Physics-informed neural networks for shock capturing in inviscid flows around an airfoil

Physics-informed neural networks (PINNs) have shown remarkable prospects in solving forward and inverse problems involving partial differential equations (PDEs).

However, PINNs still face challenges in solving fluid mechanics problems involving shocks, especially in steady inviscid flows around an airfoil, where they may even fail to capture shocks.

In this study, we first point out that the reason PINNs fail to capture shocks is that the steady Euler equations used to construct the loss function impose weak constraints, which are difficult to correct the continuous function approximation preference of neural networks, causing gradient descent converges to a smooth local optimum.

Based on this insight, we propose to strengthen the physical constraints by reconstructing steady shock capturing as temporal evolution that gradually converges to the steady state solution.

The unsteady Euler equations constructed by introducing time derivative terms into the steady equations are used to constrain PINNs.

The output of PINNs is no longer required to directly approximate a flow field with shocks by minimizing the residuals of the steady Euler equations.

Instead, shocks gradually form under the guidance of the temporal evolution law of the flow field.

This additional temporal penalty alleviates the tendency of PINNs to converge to a smooth local optimum.

Since obtaining the steady state solution requires solving the unsteady Euler equations over a long time in the time dimension, while the capability of PINNs to solve such problems is poor, we introduce a PDE loss function that embeds the concept of pseudo time-stepping to avoid this issue.

In addition, to further improve the shock capturing accuracy, we develop a simplified formulation of the Euler equations.

By solving four forward problems involving different flow conditions and geometries, we validate the effectiveness of the proposed method.