Physics-informed neural network modeling of shock waves by appropriately incorporating equation of state

Numerical analyses and surrogate models based on the compressible Euler and Navier–Stokes equations are essential for understanding and estimating nonlinear physical phenomena in fluid dynamics. Physics-informed neural networks (PINNs) approximate physical phenomena by integrating machine learning into physical models defined by partial differential equations (PDE) and initial/boundary conditions. Implementing the PINN method to estimate flow fields with discontinuities, such as shock waves, remains a challenge due to the difficulty in approximating sharp discontinuities with a neural network (NN). In this study, the influence of NN output variables selection on the accuracy of shock wave estimation was investigated. In the proposed PINN model, the loss function for the PDE is calculated not only from the Euler equations but also from the equation of state (EOS). The NN output variables consisted of density, velocity, temperature, and pressure to ensure consistency between the number of equations used to calculate the PDE loss function and the number of unknown variables. In addition, the inclusion of temperature in the energy calculations allows an assessment of the thermodynamic consistency of energy. With this consistent PINN formulation, the sharp discontinuities of the one-dimensional shock wave and the two-dimensional oblique shock wave problems were accurately captured and were in good agreement with the theoretical results.