A Novel Training Framework for Physics-informed Neural Networks: Towards Real-time Applications in Ultrafast Ultrasound Blood Flow Imaging

Ultrafast ultrasound blood flow imaging is a state-of-the-art technique for depiction of complex blood flow dynamics in vivo through thousands of full-view image data (or, timestamps) acquired per second. Physics-informed Neural Network (PINN) is one of the most preeminent solvers of the Navier-Stokes equations, widely used as the governing equation of blood flow. However, that current approaches rely on full Navier-Stokes equations is impractical for ultrafast ultrasound. We hereby propose a novel PINN training framework for solving the Navier-Stokes equations. It involves discretizing Navier-Stokes equations into steady state and sequentially solving them with test-time adaptation. The novel training framework is coined as SeqPINN. Upon its success, we propose a parallel training scheme for all timestamps based on averaged constant stochastic gradient descent as initialization. Uncertainty estimation through Stochastic Weight Averaging Gaussian is then used as an indicator of generalizability of the initialization. This algorithm, named SP-PINN, further expedites training of PINN while achieving comparable accuracy with SeqPINN. The performance of SeqPINN and SP-PINN was evaluated through finite-element simulations and in vitro phantoms of single-branch and trifurcate blood vessels. Results show that both algorithms were manyfold faster than the original design of PINN, while respectively achieving Root Mean Square Errors of 0.63 cm/s and 0.81 cm/s on the straight vessel and 1.35 cm/s and 1.63 cm/s on the trifurcate vessel when recovering blood flow velocities. The successful implementation of SeqPINN and SP-PINN open the gate for real-time training of PINN for Navier-Stokes equations and subsequently reliable imaging-based blood flow assessment in clinical practice.