Neural Preconditioned Born Series: A Metric-Matched Framework for Learning-based Preconditioners
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 over direct residual learning, with gains increasing from to as the wavenumber rises. Compared to classical CBS, learned NPBS reduces stationary iteration counts by over ; 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