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

TL;DR

This paper introduces a neural multigrid method with learned smoothers for integral equations, achieving efficient, robust solutions that outperform classical methods and generalize well without retraining.

Contribution

It proposes a novel neural multigrid scheme with level-wise spectral filtering, replacing classical smoothers for integral equations, and demonstrates its effectiveness and generalizability.

Findings

Superior efficiency over classical solvers

Robust convergence across different problem sizes

Effective spectral filtering in neural smoothers

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.