PDE Inverse Problems

PDE: Partial Differential Equation

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

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

Hierarchical Inference and Closure Learning via Adaptive Surrogates for ODEs and PDEs
Mar 4, 2026Pengyu Zhang, Arnaud Vadeboncoeur, Alex Glyn-Davies +1PDE Inverse Problems PDE Surrogate Modeling
Department of Engineering, University of Cambrdige, 7a JJ Thomson Ave, Cambridge, CB3 0FA, UK · The Alan Turing Institute, 96 Euston Rd., London, NW1 2DB, UK
Solving Inverse PDE Problems using Minimization Methods and AI
Mar 2, 2026Noura Al Helwani, Sophie Moufawad, Georges SakrPDE Solving PDE Inverse Problems
Department of Physics, American University of Beirut, Lebanon · Department of Mathematics, American University of Beirut, Lebanon · The authors would like to acknowledge
Dual-space posterior sampling for Bayesian inference in constrained inverse problems
Feb 28, 2026Ali Siahkoohi, Kamal Aghazade, Ali GholamiBayesian Inverse Problems Inverse Problems
Department of Computer Science, University of Central Florida, Orlando, FL, USA · Institute of Geophysics, Polish Academy of Sciences, Warsaw, Poland
A Stability-Aware Frozen Euler Autoencoder for Physics-Informed Tracking in Continuum Mechanics (SAFE-PIT-CM)
Feb 27, 2026Emil HovadPhysics-Informed ML PDE Inverse Problems
Section for Statistics and Data Analysis, DTU Compute, Technical University of Denmark, Kgs. Lyngby, Denmark
Conditional neural control variates for variance reduction in Bayesian inverse problems
Feb 24, 2026Ali Siahkoohi, Hyunwoo OhVariance Reduction Bayesian Inverse Problems
Department of Computer Science, University of Central Florida, Orlando, Florida, USA · Department of Physics and Maryland Center for Fundamental Physics, University of Maryland, College Park, Maryland, USA
Unlearning Noise in PINNs: A Selective Pruning Framework for PDE Inverse Problems
Feb 23, 2026Yongsheng Chen, Yong Chen, Wei Guo +1Physics-Informed ML PDE Inverse Problems
School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China. · Department of Biostatistics, Epidemiology and Informatics, University of Pennsylvania, Philadelphia, PA 19104, USA. · Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 70409, USA.
Symbolic recovery of PDEs from measurement data
Feb 17, 2026Erion Morina, Philipp Scholl, Martin HollerPDE Discovery PDE Inverse Problems
IDea Lab - The Interdisciplinary Digital Lab at the University of Graz, Austria. · Aleph Alpha Research, Germany.
Solving Inverse Parametrized Problems via Finite Elements and Extreme Learning Networks
Feb 16, 2026Erik Burman, Mats G. Larson, Karl Larsson +1PDE Surrogate Modeling PDE Inverse Problems
GenPANIS: A Latent-Variable Generative Framework for Forward and Inverse PDE Problems in Multiphase Media
Feb 16, 2026Matthaios Chatzopoulos, Phaedon-Stelios KoutsourelakisPDE Inverse Problems Physics-Informed Generative Modeling
Technical University of Munich, Professorship of Data-driven Materials Modeling, School of Engineering and Design, Boltzmannstr. 15, 85748 Garching, Germany · Munich Data Science Institute (MDSI - Core member), Garching, Germany
Learning functional components of PDEs from data using neural networks
Feb 13, 2026Torkel E. Loman, Yurij Salmaniw, Antonio Leon Villares +2PDE Inverse Problems
Mathematical Institute, University of Oxford, United Kingdom · Department of Mathematical and Physical Sciences, University of Sheffield, United Kingdom · Department of Engineering, University of Oxford, United Kingdom
Efficient Generative Transformer Operators For Million-Point PDEs
Dec 4, 2025Armand Kassaï Koupaï, Lise Le Boudec, Patrick GallinariPDE Surrogate Modeling Transformer Neural Operator
Sorbonne Université, CNRS, ISIR, 75005 Paris, France · Criteo AI Lab, Paris, France
Inverse Problem for Partial Differential Equations with Jump Discontinuities in Coefficients by Two-stage Physics-Informed Deep Learning and Statistical Mixture Models
Oct 16, 2025Zhikun Zhang, Guanyu Pan, Xiangjun Wang +2Bayesian Mixture Models Physics-Informed ML
School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an 710072, China · School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, China · SandGold AI Research, Guangzhou 510642, China +3
Physics-Constrained Fine-Tuning of Flow-Matching Models for Generation and Inverse Problems
Aug 5, 2025Jan Tauberschmidt, Sophie Fellenz, Sebastian J. Vollmer +1Physics-Informed Generative Modeling Flow Matching
DSA, German Research Center for Artificial Intelligence (DFKI) · Department of Computer Science, University of Kaiserslautern–Landau (RPTU) · Department of Mathematics, Imperial College London
Counterfactual Operator Relevance for PDE Discovery: Screening, Pruning, and Identifiability
Jun 25, 2025Ronald KatendeCounterfactual Evaluation PDE Discovery
Department of Mathematics Kabale University Kikungiri Hill, Katuna Road, 317, Kabale, Uganda
EquiReg: Equivariance Regularized Diffusion for Inverse Problems
May 29, 2025Bahareh Tolooshams, Aditi Chandrashekar, Rayhan Zirvi +4Diffusion Models for Image Restoration Diffusion-Based Inverse Problems
University of Alberta · Alberta Machine, Intelligence Institute (Amii) · Neuroscience and Mental Health Institute, University of Alberta +3
Physics-Informed Distillation of Diffusion Models for PDE-Constrained Generation
May 28, 2025Yi Zhang, Peng Wang, Difan ZouPhysics-Informed Diffusion Models Diffusion Model Distillation
Institute of Data Science · Department of Computer and Information Science · Institute of Data Science & School of Computing and Data Science
Gaussian process policy iteration with additive Schwarz acceleration for forward and inverse HJB and mean field game problems
May 1, 2025Xianjin Yang, Jingguo ZhangMean Field Games Stochastic Optimal Control
Department of Computing and Mathematical Sciences, California Institute of Technology, CA, USA. · Department of Mathematics and Risk Management Institute, National University of Singapore, Singapore.

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Hierarchical Inference and Closure Learning via Adaptive Surrogates for ODEs and PDEs

Pengyu Zhang, Arnaud Vadeboncoeur, Alex Glyn-Davies, Mark Girolami

Department of Engineering, University of Cambrdige, 7a JJ Thomson Ave, Cambridge, CB3 0FA, UK · The Alan Turing Institute, 96 Euston Rd., London, NW1 2DB, UK

PDE Inverse Problems PDE Surrogate Modeling

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Inverse problems are the task of calibrating models to match data. They play a pivotal role in diverse engineering applications by allowing practitioners to align models with reality. In many applications, engineers and scientists do not have a complete picture of i) the detailed properties of a system (such as material properties, geometry, initial conditions, etc.); ii) the complete laws describing all dynamics at play (such as friction laws, complicated damping phenomena, and general nonlinear interactions). In this paper, we develop a principled methodology for leveraging data from collections of distinct yet related physical systems to jointly estimate the individual model parameters of each system, and learn the shared unknown dynamics in the form of an ML-based closure model. To robustly infer the unknown parameters for each system, we employ a hierarchical Bayesian framework, which allows for the joint inference of multiple systems and their population-level statistics. To learn the closures, we use a maximum marginal likelihood estimate of a neural network embeded within the ODE/PDE formulation of the problem. To realize this framework we utilize the ensemble Metropolis-Adjusted Langevin Algorithm (MALA) for stable and efficient sampling. To mitigate the computational bottleneck of repetitive forward evaluations in solving inverse problems, we introduce a bilevel optimization strategy to simultaneously train a surrogate forward model alongside the inference. Within this framework, we evaluate and compare distinct surrogate architectures, specifically Fourier Neural Operators (FNO) and parametric Physics-Informed Neural Network (PINNs).