  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 introduce an
adaptive artificial viscosity reduction procedure for physics-informed neural
networks enabling approximate parametric solutions for forward problems
governed by the stationary two-dimensional Euler equations in sub- and
supersonic conditions. To the best of our knowledge, this is the first time
that the concept of artificial viscosity in physics-informed neural networks is
successfully applied to a complex system of conservation laws in more than one
dimension. Moreover, we highlight the unique ability of this method to solve
forward problems in a continuous parameter space. The presented methodology
takes the next step of bringing physics-informed neural networks closer towards
realistic compressible flow applications.
