Deep Ritz Method for Solving High-Dimensional Fractional Differential Equations
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.