Solutions of certain partial differential equations (PDEs) are often represented by the steepest descent curves of corresponding functionals. Minimizing movement scheme was developed in order to study such curves in metric spaces. Especially, Jordan-Kinderlehrer-Otto studied the Fokker-Planck equation in this way with respect to the Wasserstein metric space. In this paper, we propose a deep learning-based minimizing movement scheme for approximating the solutions of PDEs. The proposed method is highly scalable for high-dimensional problems as it is free of mesh generation. We demonstrate through various kinds of numerical examples that the proposed method accurately approximates the solutions of PDEs by finding the steepest descent direction of a functional even in high dimensions.

翻译：某些偏微分方程（PDEs）的解通常由相应泛函的最速下降曲线表示。最小移动方案被提出用于研究度量空间中的此类曲线。特别地，Jordan-Kinderlehrer-Otto利用Wasserstein度量空间以这种方式研究了Fokker-Planck方程。本文提出了一种基于深度学习的最小移动方案，用于逼近PDEs的解。由于无需网格生成，所提方法对高维问题具有高度可扩展性。我们通过多种数值实例证明，即使在较高维度下，该方法通过寻找泛函的最速下降方向也能精确逼近PDEs的解。