Physics Informed Neural Networks with strong and weak residuals for advection-dominated diffusion problems

TL;DR

This paper investigates the effectiveness of Physics Informed Neural Networks (PINN) and Variational PINN in solving challenging advection-dominated diffusion problems, comparing their performance to stabilized finite element methods.

Contribution

It introduces strong and weak residual formulations for PINN and VPINN, demonstrating their application to advection-dominated problems and comparing results with higher-order FEM methods.

Findings

PINN and VPINN can effectively solve advection-dominated diffusion problems.

Standard FEM struggles with these problems, but Petrov-Galerkin stabilization helps.

PINN and VPINN solutions are comparable or superior to stabilized FEM results.

Abstract

This paper deals with the following important research questions. Is it possible to solve challenging advection-dominated diffusion problems in one and two dimensions using Physics Informed Neural Networks (PINN) and Variational Physics Informed Neural Networks (VPINN)? How does it compare to the higher-order and continuity Finite Element Method (FEM)? How to define the loss functions for PINN and VPINN so they converge to the correct solutions? How to select points or test functions for training of PINN and VPINN? We focus on the one-dimensional advection-dominated diffusion problem and the two-dimensional Eriksson-Johnson model problem. We show that the standard Galerkin method for FEM cannot solve this problem. We discuss the stabilization of the advection-dominated diffusion problem with the Petrov-Galerkin (PG) formulation and present the FEM solution obtained with the PG method. We employ PINN and VPINN methods, defining several strong and weak loss functions. We compare the training and solutions of PINN and VPINN methods with higher-order FEM methods.