English

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

Numerical Analysis 2026-05-18 v4 Numerical Analysis

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 1.9×1.9\times over direct residual learning, with gains increasing from 1.2×1.2\times to 1.9×1.9\times as the wavenumber rises. Compared to classical CBS, learned NPBS reduces stationary iteration counts by over 20×20\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.

Cite

@article{arxiv.2603.18527,
  title  = {Neural Preconditioned Born Series: A Metric-Matched Framework for Learning-based Preconditioners},
  author = {Juntao Wang and Jiwei Jia and Xinliang Liu},
  journal= {arXiv preprint arXiv:2603.18527},
  year   = {2026}
}

Comments

22 pages, 7 figures

R2 v1 2026-07-01T11:27:31.625Z