Multilevel domain decomposition-based architectures for physics-informed neural networks

TL;DR

This paper introduces multilevel domain decomposition architectures for physics-informed neural networks, significantly enhancing their ability to solve high-frequency and multi-scale differential equation problems by improving accuracy and scalability.

Contribution

It extends FBPINNs with multiple domain decomposition levels inspired by classical multilevel Schwarz methods, improving performance and accuracy in complex problems.

Findings

Multilevel FBPINNs outperform standard PINNs and FBPINNs in high-frequency, multi-scale problems.

Multilevel FBPINNs improve accuracy with many subdomains by enhancing global communication.

Numerical tests demonstrate better scalability and solution quality for multilevel FBPINNs.

Abstract

Physics-informed neural networks (PINNs) are a powerful approach for solving problems involving differential equations, yet they often struggle to solve problems with high frequency and/or multi-scale solutions. Finite basis physics-informed neural networks (FBPINNs) improve the performance of PINNs in this regime by combining them with an overlapping domain decomposition approach. In this work, FBPINNs are extended by adding multiple levels of domain decompositions to their solution ansatz, inspired by classical multilevel Schwarz domain decomposition methods (DDMs). Analogous to typical tests for classical DDMs, we assess how the accuracy of PINNs, FBPINNs and multilevel FBPINNs scale with respect to computational effort and solution complexity by carrying out strong and weak scaling tests. Our numerical results show that the proposed multilevel FBPINNs consistently and significantly outperform PINNs across a range of problems with high frequency and multi-scale solutions. Furthermore, as expected in classical DDMs, we show that multilevel FBPINNs improve the accuracy of FBPINNs when using large numbers of subdomains by aiding global communication between subdomains.