(An) overlapping domain decomposition method using deep operator network

Abstract

The overlapping domain decomposition method is known to be an effective way to solve partial differential equations. The method splits the domain into small subdomains, solves the subproblems defined in the subdomains in parallel, and brings the solutions together to update the whole domain solution. In this thesis, we use the deep operator network as a solver for subproblems. This deep operator network is trained to approximate an operator which takes boundary values as input in a given domain and outputs a solution of a homogeneous equation inside the domain. The deep operator network allows the solution of the corresponding homogeneous problem to be obtained even if the boundary values of the subdomain change during the iteration steps of the domain decomposition method. By combining a finite difference method and the deep operator network, we design a very fast overlapping domain decomposition method and numerically check the strong and weak scalabilities of our method.