Neural network-driven domain decomposition for efficient solutions to the Helmholtz equation

TL;DR

This paper explores the use of neural network-based domain decomposition methods, specifically FBPINNs, to efficiently solve the Helmholtz equation, addressing computational challenges in high-frequency wave simulations in complex domains.

Contribution

It introduces and evaluates FBPINNs and multilevel extensions as innovative neural network approaches for domain decomposition in Helmholtz equation solutions.

Findings

FBPINNs effectively solve the Helmholtz equation in homogeneous cases.

The methods show potential to reduce computational costs compared to traditional techniques.

Multilevel extensions improve accuracy and efficiency in complex domains.

Abstract

Accurately simulating wave propagation is crucial in fields such as acoustics, electromagnetism, and seismic analysis. Traditional numerical methods, like finite difference and finite element approaches, are widely used to solve governing partial differential equations (PDEs) such as the Helmholtz equation. However, these methods face significant computational challenges when applied to high-frequency wave problems in complex two-dimensional domains. This work investigates Finite Basis Physics-Informed Neural Networks (FBPINNs) and their multilevel extensions as a promising alternative. These methods leverage domain decomposition, partitioning the computational domain into overlapping sub-domains, each governed by a local neural network. We assess their accuracy and computational efficiency in solving the Helmholtz equation for the homogeneous case, demonstrating their potential to mitigate the limitations of traditional approaches.