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A level-wise training scheme for learning neural multigrid smoothers with application to integral equations

Lingfeng Li, Yin King Chu, Raymond Chan, Justin Wan

TL;DR

This work proposes a novel neural multigrid scheme where learned neural operators replace classical smoothers, and this framework is generalizable to all kinds of problems, including PDE problems.

Abstract

Convolution-type integral equations commonly occur in signal processing and image processing. Discretizing these equations yields large and ill-conditioned linear systems. While the classic multigrid method is effective for solving linear systems derived from partial differential equations (PDE) problems, it fails to solve integral equations because its smoothers, which are implemented as conventional relaxation methods, are ineffective in reducing high-frequency components in the errors. We propose a novel neural multigrid scheme where learned neural operators replace classical smoothers. Unlike classical smoothers, these operators are trained offline. Once trained, the neural smoothers generalize to new right-hand-side vectors without retraining, making it an efficient solver. We design level-wise loss functions incorporating spectral filtering to emulate the multigrid frequency decomposition principle, ensuring each operator focuses on solving distinct high-frequency spectral bands. Although we focus on integral equations, the framework is generalizable to all kinds of problems, including PDE problems. Our experiments demonstrate superior efficiency over classical solvers and robust convergence across varying problem sizes and regularization weights.

A level-wise training scheme for learning neural multigrid smoothers with application to integral equations

TL;DR

This work proposes a novel neural multigrid scheme where learned neural operators replace classical smoothers, and this framework is generalizable to all kinds of problems, including PDE problems.

Abstract

Convolution-type integral equations commonly occur in signal processing and image processing. Discretizing these equations yields large and ill-conditioned linear systems. While the classic multigrid method is effective for solving linear systems derived from partial differential equations (PDE) problems, it fails to solve integral equations because its smoothers, which are implemented as conventional relaxation methods, are ineffective in reducing high-frequency components in the errors. We propose a novel neural multigrid scheme where learned neural operators replace classical smoothers. Unlike classical smoothers, these operators are trained offline. Once trained, the neural smoothers generalize to new right-hand-side vectors without retraining, making it an efficient solver. We design level-wise loss functions incorporating spectral filtering to emulate the multigrid frequency decomposition principle, ensuring each operator focuses on solving distinct high-frequency spectral bands. Although we focus on integral equations, the framework is generalizable to all kinds of problems, including PDE problems. Our experiments demonstrate superior efficiency over classical solvers and robust convergence across varying problem sizes and regularization weights.
Paper Structure (19 sections, 41 equations, 10 figures, 6 tables, 5 algorithms)

This paper contains 19 sections, 41 equations, 10 figures, 6 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Convolution-type integral equations
  4. Classical relaxation methods
  5. Multigrid method
  6. The neural multigrid method
  7. Neural operators
  8. Neural multigrid method for one-dimensional integral equation
  9. Neural multigrid method for two-dimensional integral equations
  10. Training strategy
  11. Numerical experiments
  12. One-dimensional problems
  13. Integral equation with the Tikhonov regularization
  14. Integral equation with the anisotropic regularization
  15. Second-order PDE
  16. ...and 4 more sections

Figures (10)

  • Figure 1: Plot of $1-\mu_\phi$ for $\phi\in[-\pi,\pi]$ when $\omega=1/2$.
  • Figure 2: Eigenvectors corresponding to different eigenvalues of the linear system discretized from the Poisson equation (first row) and the Integral equation (second row).
  • Figure 3: Error vectors when solving different linear systems using the Jacobi method.
  • Figure 4: Graph illustration of a V-cycle multigrid scheme with $L=3$
  • Figure 5: Explanation of the frequency domain decomposition (L=3).
  • ...and 5 more figures