Deep learning for gradient flows using the brezis-ekeland principle

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.