Physics-Informed Priors with Application to Boundary Layer Velocity

TL;DR

This paper introduces a Bayesian framework that incorporates PDE-based priors into neural networks to improve physical consistency and uncertainty quantification in boundary layer velocity predictions with minimal data.

Contribution

It presents a novel model-based approach that integrates PDEs as priors in Bayesian neural networks, enabling physics-informed forecasts with uncertainty quantification.

Findings

Requires few observations for accurate, physically consistent forecasts.

Outperforms non-informed priors in complex fluid systems.

Successfully applied to simulated and experimental boundary layer data.

Abstract

One of the most popular recent areas of machine learning predicates the use of neural networks augmented by information about the underlying process in the form of Partial Differential Equations (PDEs). These physics-informed neural networks 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 which regards the PDE as a prior information of a deep Bayesian neural network. The prior is calibrated without data to resemble the PDE solution in the prior mean, while our degree in confidence on the PDE with respect to the data is expressed in terms of the prior variance. The information embedded in the PDE is then propagated to the posterior yielding physics-informed forecasts with uncertainty quantification. We apply our approach to a simulated viscous fluid and to experimentally-obtained turbulent boundary layer velocity in a water tunnel using an appropriately simplified Navier-Stokes equation. Our approach requires very few observations to produce physically-consistent forecasts as opposed to non-physical forecasts stemming from non-informed priors, thereby allowing forecasting complex systems where some amount of data as well as some contextual knowledge is available.