Bayesian seismic tomography based on velocity-space Stein variational gradient descent for physics-informed neural network

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.