  In many applications, flow measurements are usually sparse and possibly
noisy. The reconstruction of a high-resolution flow field from limited and
imperfect flow information is significant yet challenging. In this work, we
propose an innovative physics-constrained Bayesian deep learning approach to
reconstruct flow fields from sparse, noisy velocity data, where equation-based
constraints are imposed through the likelihood function and uncertainty of the
reconstructed flow can be estimated. Specifically, a Bayesian deep neural
network is trained on sparse measurement data to capture the flow field. In the
meantime, the violation of physical laws will be penalized on a large number of
spatiotemporal points where measurements are not available. A non-parametric
variational inference approach is applied to enable efficient
physics-constrained Bayesian learning. Several test cases on idealized vascular
flows with synthetic measurement data are studied to demonstrate the merit of
the proposed method.
