---
title: "McMg: A Learned Phase-Space Multi-channel Multigrid Preconditioner for Helmholtz Equation"
source: "arXiv AI/CL/LG（论文）"
source_tier: "t2"
url: "https://arxiv.org/abs/2606.30495v1"
permalink: "https://feed.astralor.com/items/itm_18164b54a02994c163e77ccb"
published: "2026-06-29T16:01:44.000Z"
score: 59
channel: "tech"
tags: ["McMg","Helmholtz equation","multigrid","neural PDE operator"]
---

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

## At a glance

Solving heterogeneous Helmholtz equations at high wavenumbers remains challenging due to indefinite operators and phase accuracy degradation. This paper proposes McMg, a learned phase-space multigrid preconditioner. Its key 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. The architecture combines linear multi-channel transfer operators with locally adaptive stencils, neural PDE operators, and medium-dependent smoothers. The V-cycle is linear in the residual. Experiments show McMg substantially reduces iterations and wall-clock time compared to strong classical baselines and existing neural preconditioners, with cross-scale generalization ability.

## Key points

- Proposes McMg, a learned phase-space multichannel multigrid preconditioner for efficient solution of high-frequency heterogeneous Helmholtz equations.
- Key innovation: retains unresolved local wave information (amplitude, phase, direction, scattering) in channel dimension when coarsening physical space, instead of scalar unknowns.
- Architecture combines linear multi-channel transfer operators, locally adaptive stencils, neural PDE operators, and medium-dependent smoothers; V-cycle is linear in residual, enabling efficient online inference after precomputation.
- Experiments demonstrate McMg outperforms classical multigrid and existing neural preconditioners in iterations and wall-clock time, with cross-scale generalization and layer-by-layer progressive finetuning support.

## Full text

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.

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