  Physics-informed neural networks (PINNs) are successful machine-learning
methods for the solution and identification of partial differential equations
(PDEs). We employ PINNs for solving the Reynolds-averaged
Navier$\unicode{x2013}$Stokes (RANS) equations for incompressible turbulent
flows without any specific model or assumption for turbulence, and by taking
only the data on the domain boundaries. We first show the applicability of
PINNs for solving the Navier$\unicode{x2013}$Stokes equations for laminar flows
by solving the Falkner$\unicode{x2013}$Skan boundary layer. We then apply PINNs
for the simulation of four turbulent-flow cases, i.e., zero-pressure-gradient
boundary layer, adverse-pressure-gradient boundary layer, and turbulent flows
over a NACA4412 airfoil and the periodic hill. Our results show the excellent
applicability of PINNs for laminar flows with strong pressure gradients, where
predictions with less than 1% error can be obtained. For turbulent flows, we
also obtain very good accuracy on simulation results even for the
Reynolds-stress components.
