Model Order Reduction Method Based on Machine Learning for Parameterized Time-Dependent Partial Differential Equations

摘要

Abstract In this paper, we propose a data-driven model order reduction method to solve parameterized time-dependent partial differential equations. We describe the system with the state variable equations, and represent a class of candidate models with the artificial neural network. The discrete L2documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$L_2$$end{document} error between the output of artificial neural network and the high-fidelity solution is minimized with the state variable equations and initial conditions as constraints. Therefore, the model order reduction problem can be described as a kind of optimization problem with constraints, which can be solved by combining Levenberg–Marquardt algorithm and linear search algorithm, followed by sensitivity analysis of the artificial neural network parameters. Finally, by a number of calculating examples, compared to the model-based model order reduction method, data-driven model order reduction method is non-intrusive, is not limited to state variable degrees of freedom. We can find that the data-driven model order reduction method is better than the model-based model order reduction method in both computation time and precision, and has good approximation properties.