Discrete Gradient Flow Approximations of High Dimensional Evolution Partial Differential Equations via Deep Neural Networks

@article{Georgoulis2022DiscreteGF,
  title={Discrete Gradient Flow Approximations of High Dimensional Evolution Partial Differential Equations via Deep Neural Networks},
  author={Emmanuil H. Georgoulis and Michail Loulakis and Asterios Tsiourvas},
  journal={Commun. Nonlinear Sci. Numer. Simul.},
  year={2022},
  volume={117},
  pages={106893}
}
We consider the approximation of initial/boundary value problems involving, possibly high-dimensional, dissipative evolution partial differential equations (PDEs) using a deep neural network framework. More specifically, we first propose discrete gradient flow approximations based on non-standard Dirichlet energies for problems involving essential boundary conditions posed on bounded spatial domains. The imposition of the boundary conditions is realized weakly via non-standard functionals; the… 
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