  We formulate a physics-informed compressed sensing (PICS) method for the
reconstruction of velocity fields from noisy and sparse phase-contrast magnetic
resonance signals. The method solves an inverse Navier-Stokes boundary value
problem, which permits us to jointly reconstruct and segment the velocity
field, and at the same time infer hidden quantities such as the hydrodynamic
pressure and the wall shear stress. Using a Bayesian framework, we regularize
the problem by introducing a priori information about the unknown parameters in
the form of Gaussian random fields. This prior information is updated using the
Navier-Stokes problem, an energy-based segmentation functional, and by
requiring that the reconstruction is consistent with the $k$-space signals. We
create an algorithm that solves this reconstruction problem, and test it for
noisy and sparse $k$-space signals of the flow through a converging nozzle. We
find that the method is capable of reconstructing and segmenting the velocity
fields from sparsely-sampled (15% $k$-space coverage), low ($\sim$$10$)
signal-to-noise ratio (SNR) signals, and that the reconstructed velocity field
compares well with that derived from fully-sampled (100% $k$-space coverage)
high ($>40$) SNR signals of the same flow.
