The Deep Minimizing Movement Scheme

TL;DR

This paper introduces a deep learning-based minimizing movement scheme that efficiently approximates solutions to PDEs in high-dimensional spaces by leveraging the steepest descent of functionals without requiring mesh generation.

Contribution

It presents a novel deep learning approach to the minimizing movement scheme, enabling scalable PDE solutions in high dimensions without mesh dependency.

Findings

Accurately approximates PDE solutions in high-dimensional problems.

Demonstrates scalability and mesh-free computation.

Effective in various numerical examples.

Abstract

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.