Domain Decomposition Subspace Neural Network Method for Solving Linear and Nonlinear Partial Differential Equations

TL;DR

This paper introduces a novel domain decomposition subspace neural network approach that efficiently solves linear and nonlinear PDEs with high accuracy and reduced computational cost by combining neural basis functions and interface continuity conditions.

Contribution

It presents a new method integrating domain decomposition and neural networks to construct basis functions for PDE solutions, ensuring smoothness and high accuracy.

Findings

Achieves errors up to 10^{-13} in numerical experiments.

Reduces computational costs compared to PINNs, DGM, DRM.

Demonstrates superior accuracy and training efficiency.

Abstract

This paper proposes a domain decomposition subspace neural network method for efficiently solving linear and nonlinear partial differential equations. By combining the principles of domain decomposition and subspace neural networks, the method constructs basis functions using neural networks to approximate PDE solutions. It imposes $C^k$ continuity conditions at the interface of subdomains, ensuring smoothness across the global solution. Nonlinear PDEs are solved using Picard and Newton iterations, analogous to classical methods. Numerical experiments demonstrate that our method achieves exceptionally high accuracy, with errors reaching up to $10^{-13}$, while significantly reducing computational costs compared to existing approaches, including PINNs, DGM, DRM. The results highlight the method's superior accuracy and training efficiency.