Artificial Viscosity in Physics-Informed Neural Networks for Parametric Compressible Flows

The numerical approximation of solutions to the compressible Euler and Navier-stokes equations is a crucial but challenging task with relevance in various fields of science and engineering. Recently, methods from deep learning have been successfully employed for solving partial differential equations by incorporating the equations into a loss function that is minimized during the training of a neural network. This approach yields a so-called physics-informed neural network. It is not based upon classical discretizations, such as finite-volume or finite-element schemes, and can even address parametric problems in a straightforward manner. This has raised the question, whether physics-informed neural networks may be a viable alternative to conventional methods for computational fluid dynamics. In this article we propose a physics-informed neural network training procedures to approximate steady-state solutions of boundary-value problems for the compressible Euler equations. It turns out that the addition of artificial dissipation during the training process is important to avoid unphysical results. A method for reducing this additional numerical viscosity during the training is presented. Furthermore, we showcase how this approach can be combined with parametric boundary conditions. Our results highlight the appearance of unphysical results when solving compressible flows with physics-informed neural networks and offer a new approach to overcome this problem.