Deep Ritz Method for Solving High-Dimensional Fractional Differential Equations

Juan Yang1

Jiarong Zuo1,Email

Yu Tian1

Ming Lei2

1School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China
2School of Integrated Circuits, Beijing University of Posts and Telecommunications, Beijing 100876, China

Abstract

In this paper, we approximate the solutions of high-dimensional fractional order differential equations involving the right Riemann-Liouville fractional derivatives, left Caputo fractional derivatives and boundary value conditions. Once the problem’s variational structure has been identified, solving the equation can be stated as an optimal control problem. We introduce a deep learning-based numerical scheme for this optimal control problem. The deep Ritz method and point-taking method play an important role. The proposed numerical scheme produces accurate results of fractional order differential equations of low, medium and high dimensions.

Deep Ritz Method for Solving High-Dimensional Fractional Differential Equations