  Blood flow reconstruction in the vasculature is important for many clinical
applications. However, in clinical settings, the available data are often quite
limited. For instance, Transcranial Doppler ultrasound (TCD) is a noninvasive
clinical tool that is commonly used in the clinical settings to measure blood
velocity waveform at several locations on brain's vasculature. This amount of
data is grossly insufficient for training machine learning surrogate models,
such as deep neural networks or Gaussian process regression. In this work, we
propose a Gaussian process regression approach based on physics-informed
kernels, enabling near-real-time reconstruction of blood flow in data-poor
regimes. We introduce a novel methodology to reconstruct the kernel within the
vascular network, which is a non-Euclidean space. The proposed kernel encodes
both spatiotemporal and vessel-to-vessel correlations, thus enabling blood flow
reconstruction in vessels that lack direct measurements. We demonstrate that
any prediction made with the proposed kernel satisfies the conservation of mass
principle. The kernel is constructed by running stochastic one-dimensional
blood flow simulations, where the stochasticity captures the epistemic
uncertainties, such as lack of knowledge about boundary conditions and
uncertainties in vasculature geometries. We demonstrate the performance of the
model on three test cases, namely, a simple Y-shaped bifurcation, abdominal
aorta, and the Circle of Willis in the brain.
