Improving hp-Variational Physics-Informed Neural Networks for Steady-State Convection-Dominated Problems

TL;DR

This paper enhances hp-Variational Physics-Informed Neural Networks for convection-dominated problems by introducing stabilization techniques and adaptive boundary condition parameter learning, resulting in improved accuracy.

Contribution

It introduces a SUPG-inspired stabilization term and a neural network architecture for learning boundary indicator parameters within the FastVPINNs framework.

Findings

Enhanced accuracy over existing methods

Effective stabilization for convection-dominated problems

Adaptive boundary condition parameter learning improves solutions

Abstract

This paper proposes and studies two extensions of applying hp-variational physics-informed neural networks, more precisely the FastVPINNs framework, to convection-dominated convection-diffusion-reaction problems. First, a term in the spirit of a SUPG stabilization is included in the loss functional and a network architecture is proposed that predicts spatially varying stabilization parameters. Having observed that the selection of the indicator function in hard-constrained Dirichlet boundary conditions has a big impact on the accuracy of the computed solutions, the second novelty is the proposal of a network architecture that learns good parameters for a class of indicator functions. Numerical studies show that both proposals lead to noticeably more accurate results than approaches that can be found in the literature.