Discrete Gradient Flow Approximations of High Dimensional Evolution Partial Differential Equations via Deep Neural Networks

TL;DR

This paper introduces a deep neural network framework for approximating high-dimensional dissipative PDEs using discrete gradient flows, effectively handling boundary conditions and demonstrating strong numerical performance across dimensions.

Contribution

It proposes novel DNN-based discrete gradient flow methods for high-dimensional PDEs, incorporating weak boundary condition imposition and JKO-type flows with implicit time-stepping.

Findings

Effective Dirichlet energy approximations in low dimensions

Excellent performance of JKO-type energies in high dimensions

DNN discretization via residual networks improves training efficiency

Abstract

We consider the approximation of initial/boundary value problems involving, possibly high-dimensional, dissipative evolution partial differential equations (PDEs) using a deep neural network framework. More specifically, we first propose discrete gradient flow approximations based on non-standard Dirichlet energies for problems involving essential boundary conditions posed on bounded spatial domains. The imposition of the boundary conditions is realized weakly via non-standard functionals; the latter classically arise in the construction of Galerkin-type numerical methods and are often referred to as "Nitsche-type" methods. Moreover, inspired by the seminal work of Jordan, Kinderleher, and Otto (JKO) \cite{jko}, we consider the second class of discrete gradient flows for special classes of dissipative evolution PDE problems with non-essential boundary conditions. These JKO-type gradient flows are solved via deep neural network approximations. A key, distinct aspect of the proposed methods is that the discretization is constructed via a sequence of residual-type deep neural networks (DNN) corresponding to implicit time-stepping. As a result, a DNN represents the PDE problem solution at each time node. This approach offers several advantages in the training of each DNN. We present a series of numerical experiments which showcase the good performance of Dirichlet-type energy approximations for lower space dimensions and the excellent performance of the JKO-type energies for higher spatial dimensions.