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

TL;DR

The paper addresses the computational challenge of high-frequency open-domain wave propagation governed by the Helmholtz equation $-\Delta u - k^2 u = g$ by proposing Finite Basis PINNs (FBPINNs) with Perfectly Matched Layer (PML) absorption. The global solution is built from overlapping local networks via the ansatz $\tilde u = \sum_j \varphi_j v_j$ and trained with the PML-augmented operator $\mathcal{D}_{\text{PML}}$, minimizing $L(\boldsymbol{\theta}) = \frac{1}{N} \sum_i (\mathcal{D}_{\text{PML}}[\tilde u] - g)^2$. Numerical results show that FBPINNs yield lower $L^2$ errors than standard PINNs at higher wavenumbers, while the Energy Natural Gradient Descent (ENGD) optimizer improves convergence stability at the cost of slower training due to the energy matrix $G_E(\theta) = \nabla_\theta L(\theta)^T \nabla_\theta L(\theta)$. The framework demonstrates a mesh-free, domain-decomposed approach to scalable open-domain wave simulations, with potential extensions to multilevel FBPINNs and internal absorbing layers for more complex media.

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.

Figures (3)

• Figure 1: Schematic illustration of the computational domains for PINNs (left) and FBPINNs (right) with PML. The blue rectangle denotes the physical domain $\Omega$ where we want to find the solution. The red line indicates the outer boundary of the computational domain, including the PML region. The PML is thus located between the red and blue boundaries and ensures wave absorption. In the FBPINNs case (right), the physical domain together with the PML region is further divided into four overlapping subdomains ${\{\Omega_j\}_{j \in \{1, \dots, 4\}}}$.
• Figure 2: Normalized physics loss during training for Adam + L-BFGS (left) and ENGD (right). Rows correspond to different $k$ and curves compare PINNs vs FBPINNs across PML thicknesses.
• Figure 3: Approximation and reference solutions of the real and imaginary parts for the Helmholtz problem with $k=4.51$ comparing ADAM+L-BFGS and ENGD optimization methods for PINNs and FBPINNs.