Higher-order multi-scale deep Ritz method (HOMS-DRM) and its convergence analysis for solving thermal transfer problems of composite materials

The challenging limitations of prohibitive computation and Frequency Principle remain significantly difficult for deep learning methods to effectively resolve multi-scale problems. In this work, a higher-order multi-scale deep Ritz method (HOMS-DRM) is developed to address this issue and effectively compute thermal transfer equation of composite materials with highly oscillatory, discontinuous and high-contrast coefficients. In the computational framework of HOMS-DRM, higher-order multi-scale modeling is first employed to overcome limitations of prohibitive computation and Frequency Principle when direct deep learning simulation. Then, improved deep Ritz method is designed to high-accuracy and mesh-free simulation for lower-order and higher-order microscopic unit cell functions, and macroscopic homogenized equations of multi-scale composites, which are then assembled into higher-order multi-scale solutions for multi-scale thermal transfer problems by using automatic differentiation technology. Besides, corresponding numerical algorithm of HOMS-DRM is developed for implementing high-accuracy multi-scale simulation in periodic composite medium. Moreover, the theoretical convergence of the proposed HOMS-DRM is rigorously demonstrated under appropriate assumptions. Finally, 2D and 3D numerical experiments including high-contrast composite materials are presented to validate the computational performance of HOMS-DRM.