PDE Solving

PDE: Partial Differential Equation

Ask about this topic

Overview

PDE solving seeks approximate solutions to equations describing processes such as heat flow, fluids, waves, and diffusion, where exact solutions are often unavailable. Classical approaches include finite difference, finite element, finite volume, spectral, and domain-decomposition methods; learning-based approaches include physics-informed neural networks, neural operators, and randomized or basis-based solvers. Current work also combines these methods, using learned models for acceleration, adaptive refinement, or surrogate prediction while retaining numerical constraints such as conservation and boundary conditions. A central challenge is balancing speed and flexibility with accuracy, stability, reliable generalization, and guarantees that residual errors reflect true solution errors.

Latest papers

CardsList

255 papers30 papers in the last four weeks, up 20% on the four weeks before. 0.2% of all new papers.

Mar 19, 2026

Deep Hilbert--Galerkin Methods for Infinite-Dimensional PDEs and Optimal Control

We develop deep learning-based approximation methods for fully nonlinear second-order PDEs on separable Hilbert spaces, such as HJB equations for infinite-dimensional control, by parameterizing solutions via Hilbert--Galerkin Neural Operators (HGNOs). We prove the first Universal Approximation Theorems (UATs) which are sufficiently powerful to address these problems, based on novel topologies for Hessian terms and corresponding novel continuity assumptions on the fully nonlinear operator. These topologies are non-sequential and non-metrizable, making the problem delicate. In particular, we prove UATs for functions on Hilbert spaces, together with their Fréchet derivatives up to second order, and for unbounded operators applied to the first derivative, ensuring that HGNOs are able to approximate all the PDE terms. For control problems, we further prove UATs for optimal feedback controls in terms of our approximating value function HGNO. We develop numerical training methods, which we call Deep Hilbert--Galerkin and Hilbert Actor-Critic (reinforcement learning) Methods, for these problems by minimizing the Lμ2(H)L^2_μ(H)-norm of the residual of the PDE on the whole Hilbert space, not just a projected PDE to finite dimensions. This is the first paper to propose such an approach. The models considered arise in many applied sciences, such as functional differential equations in physics and Kolmogorov and HJB PDEs related to controlled PDEs, SPDEs, path-dependent systems, partially observed stochastic systems, and mean-field SDEs. We numerically solve examples of Kolmogorov and HJB PDEs related to the optimal control of deterministic and stochastic heat and Burgers' equations, demonstrating the promise of our deep learning-based approach.

Samuel N. Cohen, Filippo de Feo, Jackson Hebner +1

Mathematical Institute, University of Oxford · Institut für Mathematik, Technische Universität Berlin

HJB EquationsPDE SolvingNeural PDE SolversPDE Operator LearningOptimal ControlNeural Network Approximation Theory+1
AskPDF
Mar 19, 2026

Rigorous Error Certification for Neural PDE Solvers: From Empirical Residuals to Solution Guarantees

Uncertainty quantification for partial differential equations is traditionally grounded in discretization theory, where solution error is controlled via mesh/grid refinement. Physics-informed neural networks fundamentally depart from this paradigm: they approximate solutions by minimizing residual losses at collocation points, introducing new sources of error arising from optimization, sampling, representation, and overfitting. As a result, the generalization error in the solution space remains an open problem. Our main theoretical contribution establishes generalization bounds that connect residual control to solution-space error. We prove that when neural approximations lie in a compact subset of the solution space, vanishing residual error guarantees convergence to the true solution. We derive deterministic and probabilistic convergence results and provide certified generalization bounds translating residual, boundary, and initial errors into explicit solution error guarantees.

Amartya Mukherjee, Maxwell Fitzsimmons, David C. Del Rey Fernández +1

Department of Applied Mathematics, University of Waterloo, Waterloo, ON, Canada

Neural Network GeneralizationNeural PDE SolversPDE SolvingPhysics-Informed MLGeneralization Bounds
AskPDF
Mar 19, 2026

Neural Galerkin Normalizing Flow for Transition Probability Density Functions of Diffusion Models

We propose a new Neural Galerkin Normalizing Flow framework to approximate the transition probability density function of a diffusion process by solving the corresponding Fokker-Planck equation with an atomic initial distribution, parametrically with respect to the location of the initial mass. By using Normalizing Flows, we look for the solution as a transformation of the transition probability density function of a reference stochastic process, ensuring that our approximation is structure-preserving and automatically satisfies positivity and mass conservation constraints. By extending Neural Galerkin schemes to the context of Normalizing Flows, we derive a system of ODEs for the time evolution of the Normalizing Flow's parameters. Adaptive sampling routines are used to evaluate the Fokker-Planck residual in meaningful locations, which is of vital importance to address high-dimensional PDEs. Numerical results show that this strategy captures key features of the true solution and enforces the causal relationship between the initial datum and the density function at subsequent times. After completing an offline training phase, online evaluation becomes significantly more cost-effective than solving the PDE from scratch. The proposed method serves as a promising surrogate model, which could be deployed in many-query problems associated with stochastic differential equations, like Bayesian inference, simulation, and diffusion bridge generation.

Riccardo Saporiti, Fabio Nobile

CSQI, École Polytechnique Féderale de Lausanne, Switzerland

Normalizing FlowsNeural PDE SolversPDE SolvingPDE Surrogate ModelingFokker–Planck Equation
AskPDF
Mar 19, 2026

Model Order Reduction of Cerebrovascular Hemodynamics Using POD_Galerkin and Reservoir Computing_based Approach

We investigate model order reduction (MOR) strategies for simulating unsteady hemodynamics within cerebrovascular systems, contrasting a physics-based intrusive approach with a data-driven non-intrusive framework. High-fidelity 3D Computational Fluid Dynamics (CFD) snapshots of an idealised basilar artery bifurcation are first compressed into a low-dimensional latent space using Proper Orthogonal Decomposition (POD). We evaluate the performance of a POD-Galerkin (POD-G) model, which projects the Navier-Stokes equations onto the reduced basis, against a POD-Reservoir Computing (POD-RC) model that learns the temporal evolution of coefficients through a recurrent architecture. A multi-harmonic and multi-amplitude training signal is introduced to improve training efficiency. Both methodologies achieve computational speed-ups on the order of 10^2 to 10^3 compared to full-order simulations, demonstrating their potential as efficient and accurate surrogates for predicting flow quantities such as wall shear stress.

Rahul Halder, Arash Hajisharifi, Kabir Bakhshaei +1

Mathematics Area, mathLab, SISSA, via Bonomea 265, I-34136 Trieste, Italy · Biorobotics Institute, SMART Lab, Sant’Anna School of Advanced Studies, Pisa, Italy · Department of Computer Science, University of Pisa, Pisa, Italy

Reduced-Order ModelingPDE Surrogate ModelingComputational Fluid DynamicsReservoir ComputingPDE Solving
AskPDF
Mar 17, 2026

Neural Pushforward Samplers for the Fokker-Planck Equation on Embedded Riemannian Manifolds

In this paper, we extend the Weak Adversarial Neural Pushforward Method to the Fokker--Planck equation on compact embedded Riemannian manifolds. The method represents the solution as a probability distribution via a neural pushforward map that is constrained to the manifold by a retraction layer, enforcing manifold membership and probability conservation by construction. Training is guided by a weak adversarial objective using ambient plane-wave test functions, whose intrinsic differential operators are derived in closed form from the geometry of the embedding, yielding a fully mesh-free and chart-free algorithm. Both steady-state and time-dependent formulations are developed, and numerical results on a double-well problem on the two-sphere demonstrate the capability of the method in capturing multimodal invariant distributions on curved spaces.

Andrew Qing He, Wei Cai

Department of Mathematics, Southern Methodist University, Dallas, TX 75275.

Fokker–Planck EquationPDE SolvingNeural PDE Solvers
AskPDF
Mar 16, 2026

OMNIFLOW: A Physics-Grounded Multimodal Agent for Generalized Scientific Reasoning

Large Language Models (LLMs) have demonstrated exceptional logical reasoning capabilities but frequently struggle with the continuous spatiotemporal dynamics governed by Partial Differential Equations (PDEs), often resulting in non-physical hallucinations. Existing approaches typically resort to costly, domain-specific fine-tuning, which severely limits cross-domain generalization and interpretability. To bridge this gap, we propose OMNIFLOW, a neuro-symbolic architecture designed to ground frozen multimodal LLMs in fundamental physical laws without requiring domain-specific parameter updates. OMNIFLOW introduces a novel Semantic-Symbolic Alignment mechanism that projects high-dimensional flow tensors into topological linguistic descriptors, enabling the model to perceive physical structures rather than raw pixel values. Furthermore, we construct a Physics-Guided Chain-of-Thought (PG-CoT) workflow that orchestrates reasoning through dynamic constraint injection (e.g., mass conservation) and iterative reflexive verification. We evaluate OMNIFLOW on a comprehensive benchmark spanning microscopic turbulence, theoretical Navier-Stokes equations, and macroscopic global weather forecasting. Empirical results demonstrate that OMNIFLOW significantly outperforms traditional deep learning baselines in zero-shot generalization and few-shot adaptation tasks. Crucially, it offers transparent, physically consistent reasoning reports, marking a paradigm shift from black-box fitting to interpretable scientific reasoning.

Hao Wu, Yongheng Zhang, Yuan Gao +7

Tsinghua University · Tencent · Shenzhen Loop Area Institute +2

Scientific Reasoning in Language ModelsPhysics-Informed MLPDE SolvingMultimodal Reasoning
AskPDF
Mar 16, 2026

Building Trust in PINNs: Error Estimation through Finite Difference Methods

Physics-informed neural networks (PINNs) constitute a flexible deep learning approach for solving partial differential equations (PDEs), which model phenomena ranging from heat conduction to quantum mechanical systems. Despite their flexibility, PINNs offer limited insight into how their predictions deviate from the true solution, hindering trust in their prediction quality. We propose a lightweight post-hoc method that addresses this gap by producing pointwise error estimates for PINN predictions, which offer a natural form of explanation for such models, identifying not just whether a prediction is wrong, but where and by how much. For linear partial differential equations, the error between a PINN approximation and the true solution satisfies the same differential operator as the original problem, but driven by the PINN's PDE residual as its source term. We solve this error equation numerically using finite difference methods requiring no knowledge of the true solution. Evaluated on several benchmark PDEs, our method yields accurate error maps at low computational cost, enabling targeted and interpretable validation of PINNs.

Aleksander Krasowski, René P. Klausen, Aycan Celik +3

Department of Artificial Intelligence, Fraunhofer Heinrich Hertz Institute · Centre of eXplainable Artificial Intelligence, Technological University Dublin · Department of Electrical Engineering and Computer Science, Technische Universität Berlin +1

Physics-Informed MLPDE SolvingNeural Network Interpretability
AskPDF
Mar 16, 2026

A scaled TW-PINN: A physics-informed neural network for traveling wave solutions of reaction-diffusion equations with general coefficients

We propose an efficient and generalizable physics-informed neural network (PINN) framework for computing traveling wave solutions of nn-dimensional reaction-diffusion equations with various reaction and diffusion coefficients. By applying a scaling transformation with the traveling wave form, the original problem is reduced to a one-dimensional scaled reaction-diffusion equation with unit reaction and diffusion coefficients. This reduction leads to the proposed framework, termed scaled TW-PINN, in which a single PINN solver trained on the scaled equation is reused for different coefficient choices and spatial dimensions. We also prove a universal approximation property of the proposed PINN solver for traveling wave solutions. Numerical experiments in one and two dimensions, together with a comparison to the existing wave-PINN method, demonstrate the accuracy, flexibility, and superior performance of scaled TW-PINN. Finally, we explore an extension of the framework to the Fisher's equation with general initial conditions.

Seungwan Han, Kwanghyuk Park, Jiaxi Gu +1

Reaction-Diffusion SystemsPDE SolvingPhysics-Informed MLDynamical Systems
AskPDF
Mar 16, 2026

Gauge-Equivariant Intrinsic Neural Operators for Geometry-Consistent Learning of Elliptic PDE Maps

Learning solution operators of partial differential equations (PDEs) from data has emerged as a promising route to fast surrogate models in multi-query scientific workflows. However, for geometric PDEs whose inputs and outputs transform under changes of local frame (gauge), many existing operator-learning architectures remain representation-dependent, brittle under metric perturbations, and sensitive to discretization changes. We propose Gauge-Equivariant Intrinsic Neural Operators (GINO), a class of neural operators that parameterize elliptic solution maps primarily through intrinsic spectral multipliers acting on geometry-dependent spectra, coupled with gauge-equivariant nonlinearities. This design decouples geometry from learnable functional dependence and enforces consistency under frame transformations. We validate GINO on controlled problems on the flat torus (T2\mathbb{T}^2), where ground-truth resolvent operators and regularized Helmholtz--Hodge decompositions admit closed-form Fourier representations, enabling theory-aligned diagnostics. Across experiments E1--E6, GINO achieves low operator-approximation error, near machine-precision gauge equivariance, robustness to structured metric perturbations, strong cross-resolution generalization with small commutation error under restriction/prolongation, and structure-preserving performance on a regularized exact/coexact decomposition task. Ablations further link the smoothness of the learned spectral multiplier to stability under geometric perturbations. These results suggest that enforcing intrinsic structure and gauge equivariance yields operator surrogates that are more geometry-consistent and discretization-robust for elliptic PDEs on form-valued fields.

Pengcheng Cheng

School of Mathematics, Jilin University, Changchun, 130012, China

PDE Operator LearningNeural OperatorsPDE SolvingEquivariant Neural Networks
AskPDF
Mar 13, 2026

Parameterized Temporal Extrapolation of PDEs via Disentangled Latent Dynamics Manifold Fusion

Generalizing neural surrogate models across different PDE parameters remains difficult because changes in PDE coefficients often make learning harder and optimization less stable. The problem becomes even more severe when the model must also predict beyond the training time range. Existing methods usually cannot handle parameter generalization and temporal extrapolation at the same time. Standard parameterized models treat time as just another input and therefore fail to capture intrinsic dynamics, while recent continuous-time latent methods often rely on expensive test-time auto-decoding for each instance, which is inefficient and can disrupt continuity across the parameterized solution space. To address this, we propose Disentangled Latent Dynamics Manifold Fusion (DLDMF), a physics-informed framework that explicitly separates space, time, and parameters. Instead of unstable auto-decoding, DLDMF maps PDE parameters directly to a continuous latent embedding through a feed-forward network. This embedding initializes and conditions a latent state whose evolution is governed by a parameter-conditioned Neural ODE. We further introduce a dynamic manifold fusion mechanism that uses a shared decoder to combine spatial coordinates, parameter embeddings, and time-evolving latent states to reconstruct the corresponding spatiotemporal solution. By modeling prediction as latent dynamic evolution rather than static coordinate fitting, DLDMF reduces interference between parameter variation and temporal evolution while preserving a smooth and coherent solution manifold. As a result, it performs well on unseen parameter settings and in long-term temporal extrapolation. Experiments on several benchmark problems show that DLDMF consistently outperforms state-of-the-art baselines in accuracy, parameter generalization, and extrapolation robustness.

Zhangyong Liang, Jihong Zhu, Huanhuan Gao

National Center for Applied Mathematics, Tianjin University, Tianjin, 300072, PR China · School of Mechanical Engineering, Northwestern Polytechnical University, Xi’an, 710072, China · School of Mechanical and Aerospace Engineering, Jilin University, Changchun, 130025, China

Latent Dynamics ModelingPDE SolvingPDE Surrogate ModelingPhysics-Informed ML
AskPDF
Mar 11, 2026

Neural Field Thermal Tomography: A Differentiable Physics Framework for Non-Destructive Evaluation

Inverse problems for stiff parabolic partial differential equations (PDEs), such as the inverse heat conduction problem (IHCP), are severely ill-posed: the forward map rapidly damps high-frequency interior structure before it reaches the boundary. Soft-constrained physics-informed neural networks (PINNs), which embed the PDE as a residual penalty, suffer from gradient pathology in this regime and tend to fit boundary measurements while leaving the interior field essentially untouched. We propose Neural Field Thermal Tomography (NeFTY), a hard-constrained neural field framework for label-free three-dimensional inverse heat conduction. NeFTY represents the unknown diffusivity as a continuous coordinate-based neural network, and at every optimization step passes the candidate field through a differentiable implicit-Euler heat solver with harmonic-mean interface flux, so that the governing PDE holds exactly on the discretization rather than as a soft penalty. Adjoint gradients propagate the surface reconstruction error back to the network weights at solver-level memory cost, making test-time inversion tractable on a single GPU. Across synthetic 3D benchmarks, NeFTY substantially outperforms soft-constrained PINN variants and a voxel-grid baseline on label-free volumetric recovery, and it transfers to real thermography data, surpassing classical signal-processing baselines in both defect segmentation and depth estimation. Additional details at https://cab-lab-princeton.github.io/nefty/

Tao Zhong, Yixun Hu, Dongzhe Zheng +2

Princeton University

PDE Inverse ProblemsThermal ImagingPDE SolvingComputed TomographyImplicit Neural Representations
AskPDF
Mar 9, 2026

MUSA-PINN: Multi-scale Weak-form Physics-Informed Neural Networks for Fluid Flow in Complex Geometries

While Physics-Informed Neural Networks (PINNs) offer a mesh-free approach to solving fluid-flow PDEs, standard point-wise residual minimization suffers from convergence pathologies in topologically complex domains like Triply Periodic Minimal Surfaces (TPMS). The locality bias of point-wise constraints fails to propagate global information through tortuous channels, causing unstable gradients and conservation violations. To address this, we propose the Multi-scale Weak-form PINN (MUSA-PINN), which reformulates Navier-Stokes equation constraints as integral conservation laws over hierarchical spherical control volumes. We enforce continuity and momentum conservation via flux-balance residuals on control surfaces. Our method utilizes a three-scale subdomain strategy-comprising large volumes for long-range coupling, skeleton-aware meso-scale volumes aligned with transport pathways, and small volumes for local refinement-alongside a two-stage training schedule prioritizing continuity. Experiments on steady incompressible flow in TPMS geometries show MUSA-PINN outperforms state-of-the-art baselines, reducing relative errors by up to 93% and preserving mass conservation.

Weizheng Zhang, Xunjie Xie, Hao Pan +4

Shandong University, Qingdao, China · Tsinghua University, Beijing, China · Institute of Engineering Thermophysics, Chinese Academy of Sciences, Beijing, China

Neural PDE SolversPDE SolvingPhysics-Informed MLComputational Fluid Dynamics
AskPDF
Mar 8, 2026

Learning embeddings of non-linear PDEs: the Burgers' equation

Embeddings provide low-dimensional representations that organize complex function spaces and support generalization. They provide a geometric representation that supports efficient retrieval, comparison, and generalization. In this work we generalize the concept to Physics Informed Neural Networks. We present a method to construct solution embedding spaces of nonlinear partial differential equations using a multi-head setup, and extract non-degenerate information from them using principal component analysis (PCA). We test this method by applying it to viscous Burgers' equation, which is solved simultaneously for a family of initial conditions and values of the viscosity. A shared network body learns a latent embedding of the solution space, while linear heads map this embedding to individual realizations. By enforcing orthogonality constraints on the heads, we obtain a principal-component decomposition of the latent space that is robust to training degeneracies and admits a direct physical interpretation. The obtained components for Burgers' equation exhibit rapid saturation, indicating that a small number of latent modes captures the dominant features of the dynamics.

Pedro Tarancón-Álvarez, Leonid Sarieddine, Pavlos Protopapas +1

Departament de F´ısica Qu´antica i Astrof´ısica, Universitat de Barcelona Institut de Ciencies del Cosmos (ICC), Universitat de Barcelona, Marti i Franques 1, ES-08028, Barcelona, Spain. · Institute for Applied Computational Science Harvard University Cambridge, MA, USA · Institut de Ciencies del Cosmos (ICC), Universitat de Barcelona, Marti i Franques 1, ES-08028, Barcelona, Spain. +1

Physics-Informed MLPDE SolvingOrthogonal Representation Learning
AskPDF
Mar 8, 2026

Partial Differential Equations in the Age of Machine Learning: A Critical Synthesis of Classical, Machine Learning, and Hybrid Methods

Partial differential equations (PDEs) govern physical phenomena across the full range of scientific scales, yet their computational solution remains one of the defining challenges of modern science. This critical review examines two mature but epistemologically distinct paradigms for PDE solution, classical numerical methods and machine learning approaches, through a unified evaluative framework organized around six fundamental computational challenges. Classical methods are assessed for their structure-preserving properties, rigorous convergence theory, and scalable solver design; their persistent limitations in high-dimensional and geometrically complex settings are characterized precisely. Machine learning approaches are introduced under a taxonomy organized by the degree to which physical knowledge is incorporated and subjected to the same critical evaluation applied to classical methods. Classical methods are deductive -- errors are bounded by quantities derivable from PDE structure and discretization parameters -- while machine learning methods are inductive -- accuracy depends on statistical proximity to the training distribution. This epistemological distinction is the primary criterion governing responsible method selection. We identify three genuine complementarities between the paradigms and develop principles for hybrid design, including a framework for the structure inheritance problem that addresses when classical guarantees propagate through hybrid couplings, and an error budget decomposition that separates discretization, neural approximation, and coupling contributions. We further assess emerging frontiers, including foundation models, differentiable programming, quantum algorithms, and exascale co-design, evaluating each against the structural constraints that determine whether current barriers are fundamental or contingent on engineering progress.

Mohammad Nooraiepour, Jakub Wiktor Both, Teeratorn Kadeethum +1

Faculty of Mathematics and Natural Sciences, University of Oslo, 1047 Blindern, 0316 Oslo, Norway · Department of Mathematics, University of Bergen, Bergen, Norway · AI Lab, Siemens Energy, Orlando, FL 32826, USA +1

PDE SolvingPartial Differential EquationsScientific MLPhysics-Informed ML
AskPDF
Mar 6, 2026

Kinetic-based regularization: Learning spatial derivatives and PDE applications

Accurate estimation of spatial derivatives from discrete and noisy data is central to scientific machine learning and numerical solutions of PDEs. We extend kinetic-based regularization (KBR), a localized multidimensional kernel regression method with a single trainable parameter, to learn spatial derivatives with provable second-order accuracy in 1D. Two derivative-learning schemes are proposed: an explicit scheme based on the closed-form prediction expressions, and an implicit scheme that solves a perturbed linear system at the points of interest. The fully localized formulation enables efficient, noise-adaptive derivative estimation without requiring global system solving or heuristic smoothing. Both approaches exhibit quadratic convergence, matching second-order finite difference for clean data, along with a possible high-dimensional formulation. Preliminary results show that coupling KBR with conservative solvers enables stable shock capture in 1D hyperbolic PDEs, acting as a step towards solving PDEs on irregular point clouds in higher dimensions while preserving conservation laws.

Abhisek Ganguly, Santosh Ansumali, Sauro Succi

Engineering Mechanics Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bengaluru-560064, Karnataka, India · Center for Life Nano-Neuroscience at La Sapienza, Fondazione Istituto Italiano di Tecnologia, Viale Regina Elena 291, 00161 Roma, Italy

Kernel RegressionPartial Differential EquationsPDE Solving
AskPDF
Mar 6, 2026

Learning Where the Physics Is: Probabilistic Adaptive Sampling for Stiff PDEs

Modeling stiff partial differential equations (PDEs) with sharp gradients remains a significant challenge for scientific machine learning. While Physics-Informed Neural Networks (PINNs) struggle with spectral bias and slow training times, Physics-Informed Extreme Learning Machines (PIELMs) offer a rapid, closed-form linear solution but are fundamentally limited by physics-agnostic, random initialization. We introduce the Gaussian Mixture Model Adaptive PIELM (GMM-PIELM), a probabilistic framework that learns a probability density function representing the ``location of physics'' for adaptively sampling kernels of PIELMs. By employing a weighted Expectation-Maximization (EM) algorithm, GMM-PIELM autonomously concentrates radial basis function centers in regions of high numerical error, such as shock fronts and boundary layers. This approach dynamically improves the conditioning of the hidden layer without the expensive gradient-based optimization(of PINNs) or Bayesian search. We evaluate our methodology on 1D singularly perturbed convection-diffusion equations with diffusion coefficients ν=10−4ν=10^{-4}. Our method achieves L2L_2 errors up to 77 orders of magnitude lower than baseline RBF-PIELMs, successfully resolving exponentially thin boundary layers while retaining the orders-of-magnitude speed advantage of the ELM architecture.

Akshay Govind Srinivasan, Balaji Srinivasan

Indian Institute of Technology, Madras Chennai, TN 600107, India

Neural PDE SolversPDE SolvingPhysics-Informed MLAdaptive Sampling
AskPDF
Mar 3, 2026

Physics-informed post-processing of stabilized finite element solutions for transient convection-dominated problems

The numerical simulation of convection-dominated transient transport phenomena poses significant computational challenges due to sharp gradients and propagating fronts across the spatiotemporal domain. Classical discretization methods often generate spurious oscillations, requiring advanced stabilization techniques. However, even stabilized finite element methods may require additional regularization to accurately resolve localized steep layers. On the other hand, standalone physics-informed neural networks (PINNs) struggle to capture sharp solution structures in convection-dominated regimes and typically require a large number of training epochs. This work presents a hybrid computational framework that extends the PINN-Augmented SUPG with Shock-Capturing (PASSC) methodology from steady to unsteady problems. The approach combines a semi-discrete stabilized finite element method with a PINN-based correction strategy for transient convection-diffusion-reaction equations. Stabilization is achieved using the Streamline-Upwind Petrov-Galerkin (SUPG) formulation augmented with a YZbeta shock-capturing operator. Rather than training over the entire space-time domain, the neural network is applied selectively near the terminal time, enhancing the finite element solution using the last K_s temporal snapshots while enforcing residual constraints from the governing equations and boundary conditions. The network incorporates residual blocks with random Fourier features and employs progressive training with adaptive loss weighting. Numerical experiments on five benchmark problems, including boundary and interior layers, traveling waves, and nonlinear Burgers dynamics, demonstrate significant accuracy improvements at the terminal time compared to standalone stabilized finite element solutions.

Süleyman Cengizci, Ömür Uğur, Srinivasan Natesan

Department of Computer Technologies, Antalya Bilim University, Antalya 07190, Turkey · Institute of Applied Mathematics, Middle East Technical University, Ankara 06800, Turkey · Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati 781039, India

PDE SolvingPhysics-Informed MLFinite Element Method
AskPDF
Mar 3, 2026

Stabilized Adaptive Loss and Residual-Based Collocation for Physics-Informed Neural Networks

Physics-Informed Neural Networks (PINNs) have been recognized as a mesh-free alternative to solve partial differential equations where physics information is incorporated. However, in dealing with problems characterized by high stiffness or shock-dominated dynamics, traditional PINNs have been found to have limitations, including unbalanced training and inaccuracy in solution, even with small physics residuals. In this research, we seek to address these limitations using the viscous Burgers' equation with low viscosity and the Allen-Cahn equation as test problems. In addressing unbalanced training, we have developed a new adaptive loss balancing scheme using smoothed gradient norms to ensure satisfaction of initial and boundary conditions. Further, to address inaccuracy in the solution, we have developed an adaptive residual-based collocation scheme to improve the accuracy of solutions in the regions with high physics residuals. The proposed new approach significantly improves solution accuracy with consistent satisfaction of physics residuals. For instance, in the case of Burgers' equation, the relative L2 error is reduced by about 44 percent compared to traditional PINNs, while for the Allen-Cahn equation, the relative L2 error is reduced by approximately 70 percent. Additionally, we show the trustworthy solution comparison of the proposed method using a robust finite difference solver.

Divyavardhan Singh, Shubham Kamble, Dimple Sonone +1

Sardar Vallabhbhai National Institute of Technology (SVNIT), Surat, India

Physics-Informed MLPDE SolvingAdaptive Loss Weighting
AskPDF
Mar 2, 2026

Solving Inverse PDE Problems using Minimization Methods and AI

Many physical and engineering systems require solving direct problems to predict behavior and inverse problems to determine unknown parameters from measurement. In this work, we study both aspects for systems governed by differential equations, contrasting well-established numerical methods with new AI-based techniques, specifically Physics-Informed Neural Networks (PINNs). We first analyze the logistic differential equation, using its closed-form solution to verify numerical schemes and validate PINN performance. We then address the Porous Medium Equation (PME), a nonlinear partial differential equation with no general closed-form solution, building strong solvers of the direct problem and testing techniques for parameter estimation in the inverse problem. Our results suggest that PINNs can closely estimate solutions at competitive computational cost, and thus propose an effective tool for solving both direct and inverse problems for complex systems.

Noura Al Helwani, Sophie Moufawad, Georges Sakr

Department of Physics, American University of Beirut, Lebanon · Department of Mathematics, American University of Beirut, Lebanon · The authors would like to acknowledge

PDE SolvingPDE Inverse ProblemsPhysics-Informed ML
AskPDF
Mar 2, 2026

Randomized Neural Networks for Partial Differential Equation on Static and Evolving Surfaces

Surface partial differential equations arise in numerous scientific and engineering applications. Their numerical solution on static and evolving surfaces remains challenging due to geometric complexity and, for evolving geometries, the need for repeated mesh updates and geometry or solution transfer. While neural-network-based methods offer mesh-free discretizations, approaches based on nonconvex training can be costly and may fail to deliver high accuracy in practice. In this work, we develop a randomized neural network (RaNN) method for solving PDEs on both static and evolving surfaces: the hidden-layer parameters are randomly generated and kept fixed, and the output-layer coefficients are determined efficiently by solving a least-squares problem. For static surfaces, we present formulations for parametrized surfaces, implicit level-set surfaces, and point-cloud geometries, and provide a corresponding theoretical analysis for the parametrization-based formulation with interface compatibility. For evolving surfaces with topology preserved over time, we introduce a RaNN-based strategy that learns the surface evolution through a flow-map representation and then solves the surface PDE on a space--time collocation set, avoiding remeshing. Extensive numerical experiments demonstrate broad applicability and favorable accuracy--efficiency performance on representative benchmarks.

Jingbo Sun, Fei Wang

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

Neural PDE SolversPDE Solving
AskPDF
Mar 1, 2026

Adaptive-Growth Randomized Neural Networks for Level-Set Computation of Multivalued Nonlinear First-Order PDEs with Hyperbolic Characteristics

This paper proposes an Adaptive-Growth Randomized Neural Network (AG-RaNN) method for computing multivalued solutions of nonlinear first-order PDEs with hyperbolic characteristics, including quasilinear hyperbolic balance laws and Hamilton--Jacobi equations. Such solutions arise in geometric optics, seismic waves, semiclassical limit of quantum dynamics and high frequency limit of linear waves, and differ markedly from the viscosity or entropic solutions. The main computational challenges lie in that the solutions are no longer functions, and become union of multiple branches, after the formation of singularities. Level-set formulations offer a systematic alternative by embedding the nonlinear dynamics into linear transport equations posed in an augmented phase space, at the price of substantially increased dimensionality. To alleviate this computational burden, we combine AG-RaNN with an adaptive collocation strategy that concentrates samples in a tubular neighborhood of the zero level set, together with a layer-growth mechanism that progressively enriches the randomized feature space. Under standard regularity assumptions on the transport field and the characteristic flow, we establish a convergence result for the AG-RaNN approximation of the level-set equations. Numerical experiments demonstrate that the proposed method can efficiently recover multivalued structures and resolve nonsmooth features in high-dimensional settings.

Haoning Dang, Shi Jin, Fei Wang

School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China. · School of Mathematical Sciences & Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai 200240, China. · School of Mathematics and Statistics & State Key Laboratory of Multiphase Flow in Power Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China.

PDE SolvingNeural PDE Solvers
AskPDF
Feb 28, 2026

TENG-BC: Unified Time-Evolving Natural Gradient for Neural PDE Solvers with General Boundary Conditions

Accurately solving time-dependent partial differential equations (PDEs) with neural networks remains challenging due to long-time error accumulation and the difficulty of enforcing general boundary conditions. We introduce TENG-BC, a high-precision neural PDE solver based on the Time-Evolving Natural Gradient, designed to perform under general boundary constraints. At each time step, TENG-BC performs a boundary-aware optimization that jointly enforces interior dynamics and boundary conditions, accommodating Dirichlet, Neumann, Robin, and mixed types within a unified framework. This formulation admits a natural-gradient interpretation, enabling stable time evolution without delicate penalty tuning. Across benchmarks over diffusion, transport, and nonlinear PDEs with various boundary conditions, TENG-BC achieves solver-level accuracy under comparable sampling budgets, outperforming conventional solvers and physics-informed neural network (PINN) baselines.

Hongjie Jiang, Di Luo

School of Mathematical Sciences, Peking University, Beijing, China · Department of Physics, Tsinghua University, Beijing, China · Institute of Advanced Study, Tsinghua University, Beijing, China

Neural PDE SolversPDE SolvingNatural Gradient Methods
AskPDF
Feb 27, 2026

A Boundary Integral-based Neural Operator for Mesh Deformation

This paper presents an efficient mesh deformation method based on boundary integration and neural operators, formulating the problem as a linear elasticity boundary value problem (BVP). To overcome the high computational cost of traditional finite element methods and the limitations of existing neural operators in handling Dirichlet boundary conditions for vector fields, we introduce a direct boundary integral representation using a Dirichlet-type Green's tensor. This formulation expresses the internal displacement field solely as a function of boundary displacements, eliminating the need to solve for unknown tractions. Building on this, we design a Boundary-Integral-based Neural Operator (BINO) that learns the geometry- and material-aware Green's traction kernel. A key technical advantage of our framework is the mathematical decoupling of the physical integration process from the geometric representation via geometric descriptors. While this study primarily demonstrates robust generalization across diverse boundary conditions, the architecture inherently possesses potential for cross-geometry adaptation. Numerical experiments, including large deformations of flexible beams and rigid-body motions of NACA airfoils, confirm the model's high accuracy and strict adherence to the principles of linearity and superposition. The results demonstrate that the proposed framework ensures mesh quality and computational efficiency, providing a reliable new paradigm for parametric mesh generation and shape optimization in engineering.

Zhengyu Wu, Jun Liu, Wei Wang

School of Electronics and Information Engineering, Hangzhou Dianzi University, Hangzhou, China

Neural OperatorsPDE Operator LearningPDE SolvingDeformable Object Simulation
AskPDF
Feb 22, 2026

Regularity of Second-Order Elliptic PDEs in Spectral Barron Spaces

We establish a regularity theorem for second-order elliptic PDEs on Rd\mathbb{R}^{d} in spectral Barron spaces. Under mild ellipticity and smallness assumptions, the solution gains two additional orders of Barron regularity. As a corollary, we identify a class of PDEs whose solutions can be approximated by two-layer neural networks with cosine activation functions, where the width of the neural network is independent of the spatial dimension.

Ziang Chen, Liqiang Huang, Mengxuan Yang +1

PDE Solving
AskPDF
Feb 19, 2026

Weak-Form Evolutionary Kolmogorov-Arnold Networks for Solving Partial Differential Equations

Partial differential equations (PDEs) form a central component of scientific computing. Among recent advances in deep learning, evolutionary neural networks have been developed to successively capture the temporal dynamics of time-dependent PDEs via parameter evolution. The parameter updates are obtained by solving a linear system derived from the governing equation residuals at each time step. However, strong-form evolutionary approaches can yield ill-conditioned linear systems due to pointwise residual discretization, and their computational cost scales unfavorably with the number of training samples. To address these limitations, we propose a weak-form evolutionary Kolmogorov-Arnold Network (KAN) for the scalable and accurate prediction of PDE solutions. We decouple the linear system size from the number of training samples through the weak formulation, leading to improved scalability compared to strong-form approaches. We also rigorously enforce boundary conditions by constructing the trial space with boundary-constrained KANs to satisfy Dirichlet and periodic conditions, and by incorporating derivative boundary conditions directly into the weak formulation for Neumann conditions. In conclusion, the proposed weak-form evolutionary KAN framework provides a stable and scalable approach for PDEs and contributes to scientific machine learning with potential relevance to future engineering applications.

Bongseok Kim, Jiahao Zhang, Guang Lin

Neural PDE SolversPDE SolvingKolmogorov-Arnold NetworksNeuroevolution
AskPDF
Feb 19, 2026

AutoNumerics: An Autonomous, PDE-Agnostic Multi-Agent Pipeline for Scientific Computing

PDEs are central to scientific and engineering modeling, yet designing accurate numerical solvers typically requires substantial mathematical expertise and manual tuning. Recent neural network-based approaches improve flexibility but often demand high computational cost and suffer from limited interpretability. We introduce AutoNumerics, a multi-agent framework that autonomously designs, implements, debugs, and verifies numerical solvers for general PDEs directly from natural language descriptions. Unlike black-box neural solvers, our framework generates transparent solvers grounded in classical numerical analysis. We introduce a coarse-to-fine execution strategy and a residual-based self-verification mechanism. Experiments on 24 canonical and real-world PDE problems demonstrate that AutoNumerics achieves competitive or superior accuracy compared to existing neural and LLM-based baselines, and correctly selects numerical schemes based on PDE structural properties, suggesting its viability as an accessible paradigm for automated PDE solving.

Jianda Du, Youran Sun, Haizhao Yang

Department of Mathematics, University of Maryland, College Park, MD, USA · Department of Computer Science, University of Maryland, College Park, MD, USA

PDE SolvingAutomated Algorithm DiscoveryAI Agents for Scientific DiscoveryLLM-Based Program Synthesis
AskPDF
Feb 16, 2026

Parameter-Minimal Neural DE Solvers via Horner Polynomials

We propose a parameter-minimal neural architecture for solving differential equations by restricting the hypothesis class to Horner-factorized polynomials, yielding an implicit, differentiable trial solution with only a small set of learnable coefficients. Initial conditions are enforced exactly by construction by fixing the low-order polynomial degrees of freedom, so training focuses solely on matching the differential-equation residual at collocation points. To reduce approximation error without abandoning the low-parameter regime, we introduce a piecewise ("spline-like") extension that trains multiple small Horner models on subintervals while enforcing continuity (and first-derivative continuity) at segment boundaries. On illustrative ODE benchmarks and a heat-equation example, Horner networks with tens (or fewer) parameters accurately match the solution and its derivatives and outperform small MLP and sinusoidal-representation baselines under the same training settings, demonstrating a practical accuracy-parameter trade-off for resource-efficient scientific modeling.

T. Matulić, D. Seršić

Dept. of Electronic Systems and Information Processing, Faculty of Electrical Engineering and Computing, University of Zagreb, Zagreb, Croatia

Neural PDE SolversPolynomial Neural NetworksOrdinary Differential EquationsPDE SolvingPhysics-Informed ML
AskPDF
Feb 16, 2026

Pseudo-differential-enhanced physics-informed neural networks

We present pseudo-differential enhanced physics-informed neural networks (PINNs), an extension of gradient enhancement but in Fourier space. Gradient enhancement of PINNs dictates that the PDE residual is taken to a higher differential order than prescribed by the PDE, added to the objective as an augmented term in order to improve training and overall learning fidelity. We propose the same procedure after application via Fourier transforms, since differentiating in Fourier space is multiplication with the Fourier wavenumber under suitable decay. Our methods are fast and efficient. Our methods oftentimes achieve superior PINN versus numerical error in fewer training iterations, potentially pair well with few/fixed samples in collocation, and can achieve breakthrough rather than gradual effects on the error. Moreover, our methods are suitable for fractional derivatives. We establish that our methods, due to the dynamical effects, improve spectral eigenvalue decay of the neural tangent kernel (NTK), and so our methods contribute towards the learning of high frequencies in early training, mitigating the effects of frequency bias up to the polynomial order and possibly greater with smooth activations. In particular, our primary contribution can be characterized as gradient enhancement affects spectral bias, and we specialize to Fourier space, although our theory does not prove the ambient case exactly due to a result with the Plancherel theorem on the descent paths. Our methods accommodate advanced techniques in PINNs, such as Fourier feature embeddings. A pitfall of discrete Fourier transforms via the Fast Fourier Transform (FFT) is mesh subjugation, and so we demonstrate compatibility of our methods for greater mesh flexibility and invariance on alternative Euclidean and non-Euclidean domains via Monte Carlo methods and otherwise, although possessing trade-offs of their own.

Andrew Gracyk

Department of Mathematics, Purdue University, West Lafayette, IN 47907, United States

PDE SolvingSpectral BiasPhysics-Informed ML
AskPDF
Feb 14, 2026

A Unified Physics-Informed Neural Network for Modeling Coupled Electro- and Elastodynamic Wave Propagation Using Three-Stage Loss Optimization

Physics-Informed Neural Networks present a novel approach in SciML that integrates physical laws in the form of partial differential equations directly into the NN through soft constraints in the loss function. This work studies the application of PINNs to solve a one dimensional coupled electro-elastodynamic system modeling linear piezoelectricity in stress-charge form, governed by elastodynamic and electrodynamic equations. Our simulation employs a feedforward architecture, mapping space-time coordinates to mechanical displacement and electric potential. Our PINN model achieved global relative L2 errors of 2.34 and 4.87 percent for displacement and electric potential respectively. The results validate PINNs as effective mesh free solvers for coupled time-dependent PDE systems, though challenges remain regarding error accumulation and stiffness in coupled eigenvalue systems.

Suhas Suresh Bharadwaj, Reuben Thomas Thovelil

Department of Electrical and Electronics Engineering, BITS Pilani, Dubai Campus, Dubai, United Arab Emirates

Neural PDE SolversPDE SolvingPhysics-Informed MLMultiphysics Simulation
AskPDF
Feb 5, 2026

Smoothness Errors in Dynamics Models and How to Avoid Them

Modern neural networks have shown promise for solving partial differential equations over surfaces, often by discretizing the surface as a mesh and learning with a mesh-aware graph neural network. However, graph neural networks suffer from oversmoothing, where a node's features become increasingly similar to those of its neighbors. Unitary graph convolutions, which are mathematically constrained to preserve smoothness, have been proposed to address this issue. Despite this, in many physical systems, such as diffusion processes, smoothness naturally increases and unitarity may be overconstraining. In this paper, we systematically study the smoothing effects of different GNNs for dynamics modeling and prove that unitary convolutions hurt performance for such tasks. We propose relaxed unitary convolutions that balance smoothness preservation with the natural smoothing required for physical systems. We also generalize unitary and relaxed unitary convolutions from graphs to meshes. In experiments on PDEs such as the heat and wave equations over complex meshes and on weather forecasting, we find that our method outperforms several strong baselines, including mesh-aware transformers and equivariant neural networks.

Edward Berman, Luisa Li, Jung Yeon Park +1

Geometric Learning Lab, Northeastern University.

Graph Neural NetworksOver-Smoothing in GNNsPDE Solving
AskPDF