PDE Discovery

PDE: Partial Differential Equation

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Overview

Partial differential equation discovery studies how to infer the operators and coefficients governing spatially and temporally changing fields, helping explain physical systems and predict their behavior. Common approaches include sparse regression such as SINDy over candidate term libraries, weak-form methods that reduce noise from numerical differentiation, and symbolic or neural-symbolic methods that search for equations; newer systems also use language-model agents to propose and test hypotheses. Researchers test these methods on canonical PDE benchmarks and experimental or observational data, including sparse, noisy, multiscale, nonlocal, and geometry-varying settings. A central challenge is distinguishing a genuinely identifiable, physically meaningful law from a model that merely fits the available data, so evaluation increasingly considers stability, uncertainty, physical consistency, and performance on new conditions as well as predictive error.

Latest papers

CardsList

36 papers6 papers in the last four weeks, against 2 the four weeks before. 0.0% of all new papers.

  1. LegoSR: A Unified Framework for Self-improving Agent-Guided Scientific Discovery

    Oct 24, 2026Hao Mao, Xu T. Liu, Peng Zhao +2AI Agents for Scientific DiscoverySymbolic Regression

    The Hong Kong Polytechnic University, Hong Kong SAR, China · Eastern Institute of Technology, Ningbo, China · Amazon Web Services, Seattle, USA +3

  2. Machine-learned spectral PDE solvers for fluids and plasmas

    Oct 19, 2026Mellow Baixuan Chen, Alan A. KaptanogluPDE SolvingPDE 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

  3. FoundPDE: A generative foundation model for data-driven symbolic discovery of partial differential equations

    Oct 18, 2026Langchen Liu, Junlin Luo, Min Zhu +2PDE DiscoveryPDE Foundation Models

    Department of Statistics and Data Science, Yale University, New Haven, CT 06511, USA · Department of Mechanical Engineering, Stanford University, Stanford, CA 94305, USA · Department of Chemical and Environmental Engineering, Yale University, New Haven, CT 06511, USA

  4. Beyond Equation Search: Conjecture-Guided Discovery with SCOPE and ClosureBench

    Oct 13, 2026Yuxiao Hu, Qian Li, Ding Wang +4Scientific DiscoveryAI Agents for Scientific Discovery

    The Hong Kong Polytechnic University, Hong Kong, China · Eastern Institute of Technology, Ningbo, China · Shanghai Jiao Tong University, Shanghai, China +1

  5. Hyperparameter selection for equation learning with biologically-informed neural networks

    Oct 2, 2026William Lavery, Jodie A. Cochrane, John T. Nardini +1Hyperparameter OptimizationPhysics-Informed ML

    Department of Information Technology, Uppsala University, Sweden · Department of Mathematics and Statistics, The College of New Jersey, USA

  6. Geometry-physics confounding impairs PDE learning across varying domains

    Sep 29, 2026Yinghao Cheng, Gengxiang Chen, Xu Liu +6PDE DiscoveryPDE Operator Learning

    College of Mechanical & Electrical Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, China · School of Electronics, Electrical Engineering and Computer Science, Queen’s University Belfast, Belfast, United Kingdom · School of Mechanical and Aerospace Engineering, Queen’s University Belfast, Belfast, United Kingdom +3

  7. Recovering Governing Dynamics from Distributed Observations via Exact Spline Merging

    Sep 15, 2026Naveen MysorePDE DiscoveryDistributed Inference

    University of California, Santa Barbara · Dyssonance AI

  8. Learning the Geometry of Admissible Hypotheses through Inductive Bias in Training Distributions

    Aug 31, 2026James Crowley, Faez Ahmed, Anton van BeekRepresentation LearningPDE Discovery

    School of Mechanical and Material Engineering University College Dublin · Department of Mechanical Engineering Massachusets Institute of Technology

  9. See, Hypothesize, Validate: Multimodal Agentic Framework for Discovering Governing PDEs

    Aug 28, 2026Sarang Manoj Pekhale, Amartya Roy, Rajat Sarkar +1PDE DiscoveryPartial Differential Equations

    Department of Applied Mechanics Indian Institute of Technology Delhi, India Yardi School of Artificial Intelligence (ScAI) Indian Institute of Technology Delhi, India

  10. What Should a Large Language Model See? Physical Invariants as a Data Representation for PDE Discovery

    Aug 25, 2026Fan Yang, Matt ThomsonPDE DiscoveryLarge Language Model-Guided Scientific Discovery

    Division of Biology and Biological Engineering California Institute of Technology Pasadena, CA 91125

  11. Hypothesize, Evaluate, Refine: A Scientific Agent for PDE Discovery with Unknown Spatial Coefficient Fields

    Aug 19, 2026YuJie Huang, WenWu He, ZhuoEr Lin +3PDE DiscoveryPartial Differential Equations

    Fujian University of Technology · Fujian University of Technology Fujian Provincial Key Laboratory of Big Data Mining and Applications · Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences

  12. Robust data-driven discovery of fractional differential equations via weak formulations and Pareto-based subset selection

    Aug 13, 2026Pongpisit Thanasutives, Yoshinobu KawaharaSparse RecoveryPDE Discovery

    Center for Advanced Intelligence Project (AIP), RIKEN, Tokyo, Japan. · Graduate School of Information Science and Technology, The University of Osaka, Osaka, Japan.

  13. Large language models for partial differential equation workflows

    Aug 4, 2026Han Wan, Rui Zhang, Hao SunPDE DiscoveryLarge Language Model-Guided Optimization

    Gaoling School of Artificial Intelligence, Renmin University of China, Beijing 100872, China

  14. Data Driven Equation Discovery for Phase-Ordering Dynamics : From Allen Cahn to the Ising Model

    Aug 4, 2026Partha Sarathi Mondal, Manav Kumar Jalan, Anish Kumar +1Sparse Identification of Nonlinear DynamicsPDE Discovery

    Department of Physics, Indian Institute of Technology (BHU) Varanasi, India 221005

  15. Freeze, Then Select: Structured Field Adapters and Stability-Validated Weak Selection for PDE Discovery from Sparse Observations

    Jul 31, 2026Juncheng Zhong, Chenghuang Shen, Jianfeng Liu +5Sparse RecoverySparse Identification of Nonlinear Dynamics

    School of Mathematical Sciences, Fudan University, Shanghai 200433, China · Shanghai Center for Mathematical Sciences, Fudan University, Shanghai 200438, China · Alibaba Group, Hangzhou, China +3

  16. Dynamics-aware identification of governing equations from sparse and noisy data

    Jul 31, 2026Pongpisit Thanasutives, Yoshinobu KawaharaKoopman Operator LearningSparse Identification of Nonlinear Dynamics

    Center for Advanced Intelligence Project (AIP), RIKEN, Osaka, Japan · 2Graduate School of Information Science and Technology, The University of Osaka, Osaka, Japan

  17. On the post-hoc Evaluation of PDE Discovery: A Multifaceted Challenge of Scientific Advancement

    Jul 26, 2026Baptiste Mathevon, Farah Cherfaoui, Amaury Habrard +1PDE DiscoveryPhysics-Informed ML

    Université Jean Monnet Saint-Étienne, CNRS, Institut d’Optique Graduate School, Inria, Laboratoire Hubert Curien UMR 5516, F-42023, SAINT-ÉTIENNE, France · Institut Universitaire de France (IUF)

  18. A zero-one law for one-shot system identification

    Jul 17, 2026Nicolas Boullé, Diana Halikias, Samuel E. Otto +1Parameter IdentifiabilityPDE Discovery

    Department of Mathematics, Imperial College London, London, SW7 2AZ, UK · Courant Institute, New York University, New York, NY 10012, USA · Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14853, USA +1

  19. LLM-PDESR: Robust PDE Discovery via Subdomain Weighted Residuals and LLM-Guided Symbolic Hypothesis Generation

    Jul 12, 2026Jinyang Du, Hao Ma, Xiaohu Shi +4PDE DiscoveryLarge Language Model-Guided Scientific Discovery

    College of Computer Science and Technology, Jilin University, Changchun, China · School of Big Data and Artificial Intelligence, Guangdong University of Finance and Economics, Guangzhou, China · Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education, Jilin University, Changchun, China +2

  20. Joint discovery of governing partial differential equations from multi-source datasets by competitive optimization

    Jun 29, 2026Hao Xu, Siyu Lou, Yuntian Chen +1PDE DiscoverySymbolic Regression

    Zhejiang Key Laboratory of Industrial Intelligence and Digital Twin, Eastern Institute of Technology, Ningbo, Zhejiang 315200, China · Department of Electrical Engineering, Tsinghua University, Beijing 100084, P. R. China. · Shanghai Jiao Tong University, School of Computer Science, Shanghai, China +1

  21. Agentic Symbolic Search: Characterizing PDEs Beyond Hand-crafted Expressions, Meshes, and Neural Networks

    Jun 18, 2026Zongmin Yu, Liu YangAI Agents for Scientific DiscoveryPDE Solving

    National University of Singapore

  22. How Low Can You Go? Active Learning for Sparse Model Discovery in the Ultra-Low-Data Limit

    Jun 10, 2026Ana Larrañaga, Urban Fasel, Steven L. BruntonSparse Identification of Nonlinear DynamicsPDE Discovery

    Department of Mechanical Engineering, University of Washington, Seattle, WA 98195, United States · NSF AI Institute in Dynamic Systems, University of Washington, Seattle, WA 98195, United States · Department of Aeronautics, Imperial College London, SW7 2AZ, United Kingdom

  23. Data-driven discovery of governing differential equations across physical systems

    Jun 8, 2026Siyu Lou, Hao Xu, Wenguan Wang +6PDE DiscoveryAutonomous Scientific Discovery

    School of Computer Science, Shanghai Jiao Tong University, Shanghai, China · Ningbo Key Laboratory of Advanced Manufacturing Simulation, Eastern Institute of Technology, Ningbo, Ningbo, China · The State Key Lab of Brain-Machine Intelligence, Zhejiang University, Hangzhou, China. +5

  24. Data-driven sparse identification of governing PDEs via knockoff filters and multi-criteria trade-offs

    May 26, 2026Pongpisit Thanasutives, Naichang Ke, Yoshinobu KawaharaFDR ControlPDE Discovery

    1RIKEN Center for Advanced Intelligence Project (AIP) · 2The University of Osaka

  25. EqOD: Symmetry-Informed Stability Selection for PDE Identification

    May 12, 2026Gnankan Landry Regis N'guessan, Bum Jun KimPDE DiscoveryPartial Differential Equations

    1Axiom Research Group · Department of Applied Mathematics and Computational Science, NM-AIST, Tanzania · 3African Institute for Mathematical Sciences (AIMS), Research and Innovation Centre, Rwanda +1

  26. From Video-to-PDE: Data-Driven Discovery of Nonlinear Dye Plume Dynamics

    May 6, 2026Cesar Acosta-Minoli, Sayantan SarkarPDE SolvingSparse 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

  27. Physics-Informed Neural Networks for Biological 2D+t2\mathrm{D}{+}t Reaction-Diffusion Systems

    Apr 20, 2026William Lavery, Jodie A. Cochrane, Christian Olesen +3Dynamical SystemsReaction-Diffusion Systems

    Dept. of Information Technology Uppsala University Uppsala, Sweden · Dept. of Medical Genetics Oslo University Hospital Oslo, Norway · Dept. of Translational Hematology and Oncology Research, Cleveland Clinic, Cleveland, USA +1

  28. Balance-Guided Sparse Identification of Multiscale Nonlinear PDEs with Small-coefficient Terms

    Apr 20, 2026Zhenhua Dang, Lei Zhang, Long Wang +1Sparse Identification of Nonlinear DynamicsPDE Discovery

  29. Bias Inheritance in Neural-Symbolic Discovery of Constitutive Closures Under Function-Class Mismatch

    Apr 1, 2026Hanbing Liang, Ze Tao, Fujun LiuReaction-Diffusion SystemsPDE Discovery

    Nanophotonics and Biophotonics Key Laboratory of Jilin Province, School of Physics, Changchun University of Science and Technology, Changchun 130022, P.R. China

  30. Symbolic--KAN: Kolmogorov-Arnold Networks with Discrete Symbolic Structure for Interpretable Learning

    Mar 25, 2026Salah A Faroughi, Farinaz Mostajeran, Amirhossein Arzani +1Interpretable MLScientific ML

    Energy & Intelligence Lab, Department of Chemical Engineering, University of Utah, Salt Lake City, Utah, 84112, USA · Department of Mechanical Engineering, University of Utah, Salt Lake City, Utah, 84112, USA · Scientific Computing and Imaging Institute, University of Utah, Salt Lake City, Utah, 84112, USA +2