In this paper, we introduce a type of tensor neural network based machine learning method to solve elliptic multiscale problems. Based on the special structure, we can do the direct and highly accurate high dimensional integrations for the tensor neural network functions without Monte Carlo process. Here, with the help of homogenization techniques, the multiscale problem is first transformed to the high dimensional limit problem with reasonable accuracy. Then, based on the tensor neural network, we design a type of machine learning method to solve the derived high dimensional limit problem. The proposed method in this paper brings a new way to design numerical methods for computing more general multiscale problems with high accuracy. Several numerical examples are also provided to validate the accuracy of the proposed numerical methods.

翻译：本文提出了一种基于张量神经网络的机器学习方法，用于求解椭圆型多尺度问题。基于张量神经网络的特殊结构，我们能够在不依赖蒙特卡罗过程的情况下，直接且高精度地计算高维积分。借助均匀化技术，多尺度问题首先被转化为具有合理精度的高维极限问题。随后，基于张量神经网络，我们设计了一种机器学习方法来求解推导出的高维极限问题。本文提出的方法为设计高精度计算更一般多尺度问题的数值方法提供了一条新途径。文中还给出了多个数值算例，以验证所提数值方法的精度。