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.

翻译：本文探讨如下重要研究问题：能否利用物理信息神经网络（PINN）和变分物理信息神经网络（VPINN）求解具有挑战性的一维和二维对流主导扩散问题？该方法与高阶连续性有限元法（FEM）相比表现如何？如何定义PINN和VPINN的损失函数以确保其收敛于正确解？如何为PINN和VPINN选择训练点或测试函数？我们重点研究一维对流主导扩散问题和二维Eriksson-Johnson模型问题。研究表明，标准伽辽金有限元法无法求解该问题。我们讨论了采用彼得罗夫-伽辽金（PG）公式化方法对对流主导扩散问题进行稳定化处理，并给出了基于PG方法的有限元解。我们采用PINN和VPINN方法，定义了多种强残差和弱残差损失函数，并将PINN和VPINN方法的训练过程及解与高阶有限元方法进行了比较。