On the Neural Network Approximations of the Solutions of PDEs by Optimizing the Minimizing Movement Scheme

Abstract

Solutions of certain partial differential equations (PDEs) are often represented by the steepest descent curves of corresponding functionals. In the previous work of De Giorgi, the minimizing movement scheme was developed in order to study such curves in general metric spaces. Later, Jordan-Kinderlehrer-Otto showed that the Fokker-Planck equation can be realized as the Cauchy problem in the Wasserstein metric space and the minimizing movement scheme can be efficiently applied to solve the given equation. In recent years, many evolution equations have been successfully found to be the gradient flows in certain metric spaces, and the reformulation of evolution equations into gradient flows is of great interest in the mathematical community. In this paper, we propose a deep learning-based minimizing movement scheme for approximating the solutions of evolution equations. The proposed method is highly scalable for high-dimensional problems as it is free of mesh generation, and is beneficial in that time extrapolation can be easily obtained compared to previous deep learning-based methods such as the Physics Informed Neural Networks (PINNs). We demonstrate through various kinds of numerical examples that the proposed method accurately approximates the solutions of evolution equations by finding the steepest descent direction of a functional in both low and high dimensions.