Galerkin neural network approximation of singularly-perturbed elliptic systems

Abstract

We consider the neural network approximation of systems of partial differential equations exhibiting multiscale features such as the Reissner–Mindlin plate model which poses significant challenges due to the presence of boundary layers and numerical phenomena such as locking. This work builds on the basic Galerkin Neural Network approach established in Ainsworth and Dong (2021) for symmetric, positive-definite problems. The key contributions of this work are (1) the analysis and comparison of several new least squares-type variational formulations for the Reissner–Mindlin plate, and (2) their numerical approximation using the Galerkin Neural Network approach. Numerical examples are presented which demonstrate the ability of the approach to resolve multiscale phenomena such for the Reissner–Mindlin plate model for which we develop a new family of benchmark solutions which exhibit boundary layers.