Neural Networks Based on Power Method and Inverse Power Method for Solving Linear Eigenvalue Problems

TL;DR

This paper introduces neural networks inspired by power and inverse power methods to efficiently solve linear eigenvalue problems, learning eigenfunctions through optimization and automatic differentiation.

Contribution

It proposes novel neural network architectures based on classical iterative methods for eigenvalue problems, enabling accurate eigenvalue and eigenfunction approximation.

Findings

Effective in 1D, 2D, and higher dimensions

Accurate eigenvalues and eigenfunctions obtained

Applicable to largest, smallest, and interior eigenvalues

Abstract

In this article, we propose two kinds of neural networks inspired by power method and inverse power method to solve linear eigenvalue problems. These neural networks share similar ideas with traditional methods, in which the differential operator is realized by automatic differentiation. The eigenfunction of the eigenvalue problem is learned by the neural network and the iterative algorithms are implemented by optimizing the specially defined loss function. The largest positive eigenvalue, smallest eigenvalue and interior eigenvalues with the given prior knowledge can be solved efficiently. We examine the applicability and accuracy of our methods in the numerical experiments in one dimension, two dimensions and higher dimensions. Numerical results show that accurate eigenvalue and eigenfunction approximations can be obtained by our methods.