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

In this article, we propose three kinds of neural networks inspired by power method, inverse power method and shifted inverse power method to solve linear eigenvalue problem, respectively. These neural networks share similar ideas with traditional methods, in which differential operator is realized by automatic differentiation. The eigenfunction of the eigenvalue problem is learned by the neural network and the iterations are implemented by optimizing the specially defined loss function. We examine the applicability and accuracy of our methods in the numerical experiments in one dimension, two dimensions and even higher dimensions. Numerical results show that accurate eigenvalue and eigenfunction approximations can be obtained by our methods.