AI-augmented stabilized finite element method

Abstract

An artificial intelligence-augmented Streamline Upwind/Petrov-Galerkin finite element scheme (AiStab-FEM) is proposed for solving singularly perturbed partial differential equations. In particular, an artificial neural network framework is proposed to predict optimal values for the stabilization parameter. The neural network is trained by minimizing a physics-informed cost function, where the equation's mesh and physical parameters are used as input features. Further, the predicted stabilization parameter is normalized with the gradient of the Galerkin solution to treat the boundary/interior layer region adequately. The proposed approach suppresses the undershoots and overshoots in the stabilized finite element solution and outperforms the existing neural network-based partial differential equation solvers such as Physics-Informed Neural Networks and Variational Neural Networks.