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

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.

💡 Research Summary

This paper investigates the use of neural‑network‑driven domain decomposition, specifically Finite Basis Physics‑Informed Neural Networks (FBPINNs), together with perfectly matched layers (PML) to solve the two‑dimensional Helmholtz equation in free space. The Helmholtz equation, −Δu−k²u=g(x), models steady‑state wave propagation and requires the Sommerfeld radiation condition at infinity. The authors replace explicit boundary conditions with a PML that absorbs outgoing waves via a complex coordinate stretching σ(x)=σ₀·max(|x|−L,0)/L_PML, leading to a modified differential operator D_PML.

In the FBPINN framework the computational domain Ω is partitioned into J overlapping subdomains {Ω_j}. Each subdomain hosts an independent neural network v_j(x;θ_j). Overlap is ensured by a ratio η>1, and a smooth cosine‑based window function φ_j(x) provides a partition of unity, guaranteeing that the global approximation ũ(x,θ)=∑j φ_j(x)v_j(x;θ_j) is continuous across interfaces without imposing explicit continuity constraints. The loss function is purely a physics residual:
L(θ)= (1/N)∑{i=1}^N |D_PML