D3M: A Deep Domain Decomposition Method for Partial Differential Equations

TLDR

The authors propose D3M, a variational deep domain decomposition method for solving partial differential equations. They formulate PDEs as constrained optimization problems and solve them with a multi‑fidelity neural network framework that implements domain decomposition in parallel. The method converges to the exact PDE solution and demonstrates accurate, efficient performance in numerical experiments, laying groundwork for large‑scale engineering applications.

Abstract

A state-of-the-art deep domain decomposition method (D3M) based on the variational principle is proposed for partial differential equations (PDEs). The solution of PDEs can be formulated as the solution of a constrained optimization problem, and we design a multi-fidelity neural network framework to solve this optimization problem. Our contribution is to develop a systematical computational procedure for the underlying problem in parallel with domain decomposition. Our analysis shows that the D3M approximation solution converges to the exact solution of underlying PDEs. Our proposed framework establishes a foundation to use variational deep learning in large-scale engineering problems and designs. We present a general mathematical framework of D3M, validate its accuracy and demonstrate its efficiency with numerical experiments.