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.

翻译：物理信息神经网络（PINN）是求解纳维-斯托克斯方程最杰出的求解器之一，该方程被广泛用作血流的控制方程。然而，当前依赖完整纳维-斯托克斯方程的方法对于超快多普勒超声（一种通过每秒采集数千帧图像（或时间戳）来描绘体内复杂血流动力学的最先进技术）而言并不实用。本文首先提出一种新的PINN训练框架，通过将纳维-斯托克斯方程离散化为稳态，并利用迁移学习顺序求解稳态纳维-斯托克斯方程，从而求解该方程。该新训练框架被称为SeqPINN。在SeqPINN成功的基础上，我们采用平均常数随机梯度下降（SGD）作为初始化思路，并提出一种针对所有时间戳的并行训练方案。为确保初始化具有良好的泛化能力，我们借鉴随机权重平均高斯（Stochastic Weight Averaging Gaussian）的概念进行不确定性估计，以指示初始化的泛化性。该算法命名为SP-PINN，进一步加速了PINN的训练，同时达到与SeqPINN相当的精度。使用单分支和三叉血管的有限元仿真与体外模型来评估SeqPINN和SP-PINN的性能。结果表明，在恢复血流速度时，SeqPINN和SP-PINN均比原始PINN设计快数倍，在直血管上分别达到1.01 cm/s和1.26 cm/s的均方根误差（RMSE），在三叉血管上分别达到1.91 cm/s和2.56 cm/s的RMSE。