D3M: A deep domain decomposition method for partial differential equations

TL;DR

This paper introduces D3M, a deep domain decomposition method leveraging variational principles and neural networks to efficiently solve PDEs, with proven convergence and applicability to large-scale engineering problems.

Contribution

It develops a systematic deep learning framework for domain decomposition of PDEs, integrating variational principles and multi-fidelity neural networks.

Findings

D3M converges to the exact PDE solution.

Framework demonstrates high accuracy in numerical experiments.

Efficiently handles large-scale PDE problems.

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.