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Computer Methods in Applied Mechanics and Engineering

In this work we investigate the possibility of using physics-informed  neural networks    (PINNs) to approximate the  Euler equations  that model high-speed aerodynamic flows. In particular, we solve both the forward and   inverse problems     in one-dimensional and two-dimensional domains. For the forward problem, we utilize the Euler equations and the initial/boundary conditions to formulate the loss function, and solve the one-dimensional Euler equations with smooth solutions and with solutions that have a contact discontinuity as well as a two-dimensional  oblique shock  wave problem. We demonstrate that we can capture the solutions with only a few scattered points clustered randomly around the discontinuities. For the inverse problem, motivated by mimicking the   Schlieren  photography experimental technique used traditionally in high-speed aerodynamics, we use the data on   density gradient         ∇  ρ   (  x  ,  t  )     , the pressure     p   (    x    ∗    ,  t  )      at a specified point     x  =    x    ∗        as well as the conservation laws to infer all states of interest (density, velocity and pressure fields). We present illustrative benchmark examples for both the problem with smooth solutions and  Riemann problems  (Sod and Lax problems) with PINNs, demonstrating that all inferred states are in good agreement with the reference solutions. Moreover, we show that the choice of the position of the point       x    ∗      plays an important role in the learning process. In particular, for the problem with smooth solutions we can randomly choose the position of the point      x    ∗      from the computational domain, while for the Sod or Lax problem, we have to choose the position of the point      x    ∗       from the domain between the initial  discontinuous point  and the shock position of the final time. We also solve the inverse problem by combining the aforementioned data and the Euler equations in characteristic form, showing that the results obtained by using the Euler equations in chara...

Physics-informed neural networks for high-speed flows