Bayesian Neural Networks With Physics-Informed Priors With Application to Boundary Layer Velocity

One of the most popular recent areas of machine learning predicates the use of neural networks (NNs) augmented by information about the underlying process in the form of partial differential equations (PDEs). These physics-informed NNs (PINNs) are obtained by penalizing the inference with a PDE and have been cast as a minimization problem currently lacking a formal approach to quantify the uncertainty. In this work, we propose a novel model-based framework that regards the PDE as a prior information of a deep Bayesian NN (BNN), physics-informed prior (PIP)-BNN. The prior is calibrated without data to resemble the PDE solution in the prior mean, while our degree of confidence in the PDE with respect to the data is expressed in terms of the prior variance. The information embedded in the PDE are then propagated to the posterior yielding physics-informed forecasts with uncertainty quantification. We apply our approach to a simulated viscous fluid and to an experimentally obtained turbulent boundary layer velocity in a water tunnel using an appropriately simplified Navier-Stokes (NS) equation. Our approach requires very few observations to produce physically consistent forecasts as opposed to nonphysical forecasts stemming from noninformed priors, thereby allowing forecasting complex systems, where some amount of data as well as some contextual knowledge is available.