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

2026-03-01

AI Summary

This work proposes a neural multigrid method to address the challenge that classical smoothers in traditional multigrid algorithms struggle to effectively suppress high-frequency errors when solving ill-conditioned linear systems arising from the discretization of integral equations. By integrating neural operators with the spectral decomposition principle of multigrid, the approach replaces conventional smoothers with a neural smoother trained offline. A hierarchical loss function and a spectral filtering mechanism are designed to enable each grid level to target specific high-frequency error components. The method requires no retraining for new right-hand sides and demonstrates superior convergence efficiency and robustness over classical solvers across varying problem sizes and regularization parameters. Furthermore, it exhibits strong generalizability to broader contexts, including partial differential equations.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

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.

Problem

• integral equations
• multigrid method
• ill-conditioned linear systems
• high-frequency error
• smoothers

Innovation

• neural multigrid
• learned smoothers
• spectral filtering
• level-wise training
• integral equations