  The high cost of acquiring a sufficient amount of seismic data for training
has limited the use of machine learning in seismic tomography. In addition, the
inversion uncertainty due to the noisy data and data scarcity is less discussed
in conventional seismic tomography literature. To mitigate the uncertainty
effects and quantify their impacts in the prediction, the so-called Bayesian
Physics-Informed Neural Networks (BPINNs) based on the eikonal equation are
adopted to infer the velocity field and reconstruct the travel-time field. In
BPINNs, two inference algorithms including Stein Variational Gradient Descent
(SVGD) and Gaussian variational inference (VI) are investigated for the
inference task. The numerical results of several benchmark problems demonstrate
that the velocity field can be estimated accurately and the travel-time can be
well approximated with reasonable uncertainty estimates by BPINNs. This
suggests that the inferred velocity model provided by BPINNs may serve as a
valid initial model for seismic inversion and migration.
