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：某些部分差异方程式(PDEs)的解决方案往往由相应功能最陡峭的下行曲线代表。制定了最大限度地减少移动计划,以研究计量空间的曲线。特别是,约旦-Kinderle Header-Otto以这种方式研究了Fokker-Planck方程式瓦塞尔斯坦度空间。在本文件中,我们提出了一个以深层次学习为基础的最大限度地减少移动计划,以接近PDEs的解决方案。拟议方法对于高维度问题来说是高度可伸缩的,因为它是无网状一代的。我们通过各种数字例子表明,拟议方法通过在高维度找到功能最陡峭的下行方向,从而准确地接近PDEs的解决办法。