Deep Interface Alternation Method (DIAM) based on domain decomposition for solving elliptic interface problems

Abstract

The interface problem is highly challenging due to its non-smoothness, discontinuity, and interface complexity. In this paper, a new and simple Deep Interface Alternation Method (DIAM) is developed to solve elliptic interface problems to avoid dealing with interfaces. It combines the ideas of domain decomposition methods and deep learning methods. Specifically, we first transform the interface problem with discontinuous derivatives into multiple continuous subproblems based on the Dirichlet–Dirichlet algorithm of domain decomposition. Then, we establish different fully connected neural networks for each subproblem to approximate parallelly the continuous solutions in the subdomain. The interface information is especially exchanged among the different loss functions of each subdomain neural network while minimizing the loss functions of each subdomain separately to obtain solutions to the entire interface problem. Numerical experiments were conducted on two-dimensional and three-dimensional elliptical interface problems with different coefficient contrasts and interface complexity. The results indicate that the Deep Interface Alternation Method has effectiveness and accuracy.