Uniform convergence guarantees for the deep Ritz method for nonlinear problems

Patrick Dondl; Johannes Müller; Marius Zeinhofer

doi:10.1186/s13662-022-03722-8

Uniform convergence guarantees for the deep Ritz method for nonlinear problems

Adv Contin Discret Model. 2022;2022(1):49. doi: 10.1186/s13662-022-03722-8. Epub 2022 Jul 15.

Authors

Patrick Dondl¹, Johannes Müller², Marius Zeinhofer³

Affiliations

¹ Department of Applied Mathematics, University of Freiburg, Hermann-Herder-Straße 10, 79104 Freiburg i. Br., Germany.
² Max Planck Institute for Mathematics in the Sciences, Inselstraße 22, 04103 Leipzig, Germany.
³ Simula Research Laboratory, Department of Numerical Analysis and Scientific Computing, Kristian Augusts Gate 23, 0164 Oslo, Norway.

Abstract

We provide convergence guarantees for the Deep Ritz Method for abstract variational energies. Our results cover nonlinear variational problems such as the p-Laplace equation or the Modica-Mortola energy with essential or natural boundary conditions. Under additional assumptions, we show that the convergence is uniform across bounded families of right-hand sides.

Keywords: Boundary penalty method; Calculus of variations; Neural networks; Nonlinear problems; Ritz method.