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.

翻译：多尺度问题的直接深度学习模拟仍是一个具有挑战性的难题。本文针对具有高度振荡和不连续系数的真实复合材料热传导方程，提出了一种新型的高阶多尺度深度Ritz方法（HOMS-DRM）。在该方法中，首先采用高阶多尺度分析与建模来克服直接深度学习模拟中存在的计算代价高昂和频率原理限制。随后，设计了改进的深度Ritz方法，用于对无多尺度特性的宏观均匀化方程以及具有高度不连续系数的微观低阶和高阶胞元问题进行高精度无网格模拟。此外，在适当假设下，严格论证了所提HOMS-DRM的理论收敛性。最后，通过大量数值实验展示了所提HOMS-DRM的计算精度。本研究提供了一个鲁棒且高精度的多尺度深度学习框架，能够有效模拟和分析真实复合材料的多尺度问题。