  Accurate solutions to inverse supersonic compressible flow problems are often
required for designing specialized aerospace vehicles. In particular, we
consider the problem where we have data available for density gradients from
Schlieren photography as well as data at the inflow and part of wall
boundaries. These inverse problems are notoriously difficult and traditional
methods may not be adequate to solve such ill-posed inverse problems. To this
end, we employ the physics-informed neural networks (PINNs) and its extended
version, extended PINNs (XPINNs), where domain decomposition allows deploying
locally powerful neural networks in each subdomain, which can provide
additional expressivity in subdomains, where a complex solution is expected.
Apart from the governing compressible Euler equations, we also enforce the
entropy conditions in order to obtain viscosity solutions. Moreover, we enforce
positivity conditions on density and pressure. We consider inverse problems
involving two-dimensional expansion waves, two-dimensional oblique and bow
shock waves. We compare solutions obtained by PINNs and XPINNs and invoke some
theoretical results that can be used to decide on the generalization errors of
the two methods.
