An Artificial-Compressibility Physics-Informed Neural Network for the Unsteady Incompressible Navier--Stokes Equations

We study a physics-informed neural network (PINN) for the unsteady, two-dimensional incompressible Navier--Stokes equations in which the stiff divergence-free constraint is replaced by an artificial-compressibility (AC) relaxation governed by a single scalar parameter $\eps$. The relaxation reintroduces a pressure time derivative, converting a differential-algebraic constraint into an ordinary residual that a PINN can minimise directly. On the Taylor--Green vortex, which admits a closed-form unsteady solution, we quantify the effect of $\eps$: the residual divergence scales as $\eps\,|\partial_t p|$, so larger $\eps$ raises both the divergence and the velocity error, and both decrease monotonically and saturate as $\eps$ is reduced. On the $Re=100$ cylinder wake the plain forward AC-PINN collapses to the steady symmetric branch and does not reproduce von K\'arm\'an shedding; assimilating a few hundred sparse velocity sensors from a boundary-layer-resolved finite-element reference (whose Strouhal number, $0.176$, we bring close to the $0.164$--$0.172$ literature band by resolving the separating shear layer, though it remains just above it) recovers the unsteady vortex street to $7\%$ over the wake and its shedding frequency to within $3\%$ of that same reference --- a bound set by the reference's own fidelity rather than an independent validation against the true flow.