Solving High Dimensional Partial Differential Equations Using Tensor Neural Network and A Posteriori Error Estimators

TL;DR

This paper introduces a novel machine learning approach combining tensor neural networks and a posteriori error estimators to efficiently and accurately solve high-dimensional boundary value and eigenvalue problems for elliptic operators.

Contribution

It proposes a new method integrating tensor neural networks with a posteriori error estimation for high-dimensional PDEs, enhancing accuracy and computational efficiency.

Findings

High accuracy in high-dimensional integrations achieved

Adaptive loss function improves solution precision

Numerical examples validate the method's effectiveness

Abstract

In this paper, based on the combination of tensor neural network and a posteriori error estimator, a novel type of machine learning method is proposed to solve high-dimensional boundary value problems with homogeneous and non-homogeneous Dirichlet or Neumann type of boundary conditions and eigenvalue problems of the second-order elliptic operator. The most important advantage of the tensor neural network is that the high dimensional integrations of tensor neural networks can be computed with high accuracy and high efficiency. Based on this advantage and the theory of a posteriori error estimation, the a posteriori error estimator is adopted to design the loss function to optimize the network parameters adaptively. The applications of tensor neural network and the a posteriori error estimator improve the accuracy of the corresponding machine learning method. The theoretical analysis and numerical examples are provided to validate the proposed methods.