Tensor Neural Network–Based Machine Learning Method for Elliptic Multiscale Problems

In this paper, we propose a tensor neural network–based machine learning method for solving elliptic multiscale problems. Leveraging the special structure of tensor neural networks, we can perform direct and highly accurate high-dimensional integration without relying on Monte Carlo methods. Within the framework of the homogenization method, the original multiscale problem is reformulated as several cell problems and a homogenized equation with reasonable accuracy. We then develop a machine learning framework, based on tensor neural networks, to solve the derived equations, especially the high-dimensional cell problems. The proposed method offers a novel approach to designing numerical algorithms for a broader class of multiscale problems with high accuracy. Several numerical examples are presented to demonstrate the effectiveness and accuracy of the proposed method.