PDE Solving

PDE: Partial Differential Equation

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

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

Deep Hilbert--Galerkin Methods for Infinite-Dimensional PDEs and Optimal Control
Mar 19, 2026Samuel N. Cohen, Filippo de Feo, Jackson Hebner +1HJB Equations PDE Solving
Mathematical Institute, University of Oxford · Institut für Mathematik, Technische Universität Berlin
Rigorous Error Certification for Neural PDE Solvers: From Empirical Residuals to Solution Guarantees
Mar 19, 2026Amartya Mukherjee, Maxwell Fitzsimmons, David C. Del Rey Fernández +1Neural Network Generalization Neural PDE Solvers
Department of Applied Mathematics, University of Waterloo, Waterloo, ON, Canada
Neural Galerkin Normalizing Flow for Transition Probability Density Functions of Diffusion Models
Mar 19, 2026Riccardo Saporiti, Fabio NobileNormalizing Flows Neural PDE Solvers
CSQI, École Polytechnique Féderale de Lausanne, Switzerland
Model Order Reduction of Cerebrovascular Hemodynamics Using POD_Galerkin and Reservoir Computing_based Approach
Mar 19, 2026Rahul Halder, Arash Hajisharifi, Kabir Bakhshaei +1Reduced-Order Modeling PDE Surrogate Modeling
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
Neural Pushforward Samplers for the Fokker-Planck Equation on Embedded Riemannian Manifolds
Mar 17, 2026Andrew Qing He, Wei CaiFokker–Planck Equation PDE Solving
Department of Mathematics, Southern Methodist University, Dallas, TX 75275.
OMNIFLOW: A Physics-Grounded Multimodal Agent for Generalized Scientific Reasoning
Mar 16, 2026Hao Wu, Yongheng Zhang, Yuan Gao +7Scientific Reasoning in Language Models Physics-Informed ML
Tsinghua University · Tencent · Shenzhen Loop Area Institute +2
Building Trust in PINNs: Error Estimation through Finite Difference Methods
Mar 16, 2026Aleksander Krasowski, René P. Klausen, Aycan Celik +3Physics-Informed ML PDE Solving
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
A scaled TW-PINN: A physics-informed neural network for traveling wave solutions of reaction-diffusion equations with general coefficients
Mar 16, 2026Seungwan Han, Kwanghyuk Park, Jiaxi Gu +1Reaction-Diffusion Systems PDE Solving
Gauge-Equivariant Intrinsic Neural Operators for Geometry-Consistent Learning of Elliptic PDE Maps
Mar 16, 2026Pengcheng ChengPDE Operator Learning Neural Operators
School of Mathematics, Jilin University, Changchun, 130012, China
Parameterized Temporal Extrapolation of PDEs via Disentangled Latent Dynamics Manifold Fusion
Mar 13, 2026Zhangyong Liang, Jihong Zhu, Huanhuan GaoLatent Dynamics Modeling PDE Solving
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
Neural Field Thermal Tomography: A Differentiable Physics Framework for Non-Destructive Evaluation
Mar 11, 2026Tao Zhong, Yixun Hu, Dongzhe Zheng +2PDE Inverse Problems Thermal Imaging
Princeton University
MUSA-PINN: Multi-scale Weak-form Physics-Informed Neural Networks for Fluid Flow in Complex Geometries
Mar 9, 2026Weizheng Zhang, Xunjie Xie, Hao Pan +4Neural PDE Solvers PDE Solving
Shandong University, Qingdao, China · Tsinghua University, Beijing, China · Institute of Engineering Thermophysics, Chinese Academy of Sciences, Beijing, China
Learning embeddings of non-linear PDEs: the Burgers' equation
Mar 8, 2026Pedro Tarancón-Álvarez, Leonid Sarieddine, Pavlos Protopapas +1Physics-Informed ML PDE Solving
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
Partial Differential Equations in the Age of Machine Learning: A Critical Synthesis of Classical, Machine Learning, and Hybrid Methods
Mar 8, 2026Mohammad Nooraiepour, Jakub Wiktor Both, Teeratorn Kadeethum +1PDE Solving Partial Differential Equations
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
Kinetic-based regularization: Learning spatial derivatives and PDE applications
Mar 6, 2026Abhisek Ganguly, Santosh Ansumali, Sauro SucciKernel Regression Partial Differential Equations
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
Learning Where the Physics Is: Probabilistic Adaptive Sampling for Stiff PDEs
Mar 6, 2026Akshay Govind Srinivasan, Balaji SrinivasanNeural PDE Solvers PDE Solving
Indian Institute of Technology, Madras Chennai, TN 600107, India
Physics-informed post-processing of stabilized finite element solutions for transient convection-dominated problems
Mar 3, 2026Süleyman Cengizci, Ömür Uğur, Srinivasan NatesanPDE Solving Physics-Informed ML
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
Stabilized Adaptive Loss and Residual-Based Collocation for Physics-Informed Neural Networks
Mar 3, 2026Divyavardhan Singh, Shubham Kamble, Dimple Sonone +1Physics-Informed ML PDE Solving
Sardar Vallabhbhai National Institute of Technology (SVNIT), Surat, India
Solving Inverse PDE Problems using Minimization Methods and AI
Mar 2, 2026Noura Al Helwani, Sophie Moufawad, Georges SakrPDE Solving PDE Inverse Problems
Department of Physics, American University of Beirut, Lebanon · Department of Mathematics, American University of Beirut, Lebanon · The authors would like to acknowledge
Randomized Neural Networks for Partial Differential Equation on Static and Evolving Surfaces
Mar 2, 2026Jingbo Sun, Fei WangNeural PDE Solvers PDE Solving
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.
Adaptive-Growth Randomized Neural Networks for Level-Set Computation of Multivalued Nonlinear First-Order PDEs with Hyperbolic Characteristics
Mar 1, 2026Haoning Dang, Shi Jin, Fei WangPDE Solving Neural PDE Solvers
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.
TENG-BC: Unified Time-Evolving Natural Gradient for Neural PDE Solvers with General Boundary Conditions
Feb 28, 2026Hongjie Jiang, Di LuoNeural PDE Solvers PDE Solving
School of Mathematical Sciences, Peking University, Beijing, China · Department of Physics, Tsinghua University, Beijing, China · Institute of Advanced Study, Tsinghua University, Beijing, China
A Boundary Integral-based Neural Operator for Mesh Deformation
Feb 27, 2026Zhengyu Wu, Jun Liu, Wei WangNeural Operators PDE Operator Learning
School of Electronics and Information Engineering, Hangzhou Dianzi University, Hangzhou, China
Regularity of Second-Order Elliptic PDEs in Spectral Barron Spaces
Feb 22, 2026Ziang Chen, Liqiang Huang, Mengxuan Yang +1PDE Solving
Weak-Form Evolutionary Kolmogorov-Arnold Networks for Solving Partial Differential Equations
Feb 19, 2026Bongseok Kim, Jiahao Zhang, Guang LinNeural PDE Solvers PDE Solving
AutoNumerics: An Autonomous, PDE-Agnostic Multi-Agent Pipeline for Scientific Computing
Feb 19, 2026Jianda Du, Youran Sun, Haizhao YangPDE Solving Automated Algorithm Discovery
Department of Mathematics, University of Maryland, College Park, MD, USA · Department of Computer Science, University of Maryland, College Park, MD, USA
Parameter-Minimal Neural DE Solvers via Horner Polynomials
Feb 16, 2026T. Matulić, D. SeršićNeural PDE Solvers Polynomial Neural Networks
Dept. of Electronic Systems and Information Processing, Faculty of Electrical Engineering and Computing, University of Zagreb, Zagreb, Croatia
Pseudo-differential-enhanced physics-informed neural networks
Feb 16, 2026Andrew GracykPDE Solving Spectral Bias
Department of Mathematics, Purdue University, West Lafayette, IN 47907, United States
A Unified Physics-Informed Neural Network for Modeling Coupled Electro- and Elastodynamic Wave Propagation Using Three-Stage Loss Optimization
Feb 14, 2026Suhas Suresh Bharadwaj, Reuben Thomas ThovelilNeural PDE Solvers PDE Solving
Department of Electrical and Electronics Engineering, BITS Pilani, Dubai Campus, Dubai, United Arab Emirates
Smoothness Errors in Dynamics Models and How to Avoid Them
Feb 5, 2026Edward Berman, Luisa Li, Jung Yeon Park +1Graph Neural Networks Over-Smoothing in GNNs
Geometric Learning Lab, Northeastern University.

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

Samuel N. Cohen, Filippo de Feo, Jackson Hebner, Justin Sirignano

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

HJB Equations PDE Solving Neural PDE Solvers PDE Operator Learning Optimal Control Neural Network Approximation Theory+1

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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

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.