  Cardiovascular diseases (CVDs) are the leading cause of death worldwide, with
blood pressure serving as a crucial indicator. Arterial blood pressure (ABP)
waveforms provide continuous pressure measurements throughout the cardiac cycle
and offer valuable diagnostic insights. Consequently, there is a significant
demand for non-invasive and cuff-less methods to measure ABP waveforms
continuously. Accurate prediction of ABP waveforms can also improve the
estimation of mean blood pressure, an essential cardiovascular health
characteristic.
  This study proposes a novel framework based on the physics-informed DeepONet
approach to predict ABP waveforms. Unlike previous methods, our approach
requires the predicted ABP waveforms to satisfy the Navier-Stokes equation with
a time-periodic condition and a Windkessel boundary condition. Notably, our
framework is the first to predict ABP waveforms continuously, both with
location and time, within the part of the artery that is being simulated.
Furthermore, our method only requires ground truth data at the outlet boundary
and can handle periodic conditions with varying periods. Incorporating the
Windkessel boundary condition in our solution allows for generating natural
physical reflection waves, which closely resemble measurements observed in
real-world cases. Moreover, accurately estimating the hyper-parameters in the
Navier-Stokes equation for our simulations poses a significant challenge. To
overcome this obstacle, we introduce the concept of meta-learning, enabling the
neural networks to learn these parameters during the training process.
