Deep learning for gradient flows using the Brezis–Ekeland principle

Laura Carini, Max Jensen, and Robert Nürnberg

Address:
Dipartimento di Mathematica, Università di Trento, 38123 Trento, Italy
Mathematics Department, University College London, 25 Gordon Street, London, WC1H 0AY, United Kingdom
Dipartimento di Mathematica, Università di Trento, 38123 Trento, Italy

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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.

AMSclassification: primary 35K15; secondary 35A15, 68T07.

Keywords: machine learning, deep neural networks, gradient flows, Brezis–Ekeland principle, adversarial networks, differential equations.

DOI: 10.5817/AM2023-3-249