math.NANov 19, 2025

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

Authors: Victorita Dolean, Daria Hrebenshchykova, Stéphane Lanteri, Victor Michel-Dansac

Organizations: Department of Mathematics and Computer Science, Eindhoven University of Technology, The Netherlands · Atlantis project-team, Inria Research Center at Université Côte d’Azur, France · MACARON project-team, Université de Strasbourg, CNRS, Inria, IRMA, France

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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 & tables

Figure 1: Schematic illustration of the computational domains for PINNs (left) and FBPINNs (right) with PML. The blue rectangle denotes the physical domain Ω\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 {Ωj}j∈{1,…,4}\smash{\{\Omega_{j}\}_{j\in\{1,\dots,4\}}} .
(a) Adam + L-BFGS optimizer Method kk 0.25 λ\lambda 0.5 λ\lambda 1.0 λ\lambda PINNs 0.57 5.6e–2 1.6e–2 4.0e–3 1.59 4.0e–1 2.9e–2 2.2e–2 4.51 9.1e–1 9.7e–1 9.7e–1 FBPINNs 0.57 4.1e–2 1.5e–2 7.3e–3 1.59 3.0e–1 1.6e–2 1.1e–2 4.51 9.3e–1 9.3e–1 9.5e–1(b) ENGD optimizer Method kk 0.25 λ\lambda 0.5 λ\lambda 1.0 λ\lambda PINNs 0.57 1.7e–1 1.0e–2 5.4e–3 1.59 6.7e–1 1.8e–2 4.9e–3 4.51 6.7e–1 3.0e–1 1.4e–2 FBPINNs 0.57 7.3e–1 6.1e–3 3.3e–3 1.59 8.7e–1 1.5e–2 4.1e–3 4.51 6.4e–1 1.7e–1 1.1e–2
Table 1: Relative L2L_{2} error between real part of approximate and reference solutions for different LPMLL_{\mathrm{PML}} , comparing PINNs and FBPINNs performance using (a) Adam + L-BFGS and (b) ENGD.
(a) Adam + L-BFGS optimizer Method kk 0.25 λ\lambda 0.5 λ\lambda 1.0 λ\lambda PINNs 0.57 1.0e–1 2.0e–2 6.2e–3 1.59 6.4e–1 2.7e–1 8.8e–1 4.51 1.0e+0 1.0e+0 1.0e+0 FBPINNs 0.57 5.6e–1 2.7e–2 1.8e–3 1.59 5.1e–1 5.9e–2 3.3e–2 4.51 1.0e+0 1.0e+0 1.0e+0(b) ENGD optimizer Method kk 0.25 λ\lambda 0.5 λ\lambda 1.0 λ\lambda PINNs 0.57 2.7e–1 1.4e–2 1.8e–3 1.59 8.5e–1 2.4e–2 1.4e–3 4.51 9.0e–1 4.7e–1 8.2e–2 FBPINNs 0.57 5.9e–1 1.3e–2 1.3e–3 1.59 6.1e–1 2.0e–2 1.3e–3 4.51 9.0e–1 3.7e–1 1.6e–2
Table 2: Relative L2L_{2} error between imaginary part of approximate and reference solutions for different LPMLL_{\mathrm{PML}} , comparing PINNs and FBPINNs performance using (a) Adam + L-BFGS and (b) ENGD.
Figure 2: Normalized physics loss during training for Adam + L-BFGS (left) and ENGD (right). Rows correspond to different kk 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=4.51k=4.51 comparing Adam+L-BFGS and ENGD optimization methods for PINNs and FBPINNs.

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