signal processing, and many other applications. State-of-the-art approaches to this problem either rely on spatial statistics that ignore the governing physics, integrate the physics into a multiple-objective optimization process, or require examples of the complete, fully-resolved simulation state during training, which are frequently unavailable outside of synthetic benchmarks. Here, we present a novel alternative that leverages recent advances in the integration of numerical simulators with data-driven models. Namely, we propose a hybrid modeling pipeline that couples Radial Basis Function (RBF) reconstruction with a Neural Network (NN) correction and a Partial Differential Equation (PDE) solver, so that the numerical simulator itself is embedded directly in the training loop of the learned component. Notably, the NN is trained without assuming availability of examples of the fully-resolved simulation state. This is made possible by implementing the PDE solver so that it is end-to-end differentiable, allowing gradients to be backpropagated through the simulation step during training. This grey-box methodology is evaluated on three standard benchmarks from fluid mechanics, where it achieves superior results over statistical and machine-learning-based reconstruction methods.

Ofek Aloni, Barak Fishbain

Department of Mathematics, Technion, Israel · Department of Civil and Environmental Engineering, Technion, Israel

Neural Partial Differential Equation Solvers Partial Differential Equations Neural Reconstruction Spatiotemporal Fields Scalar Field Radial Basis Function