Numerical solution of inverse problems by weak adversarial networks

Abstract

Abstract In this paper, a weak adversarial network approach is developed to numerically solve a class of inverse problems, including electrical impedance tomography and dynamic electrical impedance tomography problems. The weak formulation of the partial differential equation for the given inverse problem is leveraged, where the solution and the test function are parameterized as deep neural networks. Then, 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. Theoretical justifications are provided on the convergence of the proposed algorithm. The proposed 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 this approach.