While physics-informed neural networks have been shown to accurately solve a wide range of fluid dynamics problems, their effectivity on highly compressible flows is so far limited. In particular, they struggle with transonic and supersonic problems that involve discontinuities such as shocks. While there have been multiple efforts to alleviate the nonphysical phenomena that arise on such problems, current work does not identify and address the underlying failure modes sufficiently. This thesis shows that physics-informed neural networks struggle with highly compressible problems for two independent reasons. Firstly, the differential Euler equations conserve entropy along streamlines, so that physics-informed neural networks try to find an isentropic solution to a non-isentropic problem. Secondly, conventional slip boundary conditions form strong local minima that result in fictive objects that simplify the flow. In response to these failure modes, two new adaptations are introduced, namely a local viscosity method and a streamline output representation. The local viscosity method includes viscosity as an additional network output and adds a viscous loss term to the loss function, resulting in localized viscosity that facilitates the entropy change at shocks. Furthermore, the streamline output representation provides a more natural formulation of the slip boundary conditions, which prevents zero-velocity regions while promoting shock attachment. To the author's best knowledge, this thesis provides the first inviscid steady solutions of curved and detached shocks by physics-informed neural networks.