On collocation points for physics-informed neural networks applied to convection-dominated convection-diffusion problems

Abstract

In recent years physics-informed neural networks (PINNs) for approximating the solution to (initial-)boundary value problems gained a lot of interest. PINNs are trained to minimize several residuals of the problem in collocation points. In this work we tackle convection-dominated convection-diffusion problems, whose solutions usually possess layers, which are small regions where the solution has a steep gradient. Inspired by classical Shishkin meshes, we compare hard- constrained PINNs trained with layer-adapted collocation points with ones trained with equispaced and uniformly randomly chosen points. We observe that layer-adapted points work the best for a problem with an interior layer and the worst for a problem with boundary layers. For both problems at most acceptable solutions can be obtained with PINNs.