Part of International Conference on Representation Learning 2025 (ICLR 2025) Conference
Yan Xie, Minrui Lv, Chen-Song Zhang
Solving high wavenumber Helmholtz equations is notoriously challenging. Traditional solvers have yet to yield satisfactory results, and most neural network methods struggle to accurately solve cases with extremely high wavenumbers within heterogeneous media. This paper presents an advanced multigrid-hierarchical AI solver, tailored specifically for high wavenumber Helmholtz equations. We adapt the MGCNN architecture to align with the problem setting and incorporate a novel Fourier neural network (FNN) to match the characteristics of Helmholtz equations. FNN, mathematically akin to the convolutional neural network (CNN), enables faster propagation of source influence during the solve phase, making it particularly suitable for handling large size, high wavenumber problems. We conduct supervised learning tests against numerous neural operator learning methods to demonstrate the superior learning capabilities of our solvers. Additionally, we perform scalability tests using an unsupervised strategy to highlight our solvers' significant speedup over the most recent specialized AI solver and AI-enhanced traditional solver for high wavenumber Helmholtz equations. We also carry out an ablation study to underscore the effectiveness of the multigrid hierarchy and the benefits of introducing FNN. Notably, our solvers exhibit optimal convergence of $\mathcal{O}(k)$ up to $k \approx 2000$.