---
title: "McMg：用于亥姆霍兹方程的相位空间多通道多重网格预处理器"
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","亥姆霍兹方程","多重网格","神经PDE算子"]
---

# McMg：用于亥姆霍兹方程的相位空间多通道多重网格预处理器

## 速览

求解高频异质亥姆霍兹方程极具挑战，传统多重网格因忽略振荡误差的相位与传播方向而效果不佳。本文提出McMg，一种基于学习的相位空间多重网格预处理器。其核心思想是在粗化物理空间的同时，通过通道维度保留未解析的局部波信息——每个粗网格节点携带一个学习到的包，包含振幅、相位、方向和散射系数。McMg结合线性多通道转移算子、自适应模板、神经PDE算子以及介质相关平滑器，整个V-cycle为线性算子。实验表明，McMg在迭代次数和耗时上显著优于经典基线和现有神经预处理器，且支持跨尺度泛化与小域到大域的直接迁移。

## 关键点

- 提出McMg，一种基于学习的相位空间多通道多重网格预处理器，用于高效求解高频异质亥姆霍兹方程。
- 核心创新：在粗化物理空间时，通过通道维度保留振荡误差的振幅、相位、方向和散射系数，而非仅用标量未知。
- 架构结合线性多通道算子、自适应模板、神经PDE算子及介质相关平滑器，V-cycle对残差为线性，预计算固定介质后可高效在线推理。
- 实验验证：McMg在迭代次数和实际耗时上优于经典多重网格和现有神经预处理器，并支持跨尺度泛化与渐进微调扩展。

## 全文

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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引用前请核对原文;本文由星际流动整理:https://feed.astralor.com/items/itm_18164b54a02994c163e77ccb