AI-augmented stabilized finite element method

TL;DR

This paper introduces AiStab-FEM, an AI-augmented finite element method that uses neural networks to predict optimal stabilization parameters, significantly improving solution accuracy for singularly perturbed PDEs.

Contribution

It presents a novel neural network framework for predicting stabilization parameters in finite element methods, enhancing solution stability and accuracy over existing neural PDE solvers.

Findings

Reduces undershoots and overshoots in solutions.

Outperforms Physics-Informed Neural Networks.

Effective for singularly perturbed PDEs.

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.