Deep learning for gradient flows using the Brezis-Ekeland principle
This work addresses computational challenges in solving high-dimensional PDEs for researchers in applied mathematics and scientific computing, but it appears incremental as it adapts an existing principle to deep learning.
The authors tackled the numerical solution of gradient flow PDEs by proposing a deep learning method based on the Brezis-Ekeland principle, which defines an objective function for minimization, and demonstrated it with an example implementation for the heat equation in dimensions two to seven.
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.