Deep learning for gradient flows using the Brezis-Ekeland principle
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Abstract (arXiv)
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.
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References · 8
- 2014Adam: A Method for Stochastic Optimizationcited by 37,240
- 2017DGM: A deep learning algorithm for solving partial differential equationscited by 678
- 2017The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problemscited by 583
- 2017Solving high-dimensional partial differential equations using deep learningcited by 556
- 2017Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equationscited by 208
- 2016Deep Learning Approximation for Stochastic Control Problemscited by 55
- 2018Forward-Backward Stochastic Neural Networks: Deep Learning of High-dimensional Partial Differential Equationscited by 55
- 2022The Deep Ritz Method for Parametric p-Dirichlet Problemscited by 3
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