In this study, we propose a Bayesian seismic tomography inference method
using physics-informed neural networks (PINN). PINN represents a recent advance
in deep learning, offering the possibility to enhance physics-based simulations
and inverse analyses. PINN-based deterministic seismic tomography uses two
separate neural networks (NNs) to predict seismic velocity and travel time.
Naive Bayesian NN (BNN) approaches are unable to handle the high-dimensional
spaces spanned by the weight parameters of these two NNs. Hence, we reformulate
the problem to perform the Bayesian estimation exclusively on the NN predicting
seismic velocity, while the NN predicting travel time is used only for
deterministic travel time calculations, with the help of the adjoint method.
Furthermore, we perform BNN by introducing a function-space Stein variational
gradient descent (SVGD), which performs particle-based variational inference in
the space of the function predicted by the NN (i.e., seismic velocity), instead
of in the traditional weight space. The result is a velocity-space SVGD for the
PINN-based seismic tomography model (vSVGD-PINN-ST) that decreases the
complexity of the problem thus enabling a more accurate and physically
consistent Bayesian estimation, as confirmed by synthetic tests in one- and
two-dimensional tomographic problem settings. The method allows PINN to be
applied to Bayesian seismic tomography practically for the first time. Not only
that, it can be a powerful tool not only for geophysical but also for general
PINN-based Bayesian estimation problems associated with compatible NNs
formulations and similar, or reduced, complexity