The so-called "grey-box modeling" approach cleverly combines insights from simulations ("white-box modeling?) on high-performance computing (HPC) architectures with data-driven approaches ("black-box modeling?) using artificial intelligence (AI) methods. As an example, we consider here the so-called "field inversion" method in data-driven turbulence modeling for the Navier-Stokes equations. For the field inversion, classical methods from the area of optimization with partial differential equations are used.