Physics-informed neural networks for inviscid transonic flows around an airfoil

Physics-informed neural networks (PINNs) have gained popularity as a deep-learning-based parametric partial differential equation solver. Especially for engineering applications, this approach is promising because a single neural network (NN) could substitute many classical simulations in multi-query scenarios. In aerodynamics, transport equations, such as the Euler equations, need to be solved. These equations model an inviscid, compressible fluid and can pose a significant challenge for the PINN approach. Only recently, researchers have successfully solved subsonic flows around airfoils by utilizing mesh transformations to precondition the training of the NN. However, compressible flows in the transonic regime could not be accurately approximated due to shock waves resulting in local discontinuities. In this article, we propose techniques to successfully approximate solutions of the compressible Euler equations for sub- and transonic flows with PINNs. Inspired by classical numerical algorithms for solving conservation laws, the presented method locally introduces artificial dissipation to stabilize shock waves. We compare different viscosity variants, such as scalar- and matrix-valued artificial viscosity, and validate the method at transonic flow conditions for an airfoil, obtaining good agreement with finite-volume simulations. Finally, the suitability for parametric problems is showcased by approximating transonic solutions at varying angles of attack with a single network. The presented work proposes a solution to the previously encountered difficulties for PINNs in transonic flow conditions, enabling the application as parametric solvers to a new class of industrially relevant flow conditions in aerodynamics and beyond.