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

TL;DR

This paper introduces a higher-order multi-scale deep Ritz method (HOMS-DRM) that effectively simulates multi-scale thermal transfer problems in composite materials with high accuracy and efficiency, overcoming computational challenges.

Contribution

The paper develops a novel HOMS-DRM combining higher-order multi-scale analysis with an improved deep Ritz method for complex composite materials.

Findings

Achieves high-accuracy, mesh-free simulation of homogenized equations.

Demonstrates rigorous convergence of the proposed method.

Validates effectiveness through extensive numerical experiments.

Abstract

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.