  Physics-informed Neural Network (PINN) is one of the most preeminent solvers
of Navier-Stokes equations, which are widely used as the governing equation of
blood flow. However, current approaches, relying on full Navier-Stokes
equations, are impractical for ultrafast Doppler ultrasound, the
state-of-the-art technique for depiction of complex blood flow dynamics
\emph{in vivo} through acquired thousands of frames (or, timestamps) per
second. In this article, we first propose a novel training framework of PINN
for solving Navier-Stokes equations by discretizing Navier-Stokes equations
into steady state and sequentially solving steady-state Navier-Stokes equations
with transfer learning. The novel training framework is coined as SeqPINN. Upon
the success of SeqPINN, we adopt the idea of averaged constant stochastic
gradient descent (SGD) as initialization and propose a parallel training scheme
for all timestamps. To ensure an initialization that generalizes well, we
borrow the concept of Stochastic Weight Averaging Gaussian to perform
uncertainty estimation as an indicator of generalizability of the
initialization. This algorithm, named SP-PINN, further expedites training of
PINN while achieving comparable accuracy with SeqPINN. Finite-element
simulations and \emph{in vitro} phantoms of single-branch and trifurcate blood
vessels are used to evaluate the performance of SeqPINN and SP-PINN. Results
show that both SeqPINN and SP-PINN are manyfold faster than the original design
of PINN, while respectively achieving Root Mean Square Errors (RMSEs) of 1.01
cm/s and 1.26 cm/s on the straight vessel and 1.91 cm/s and 2.56 cm/s on the
trifurcate blood vessel when recovering blood flow velocities.
