Deep learning for gradient flows using the Brezis-Ekeland principle

TL;DR

This paper introduces a deep learning approach for solving gradient flow PDEs based on the Brezis--Ekeland principle, enabling efficient numerical solutions across multiple spatial dimensions.

Contribution

It presents a novel deep neural network framework leveraging the Brezis--Ekeland principle for gradient flow PDEs, expanding applicability to higher dimensions.

Findings

Successfully applied to heat equation in 2-7 dimensions

Demonstrates effectiveness of the method for high-dimensional PDEs

Provides a general framework for neural network-based gradient flow solutions

Abstract

We propose a deep learning method for the numerical solution of partial differential equations that arise as gradient flows. The method relies on the Brezis--Ekeland principle, which naturally defines an objective function to be minimized, and so is ideally suited for a machine learning approach using deep neural networks. We describe our approach in a general framework and illustrate the method with the help of an example implementation for the heat equation in space dimensions two to seven.