Numerical Solution of Inverse Problems by Weak Adversarial Networks

Abstract

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.

PFNN-2: A Domain Decomposed Penalty-Free Neural Network Method for Solving Partial Differential Equations

“…It is also possible to take compactly supported piecewise linear functions to construct the test space with low cost [31], which however requires a grid to arrange the locations of the test functions and is again hard to adapt to complex geometries. In some recent works [34,66], an adversarial network was introduced to evaluate the maximum weighted loss in an adaptive manner, but the method is often hard to converge because of the difficulty in balancing the efforts devoted to training the generative network for the approximate solution and the adversarial network for the test function.…”

hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition

“…Other formulations based on the variational form of the problem have been developed. VarNet [26,33] takes the test functions to be the piece-wise linear shape functions of finite element method, D3M [8] formulation includes the reformulation of problem (1.1)-(1.2) into a system of first-order equations, and WAN [34,35] develops an adversarial framework by taking the test function to be a separate network.…”