Fractional physical-inform neural networks (fPINNs) for turbulent flows

We employ fractional operators in conjunction with physics-informed neural networks (PINNs) to discover new governing equations for modeling and simulating the Reynolds stresses in the Reynolds Averaged Navier-Stokes equations (RANS) for wall-bounded turbulent flows at high Reynolds number. In particular, we develop a simple one-dimensional model for fully-developed wall-turbulence that involves a fractional operator with fractional order that varies with the distance from the wall. We use available DNS data bases to infer the function that describes the fractional order, which has an integer value at the wall and decays monotonically to an asymptotic value at the centerline. We show that this function is universal upon re-scaling and hence it can be used to predict the mean velocity profile at all Reynolds numbers. We also extend the fractional RANS for fully-developed turbulent channel flow to a turbulent boundary layer and infer the fractional order in the wake region.