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.

Publication:

MULTISCALE MODELING & SIMULATION

http://dx.doi.org/10.1137/24M1648338

Author:

ZHONGSHUO LIN

SKLMS, NCMIS, Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

linzhongshuo@lsec.cc.ac.cn

HAOCHEN LIU

SKLMS, NCMIS, Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

liuhaochen@lsec.cc.ac.cn

HEHU XIE

SKLMS, NCMIS, Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

hhxie@lsec.cc.ac.cn