Born-Series-Inspired Residual Metric for Learned Preconditioners

Juntao Wang, Xinliang Liu, Jiwei Jia

Keywords: Neural Preconditioner, Born-Series-Inspired, Preconditioned Loss, Helmholtz, CDR, Nonlinear PDE

TL;DR: We propose a neural preconditioned Born-Series solver and a Born-Series-inspired preconditioned loss, reducing iterations across diverse PDEs.

Abstract:

Learned PDE preconditioners typically train by minimizing an unpreconditioned residual norm, yet iterate in preconditioned coordinates at inference.
For indefinite operators such as the high-frequency Helmholtz equation, this objective--geometry mismatch impedes convergence of near-resonant error modes.
We close this gap by recasting the training objective as a \emph{preconditioned residual metric}: the standard Born-series / shifted-Laplacian relation $I-G_\eta V_\eta = G_\eta A = L_\eta^{-1}A$ shows that measuring the residual in the $G_\eta^\ast G_\eta$-weighted inner product (a Riesz-map metric) inherits the conditioning benefits of classical shifted-Laplacian preconditioning, aligning the training geometry with the inference geometry.
Building on this, we propose (i) a \emph{neural preconditioned Born-series} solver that replaces the CBS scalar correction with a learned operator, and (ii) a \emph{Born-series-inspired loss} that trains in the matched preconditioned metric.
The framework is architecture-agnostic: any learned preconditioner that is linear in the residual can be plugged in.
We validate on three PDE classes---high-frequency Helmholtz, convection--diffusion--reaction, and linearized Newton systems from nonlinear PDEs---with controlled experiments that separate the solver-level benefit of replacing CBS with a learned operator (up to $68.6\times$ fewer iterations on OpenFWI) from the pure loss-design benefit of switching to the preconditioned metric ($\sim 28$% fewer iterations on high-contrast media at fixed architecture).

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Submission Number: 117