Higher-order multi-scale physics-informed neural network (HOMS-PINN) method and its convergence analysis for solving elastic problems of authentic composite materials

The limitations of prohibitive computation and Frequency Principle remain difficult for deep learning methods to effectively resolve multi-scale problems. In this work, a novel higher-order multi-scale physics-informed neural network (HOMS-PINN) framework is developed to compute the elastic problems of authentic composite materials with strongly discontinuous and high-contrast material parameters, which inherits the advantages of higher-order multi-scale method and physics-informed neural network, and halves the computational expense remarkably in comparison with direct PINN simulation. In the numerical framework of HOMS-PINN method, two types of network structures including single neural networks and independent neural networks are designed to mesh-free solve lower-order and higher-order microscopic unit cell (UC) functions, and macroscopic homogenization equations of multi-scale composites, which are then assembled into higher-order multi-scale solutions for multi-scale elastic problems by using automatic differentiation technology. Moreover, transfer learning is introduced to extend HOMS-PINN framework accelerating simulation of multi-scale elastic problems for composite materials with high-contrast mechanical parameters. Furthermore, the error estimate of the proposed HOMS-PINN method is demonstrated under some assumptions. Finally, numerous numerical experiments including high-contrast, multi-inclusion and three-dimensional composites are presented to validate the accuracy and effectiveness of HOMS-PINN method, especially for accurately capturing the sharply oscillatory behaviors in micro-scale. This study offers a robust HOMS-PINN computational framework that enables the simulation and analysis of large-scale mechanical problems of authentic composite materials.