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

TL;DR

This work introduces SeqPINN and SP-PINN to accelerate physics-informed neural networks solving steady-state Navier–Stokes equations for ultrafast ultrasound blood flow imaging. SeqPINN enables online, frame-by-frame adaptation by leveraging a steady-state prior and reduced input dimensionality, while SP-PINN uses constant-SGD sampling and SWAG-like uncertainty to initialize and train across timestamps in parallel. In simulations and vessel-phantom experiments, both methods achieve substantially faster training and improved velocity-field accuracy compared with Vanilla PINN and X-PINN, with SeqPINN enabling near real-time adaptation and SP-PINN providing robust generalization across frames. The results demonstrate a viable path toward clinically applicable, physics-informed, real-time blood flow imaging from ultrafast ultrasound data.

Abstract

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.

Figures (9)

• Figure 1: Our proposed sequential and parallel training frameworks of PINN with state-state Navier-Stokes equations as an example.
• Figure 2: Illustration of blood flow (only the lateral (i.e., horizontal) velocity component displayed here) in (a) a single-branch vessel and (b) a three-branch vessel with prescribed temporal profiles of the (c) flow velocity at the inlet and (d) blood pressure at the outlet. Black arrows in (a) and (b) represent blood flow velocity vectors.
• Figure 3: Learning the full fluid velocity field from sparse measurements on a single-branch vessel. (a) SeqPINN and SP-PINN predictions and error maps of the velocity fields at five representative phases in a cardiac cycle: End diastole, Peak systole, End systole, Dicrotic notch, Mid diastole. Error maps show absolute errors. (b) Convergence of loss in the initialization of SeqPINN and SP-PINN. (c) Comparison of SeqPINN training time and accuracy as the number of training epochs per frame, $m$, increases in the simulated single-branch vessel case.
• Figure 4: Learning the full velocity field from sparse measurements on a three-branch vessel. (a) SeqPINN and SP-PINN predictions and error maps of the velocity fields at five representative phases in a cardiac cycle: End diastole, Peak systole, End systole, Dicrotic notch, Mid diastole. Error maps show the absolute errors. (b) Convergence of loss in the initialization of SeqPINN and SP-PINN. (c) Comparison of SeqPINN training time and accuracy as the number of training epochs per frame $m$ increases in the simulated three-branch vessel case.
• Figure 5: Comparison of RMSEs among Vanilla PINN, X-PINN, SeqPINN, and SP-PINN across 608 timestamps in the simulated (a) single-branch and (b) three-branch vessels. (c) and (d) show the comparison of accuracy and time-steps for SeqPINN on the simulated single-branch and three-branch vessels, respectively. Mean and standard deviation were calculated using RMSEs at each timestamp. Time-step sizes of 5, 10, 20, 30, 40, and 50 ms were tested.