McMg: A Learned Phase-Space Multi-channel Multigrid Preconditioner for Helmholtz Equation

2026-06-29

AI Summary

This work addresses the challenge of efficiently solving high-wavenumber heterogeneous Helmholtz equations, which are hindered by the indefiniteness of discretized operators and severe phase pollution. Conventional multigrid methods fail because scalar coarsening discards essential local wave information. To overcome this, the authors propose a Multi-channel Multigrid (McMG) preconditioner that preserves unresolved local wave structures during spatial coarsening by representing amplitude, phase, propagation direction, and scattering characteristics through multiple channels. The method integrates a learnable Green’s function, a wavespeed-aware medium-dependent smoother, linear multi-channel transfer operators, and locally adaptive stencils, complemented by a Layer-wise Progressive Fine-tuning (LLPF) strategy to enhance cross-scale generalization. Numerical experiments on large-scale three-dimensional problems with high frequencies and high contrast demonstrate significantly reduced iteration counts and runtime, outperforming both classical and state-of-the-art neural preconditioners.

Abstract

Solving heterogeneous Helmholtz equations at high wavenumbers remains challenging because the discretized operator is indefinite, pollution degrades phase accuracy, and scalar coarse-grid correction can discard the local phase and propagation-direction information carried by oscillatory errors. We propose Multi-channel Multigrid (McMg), a learned phase-space multigrid preconditioner for heterogeneous Helmholtz equations. Rather than predicting the solution directly, McMg maps residuals to corrections within an iterative framework. Its central idea is to coarsen physical space while retaining unresolved local wave information in the channel dimension: each coarse node carries a learned packet of amplitude, phase, direction, and scattering coefficients rather than a single scalar unknown. The architecture combines linear multi-channel transfer operators with locally adaptive stencils, neural PDE operators, and medium-dependent smoothers whose coefficients are generated from the wave speed. For a fixed medium, the V-cycle is linear in the residual; nonlinear physical features are computed once in a setup phase and cached, so each online iteration reduces to convolutions with fixed coefficients. We further study generalization across scales. Models trained on small domains transfer directly to larger domains and higher effective wavenumbers, and a Layer-by-Layer Progressive Finetuning (LLPF) strategy extends the support of the learned Green's operator by adding and finetuning only new coarse levels. Numerical experiments on high-frequency, high-contrast, and large-scale three-dimensional problems demonstrate that McMg requires substantially fewer iterations and less wall-clock time than strong classical baselines, while consistently outperforming existing neural preconditioners.

Problem

• Helmholtz equation
• high wavenumber
• phase accuracy
• multigrid preconditioner
• oscillatory errors

Innovation

• learned preconditioner
• multi-channel multigrid
• phase-space representation
• neural PDE operators
• cross-scale generalization