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

  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.},
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… 

Tables from this paper

A Deep Learning Approach to Nonconvex Energy Minimization for Martensitic Phase Transitions

A mesh-free method to solve nonconvex energy minimization problems for martensitic phase transitions and twinning in crystals, using the deep learning approach using the Deep Ritz method, whereby candidates for minimizers are represented by parameter-dependent deep neural networks, and the energy is minimized with respect to network parameters.



Deep neural network approximation for high-dimensional elliptic PDEs with boundary conditions

It is shown that deep neural networks are capable of representing solutions of the Poisson equation without incurring the curse of dimension and the proofs are based on a probabilistic representation of the solution to thePoisson equation as well as a suitable sampling method.

Solving Nonlinear and High-Dimensional Partial Differential Equations via Deep Learning

The main goals of this paper are to elucidate the features, capabilities and limitations of DGM by analyzing aspects of its implementation for a number of different PDEs and PDE systems.

Numerical Approximations of Stochastic Differential Equations With Non-globally Lipschitz Continuous Coefficients

Many stochastic differential equations (SDEs) in the literature have a superlinearly growing nonlinearity in their drift or diffusion coefficient. Unfortunately, momentsof the computationally

Implicit bias with Ritz-Galerkin method in understanding deep learning for solving PDEs

DNNs solve a particular Poisson problem, where the information of the right-hand side of the equation f is only available at n sample points while the bases (neuron) number is much larger than n, which is common in DNN-based methods, with a much smoother function.

Artificial neural networks for solving ordinary and partial differential equations

This article illustrates the method by solving a variety of model problems and presents comparisons with solutions obtained using the Galekrkin finite element method for several cases of partial differential equations.

The Deep Ritz Method: A Deep Learning-Based Numerical Algorithm for Solving Variational Problems

A deep learning-based method, the Deep Ritz Method, for numerically solving variational problems, particularly the ones that arise from partial differential equations, which is naturally nonlinear, naturally adaptive and has the potential to work in rather high dimensions.