Predictive uncertainty quantification for Bayesian Physics-Informed Neural Network (PINN) in hypocentre estimation problem

Physics-informed neural networks (PINNs) have appeared on the scene as a flexible and a versatile framework for solving partial differential equations (PDEs), along with any initial or boundary conditions. An important component of these solutions, especially when using the data as a boundary condition, is our confidence in their accuracy. There has been little study of PINN accuracy as an inversion tool. We introduce an approximate Bayesian framework for estimating predictive uncertainties in Physics-Informed Neural Network (PINN). This work investigates propagation of uncertainties from the random realisations of Eikonal PINN's weights and biases using the Laplace approximation. Laplace approximation is arguably the simplest family of approximations for the intractable posteriors of deep neural networks. We specifically use a hypocenter estimation problem based on the eikonal equation to demonstrate the approach effectiveness in measuring the predictive uncertainty in the PINN hypocenter estimation. The uncertainties estimation from this approach is called predictive uncertainty or, simply, forward modelling uncertainty in the context of PINN.