Machine Learning-Enhanced High-Order Stochastic Multiscale Method for Random Materials

Source abstract

This study proposes a novel machine learning-enhanced high-order stochastic multiscale (HOSM) method, designed to efficiently solve the multiscale diffusion equation while capturing local oscillations in both stationary and non-stationary random materials. The key feature of this method is its ability to circumvent the large-scale sampling and fine meshes required by direct numerical simulations. Moreover, conventional homogenization techniques often fail to accurately capture local oscillations in solutions. Multiscale asymptotic expansion formulas are derived for both stationary and non-stationary random problems, including homogenized equations and cell functions. The homogenized equation is efficiently solved using a two-stage stochastic homogenization method induced by the normalizing flow and multi-modes Monte Carlo strategy. Subsequently, a limited number of cell problems at various locations within the macroscopic structure are solved using the finite element method, and the expected values of the corresponding cell functions are computed. A neural network model is then developed to establish a mapping between macroscopic and microscopic coordinates and the expected values of the cell functions, enabling rapid prediction of these values. The HOSM solutions are then constructed by integrating the two-stage stochastic homogenization solutions with the neural network cell functions. Besides, the convergence results for the HOSM solutions with an explicit convergence rate is given. Numerical examples demonstrate the accuracy and efficiency of the proposed machine learning-enhanced HOSM method.