Physics-informed neural networks for inverse problems in supersonic flows

In particular, PINNs do not require meshes and can efficiently solve forward problems and even ill-posed inverse problems, which are otherwise difficult or sometimes even impossible to solve using traditional numerical methods. Moreover, PINN can easily handle noisy, sparse and multi-fidelity data sets. The main advantage of the PINN methodology is that it can seamlessly incorporate all the given information like governing equation, experimental data, initial/boundary conditions into the loss function, thereby recasting the original problem into an equivalent optimization problem. Recently, Shin et al. [16] established the mathematical foundation of PINNs for linear partial differential equations, whereas in [17], Mishra and Molinaro presented estimate on the generalization error of the PINN methodology. In this work, we consider inverse problems of the shock wave problems in supersonic compressible flows. The governing equations of such flows are compressible Euler equations, which admit discontinuities or shocks, even though the initial states are smooth. Such ill-posed inverse problems are difficult or even sometimes impossible to solve using the traditional numerical solvers. Moreover, for the shock wave problem, the traditional numerical methods usually require boundary conditions (BCs) for all field variables.