Deep learning for gradient flows using the Brezis-Ekeland principle. (English) Zbl 07675595

Carini, Laura; Jensen, Max; Nürnberg, Robert

Arch. Math. (Brno) 59, No. 3, 249-261 (2023).

Summary: 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.

MSC:

35K15	Initial value problems for second-order parabolic equations
35A15	Variational methods applied to PDEs
68T07	Artificial neural networks and deep learning

Keywords:

machine learning; deep neural networks; gradient flows; Brezis-Ekeland principle; adversarial networks; differential equations

Software:

DGM; BENNO