Solution multiplicity and effects of data and eddy viscosity on Navier-Stokes solutions inferred by physics-informed neural networks

Physics-informed neural networks (PINNs) have emerged as a new simulation paradigm for fluid flows and are especially effective for inverse and hybrid problems. However, vanilla PINNs often fail in forward problems, especially at high Reynolds (Re) number flows. Herein, we study systematically the classical lid-driven cavity flow at Re=1000, 2000,3000 and 5000. We observe that vanilla PINNs obtain two classes of solutions, one class that agrees with DNS and another one that is an unstable solution to the Navier-Stokes equations and not physically realizable. We attribute this solution multiplicity to singularities and unbounded vorticity, and we propose regularization methods that restore a unique solution within 1% difference from the DNS solution. In particular, we introduce a parametrized entropy-viscosity method as artificial eddy viscosity and identify suitable parameters that drive the PINNs solution to the DNS solution. Furthermore, we solve the inverse problem (by subsampling the DNS solution), and we identify a new eddy viscosity distribution that leads to velocity and pressure fields almost identical with their DNS counterparts. Surprisingly, a single measurement at a random point suffices to obtain a unique PINNs DNS-like solution even without artificial viscosity, which suggests possible pathways in simulating high Reynolds number turbulent flows using vanilla PINNs.