McMg is a phase-space multi-channel multigrid preconditioner that maps residuals to corrections while retaining unresolved wave information in extra channels, showing fewer iterations and lower runtime than classical and neural baselines on high-wavenumber 3D Helmholtz problems.
Neural Preconditioned Born Series: A Metric-Matched Framework for Learning-based Preconditioners
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abstract
High-frequency Helmholtz problems in heterogeneous media remain challenging for both classical iterative methods and end-to-end neural PDE solvers. We propose Neural Preconditioned Born Series (NPBS), a learned iterative preconditioning framework that operates in preconditioned residual coordinates induced by the Convergent Born Series (CBS). Existing learned Born-series methods primarily use Born-style unrolling for forward wavefield prediction, while learned Helmholtz preconditioners are usually formulated in physical residual coordinates. NPBS fills this gap by recasting Born-series iteration as shifted-Laplacian left preconditioning, and replacing the CBS preconditioner with a learned residual-to-correction map in the Born-preconditioned coordinates. The left preconditioner further induces a residual metric, which yields a metric-matched training objective that aligns optimization with the preconditioned geometry used at inference. On heterogeneous Helmholtz benchmarks, metric-matched NPBS reduces iteration counts by up to $1.9\times$ over direct residual learning, with gains increasing from $1.2\times$ to $1.9\times$ as the wavenumber rises. Compared to classical CBS, learned NPBS reduces stationary iteration counts by over $20\times$; when used as a preconditioner for FGMRES, it further achieves the lowest wall-clock time among all evaluated methods. The same metric-matched formulation also improves convergence on convection--diffusion--reaction systems and Newton linear systems for nonlinear PDEs, indicating that residual-metric matching is a general design principle for neural preconditioners.
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math.NA 1years
2026 1verdicts
UNVERDICTED 1representative citing papers
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McMg: A Learned Phase-Space Multi-channel Multigrid Preconditioner for Helmholtz Equation
McMg is a phase-space multi-channel multigrid preconditioner that maps residuals to corrections while retaining unresolved wave information in extra channels, showing fewer iterations and lower runtime than classical and neural baselines on high-wavenumber 3D Helmholtz problems.