Deep Ritz revisited
This provides a theoretical foundation for using neural networks in numerical PDE analysis, which is incremental as it builds on existing variational methods.
The paper tackles the problem of proving convergence for neural networks solving partial differential equations (PDEs) by using Γ-convergence to show that ReLU networks trained with regularized Dirichlet energies converge to the true solution of the Poisson problem, and it generalizes this approach to nonlinear stationary PDEs like the p-Laplace.
Recently, progress has been made in the application of neural networks to the numerical analysis of partial differential equations (PDEs). In the latter the variational formulation of the Poisson problem is used in order to obtain an objective function - a regularised Dirichlet energy - that was used for the optimisation of some neural networks. In this notes we use the notion of $Γ$-convergence to show that ReLU networks of growing architecture that are trained with respect to suitably regularised Dirichlet energies converge to the true solution of the Poisson problem. We discuss how this approach generalises to arbitrary variational problems under certain universality assumptions of neural networks and see that this covers some nonlinear stationary PDEs like the $p$-Laplace.