Neural PDE Solvers

PDE: Partial Differential Equation

Follow this topic Ask about this topic

Momentum

23 papers in the last four weeks, up 109% on the four weeks before. 0.2% of all new papers.

Jul 13Week of Sep 28

Latest papers 166

Jul 21, 2026cs.CV

Image Editing Models are Numerical Solvers

We investigate whether a pretrained generative image-editing model can provide a common interface for numerical simulation. Physical inputs and solutions are rendered as images, while scalar quantities such as material properties, diffusivity, and loading parameters enter through lightweight adapters. Using established numerical and analytic solvers for supervision, we apply the same architecture and training protocol to heterogeneous elliptic equations, forced heat and Burgers evolution, complex Ginzburg-Landau dynamics, two-dimensional Navier-Stokes prediction, potential flow, elasticity, eikonal travel time, phase-field fracture, and entropic optimal transport. The results show that a pretrained image model can represent diverse static and time-dependent physical mappings, including unstable and shock-like behavior, when each task is expressed through a suitable visual encoding. This work is a capability study rather than an attempt to surpass specialized solvers. It also identifies fundamental constraints: image and latent representations complicate numerical range selection and direct enforcement of governing equations or invariants, while a failed Kuramoto-Sivashinsky experiment indicates that representation errors prevent meaningful long-horizon simulation of chaotic systems.

Ulysse Mizrahi

Tel Aviv University

Neural PDE Solvers PDE Solving PDE Surrogate Modeling

Jul 20, 2026cs.CV

Physics Closure Matters for Machine Olfaction: A Maxwell-Stefan Graph Solver for Identifiable Dynamic Gas Unmixing

Machine olfaction for gas unmixing is an underconstrained inverse problem in which gas compositions must be inferred from low-dimensional, delayed, and entangled sensor responses produced by interacting chemical transport, surface adsorption, and sensor transduction. One of the key obstacles is physics closure misspecification, where a neural network is designed to fit sensor traces rather than infer a physically closed olfactory process. In this work, we formulate gas unmixing as a multi-physics-constrained inverse problem governed by Maxwell--Stefan multicomponent transport PDEs, competitive adsorption ODEs, and nonlinear sensor transduction ODEs. Directly solving such a high-dimensional coupled system is computationally expensive and often numerically unstable. To this end, we propose UnMixNet, a physics-closed graph neural solver that embeds this multi-physics forward process into end-to-end gas unmixing. UnMixNet discretizes Maxwell--Stefan cross-diffusion on spatial graphs and formulates the multicomponent flux on each edge. This design enables local, differentiable, and flux-conservative inference for multicomponent cross-diffusion. Evaluations on SmellNet show improved single-odor recognition, seen-mixture unmixing, and unseen-mixture generalization. In addition, an external validation on UCI Dynamic Gas Mixtures shows that the inferred concentration process agrees with ground truth concentration set points under dynamic transitions. Process-consistency diagnostics further show that the proposed model learns transferable dynamic physical fingerprints that better satisfies transport, conservation, adsorption, and readout closure.

Yue Shi, Liangxiu Han, Xin Zhang +1

Department of Computing and Mathematics, Faculty of Science and Engineering, Manchester Metropolitan University, Manchester M1 5GD, U.K.

Graph Neural Networks Neural PDE Solvers Physics-Informed ML

Jul 20, 2026cs.LG

FlashPDE: A Drop-In Fused Triton Operator Library for Neural PDE Solvers

Physics-Informed Neural Networks (PINNs) solve PDEs by incorporating physical constraints into neural-network training, but large-scale problems are limited by automatic-differentiation memory overhead and inefficient execution of grid-based PDE operators. We present FlashPDE, a drop-in fused operator library for grid-based scientific machine learning. FlashPDE replaces fragmented PyTorch finite-difference execution with differentiable Triton kernels. Each operator integrates fused stencil evaluation, an analytic discrete-adjoint backward pass, and boundary-gradient correction within a unified PyTorch autograd Function interface. The library provides 14 differentiable PDE operators covering 17 configurations across 1D--3D elliptic, parabolic, and Navier--Stokes systems, while remaining independent of neural architectures and training strategies. Experiments on an NVIDIA A100 GPU show that FlashPDE reduces peak memory usage by up to 37.0x compared with coordinate-based automatic differentiation and reduces CUDA kernel launches by up to 3.5x compared with eager PyTorch finite-difference implementations. Across six representative PDE benchmarks, FlashPDE achieves up to 2.30x end-to-end time-to-solution speedup and up to 19.2x kernel-level acceleration while maintaining numerical agreement with PyTorch finite-difference references. FlashPDE provides a hardware-efficient execution layer that bridges differentiable PDE solvers and GPU-optimized numerical computation within the PyTorch ecosystem.

Peiyu Zang, Bosen Xie, Ruoxiang Xu +1

School of Mathematical Sciences, Beijing Normal University · 2Beijing University of Posts and Telecommunications

Automatic Differentiation Neural PDE Solvers GPU Kernel Optimization PDE Solving Differentiable Programming Physics-Informed ML

Jul 17, 2026math.NA

Non-Asymptotic Variational Learning for Monotone Nonlinear Multiscale Elliptic Equations: Scale-Robust Primal-Dual Bounds and Strong-Form Statistical Ill-Conditioning

We develop a non-asymptotic approximation, sampling, and finite-iteration optimization theory for variational physics-informed approximation of uniformly monotone nonlinear multiscale elliptic equations. For boundary-compatible neural feature classes, the population error splits into approximation, empirical quadrature, and projected-gradient terms, with all non-approximation constants uniform in the microscopic scale

\varepsilon

. Assuming a quantitative corrected

H^1

-estimate, a two-scale state class yields

\mathcal A_m^\varepsilon \le C\bigl(\varepsilon+Φ_{0,m_0}^2+Φ_{1,m_1}^2\bigr)

in arbitrary dimension. We further introduce a convex primal-dual physics loss whose population value is a computable upper certificate for the state error. With additional flux-corrector regularity, a divergence-compatible two-scale flux class gives a certified state-flux bound combining

O(\varepsilon)

approximation, state and flux feature errors, empirical sampling error, and an

O(K^{-1})

optimization term. In contrast, for general periodic nonlinear fluxes satisfying a natural nondegeneracy condition, the empirical Rademacher complexities of strong-residual and squared-residual classes are bounded below by constant multiples of

(\varepsilon\sqrt N)^{-1}

and

(\varepsilon^2\sqrt N)^{-1}

, respectively. These optimizer-independent lower bounds hold in every spatial dimension. Numerical experiments confirm the predicted

\varepsilon

- and

N

-scalings for nonlinear fluxes in

d=1,2,3

, validate every computed primal-dual certificate, and show that corrector-enriched classes substantially reduce energy and

H^1

errors as the microscopic scale is refined.

Ronald Katende

Department of Mathematics Kabale University Kikungiri Hill, Katuna Road, Kabale, Uganda

Primal-Dual Optimization Neural PDE Solvers Neural Network Approximation Theory Physics-Informed ML Statistical Learning Theory

Jul 17, 2026cs.NE

Evolutionary Algorithm-Guided LLMs for Physics-Informed Neural Network Design

Physics-informed neural networks (PINNs) are unusually sensitive to interacting choices of architecture, activation, loss weighting, collocation, optimization, and constraint enforcement. Large language models (LLMs) can propose these choices, but independent recommendations do not accumulate experience from previously trained PINNs. We propose a closed-loop evolutionary algorithm that guides an LLM to generate complete, executable PINN configurations across generations, using measured training outcomes to determine subsequent search decisions. The algorithm maintains an evaluated population and lineage, applies parent-conditioned mutation and crossover, preserves elite and diverse solutions, rejects effective duplicates, and converts parent-relative successes and failures into the next-generation context supplied to the LLM. Every proposed configuration is executed directly under an exact optimizer-step budget. On a one-dimensional multiscale wave equation, two independent ten-generation runs trained 60 PINNs for 600,000 optimizer steps. In both runs, the best configuration appeared in the final generation, with best mean-squared error reduced by 2.97% and 95.38% relative to the initial population. The stronger run validated residual connections and increased depth on separate branches, combined them in a later generation, and then refined width and collocation density. It also revealed that low solution error can coexist with a high PDE residual. These results demonstrate the feasibility of evolutionary-algorithm-guided LLMs for PINN design on a controlled PDE while motivating broader, physics-aware evaluation.

Xu Yang, Mingyang Yu, Jing Xu +1

Shanghai Institute of Intelligent Education, East China Normal University, Shanghai 200062, China · College of Artificial Intelligence, Nankai University, Tianjin 300350, China

Neural PDE Solvers Evolutionary Optimization Large Language Model-Guided Optimization Physics-Informed ML

Jul 15, 2026cs.LG

LIGO-PINN: Learned Initialization via Gated Optimization to Alleviate Convergence Failures in Physics Informed Neural Networks

Physics-informed neural networks (PINNs) have had a broad research impact in modeling domains governed by partial differential equations (PDE). However, PINNs have been shown to perform poorly, sometimes even converging to trivial solutions, in challenging PDE domains, or when generalizing to unseen but related PDE domains. Previously proposed solutions detail hyperparameter tuning to reduce loss imbalance between data-driven and physics guided losses, curriculum learning based training strategies, or dynamic re-sampling of hard collocation points. These methods face certain pitfalls: hyperparameter tuning is expensive, designing a training curriculum is ambiguous in multi-parameter PDE settings, and dynamic resampling still fails in complex PDE settings. Complementary to this line of thinking, we believe the initial PINN network weights also play a crucial role in the emergence of catastrophic failures during training, yet the effect of PINN weight initialization has been surprisingly under-investigated. To this end, we propose a framework for Learned Initialization via Gated Layerwise Optimization (LIGO-PINN) to overcome PINN convergence failures. Through rigorous evaluation on 1D and 2D PDE domains, including a challenging 2D fluid dynamics setting, we demonstrate that our methodology outperforms state-of-the-art methods designed to alleviate PINN failures, achieving a 91.5% average performance improvement across six baselines and 81% over the strongest baseline. We also verify that LIGO-PINN generalizes to 3D unstructured domains. Finally, we analyze training dynamics across all three PDE domains to explain both LIGO-PINN's improvement and the convergence failure of traditional PINNs. Code: https://github.com/scailab/ligo-pinn Keywords: Machine Learning, Physics-Informed Neural Networks, Deep Learning, PDE Modeling

Nilay Anurag, Shital Adhikari, Taniya Kapoor +1

Department of Computer Science, Stevens Institute of Technology · Artificial Intelligence Group, Wageningen University & Research

Neural PDE Solvers Neural Network Optimization Physics-Informed ML

Jul 13, 2026cs.LG

Multi-dimensional training-priority weighting based on physical information propagation paths: a unified residual-weighting framework for physics-informed neural networks

Physics-informed neural networks (PINNs) have shown promise for solving partial differential equations (PDEs); however, their synchronous optimization treats residuals of different regions and constraints equally, which is inconsistent with the progressive "from source to response" physical information propagation path, degrading training stability and accuracy. Existing causal training methods focus mainly on the temporal dimension, lacking a unified characterization of spatial and boundary dimensions. To address this, we define a unified class of training priorities according to the physical information propagation path: premise regions should be learned before dependent regions; temporal, spatial, and boundary priorities are instances of this principle. Using neural tangent kernel (NTK) dynamics, we theoretically analyze why standard PINNs do not obey this priority: their residual convergence order is governed by the NTK spectrum and is independent of the propagation path. Accordingly, we propose a unified multi-dimensional priority-constraint framework that partitions the domain along the propagation path and constructs negative-exponential residual weights, converting the physical propagation order into a training priority. For cases with coexisting priorities, we introduce a directional compatibility coefficient to clarify that "orthogonal directions can be coupled multiplicatively in synergy, whereas coaxial opposite directions cannot." Benchmark cases show that this method consistently improves the convergence behavior and prediction accuracy of PINNs on problems with clear propagation paths or constraint-dominated structures, without modifying the network architecture and with controllable additional computational cost.

Zhangyi Lian, Xinda Dong, Wenxuan Huo +3

College of Mechanical and Vehicle Engineering Chongqing University Chongqing 400030, China · State Key Laboratory of Tribology in Advanced Equipment Department of Mechanical Engineering Tsinghua University Beijing 100084, China

Neural PDE Solvers Adaptive Loss Weighting Physics-Informed ML

Jul 13, 2026cs.LG

Learning to Discretize: Diffusion-Based Adaptive Mesh with Spectral Guidance

Most neural partial differential equation (PDE) surrogates learn how fields evolve after a grid has already been chosen. However, before any operator is applied, the grid has already determined how modeling capacity is allocated across space, resolution, and spectral bandwidth. We argue that this hidden design choice should itself be learnable, leading to a question different from standard operator learning: can a surrogate learn where resolution should exist before predicting field evolution? We formulate adaptive discretization as a physics-constrained conditional generation problem over valid mesh displacements. The success of diffusion models in PDE field prediction suggests their potential for learning adaptive discretizations under similar structured constraints. This leads to a two-stage diffusion framework: Stage 1 learns an r-adaptive displacement mesh conditioned on the observed dynamics, while Stage 2 predicts the solution evolution from the mesh-informed representation. The mesh generator is regularized by physics-aware proxy channels, geometric validity constraints, and local spectral concentration so that adaptation remains physically interpretable and numerically legal. Across five PDE regimes, the results show that diffusion-based learned discretization is competitive with adaptive-mesh and reduced-order baselines, with particularly strong gains in regimes where fixed or handcrafted allocation is insufficient. The main conclusion is not that there exists a universal optimal mesh rule, but that discretization should be learned in a regime-dependent manner: different spatial and spectral structures favor different allocation behaviors. This reframes adaptive meshing for neural PDE solvers from a solver-specific heuristic into a generative representation-learning problem.

Zixuan Shen, Bingchuan Wang, Zhi Wang +1

Central South University · Central South University, Changsha, Hunan 410083, China · Nanjing University +1

Neural PDE Solvers PDE Surrogate Modeling Physics-Informed ML Adaptive Mesh Refinement Diffusion Models

Jul 11, 2026cs.LG

The Differential Neural Tangent Kernel and Its Positivity

The Neural Tangent Kernel (NTK) is one powerful tool for analyzing the training dynamics of neural networks in the over-parameterized regime. Recently, the theoretical framework has been extended to physics-informed neural networks (PINNs) for solving linear PDEs, one highly popular class of neural PDE solvers. In the analysis, the positivity of the associated NTK plays a fundamental role. However, establishing the positivity of the NTK for PINNs is highly challenging, due to the presence of multiple differential operators. In this work, we propose a new theoretical framework, called Differential Neural Tangent Kernel (DNTK), for analyzing PINNs through the lens of the NTK, and establish the positivity of the infinite width DNTK for both shallow and deep neural networks for a wide class of activation functions, including RePU and smooth but non-polynomial activations, for all linear differential operators. These theoretical results lay the foundation for the analysis of gradient type algorithms for training PINNs.

Bangti Jin, Longjun Wu

Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong, P.R. China

Neural PDE Solvers Physics-Informed ML Neural Network Training Dynamics Neural Tangent Kernel

Jul 7, 2026cs.LG

SplineNet: An Isogeometric Deep Learning Method for Complex Shells

We present a novel isogeometric deep learning method, termed SplineNet, for the seamless design and analysis of shell structures with complex geometries. The proposed approach is built upon watertight spline representations, e.g., analysis-suitable unstructured T-splines, and features exact geometric descriptions of Computer-Aided Design (CAD) models in neural networks. Bézier extraction is used to build the network architecture, where Bernstein polynomials serve as the nonlinear activation functions. SplineNet can be applied in a data-free or data-driven way. In the data-free case, energy-based formulations can be naturally incorporated as loss terms, which fulfill the need of Computer-Aided Engineering (CAE) and can be accurately calculated. In particular, the Kirchhoff--Love (KL) model is adopted to solve for the mechanical behaviors of shell structures. This way, CAD and CAE can be tightly integrated in a deep neural network without the time-consuming model/data exchange process. In the data-driven case, SplineNet can be used as the trunk net of Deep Operator Networks (DeepONet) to provide interpretability. Given such a trained network and unseen input data, results can be immediately obtained without retraining the network or repeatedly performing the traditional workflow for analysis. In the end, a variety of numerical examples are studied to demonstrate the effectiveness of the proposed method, especially when real-world complex geometries are involved.

Shizhou Luo, Xiaodong Wei

Global College, Shanghai Jiao Tong University, Shanghai, 200240, China.

Neural PDE Solvers

Jul 4, 2026math.AP

LRX-PINN: A Layer-Resolving XNet Physics-Informed Neural Network with Integrated Cauchy Activations for Convection-Dominated Problems

Convection-dominated convection-diffusion problems often develop thin layers, where the solution has sharp transition profiles and its derivatives are highly localized. This creates a structural mismatch for standard physics-informed neural networks (PINNs), whose trial spaces are not designed to match the value--derivative structure of such layers. We propose a Layer-Resolving XNet Physics-Informed Neural Network (LRX-PINN) based on integrated Cauchy activations. The proposed basis is transition-type at the solution level, while its derivative recovers a localized Cauchy kernel. We show that this structure matches the scaling of convection-dominated layers, inherits the Cauchy approximation mechanism at the derivative-profile level, and identifies

d/\|w\|

as the effective physical width of a ridge neuron. For analytic layer profiles, this yields derivative-stable exponential approximation in the stretched coordinate and a layer-scaled estimate for the strong residual of the singularly perturbed operator. Numerical experiments on several convection-dominated benchmarks show that LRX-PINN achieves higher accuracy than PIKAN and Fourier-feature PINNs while using less than

30\%

of their trainable parameters. On more challenging benchmarks, embedding the proposed representation into hp-VPINN-based frameworks further improves the best results obtained by existing hp-VPINN-based baselines without changing their original loss functionals or stabilization strategies. These results show that neural representations aligned with layer structure provide a compact and effective approach for convection-dominated problems.

Zihao Guo, Xin Li, Zhihong Xia

Institute for Advanced Research, Great Bay University, Dongguan, China · College of Computer Science and Technology, Dongguan University of Technology, Dongguan, China · Department of Mathematics, Northwestern University, Evanston, USA

Neural PDE Solvers PDE Solving Neural Network Activation Functions Neural Network Approximation Theory Physics-Informed ML

Jul 1, 2026cs.CE

A Multi-Resolution Finite-Volume Inspired Deep Learning Framework for Spatiotemporal Dynamics Prediction

Predicting complex spatiotemporal dynamics in physical processes often demands computationally expensive numerical methods or data-driven neural networks that suffer from high training costs, error accumulation, and limited generalizability to unseen parameters. An effective approach to address these challenges is leveraging physics priors in training neural networks, known as physics-informed deep learning (PiDL). In this work, we introduce the Multi-Resolution Finite-Volume-inspired network, MuRFiV, designed to capitalize on the conservative property of finite volume on the global scale and the expressive power of deep learning on the local scale. We demonstrate the effectiveness of MuRFiV on several spatio-temporal systems governed by partial differential equations (PDEs), including Burgers' equation, shallow water equations, and incompressible Navier-Stokes equations. By embedding PDE information into the deep learning architecture, MuRFiV achieves strong long-term prediction accuracy and remains stable over very long autoregressive rollouts, significantly outperforming data-driven neural network baselines. This result highlights the promise of combining multiresolution learning with finite-volume-inspired inductive bias for accurate and robust long-term prediction of complex dynamics.

Xin-Yang Liu, Xiantao Fan, Jian-Xun Wang

Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, IN, USA · Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY, USA

Neural PDE Solvers PDE Surrogate Modeling Physics-Informed ML Spatiotemporal Forecasting

Jun 30, 2026math.NA

Domain-Decomposed Randomized Neural Networks for Partial Differential Equations in Unbounded Domains

Partial differential equations on unbounded domains are challenging because the exterior region must be represented without excessive truncation error. Truncation-based methods often require problem-dependent artificial boundary conditions, while global spectral bases may be inefficient for localized structures, irregular geometries, or solutions with different near-field and far-field behaviors. We propose a domain-decomposed randomized neural network framework for such problems. Different randomized subnetworks are assigned to different spatial regimes: a near-field subnetwork captures local and geometric features, whereas a far-field subnetwork represents exterior decay. The subnetworks are coupled by boundary and interface conditions, and only the output-layer coefficients are solved from linear least-squares systems arising from Petrov--Galerkin or collocation formulations. We develop a Petrov--Galerkin method for semi-unbounded elliptic problems and a collocation method for fully unbounded, perforated, and time-dependent problems. A conditional bounded-parameter approximation result is proved in a broken Sobolev norm, together with an error decomposition covering approximation, empirical-consistency/quadrature, and least-squares optimization errors. Numerical experiments for Poisson and time-dependent Schrödinger equations demonstrate the accuracy and flexibility of the proposed method.

Haixin Wang, Haoning Dang, Fei Wang +1

School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, P.R. China. · School of Mathematics and Statistics & State Key Laboratory of Multiphase Flow in Power Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China.

Neural PDE Solvers PDE Solving Neural Network Approximation Theory

Jun 29, 2026math.NA

McMg: A Learned Phase-Space Multi-channel Multigrid Preconditioner for Helmholtz Equation

Solving heterogeneous Helmholtz equations at high wavenumbers remains challenging because the discretized operator is indefinite, pollution degrades phase accuracy, and scalar coarse-grid correction can discard the local phase and propagation-direction information carried by oscillatory errors. We propose Multi-channel Multigrid (McMg), a learned phase-space multigrid preconditioner for heterogeneous Helmholtz equations. Rather than predicting the solution directly, McMg maps residuals to corrections within an iterative framework. Its central idea is to coarsen physical space while retaining unresolved local wave information in the channel dimension: each coarse node carries a learned packet of amplitude, phase, direction, and scattering coefficients rather than a single scalar unknown. The architecture combines linear multi-channel transfer operators with locally adaptive stencils, neural PDE operators, and medium-dependent smoothers whose coefficients are generated from the wave speed. For a fixed medium, the V-cycle is linear in the residual; nonlinear physical features are computed once in a setup phase and cached, so each online iteration reduces to convolutions with fixed coefficients. We further study generalization across scales. Models trained on small domains transfer directly to larger domains and higher effective wavenumbers, and a Layer-by-Layer Progressive Finetuning (LLPF) strategy improves large-domain scalability by adding new coarse levels while finetuning only the newly introduced parameters. Numerical experiments on high-frequency, high-contrast, and large-scale three-dimensional problems demonstrate that McMg requires substantially fewer iterations and less wall-clock time than strong classical baselines, while consistently outperforming existing neural preconditioners.

Jiwei Jia, Xinliang Liu, Juntao Wang +1

School of Mathematics, Jilin University, Changchun 130012, China · Shenzhen Loop Area Institute, Shenzhen 518038, China · Ocean University of China, Qingdao 266100, China +1

Neural PDE Solvers PDE Solving PDE Operator Learning

Jun 28, 2026math.OC

A Posteriori Error Analysis for Decoupled Neural Approximations of Fully Coupled FBSDEs with Control Mismatch

This paper develops an a posteriori error analysis framework for decoupled neural approximations of fully coupled forward--backward stochastic differential equations (FBSDEs). It provides an a posteriori error-analysis for the idealized discrete adapted trajectory. The main feature of the proposed formulation is the use of an auxiliary control process in the forward coefficients, which may differ from the backward component approximated by the neural network. This decoupling is useful in practical deep learning implementations, but it creates a control mismatch that must be included in the error analysis. We first establish a continuous-time stability estimate for fully coupled FBSDEs under perturbations of the drift, diffusion, generator, terminal condition, and auxiliary control input. We then transfer this estimate to the discrete-time setting and derive computable a posteriori error bounds depending only on the terminal defect, the pathwise residual, and the control mismatch. When the auxiliary control is identified with the backward approximation, the mismatch term vanishes and the bound reduces to the standard two-term form. Numerical experiments on a linear--quadratic FBSDE with an explicit reference solution and a multidimensional Burgers-type FBSDE without a reference solution illustrate the diagnostic role of the proposed indicators and the contribution of the mismatch penalty to the consistency and reproducibility of the numerical approximations.

Xichuan Zhang

Intelligent Game and Decision Lab, Beijing 100091, China

Stochastic Differential Equations Neural PDE Solvers

Jun 26, 2026cs.CE

Higher-Order Fourier Neural Operator: Explicit Mode Mixer for Nonlinear PDEs

Neural operators provide deep neural networks for learning mappings between function spaces. Among them, the Fourier Neural Operator (FNO) is particularly effective: its spectral convolution relies on low-dimensional Fourier-domain representations and can handle inputs at different resolutions. This design aligns well with settings where the Fourier basis diagonalizes the underlying operator, such as linear, constant-coefficient PDEs on periodic domains, in which Fourier modes evolve independently. However, nonlinear PDEs may benefit from an additional inductive bias, as they exhibit structured interactions between modes, governed by polynomial nonlinearities. To capture this inductive bias, we introduce the Higher-Order Spectral Convolution, a spectral mixer that extends FNO from diagonal modulation to explicit n-linear mode mixing, aligned with the dynamics of nonlinear PDEs. Our experiments on standard benchmarks show that the proposed Higher-Order FNO (HO-FNO) retains the efficiency of FNO-based architectures and consistently improves over other spectral neural operators. HO-FNO also performs on par with or better than state-of-the-art transformers and state-space models on several datasets, with stronger gains in highly nonlinear regimes, such as the Poisson equation with polynomial forcing, where a single HO-FNO layer outperforms FNO models with up to 16 layers. We open-source our code for reproducibility at: https://github.com/AlexColagrande/HO-FNO.

Alex Colagrande, Paul Caillon, Eva Feillet +1

Miles Team, LAMSADE, Université Paris Dauphine-PSL, Paris, France · Université Paris-Saclay, CNRS, LISN, 91400, Orsay, France · ESPCI PSL, Paris, France

Fourier Neural Operator Neural PDE Solvers PDE Operator Learning

Jun 25, 2026cs.LG

Error-Conditioned Neural Solvers

Neural surrogate models offer fast approximate mappings from PDE parameters to solutions, but they typically treat solving as a purely statistical task: once trained, they struggle to correct their own constraint violations and extrapolate beyond the training distribution. Recent hybrid methods promote physical correctness by targeting the PDE residual via gradient descent or Gauss--Newton steps, but inherit the compute cost and instability of the underlying classical optimizers. We show, theoretically and empirically, that numerically minimizing the PDE residual can be an unreliable proxy for reconstruction accuracy in ill-conditioned systems, explaining why these methods often do not make accurate predictions despite achieving low residuals. We propose error-conditioned Neural Solvers (ENS), built on a different principle: rather than an optimization target, the PDE residual field is passed as a direct input to the network at each iteration, enabling it to read the spatial structure of its own errors and learn an update policy to iteratively correct its predictions. Across four PDE families, ENS attains the highest prediction accuracy in the large majority of settings, with gains reaching

10\times

on turbulent Kolmogorov flow, while avoiding the expensive compute cost of hybrid methods. ENS's learned correction policy generalizes under distribution shift, including zero-shot parameter changes and cross-equation transfer, where its relative advantage is largest in the ill-conditioned regimes where residual minimization is least reliable. Project website: https://neuralsolver.github.io/.

Haina Jiang, Liam Wang, Peng-Chen Chen +4

University of Michigan · KAIST AI · Los Alamos National Laboratory

Neural PDE Solvers PDE Surrogate Modeling

Jun 25, 2026cs.LG

fTNN: a tensor neural network for fractional PDEs

We develop the fTNN, a deterministic tensor neural network subspace method for problems involving the fractional Laplacian on bounded domains, taking the fractional Poisson equation and time-dependent fractional advection-diffusion equation as typical representatives. The work employs a geometry-adapted integration split featuring a spatially dependent near-field radius, which decomposes the fractional Laplacian into three contributions: a singular near field, a regular interior far field, and an analytical exterior far field. Then the singular radial integrals are treated by Gauss-Jacobi quadrature, the regular radial integrals by Gauss quadrature, and the angular variables by deterministic angular quadrature, yielding a fully deterministic integration framework of the fractional Laplacian operator. To accurately resolve low-regularity solutions and the associated loss functional, we construct boundary-singularity-aware trial functions enriched with explicit boundary features, and propose two strategies for automatically selecting the leading exponent and evaluating the loss function from the singularity structure induced by the fractional operator, or jointly by the fractional operator and the source term. For time-dependent fractional PDEs, we design a spatiotemporally separable neural network that factorizes the time-space residual into a sum of low-dimensional temporal and spatial integrals, and we integrate this representation with an alternating neural network subspace optimization strategy for efficient training. Numerical experiments show that the proposed framework attains high accuracy on the tested benchmarks and improves substantially over existing fPINN and Monte Carlo baselines, particularly for problems with strong boundary singularities and long-time simulations.

Qingkui Ma, Hehu Xie, Xiaobo Yin

School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan 430079, China · SKLMS, NCMIS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, No.55, Zhongguancun Donglu, Beijing 100190, China · School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China +1

Tensor Networks Neural PDE Solvers PDE Solving

Jun 23, 2026math.NA

Dirac-Frenkel dynamics with inertia for nonlinearly parametrized solutions of evolution problems

Even when Dirac-Frenkel dynamics determine a well-defined evolution in function space, the corresponding parameter dynamics can be non-unique or ill-conditioned for redundant nonlinear parametrizations such as neural networks or mixture models. We propose to add inertia to the Dirac-Frenkel dynamics and show that this allows useful parameter velocity information to persist from the past trajectory in directions that are weakly informed, while well-informed parameter velocity directions continue to follow the Dirac-Frenkel dynamics. We prove that the inertial formulation yields well-posed parameter dynamics and provide a posteriori error bounds. After time discretization, the method requires the solution of the same type of regularized linear least-squares problem as standard Dirac-Frenkel dynamics, but with the previous velocity appearing as an anchor. Numerical experiments demonstrate the increased robustness obtained with inertia.

Matteo Raviola, Benjamin Peherstorfer

Scientific Computing and Uncertainty Quantification - CADMOS Chair, EPFL · Courant Institute of Mathematical Sciences, New York University

Dynamical Systems Neural PDE Solvers

Jun 23, 2026cs.LG

A Zeroth-Order Deep Learning Method for Fully Nonlinear Parabolic Partial Differential Equations with Unknown Coefficients

High-dimensional partial differential equations (PDEs) with unknown coefficients arise widely in scientific machine learning, including continuous-time reinforcement learning, yet solving them efficiently in a data-driven way remains challenging. Existing deep learning solvers often rely on repeated automatic differentiation to evaluate differential operators, which can cause instability and amplify derivative errors in high dimensions, while probabilistic methods based on stochastic representations require explicit knowledge of the data-generating dynamics and therefore do not apply to black-box environments. We introduce two types of simulators as data-generating mechanisms, and take a ``representing-then-learning" approach that learns the solutions and their derivatives under settings where the underlying PDE operators are accessible only through simulations and pointwise evaluations. Our representation of derivatives relies on the zeroth-order derivative (ZOD) estimators derived from perturbed Monte Carlo trajectories. This fully model-free approach generates targets for the gradient and Hessian networks using only function evaluations. We provide a statistical learning analysis of the proposed approach, including a bias--variance tradeoff for ZODs. Assuming a standard contraction property of the underlying operator, we establish a non-asymptotic error bound that decomposes the total error into discretization error, approximation error, statistical error, and ZOD bias. Crucially, we derive the sample complexity of the learned representations in (weighted) Sobolev space, characterizing the error up to second-order derivatives. Numerical experiments illustrate the competitive performance of the method in moderate and high dimensions.

Yanwei Jia, Du Ouyang, Huyên Pham +1

Department of Systems Engineering and Engineering Management The Chinese University of Hong Kong Shatin, New Territories, Hong Kong SAR · Department of Mathematical Sciences Tsinghua University Beijing, Beijing 100084, China · CMAP Ecole Polytechnique Route de Saclay, Palaiseau 91128, France +1

Neural PDE Solvers PDE Solving

Jun 23, 2026math.NA

Deep numerical schemes for systems of Ergodic BSDEs with applications to regime-switching forward utilities

In this paper, we introduce two neural-network-based numerical schemes for solving systems of coupled ergodic Backward Stochastic Differential Equations (eBSDEs), motivated by the approximation of optimal strategies within the framework of forward utilities in a regime-switching stochastic factor model. Our approach builds on the representation of such models through systems of eBSDEs introduced in [HLT20]. We first establish a link between the solution of the system of ergodic BSDEs and that of an associated multidimensional BSDE with random terminal time, given by the hitting time of the positive recurrent stochastic factor. Building on this representation, we introduce a locally additive deep learning scheme obtained by minimizing aggregated local error terms. We then present a new Deep Galerkin Method (DGM) inspired algorithm that minimizes the residual of the associated ergodic PDE system, relying on a representation of the ergodic cost. Finally, we apply this framework to regime-switching forward utilities in a stochastic factor model. We first derive a general consistency SPDE that characterizes regime-switching forward utilities and retrieve their representation with systems of ergodic BSDEs in the homothetic case. Numerical experiments demonstrate the performance of the proposed methods, with a particular focus on the impact on forward preferences of taking into account regime switches.

Guillaume Broux-Quemerais, Sarah Kaakai, Anis Matoussi +1

LMM · 1Laboratoire Manceau de Mathématiques & FR CNRS No 2962, Institut du Risque et de l’Assurance, Le Mans Université · LAGA +1

Stochastic Differential Equations Neural PDE Solvers Quantitative Finance Regime-Switching Dynamical Systems

Jun 20, 2026cs.LG

Parameterized Representations via Implicit Stochastic Modulation for High-Dimensional and High-Order Neural PDE Solvers

Solving high-dimensional and high-order PDEs is challenged by the coupled growth of spatial dimensionality and derivative order. Recent stochastic derivative estimators reduce this cost by replacing full derivative tensors with randomized dimension or Taylor estimators, but they are mostly designed for fixed physical parameters and require retraining for each new parameter. We show that direct conditional parameterization of such solvers entangles physical parameters with the high-order automatic differentiation graph, causing extra memory growth and parameter-induced variance amplification. We propose Parameterized Representations via Implicit Stochastic Modulation (PRISM), a plug-and-play framework for parameterized high-dimensional and high-order stochastic neural PDE solvers. PRISM uses a hyper-generator to map physical parameters to affine modulators that scale and shift a purely spatial latent manifold, while keeping parameter branches value-connected but spatial-tangent-disconnected. This design preserves unbiased stochastic dimension and Taylor estimators, removes the parameter encoder from high-order spatial AD, and provides a variance-aware Lipschitz envelope over the parameter space. We prove parameterized unbiasedness, estimation-error bounds, and convergence under bounded stochastic variance. Experiments with PRISM-STDE and PRISM-SDGD on nonlinear parameterized PDEs show stable zero-shot generalization, reduced memory usage, and scalability up to 100,000 dimensions on a single GPU, with efficient low-rank SVD adaptation for unseen parameters.

Zhangyong Liang, Huanhuan Gao

National Center for Applied Mathematics, Tianjin University, Tianjin, 300072, PR China · School of Mechanical and Aerospace Engineering, Jilin University, Changchun, 130025, PR China

Neural PDE Solvers PDE Solving

Jun 18, 2026cs.AI

Modularity-Free Conflict-Averse Training for Generalized PINNs

Physics-informed neural networks (PINNs) have become a powerful framework for solving PDEs by embedding physical laws into differentiable objectives. Despite their advances, training PINNs remains fragile: recent conflict-averse optimization schemes alleviate gradient interference between residual and boundary losses, but we show that their effectiveness deteriorates as model capacity increases. In this paper, we identify a capacity-induced failure mode, where overparameterized networks undergo functional modularity, self-partitioning into task-exclusive modules that suppress cross-objective interaction and hinder convergence toward Pareto-stationary points. To address this issue, we propose a novel framework, Modular-Sparsity Synchronization (ModSync), which integrates structural optimization into conflict-averse training by penalizing task-exclusive connections while preserving interaction-promoting pathways. Extensive experiments across diverse PDE benchmarks demonstrate that ModSync consistently prevents capacity-driven failures, sustains robust cross-objective coupling, and achieves state-of-the-art accuracy. Codes are available at https://github.com/heejokong/ModSync.

Heejo Kong, Beomchul Park, Sung-Jin Kim +1

Department of Brain and Cognitive Engineering, Korea University, Seoul, Republic of Korea · Department of Artificial Intelligence, Korea University, Seoul, Republic of Korea

Neural PDE Solvers Gradient Interference Physics-Informed ML Multi-Objective Optimization Modular Neural Networks

Jun 18, 2026math.NA

Structure-Oriented Randomized Neural Networks for Poisson-Nernst-Planck and Poisson-Nernst-Planck-Navier-Stokes Systems

We develop a structure-oriented randomized neural network framework, termed SO-RaNN, for the Poisson-Nernst-Planck (PNP) system and the Poisson-Nernst-Planck-Navier-Stokes (PNP-NS) system. The decoupled linearized subproblems are solved iteratively by randomized neural networks in a space-time framework. For the concentration variables, a pointwise cut-off is used to enforce positivity at the value level, and discrete mass-scaling factors are computed at selected correction instants and interpolated in time, so as to ensure exact mass matching at those instants and to promote approximate mass preservation between them. To introduce an auxiliary discrete dissipation mechanism, we further employ an SAV-type post-processing correction, which yields monotonicity of the SAV auxiliary variable under the ideal SAV update. For the PNP-NS system, a structure-preserving randomized neural network (SP-RaNN) is used for the velocity field, so that the velocity approximation satisfies the incompressibility constraint pointwise by construction. On the theoretical side, we derive residual-based estimates for the raw, uncorrected RaNN solvers of the linearized subproblems, formulate a conditional local-in-time convergence result for the raw outer Picard iteration of the PNP system, and analyze the value-level positivity correction together with the mass-correction and SAV post-processing steps. For the PNP-NS system, we establish an approximation result for the SP-RaNN space and provide a conditional error statement for the corresponding linearized Oseen-type problem. Numerical experiments demonstrate approximation accuracy in the source-driven manufactured tests and illustrate the intended value-level positivity correction, selected-time mass matching, computed free-energy curves based on the final gauge-fixed potential, and divergence-free approximation in benchmark tests.

Yunlong Li, Fei Wang

School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, P.R. China. · School of Mathematics and Statistics & State Key Laboratory of Multiphase Flow in Power Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China.

Neural PDE Solvers Computational Fluid Dynamics

Jun 18, 2026cs.LG

Physics-Informed Neural Network with Squeeze-Excitation-like Attention

We introduce SEA-PINN, a novel architecture that incorporates a Squeeze-Excitation-like attention mechanism into physics-informed neural networks to dynamically recalibrate the importance of neurons across layers. A key feature of SEA-PINN is its highly stable initialization. On 17 out of 20 benchmark problems, SEA-PINN exhibit nearly negligible variance and significantly reduced initial loss, establishing a quasi-deterministic and favorable starting point for optimization. Notably, without employing Fourier feature embeddings or periodic activation functions, SEA-PINN attained competitive accuracy (83% vs. 90% improvement relative to FNN-PINN on the high-frequency case 7) as compared with TSA-PINN-a model specifically engineered for high-frequency problems via learnable frequencies in sinusoidal activations. Furthermore, integrating SEA-PINN into TSA-PINN boosted performance by 42.49%. These results underscore SEA-PINN as a lightweight plug-in module that enhances nonlinear representation power, promotes more robust and efficient convergence, and strengthens the overall reliability of physics-informed learning.

Yun-Fei Song, Long-Gang Pang, Fu-Peng Li +1

Key Laboratory of Quark and Lepton Physics (MOE) & Institute of Particle Physics, Central China Normal University, Wuhan 430079, China · Artificial Intelligence and Computational Physics Research Center, Central China Normal University, Wuhan 430079, China · Key Laboratory of Nuclear Physics and Ion-beam Application (MOE) & Institute of Modern Physics, Fudan University, Shanghai 200433, China +2

Neural PDE Solvers Physics-Informed ML Neural Network Training Dynamics

Jun 17, 2026cs.LG

Trainable Photonic Measurement for Physics-Informed PDE Learning

Photonic quantum machine learning offers a route to trainable physical representations built from phase, interference and measurement. However, its role in scientific machine learning remains largely unexplored. Physics-informed neural fields provide a natural setting, because differential equations require trial spaces that preserve phase, frequency and derivative structure. Here we introduce a photonic quantum neural field in which coordinates become trainable optical phases, are mixed by multi-photon Fock-space interference and are decoded from photon-number measurements. The photonic circuit is optimized as the neural-field representation itself, not as a fixed feature map or hardware accelerator. Photonic measurement is therefore a trainable representation on which the physics-informed residual is minimized. Across seven elliptic, wave, nonlinear dispersive and inverse PDE benchmarks, we observe a phase-complexity transition: classical coordinate and Fourier-feature networks suffice in smooth regimes, whereas the photonic field is most accurate when residual derivatives amplify phase mismatch. In the hardest regimes it gives the lowest errors, with margins reaching an order of magnitude and about one quarter of the trainable parameters of classical baselines. Frozen and shuffled controls, together with noise stress tests, attribute this gain to learned interference and stable Fock-probability readout under compound perturbations. These results identify photonic quantum measurement as a representation-learning principle for scientific machine learning.

Jiale Linghu, Hao Dong, Yangshuai Wang

School of Mathematics and Statistics, Xidian University, Xi’an, Shaanxi, 710071, China. · Department of Mathematics, Faculty of Science, National University of Singapore, Singapore, 119077, Singapore.

Neural PDE Solvers Quantum Machine Learning Physics-Informed ML Photonic Computing Quantum Neural Networks

Jun 16, 2026math.NA

INI-VPINN: A Variational Physics-Informed Neural Network with Implicit Neumann and Interface Handling for Multi-Material Domains with Geometric Singularities

We propose a new weak-form Physics-Informed Neural Network approach (named INI-VPINN). INI-VPINN naturally incorporates Neumann boundary and interface conditions into the variational formulation. It removes the need for additional loss terms or multiple subdomain networks. This framework employs compact support weighting functions and integration by parts to implicitly impose flux and continuity constraints. In this way, it implicitly ensures physical consistency across material boundaries. The proposed method is tested on Poisson and Laplace problems with sharp interfaces and complex geometries. Results show that, compared with several other Physics Informed Neural Networks-based formulations, the INI-VPINN consistently achieves higher accuracy, smoother and faster convergence. The proposed framework provides a general approach for solving multimaterial problems with complex geometries and mixed Neumann-Dirichlet boundary conditions using neural networks. The implementation is publicly available in a GitHub repository.

Shayan Dodge, Alessandro Formisano, Sami Barmada

DESTeC, University of Pisa, Pisa, Italy · DESTeC University of Pisa Pisa, Italy · Department of Engineering, University of Campania Luigi Vanvitelli, Aversa, Italy +1

Neural PDE Solvers PDE Solving Physics-Informed ML

Jun 15, 2026math.NA

Petrov-Galerkin Variational Physics-Informed Neural Network Framework for Two-Dimensional Singularly Perturbed Problems

This study proposes a Petrov-Galerkin based Variational Physics-Informed Neural Network (VPINN) for efficiently solving two-dimensional singularly perturbed problems (SPPs) with one and two small perturbation parameters. The approach employs neural networks to construct the trial solution space, while tensor-product hat functions are adopted as test functions to enforce the variational form. To accurately resolve of sharp boundary layers, the variational form is implemented using a Petrov-Galerkin formulation. Dirichlet boundary conditions are imposed directly, while the source terms are computed using automatic differentiation. Computational experiments on standard two-dimensional problems demonstrate that the proposed method achieves high accuracy in both the maximum and L_2 norms. These results confirm the efficiency and robustness of the Petrov-Galerkin VPINN approach in accurately capturing the multiscale features of two-dimensional SPPs.

Vijay Kumar, Gautam Singh

Department of Mathematics, National Institute of Technology Tiruchirappalli, Tiruchirappalli, 620015, Tamil Nadu, India.

Neural PDE Solvers Physics-Informed ML

Jun 12, 2026cs.LG

Separable Neural Architectures as Physical World Models: from Mathematical Theory to Applications

This work introduces the Separable Neural Architecture (SNA), a function representational class combining neural approximation with tensor decomposition. The SNA decouples localized coordinate functions (atoms) from global interactions governed by a sparse, low-rank interaction object. This architecture possesses a compact and smooth inductive bias well-suited for solving partial differential equations (PDEs). When viewed as a Galerkin trial space under the variational SNA (VSNA) framework, the formulation satisfies classical variational guarantees under Lax-Milgram: well-posedness, quasi-optimality, convergence, and stability. In high-dimensional spatiotemporal--parametric PDEs, the VSNA mitigates the curse of dimensionality by scaling algebraically rather than exponentially. Exploiting an entirely factorized, tensor-native alternating least squares (ALS) optimization framework reduces this cost to linear in dimension. The VSNA is validated across elliptic, hyperbolic, and parabolic systems, demonstrating close alignment with predicted algebraic and spectral scaling rates. We showcase the SNA as a "solve once, query anywhere" physical world model via two engineering case studies: a 7D parametric manufacturing simulation and an experimental thermal-to-property inversion pipeline for Inconel 718. The VSNA executes a 1,000,000-query Monte Carlo sweep in 102s on a standard laptop CPU, yielding a 150,000x speedup over a full-grid finite element baseline hosted on an NVIDIA A100 GPU. It further enables real-time generative inverse-mode reconstructions under 100ms. These results demonstrate that the SNA serves as a compact mathematical substrate for continuous parameter manifolds to enable real-time inversion, optimization loops, and rapid uncertainty propagation.

Reza T Batley, Andrew Kichline, Sourav Saha

Kevin T. Crofton Department of Aerospace and Ocean Engineering, Virginia Polytechnic Institute and State University, 1600 Innovation Drive, Blacksburg, 24060, VA, United States

Neural PDE Solvers PDE Solving PDE Surrogate Modeling

Jun 10, 2026cs.LG

HAMNO: A Hierarchical Adaptive Multi-scale Neural Operator with Physics-Informed Learning for Dynamical Systems

Neural operators provide a powerful framework for learning solution mappings of partial differential equations directly in function space. However, many existing architectures still struggle to represent nonlinear time-dependent systems that involve multi-scale structures, long-range interactions, and stable long-time evolution. In this work, we introduce the Hierarchical Adaptive Multi-scale Neural Operator (HAMNO), a neural-operator architecture that combines local convolutional representations, global spectral operators, and hierarchical encoder-decoder processing. The central component of HAMNO is a data-dependent gating mechanism that adaptively balances local and global information at each spatial location, allowing the model to resolve fine-scale features while preserving long-range dependencies. We further develop a physics-informed extension, PI-HAMNO, based on a multi-objective loss strategy that combines data fitting with strong- and weak-form physics constraints. The strong-form term penalizes the domain-integrated squared PDE residual in physical coordinates, while the weak-form term is constructed by multiplying the governing residual by finite-element test functions and evaluating the resulting element integrals using centroid-based tetrahedral quadrature. The framework is evaluated on non-periodic Allen-Cahn (AC), Cahn-Hilliard (CH), and Swift-Hohenberg (SH) equations defined on cubic domains. Across long-horizon rollout, data-limited training, out-of-distribution initial-condition shifts, and random-seed variations, HAMNO improves predictive accuracy over standard neural-operator baselines, while PI-HAMNO further enhances stability, physical consistency, and data efficiency. The implementation is publicly available at https://github.com/MBamdad/HAMNO .

Mostafa Bamdad, Mohammad Sadegh Eshaghi, Timon Rabczuk

Chair of Computational Mechanics, Institute of Structural Mechanics, Bauhaus-Universität Weimar, 99423 Weimar, Germany · Chair of Computational Science and Simulation Technology, Institute of Photonics, Department of Mathematics and Physics, Leibniz University Hannover, 30167 Hannover, Germany

Neural PDE Solvers Physics-Informed Neural Operators Neural Operators

Jun 9, 2026physics.comp-ph

An adaptive framework for the axisymmetric pulsar magnetosphere using physics-informed Kolmogorov-Arnold networks

The pulsar magnetosphere has only recently been addressed using Physics-Informed Neural Networks (PINNs), by deploying a domain-decomposition approach and treating the separatrix and equatorial current sheet as infinitesimally thin discontinuities. However, this baseline requires extensive manual hyperparameter tuning, achieves limited final accuracy and demands several hours of training. We refine this framework by introducing domain-specific neural architectures based on Kolmogorov-Arnold networks, an automated adaptive training pipeline and a physics-based convergence criterion that eliminate the need for manual calibration. The proposed methodology delivers self-consistent axisymmetric magnetosphere solutions with mean squared errors of the PDE residuals at O(1e-6) in double precision - an improvement of two orders of magnitude over the baseline - while achieving convergence in under 20 minutes in single precision. Importantly, the method reliably resolves stellar radii reduced by up to 80% compared to the baseline, overcoming the severe spatial scale disparities that also challenge traditional solvers. Furthermore, by varying the flux that opens to infinity, we provide a correction to the equation that connects it to the equatorial T-point's position. The complete framework is released as the open-source library PulsarX.

Spyros Rigas, Ioannis Contopoulos, Georgios Alexandridis +1

Department of Digital Industry Technologies, School of Science, National and Kapodistrian University of Athens, 34400 Psachna, Greece · Research Center for Astronomy and Applied Mathematics, Academy of Athens, 11527 Athens, Greece

Neural PDE Solvers Physics-Informed ML

Jun 5, 2026cs.LG

Architecture Shapes Transfer Specificity in Implicit Neural Representations

Transfer in coordinate networks is often measured by warm-start gain, but whether that gain reflects source-specific structure or generic weight reuse is less clear. We study this question across three implicit neural representation (INR) families, SIREN, ReLU MLPs, and Fourier-feature MLPs, using controlled analytic tests, a 2D lid-driven-cavity Navier--Stokes benchmark, and 1D PDE reference-solution suites for heat, viscous Burgers, and focusing cubic NLS. The analytic tests use independent-seed random controls, while the PDE benchmarks use alternate same-family source controls and auxiliary ablations. Across settings, transfer magnitude and transfer specificity separate clearly. In a 10-seed controlled 1D geometric test, Fourier Features show the largest structured transfer (

33.1\times

), followed by SIREN (

23.0\times

) and ReLU (

10.7\times

), but ReLU is far more selective: random-control transfer is

0.41\times

for ReLU versus

14.24\times

for SIREN. On a controlled two-parameter 1D family, the ranking changes: ReLU gives the clearest structured-versus-control separation at default settings, whereas Fourier Features improve only after bandwidth retuning. In Navier--Stokes and the broader 1D PDE suite, no single architecture dominates every equation, yet the same pattern remains: SIREN often reuses weights broadly, whereas ReLU and, in some equations, Fourier Features are more source-selective. Static diagnostics remain weak, and the heuristic scaling law

A_{\text{transfer}} \propto 1/Δt^2

is rejected in the implemented 1D audit. These results position transfer specificity as a useful diagnostic for coordinate networks and suggest that architecture selection in scientific machine learning should be evaluated under explicit control conditions, not by transfer magnitude alone.

D Yang Eng

Neural PDE Solvers Implicit Neural Representations Transfer Learning

Jun 3, 2026cs.LG

Curvature-aware dynamic precision approach for physics-informed neural networks

Physics-informed neural networks (PINNs) have become a promising framework for simulating partial differential equations (PDEs) by embedding physical laws directly into neural network training. However, recent studies show that PINN optimisation is sensitive to numerical precision. Existing implementations commonly use either single precision (FP32), which is computationally efficient but prone to failure modes, or double precision (FP64), which is robust but substantially expensive. This creates a trade-off between computational efficiency and numerical accuracy. To reduce the computational cost of double-precision training while retaining prediction accuracy, we propose a curvature-aware precision controller that adapts numerical precision during training rather than treating it as a fixed implementation choice. The proposed method reuses curvature information derived from the limited-memory BFGS (L-BFGS) optimiser to construct a precision controller, retaining FP32 when lower precision is sufficient and promoting computation to FP64 when the training dynamics indicate numerical sensitivity or precision-limited stagnation. We evaluate the proposed approach on four canonical PINN failure-mode benchmarks and an irradiance-driven ordinary differential equation example. We further test the proposed approach across different neural network architectures. The method consistently matches or even slightly exceeds full FP64 solution accuracy while reducing training time relative to full double-precision training on all benchmark equations. The obtained results indicate that precision sensitivity in PINN optimisation is phase-dependent, and that selectively applying higher precision only during numerically critical stages can lower computational cost without sacrificing predictive accuracy.

Yingjie Shao, Ioannis N. Athanasiadis, George van Voorn +1

Mathematical & Statistical Methods Group (Biometris), Wageningen University & Research, Wageningen, 6700AA, The Netherlands · Artificial Intelligence Group, Wageningen University & Research, Wageningen, 6700AA, The Netherlands

Neural PDE Solvers Neural Network Optimization Physics-Informed ML

Jun 3, 2026cs.LG

MeshTok: Efficient Multi-Scale Tokenization for Scalable PDE Transformers

Conventional patchified Transformers operate on uniform spatial partitions, distributing computational effort evenly across the domain irrespective of local features. This inflexible tokenization scheme is inherently limited in its ability to efficiently represent and process solutions to complex PDEs. To address this, we propose MeshTok, an adaptive mesh refinement (AMR)-inspired tokenization and sequence modeling framework. This method selectively refines spatial regions exhibiting sharp gradients, transient features, or multiscale structures, generating a heterogeneous set of multiscale tokens defined on a fixed simulation grid. These tokens are processed within a unified Transformer sequence, enabling the model to simultaneously capture coarse-grained global context and fine-grained local details without requiring specialized architectural components. Although adaptive refinement moderately increases token count, it promotes a more targeted allocation of computational resources to physically informative regions, which we view as a practical inductive bias rather than a formal optimality guarantee. Experimental evaluations across multiple PDE families and benchmark datasets demonstrate that MeshTok consistently improves the efficiency-accuracy trade-off compared to uniform-grid baselines. This suggests adaptive multiscale tokenization as a scalable and generalizable design principle for neural PDE modeling. Code is available at https://github.com/SCAILab-USTC/MeshTok.

Yanshun Zhao, Xiaoyu Peng, Jiamin Jiang +2

School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China · Suzhou Institute for Advanced Research, University of Science and Technology of China, Suzhou 215123, China.

Neural PDE Solvers PDE Surrogate Modeling Adaptive Mesh Refinement

Jun 3, 2026cs.LG

Neural Galerkin Normalizing Flows for Bayesian Inference of Diffusions with Inaccessible Boundaries

One of the primary challenges in Bayesian inference on the parameters of a diffusion model from discrete observations is the unavailability of an analytical expression for the transition density function between consecutive observation times, which is needed to derive the likelihood function. Extending previous studies that solve Fokker-Planck (FP) type partial differential equations with Normalizing Flows, we propose a new Normalizing Flow architecture to learn the transition density function of the diffusion process between two observation times. We do so by solving in a Neural Galerkin framework the associated FP equation with a Dirac mass as initial condition, over a specified training distribution of the initial datum and the coefficients of the diffusion. We specifically focus on processes whose diffusion matrix vanishes in certain inaccessible boundary regions, such as Stochastic Volatility models that satisfy a Feller condition. The product of the obtained transition densities evaluated along the observed trajectory approximates the likelihood function, thereby enabling cheap posterior sampling via Markov chain Monte Carlo (MCMC). After the offline training phase, inference becomes significantly more efficient, as it avoids the need to solve the FP equation in real time for each parameter proposed by the MCMC sampler or to rely on other likelihood-free methods for Bayesian inference that involve repeated simulation of diffusion bridges.

Riccardo Saporiti, Fabio Nobile

CSQI École Polytechnique Fédérale de Lausanne Lausanne, Switzerland 1015

Normalizing Flows Neural PDE Solvers Bayesian Inference

May 31, 2026cs.LG

A Per-Component Diagnostic Protocol for Neural HJB-PIDE Solvers under Control-Dependent Lévy Jumps

We propose a five-step diagnostic protocol for residual-trained neural HJB-PIDE solvers with control-dependent Lévy jumps, targeting a general failure mode of neural PDE methods: a learned solution can match headline scalar diagnostics while miscomputing an operator inside its training loss. The protocol pairs each neural solve with at least one from-scratch independent reference, decomposes the Hamiltonian into drift, diffusion, compensator, and nonlocal-integral components across a u-grid, and compares the value function and its low-order derivatives over a (t,x) grid before any argmax comparison. Applied to a standard CRRA-Merton-Variance-Gamma benchmark, it isolates a missing 1/2-mixture factor in the neural method's importance-proposal density that scaled the nonlocal integral by exactly half - a textbook signature of a constant proposal scale error, invisible to longer training, grid refinement, and truncation sweeps. With the bug corrected, four references - two finite-difference solvers with disjoint discretizations, the neural solver, and a semi-analytic scalar baseline obtained from CRRA homogeneity - agree on the optimal control to within ~2%. The constant-coefficient CRRA benchmark collapses by homogeneity to a scalar maximization, so the scalar baseline is the efficient method here; the contribution is the protocol, applicable in principle to non-homogeneous and higher-dimensional settings where neural HJB-PIDE solvers are genuinely needed. The episode is a concrete instance of a broader neural-PDE verification failure: pointwise agreement of a learned value or control can coexist with a systematically wrong nonlocal operator, so per-component and surface-level checks are needed before trusting the argmax policy.

R. Drissi

Neural PDE Solvers Stochastic Optimal Control

May 29, 2026math.NA

A holomorphic neural network framework for 3D boundary value problems governed by harmonic potentials

We present a neural-network-based framework for the solution of three-dimensional boundary value problems where the solution is expressible in terms of harmonic potentials. The approach leverages the Whittaker integral formula, which allows representing the solution through functions that are holomorphic with respect to a suitable complex variable. These functions are subsequently approximated using holomorphic neural networks, which guaranty fulfillment of the holomorphicity requirement. A key feature of the proposed formulation is that the governing partial differential equations (PDEs) are satisfied exactly by construction. Therefore, in contrast to standard physics-informed neural networks, no residual minimization of PDEs is required in the interior of the domain, and training is based exclusively on boundary collocation points. The method is validated against three-dimensional Laplace and linear elasticity problems, where, in the latter case, displacement and stress fields are expressed via the Papkovich-Neuber potentials. The numerical results show an accurate approximation of both scalar and vector fields, with errors remaining controlled throughout the domain. Overall, the work demonstrates that the incorporation of analytical structures into neural network architectures provides a natural and effective framework for the meshless approximation of three-dimensional boundary value problems while preserving the underlying properties of the governing equations.

Enrico Ballini, Allan Peter Engsig-Karup, Tito Andriollo

Department of Mechanical and Production Engineering, Aarhus University, Aarhus, Denmark · Department of Applied Mathematics and Computer Science, Technical University of Denmark, Kongens Lyngby, Denmark

Neural PDE Solvers PDE Solving

May 29, 2026cs.NE

Oscillatory State-Space Models as Inductive Biases for Physics-Informed Neural PDE Solvers

Solving time-dependent partial differential equations (PDEs) is an important problem in computational science and engineering. Physics-informed neural networks (PINNs) learn PDE solutions from governing equations. However, accurately capturing temporal evolution remains challenging. Recent sequence-model-based approaches parameterize time evolution using general-purpose sequence models, which capture temporal dependencies but do not explicitly encode the structured dynamics of PDE solutions. In addition, their memory requirements can scale unfavorably with sequence length and resolution, limiting applicability in large-scale or high-dimensional settings. This work introduces a PINN approach that incorporates oscillatory state-space dynamics to represent the modal structure of PDE solutions. The proposed method leverages a linear-oscillator-based temporal evolution, together with a PDE-aware spectral basis in space. This design enables closed-form spatial differentiation and consistent enforcement of boundary conditions. The method is evaluated on forward, inverse, and high-dimensional PDE problems, including cases up to 100 spatial dimensions. The results show improved accuracy and reduced memory usage compared to recent sequence-model-based PINN approaches. Overall, this work highlights the benefits of incorporating structured dynamical priors into the temporal evolution of neural PDE solvers and suggests designing more physics-aligned and computationally efficient PINN architectures.

Abhishek Chandra, Taniya Kapoor

KTH Royal Institute of Technology Stockholm, Sweden · Wageningen University & Research Wageningen, The Netherlands

Neural PDE Solvers Physics-Informed ML PDE Inverse Problems State Space Models

May 29, 2026cs.LG

Multi-Scale Separable Fourier Neural Networks for Solving High-Frequency PDEs

Solving high-frequency partial differential equations (PDEs) with neural networks is notoriously difficult due to the spectral bias of conventional architectures. We propose the Multi-Scale Separable Fourier Neural Network (MS-SFNN), a framework designed to overcome this limitation by explicitly encoding multi-scale Fourier features within a separable representation. The network factorizes the solution into

d

single-coordinate subnetworks with fixed, randomly initialized weights; these subnetworks are combined via element-wise products to form a rich set of basis functions. This separable construction scales linearly with the problem dimension, thus inherently alleviating the curse of dimensionality. Crucially, each subnetwork is equipped with trainable scaling factors coupled with cosine activations, providing an adaptive mechanism for multi-scale frequency selection that endows the model with strong spectral approximation capability. The PDE solution is expressed as a linear combination of these learned basis functions, and the combination coefficients are determined by solving a large-scale least-squares system. To resolve the memory bottleneck in high-frequency or three-dimensional settings, we replace automatic differentiation (AD) with analytical derivatives of the basis functions and use a memory-efficient batched QR decomposition for solving the least-squares systems efficiently. Extensive numerical experiments demonstrate that MS-SFNN achieves superior accuracy and substantially outperforms state-of-the-art methods, including Physics-Informed Neural Network (PINN) and the Separated-Variable Spectral Neural Network (SV-SNN).

Qihong Yang, Qiaolin He

School of Mathematics, Sichuan University, city=Chengdu, postcode=610065, country=China

Neural PDE Solvers PDE Solving Frequency-Domain Feature Learning

May 29, 2026cs.LG

Physics-Informed Coarsening for Multigrid Graph Neural Surrogates

Learning-based surrogates for partial differential equations have recently matched the accuracy of classical solvers while achieving orders-of-magnitude speedups, predominantly in fluid settings and structured geometries. In contrast, robust surrogates for deformable solids remain underexplored, despite the presence of nonlinear elasticity, plasticity, and transient behavior that challenge standard architectures. We introduce a multigrid graph neural network for solid mechanics that couples an encoder-processor-decoder backbone with a physics-informed coarsening strategy. Instead of downsampling via geometric heuristics, our method scores nodes using a residual-based measure of local physical activity and preferentially retains regions of high strain or stress concentration, allocating multiscale capacity where it is most needed. This preserves long-range interactions through hierarchical message passing while improving stability over long rollouts. We evaluate on multiple datasets covering linear, nonlinear, and transient regimes, and observe consistent gains in accuracy and rollout stability compared to standard sampling baselines. Our results highlight the importance of physics-informed coarsening for scalable surrogate modeling in solid mechanics.

Amir Bazzi, David Cardinaux, Ramy Nemer +3

1CEMEF, Mines Paris – PSL, Sophia Antipolis, France · 2Aubert et Duval, France · 3Transvalor, Mougins, France.

Graph Neural Networks Neural PDE Solvers Structural Mechanics PDE Surrogate Modeling Physics-Informed ML

May 29, 2026cs.LG

LFNO: Bridging Laplace and Fourier via Transient-Steady Decomposition

We introduce the Laplace-Fourier Neural Operator (LFNO), a unified framework for modeling dynamical systems across transient and steady-state regimes by integrating the spectral advantages of Laplace and Fourier Neural Operators. LFNO employs a dual-branch architecture that explicitly decomposes system dynamics into transient and steady-state components. We evaluate LFNO on nine benchmarks, including three ODE systems (Duffing, Lorenz, and Pendulum) and six PDE systems (Euler-Bernoulli beam, Heat, Reaction-diffusion, Brusselator, Burgers, and Navier-Stokes). LFNO significantly outperforms existing operators on ODE systems, where transient dynamics dominate, and consistently surpasses LNO while achieving performance competitive with FNO on PDE benchmarks. Furthermore, LFNO offers improved stability and physical interpretability through its component-wise decomposition. These results demonstrate that LFNO provides a robust and unified approach for learning complex dynamical systems across multiple temporal scales.

Jeongun Ha, Sanga Yoon, Donghun Lee

Department of Mathematics, Korea University, 145 Anam-ro Seongbuk-gu Seoul, Republic of Korea.

Dynamical Systems Neural PDE Solvers PDE Operator Learning Neural Operators

May 29, 2026cs.LG

PINNs Failure Modes are Overfitting

Physics-Informed Neural Networks (PINNs) are a common class of machine learning-based partial differential equation (PDE) solvers which train a network to represent a solution by minimizing a residual loss that encodes the PDE. Despite their successes, they are known to fail on certain simple equations, converging to an incorrect solution despite low loss. These failure modes have garnered significant attention in the literature over the past several years, motivating both architectural and optimization based solutions. By directly visualizing the residual, we show that failure modes are the result of overfitting: the loss is minimized on the collocation points, but not elsewhere. Applying regularization causes the failure modes to vanish. Finally, we extend double backpropagation over the full set of residuals, and use it to achieve state-of-the-art performance on four standard failure mode equations with up to

23\times

fewer collocation points and a vanilla architecture.

Nigel T. Andersen, Takashi Matsubara

Graduate School of Information Science and Technology Hokkaido University Sapporo, Japan · RIKEN Center for Advanced Intelligence Project (AIP) Tokyo, Japan

Neural Network Generalization Neural PDE Solvers Backpropagation Physics-Informed ML

May 28, 2026cs.LG

Striding Across Reynolds Numbers: Representation Geometry in Neural PDE Generalisation

Cross-Reynolds generalisation in neural PDE solvers remains poorly characterised. On the canonical forced 2D Navier-Stokes benchmark, a trained Fourier Neural Operator reaches 46.68% relative L2 error under a 10x Reynolds-number shift, yet zero-forward-model retrieval baselines already improve to 41-42%. This suggests representation geometry as a major organising variable among the tested methods. We test this hypothesis through ConvAE-Relay, which matches states in a source-trained convolutional autoencoder latent space and borrows dynamics from a source-regime database, achieving 38.34+/-0.07% using only a source-regime database and no target-regime fitting, labels, or database entries. A 2x2 ablation isolates matching quality as dominant over the update rule. Oracle experiments confirm that source-regime dynamics directions remain transferable (cosine similarity ~0.84) when matching stays on-manifold; autoregressive drift is the primary bottleneck (~12 percentage points). From the learned-prediction side, a U-Net with multi-scale skip connections achieves 34.72+/-0.60%, consistent with the retrieval-side finding that local, multi-scale representations organise cross-Reynolds transfer among tested methods. All claims are scoped to this benchmark.

Jianing Shi

London School of Economics and Political Science.

Neural PDE Solvers Neural Representation Geometry Computational Fluid Dynamics

May 28, 2026cs.LG

A Novel Tensor Product-Based Neural Network for Solving Partial Differential Equations

This paper presents the Tensor Product Network (TPNet), a novel neural architecture for efficient and accurate function approximation and PDE solving. The core of the proposal involves constructing the solution explicitly as a linear combination of basis functions integrated into the network, with coefficients determined by a direct least-squares solve, thereby bypassing traditional gradient-based training. The key methodological contribution include: (1) an efficient tensor-product scheme that generates multi-dimensional basis functions from combinations of two sets of subnetwork outputs, significantly reducing model complexity and parameter count while maintaining expressivity; (2) a block time-marching strategy to improve computational efficiency in long-time simulations; and (3) a linear reformulation strategy for handling nonlinear PDEs by treating known nonlinear terms as sources. TPNet achieves superior accuracy and shorter training times than conventional neural network solvers. This performance gain stems from its structured design and deterministic least-squares fitting, which contrast with the iterative, often computationally intensive optimization required by mainstream methods like Physics-Informed Neural Networks (PINNs).

Qihong Yang, Yangtao Deng, Qiaolin He +1

School of Mathematics, Sichuan University, Chengdu, 610065, China

Neural PDE Solvers PDE Solving

May 25, 2026cs.LG

Small Models, Strong Priors: Architectural Inductive Bias for Parameter-Efficient Neural PDE Solvers

Neural PDE solvers have followed the scaling trajectory of vision and language, with recent foundation models reaching billions of parameters. We argue that scale is a poor substitute for architectural inductive bias in this domain: structured priors deliver outsized parameter efficiency, and the pattern of where they succeed and fail is itself informative about what they capture. We instantiate this argument in WaveLiT, an architecture combining a discrete wavelet transform for lossless multi-resolution tokenization, an augmented linear attention block, a shared-weight multiscale feature pyramid, and a wavelet-domain auxiliary loss. Bespoke 1-10M-parameter WaveLiT models compete with foundation models of 100-1000

\times

their size across eight TheWell benchmarks, with the largest gains on wave and acoustic-dominated benchmarks where the wavelet-multiscale prior fits the dominant dynamical structure and small per-step errors do not compound geometrically under rollout. Trained jointly across all eight benchmarks, a 10M-parameter foundation variant exhibits a structured, physically interpretable transfer pattern -- strongest where the wavelet-multiscale prior matches the dynamics, weakest on chaotic advection-dominated flows. The entire pipeline trains on a single GPU. The results suggest that small-model PDE performance is shaped by architectural inductive bias rather than scale, and that the structure of a prior's failures is a useful empirical signal about its content.

Shyam Sankaran, Hanwen Wang, Paris Perdikaris

Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania · Graduate Program in Applied Mathematics and Computational Science, University of Pennsylvania

Neural PDE Solvers PDE Operator Learning

May 25, 2026cs.LG

NPSolver: Neural Poisson Solver with Iterative Physics Supervision

Efficiently solving Poisson equations on complex, irregular domains remains a fundamental challenge in scientific computing, as classical iterative solvers often suffer from prohibitive runtime due to ill-conditioned systems. While neural operators offer a fast alternative, they typically rely on large-scale labeled datasets or struggle with unstable training dynamics when using physics-informed residual losses. We propose \textsc{NPSolver}, a neural Poisson solver trained without solution labels via iterative physics supervision. Instead of relying on fully converged numerical solutions or raw PDE residuals, \textsc{NPSolver} utilizes a small number of preconditioned conjugate gradient (PCG) steps to refine its own predictions, providing a more stable and well-scaled training signal. Theoretical analysis confirms that this iterative supervision serves as a well-conditioned error proxy and that a stop-gradient design is essential for optimization stability. To better capture boundary-driven features under mixed boundary conditions, we further introduce the Boundary-Aware Transolver (\textsc{BA-Transolver}) architecture that explicitly separates interior and boundary tokenization. Extensive evaluations on 2D and 3D irregular geometries demonstrate that \textsc{NPSolver} outperforms both physics-informed and data-driven baselines. Furthermore, a downstream thermal control task highlights the model's capability for conducting efficient and reliable gradient-based boundary control. We will release our codes and data at https://github.com/intell-sci-comput/NPSolver.

Bocheng Zeng, Rui Zhang, Runze Mao +5

Gaoling School of Artificial Intelligence Renmin University of China Beijing, China · School of Engineering Science University of Chinese Academy of Sciences Beijing, China

Neural PDE Solvers PDE Solving Physics-Informed Neural Operators

May 24, 2026math.NA

Random Neural Network Expressivity for Non-Linear Partial Differential Equations

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, showing that the obtained convergence rates extend beyond the considered setting.

Muhammed Ali Mehmood, Lukas Gonon

Department of Mathematics Imperial College London UK · School of Computer Science University of St. Gallen Switzerland

Neural PDE Solvers PDE Solving Neural Network Approximation Theory Neural Network Expressivity

May 24, 2026cs.LG

Mitigating Gradient Pathology in PINNs through Aligned Constraint

While Physics-Informed Neural Networks (PINNs) are powerful for solving Partial Differential Equations (PDEs), their training is often paralyzed by gradient pathology. The gradients from the PDE residuals and boundary constraints oppose each other, trapping the model in local minima. Current solutions, such as adaptive weighting or hard constraints, either fail to fundamentally resolve this ill-conditioning or are limited to simple geometries. In this study, we systematically analyze the possible causes of this gradient pathology from the perspectives of loss landscapes and optimization dynamics. Based on the obtained conclusion, we propose Constraint-Aligned loss with Manifold Lifting (CAML). By reformulating all zeroth-order terms into aligned constraints, our method effectively mitigates gradient conflicts. In addition, we introduce a delay factor to help the optimizer skip the high-curvature area. Experiments demonstrate that our CAML significantly enhances numerical stability and efficiency in highly complex PINN problems. Our code is open-sourced on https://github.com/YichenLuo-0/CAML.

Yichen Luo, Peiyu Zhu, Dongxiao Hu +5

Department of Information Science and Engineering, KTH Royal Institute of Technology, Stockholm, Sweden · School of Advanced Manufacturing and Robotics, Peking University, Beijing, China · School of Advanced Technology, Xi’an Jiaotong-Liverpool University, Suzhou, China +2

Neural PDE Solvers Gradient Interference Constrained Optimization Physics-Informed ML Neural Network Training Dynamics

May 24, 2026math.NA

IV-Net: A neural network for elliptic PDEs with random and highly varying coefficients

We introduce a novel neural operator architecture designed to approximate solutions of linear elliptic partial differential equations with high-contrast, spatially varying coefficients. The network, termed the Iterated V-shaped Net (IV-Net), realizes a mapping from the input coefficients and righthand side to the corresponding solution field. The architecture of IV-Net is informed by, and closely resembles, a V-cycle multigrid solver. The IV-Net model is parameterized via convolutional layers defined in the physical domain. For coercive problems with highly heterogeneous coefficients, the proposed network exhibits superior performance relative to a proper orthogonal decomposition (POD) approach and several existing neural operator architectures. For low-frequency oscillatory Helmholtz problems with smooth coefficients, its performance is similar to that of a Fourier neural operator. We analyze the approximation error and convergence behavior of IV-Net, its data efficiency, and its dependence on the underlying discretization mesh. Furthermore, we demonstrate the practical effectiveness of the architecture through a series of numerical experiments, including applications to uncertainty quantification, inverse problems, and prediction of quantities of interest.

Shan Zhong, George Biros

Oden Institute for Computational Science and Engineering, The University of Texas at Austin, 201 E 24th St, Austin, 78712, TX, USA · Walker Department of Mechanical Engineering, The University of Texas at Austin, 3012026 E Dean Keeton St, Austin, 78712, TX, USA

Neural PDE Solvers PDE Solving PDE Operator Learning Neural Network Approximation Theory Neural Operators

May 23, 2026cs.LG

Hermite-NGP: Gradient-Augmented Hash Encoding for Learning PDEs

We propose Hermite-NGP, a gradient-augmented multi-resolution hash encoding designed to enable fast and accurate computation of spatial derivatives for neural PDE solvers. Unlike existing NGP-based approaches that rely on automatic differentiation or finite differences and suffer from instability or high cost, Hermite-NGP explicitly stores function values and mixed partial derivatives at hash grid vertices, allowing fully analytic evaluation of gradients, Jacobians, and Hessians via Hermite interpolation. This design preserves the efficiency and spatial adaptivity of NGP while supporting analytic differential operators up to second order. We further introduce a multi-resolution curriculum training strategy analogous to multigrid V-cycles to enable coarse-to-fine optimization. Across a range of 2D and 3D PDE benchmarks, Hermite-NGP achieves up to approximately 20 times lower error than prior neural PDE methods, and reduces wall-clock convergence time by 2 to 10 times compared to other solvers, with per-epoch training times as low as 3.5 ms for models with up to 17M parameters.

Jinjin He, Zhiqi Li, Sinan Wang +1

Georgia Institute of Technology, Atlanta, GA, USA.

Neural PDE Solvers PDE Solving PDE Surrogate Modeling

May 23, 2026math.NA

WINO: A Weak-Form Physics Informed Neural Operator for Hyperelasticity on Variable Domains

We propose a Weak-form Physics-Informed Neural Operator (WINO), a data-free framework that combines the efficiency of neural operators with the geometric flexibility of the

\varphi

-finite element method (

\varphi

-FEM).

\varphi

-FEM is an unfitted method that accommodates geometric variations without body-fitted meshes, where the domain geometry is represented by the level-set function

\varphi

. To impose the boundary conditions, Dirichlet problems adopt the

\varphi

-FEM lifting so only the homogeneous displacement contribution is learned, whereas traction-driven Neumann problems additionally predict the auxiliary fields necessary for the unfitted weak formulation. Parameters are trained by minimizing squared weak-form residuals aligned with

\varphi

-FEM together with squared penalties on the cut-cell auxiliary equations, which removes the need for large paired datasets of converged reference solutions. After training, WINO outputs can seed the nonlinear

\varphi

-FEM solvers as neural operator warm starts (NOWS), which reduce iteration counts relative to traditional cold-started solvers. Numerical benchmarks show that WINO achieves high accuracy below 0.04 across all benchmarks, while reducing total computational time by 50--80% compared with purely data-driven methods.

Bokai Zhu, Qinghui Zhang, Timon Rabczuk

School of Science, Harbin Institute of Technology, Shenzhen, P. R. China. · School of Science, Harbin Institute of Technology, Shenzhen, Guangdong. · Institute of Structural Mechanics, Bauhaus-Universität Weimar, Weimar, 99423, Germany.

Neural PDE Solvers PDE Solving Structural Mechanics Hyperelasticity Finite Element Method Physics-Informed Neural Operators

May 21, 2026cs.LG

Iterative Refinement Neural Operators are Learned Fixed-Point Solvers: A Principled Approach to Spectral Bias Mitigation

Neural operators serve as fast, data-driven surrogates for scientific modeling but typically rely on a monolithic, single-pass inference procedure that struggles to resolve high-frequency details, a limitation known as spectral bias. We introduce the Iterative Refinement Neural Operator (IRNO), which augments pre-trained operators with a learned refinement module iteratively applied via fixed-point iteration. IRNO decomposes the prediction into a coarse initialization followed by successive residual corrections, paralleling classical numerical solvers. Under local assumptions, we establish contraction of the induced operator, ensuring convergence to a unique fixed point. To explicitly target high-frequency errors, we propose a progressive spectral loss that adaptively increases penalty on high-frequency components over refinement steps during training. Across physical systems, IRNO consistently lowers error, with up to 56.05% improvement on turbulent flow. On Active Matter, spectral analysis reveals that, relative to base operator, the normalized error ratios decrease to 27.72-36.10% in low-, 5.07-6.68% in mid-, and 1.48-2.04% in high-frequencies, remaining stable beyond the trained iteration count. Code is available at https://github.com/xiaotianliu-dartmouth/Iterative_Refinement_Neural_Operator

Xiaotian Liu, Shuyuan Shang, Xiaopeng Wang +2

1Dartmouth College · 2CUHK Shenzhen · 3Lawrence Berkeley National Lab

Neural PDE Solvers Spectral Bias Iterative Refinement Neural Operators

May 19, 2026cs.LG

Fast Reconstruction of Exact Maxwell Dynamics from Sparse Data

We introduce FLASH-MAX, a shallow, exact-by-construction neural network architecture for predicting homogeneous electromagnetic fields from sparse pointwise observations. Each hidden neuron represents a separate exact solution to Maxwell's equations, so that the network satisfies the governing equations symbolically by construction and can be trained end-to-end from sparse data within seconds. We prove a universal approximation result showing that this exact model class remains universal on arbitrary domains. FLASH-MAX reaches sub-1% relative validation error from about 1K sparse pointwise observations in seconds, all while maintaining a zero PDE residual, and keeps single-digit errors even for only 100 observations sampled from 3D space. These results suggest that moving governing structure from the loss into the hypothesis class can dramatically improve the trade-off between precision and optimization speed in scientific machine learning.

Dan DeGenaro, Xin Li, Obed Amo +4

Department of Computer Science, Georgetown University · Department of Mathematics, Georgetown University · Department of Mathematics and Statistics, Old Dominion University +2

Neural PDE Solvers Neural Network Approximation Theory Physics-Informed ML

May 14, 2026cs.LG

Breakeven complexity: A new perspective on neural partial differential equation solvers

Neural surrogate solvers of partial differential equations (PDEs) promise dramatic speedups over numerical methods, especially in scenarios requiring many solves. However, current accuracy-based evaluations do not fully consider two central issues: (1) neural solvers incur substantial up-front costs for data generation, training, and tuning; and (2) classical solvers can also generate low-fidelity solutions at a sufficiently low simulation cost. To explicitly account for these realities and fully incorporate end-to-end costs, we propose an evaluation framework centered on breakeven complexity, a metric that counts the forward solves before a learned solver is cost-effective relative to an error-equivalent traditional solver. To evaluate this measure, we apply scaling laws to determine how much training budget to allocate to data generation and discuss how to achieve smooth error-matching in diverse settings. We evaluate the breakeven complexity of multiple neural PDE solvers on three PDEs on 2D periodic domains from APEBench and a novel benchmark of flows past multiple obstacles generated by the GPU-native PyFR code. Among other findings, our results suggest that neural PDE solvers become more effective as problems get harder in terms of cost, dimension, rollout, physics regime (e.g. higher Reynolds number), etc.

Yijing Zhang, Nicholas Roberts, Tanya Marwah +1

University of Wisconsin–Madison · Google DeepMind

Neural PDE Solvers Benchmark Design PDE Surrogate Modeling

May 14, 2026cs.LG

Unbiased and Second-Order-Free Training for High-Dimensional PDEs

Deep learning methods based on backward stochastic differential equations (BSDEs) have emerged as competitive alternatives to physics-informed neural networks (PINNs) for solving high-dimensional partial differential equations (PDEs). By leveraging probabilistic representations, BSDE approaches can avoid the curse of dimensionality and often admit second-order-free training objectives that do not require explicit Hessian evaluations. It has recently been established that the commonly used Euler-Maruyama (EM) time discretization induces an intrinsic bias in BSDE training losses. While high-order schemes such as Heun can fully eliminate this bias, such schemes re-introduce second-order spatial derivatives and incur substantial computational overhead. In this work, we provide a principled analysis of EM-induced loss bias and propose an unbiased, second-order-free training framework that preserves the computational advantages of BSDE methods. Our code is available at https://github.com/seojaemin22/Un-EM-BSDE.

Jaemin Seo, Surin Lee, Jae Yong Lee

Department of Artificial Intelligence, Chung-Ang University, Seoul, Republic of Korea.

Stochastic Differential Equations Neural PDE Solvers PDE Solving

May 13, 2026cs.LG

Di-BiLPS: Denoising induced Bidirectional Latent-PDE-Solver under Sparse Observations

Partial differential equations (PDEs) are fundamental for modeling complex natural and physical phenomena. In many real-world applications, however, observational data are extremely sparse, which severely limits the applicability of both classical numerical solvers and existing neural approaches. While neural methods have shown promising results under moderately sparse observations, their inference efficiency at high resolutions is limited, and their accuracy degrades substantially in the extremely sparse regime. In this work, we propose the Di-BiLPS, a unified neural framework that effectively handle both forward and inverse PDE problems under extremely sparse observations. Di-BiLPS combines a variational autoencoder to compress high-dimensional inputs into a compact latent space, a latent diffusion module to model uncertainty, and contrastive learning to align representations. Operating entirely in this latent space, the framework achieves efficient inference while retaining flexible input-output mapping. In addition, we introduce a PDE-informed denoising algorithm based on a variance-preserving diffusion process, which further improves inference efficiency. Extensive experiments on multiple PDE benchmarks demonstrate that Di-BiLPS consistently achieves SOTA performance under extremely sparse inputs (as low as 3%), while substantially reducing computational cost. Moreover, Di-BiLPS enables zero-shot super-resolution, as it allows predictions over continuous spatial-temporal domains.

Zhonghao Li, Chaoyu Liu, Qian Zhang

School of Science, Harbin Institute of Technology, Shenzhen, China · Department of Theoretical Physics and Applied Mathematics, University of Cambridge, Cambridge, United Kingdom

Neural PDE Solvers Latent Diffusion Models PDE Solving Diffusion-Based Inverse Problems Diffusion Model Inference Acceleration PDE Inverse Problems

May 12, 2026cs.LG

MetaColloc: Optimization-Free PDE Solving via Meta-Learned Basis Functions

Solving partial differential equations (PDEs) with machine learning typically requires training a new neural network for every new equation. This optimization is slow. We introduce MetaColloc. It is an optimization-free and data-free framework that removes this bottleneck completely. We decouple basis discovery from the solving process. We meta-train a dual-branch neural network on diverse Gaussian Random Fields. This offline process creates a universal dictionary of neural basis functions. At test time, we freeze the network. We solve the PDE by assembling a collocation matrix. We find the solution through a single linear least squares step. For non-linear PDEs, we apply the Newton-Raphson method to achieve fast quadratic convergence. Our experiments across six 2D and 3D PDEs show massive improvements. MetaColloc reaches state-of-the-art accuracy on smooth and non-linear problems. It also reduces test-time computation by several orders of magnitude. Finally, we provide a detailed frequency sweep analysis. This analysis reveals a critical mismatch between function approximation and operator stability at extremely high frequencies. This profound finding opens a clear path toward future operator-aware meta-learning.

Zichuan Yang

School of Mathematical Sciences Tongji University Shanghai, P.R. China

Neural PDE Solvers Meta-Learning PDE Solving

May 12, 2026cs.LG

Neural-Schwarz Tiling for Geometry-Universal PDE Solving at Scale

Most learned PDE solvers follow a global-surrogate paradigm: a neural operator is trained to map full problem descriptions to full solution fields for a prescribed distribution of geometries, boundary conditions, and coefficients. This has enabled fast inference within fixed problem families, but limits reuse across new domains and makes large-scale deployment dependent on expensive problem-specific data generation. We introduce

\textbf{NEST}

(

\textbf{Ne}

ural-

\textbf{S}

chwarz

\textbf{T}

iling), a local-to-global framework that shifts learning from full-domain solution operators to reusable local physical solvers. The central premise is that, although global PDE solutions depend on geometry, scale, and boundary conditions, the physical response on small neighborhoods can be learned locally and composed into global solutions through classical domain decomposition. NEST learns a neural operator on minimal voxel patches (

3 \times 3 \times 3

) with diverse local geometries and boundary/interface data. At inference time, an unseen voxelized domain is tiled into overlapping patches, the learned local solver is applied patchwise, and global consistency is enforced through iterative Schwarz coupling with partition-of-unity assembly. In this way, generalization is shifted from a monolithic neural model to the combination of local physics learning and algorithmic global assembly. We instantiate NEST on nonlinear static equilibrium in compressible neo-Hookean solids and evaluate it on large, geometrically complex 3D domains far outside the scale of the training patches. Our results show that local neural building blocks, coupled through Schwarz iteration, offer a reusable local-training path toward scalable learned PDE solvers that generalize across domain size, shape, and boundary-condition configurations.

Paolo Secchi, Daniel S. Balint, Marco Maurizi

Imperial College London · Italian Institute of Artificial Intelligence

Neural PDE Solvers PDE Solving Neural Operators

May 11, 2026cs.LG

Chebyshev Center-Based Direction Selection for Multi-Objective Optimization and Training PINNs

Physics-informed neural networks (PINNs) are a promising approach for solving partial differential equations (PDEs). Their training, however, is often difficult because multiple loss terms induced by PDE residuals and boundary or initial conditions must be optimized simultaneously. To address this difficulty, existing approaches often construct update directions by explicitly enforcing particular desirable properties, such as scale robustness and simultaneous descent. While effective in many cases, such property-by-property designs can make it unclear which conditions are essential, what geometric principle determines the selected update direction, and how different methods are structurally related. In this work, we formulate update-direction selection for PINN training as a Chebyshev-center problem in the dual cone. The proposed formulation selects a normalized direction that maximizes the minimum distance to the cone facets. The resulting formulation admits an efficient dual problem in a much lower-dimensional space and yields a convergence guarantee in the nonconvex setting. It also recovers the key desirable properties targeted by existing approaches without imposing them separately; rather, they follow from the single geometric criterion underlying the formulation. This makes the selected direction interpretable through a single geometric rule and provides a unified basis for systematically comparing related direction-selection methods. Experiments on several PINN benchmarks further demonstrate strong empirical performance of the proposed method.

Hoyeol Yoon, Seoungbin Bae, Nam Ho-Nguyen +1

Department of Industrial & Systems Engineering, KAIST · Discipline of Business Analytics, The University of Sydney · Department of Mathematical Sciences, Seoul National University

Neural PDE Solvers Neural Network Optimization Physics-Informed ML Multi-Objective Optimization

May 10, 2026cs.LG

MC $^2$ : Monte Carlo Correction for Fast Elliptic PDE Solving

Partial differential equation (PDE) solvers underpin scientific computing, but real-world deployment is bounded by compute. Classical Monte Carlo solvers such as Walk-on-Spheres (WoS) are unbiased and geometry-agnostic but are slow. Learned solvers are fast but biased and brittle under distribution shift. We present \textbf{MC

^2

}, a hybrid WoS-Neural Network (WoS-NN) PDE solver that treats a low-budget Monte Carlo solution as a structured estimator of the true field and learns a single-pass neural correction to recover a high-fidelity solution. MC

^2

matches the accuracy of solutions using over

1000\times

more Monte Carlo compute, outperforming all evaluated classical, denoising, and neural-operator baselines. To enable reproducible study of finite-compute PDE solving, we additionally release \textbf{PDEZoo}, the largest standardized elliptic PDE benchmark to date: 2M PDEs spanning five elliptic families and unlimited geometric compositions, with analytic ground truth and multi-budget Monte Carlo trajectories. Together \textbf{MC

^2

} and \textbf{PDEZoo} (1) empirically establish that finite-sample Monte Carlo error is structured, learnable, and correctable in a single forward pass, (2) show that we can solve PDEs

\sim

\textbf{1000x} faster than with just WoS, and (3) provide the evaluation infrastructure the field has so far lacked.

Ethan Hsu, Hong Meng Yam, Ivan Ge

Stanford University

Neural PDE Solvers PDE Solving