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The direct deep learning simulation for multi-scale problems remains achallenging issue. In this work, a novel higher-order multi-scale deep Ritzmethod (HOMS-DRM) is developed for thermal transfer equation of authenticcomposite materials with highly oscillatory and discontinuous coefficients. Inthis novel HOMS-DRM, higher-order multi-scale analysis and modeling are firstemployed to overcome limitations of prohibitive computation and FrequencyPrinciple when direct deep learning simulation. Then, improved deep Ritz methodare designed to high-accuracy and mesh-free simulation for macroscopichomogenized equation without multi-scale property and microscopic lower-orderand higher-order cell problems with highly discontinuous coefficients.Moreover, the theoretical convergence of the proposed HOMS-DRM is rigorouslydemonstrated under appropriate assumptions. Finally, extensive numericalexperiments are presented to show the computational accuracy of the proposedHOMS-DRM. This study offers a robust and high-accuracy multi-scale deeplearning framework that enables the effective simulation and analysis ofmulti-scale problems of authentic composite materials.

Higher-order multi-scale deep Ritz method for multi-scale problems of authentic composite materials