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

arXiv: https://arxiv.org/abs/2606.30495 · HTML: https://mlsanity.com/paper/2606.30495/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 17 of its 49 references that are in the map.
- First posted on arXiv 29 Jun 2026.

## Builds on (most cited first, 17 of 17)

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- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
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- [Blending Neural Operators and Relaxation Methods in PDE Numerical Solvers](https://mlsanity.com/paper/2208.13273/) (2022)
- [OpenFWI: Large-Scale Multi-Structural Benchmark Datasets for Seismic Full Waveform Inversion](https://mlsanity.com/paper/2111.02926/) (2021)
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- [MgNO: Efficient Parameterization of Linear Operators via Multigrid](https://mlsanity.com/paper/2310.19809/) (2023)
- [A Helmholtz equation solver using unsupervised learning: Application to transcranial ultrasound](https://mlsanity.com/paper/2010.15761/) (2020)
- [Multigrid-Augmented Deep Learning Preconditioners for the Helmholtz Equation using Compact Implicit Layers](https://mlsanity.com/paper/2306.17486/) (2023)
- [Fast meta-solvers for 3D complex-shape scatterers using neural operators trained on a non-scattering problem](https://mlsanity.com/paper/2405.12380/) (2024)
- [Leveraging Operator Learning to Accelerate Convergence of the Preconditioned Conjugate Gradient Method](https://mlsanity.com/paper/2508.00101/) (2025)
- [Hybrid Iterative Solvers with Geometry-Aware Neural Preconditioners for Parametric PDEs](https://mlsanity.com/paper/2512.14596/) (2025)
- [OpenBreastUS: Benchmarking Neural Operators for Wave Imaging Using Breast Ultrasound Computed Tomography](https://mlsanity.com/paper/2507.15035/) (2025)

## Abstract (arXiv)

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 improves large-domain scalability by adding new coarse levels while finetuning only the newly introduced parameters. 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.