Singular layer PINN methods for Burgers' equation

Abstract

In this article, we present a new learning method called sl-PINN to tackle the one-dimensional viscous Burgers problem at a small viscosity, which results in a singular interior layer. To address this issue, we first determine the corrector that characterizes the unique behavior of the viscous flow within the interior layers by means of asymptotic analysis. We then use these correctors to construct sl-PINN predictions for both stationary and moving interior layer cases of the viscous Burgers problem. Our numerical experiments demonstrate that sl-PINNs accurately predict the solution for low viscosity, notably reducing errors near the interior layer compared to traditional PINN methods. Our proposed method offers a comprehensive understanding of the behavior of the solution near the interior layer, aiding in capturing the robust part of the training solution.

Figures (16)

• Figure 4.1: Smooth initial data case: sl-PINN training data distribution.
• Figure 4.2: Smooth initial data: sl-PINN's training loss during the training process.
• Figure 4.3: Smooth initial data: PINN's training loss during the training process.
• Figure 4.4: Smooth initial data case when $\varepsilon=10^{-2}/\pi$: PINN vs sl-PINN solution plots with small NN size $4\times 20$.
• Figure 4.5: Smooth initial data case when $\varepsilon=10^{-3}/\pi$: PINN vs sl-PINN solution plots with small NN size $4\times 20$.