Physics-informed neural networks for inverse problems in supersonic\n flows

Accurate solutions to inverse supersonic compressible flow problems are often\nrequired for designing specialized aerospace vehicles. In particular, we\nconsider the problem where we have data available for density gradients from\nSchlieren photography as well as data at the inflow and part of wall\nboundaries. These inverse problems are notoriously difficult and traditional\nmethods may not be adequate to solve such ill-posed inverse problems. To this\nend, we employ the physics-informed neural networks (PINNs) and its extended\nversion, extended PINNs (XPINNs), where domain decomposition allows deploying\nlocally powerful neural networks in each subdomain, which can provide\nadditional expressivity in subdomains, where a complex solution is expected.\nApart from the governing compressible Euler equations, we also enforce the\nentropy conditions in order to obtain viscosity solutions. Moreover, we enforce\npositivity conditions on density and pressure. We consider inverse problems\ninvolving two-dimensional expansion waves, two-dimensional oblique and bow\nshock waves. We compare solutions obtained by PINNs and XPINNs and invoke some\ntheoretical results that can be used to decide on the generalization errors of\nthe two methods.\n