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

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