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

TL;DR

This paper proposes and studies two extensions of applying hp-variational physics-informed neural networks to convection-dominated convection-diffusion-reaction problems, more precisely the FastVPINNs framework, and proposes a network architecture that predicts spatially varying stabilization parameters.

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.

Figures (11)

• Figure 1: Tensor-based loss computation schematic for FastVPINNs
• Figure 2: hp-VPINNs architecture for convection-dominated problems with SUPG stabilization, prescribed stabilization parameter, and hard constraints
• Figure 3: Exact Solution of $\text{P}_{\text{EJ}}$ for (a) $\varepsilon=0.1$ , (b) $\varepsilon=0.01$ and (c) $\varepsilon=0.001$
• Figure 4: Exact solution for (a) $\text{P}_{\text{out}}$ and (b) $\text{P}_{\text{para}}$
• Figure 5: Best results for $\text{P}_{\text{out}}$ (top) and $\text{P}_{\text{para}}$ (bottom). The exact solutions for $\text{P}_{\text{out}}$ and $\text{P}_{\text{para}}$ are shown in (a) and (d). The predicted solutions are shown in (b) and (e) and the point-wise errors in (c) and (f)