Neural network augmented inverse problems for PDEs

In this paper we show how to augment classical methods for inverse problems with artificial neural networks. The neural network acts as a parametric container for the coefficient to be estimated fr ...

stat.ML math.NA Computer Science Mathematics Engineering Physics

View graph Source

Paper

References (39)

01A unified deep artificial neural network approach to partial differential equations in complex geometriesJens Berg, K. Nyström2017 · Neurocomputing · 685 citations In Library

02Numerical Gaussian Processes for Time-Dependent and Nonlinear Partial Differential EquationsM. Raissi, P. Perdikaris, G. Karniadakis2017 · SIAM Journal on Scientific Computing · 316 citations In Library

03Bayesian Probabilistic Numerical MethodsJ. Cockayne, C. Oates, T. Sullivan et al.2017 · SIAM Review · 185 citations

04Machine learning of linear differential equations using Gaussian processesM. Raissi, P. Perdikaris, G. Karniadakis2017 · Journal of Computational Physics · 654 citations In Library

05An iteration count estimate for a mesh-dependent steepest descent method based on finite elements and Riesz inner product representationTobias Schwedes, S. Funke, D. Ham2016 · 9 citations

06Probabilistic Meshless Methods for Partial Differential Equations and Bayesian Inverse ProblemsJ. Cockayne, C. Oates, T. Sullivan et al.2016 · arXiv.org · 47 citations

07The FEniCS Project Version 1.5M. Alnæs, Jan Blechta, J. Hake et al.2015 · 2.5k citations

08Probabilistic numerics and uncertainty in computationsPhilipp Hennig, Michael A. Osborne, M. Girolami2015 · Proceedings of the Royal Society A · 334 citations In Library

09Solving large-scale PDE-constrained Bayesian inverse problems with Riemann manifold Hamiltonian Monte CarloT. Bui-Thanh, M. Girolami2014 · 79 citations

10Finite element methods for coefficient identification in an elliptic equationD. Hào, Tran Nhan Tam Quyen2014 · 7 citations

11Fast Probabilistic Optimization from Noisy GradientsPhilipp Hennig2013 · International Conference on Machine Learning · 35 citations

12Improving neural networks by preventing co-adaptation of feature detectorsGeoffrey E. Hinton, Nitish Srivastava, A. Krizhevsky et al.2012 · arXiv.org · 8.0k citations In Library

Scroll for more · 27 remaining

Similar papers

Neural networks as smooth priors for inverse problems for PDEsJ. Berg, K. Nyström2021 · Journal of Computational Mathematics and Data Science · 28 citations In Library

The Neural Network Approach to Inverse Problems in Differential EquationsKailai Xu, Eric F Darve2019 · arXiv: Numerical Analysis · 34 citations In Library

Inverse Evolution Data Augmentation for Neural PDE SolversChaoyu Liu, Chris Budd, Carola‐Bibiane Schönlieb2025 · arXiv (Cornell University) · 1 citation In Library

Solving inverse-PDE problems with physics-aware neural networksSamira Pakravan, Pouria A. Mistani, M. Aragon-Calvo et al.2020 · Journal of Computational Physics · 67 citations In Library

BIAN: A Deep Learning Method to Solve Inverse Problems Using Only Boundary InformationChen, Feng, Li, Kegan, Meng, Yiran et al.2025 · arXiv (Cornell University) · 0 citations In Library

A Method for Computing Inverse Parametric PDE Problems with Random-Weight Neural NetworksS. Dong, Yiran Wang2022 · Journal of Computational Physics · 36 citations In Library

Solving the inverse potential problem in the parabolic equation by the deep neural networks methodMengmeng Zhang, Zhidong Zhang2023 · arXiv.org · 1 citation In Library

Solving parametric PDE problems with artificial neural networksY. Khoo, Jianfeng Lu, Lexing Ying2017 · European journal of applied mathematics · 409 citations In Library

General Covariance Data Augmentation for Neural PDE SolversV. Fanaskov, Tianchi Yu, A. Rudikov et al.2023 · International Conference on Machine Learning · 4 citations In Library

Semi-supervised Invertible Neural Operators for Bayesian Inverse ProblemsSebastian Kaltenbach, P. Perdikaris, P. Koutsourelakis2022 · Computational Mechanics · 44 citations In Library

Patents near this paper

Stochastic Realization of Parameter Inversion in Physics-Based Empirical ModelsUS 11,668,684 · LANDMARK GRAPHICS CORPORATION · 2023 NEURAL NETWORK TRAINING BY INTEGRATION OF ADJOINT SYSTEMS OF EQUATIONS FORWARD IN TIMEUS 5,930,781 · CALIFORNIA INSTITUTE OF TECHNOLOGY, THE · 1999 EFFICIENT SOLUTIONS OF INVERSE PROBLEMSUS 11,143,788 · STATOIL PETROLEUM AS · 2021 SYSTEM AND METHOD OF GLOBAL OPTIMIZATION USING ARTIFICIAL NEURAL NETWORKSUS 5,377,307 · SCHLUMBERGER TECHNOLOGY CORPORATION · 1994 NEURAL RENDERING FOR INVERSE GRAPHICS GENERATIONUS 11,494,976 · NVIDIA CORPORATION · 2022

Source of record

arXiv