PDE Inverse Problems

PDE: Partial Differential Equation

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Overview

PDE inverse problems use measured data—often sparse or noisy—to infer unknown coefficients, initial or boundary conditions, sources, or entire physical fields. Classical approaches include regularization, adjoint-based optimization, and Bayesian sampling; newer methods use physics-informed neural networks, neural operators, differentiable solvers, and generative models such as diffusion and flow matching. A central challenge is ill-posedness: a method may reconstruct the observed field well while failing to identify the true parameters or quantify uncertainty reliably. Current work seeks more scalable posterior inference and stronger evaluation of identifiability, robustness to changing observations, and whether predictions support scientific conclusions.

Latest papers

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107 papers17 papers in the last four weeks, up 55% on the four weeks before. 0.1% of all new papers.

  1. Physics Informed Neural Networks for Solving the Drift-Diffusion Equations

    Oct 23, 2026Luke Ragan, Matteo G. C. Alasio, Enrico Bellotti +1PDE SolvingPDE 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

  2. Automatic Differentiation with Constraints in Forward and Inverse Problems

    Oct 20, 2026Nikolay Yavich, Alexander Ryabov, Viacheslav Naumov +4PDE SolvingPDE 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

  3. SALTO: Solver-Anchored Lifted Transfer Operators for Nonlinear Event Inference

    Oct 15, 2026Qi Wang, Rylie FosterPDE Inverse ProblemsKoopman Operator Learning

    Department of Aerospace Engineering San Diego State University San Diego, CA 92182, USA · Computational Science Research Center San Diego State University San Diego, CA 92182, USA · Department of Mechanical and Aerospace Engineering University of California, Irvine, CA 92697, USA

  4. Sampling from Neural Operator Priors to Solve Partial Differential Equations

    Oct 12, 2026Andrew Millard, Joshua Murphy, Zheng ZhaoPDE SolvingPartial Differential Equations

    Department of Computer and Information Science, Linköping University, Sweden. · Department of Political Science, Trinity College Dublin, Ireland.

  5. 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 ModelingPDE Inverse Problems

    The School of Civil and Architectural Engineering, Hainan University, HaiKou, 570000, Chain.

  6. SBNO : Schrödinger Bridge Neural Operator for the Forward and Inverse Problems under Missing Information

    Oct 9, 2026Jaehyeon Park, Inhyeok Jeong, Noseong ParkPDE Operator LearningPDE Surrogate Modeling

    SolverX · KAIST

  7. IGA-KAN: Isogeometric Analysis with Physics-Informed Closed-Form Kolmogorov-Arnold Networks for Forward and Inverse PDEs

    Oct 5, 2026Sima Naraghi, Kourosh Parand, Amirhossein Sadr +1Neural PDE SolversKolmogorov-Arnold Networks

    Department of Applied Mathematics, Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran, Iran · Department of Computer and Data Sciences, Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran, Iran · Department of Cognitive Modeling, Institute for Cognitive and Brain Sciences, Shahid Beheshti University, Tehran, Iran +1

  8. OCL-PDE: A Generative Framework for PDE Inverse Problems with Observation-Complementary Latents

    Oct 5, 2026Ding Yang, Chuqi Chen, Chang Ma +1Physics-Informed Generative ModelingPDE Operator Learning

    Department of Mathematics, The Hong Kong University of Science and Technology, Hong Kong · Department of Mathematics, University of Michigan, Ann Arbor, USA · Department of Materials, The Hong Kong University of Science and Technology, Hong Kong +1

  9. Inferring physical fields in coupled systems with unknown parameters from incomplete observations using physics-constrained attentive neural operators

    Oct 5, 2026Shilun Wei, Xiaoqiang Sun, Wei Li +1Parameter IdentifiabilityPhysics-Informed Neural Operators

    School of Mathematics, Sun Yat-sen University, Guangzhou, China · School of Sciences, Great Bay University, Dongguan, China · John Hopcroft Center for Computer Science, School of Computer Science, Shanghai Jiao Tong University, Shanghai, China

  10. Robust Ensemble Guidance for Scientific Inverse Problems

    Oct 4, 2026Zixiang Li, Wei Wang, Yunchao Wei +2Diffusion Model GuidanceDiffusion-Based Inverse Problems

    Beijing Jiaotong University · Tsinghua University

  11. Backward-Consistent Diffusion Sampling for Sparsely Observed PDE Inverse Problems

    Oct 3, 2026Yida Pan, Muhammad H. Ashiq, Chanyong Jung +4PDE Inverse ProblemsDiffusion-Based Inverse Problems

    University of Michigan · Los Alamos National Laboratory

  12. Flow Matching for Fast Posterior Sampling in Bayesian Inverse Problems

    Oct 1, 2026Jan Blechschmidt, Oliver G. Ernst, Moritz Poguntke +1Posterior SamplingBayesian Inverse Problems

  13. PhysDEM: Physics-Defined Energy-Matching Diffusion for Spatiotemporal Field Generation under Scarce Measurements

    Oct 1, 2026Zhenyu Liang, Yining Huang, Yubo Zhao +1Physics-Informed Diffusion ModelsPhysics-Informed Generative Modeling

    HKUST · UT Austin

  14. Initial condition recovery in nonlinear damped viscous photoacoustic tomography using a convolutional neural network-guided gradient-free optimization framework

    Oct 1, 2026Madhu Gupta, Anwesa Dey, Prapti Tala +1Medical ImagingGradient-Free Optimization

    Department of Mathematics, Indian Institute of Technology, Gandhinagar, Gujarat, India. · Department of Mathematics, University of Utah, Salt Lake City, Utah, UT, USA. · Department of Mathematics, The University of Texas at Arlington, Arlington, TX, USA.

  15. SCOPE: Observation-Conditioned Full-Target Prediction for Sparse PDE Inference

    Sep 29, 2026Ruichen Xu, Siyao Wang, Fang Wan +8PDE Surrogate ModelingPDE Inverse Problems

    Stony Brook University · University of California, Davis · Independent Research +3

  16. PDE-OBS: Controlled Evaluation Across Observation Patterns

    Sep 29, 2026Ruichen Xu, Siyao Wang, Fang Wan +7Benchmark DesignPDE Surrogate Modeling

    Stony Brook University · University of California, Davis · Independent Research +3

  17. Beyond Gradient Flow: Identifiability and Recovery from Distribution Snapshots

    Sep 28, 2026Nam D. Nguyen, Valeriya MalyshevaFokker–Planck EquationSystem Identification

    VIB, Center for Molecular Neurology, Antwerp, Belgium · VIB, Center for AI and Computational Biology, Leuven, Belgium · Faculty of Pharmaceutical, Biomedical and Veterinary Sciences, University of Antwerp, Antwerp, Belgium +2

  18. PI-NOMT: Physics-Informed Neural Optimal Mass Transport for Brain Fluid Dynamics

    Sep 27, 2026Mehmet Emin Acar, Vahit Bugra Yesilkaynak, Helene Benveniste +1Physics-Informed MLPDE Inverse Problems

    Istanbul Technical University, Faculty of Computer and Informatics Engineering, AI and Data Engineering Department, Istanbul, Turkiye · Yale University, School of Medicine, New Haven, CT, USA

  19. NEXT: Physics-Informed Neuro-Spectral Exponential Time Differencing Architectures

    Sep 25, 2026Márcio Marques, Leonardo Mendonça, Leonardo M. Moreira +6Neural PDE SolversPDE Solving

    Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, Brasil · Universidade do Estado do Rio de Janeiro (UERJ), Rio de Janeiro, Brasil · Universidade Federal Fluminense (UFF), Niterói, Brasil

  20. Variational objectives for amortized Bayesian inference in inverse problems: The role of posterior conditioning

    Sep 21, 2026Abhishek Srivastava, Arijit Hazra, Rajesh DubbakuAmortized InferenceBayesian Inverse Problems

    Department of Mechanical Engineering, Indian Institute of Technology Palakkad · Department of Mathematics, Indian Institute of Technology Palakkad

  21. PosteriorBench: From Point Estimates to Posterior Matching in Evaluating Generative Inverse Solvers

    Sep 17, 2026Jiachen Yao, Zi-Siang Hsu, Xi Deng +5Uncertainty QuantificationBayesian Inverse Problems

    California Institute of Technology · National Taiwan University · Lawrence Berkeley National Laboratory +3

  22. Linearized PINN with pretrained nonlinear layers

    Sep 14, 2026Wenhao Chen, Alexandre M. TartakovskyEfficient Neural Network InferencePDE Solving

    Civil and Environmental Engineering Department University of Illinois Urbana-Champaign Urbana, IL 61801, USA · Pacific Northwest National Laboratory Richland, WA 99352, USA

  23. Physical-State-Guided Diffusion Sampling for Full-Waveform Inversion

    Sep 11, 2026Chen Min, Haowen Jiang, Zheng Ma +1Physics-Informed Diffusion ModelsSeismic Imaging

    School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, China · CMA-Shanghai, Shanghai Jiao Tong University, Shanghai, China · School of Mathematics and Statistics, Lanzhou University, Lanzhou, China

  24. Physics-Informed Neural Networks to Infer the Perpendicular Energy Conductivity in the Scrape-Off Layer of Stellarator Devices

    Sep 10, 2026J. Gallego, P. Protopapas, A. Bustos +6Physics-Informed MLPDE Inverse Problems

    Departamento de Tecnología, CIEMAT, Spain · Harvard John A. Paulson School of Engineering and Applied Sciences, USA · Laboratorio Nacional de Fusión, CIEMAT, Spain

  25. Why Learning Rediscovers the Closed-Form Diagonal Regularizer

    Sep 9, 2026Jeahn Han, Pyojin KimSpectral RegularizationPDE Inverse Problems

    1GIST

  26. Local gradient neural operator

    Sep 7, 2026Baiming Zhang, Jinsong Tang, Ying Xu +2PDE Surrogate ModelingPDE Operator Learning

    Zhejiang University · Nanjing University of Science and Technology · Hefei University of Technology

  27. Bi-HYCO: Bi-Objective Cooperative Learning for PDE Parameter Identification under Fragmented Observations

    Sep 6, 2026Umberto Biccari, Jun Chen, Roberto Morales +1Parameter EstimationPDE Inverse Problems

    University of Madrid · China Scholarship Council

  28. Learning Informative Prior with Infinite-Dimensional Continuous Normalizing Flow for Bayesian Inverse Problem

    Sep 3, 2026Yang Zhao, Junxiong Jia, Tao ZhouNormalizing FlowsBayesian Inverse Problems

    School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, China · Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing, 100190, China

  29. Sensitivity-Constrained Neural Operators for Data-Efficient Forward and Inverse Modeling of Partial Differential Equation Systems

    Aug 30, 2026Abdolmehdi Behroozi, Chaopeng Shen, Daniel Kifer +1PDE Surrogate ModelingJacobian Regularization

    Department of Civil and Environmental Engineering, Penn State University, University Park, PA 16802, USA · School of Electrical Engineering and Computer Science, Penn State University, University Park, PA 16802, USA

  30. Physics-Informed Stochastic Configuration Machine: A Backpropagation-Free Neural Network with Fast Training for Nonlinear Differential Equations

    Aug 27, 2026Yuehao Song, Zhong Chen, Lihui Cen +2Physics-Informed MLNeural Network Optimization

    School of Automation, Central South University, Changsha, China · School of Automation, Central South University, 932 South Lushan Road, Changsha 410083, China · Johns Hopkins University, Baltimore, USA +3