  The direct deep learning simulation for multi-scale problems remains a
challenging issue. In this work, a novel higher-order multi-scale deep Ritz
method (HOMS-DRM) is developed for thermal transfer equation of authentic
composite materials with highly oscillatory and discontinuous coefficients. In
this novel HOMS-DRM, higher-order multi-scale analysis and modeling are first
employed to overcome limitations of prohibitive computation and Frequency
Principle when direct deep learning simulation. Then, improved deep Ritz method
are designed to high-accuracy and mesh-free simulation for macroscopic
homogenized equation without multi-scale property and microscopic lower-order
and higher-order cell problems with highly discontinuous coefficients.
Moreover, the theoretical convergence of the proposed HOMS-DRM is rigorously
demonstrated under appropriate assumptions. Finally, extensive numerical
experiments are presented to show the computational accuracy of the proposed
HOMS-DRM. This study offers a robust and high-accuracy multi-scale deep
learning framework that enables the effective simulation and analysis of
multi-scale problems of authentic composite materials.
