Neural Preconditioned Born Series: A Metric-Matched Framework for Learning-based Preconditioners

TL;DR

The paper introduces Neural Preconditioned Born Series (NPBS), a novel learned preconditioning framework for high-frequency Helmholtz problems that improves convergence and reduces iteration counts by aligning training with the preconditioned geometry.

Contribution

NPBS recasts Born-series iteration as shifted-Laplacian preconditioning and introduces a metric-matched training objective, advancing neural preconditioners for PDEs.

Findings

NPBS reduces iteration counts by up to 1.9× on Helmholtz benchmarks.

Compared to classical CBS, NPBS reduces stationary iterations by over 20×.

The metric-matched formulation improves convergence on various PDE systems.

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.