Iterative algorithms for partitioned neural network approximation to partial differential equations

TL;DR

This paper introduces iterative algorithms inspired by Schwarz domain decomposition to improve convergence and parallelization in partitioned neural network methods for solving PDEs, supported by numerical experiments.

Contribution

It provides the first rigorous analysis of convergence and parallel computing benefits for partitioned neural networks solving PDEs using iterative Schwarz-based algorithms.

Findings

Enhanced convergence of partitioned neural networks

Improved parallel computing performance

Numerical results demonstrate algorithm effectiveness

Abstract

To enhance solution accuracy and training efficiency in neural network approximation to partial differential equations, partitioned neural networks can be used as a solution surrogate instead of a single large and deep neural network defined on the whole problem domain. In such a partitioned neural network approach, suitable interface conditions or subdomain boundary conditions are combined to obtain a convergent approximate solution. However, there has been no rigorous study on the convergence and parallel computing enhancement on the partitioned neural network approach. In this paper, iterative algorithms are proposed to address these issues. Our algorithms are based on classical additive Schwarz domain decomposition methods. Numerical results are included to show the performance of the proposed iterative algorithms.