An Accelerated Training Framework for Physics-Informed Neural Networks: Applications in Ultrafast Ultrasound Blood...

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, the current approaches which implement time-dependent Navier-Stokes equations within the loss function are impractical for ultrafast ultrasound. We hereby propose an accelerated PINN training framework for solving the Navier-Stokes equations. It involves discretizing the time domain in Navier-Stokes equations 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 PINN training while achieving comparable accuracy to 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 1.07 cm/s and 1.41 cm/s on the trifurcate vessel when recovering blood flow velocities. The successful implementation of SeqPINN and SP-PINN opens the gate for real-time training of PINNs for Navier-Stokes equations and subsequently reliable imaging-based blood flow assessment in clinical practice.