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.

Probabilistic Meshless Methods on Manifolds
Oct 29, 2026Federico Pavesi, Noémie Jaquier, Antonio CandelieriPDE Solving PDE Surrogate Modeling
Department of Economics, Management and Statistics University of Milano-Bicocca Piazza dell’Ateneo Nuovo 1, 20126 Milan, Italy · Department of Robotics, Perception and Learning KTH Royal Institute of Technology Lindstedtsvägen 24, 10044 Stockholm, Sweden
CFKS: Continuum-Feature Kernel PDE Solvers
Oct 28, 2026Vikas Dwivedi, Balaji Srinivasan, Monica Sigovan +1PDE Solving Kernel Methods
CREATIS Biomedical Imaging Laboratory INSA, CNRS UMR 5220, Inserm, Université Lyon 1 Lyon 69621, France · Wadhwani School of Data Science and AI IIT Madras, India
A Hybrid Physics-informed holomorphic neural networks Method
Oct 28, 2026Zhijie Su, Qiaolin He, Zhiping Mao +1PDE Solving Neural PDE Solvers
Department of Mathematics, Sichuan University, No. 24, Section 1, South Yihuan Road, Chengdu, 610065, China · Eastern Institute of Technology, Ningbo, No. 2911, Haijiang Avenue, Zhenhai District, Ningbo, 315200, China
HotPOT: Operator Transformer with Heterogeneous Physics Experts for Large-Scale PDE Pre-training
Oct 28, 2026Jiayi Li, Guancheng Zhou, Yisi Luo +5PDE Foundation Models PDE Operator Learning
University of New South Wales · Xi’an Jiaotong University · Commonwealth Scientific and Industrial Research Organisation
Physics Informed Neural Networks for Solving the Drift-Diffusion Equations
Oct 23, 2026Luke Ragan, Matteo G. C. Alasio, Enrico Bellotti +1PDE Solving PDE Inverse Problems
Department of Electrical & Computer Engineering, Boston University, Boston, MA, USA · Dipartimento di Elettronica e Telecomunicazioni, Politecnico di Torino, Torino, TO, Italy · Department of Electrical & Computer Engineering, Division of Materials Science & Engineering, Boston University, Boston, MA, USA +1
GAV: A scientific experimentation environment for PDE agents
Oct 21, 2026Muheng Li, Wenlong MouPDE Solving AI Agent Evaluation
Department of Statistical Sciences, University of Toronto · Vector Institute
Automatic Differentiation with Constraints in Forward and Inverse Problems
Oct 20, 2026Nikolay Yavich, Alexander Ryabov, Viacheslav Naumov +4PDE Solving PDE Inverse Problems
Artificial Intelligence Center, Skolkovo Institute of Science and Technology, 121205 Moscow, Russia · Yardi School of Artificial Intelligence, Indian Institute of Technology Delhi, Hauz Khas, New Delhi, India 110016 · Learnable Intelligence Group, Autonomous Non-Profit Organization Artificial Intelligence Research Institute (AIRI), 105064 Moscow, Russia
Soliton Profiles in 1D: Enhanced Neural-Network Solvers and Constrained Deep Ritz Methods
Oct 20, 2026Chandler Haight, Alex D Rodriguez, Svetlana RoudenkoNeural PDE Solvers PDE Solving
Machine-learned spectral PDE solvers for fluids and plasmas
Oct 19, 2026Mellow Baixuan Chen, Alan A. KaptanogluPDE Solving PDE Discovery
Courant Institute of Mathematical Sciences, New York University, New York, 10012, NY, USA · William E. Boeing Department of Aeronautics and Astronautics, University of Washington, Seattle, 98195, Washington, USA
Low Residuals, Wrong Value: Learning the Viscosity Solution of Hamilton-Jacobi-Isaacs Equations with PINNs
Oct 19, 2026Raul Jimenez, Pablo Tejerina-Pérez, Leonid Sarieddine +6PDE Solving Partial Differential Equations
Institut de Ciències del Cosmos (ICCUB), Universitat de Barcelona, Barcelona, Spain · Institució Catalana de Recerca i Estudis Avançats (ICREA), Barcelona, Spain · Departament de Física Quàntica i Astrofísica, Universitat de Barcelona, Barcelona, Spain +3
Residual-Based Adaptive Refinement in Hybrid Variational-Collocation PINNs for Singular Elliptic PDEs with Mixed Dirichlet-Neumann Boundary Conditions
Oct 19, 2026Somnath Giri, Muslim Malik, Rishikesh YadavPDE Solving Partial Differential Equations
School of Mathematical & Statistical Sciences, Indian Institute of Technology Mandi, Mandi, 175005, Himachal Pradesh, India
LowNet: Looped Wavelet Neural Networks with 24 Learnable Parameters for Solving Poisson Equations
Oct 18, 2026Yuanhao GongPDE Solving Neural PDE Solvers
Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences
A Method for Exact Enforcement of Boundary Conditions on General Curved Triangular Domains: Algorithm and Application to Scientific Machine Learning
Oct 12, 2026Suchuan Dong, Yuchuan ZhangPDE Solving Neural PDE Solvers
Center for Computational and Applied Mathematics Department of Mathematics Purdue University, USA
Neural Network Preconditioning: A Case Study for the Solution of the Parametric Helmholtz Equation
Oct 12, 2026Luc Giraud, Carola Kruse, Paul Mycek +2PDE Solving Neural PDE Solvers
Inria, Concace joint team between Airbus CR&T, Cerfacs and Inria · Cerfacs, Concace joint team between Airbus CR&T, Cerfacs and Inria · Current affiliation: Opticalp +1
Sampling from Neural Operator Priors to Solve Partial Differential Equations
Oct 12, 2026Andrew Millard, Joshua Murphy, Zheng ZhaoPDE Solving Partial Differential Equations
Department of Computer and Information Science, Linköping University, Sweden. · Department of Political Science, Trinity College Dublin, Ireland.
An Attention-Guided Graph Neural Network Surrogate for Boundary Integral Computation in Physical Field Reconstruction
Oct 11, 2026RongZe Gao, YuWei Wu, XinRan Ji +1PDE Surrogate Modeling PDE Inverse Problems
The School of Civil and Architectural Engineering, Hainan University, HaiKou, 570000, Chain.
Energy-Guided Learning for Heat Equation with Noisy Source: Optimal Estimation and Generalization Bounds
Oct 10, 2026Zixiang Zhou, Xiaohui Chen, Shanghua TengPhysics-Informed ML PDE Solving
Department of Mathematics, University of Southern California · Thomas Lord Department of Computer Science, University of Southern California · Computing and Mathematical Sciences, California Institute of Technology
Constrained Probabilistic Numerics: Gaussian Process Solvers for Linear PDEs under Linear Inequality Constraints
Oct 9, 2026Boyuan Deng, Kshitiz Upadhyay, Michael ShieldsPDE Solving Gaussian Processes
Department of Civil and Systems Engineering, Johns Hopkins University · Department of Aerospace Engineering and Mechanics, University of Minnesota
PDE-PFN: Prior-Data Fitted Neural PDE Solver
Oct 9, 2026Jaehyeon Park, Mingu Kang, Dongseok Lee +5Neural PDE Solvers PDE Solving
SolverX · KAIST · Seoul National University +3
Enriched Holomorphic Neural Networks for 2D and 3D Laplace Problems
Oct 8, 2026Kalp Shah, Andrei Vlad Nicula, Mihnea Andrei Bârsan +4Neural PDE Solvers PDE Solving
University of Amsterdam · Technical University of Denmark
An Interpretable Approach to PDE Solution Discovery via Structural Experience Distillation
Oct 8, 2026Yunpeng Gong, Huolong Wu, Can Yang +1PDE Solving Symbolic Regression
School of Informatics, Xiamen University
Gen-PINNs: Generative Adversarial Physics Informed Neural Networks for solving partial differential equations
Oct 7, 2026Muhammad M. Akmal, Kamy Sepehrnoori, Michael J. PyrczPDE Solving Physics-Informed ML
Hilderbrand Department of Petroleum and Geosystems Engineering, The University of Texas at Austin · Department of Earth and Planetary Sciences, The University of Texas at Austin
Physics-Informed Neural Plasticity: PDE Solvers That Reshape Themselves
Oct 7, 2026Chun-Wun Cheng, Bingcheng Hu, Angelica I. Aviles-RiveroNeural PDE Solvers PDE Solving
Shanghai Normal University · YMSC, Tsinghua University
A Neural JKO Scheme for Hellinger-Kantorovich Gradient Flows via Monge-Growth Pairs
Oct 6, 2026Geuntaek Seo, Cheolhyeong Kim, Hwijae Son +1Neural PDE Solvers PDE Solving
Department of Mathematics, Pohang University of Science and Technology · Samsung Electronics · Department of Mathematics, Konkuk University
Low-Fidelity FDM Spectral Guidance for Neural Eigenvalue Solvers
Oct 3, 2026Aryan Chaudhary, Manikandan Padmanaban, Jagabondhu HazraNeural PDE Solvers PDE Solving
BITS Pilani · IBM Research
Efficient Neural Surrogates for Linear Radiation Transport on the Lattice and Hohlraum benchmarks
Oct 3, 2026Carmelo Gonzales, Steffen Schotthöfer, Cory D. HauckNeural Surrogate Modeling PDE Surrogate Modeling
NVIDIA, Santa Clara, US · Seamless Labs, Mountain View, US · Oak Ridge National Laboratory
CEENs: Causality-enforced evolutional networks for solving time-dependent partial differential equations
Oct 3, 2026Jeahan Jung, Heechang Kim, Hyomin Shin +1Neural PDE Solvers PDE Solving
Department of Mathematics, Pohang University of Science and Technology (POSTECH), Pohang 37673, Republic of Korea
Geometry Meets Physics: Data-Efficient Pre-Training for Unstructured Neural PDE Solvers
Oct 2, 2026Luis Medrano-Navarro, Giacomo Baldan, Qiang Liu +4Synthetic Data Pretraining Neural PDE Solvers
Technical University of Munich · University of Stuttgart
Kolmogorov-Arnold Networks for Free-Boundary Partial Differential Equations
Oct 1, 2026Tan Phuong Dong LeNeural PDE Solvers PDE Solving
Department of Applied Mathematics, University of Waterloo, Canada · Department of Statistics and Actuarial Science, University of Waterloo, Canada
Learning PDE Dynamics between Submanifolds Using Green's Observation Operators
Oct 1, 2026Jan Tauberschmidt, Jephte Abijuru, Naukshatro Bose +5PDE Solving PDE Surrogate Modeling
German Research Center for Artificial Intelligence (DFKI), Kaiserslautern, Germany · Rheinland-Pfälzische Technische Universität (RPTU), Kaiserslautern, Germany · University of Manchester, Manchester, United Kingdom

Oct 29, 2026J/K move · Enter open · S save

Probabilistic Meshless Methods on Manifolds

Federico Pavesi, Noémie Jaquier, Antonio Candelieri

Department of Economics, Management and Statistics University of Milano-Bicocca Piazza dell’Ateneo Nuovo 1, 20126 Milan, Italy · Department of Robotics, Perception and Learning KTH Royal Institute of Technology Lindstedtsvägen 24, 10044 Stockholm, Sweden

PDE Solving PDE Surrogate Modeling Gaussian Processes

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Probabilistic meshless methods solve partial differential equations (PDEs) by placing a Gaussian process prior on the solution and conditioning it on the PDE at a finite set of collocation nodes. The posterior provides both an approximation of the solution and a Bayesian quantification of its error. While these methods are well established on Euclidean domains, their extension to manifolds lacks, to our knowledge, any convergence analysis. We extend them to compact manifolds and study their convergence in two settings. First, for known manifolds whose Laplace-Beltrami eigenpairs are available, the prior can be evaluated exactly, but the PDE operator generally cannot be applied to it in closed form. We replace this operator by a standard generalized finite difference discretization on a point cloud sampled from the manifold, and show that the resulting method is consistent at the points of the cloud, at a rate governed by their fill distance. Second, for implicit manifolds, known only through a point cloud, both the prior and the operator must be approximated. We propose a new generalized finite difference scheme with a free stencil radius, and show that it retains the consistency of the standard construction while controlling the growth of its coefficients, and that an appropriate radius yields uniform convergence of the discrete Gram matrix. In both settings, we derive error bounds for the posterior mean which separate the collocation error intrinsic to the method from the error due to approximating the operator and, for implicit manifolds, the prior. Experiments on the circle, the sphere and the flat torus confirm our theoretical claims. On real-world cortical thickness data, our approach attains the accuracy of established discretizations, while conditioning on fewer than a tenth of the observations and providing a credible interval at every vertex.