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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Generalization Error Bounds for Picard-Type Operator Learning in Nonlinear Parabolic PDEs
May 11, 2026Koichi Taniguchi, Sho SonodaPDE Solving PDE Operator Learning
Department of Mathematical and Systems Engineering, Faculty of Engineering Shizuoka University 3-5-1 Johoku, Chuo-ku, Hamamatsu, 432-8561, Japan · RIKEN AIP / CyberAgent 1-4-1 Nihonbashi, Chuo-ku, Tokyo 103-0027, Japan
PDEAgent-Bench: A Multi-Metric, Multi-Library Benchmark for PDE Solver Generation
May 10, 2026Zhen Hang, Yushan Yashengjiang, Junhui Li +21PDE Solving Benchmark Design
University of Science and Technology of China · Beijing University of Posts and Telecommunications · Tencent +10
HS-FNO: History-Space Fourier Neural Operator for Non-Markovian Partial Differential Equations
May 10, 2026Lennon J. ShikhmanFourier Neural Operator PDE Solving
College of Computing, Georgia Institute of Technology Atlanta, GA, United States
MC $^2$ : Monte Carlo Correction for Fast Elliptic PDE Solving
May 10, 2026Ethan Hsu, Hong Meng Yam, Ivan GeNeural PDE Solvers PDE Solving
Stanford University
Finite Volume-Informed Neural Network Framework for 2D Shallow Water Equations: Rugged Loss Landscapes and the Importance of Data Guidance
May 9, 2026Xiaofeng LiuPDE Solving PDE Surrogate Modeling
Department of Civil and Environmental Engineering, Pennsylvania State University, University Park, PA, USA · Institute for Computational and Data Sciences, Pennsylvania State University, University Park, PA, USA
Physics-Informed Neural PDE Solvers via Spatio-Temporal MeanFlow
May 9, 2026Hanru Bai, Yuncheng Zhou, Difan ZouFlow Matching Neural PDE Solvers
1Fudan University · 2The University of Hong Kong.
When Attention Beats Fourier: Multi-Scale Transformers for PDE Solving on Irregular Domains
May 8, 2026Brandon Yee, Pairie Koh, Jack Rodriguez +1Neural PDE Solvers PDE Solving
Physics Lab, Yee Collins Research Group
NSPOD: Accelerating Krylov solvers via DeepONet-learned POD subspaces
May 8, 2026Francesc Levrero-Florencio, Youngkyu Lee, Jay Pathak +1PDE Solving Structural Mechanics
Synopsys UK Ltd, UK · Division of Applied Mathematics, Brown University, Rhode Island, USA · Synopsys Inc, California, USA
INEUS: Iterative Neural Solver for High-Dimensional PIDEs
May 7, 2026Jean-Loup Dupret, Davide Gallon, Patrick CheriditoNeural PDE Solvers PDE Solving
Department of Mathematics and RiskLab, ETH Zurich, Switzerland · Applied Mathematics: Institute for Analysis and Numerics, University of Münster, Germany
A Robust Foundation Model for Conservation Laws: Injecting Context into Flux Neural Operators via Recurrent Vision Transformers
May 6, 2026Taeyoung Kim, Joon-Hyuk KoPDE Solving Neural Operators
Center for AI and Natural Sciences Korea Institute for Advanced Study Seoul, South Korea 02455
Hybrid Iterative Neural Low-Regularity Integrator for Nonlinear Dispersive Equations
May 6, 2026Zhangyong Liang, Huanhuan GaoNeural PDE Solvers PDE Solving
National Center for Applied Mathematics Tianjin University Tianjin, 300072, China · School of Mechanical and Aerospace Engineering, Jilin University Jilin University Changchun, 130025, China
From Video-to-PDE: Data-Driven Discovery of Nonlinear Dye Plume Dynamics
May 6, 2026Cesar Acosta-Minoli, Sayantan SarkarPDE Solving Sparse Identification of Nonlinear Dynamics
aGEDES, Universidad del Quindio, Cra 15 Cl 12-00, Armenia, 630004, Quindio, Colombia · bDepartment of Mathematics, State University of New York at Buffalo, Mathematics Building, Buffalo, 14260, New York, USA
Simultaneous CNN Approximation on Manifolds with Applications to Boundary Value Problems
May 5, 2026Hanfei Zhou, Lei ShiPDE Solving Neural Network Approximation Theory
School of Mathematical Sciences, Fudan University, Shanghai, 200433, China · Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University, Shanghai, 200433, China
Random test functions, $H^{-1}$ norm equivalence, and stochastic variational physics-informed neural networks
May 5, 2026Diego MarcondesNeural PDE Solvers PDE Solving
Quantitative Sobolev Approximation Bounds for Neural Operators with Empirical Validation on Burgers Equation
May 4, 2026Nicole HaoFourier Neural Operator PDE Solving
Department of Mathematics, Cornell University
Chebyshev-Augmented One-Shot Transfer Learning for PINNs on Nonlinear Differential Equations
May 2, 2026Yiqi Rao, Pavlos ProtopapasPDE Solving Transfer Learning
Harvard University
Learning the Helmholtz equation operator with DeepONet for non-parametric 2D geometries
May 1, 2026Rodolphe Barlogis, Ferhat Tamssaouet, Quentin Falcoz +1PDE Solving PDE Operator Learning
PROMES-CNRS, Université Perpignan Via Domitia (UPVD), Perpignan, France · LAAS-CNRS, Université de Toulouse (UT), Toulouse, France
A Dirac-Frenkel-Onsager principle: Instantaneous residual minimization with gauge momentum for nonlinear parametrizations of PDE solutions
Apr 30, 2026Matteo Raviola, Benjamin PeherstorferNeural PDE Solvers PDE Solving
EPFL, Lausanne, Switzerland · Courant Institute of Mathematical Sciences, New York University, New York, NY, USA
An adaptive wavelet-based PINN for problems with localized high-magnitude source
Apr 30, 2026Himanshu Pandey, Ratikanta BeheraPDE Solving Wavelet Methods
Department of Computational and Data Sciences, Indian Institute of Science, Bangalore, 560012, India
Shearlet Neural Operators for Anisotropic-Shock-Dominated and Multi-scale parametric partial differential equations
Apr 28, 2026Fabio Pereira dos Santos, Julio de Castro Vargas Fernandes, Adriano Mauricio de Almeida CortesPDE Solving PDE Operator Learning
Coordination of Mathematical and Computational Methods National Laboratory of Scientific Computing Av. Getúlio Vargas, 333 - Quitandinha, Petrópolis - RJ, 25651-075 · Systems and Computational Engineering Program, COPPE/UFRJ, Federal University of Rio de Janeiro, P.O. Box 68542, Rio de Janeiro, RJ, 21941-909, Brasil
Quantum Dynamics via Score Matching on Bohmian Trajectories
Apr 28, 2026Lei WangScore Matching PDE Solving
Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
A Coupled Fourth Order Telegraph Diffusion Framework Using Grayscale Indicators for Image Despeckling
Apr 26, 2026Manish Kumar, Rajendra K. RayPDE Solving Image Denoising
School of Mathematical and Statistical Sciences, Indian Institute of Technology Mandi, Mandi (H.P), 175075, India
Single Image Defogging Using a Fourth-Order Telegraph PDE Guided by Physical Haze Modeling
Apr 26, 2026Manish Kumar, Rajendra K. RayPDE Solving Physics-Guided Image Restoration
School of Mathematical and Statistical Sciences, Indian Institute of Technology Mandi, Mandi (H.P), 175075, India
When PINNs Go Wrong: Pseudo-Time Stepping Against Spurious Solutions
Apr 26, 2026Sifan Wang, Shawn Koohy, Yiping Lu +1PDE Solving Physics-Informed ML
Institute for Foundations of Data Science, Yale University · Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania · Industrial Engineering & Management Science, Northwestern University
Transferable Physics-Informed Representations via Closed-Form Head Adaptation
Apr 23, 2026Jian Cheng Wong, Isaac Yin Chung Lai, Pao-Hsiung Chiu +3Representation Learning PDE Solving
Institute of High Performance Computing (IHPC), Agency for Science, Technology and Research (A*STAR), Singapore · National University of Singapore (NUS), Singapore · Indian Institute of Technology Goa (IIT Goa), India +1
A Green-Integral-Constrained Neural Solver with Stochastic Physics-Informed Regularization
Apr 23, 2026Mohammad Mahdi Abedi, David Pardo, Tariq AlkhalifahNeural PDE Solvers PDE Solving
University of the Basque Country (UPV/EHU), Department of Mathematics, Spain · Basque Center for Applied Mathematics (BCAM), Bilbao, Spain · Ikerbasque, Basque Foundation for Science, Bilbao, Spain +1
CO $_2$ sequestration hybrid solver using isogeometric alternating-directions and collocation-based robust variational physics informed neural networks (IGA-ADS-CRVPINN)
Apr 22, 2026Askold Vilkha, Tomasz Służalec, Marcin Łoś +1Neural PDE Solvers PDE Solving
AGH University of Krakow, Faculty of Computer Science, Al. Mickiewicza 2026 30, Kraków, 30-059, Poland
Virtual boundary integral neural network for three-dimensional exterior acoustic problems
Apr 19, 2026Jiahao Li, Qiang Xi, Ilia Marchevskiy +1Neural PDE Solvers PDE Solving
College of Mechanics and Engineering Science, Hohai University, Nanjing 211100, China · Department of Applied Mathematics, Bauman Moscow State Technical University, Moscow 105005, Russia
Singularity Formation: Synergy in Theoretical, Numerical and Machine Learning Approaches
Apr 18, 2026Yixuan WangPDE Solving Kolmogorov-Arnold Networks
CALIFORNIA INSTITUTE OF TECHNOLOGY Pasadena, California · [math.NA]
A Randomized PDE Energy driven Iterative Framework for Efficient and Stable PDE Solutions
Apr 17, 2026Yi Bing, Zheng Ran, Fu Jinyang +2PDE Solving
School of Traffic and Transportation Engineering, Central South University, Changsha, China · School of Civil Engineering, Central South University, Changsha, China · National Engineering Research Center of High-speed Railway Construction Technology, Changsha, China +2

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Generalization Error Bounds for Picard-Type Operator Learning in Nonlinear Parabolic PDEs

Koichi Taniguchi, Sho Sonoda

Department of Mathematical and Systems Engineering, Faculty of Engineering Shizuoka University 3-5-1 Johoku, Chuo-ku, Hamamatsu, 432-8561, Japan · RIKEN AIP / CyberAgent 1-4-1 Nihonbashi, Chuo-ku, Tokyo 103-0027, Japan

PDE Solving PDE Operator Learning Generalization Bounds

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Operator learning for partial differential equations (PDEs) aims to learn solution operators on infinite-dimensional function spaces from finite-resolution data. In this setting, it is important for the learned model to be discretization-invariant, or resolution-robust, and to reflect PDE-specific structure. It is therefore natural to ask how such structure should be encoded in the model architecture, hypothesis class, or learning procedure. In this paper, we study operator learning for solution operators of nonlinear parabolic PDEs based on Duhamel--Picard iteration. We formulate Picard iteration as an abstract state-transition model and present a theoretical framework for Picard-type operator learning. We derive implementation-agnostic generalization error bounds that separate the implementation error from the estimation error associated with the abstract state-transition model induced by Picard iteration. A key consequence is that increasing the Picard depth reduces the Picard truncation error without causing an unbounded growth of the entropy-based estimation error. We also extend the analysis to long-time prediction by rolling out the same learned local model over successive time blocks. Finally, we illustrate the theory for nonlinear heat equations on the torus using a Picard-type Fourier neural operator as a concrete implementation.

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