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

Haotian Guan, Jinping Dong+1 more|arXiv (Cornell University)|Sep 9, 2023

Model Reduction and Neural Networks4 citations

TL;DR

This paper proposes SeqPINN and SP-PINN, a novel training framework for physics-informed neural networks that accelerates real-time ultrafast ultrasound blood flow imaging by solving steady-state Navier-Stokes equations sequentially and enabling parallel training via uncertainty-aware initialization. The method reduces training time by orders of magnitude while achieving low RMSEs of 1.01 cm/s (SeqPINN) and 1.26 cm/s (SP-PINN) on straight vessels, enabling faster, scalable inference for clinical hemodynamics modeling.

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.

Motivation & Objective

• To address the slow training speed of traditional PINNs in real-time hemodynamics modeling for ultrafast Doppler ultrasound.
• To enable fast, scalable training of PINNs for high-frame-rate blood flow imaging by leveraging steady-state Navier-Stokes equations.
• To develop a parallel training scheme (SP-PINN) that maintains accuracy while significantly reducing training time.
• To improve model generalization through uncertainty estimation using Stochastic Weight Averaging Gaussian for initialization.
• To achieve real-time inference capability for patient-specific blood flow velocity reconstruction in clinical settings.

Proposed method

• Proposes SeqPINN, which discretizes time into sequential steady-state Navier-Stokes problems and trains models one timestamp at a time using transfer learning.
• Introduces SP-PINN, a parallel training scheme that applies averaged constant stochastic gradient descent (SGD) for initialization across all timestamps.
• Employs Stochastic Weight Averaging Gaussian (SWAG) to estimate uncertainty and guide initialization, enhancing generalization and convergence.
• Uses finite-element simulations and in vitro phantoms of single- and trifurcate-vessel models to validate the framework.
• Applies PDE-constrained loss functions based on Navier-Stokes equations and incompressibility, with boundary conditions enforced via physics-informed regularization.
• Leverages lightweight, mesh-free neural networks to avoid the computational burden of traditional CFD and enable fast inference.

Experimental results

Research questions

• RQ1 Can sequential training of steady-state Navier-Stokes solutions significantly reduce PINN training time for ultrafast ultrasound imaging?
• RQ2 Can parallel training of PINNs across timestamps be achieved with stable and generalizable initialization?
• RQ3 How does uncertainty estimation via SWAG improve the generalization and convergence of PINN initialization?
• RQ4 What is the trade-off between training speed, model size, and accuracy in real-time blood flow reconstruction?
• RQ5 Can the proposed framework achieve real-time inference for high-frame-rate Doppler ultrasound data?

Key findings

• SeqPINN reduced training time by multiple orders of magnitude compared to vanilla PINN, achieving an RMSE of 1.01 cm/s on a straight vessel and 1.91 cm/s on a trifurcate vessel.
• SP-PINN achieved comparable accuracy to SeqPINN with an RMSE of 1.26 cm/s on the straight vessel and 2.56 cm/s on the trifurcate vessel, while enabling parallel training.
• Uncertainty estimation via SWAG successfully identified stable initializations, improving model generalization and reducing training variance.
• The sequential training approach in SeqPINN traded storage (N models) for speed, with individual models being small (~461 kB), making them feasible for off-GPU storage.
• SP-PINN outperformed SeqPINN in training speed on GPU, even under limited hardware, due to better utilization of parallel computation.
• Despite improvements, real-time training remains unachieved due to GPU computation bottlenecks, suggesting a need for specialized hardware acceleration.