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Numerical Solution of Inverse Problems by Weak Adversarial Networks

arXiv 2002.11340 ↗First posted 26 Feb 2020math.NA cs.LG cs.NAcited by 29

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Abstract (arXiv)

We consider a weak adversarial network approach to numerically solve a class of inverse problems, including electrical impedance tomography and dynamic electrical impedance tomography problems. We leverage the weak formulation of PDE in the given inverse problem, and parameterize the solution and the test function as deep neural networks. The weak formulation and the boundary conditions induce a minimax problem of a saddle function of the network parameters. As the parameters are alternatively updated, the network gradually approximates the solution of the inverse problem. We provide theoretical justifications on the convergence of the proposed algorithm. Our method is completely mesh-free without any spatial discretization, and is particularly suitable for problems with high dimensionality and low regularity on solutions. Numerical experiments on a variety of test inverse problems demonstrate the promising accuracy and efficiency of our approach.

Key facts. Cited by 29 papers in the map (data through 14 Oct 2026). First posted on arXiv 26 Feb 2020.

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Citing papers per year: 2021: 1; 2022: 10; 2023: 8; 2024: 7; 2025: 2; 2026: 1.

29 papers.

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  1. 2022Physics-Informed Machine Learning: A Survey on Problems, Methods and Applicationscited by 92
  2. 2022Deep Learning Methods for Partial Differential Equations and Related Parameter Identification Problemscited by 34
  3. 2022Partial Differential Equations Meet Deep Neural Networks: A Surveycited by 25
  4. 2022Transformer Meets Boundary Value Inverse Problemscited by 22
  5. 2021DeepParticle: learning invariant measure by a deep neural network minimizing Wasserstein distance on data generated from an interacting particle methodcited by 9
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  7. 2023Neural Control of Parametric Solutions for High-dimensional Evolution PDEscited by 5
  8. 2023Approximating High-Dimensional Minimal Surfaces with Physics-Informed Neural Networkscited by 5
  9. 2023Multi-Grade Deep Learning for Partial Differential Equations with Applications to the Burgers Equationcited by 5
  10. 2024Weak neural variational inference for solving Bayesian inverse problems without forward models: applications in elastographycited by 5
  11. 2025DGenNO: A Novel Physics-aware Neural Operator for Solving Forward and Inverse PDE Problems based on Deep, Generative Probabilistic Modelingcited by 5
  12. 2022Weak Collocation Regression method: fast reveal hidden stochastic dynamics from high-dimensional aggregate datacited by 4
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  2. 2024Self-test loss functions for learning weak-form operators and gradient flowscited by 4
  3. 2022A Deep Double Ritz Method (D2RM) for solving Partial Differential Equations using Neural Networkscited by 3
  4. 2024WANCO: Weak Adversarial Networks for Constrained Optimization problemscited by 3
  5. 2022EgPDE-Net: Building Continuous Neural Networks for Time Series Prediction with Exogenous Variablescited by 2
  6. 2022A variational neural network approach for glacier modelling with nonlinear rheologycited by 2
  7. 2023Conductivity Imaging from Internal Measurements with Mixed Least-Squares Deep Neural Networkscited by 2
  8. 2023Electrical Impedance Tomography: A Fair Comparative Study on Deep Learning and Analytic-based Approachescited by 2
  9. 2024ODE-DPS: ODE-based Diffusion Posterior Sampling for Inverse Problems in Partial Differential Equationcited by 2
  10. 2024A Natural Primal-Dual Hybrid Gradient Method for Adversarial Neural Network Training on Solving Partial Differential Equationscited by 2
  11. 2025A unified physics-informed generative operator framework for general inverse problemscited by 2
  12. 2022Bayesian Inversion with Neural Operator (BINO) for Modeling Subdiffusion: Forward and Inverse Problemscited by 1
  13. 2023Laplace-fPINNs: Laplace-based fractional physics-informed neural networks for solving forward and inverse problems of subdiffusioncited by 1
  14. 2024Physics-aware deep learning framework for the limited aperture inverse obstacle scattering problemcited by 1
  15. 2026Harnessing AI for Inverse Partial Differential Equation Problems: Past, Present, and Prospectscited by 1
  16. 2023Consensus-based adaptive sampling and approximation for high-dimensional energy landscapescited by 0
  17. 2024Approximation of Solution Operators for High-dimensional PDEscited by 0