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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Operator learning for solving Fokker-Planck equations with various initial conditions
Jun 8, 2026Li Zeng, Xiaoliang Wan, Yaobin Wang +2PDE Solving PDE Operator Learning
School of Mathematics and Statistics, Fuzhou University, Fuzhou, China · Department of Mathematics and Center for Computation and Technology, Louisiana State University, Baton Rouge 70803, USA · Faculty of Science and Technology, Beijing Normal-Hong Kong Baptist University, Zhuhai 519087, China. +2
Overcoming the Limits of Finite Difference Method; Physics-Informed Neural Network for Noisy High-Dimensional Heat Diffusion
Jun 6, 2026Shreesh Bhattarai, Harish Chandra BhandariPDE Solving Physics-Informed ML
Department of Mathematics, Kathmandu University Dhulikhel, Nepal
A Data-Free Symbolic Regression Approach for Solving Equations
Jun 5, 2026Sergei Garmaev, Vinay Sharma, Olga FinkPDE Solving Symbolic Regression
Intelligent Maintenance and Operations Systems Laboratory, EPFL, Lausanne, CH-1015, Switzerland
DAS-PINNs for high-dimensional partial differential equations: extending deep adaptive sampling to spacetime domains
Jun 4, 2026Anshima Singh, David J. SilvesterPDE Solving Adaptive Sampling
Department of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK
Loss-Conditional PINNs for Parametric PDE Families
Jun 3, 2026Anna Lazareva, Alexander TarakanovPDE Solving PDE Operator Learning
Faculty of Computer Science HSE University, Moscow, Russia · VK and HSE University Moscow, Russia
Physics-Informed Residuals for Adaptive Mesh Refinement in Finite-Difference PDE Solvers
Jun 1, 2026Henry Kasumba, Ronald KatendePDE Solving Adaptive Mesh Refinement
Department of Mathematics · Makerere University · 7062, Kampala, Uganda +2
A holomorphic neural network framework for 3D boundary value problems governed by harmonic potentials
May 29, 2026Enrico Ballini, Allan Peter Engsig-Karup, Tito AndriolloNeural PDE Solvers PDE Solving
Department of Mechanical and Production Engineering, Aarhus University, Aarhus, Denmark · Department of Applied Mathematics and Computer Science, Technical University of Denmark, Kongens Lyngby, Denmark
Multi-Scale Separable Fourier Neural Networks for Solving High-Frequency PDEs
May 29, 2026Qihong Yang, Qiaolin HeNeural PDE Solvers PDE Solving
School of Mathematics, Sichuan University, city=Chengdu, postcode=610065, country=China
A Novel Tensor Product-Based Neural Network for Solving Partial Differential Equations
May 28, 2026Qihong Yang, Yangtao Deng, Qiaolin He +1Neural PDE Solvers PDE Solving
School of Mathematics, Sichuan University, Chengdu, 610065, China
Faster Thermal Profiling of a Lunar Rover with Machine Learning Adapted Finite Difference Model
May 26, 2026Samuel Weber, Zaki Hasnain, Souma ChowdhuryPDE Solving PDE Surrogate Modeling
University at Buffalo · Department of Mechanical and Aerospace Engineering · Zij Sciences +1
Minimal surfaces, Knots, and Neural Networks
May 25, 2026Tancredi Schettini Gherardini, Marco UsulaPDE Solving
University of Bonn and Max Planck Institute for Mathematics, Bonn, Germany. · Mathematical Institute, University of Oxford, Oxford, United Kingdom.
NPSolver: Neural Poisson Solver with Iterative Physics Supervision
May 25, 2026Bocheng Zeng, Rui Zhang, Runze Mao +5Neural PDE Solvers PDE Solving
Gaoling School of Artificial Intelligence Renmin University of China Beijing, China · School of Engineering Science University of Chinese Academy of Sciences Beijing, China
Guided Flow Matching for Forward and Inverse PDE Problems with Sparse Observations: Algorithm and Theory
May 25, 2026Xifeng Zhang, Jin ZhaoFlow Matching PDE Solving
School of Mathematical Science Capital Normal University Beijing, 100048, China · Academy for Multidisciplinary Studies Capital Normal University2026 Beijing, 100048, China
Random Neural Network Expressivity for Non-Linear Partial Differential Equations
May 24, 2026Muhammed Ali Mehmood, Lukas GononNeural PDE Solvers PDE Solving
Department of Mathematics Imperial College London UK · School of Computer Science University of St. Gallen Switzerland
IV-Net: A neural network for elliptic PDEs with random and highly varying coefficients
May 24, 2026Shan Zhong, George BirosNeural PDE Solvers PDE Solving
Oden Institute for Computational Science and Engineering, The University of Texas at Austin, 201 E 24th St, Austin, 78712, TX, USA · Walker Department of Mechanical Engineering, The University of Texas at Austin, 3012026 E Dean Keeton St, Austin, 78712, TX, USA
Hermite-NGP: Gradient-Augmented Hash Encoding for Learning PDEs
May 23, 2026Jinjin He, Zhiqi Li, Sinan Wang +1Neural PDE Solvers PDE Solving
Georgia Institute of Technology, Atlanta, GA, USA.
WLNO: Wavelet-Laplace Neural Operator for Solving Partial Differential Equations
May 23, 2026Muhammad Abid, Arth Sojitra, Omer SanPDE Solving Multi-Scale Feature Fusion
Department of Mechanical and Aerospace Engineering, University of Tennessee, Knoxville Knoxville, TN 37996, USA.
WINO: A Weak-Form Physics Informed Neural Operator for Hyperelasticity on Variable Domains
May 23, 2026Bokai Zhu, Qinghui Zhang, Timon RabczukNeural PDE Solvers PDE Solving
School of Science, Harbin Institute of Technology, Shenzhen, P. R. China. · School of Science, Harbin Institute of Technology, Shenzhen, Guangdong. · Institute of Structural Mechanics, Bauhaus-Universität Weimar, Weimar, 99423, Germany.
Fourier Feature Pyramids for Physics-Informed Neural Networks
May 22, 2026Brandon Zhao, Yixuan Wang, Jonathan T. Barron +3Fourier Feature Embeddings PDE Solving
Department of Computing and Mathematical Sciences, California Institute of Technology · Google DeepMind · Cahill Center for Astronomy and Astrophysics, California Institute of Technology
Lower Bounds for Advection-Diffusion Equations: An Exploration with AI-Generated Proofs
May 20, 2026Chenyang An, Xiaoqian XuPDE Solving
An End-to-End PyTorch Interface for Differentiable PDE Solvers: A RANS Model-Correction Study
May 19, 2026Luca Saverio, Michele Alessandro Bucci, Gianmarco Farro +2PDE Solving Differentiable Optimization
MONHADE · Digital Sciences & Technologies Department, Safran Tech, Magny-Les-Hameaux, 78114, France. · DAAA, ONERA, Institut Polytechnique de Paris, Meudon, 92190, France. +1
From Simple to Complex: Curriculum-Guided Physics-Informed Neural Networks via Gaussian Mixture Models
May 19, 2026Jianan Yang, Yiran Wang, Shuai Li +3PDE Solving Curriculum Learning
Xi’an Jiaotong University, Xi’an, China · College of International Business, Zhejiang Yuexiu University, Shaoxing, China · Shaoxing Key Laboratory of Intelligent Monitoring and Prevention of Smart Society, Shaoxing, China +1
fPINN-DeepONet: A Physics-Informed Operator Learning Framework for Multi-term Time-fractional Mixed Diffusion-wave Equations
May 15, 2026Binghang Lu, Zhaopeng Hao, Christian Moya +1PDE Solving Deep Operator Networks
Department of Mathematics, Purdue University West Lafayette, IN 47907, USA · School of Mechanical Engineering, Purdue University West Lafayette, IN 47907, USA
Curriculum Learning of Physics-Informed Neural Networks based on Spatial Correlation
May 14, 2026Xujia Chen, Xinyue Hu, Letian Chen +2PDE Solving Curriculum Learning
Department of Automation, Tsinghua University, Beijing 100084, P.R. China
Unbiased and Second-Order-Free Training for High-Dimensional PDEs
May 14, 2026Jaemin Seo, Surin Lee, Jae Yong LeeStochastic Differential Equations Neural PDE Solvers
Department of Artificial Intelligence, Chung-Ang University, Seoul, Republic of Korea.
Di-BiLPS: Denoising induced Bidirectional Latent-PDE-Solver under Sparse Observations
May 13, 2026Zhonghao Li, Chaoyu Liu, Qian ZhangNeural PDE Solvers Latent Diffusion Models
School of Science, Harbin Institute of Technology, Shenzhen, China · Department of Theoretical Physics and Applied Mathematics, University of Cambridge, Cambridge, United Kingdom
Frequency Bias and OOD Generalization in Neural Operators under a Variable-Coefficient Wave Equation
May 13, 2026Runlong Xie, An LuoFourier Neural Operator PDE Solving
Independent Researcher · School of Statistics, University of Minnesota, MN, USA.
MetaColloc: Optimization-Free PDE Solving via Meta-Learned Basis Functions
May 12, 2026Zichuan YangNeural PDE Solvers Meta-Learning
School of Mathematical Sciences Tongji University Shanghai, P.R. China
Neural-Schwarz Tiling for Geometry-Universal PDE Solving at Scale
May 12, 2026Paolo Secchi, Daniel S. Balint, Marco MauriziNeural PDE Solvers PDE Solving
Imperial College London · Italian Institute of Artificial Intelligence
Multifidelity Gaussian process regression for solving nonlinear partial differential equations
May 11, 2026Fatima-Zahrae El-Boukkouri, Josselin Garnier, Olivier RoustantSurrogate Modeling PDE Solving
Institut de Mathématiques de Toulouse, Université de Toulouse, INSA, Toulouse, 31077, France · CMAP, CNRS, Ecole polytechnique, Institut Polytechnique de Paris, Palaiseau, 91120, France

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Operator learning for solving Fokker-Planck equations with various initial conditions

Li Zeng, Xiaoliang Wan, Yaobin Wang, Fabio Nobile+1

School of Mathematics and Statistics, Fuzhou University, Fuzhou, China · Department of Mathematics and Center for Computation and Technology, Louisiana State University, Baton Rouge 70803, USA · Faculty of Science and Technology, Beijing Normal-Hong Kong Baptist University, Zhuhai 519087, China.+2

PDE Solving PDE Operator Learning Physics-Informed ML

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The Fokker-Planck equation (FPE) plays a pivotal role in describing the time evolution of probability density functions (PDFs) for systems governed by stochastic dynamics. In this work, we propose a conditional normalizing flow-based physics-informed neural network (PINN) framework for efficiently approximating the solution operator of the FPE for a whole range of initial conditions. Leveraging the Chapman-Kolmogorov equation for Markovian stochastic processes, the problem is reformulated into approximating a transition PDF starting at initial time from a Dirac mass centered at an arbitrary point. The PDF of an associated linearized stochastic differential equation (SDE) is employed as the base distribution for the normalizing flow, providing a good approximation of the target PDF, especially for small times, and thereby avoiding the singularity of the map associated with the Dirac delta initial distribution. Furthermore, a time-weighted loss function is introduced to mitigate numerical instabilities arising at small times, achieving a balance between causality and training difficulty as time progresses. A variety of numerical experiments are presented to illustrate the effectiveness and robustness of the proposed method.

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Latest papers 170