PDE Solving

PDE: Partial Differential Equation

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19 papers in the last four weeks, up 111% on the four weeks before. 0.2% of all new papers.

Jul 13Week of Sep 28

Latest papers 170

Oct 8, 2026cs.AI

An Interpretable Approach to PDE Solution Discovery via Structural Experience Distillation

PDE solution discovery aims to identify explicit symbolic expressions for unknown physical fields from observations under known physical constraints. Existing methods, however, collapse data fidelity and physical consistency into a single terminal score used as the sole feedback signal, providing little information about which subexpressions are responsible for a candidate's final performance. This opaque terminal feedback severely limits the interpretability of the search process itself, offering no insight into why a candidate succeeds or fails. Consequently, reusable structures in otherwise suboptimal candidates are often discarded, whereas incidental syntax along successful search trajectories may be repeatedly reinforced. We propose SED-MCTS, a Monte Carlo tree search approach that distills structural experience from evaluated expressions and reuses it to guide subsequent symbolic solution search. Through counterfactual subtree interventions, SED-MCTS estimates local structural contributions, routes reliable evidence to the responsible construction edges, and preserves useful components in a refined structural archive. The approach naturally extends to coupled multiphysics systems. Across a diverse suite of PDE benchmarks, SED-MCTS achieves strong performance under a fixed evaluation budget and improves search efficiency and robustness under noisy or scarce observations.

Yunpeng Gong, Huolong Wu, Can Yang +1

School of Informatics, Xiamen University

PDE Solving Symbolic Regression Monte Carlo Tree Search

Oct 7, 2026cs.LG

Gen-PINNs: Generative Adversarial Physics Informed Neural Networks for solving partial differential equations

Physics-Informed Neural Networks (PINNs) are a widely used data-free method for solving Partial Differential Equations (PDEs) using machine learning. With recent advances in Generative Adversarial Networks (GANs), adversarial learning has shown strong capabilities for modeling complex data-driven problems; however, the use of GANs in deterministic physics-informed PDE solutions remains limited. In this work, we first identify limitations of standard PINNs for solving PDEs, including spectral bias, loss imbalance, and optimizer stagnation. We then propose Generative Adversarial Physics-Informed Neural Networks (Gen-PINNs), a unified deterministic residual-adversarial framework designed to improve data-free solutions of PDEs with sharp or shock-front behavior. The generator learns the underlying PDE solution using dynamically weighted physics-informed loss components, while separate discriminators evaluate complementary PDE residual features against ideal zero-residual states. The framework further develops and adapts several methodological components, including a Fourier representation for resolving high-frequency spatial content, an orthonormal spectral diagnostic for quantifying frequency-dependent solution errors, and a modified gradient-based dynamic weighting system for physics, initial-condition, boundary-condition, and adversarial loss objectives. Gen-PINNs is tested against standard PINNs on nonlinear and higher-order PDEs, including the Burgers, Allen-Cahn, and Kuramoto-Sivashinsky equations. The results demonstrate substantial improvements in accuracy and convergence across sharp-front, stiff, and higher-order PDE solutions, highlighting the potential of deterministic residual-adversarial learning as an effective approach for solving challenging nonlinear PDEs.

Muhammad M. Akmal, Kamy Sepehrnoori, Michael J. Pyrcz

Hilderbrand Department of Petroleum and Geosystems Engineering, The University of Texas at Austin · Department of Earth and Planetary Sciences, The University of Texas at Austin

PDE Solving Physics-Informed ML Generative Adversarial Networks Adaptive Loss Weighting

Oct 7, 2026cs.LG

Physics-Informed Neural Plasticity: PDE Solvers That Reshape Themselves

Physics-informed neural PDE solvers adapt their parameters to satisfy governing equations, yet their representational structure typically remains fixed throughout training. This rigidity is poorly matched to PDE solutions with strongly heterogeneous complexity across space and space--time, leaving capacity insufficient where the physics is difficult and redundant where it is simple. We introduce physics-informed neural plasticity, a paradigm in which the representation itself reshapes during optimization in response to unresolved physics. We instantiate this principle with Representation Capacity Adaptation for PDEs (ReCAP), a Gaussian-localized solver that dynamically redistributes capacity through local enrichment, residual-directed splitting, gate-based pruning, and function-aware merging. ReCAP uses responsibility-weighted error indicators and the geometry of residual energy to determine where and how to refine. To limit the disturbance introduced by splitting, we introduce quiet-child refinement, which initializes new components by transporting the parent representation while controlling instantaneous functional perturbation. We further establish conditional a posteriori reliability and structural-stability guarantees linking localized physics residuals to solution error and stable refinement. Across five challenging 3D and 4D PDE benchmarks against 11 physics-informed solvers, ReCAP achieves the lowest relative

L^2

error on every problem, reducing error by

10.7\%

--

27.5\%

relative to the strongest competing result. These results suggest that physics-informed solvers need not merely learn their parameters---they can learn how their representational capacity should be organized.

Chun-Wun Cheng, Bingcheng Hu, Angelica I. Aviles-Rivero

Shanghai Normal University · YMSC, Tsinghua University

Neural PDE Solvers PDE Solving Physics-Informed ML Dynamic Neural Networks

Oct 6, 2026math.NA

A Neural JKO Scheme for Hellinger-Kantorovich Gradient Flows via Monge-Growth Pairs

We develop a mesh-free neural JKO scheme for advection-reaction-diffusion equations with a gradient-flow structure in the Hellinger-Kantorovich (HK) geometry of unbalanced optimal transport. Each update is parametrized by a spatial map and a mass-changing factor, allowing spatial redistribution and local mass creation or loss to be treated jointly within a single variational step. Their cone action bounds the squared HK distance from above, yielding a sufficient condition for discrete energy dissipation through comparison with the identity pair. Minimizing the pair objective over all admissible pairs recovers the exact JKO minimum when the source and a minimizer have positive densities. We establish existence and mass bounds for JKO minimizers and, under additional assumptions, obtain positivity and regularity together with a discrete Euler-Lagrange equation and a metric-dissipation identity. The self-consistent chemical potential is then nonincreasing along an optimal map. There exist parametric pairs whose endpoint densities and objective values converge to those of an exact JKO minimizer, provided a regular-pair approximation hypothesis holds. Finally, we show that a primal-dual gap controls objective suboptimality and, for Boltzmann entropy, the

L^1

density error, assuming exact-step regularity, positive-semidefinite interactions, and global dual feasibility. Numerical experiments examine pointwise agreement with the PDE, energy dissipation, and the roles of transport, reaction, and fully implicit interactions.

Geuntaek Seo, Cheolhyeong Kim, Hwijae Son +1

Department of Mathematics, Pohang University of Science and Technology · Samsung Electronics · Department of Mathematics, Konkuk University

Neural PDE Solvers PDE Solving Optimal Transport

Oct 2, 2026cs.AI

Geometry Meets Physics: Data-Efficient Pre-Training for Unstructured Neural PDE Solvers

Neural surrogate models for Partial Differential Equations (PDEs) on unstructured 3D geometries are often limited by poor generalization and the high cost of generating large-scale training datasets. Consequently, pre-training on massive datasets of related PDE dynamics has emerged as a critical alternative to enhance the robustness and scalability of these models. However, this strategy is neither compute- nor data-efficient, as it relies on massive pre-computed data that is very costly to generate. In this work, we introduce a disk-data-free pre-training framework tailored to both steady-state and transient regimes. For steady-state problems, we propose a geometry-driven strategy that leverages intrinsic shape descriptors to learn representations of complex 3D domains. For transient problems, we introduce a physics-driven approach based on online generation of synthetic PDE data, enabling scalable pre-training without reliance on expensive datasets. Across multiple experiments, our approach achieves faster convergence, greater data efficiency, and higher accuracy during fine-tuning, particularly under realistic low-data regimes. This methodology provides a practical pathway toward data-efficient neural emulators for large-scale simulations.

Luis Medrano-Navarro, Giacomo Baldan, Qiang Liu +4

Technical University of Munich · University of Stuttgart

Synthetic Data Pretraining Neural PDE Solvers PDE Solving PDE Surrogate Modeling Data-Efficient Learning

Oct 1, 2026math.NA

Kolmogorov-Arnold Networks for Free-Boundary Partial Differential Equations

We study free-boundary problems within a physics-informed framework using Kolmogorov-Arnold network (KAN) approximations. The proposed approach incorporates obstacle constraints, partial differential equation (PDE) inequalities, complementarity conditions, and boundary conditions through residual-based loss functions. We consider a linear elliptic obstacle problem, a nonlinear

p

-Laplacian obstacle problem, and a time-dependent one-phase Stefan problem. The proposed KAN solver is compared with physics-informed neural network (PINN) and residual-network baselines. Numerical experiments show that KANs achieve low relative

L^2

and

L^\infty

errors while accurately resolving contact regions and moving interfaces. The results indicate that KAN representations provide an effective alternative for solving free-boundary PDEs.

Tan Phuong Dong Le

Department of Applied Mathematics, University of Waterloo, Canada · Department of Statistics and Actuarial Science, University of Waterloo, Canada

Neural PDE Solvers PDE Solving Kolmogorov-Arnold Networks

Oct 1, 2026cs.LG

Learning PDE Dynamics between Submanifolds Using Green's Observation Operators

Many physical systems are driven and observed only on lower-dimensional submanifolds of a larger spatial domain, while their dynamics are governed by the ambient medium occupying that domain. Examples include laser-heated parts imaged by an infrared camera, and ground-level emissions measured on a sensor plane. Full-domain solvers, however, compute the entire volume for every new source although only the observation submanifold is needed, and black-box surrogates do not exploit that the ambient medium remains fixed. We introduce the \emph{Green's Observation Operator (GObO)}, which maps the ambient medium once to the Green's kernel of a linear PDE restricted to the source and observation submanifolds. New sources then cost one lower-dimensional integral and no network evaluation. Exponential rates in the kernel yield an exact finite streaming state with horizon-independent memory; we prove its stability and an approximation rate for the restricted heat kernel. On three-dimensional heat conduction and advection--diffusion with collocated and distinct source and observation geometries, GObO trained on static sources predicts responses to moving sources zero-shot with 4--8

\times

lower error than black-box surrogates, at 1.4,ms per query after a single conditioning pass. The same kernel transfers across resolutions and admits corrections for mild nonlinearities, including radiative losses and temperature-dependent conductivity, without retraining, at the cost of lower in-distribution accuracy.

Jan Tauberschmidt, Jephte Abijuru, Naukshatro Bose +5

German Research Center for Artificial Intelligence (DFKI), Kaiserslautern, Germany · Rheinland-Pfälzische Technische Universität (RPTU), Kaiserslautern, Germany · University of Manchester, Manchester, United Kingdom

PDE Solving PDE Surrogate Modeling PDE Operator Learning

Sep 30, 2026math-ph

Cluster Attention Neural Operators for Solving Parametric Partial Differential Equations

Traditional simulations of parametric partial differential equations (PDEs) rely on repetitive computations for each parameter, which makes high-fidelity design impractical. Neural operators address this issue by learning solution operators, accelerating parameter-space mapping by orders of magnitude. Recent Transformer-based neural operators attempt to capture global dependencies, but often at the cost of quadratic attention complexity. Transolver resolves this problem by projecting physical states into a reduced slice space for attention computation. Although fast, this projection sacrifices fine spatial information. Moreover, by operating in this reduced space with shared weights across attention heads, it may constrain the model's flexibility, thereby limiting its capacity to capture complex phenomena. To address these issues, we propose the Cluster Attention Neural Operator (CANO), which reformulates attention via a novel cross-attention mechanism that dynamically clusters queries while preserving full-resolution keys and values. This avoids slice compression loss and removes weight-sharing limits. At the same time, the model remains fast without losing global interactions. Empirically, CANO achieves state-of-the-art performance across canonical PDE benchmarks, covering fluid and solid dynamics (e.g., Navier-Stokes, Airfoil, Plasticity), irregular unstructured geometries (e.g., Pipe Turbulence, Composites), and long-term temporal rollouts. Across solid deformation and turbulent flow benchmarks, CANO achieves lower errors than baselines and exhibits strong geometric adaptability and temporal consistency.

Ming Zhong, Antonio Colanera, Gianluigi Rozza +1

School of Advanced Interdisciplinary Sciences, University of Chinese Academy of Sciences, Beijing 100049, China. · State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China. · International School for Advanced Studies (SISSA), Trieste 34136, Italy. +1

PDE Solving Attention Mechanisms Neural Operators

Sep 28, 2026cs.LG

Neural Harmonic Measure Operator

We introduce Neural Harmonic Measure Operator (NHMO), a neural solver for elliptic PDE problems on variable-shape domains. The harmonic measure of a domain is the boundary probability distribution that, integrated against any boundary data, returns the Dirichlet Laplace solution. It depends only on the geometry, not on the boundary data. NHMO parameterizes the density of this measure as a transformer-based boundary kernel supervised by Walk-on-Spheres exit samples, so one trained kernel handles different boundary values on a shape with no retraining. We extend it to Poisson via a classical decomposition, with an auxiliary network amortizing the source-induced correction and avoiding the singular volume quadrature that breaks direct evaluation. At inference, new boundary values and new sources both yield PDE solutions by re-integration against the fitted kernel and lift, with no retraining. NHMO improves over four prior baselines on the MCB-B 3D variable-shape Poisson benchmark across all five categories, and is competitive with major neural-operator baselines on a controlled 2D testbed.

Jinjin He, Sinan Wang, Yuchen Sun +1

Georgia Institute of Technology

Neural PDE Solvers PDE Solving PDE Operator Learning

Sep 23, 2026math.NA

A Hybrid Iterative Deep Ritz Method for Elliptic Interface Problems

In this work, we propose a hybrid iterative deep Ritz method (H-IDRM) for a class of interface problems for second-order elliptic operators. It is based on a new mixed formulation of the problem and involves solving a sequence of convex minimization problems. We employ a level-set neural network architecture, featuring a level-set representation of the interface, to accommodate the piecewise smoothness of the solution and the flux. The approach involves only volumetric representations instead of duality pairing on the interface and avoids explicit interface sampling that is inconvenient for complex interface geometries. Further, we present an analysis of the method, including the errors arising from the neural network approximation, Monte Carlo approximation, iterative scheme, and penalty parameters. Numerical experiments indicate that the H-IDRM outperforms existing neural solvers on problems with high-dimensional domains, intricate interface geometries, and lower subdomain regularity.

Tianhao Hu, Bangti Jin, Fengru Wang +1

Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong · Department of Mathematics & Scientific Computing Key Laboratory of Shanghai Universities, Shanghai Normal University, Shanghai 200234, China.

Neural PDE Solvers PDE Solving

Sep 23, 2026cs.CV

GaussPDE: Graph-Based Partial Differential Equation-Driven Rendering for 3D Gaussian Splatting

We present GaussPDE, a framework that injects physically structured partial differential equation (PDE) dynamics into pretrained 3D Gaussian scenes without mesh extraction, voxelization, or retraining. Our key observation is that PDE rendering requires not only accurate appearance, but also a reliable discrete computational domain. We therefore first introduce camera-aware regularization during 3DGS reconstruction to suppress camera-near floaters and oversized primitives that would create unstable graph topology. We then construct an active Gaussian graph using covariance-aware distances and opacity, appearance, and boundary-aware conductance, enabling mass-weighted graph Laplacian PDE evolution directly over Gaussian primitives. The evolving scalar PDE state is coupled back to rendering by modifying the direct-current spherical harmonic color coefficients while preserving geometry, opacity, and view-dependent rendering behavior. Experiments on real and synthetic scenes show that GaussPDE produces stable, controllable, and spatially coherent dynamic visualizations, with reduced cross-boundary leakage compared with baselines.

Haoyuan Yue, Fengyuan Ye, Ziyin Li

Department of Physics, Westlake University, Hangzhou, China · The Chinese University of Hong Kong, Shenzhen, China · Department of Accounting, Xi’an Jiaotong-Liverpool University, Suzhou, China

3D Gaussian Splatting PDE Solving 3D Scene Representation

Sep 21, 2026math.NA

Cost-Accuracy Trade-offs: Neural Operator vs Classical Numerical Solver

Neural operators are data-driven models that learn mappings from inputs that parameterize partial differential equations, such as spatially varying coefficients, initial conditions, forcing terms, boundary conditions, or geometries, to solution fields or quantities of interest. Once trained, they can serve as surrogates for classical numerical solvers in many-query settings that require repeated evaluations for varying inputs. We address the question of when, and then why, neural operator surrogates outperform classical numerical solvers, in terms of cost for a given accuracy. We focus on the post-training, many-query limit, in which data-acquisition and training costs are treated as fixed and fully amortized. Even in this deliberately favorable regime for neural operators, there are regimes in which classical solvers outperform the surrogate models. We compare the cost-accuracy performance of neural operator surrogates and classical numerical solvers through a reproducible benchmark study comparing neural operators with problem-matched classical solvers on representative problems in computational science and engineering, focusing on prediction error, per-query floating-point cost, and wall-clock runtime. Neural operators are most competitive at low-to-moderate accuracy requirements. Their floating-point cost advantage depends strongly on the problem structure, arising when they avoid temporal or nonlinear iterations or predict a reduced quantity of interest rather than a full solution field. Additional wall-clock speedups result from dense tensor operations that are well suited to modern hardware. As the target accuracy is tightened, achieving the required accuracy with neural operators becomes increasingly challenging, and classical solvers outperform surrogates in this regime; thus classical solvers will remain important for verification and high-accuracy computation.

Daniel Zhengyu Huang, Andrew M. Stuart

Beijing International Center for Mathematical Research, Center for Machine Learning Research, Peking University, Beijing, China · California Institute of Technology, Pasadena, CA

Neural Surrogate Modeling Neural PDE Solvers PDE Solving Neural Operators

Sep 17, 2026math.NA

Beyond PINNs: A Unified Gauss--Newton and Petrov--Galerkin Framework for Neural and Hybrid PDE Solvers

Physics-informed neural networks and finite element methods provide two different paradigms for the numerical approximation of partial differential equations: the former are commonly trained by minimizing pointwise strong residuals, whereas the latter are naturally built from weak variational formulations and the finite-dimensional systems obtained after discretization. In this work, we introduce a common framework based on the discretization of functional Gauss--Newton problems by finite families of linear measurements. We show that, through an appropriate duality pairing, the linear measurements can be represented by test functions. The resulting Gauss--Newton system is then precisely a Petrov--Galerkin discretization of the linearized functional problem. This perspective recovers pointwise collocation and natural-gradient constructions as particular cases, while making the choice of test functions an explicit algorithmic design choice. We specialize this framework to elliptic problems, where it naturally leads to weak residual formulations and to a hybrid finite element--neural construction acting on complementary approximation spaces. Numerical experiments support the proposed framework and demonstrate the effectiveness of weak Gauss--Newton formulations and hybrid finite element--neural approximations.

Nilo Schwencke, Roland Maier

Gauss-Newton Optimization Neural PDE Solvers PDE Solving Second-Order Optimization Finite Element Method

Sep 17, 2026cs.LG

Amortizing Physics-Informed Neural Solvers via Graph Hypernetworks

Amortizing physics-informed neural networks (PINNs) across related PDEs requires describing each equation to a reusable solver. Coefficient vectors encode numerical parameters in predefined slots, leaving operator and cross-field assignments implicit. We make these relationships explicit in an operator graph, with nodes for fields, derivatives, terms, and residuals and coefficients retained as term attributes. A graph hypernetwork generates diagonal codes that initialize a meta-trained factorized PINN for each target equation. Meta-training and target-specific adaptation use governing equations and prescribed conditions without solution labels. We compare coefficient-vector, DeepSets-based term-set, and graph conditioning by solution accuracy within a fixed adaptation budget. In scalar convection-diffusion-reaction problems, both term-based descriptors improve high-reaction accuracy, with similar performance. In two-field Fisher-KPP, meta-training sees uncoupled and one-way systems; after 3,000 adaptation steps on unseen two-way coupling, the graph's mean final error is 35.7% below the term set and 67.7% below the coefficient vector. In a fixed-structure capacitively coupled plasma model, the coefficient vector performs best. These results support extending coefficient conditioning with explicit equation relationships for physics-based solver adaptation.

Cheng Jing, Abhishek Verma, Kallol Bera +2

Arizona State University · Applied Materials Inc.

Amortized Inference Meta-Learning PDE Solving Hypernetworks Physics-Informed ML

Sep 15, 2026cs.LG

Hybrid coupling with numerics-informed neural networks and the overlapping Schwarz alternating method

We develop a hybrid modeling framework for coupling pre-trained numerics-informed neural networks (NINNs) with classical full order models (FOMs) using the overlapping Schwarz alternating method. We consider the two-dimensional advection-diffusion equation in the advection-dominated, Peclet-number 10^6 regime. We first demonstrate that, unlike the corresponding physics-informed neural network (PINN), a monolithic NINN can be accurately trained on our model problem without domain decomposition. We then employ overlapping multiplicative Schwarz as a deployment mechanism for coupling a pre-trained, subdomain-local NINN with a neighboring FOM, with the NINN weights held fixed throughout the Schwarz iteration. We consider two training approaches for the subdomain-local NINNs: a top-down approach, in which boundary data are obtained from a coupled Schwarz solve on the full domain with a FOM on each subdomain (FOM-FOM Schwarz), and a bottom-up approach, in which boundary traces are generated synthetically on the NINN subdomain without requiring any full-domain solves. The resulting hybrid NINN-FOM solutions agree closely with the corresponding FOM-FOM Schwarz solutions, with the top-down and bottom-up training approaches yielding comparable accuracy.

George Chumbipuma, Irina Tezaur, Alejandro Diaz +1

Rice University · Sandia National Laboratories

Neural PDE Solvers PDE Solving

Sep 14, 2026math.NA

Physics Informed Random Feature Neural Networks for Solving PDEs

Machine learning-based partial differential equations (PDEs) solvers have attracted significant attention in recent years. Most progress in this area has been driven by deep neural networks such as physics-informed neural networks (PINNs) and kernel method (such as physics-informed Gaussian Processes). We introduce a physics-informed random feature method for countering part of the spectral bias which PINN-based solvers are facing for a certain class of PDEs. Random feature method was originally proposed to approximate large-scale kernel machines and can be viewed as a specialized randomized neural network. Compared to other state-of-the-art PINN-based solvers which require a large number of collocation points, our proposed method reduces the computational complexity. In this paper, we develop a rigorous approximation error analysis and derive high-probability error bounds on the

H^1

norm. We provide extensive numerical tests for verifying our theoretical guarantees on error decay rates, as well as several comparison tests to showcase our claimed capability for combating spectral bias in these deep learning based methods.

Chi-An Chen, Chunyang Liao, Ming Zhong

Department of Applied Mathematics, Mathematics, Illinois Institute of Technology, Chicago, IL, 60616 · Department of Mathematical Sciences, University of Arkansas, Fayetteville, AR, 72701 · Department of Mathematics, University of Houston, Houston, TX, 77204

PDE Solving Random Feature Methods Neural Network Approximation Theory Physics-Informed ML

Sep 14, 2026cs.LG

Single-condition neural solvers encode transferable response spaces for parametric differential equations

Operator learning for parametric partial differential equations (PDEs) typically builds global models over prescribed domains, requiring cross-condition data or costly physics-constrained training. Here we show that the output Jacobian of a neural solution model trained at one condition defines a reusable response space for cross-condition solution variations. We introduce Linearized Subspace Transfer (LST) to exploit this space and recover target solutions by minimizing the target PDE-system residual over response-space coordinates. Because any single response space has finite coverage, Active Transfer Modeling (ATM) uses post-transfer residuals as coverage indicators to selectively acquire response spaces from additional single-condition models. Across six systems, single-condition response spaces supported cross-condition transfer, with enrichment improving accuracy when added spaces expanded representation capacity. Relative to evaluated physics-informed operator baselines, ATM reduced error and offline construction cost, with orders-of-magnitude accuracy gains in representative cases and millisecond-to-second target adaptation. These results establish neural solvers as reusable local parametric models.

Wenbo Cao, Weiwei Zhang

Institute of AI for Industries, Chinese Academy of Sciences, Nanjing 211135, China · Institute of Computing Technology, Chinese Academy of Sciences, Beijing 100190, China · School of Aeronautics, Northwestern Polytechnical University, Xi’an 710072, China +2

PDE Solving PDE Surrogate Modeling Active Learning Transfer Learning PDE Operator Learning

Sep 14, 2026math.AP

Linearized PINN with pretrained nonlinear layers

We propose a linearized Physics-Informed Neural Network (lPINN), a reduced-order neural basis method for forward and inverse differential equations. In an offline stage, lPINN learns operator-compatible continuous neural basis functions from an ensemble of numerical solutions. The basis functions are differentiable through automatic differentiation and are pretrained using solution data together with either derivative information or physics residuals. For each new problem instance, the basis functions are frozen and the solution is obtained by minimizing the governing-equation residual together with applicable initial, boundary, regularization, and observational terms. Unlike surrogate and operator-learning methods, the training data define the trial space offline, while the instance-specific solution is computed online by enforcing the governing physics. Relative to vanilla PINNs, lPINN pretrains the nonlinear hidden-layer representation offline and performs online inference only in the final linear layer. We evaluate lPINN on forward and inverse problems for the advection-diffusion equation, Burgers' equation, and the nonlinear pendulum equation. Compared with vanilla PINNs, lPINN achieves lower solution and parameter errors while reducing online inference times by approximately one to more than three orders of magnitude, with the largest gains generally observed for limited residual or measurement data. Cross-resolution experiments show that the learned continuous representation can be evaluated on finer meshes without retraining and with nearly unchanged accuracy.

Wenhao Chen, Alexandre M. Tartakovsky

Civil and Environmental Engineering Department University of Illinois Urbana-Champaign Urbana, IL 61801, USA · Pacific Northwest National Laboratory Richland, WA 99352, USA

Efficient Neural Network Inference PDE Solving Reduced-Order Modeling Physics-Informed ML PDE Inverse Problems

Sep 14, 2026math.AP

An explicit solution of the five-expert prediction PDE and the exact optimality set of COMB

In this paper, we derive an explicit solution of the stationary prediction with expert advice PDE for five experts. The formula is given in three regions. In the first two regions, it is the four-expert solution plus a single integral with an elementary positive density. In the third region, it is a finite sum of hyperbolic products whose coefficients are determined by one scalar quadrature. Our formula establishes that the direction

(1,0,1,0,0)

is optimal throughout the ordered sector, and that the COMB strategy

(1,0,1,0,1)

is optimal only on a lower dimensional subset of the sector (where

x_1=x_2

and

x_3=x_4

). This disproves the COMB optimality conjecture of Gravin, Peres and Sivan. The verification of the Hamiltonian inequalities is a tedious task, part of which is completed with a computer assisted proof. The verification reduces to 21 scalar inequalities, which we prove using 147 exact rational Bernstein polynomial certificates. The exact certificates and their independent arithmetic checks are included in a supplement to this paper, and a Lean 4 formalization machine-checks the verification and both main theorems, apart from the viscosity characterization.

Jeff Calder, Nadejda Drenska

School of Mathematics, University of Minnesota · Department of Mathematics, Louisiana State University

Sep 13, 2026cs.LG

Tackling Failure Modes of PINNs and PIKANs Using Conflict-Free Gradients

Scientific machine learning methods such as physics-informed neural networks (PINNs) increasingly rely on domain decomposition for better scalability while solving partial differential equations (PDEs) over complex geometries, yet the resulting composite loss comprising residual, boundary, and interface terms is highly susceptible to conflicting gradients that degrade training. This work bridges domain decomposition with projection-based gradient surgery to systematically mitigate such conflicts in 2D and 3D settings. We evaluate two existing projection-based algorithms, PCGrad and ConFIG, and identify their performance degradation in specific scenarios such as 3D domains with multiple overlapping interfaces. To address this limitation, we propose Norm-PCGrad, a normalized variant that achieves state-of-the-art accuracy across a range of 2D and 3D domain decomposition problems. Across the benchmarks considered, Norm-PCGrad consistently achieves the lowest relative

L_2

error compared to training without gradient surgery as well as to existing algorithms such as PCGrad and ConFIG, while incurring negligible additional computational overhead. To improve computational efficiency of domain decomposition frameworks such as Extended PINN (XPINN), we propose replacing vanilla PINNs in selected subdomains with separable architectures such as Separable PINN (SPINN), reducing the computational cost from quadratic (or cubic) to linear. We additionally demonstrate that gradient surgery extends to physics-informed Kolmogorov-Arnold Networks (PIKANs), yielding substantial accuracy improvements for 3D domain decomposition and confirming the generality of the proposed approach across network architectures.

Sidharth S. Menon, Irina Tezaur, Ameya D. Jagtap

Aerospace Engineering Department, Worcester Polytechnic Institute, Worcester, MA 01609, USA. · Sandia National Laboratories, 7011 East Ave, Livermore, CA 94550, USA.

PDE Solving Gradient Interference Gradient Projection Physics-Informed ML

Sep 11, 2026cs.LG

Attention Is All You Need (to Avoid Spurious Oscillations)

Can attention move a shock across several cells in one update without breaking it? We develop a conservative, fixed grid finite-volume scheme in which a CFL-conditioned attention flux selects upstream information according to the transport required by the current time step. One-dimensional inviscid Burgers transport is used as the central mechanism test: the same learned flux remains reliable in the conventional small-step regime and, with a time step four times larger, preserves sharp shocks while using one stage per update. A standard fifth-order WENO scheme with third-order strong-stability-preserving Runge-Kutta time integration (WENO-5+SSP-RK3) is included alongside controlled Forward Euler comparisons to separate flux selection from time integration. The learned attention shifts upstream with the local transport reach and becomes more selective near shocks; inference-time interventions and retrained ablations show that transport-scale information and state-dependent selection contribute directly to performance. Directional two-dimensional scalar Burgers transport and the one-dimensional shallow-water system then test whether the conservation-scale-selection principle transfers beyond the original scalar setting. The results support attention as a learnable information stencil for conservative large-step shock transport, while identifying finite candidate reach and problem-dependent robustness as the present limits.

Jinyoung Jeong, Joseph B. Choi, Xinlun Cheng +3

School of Mechanical Engineering & IEDT, Kyungpook National University, Daegu, South Korea · School of Data Science, University of Virginia, Charlottesville, VA 22903, USA · Department of Mechanical Engineering, University of Iowa, Iowa City, Iowa 52242, USA +1

Neural PDE Solvers PDE Solving Computational Fluid Dynamics

Sep 7, 2026cs.LG

Structured Extrema Errors in Classical Surrogates for Viscous Burgers: A Physics-Consistent Interpretation

We study the local errors of classical machine-learning surrogate models, which approximate the time evolution of the one-dimensional viscous Burgers equation. Four models are compared on the same prediction task, using the spatial grid values directly: radial basis function (RBF) kernel ridge regression (KRR), linear Ridge, ExtraTrees, and Random Forests. Across all four models, the one-step residual, defined here as the true value minus the predicted value at each grid point, forms clear curved branches near predicted maxima and minima. A more detailed analysis of KRR shows that these errors are much more strongly related to the second spatial derivative, which measures local curvature, than to the first spatial derivative. Near a smooth extremum, predicted value and curvature form a local two-branch fold. Under our local curvature-based model of the residual, this fold predicts a leading-order near-parabolic relation between predicted value and residual. This geometric result motivates a direct test of the Burgers advection (transport) and diffusion (smoothing) terms. For KRR and Ridge, regression tests on held-out trajectories, a control that breaks the spatial alignment of the diffusion term, and a spectral test of high-frequency content are consistent with insufficient viscous smoothing at moderate and high viscosity. In this case, the surrogate retains more small-scale structure than the true future state. The same physical explanation is much weaker for the tree models. Finally, a correction that uses only predicted quantities reduces both one-step error and error during recursive rollout, where each prediction is used as the next input.

Youssef Oubari

IMT Atlantique, France

Kernel Ridge Regression Surrogate Modeling PDE Solving Residual Learning PDE Surrogate Modeling

Sep 7, 2026math.NA

A Systematic Analysis of Automatic Differentiation versus Discretization-based Constraints for Physics-Informed PDE Solvers

Physics-informed neural networks (PINNs) represent a growing frontier in using artificial intelligence to solve partial differential equations (PDEs). Automatic differentiation (AD) plays a central role in this paradigm, which is mesh-free and replaces traditional iterative solvers with gradient-based optimization in continuous space. However, the inherent limitations of AD, particularly in handling higher-order derivatives and discontinuous solutions, pose significant challenges for complex problems. This has motivated a growing number of researchers to explore discretization-based constraints as an alternative path. Yet, the respective applicability of these two paradigms remains largely unexplored. In this work, we conduct systematic experiments across a wide spectrum of problems, from simple linear Poisson to high-Mach hypersonic flows with strong discontinuities. Through a rigorous decomposition of approximation, optimization, and truncation errors, we systematically elucidate the fundamental trade-offs and error-governing mechanisms of both paradigms, as well as two representative network architectures: multi-layer perceptron (MLP) and graph neural network (GNN). Our results reveal a consistent trend: as nonlinearity strengthens, the accuracy advantage of discretization-based constraints becomes increasingly pronounced, with smaller optimization errors compensating for the truncation errors. Moreover, the more complex the nonlinearity and boundary conditions, the greater the advantage of GNN over MLP. These insights offer a robust practical guideline for configuring neural PDE solvers in demanding engineering applications. Our source data and code are available at https://github.com/guoxing0809/neuropde_analysis.

Xing Guo, Hongwei Tang, Zewei Meng +3

School of Systems Science and Engineering, Sun Yat-sen University, Guangzhou, China

Automatic Differentiation Neural PDE Solvers PDE Solving Physics-Informed ML

Sep 3, 2026math.NA

Residual neural networks overcome the curse of dimensionality for semilinear heat equations

Rigorous results show that feedforward neural networks can overcome the curse of dimensionality in the numerical approximation of high-dimensional partial differential equations (PDEs), but comparatively little is known about residual neural networks (ResNets) in the nonlinear PDE setting. We prove that ResNets overcome the curse of dimensionality in the numerical approximation of solutions of semilinear heat equations with globally Lipschitz continuous, gradient-independent nonlinearities: under polynomial growth and network approximability hypotheses on the PDE data, there exist

η\in(0,\infty)

and ResNets

Ψ_{d,\varepsilon}

,

d\in\mathbb{N}

,

\varepsilon\in(0,1]

, with at most

ηd^η\varepsilon^{-η}

parameters whose realizations approximate the solution in dimension

d

with an

L^2

-error of at most

\varepsilon

. The proof represents one deterministic realization of a multilevel Picard estimator by a ResNet whose shortcut connections transmit the spatial variable and a scalar accumulator, while the residual branches successively add the summands of the estimator. For ridge-sum initial conditions, admissible sigmoidal activations, and globally Lipschitz truncations of the nonlinearity, we obtain, for every

ξ>0

, the explicit bound

C_ξd^{4+ξ}\varepsilon^{-(3+ξ)}

on the number of parameters.

Ilkhom Mukhammadiev, Diyora Salimova

Department of Applied Mathematics, University of Freiburg, Germany

Neural PDE Solvers PDE Solving Residual Learning Neural Network Approximation Theory

Sep 3, 2026math.NA

Spectral Convergence of Random Feature Method in Multiple Dimensions

We first prove spectral convergence of the random feature method (RFM) for multidimensional targets in Sobolev, Gevrey, ultra-analytic, and bandlimited classes. The analysis establishes general high-probability approximation estimates in the interpolation scale generated by a kernel integral operator. On a single event determined only by the sampled features, one random space approximates every target in a prescribed source ball; moreover, for each target, a single coefficient vector defines an approximant that attains spectral accuracy simultaneously in all admissible error norms. For both regularity-adapted frequency distributions and uniform distributions on growing frequency windows, the resulting rates range from super-exponential to algebraic, depending on the regularity of the target. Second, we establish abstract error estimates for strong- and weak-form RFM discretizations, thereby converting the preceding approximation bounds into convergence estimates for multidimensional second-order elliptic boundary value and eigenvalue problems. Finally, for random feature matrices (RFMtxs), we prove super-exponential singular-value decay with Fourier features and exponential decay with

\tanh

features, together with corresponding condition-number lower bounds. The analysis identifies a common mechanism: the same spectral approximation that yields high accuracy also drives severe ill-conditioning.

Pingbing Ming, Hao Yu

PDE Solving Random Feature Methods Neural Network Approximation Theory

Sep 2, 2026physics.comp-ph

Learning Spectral-Like Mesh-Free Discretisations

Meshfree methods such as smoothed particle hydrodynamics (SPH) with kernel corrections, radial basis function-generated finite differences (RBF-FD), and the local anisotropic basis function method (LABFM) construct discrete differential operators by imposing polynomial consistency on a local stencil. For stencils containing more nodes than there are consistency constraints, the resulting linear system is underdetermined, and the remaining degrees of freedom are fixed implicitly by the choice of kernel, basis preconditioning, or a minimum-norm condition. Polynomial consistency constrains the operator only in the low-wavenumber limit, and no part of the construction selects for accuracy at the wavenumbers where fine-scale content resides. We introduce Spectral-like Neural Discretisation (SpeND), in which the choice of those degrees of freedom is cast as a learning problem: stencil weights are parametrised by a neural network conditioned on the local node geometry, trained to approximate the modal response of a spectral operator over the resolvable band. A hard-constrained projection layer maps the network output onto the affine subspace of consistent weights, so that polynomial consistency holds exactly by construction rather than as a penalty. Training is self-supervised and physics-agnostic, requiring no reference solutions; the objective minimises dispersion and dissipation error over a prescribed band-limited function space. Modal analysis on disordered two-dimensional node distributions shows that the learned fourth-order operator follows the exact response over a substantially wider band than either explicit LABFM at equal stencil size or fourth-order finite differences on a structured grid, whilst recovering the expected fourth-order convergence rate under refinement.

Lucas Gerken Starepravo, Henry Broadley, Steven Lind +1

School of Engineering, University of Manchester Manchester, UK · School of Engineering, Cardiff University Cardiff, UK

Aug 31, 2026cs.AI

DiffPDE: Masked Diffusion Language Models as PDE Solver

Existing approaches for synthesizing Partial Differential Equation (PDE) solvers predominantly rely on autoregressive models, yet their global left-to-right decoding incurs substantial redundancy when addressing inherently localized bugs. In this work, we challenge this inefficient paradigm and propose DiffPDE, a framework leveraging discrete diffusion language models for targeted code repair. By introducing a localized re-masking and infilling strategy, DiffPDE regenerates only erroneous regions while preserving correct context, naturally aligning generation with the sparse nature of PDE errors. Furthermore, to handle coupled bugs requiring sequential interventions, we present Iterative Debugging GRPO (ID-GRPO), a reinforcement learning scheme that enables multi-round debugging within single trajectories via intermediate rewards. Experiments on PDEBench show that DiffPDE achieves competitive accuracy, outperforms same-scale AR models, and significantly accelerates repair.

Wenxuan Guo, Yuyang Hong, Lubin Fan +4

School of Artificial Intelligence, University of Chinese Academy of Sciences · MAIS, Institute of Automation, Chinese Academy of Sciences · Independent Researcher

RL for Code Generation Masked Diffusion Models Automated Program Repair PDE Solving

Aug 31, 2026cs.LG

Learning PDE Time-Stepping with Neural Cellular Automata

Classical numerical solvers for partial differential equations (PDEs) are computationally expensive to solve repeatedly across varying initial conditions, motivating the need for learned surrogates. In this paper, we propose a trainable Neural Cellular Automata (NCA) based surrogate model for learning long time PDE dynamics. Rather than mapping an entire initial field to a full trajectory in one shot, our proposed model learns a small, local, homogeneous update rule that is applied identically and repeatedly at every grid cell, mirroring the locality of differential operators. We benchmark this framework against three baselines: PDE - Net, a modified physics-informed neural network (PINN), and a Fourier Neural Operator (FNO), on five canonical PDEs (heat, advection, Burgers, Allen - Cahn, and Fisher - KPP), evaluated at temporal domain two times beyond the training temporal domain. The proposed model achieves the lowest long-horizon relative errors on the majority of the experiments.

Esha Saha, Hao Wang

1Interdisciplinary Lab for Mathematical Ecology and Epidemiology (ILMEE) & The Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton (AB), T6G 2J5, Canada

Neural Cellular Automata PDE Solving PDE Surrogate Modeling

Aug 12, 2026cs.LG

RECAST: A Machine-Learning Framework for Correction and Super-Resolution of Coarse-Grid PDE Solvers

Coarse-grid numerical solvers can substantially reduce the computational cost of time-dependent PDE simulation, but under-resolution often degrades both the trajectory and the spatial fidelity of the solution. We introduce RECAST (Recurrent Error Correction And Super-resolution of coarse-grid Trajectories), a machine-learning framework designed to restore this lost accuracy while retaining coarse-grid evolution. RECAST combines learned correction within the numerical time-stepping loop with reconstruction of the corresponding fine-grid state from the corrected coarse history. We evaluate the framework on six one-dimensional PDE systems spanning transport, diffusion, dispersion, reaction, and wave dynamics, using spatial grids coarsened by factors of 8-16 and 1000-step closed-loop rollouts from unseen initial conditions. Across the test cases, RECAST remains closely aligned with the fine-grid reference solutions and reduces time-averaged relative error by approximately 50-92% compared with the corresponding uncorrected coarse-grid solvers. Additional tests show generalization to unseen PDE parameter values, while comparison with a contemporary coarse-correction architecture shows that RECAST achieves lower error and better long-horizon agreement with the fine-grid reference over 5000-step rollouts. These results demonstrate that the learned correction and reconstruction capabilities of RECAST can enable substantially coarser PDE evolution without the corresponding loss of solution fidelity, providing a proof-of-concept route toward machine-learning acceleration of higher-dimensional numerical simulations across science and engineering.

Maryam Reza, Farbod Faraji

Independent Researcher · Department of Computing, Huxley Building, Imperial College London, London SW7 2RH, United Kingdom

Neural PDE Solvers PDE Solving PDE Surrogate Modeling

Aug 11, 2026cs.LG

Derivative Computation in PINNs: Automatic Differentiation, Finite Differences and Beyond

We systematically investigate finite-difference (FD) derivative computation in Physics-Informed Neural Networks (PINNs) as an alternative to automatic differentiation (AD). On three benchmark PDEs we show that, with a properly calibrated step size, FD matches AD in accuracy on every problem while running faster across the full tested batch-size range and using substantially less GPU memory, and that a stochastic variant we propose outperforms AD on a stationary problem. We further show that for neural architectures with inter-sample dependencies (e.g. BatchNorm, self-attention) the standard PyTorch autograd idiom is silently incorrect; the correct per-sample alternative is computationally infeasible at PINN-relevant batch sizes, while FD provides a forward-only approximation that is empirically an order of magnitude closer to the true per-sample derivative.

Maciej J. Mikulski, Tadeusz Uhl

AGH University of Krakow

Automatic Differentiation Neural PDE Solvers PDE Solving Physics-Informed ML