The numerical approximation of solutions to the compressible Euler and Navierstokes 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 which does not rely on a classical discretization and can address parametric problems in a straightforward manner. Therefore, it avoids characteristic difficulties of traditional approaches, such as finite volume methods. This has raised the question, whether physics-informed neural networks may be a viable alternative to conventional methods for computational fluid dynamics. Here, we show a new physics-informed neural network training procedure to approximately solve the two-dimensional compressible Euler equations, which makes use of artificial dissipation during the training process. We demonstrate how additional dissipative terms help to avoid unphysical results and how the additional numerical viscosity can be reduced during training while iterating towards a solution. 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. We therefore expect that the presented methods enable the application of physics-informed neural networks for previously difficult to solve problems.