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Born-Series-Inspired Residual Metric for Learning-based Preconditioners

Juntao Wang, Xinliang Liu, Jiwei Jia

Abstract

Loss functions for learning-based PDE preconditioners implicitly choose a \emph{metric} in which residuals are matched, yet most approaches still optimize an unpreconditioned Euclidean residual norm. For indefinite operators such as the high-frequency Helmholtz equation, this default metric can make both learning and iterative correction overly sensitive to near-resonant spectral components, while classical preconditioning succeeds precisely by reshaping the residual geometry. We show that the Born Series and shifted-Laplacian left preconditioning are linked by the identity $ I-G_ηV_η= G_ηA = L_η^{-1}A, $ which turns the reference Green operator $G_η$ into a natural Riesz-map residual metric $ R_η= G_η^\ast G_η$ and suggests measuring the physical residual via $ \|r\|_{R_η}=\|G_ηr\|_2. $ Building on this viewpoint, we propose a \emph{Neural Preconditioned Born Series} (NPBS) iteration that replaces the scalar CBS relaxation with a residual-driven neural operator, together with a metric-matched Born-series-inspired loss $\mathcal{L}_{\mathrm{bs}}^{R_η}$. The framework is architecture-agnostic and supports fast $\mathcal{O}(N\log N)$ evaluation via FFT/DST/DCT. Numerical experiments on heterogeneous Helmholtz problems demonstrate the effectiveness of our method, and its advantage becomes more pronounced as the systems grow more ill-conditioned; we then extend the framework to other PDE classes, including convection--diffusion--reaction equations and linearized Newton systems for nonlinear PDEs, where it also yields substantial iteration reductions.

Born-Series-Inspired Residual Metric for Learning-based Preconditioners

Abstract

Loss functions for learning-based PDE preconditioners implicitly choose a \emph{metric} in which residuals are matched, yet most approaches still optimize an unpreconditioned Euclidean residual norm. For indefinite operators such as the high-frequency Helmholtz equation, this default metric can make both learning and iterative correction overly sensitive to near-resonant spectral components, while classical preconditioning succeeds precisely by reshaping the residual geometry. We show that the Born Series and shifted-Laplacian left preconditioning are linked by the identity which turns the reference Green operator into a natural Riesz-map residual metric and suggests measuring the physical residual via Building on this viewpoint, we propose a \emph{Neural Preconditioned Born Series} (NPBS) iteration that replaces the scalar CBS relaxation with a residual-driven neural operator, together with a metric-matched Born-series-inspired loss . The framework is architecture-agnostic and supports fast evaluation via FFT/DST/DCT. Numerical experiments on heterogeneous Helmholtz problems demonstrate the effectiveness of our method, and its advantage becomes more pronounced as the systems grow more ill-conditioned; we then extend the framework to other PDE classes, including convection--diffusion--reaction equations and linearized Newton systems for nonlinear PDEs, where it also yields substantial iteration reductions.
Paper Structure (41 sections, 3 theorems, 82 equations, 7 figures, 5 tables)

This paper contains 41 sections, 3 theorems, 82 equations, 7 figures, 5 tables.

Table of Contents

  1. Introduction
  2. From Born Series to Preconditioned Coordinates
  3. Operator Splitting and Integral Equation
  4. Discrete realization.
  5. Operator splitting and Lippmann--Schwinger form.
  6. Key Identity: Integral Residual = Shifted-Laplacian Left-Preconditioned Residual
  7. CBS as Richardson Iteration in Preconditioned Coordinates
  8. Role of $\eta$ in metric and iteration coordinates.
  9. Neural Preconditioned Born Series Method
  10. Direct-residual neural iteration and training loss
  11. Neural Preconditioned Born Series (NPBS) iteration
  12. Efficient integral-form implementation.
  13. Equivalent left-preconditioned (differential) form.
  14. Born-series-inspired loss as a preconditioned residual metric
  15. Why this mitigates training difficulty.
  16. ...and 26 more sections

Key Result

Proposition 2.1

Let the discrete Helmholtz operator admit the split $A=L_\eta - V_\eta$, and assume $L_\eta$ is invertible so that $G_\eta := L_\eta^{-1}$ is well-defined. Then Hence eq:LS_discrete is algebraically identical to the left-preconditioned system Moreover, if $\|G_\eta V_\eta\|_2\le \rho<1$, then and

Figures (7)

  • Figure 1: Representative heterogeneous velocity models from the OpenFWI datasets. Top row: CurveVel-A, featuring relatively smooth, curved velocity transitions. Bottom row: CurveFault-B, characterized by sharp discontinuities, strong contrasts, and complex fault structures.
  • Figure 2: Iteration comparison of different stratgies on a representative instance from the OpenFWI CurveFault-B dataset.
  • Figure 3: CDR dataset samples: diffusion $\kappa$, velocity field $\mathbf{v}$, and source term $f$.
  • Figure 4: CDR residual histories on the synthetic dataset: Direct+$\mathcal{L}_{\mathrm{dir}}$ versus NPBS+$\mathcal{L}_{\mathrm{bs}}^{\ell_2}$ and NPBS+$\mathcal{L}_{\mathrm{bs}}^{R_\eta}$.
  • Figure 5: Multiple solutions of the nonlinear PDE under different initial guesses.
  • ...and 2 more figures

Theorems & Definitions (7)

  • Proposition 2.1: Shifted-preconditioned equivalence and spectral motivation
  • proof
  • Proposition 3.1: Riesz-map equivalence
  • proof
  • Remark 3.2: Training/inference consistency
  • Proposition B.1: Transform-diagonalizable Green operators
  • proof : Constructive verification