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.

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Latest papers 170

Aug 11, 2026math.NA

Efficient Weak-Entropy PINN for Solving Hyperbolic Conservation Laws

In recent years, neural networks have significantly advanced numerical solutions of partial differential equations (PDEs). However, solving PDEs with discontinuous solutions, such as hyperbolic conservation laws, remains challenging for neural network-based methods such as physics-informed neural networks (PINNs). Existing methods often rely on strong prior assumptions such as knowledge of discontinuity locations, or they introduce artificial smoothing terms that degrade accuracy. However, accurately solving these conservation laws and predicting the formation and propagation of discontinuities in solutions is crucial in many practical applications, including gas dynamics and traffic flow modeling. In this paper, we introduce a novel Weak-Entropy PINN (WEPINN) framework for hyperbolic conservation laws with discontinuous solutions. The method enforces the governing equations in their weak (integral) formulation and incorporates the entropy condition to select the physically admissible solution, while employing the discrete fast Fourier transform (DFFT) for efficient numerical integration. Our method is tested through extensive numerical experiments on a variety of scalar conservation laws and systems of conservation laws in one and two dimensional spaces. These experiments demonstrate that our method can accurately resolve sharp discontinuities while effectively capturing interactions between multiple shock and rarefaction waves.

Qi Gao, Kuang Huang, Xuan Di

Department of Civil Engineering and Engineering Mechanics, Columbia University · Department of Mathematics, The Chinese University of Hong Kong · Center for Smart Cities, Data Science Institute, Columbia University

Neural PDE Solvers PDE Solving Physics-Informed ML

Aug 10, 2026math.NA

Walk-on-Spheres Monte Carlo and deep neural network approximations of elliptic PDEs with drift and killing

In this paper we provide Monte Carlo and deep neural network approximations for stochastic representations of solutions to linear elliptic partial differential equations with constant diffusion, drift and killing. Building on the modified Walk-on-Spheres algorithm of Beznea et al. (arXiv:2209.01432), we introduce Monte Carlo estimators that explicitly incorporate sampled random times arising in the analyzed stochastic representations. We establish uniform error bounds for these estimators and show that, under suitable assumptions, a prescribed approximation accuracy is achieved with sample complexities growing at most polynomially in both the inverse accuracy and the problem dimension. Furthermore, we prove a deep neural network approximation result for the stochastic representations. Assuming suitable neural network representations of the boundary data and the distance function to the boundary, we use the constructed Monte Carlo to design deep neural networks that approximate the representation uniformly with a number of parameters growing at most polynomially in the inverse accuracy and the problem dimension. These results extend previous complexity analyses to a broader class of elliptic equations involving drift and killing.

Konrad Kleinberg, Thomas Kruse

Department of Mathematics & Informatics, University of Wuppertal, Germany

Neural PDE Solvers PDE Solving

Aug 9, 2026cs.LG

Quantum-Classical Physics-Informed Kolmogorov-Arnold Networks for Solving Fuzzy Differential Equations

In this study, we propose a quantum-classical physics-informed Kolmogorov-Arnold network (QCPIKAN) dedicated to the solution of fuzzy differential equations. The network takes the spatiotemporal coordinates and membership level as joint inputs and employs ChebyKAN modules and a parameterized quantum circuit to construct a hybrid function approximator. It simultaneously approximates the lower and upper endpoint functions associated with the α-cuts and incorporates the governing equations, initial-boundary conditions, and fuzzy-structural constraints into the training objective. Theoretically, a unified error-analysis framework is established for QCPIKAN and PIKAN, in which the endpoint-solution error is decomposed into approximation, sampling, optimization, and fuzzy-structure constraint errors. Under the assumptions of well-posedness and residual stability, it is proved that QCPIKAN has a smaller a priori error bound when the representational gain introduced by quantum entanglement features exceeds the additional computational error. Numerical experiments are conducted for elliptic, parabolic, and hyperbolic equations in an ideal quantum-simulation environment. The results show that QCPIKAN captures the overall contraction of the solution interval as increases. At most tested membership levels, the mean relative L2 error of PIKAN is approximately 1.1-2.7 times that of QCPIKAN. In the fuzzy convection example, the mean wavefront-position error of PIKAN is approximately 1.77 times that of QCPIKAN. Nevertheless, both models still exhibit local fuzzy-structure violations near boundaries, in high-gradient regions, and around the wavefront. These results indicate that QCPIKAN provides a quantum-classical hybrid physics-informed computational framework with comparatively high predictive accuracy for solving fuzzy partial differential equations represented by α-cuts.

Xiang Rao, Yuxuan Shen

1. School of Petroleum Engineering, Yangtze University, Wuhan 430100, China · 2. College of Future Technology, Yangtze University, Wuhan 430100, China · 3. School of Computer Science, Yangtze University, Jingzhou 434023, China +2

PDE Solving Hybrid Quantum-Classical ML Physics-Informed ML

Aug 6, 2026cs.SD

Explicit and Stable Pseudospectral Time-Domain Method for the Föppl-von Kármán Equations

Modal synthesis is a widely-used technique for simulation of musical instrument dynamics. In the linear case, a modal decomposition leads to an uncoupled system of damped and forced harmonic oscillators which can be efficiently solved by standard time-stepping methods. However, extensions to nonlinear problems are challenging due to the presence of products of modal expansions in the governing equations. In the case of the Föppl-von Kármán plate, the nonlinear coupling between the modes is described by a fourth-order tensor and is prohibitively expensive to evaluate in the modal domain. In this work, we propose a pseudospectral method in which the products are evaluated on a grid in the spatial domain while spatial derivatives are computed exactly in the modal domain. Discrete sine and cosine transforms between the modal and spatial domains are used to impose simply supported boundary conditions for the plate. Finally, we prove non-negativity of the nonlinear potential energy of the system and employ a scalar auxiliary variable technique for explicit and stable time integration in the modal domain. As a result, we reduce the computational cost of modal synthesis while preserving its advantages like a precise control over the simulated frequency range. Sound examples are presented.

Victor Zheleznov, Stefan Bilbao

Acoustics and Audio Group, University of Edinburgh, Edinburgh, UK · STMS (UMR9912), IRCAM, CNRS, Sorbonne Universit´e, Paris, France

PDE Solving Structural Mechanics Spectral Methods Physics-Based Simulation

Aug 1, 2026cs.LG

Adaptive Quantum Physics-Informed Neural Networks for Differential Equations with Applications to Fluid Dynamics

Physics-informed neural networks (PINNs) have emerged as a versatile approach for solving nonlinear partial differential equations (PDEs), yet achieving high accuracy efficiently using these techniques remains challenging for high-dimensional or multiscale systems. Here, we present a hybrid quantum-classical framework that enhances Quantum PINNs (QPINNs) through adaptive collocation point sampling and loss-aware attention mechanisms. By dynamically prioritizing points in regions with large PDE residuals or steep solution gradients, our method mitigates the spectral bias inherent in conventional PINNs. Current Quantum Physics-Informed Neural Networks are commonly assumed to be limited by the expressive power of quantum circuits. In our work, we observed that, across diverse differential equations, optimization - not only expressivity - can be an important bottleneck. Furthermore, a trainable loss-weighting scheme balances contributions from physics residuals, boundary conditions, and data fidelity during training. Integrating these strategies with quantum computing techniques (including variational quantum circuits and quantum gradient estimation) can yield at least a 60% improvement in solution accuracy under specific regimes for benchmark fluid flows and reaction-diffusion systems. Finally, we argue that merely increasing model expressivity is insufficient for resolving complex PDEs via QPINNs, as they remain constrained by the structural optimization limitations of classical PINNs. This framework provides a scalable pathway for quantum-enhanced scientific machine learning, bridging physics-based modeling with emerging quantum computational capabilities.

Fabio Pereira dos Santos, Renato Portugal, Júlio de Castro Vargas Fernandes +1

Coordination of Mathematical and Computational Methods National Laboratory of Scientific Computing Av. Getúlio Vargas, 333 - Quitandinha, Petrópolis - RJ, 25651-075 · Center for Computation and Technology - Louisiana State University, 888 S. Stadium Dr., Baton Rouge, LA 70808, USA

PDE Solving Hybrid Quantum-Classical ML Adaptive Sampling Physics-Informed ML

Jul 30, 2026math.CA

Reflected diffusion, no-flux continuity equations and confined Lagrangian flows in bounded domains

Motivated by marginal distribution flows of reflected diffusions in bounded domains, we investigate when a density/flux pair solving a no-flux continuity equation admits a regular Lagrangian flow that remains in the closed domain and generates the prescribed density flow. We give sufficient conditions in terms of interior bounded-variation regularity, bounded-variation control on a boundary collar, a one-sided bound on an absolutely continuous divergence, and vanishing normal trace of the velocity. The proof uses the fact that tangency removes the singular boundary contribution to the divergence of the zero extension, thereby making the extended velocity admissible for the Ambrosio-DiPerna-Lions theory. We show that these boundary assumptions cannot be jointly relaxed so as to admit a boundary current mechanism. We construct an explicit smooth density/flux pair carrying a boundary current. Its density evolution is unique in a weighted class and its characteristics are unique, confined and transport the marginals, yet it admits no regular Lagrangian flow because the compressibility bound fails arbitrarily close to the initial time. We also establish two uniqueness results for no-flux Fokker-Planck equations: a duality result for bounded measurable drifts and a weighted energy result for entrance-type drifts singular at the boundary. Our results provide a rigorous mathematical justification for using the ODE-based sampling of reflected diffusion models under minimal regularity assumptions on the coefficients, and also indicate when such ODE-based samplers may fail.

Rama Cont

Dynamical Systems Stochastic Differential Equations PDE Solving

Jul 30, 2026cs.LG

Learning features from Newton's algorithm: a way to accelerate nonlinear parametrized PDE solvers

It is well known that Newton's method converges faster when the initial guess is closer to a root of a system of nonlinear equations. In this paper, a two-stage Newton initial guess strategy is proposed by learning features from a parameter-space sampling and a database of precomputed solutions. The method uses discrete Newton trajectories to construct two complementary reduced spaces: a solution feature space, built from converged states, and a corrective search direction feature space, built from intermediate Newton increments. For an unseen parameter, a regression model is used to predict a surrogate solution approximation. Then, in a second step, a residual-minimizing correction is computed using a dedicated GMRES-based approach. The resulting state is then used as an initial guess for the high-fidelity Newton method, which completes convergence. The corrective step is computationally inexpensive since it only requires residual evaluations and the solution of a small least-squares problem. The methodology is weakly intrusive once the high-fidelity residual fields and a script-based programming interface are available. This strategy reduces the number of Newton iterations and decreases the overall CPU time. Numerical experiments on representative PDE problems show quantifiable speedups compared with standalone surrogate initialization. Significant speedups are observed. This generic approach can be applied to a broad class of large-scale nonlinear problems.

Rémy Vallot, Florian de Vuyst, Thibault Dairay +1

CB, Michelin · Centre Borelli UMR 9010, ENS Paris-Saclay, Université Paris-Saclay, CNRS, France · Michelin, Centre de Recherche de Ladoux, Cébazat, France +4

PDE Solving PDE Surrogate Modeling

Jul 30, 2026math.AP

Fractional Parabolic Partial Differential Equations in Anisotropic Spectral Barron Spaces: Regularity and Neural Approximation

We study fractional parabolic initial-value problems with lower-order drift and potential terms in anisotropic spectral Barron spaces, defined by weighted space--time Fourier

L^1

norms adapted to parabolic scaling. We prove existence, uniqueness, and maximal regularity with a gain of one derivative in time and

γ

derivatives in space, where

γ>0

is the order of the fractional Laplacian. The evolution is defined only for

t\geq0

, whereas the finite-time norm requires a global extension with sufficient temporal Fourier decay. We construct a finite reflected semigroup extension using a Vandermonde system to match derivatives at

t=0

, obtaining temporal Fourier estimates uniform in the semigroup parameter. Combined with Fourier multiplier estimates for the damped principal operator, it yields maximal regularity. Dimension-independent multiplication estimates support a finite regularity bootstrap, while interpolation and sufficient damping absorb the lower-order terms in the base estimate. The a priori estimate and the method of continuity yield maximal regularity without smallness assumptions on the lower-order coefficients. A frequency-localized counterexample shows that a uniform-in-time spatial Barron bound on the forcing does not imply the corresponding two-derivative solution bound, even for the one-dimensional heat equation. Using this regularity, Fourier sampling yields

n^{-1/2}

approximation rates for the solution in mixed space--time Sobolev norms using shallow networks with suitable activations. Sampling in a product Hilbert space yields a population-level PINN consistency estimate for shallow cosine networks on a bounded cylinder. There exists a single width-

n

network for which the sum of the squared mixed-Sobolev solution error, the squared

L^2

-norm of the residual for the whole-space fractional equation, and the squared initial-data error is

O(n^{-1})

.

Jae-Hwan Choi, Hyojae Lim, Jinsol Seo +2

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

Jul 28, 2026cs.LG

Physics-Informed Broad Learning System: An Efficient Backpropagation-Free Framework for Solving Partial Differential Equations

Physics-informed neural networks (PINNs) have emerged as a powerful paradigm for solving partial differential equations (PDEs) by embedding governing physical laws into deep neural networks. However, their reliance on computationally expensive gradient-based optimization and deep architectures often results in slow training, high computational cost, and limited scalability. In this work, we propose a novel physics-informed broad learning system (PI-BLS), the first physics-informed learning framework based on broad RdNNs. The proposed formulation embeds the governing differential operator and the associated initial and boundary constraints directly into a linear output-layer optimization problem, thereby replacing nonlinear gradient-based training with a deterministic least-squares solution obtained via the pseudoinverse. Consequently, the entire learning process is reduced to a single linear optimization stage while preserving the underlying physical constraints. As a result, PI-BLS offers an efficient learning paradigm for a physics-informed learning framework for solving PDEs that eliminates iterative backpropagation while preserving the underlying physical constraints. Experimental results on representative forward PDE benchmarks demonstrate that PI-BLS achieves competitive and often superior performance with reduced training time and model parameters compared with conventional PINNs.

Pinki Khatun, M. Sajid, Abhinav Jha +1

Department of Industrial Engineering, Università degli Studi di Firenze, Viale Morgagni 40/44, 50134 Firenze, Italy · Department of Mathematics, Indian Institute of Technology Indore, Indore 453552, India · Department of Mathematics, Indian Institute of Technology Gandhinagar, Palaj, Gandhinagar, 382055, Gujarat, India

Neural PDE Solvers PDE Solving Physics-Informed ML

Jul 27, 2026cs.LG

Score-Based Stabilization for Time-Dependent Problems

We propose a score-based stabilization framework for numerical simulation of partial differential equations, in which a learned score model defines a stabilization operator applied to provisional numerical updates. This operator augments standard time-stepping schemes by enforcing structure and physical consistency through a correction that drives iterates toward the manifold of admissible states. We show that the stabilization operator acts as a contraction toward this manifold, yielding a correction mechanism with basin-conditional stability. Numerical experiments on Advection, Korteweg-de Vries (KdV), Nonlinear Schrodinger (NLS), and Burgers' equations demonstrate improved robustness, suppression of nonphysical instabilities, and preservation of qualitative dynamics.

Eshed Gal, Eldad Haber, Uri Ascher

University of British Columbia Vancouver, British Columbia, Canada

Neural PDE Solvers PDE Solving

Jul 27, 2026cs.LG

Global Convergence of DGM and PINN Algorithms for Solving Nonlinear PDEs

The Deep Galerkin Method (DGM) and Physics Informed Neural Networks (PINNs) have become widely-used methods for solving partial differential equations (PDEs) in the rapidly growing field of scientific machine learning. In these methods, a neural network is trained to approximate the PDE solution by using (stochastic) gradient descent to minimize the PDE residual of the neural network. Due to the non-convexity of the PDE residual objective function, the trained neural network may, in principle, only converge to a local minimizer of the objective function (which would not be a solution of the PDE). Therefore, there is a longstanding question regarding the mathematical foundations of these algorithms, and it is highly valuable to establish that the trained neural network will converge to the PDE solution. In this paper, we consider a class of semilinear PDEs with nonlinearities in the solution and its first derivative. For this class of PDEs, we prove that neural networks trained with gradient descent to minimize the PDE residual objective function will converge to the PDE solution as the network width and training time

\rightarrow \infty

.

Justin Sirignano, Konstantinos Spiliopoulos, Samuel Cohen

Mathematical Institute, University of Oxford · Department of Mathematics and Statistics, Boston University

Neural PDE Solvers Optimization Convergence Analysis PDE Solving Physics-Informed ML

Jul 27, 2026cs.LG

Perturbative-NeuSA: A Structured Spectral Framework for Time-Dependent PDEs

Neural spectral PDE solvers often learn an entire unresolved vector field even when an inexpensive approximate model can already capture most of the trajectory. Here we introduce Perturbative-NeuSA, a residual formulation that decomposes the target solution into a low-fidelity background and a high-resolution perturbation, so that only the unresolved dynamics is learned. Starting from the exact perturbation equation, the method combines a fixed spectral operator, a background-dependent correction, the background defect in the target PDE, and an optional neural closure. This construction makes the roles of physical structure and neural closure separately measurable. Across 2D Burgers, Klein-Gordon, and heterogeneous 2D wave equations, the deterministic structured solver outperforms the trained NeuSA baseline while requiring no neural-network training. The largest gains occur on Burgers, where the deterministic correction reduces training and extrapolation errors by factors of 24 and 44, respectively. In addition, a Klein-Gordon sweep over seven background resolutions shows that the effect of the closure is conditional: it improves a poor background by 3.6 times, becomes neutral at intermediate resolutions, and degrades a well-resolved background. For the wave equation, however, the closure provides an additional 18% reduction when the remaining residual is interface-localized. Multi-initial-condition diagnostics further show that the useful closure regime depends on the initial-condition spectrum and can disappear in extrapolation when structured correction already captures the dominant Burgers dynamics. Perturbative-NeuSA therefore reframes neural closure as a conditional, diagnosable correction governed by background fidelity, residual organization, and compatibility with the closure model.

Xianli Zhu, Jia Yin

School of Mathematical Sciences, Fudan University Shanghai, China

Neural Surrogate Modeling Neural PDE Solvers PDE Solving Spectral Methods

Jul 27, 2026cs.LG

Variational Boosting for Physics-Informed Neural Networks

Physics-Informed Neural Networks (PINNs) solve differential equations by minimizing the residual of a nonlinear operator over a neural parameterization of the solution. However, monolithic PINNs often suffer from ill-conditioning, spectral bias, and optimization instability. We introduce a variational boosting framework in which solutions are constructed additively in function space. Each stage trains a weak learner whose converged correction satisfies a local orthogonality condition, equivalent to a projected functional gradient descent step onto the tangent space of the network's function manifold. Because each correction network is deliberately small, the restricted minimization admits full Newton or conjugate gradient updates, which are typically infeasible in large PINNs. The resulting method separates global nonlinear refinement into a sequence of well-conditioned subproblems while preserving the full variational structure of the operator. This framework provides a geometric interpretation of multi-stage PINNs as projected functional gradient descent and enables stable second-order optimization for nonlinear differential equations.

Kaylee Vo, Pavlos Protopapas

Harvard University

Ensemble Learning PDE Solving Second-Order Optimization Physics-Informed ML

Jul 26, 2026cs.LG

Extending Fourier Neural Operators for Modeling Parameterized and Coupled PDEs

Parameterized and coupled partial differential equations (PDEs) are central to modeling phenomena in science and engineering, yet neural operator methods that address both aspects remain limited. We extend Fourier neural operators (FNOs) with minimal architectural modifications along two directions. For parameterized dynamics, we propose a hypernetwork-based modulation that conditions the operator on physical parameters. For coupled systems, we conduct a systematic exploration of architectural choices, examining how operator components can be adapted to balance shared structure with cross-variable interactions while retaining the efficiency of standard FNOs. Evaluations on benchmark PDEs, including the one-dimensional capacitively coupled plasma equations and the Gray-Scott system, show that our methods achieve up to 55-72% lower errors than strong baselines, demonstrating the effectiveness of principled modulation and systematic design exploration.

Cheng Jing, Uvini Balasuriya Mudiyanselage, Abhishek Verma +3

Arizona State University · Applied Materials Inc.

Fourier Neural Operator PDE Solving PDE Operator Learning

Jul 24, 2026cs.CE

Generalized Neural Operator for Parametric and Boundary-Value Problems

Developing foundational neural simulators for Partial Differential Equations (PDEs) requires robust generalization across diverse physical parameters and boundary conditions. However, current deep learning approaches largely face a structural trade-off between condition-agnostic deployment and physical fidelity. Purely data-driven operators infer the underlying physics implicitly and thus lack the explicit constraints needed to ensure physically valid solutions across varying domains, rendering the learning problem ill-posed. On the other hand, Physics-Informed Neural Networks (PINNs) enforce rigorous physical constraints but necessitate costly, instance-specific optimization. Furthermore, the massive scale of emerging foundational operators has severely degraded their inference speeds, making them computationally uncompetitive with traditional numerical solvers. To address this bottleneck between condition-agnostic deployment, physical rigor, and inference efficiency, we propose a \textit{Generalized Neural Operator}. By formalizing the classical conditions for well-posedness within neural operators, our framework demonstrates the theoretical benefits of explicitly conditioning on PDE parameters and boundary conditions. To implement this synthesis without compromising computational speed, we introduce three novel architectural components: a parameter-gated mixture of kernels for efficient parameter generalization, a generalized boundary transfer operator that projects arbitrary boundary constraints into a unified latent Dirichlet representation, and a specialized training objective to ensure stability. Extensive experiments demonstrate that our theoretically grounded approach achieves superior generalization across heterogeneous physical regimes while maintaining strict inference efficiency comparable to conventional numerical baselines.

Ruoyan Li, Yizhou Sun, Wei Wang

University of California, Los Angeles

Efficient Neural Network Inference PDE Solving PDE Operator Learning Physics-Informed Neural Operators Neural Operators

Jul 22, 2026physics.flu-dyn

Label-Free Finite-Volume-Residual Training of Attention Graph Neural Networks for Coupled Thermo-Fluid Fields

Neural surrogates are widely used in scientific machine learning for fast prediction of three-dimensional (3D) thermo-fluid fields. However, generating training data using conventional numerical solvers often incurs substantial computational and storage costs. We propose to train an attention graph neural network by minimizing the finite-volume method (FVM) residuals of the governing equations. These residuals are evaluated directly on the mesh, requiring no labeled data. We evaluate the trained surrogates against computational fluid dynamics (CFD) references and a data-supervised baseline across four scenarios. On the two steady-state benchmarks, the FVM-loss model achieves an all-field normalized root-mean-square error (nRMSE) of 2.3-2.8%. It demonstrates close agreement with the CFD references, including the buoyancy-energy coupling. On the two parametric transient cases, the FVM-loss model outperforms the supervised baseline in terms of accuracy, while avoiding the data-generation cost entirely. These results indicate that the FVM loss can provide a practical training signal for neural surrogates and reduce the model development cost.

Tianyu Li, Zhiwei Cao, Qingang Zhang +3

Interdisciplinary Graduate Programme, Nanyang Technological University, Singapore, Singapore. · School of Future Technology, Fuzhou University, Fuzhou, China. · College of Computing and Data Science, Nanyang Technological University, Singapore, Singapore. +2

Neural Surrogate Modeling Graph Neural Networks PDE Solving Physics-Informed ML Physics-Informed GNNs Computational Fluid Dynamics

Jul 22, 2026math.NA

A Structure-Adaptive Random Feature Method for High-Dimensional Elliptic PDEs

Random-feature methods reduce high-dimensional elliptic PDE collocation to linear coefficient problems, but full-dimensional trial spaces overlook lower-dimensional structure. We introduce the Hierarchical Analysis-of-Variance Random Feature Method (HA-RFM), which selects coordinate blocks using closed Sobol indices of the PDE residual, identifies oblique low-rank features from fitted-predictor gradients, and couples all retained features in one regularized least-squares solve. Under structural and stability hypotheses, we establish an

L^2

error bound that links solution and residual truncation to finite-width approximation and regularized finite-sample fitting, and we derive guarantees for width and structure recovery. The resulting width is polynomial in the dimension at fixed interaction order, with dimension-independent higher-order contributions under uniform structural control. Residual screening achieves exact recovery of the prescribed three-pair support, while fitted-predictor gradients recover oblique directions through dimension

50

. In random-ridge tests, less than

1\%

additional width reduces errors by factors of

14

-

39

over coordinate blocks and

34

-

100

over equal-width full-dimensional RFM. Semilinear computations extend HA-RFM through dimension

100

, while dense and distributed interactions delineate the coordinate families required for broader structure.

Jiale Linghu, Hao Dong, Yangshuai Wang

School of Mathematics and Statistics, Xidian University, Xi’an 710071, China. · Department of Mathematics, National University of Singapore, 10 Lower Kent Ridge Road, 119076, Singapore

PDE Solving Random Feature Methods Kernel Methods

Jul 21, 2026math.NA

Boundary-Adapted PINNs for Elliptic Dirichlet Problems: $H^2(Ω)$ A Priori Error Bounds with Application to Mean Escape Time Computation

Motivated by the numerical computation of the Mean Escape Time (MET)

τ:Ω\to\mathbb{R}

of a stochastic process from a bounded domain

Ω\subseteq\mathbb{R}^d

, we study elliptic Dirichlet boundary value problems (BVPs) using boundary-enforced Physics-Informed Neural Networks (PINNs), in which the Dirichlet condition is imposed exactly by multiplying the network output with a predefined distance-to-boundary approximation

ρ

. Combining approximation-theoretic and statistical-learning arguments for Rectified Quadratic Unit (ReQU) and hyperbolic tangent (tanh) networks, we derive a priori error bounds that make explicit the dependence on

ρ

. In particular, we show that exact boundary enforcement alone is not enough for

H^2(Ω)

error bounds, and that a sufficient and essentially necessary condition is for

ρ

to be a smooth distance approximation

\textit{normalized to first order}

, of the kind constructed in arXiv:2104.08426 [math.NA]. We thereby identify this subclass of

\textit{boundary-adapted}

PINNs as the appropriate neural network ansatz for solving Dirichlet BVPs. Numerical experiments support the theory, showing that appropriate choices of

ρ

improve accuracy and convergence, while poorly chosen distance functions can substantially degrade the solution. Our proof also yields new VC-dimension bounds for hypothesis spaces of higher-order derivatives of ReQU and tanh networks, together with new approximation bounds for shallow ReQU networks in higher-order Sobolev norms, all of which are of important independent interest.

Nathanael Tepakbong, Jun Fan, Xiang Zhou +1

Department of Data Science, City University of Hong Kong · Department of Mathematics, Hong Kong Baptist University · Department of Mathematics, City University of Hong Kong +1

PDE Solving Neural Network Approximation Theory Physics-Informed ML Generalization Bounds

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.LG

Adaptive Mamba Neural Operators

Accurately solving partial differential equations (PDEs) on arbitrary geometries and a variety of meshes is an important task in science and engineering applications. In this paper, we propose Adaptive Mamba Neural Operators (AMO), which integrates reproducing kernels for state-space models (SSMs) rather than the kernel integral formulation of SSMs. This is achieved by constructing Takenaka-Malmquist systems for the PDEs. AMO offers new representations that align well with the adaptive Fourier decomposition (AFD) theory and can approximate the solution manifold of PDEs on a wide range of geometries and meshes. In several challenging benchmark PDE problems in the fields of fluid physics, solid physics, and finance on point clouds, structured meshes, regular grids, and irregular domains, AMO consistently outperforms state-of-the-art solvers in terms of relative

L^2

error. Overall, this work presents a new paradigm for designing explainable neural operator frameworks.

Zeyuan Song, Zheyu Jiang

School of Chemical Engineering Oklahoma State University Stillwater, OK 74078, USA

PDE Solving PDE Operator Learning Neural Operators State Space Models

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, 2026cs.LG

Trainable Spline Representations for Physics-Informed Learning

This work introduces Physics-Informed Splines (PI-Splines), a structured spline-based architecture for physics-informed learning. Instead of representing the solution of a differential equation with a neural network, PI-Splines directly parametrize the unknown field through a tensor-product B-spline expansion with trainable control coefficients. This formulation preserves the residual-based training paradigm of Physics-Informed Neural Networks while providing compact support, explicit smoothness control, analytical derivatives, and a direct geometric interpretation of the trainable parameters. When compatible with the spline representation, boundary conditions can be imposed strongly by fixing suitable boundary control coefficients. The proposed method is evaluated on several benchmark problems of increasing difficulty and compared with standard physics-informed frameworks under matched governing equations, collocation sets, loss terms, and optimization procedures, so as to isolate the effect of the approximation architecture. Numerical experiments show that PI-Splines provide a competitive and stable alternative to neural physics-informed architectures, particularly in settings where structured representations, locality, and parameter efficiency are desirable.

Giovanni Canali, Nicola Demo, Gianluigi Rozza

Mathematics Area, mathLab, SISSA, via Bonomea 265, I-34136, Trieste, Italy · FAST Computing Srl, via Mazzini 20, I-34136, Trieste, Italy

PDE Solving Physics-Informed ML

Jul 16, 2026math.NA

Subgrid-Scale Parameterization in Burgers' Equation Using Structure-Preserving Neural Networks and Entropy Variables

We present a machine learning approach for developing subgrid-scale (SGS) parametrizations in coarse simulations of partial differential equations. We utilize structure-preserving neural networks and entropy variables to learn subgrid fluxes in coarse simulations of the Burgers' equation. In particular, we employ a decoupled neural network architecture explicitly separating the subgrid corrections into two distinct components: a conservative Flux Potential network and an Eddy Viscosity network. We demonstrate that this reduced-order framework maintains high physical fidelity, accurately reproducing the energy spectrum, spatial and temporal correlation functions, and dynamical characteristics of the full-scale system. Furthermore, we show that our approach is robust and applicable to parameters outside the training regime.

Aijaz Nazir, Ilya Timofeyev

Dept. of Mathematics, University of Houston, Houston, TX 77204

Neural Surrogate Modeling PDE Solving Computational Fluid Dynamics

Jul 15, 2026stat.ML

Operator-Informed Gaussian Processes for Complex Helmholtz Wavefields: From Synthetic Benchmarks to In Vivo Brain Elastography

The Helmholtz equation governs time-harmonic wave propagation, and in dissipative media a complex modulus renders its squared wavenumber

κ^2

complex. Inferring such fields from sparse, noisy data calls for solvers that also quantify their own uncertainty. Physics-informed Gaussian-process (GP) regression supplies this by returning a posterior over the solution, yet operator-conditioned formulations have been developed almost exclusively for real-valued fields. We extend operator-informed GP regression to complex-valued Helmholtz problems by realifying the complex operator into an equivalent coupled real block, which enables inference with standard real-valued GP conditioning. The construction admits a family of priors, from a proper diagonal prior to coregionalized and multiscale variants, and conditions on PDE residuals and boundary traces. On benchmark problems in one to three dimensions, the solver is competitive with finite-difference and neural-network baselines at a far smaller interior-constraint budget. Unlike those deterministic baselines, it returns a posterior over the complex wavefield rather than a point estimate. Applied to \textit{in vivo} brain magnetic resonance elastography, a proper multiscale prior reconstructs the shear curl field to a correlation of

0.77

with measurement, above a

0.75

target. The gain arises from the multiscale kernel rather than from real--imaginary coupling. We further identify a low-frequency accuracy ceiling set by model mismatch and a posterior uncertainty that is not yet calibrated. Calibrated uncertainty therefore emerges as the central next step for probabilistic wavefield inference in dissipative media.

Boyuan Deng, Kshitiz Upadhyay, Michael Shields

Department of Civil and Systems Engineering, Johns Hopkins University · Department of Aerospace Engineering and Mechanics, University of Minnesota

PDE Solving Physics-Informed ML PDE Inverse Problems

Jul 15, 2026math.NA

Spectral-Informed Neural Networks Outperform Spectral Methods in High-dimensional PDEs

For low-dimensional problems (

d\leq3

), spectral methods can achieve exceptionally high accuracy. For middle-dimensional problems (

4 \leq d \lesssim 10

), spectral methods remain feasible through specific techniques such as sparse grids or hyperbolic cross. However, for high-dimensional problems (

d\gg 10

), spectral methods suffer frome the curse of dimensionality. Physics-informed neural networks (PINNs) have emerged as a promising approach to overcome this challenge, offering scalability to high dimensions, but often suffer from limited accuracy and efficiency. Recently proposed spectral-informed neural networks (SINNs) combine spectral methods with PINNs, operating directly in the spectral domain to avoid spatial derivative computations and to reduce memory consumption. In this work, we introduce Modified SINNs, which integrate coefficient decay scaling and basis embeddings motivated by harmonic analysis to enhance accuracy in high-dimensional problems and enable accurate approximation of unknown spectral coefficients. Numerical experiments on steady and time-dependent partial differential equations demonstrate that Modified SINNs outperform sparse grid spectral methods on middle-dimensional problems with incomplete spectral information and achieve superior accuracy compared to PINNs on high-dimensional problems.

Tianchi Yu, Ivan Oseledets

1AXXX · 2Applied AI Institute · 3INM

PDE Solving Physics-Informed ML Spectral Methods

Jul 14, 2026math.NA

Deep Learning-based Surrogate Modelling of the LOD Method for Multiscale Problems

Multiscale problems are notoriously difficult to tackle using traditional numerical methods, as accurately resolving fine-scale features often requires prohibitively fine discretizations. This challenge is particularly pronounced in applications such as materials science, fluid dynamics, climate systems, chemical processes, and complex networks. Recent neural operator models provide a promising data-driven alternative, but frequently struggle to achieve sufficient accuracy in the presence of strongly heterogeneous or oscillatory coefficients. In this work, we focus on the solution of elliptic PDEs with rough and high-contrast inputs. The Localized Orthogonal Decomposition (LOD) method is a well-established numerical approach for such problems, but it comes, however, at a substantial computational cost. We investigate the performance of popular neural operator architectures on these challenging multiscale problems and identify key limitations in their ability to resolve fine-scale structure. To overcome these challenges, we introduce LOD-MSNO (LOD-Multiscale Neural Operator), a hybrid approach that leverages the LOD method as a strong multiscale prior by building on its representation of the solution as a linear combination of problem-adapted basis functions, while addressing its main computational bottlenecks through data-driven operator learning. We further provide theoretical error estimates for the proposed coefficient-learning framework. Lastly, we demonstrate the potential of our proposed method to outperform current neural operator baselines in terms of accuracy for challenging multiscale inputs, while mainly retaining the computational efficiency of neural operator models.

Marc Haltmayer, Jaemin Seo, Yuseung Lee +3

Department of Artificial Intelligence, Chung-Ang University, 84 Heukseok-ro, Dongjak-gu, Seoul, 06974, Republic of Korea · Computational Science and Engineering Team, Samsung Electronics, 41 Samsung-ro, Giheung-gu, Yongin-si, 17113, Republic of Korea

PDE Solving PDE Surrogate Modeling PDE Operator Learning Neural Operators

Jul 13, 2026cs.LG

SPARC-Net: A Spectral, Causality-Aware, and Hard-Constrained Physics-Informed Architecture for Stiff and Shock-Dominated Partial Differential Equations

Physics-Informed Neural Networks (PINNs) provide a meshless approach for solving partial differential equations (PDEs), but suffer severe degradation in stiff and shock-dominated problems, where small PDE residuals can correspond to globally inaccurate solutions. We show these failures are multi-causal, arising from the concurrent interplay of (i) spectral bias against sharp features, (ii) imbalanced multi-term optimization and loss-weight collapse, (iii) violation of temporal causality, and (iv) under-resolved collocation. We present SPARC-Net, a unified architecture and training framework that jointly addresses all four pathologies. SPARC-Net leverages an adaptive multi-scale spectral encoder with a learnable spectral gate, a gated residual backbone, adaptive activations, and a hard-constraint output ansatz that exactly enforces initial and boundary conditions, structurally eliminating loss-weight collapse. Training employs stabilized gradient-norm loss balancing, floored causality-respecting residual weighting, and residual-based adaptive collocation (RAD). Validated against exact analytic and high-order spectral reference solutions across four canonical benchmarks -- viscous Burgers', Allen-Cahn, convection (beta=30), and reaction -- SPARC-Net yields substantial improvements over vanilla PINNs: relative L2 error drops from 1.47e-1 to 1.14e-1 on Burgers' (22% reduction), 9.93e-1 to 5.78e-2 on Allen-Cahn (94% reduction), and 9.82e-1 to 3.54e-3 on reaction (100% reduction). A characteristic-coordinate encoder for hyperbolic transport further reduces convection error from 5.14e-1 to 9.88e-5 (100% reduction). We report five-seed mean +/- standard deviation errors, Wilcoxon significance tests, full ablation studies, hyperparameter sensitivities, an extension to the 2D heat equation, and comparisons against parameter-matched baselines.

Divyavardhan Singh, Dimple Sonone, Hammad Mohammad +1

Dept. of Electronics Engg. SVNIT Surat, India

PDE Solving Spectral Bias Physics-Informed ML

Jul 7, 2026cs.LG

Physics-Informed Neural Embeddings of PDE Solution Families

We introduce a physics-informed framework for learning finite-dimensional embeddings of solution families of partial differential equations. The method uses a multihead Physics-Informed Neural Network in which a shared body learns a latent manifold representing the solution space, while linear heads reconstruct individual solutions associated with different initial conditions. A head-orthogonalization penalty removes degeneracies in the latent representation and stabilizes the principal-component spectrum across training realizations. Because the initial condition is built into the network output by construction, these principal components measure the additional variability the network learns on top of the initial profile, not the full solution itself. We apply the method to the one-dimensional viscous Burgers equation, with the heat and wave equations as robustness checks. For a latent dimension

n_b=20

, the learned manifolds exhibit pronounced effective dimensional reduction: for Burgers dynamics, only

2

-

4

principal components capture about

95\%

of the latent-space variance, while

4

-

7

capture about

99\%

, depending on the initial-condition family; the same qualitative compression holds for the heat and wave equations. We also split the wavenumber axis into bands (``Fourier shells'') and measure how much each band contributes to every principal component. The resulting frequency profile is invariant under the change-of-basis freedom that the orthogonalization penalty leaves in the latent space, and is therefore reproducible across independent training runs. More broadly, this establishes the learned spectral profiles and principal components as robust observables of solution-manifold geometry.

Raul Jimenez, Svitlana Mayboroda, Pavlos Protopapas +3

Institute of Cosmos Sciences (ICC), University of Barcelona, Martí i Franquès 1, ES-08028, Barcelona, Spain. · Department of Mathematics, ETH Zurich, Rämistrasse 101, 8092, Zürich, Switzerland. · Institute for Applied Computational Science, Harvard University, Cambridge, MA, USA. +1

Intrinsic Dimensionality PDE Solving Manifold Learning Physics-Informed ML

Jul 6, 2026gr-qc

Black Hole Black Boxes: Numerical Black Hole Metrics via AInstein Neural Networks

The AInstein architecture introduced an unsupervised neural method for solving the Riemannian Einstein equations on arbitrary manifolds. This Physics Informed Neural Network approach (PINN) is extended here to Lorentzian signature, validated by recovering the maximally extended Schwarzschild geometry, and tested as novel search method for arbitrary black hole solutions. The topology is built into the architecture by treating

S^{2}

globally through its standard embedding, such that the network learns an ambient metric on the manifold

\mathbb{R}^{2} \times \mathbb{R}^{3}

, where Penrose coordinates are chosen for

\mathbb{R}^2

and the metric on

S^{2}

is obtained by pullback. The architecture is first trained with the objective of recovering the Schwarzschild metric via losses encoding the vacuum Einstein equation, a quadratic Weyl scalar constraint, and the

SO(3)

symmetry of the resultant metric; directly motivated by the Birkhoff--Jebsen theorem. Following this, the objective is generalised to use the Petrov speciality index, a horizon curvature anchor, and a trapped-surface constraint, to allow search for algebraically general Petrov type I solutions, finding potentially novel general-type Lorentzian Einstein metrics with a genuinely trapped interior.

Tancredi Schettini Gherardini, Edward Hirst, Alexander George Stapleton

Instituto de Matemática, Estatística e Computação Científica (IMECC) da Universidade Estadual de Campinas (UNICAMP), 13083-859, Brasil · Mathematisches Institut (MI), University of Bonn, Bonn, Germany · Max Planck Institute for Mathematics, Bonn, Germany +1

PDE Solving Physics-Informed ML

Jul 6, 2026cs.LG

MeGA-MP: Metric Graph Advection Message Passing -- A Physics-Informed Message Passing Operator for Advection-Dominated Metric Graphs

Many real-world systems are organized as networks where spatio-temporal dynamics unfold along connections and not discretely between nodes. Examples include utility networks such as water distribution systems or gas networks, electrical grids, and traffic flow networks. Such systems are naturally modeled as metric graphs, where edges correspond to one-dimensional Euclidean subspaces connected at vertices. Metric graphs are independent of an underlying global Euclidean space, limiting direct application of typical PINNs and operator-learning methods. Especially transport dynamics like advection require a methodology able to capture antisymmetric and long-range dependencies on graphs, which is itself a challenge. We propose a novel physics-informed message passing operator that encodes linear advection on metric graphs as an inductive bias. In the purely advective setting, the operator provably recovers the exact dynamics up to a theoretically derived discretization error without any training. Combined with trainable components like MLPs, our message passing operator extends to realistic advection-reaction dynamics in water distribution systems, where we achieve superior performance compared to baselines and zero-shot generalization across different graph topologies.

Janine Strotherm, Luca Hermes, André Artelt +1

Bielefeld University

Graph Representation Learning PDE Solving Message Passing Neural Networks Physics-Informed GNNs Physics-Informed Neural Operators