  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 $Re=2,000$, $3,000$ and $5,000$. We observe that
vanilla PINNs obtain two classes of solutions, one class that agrees with
direct numerical simulations (DNS), and another 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 parameterized
entropy-viscosity method as artificial eddy viscosity and identify suitable
parameters that drive the PINNs solution towards the DNS solution. Furthermore,
we solve the inverse problem by subsampling the DNS solution, and identify a
new eddy viscosity distribution that leads to velocity and pressure fields
almost identical to 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.
