Solving Convolution-type Integral Equations using Preconditioned Neural Operators

TL;DR

This paper introduces a hybrid neural operator-based method with preconditioning to efficiently solve convolution-type integral equations, outperforming traditional multigrid and conjugate gradient methods in speed and iterations.

Contribution

It proposes a novel preconditioning training strategy for neural operators and combines them with classical iterative methods for improved efficiency.

Findings

The hybrid algorithm outperforms multigrid in iteration count and time.

The method effectively handles large-scale, ill-conditioned systems.

The approach demonstrates superior convergence and generalization.

Abstract

Convolution-type integral equations arise from various fields, \textit{e.g.}, finite impulse response filters in signal processing and deblurring problems in image processing. When solving these equations, conventional numerical methods, like the multigrid method, can only efficiently solve the low-frequency components in the error, but not the high-frequency components. In this paper, we apply neural operators to address this issue. By adopting a preconditioning approach, we propose a novel training strategy that trains neural operators to solve the high-frequency components efficiently. Then, we combine the neural operators with some classical iterative solvers, like the weighted Jacobi method, to obtain an efficient hybrid iterative algorithm for the integral equations. We analyze the generalization error of our training strategy and the convergence of the hybrid iterative algorithm. We test our algorithms on large-scale and ill-conditioned linear systems discretized from one- and two-dimensional convolution-type integral equations. Our proposed algorithm significantly outperforms the multigrid method and the preconditioned conjugate gradient method in both iteration numbers and computational time.