Computing Multi-Eigenpairs of High-Dimensional Eigenvalue Problems Using Tensor Neural Networks

TL;DR

This paper introduces a tensor neural network approach combined with the deep Ritz method to accurately compute multiple eigenpairs of high-dimensional eigenvalue problems without relying on Monte Carlo methods.

Contribution

It presents a novel tensor neural network-based machine learning method for high-dimensional eigenvalue problems, avoiding Monte Carlo procedures and improving integration accuracy.

Findings

High accuracy in high-dimensional integrations achieved

Effective computation of multiple eigenpairs demonstrated

Method validated with extensive numerical examples

Abstract

In this paper, we propose a type of tensor-neural-network-based machine learning method to compute multi-eigenpairs of high dimensional eigenvalue problems without Monte-Carlo procedure. Solving multi-eigenvalues and their corresponding eigenfunctions is one of the basic tasks in mathematical and computational physics. With the help of tensor neural network and deep Ritz method, the high dimensional integrations included in the loss functions of the machine learning process can be computed with high accuracy. The high accuracy of high dimensional integrations can improve the accuracy of the machine learning method for computing multi-eigenpairs of high dimensional eigenvalue problems. Here, we introduce the tensor neural network and design the machine learning method for computing multi-eigenpairs of the high dimensional eigenvalue problems. The proposed numerical method is validated with plenty of numerical examples.