Bayesian Proper Orthogonal Decomposition for Learnable Reduced-Order Models With Uncertainty Quantification

Designing and/or controlling complex systems in science and engineering relies on appropriate mathematical modeling of systems dynamics. Classical differential equation-based solutions in applied/computational mathematics are often computationally demanding. Recently, the connection between reduced-order models of high-dimensional differential equation systems and surrogate machine learning models has been explored. However, the focus of both existing reduced-order and machine learning models has been how to best approximate the high-fidelity model of choice. Due to high complexity and often limited training data, it is critical for the models to have reliable uncertainty quantification. In this paper, we propose such a novel framework of Bayesian reduced-order models naturally equipped with uncertainty quantification as it learns the distributions of the parameters of the reduced-order models instead of their point estimates. Specifically, we develop
learnable Bayesian proper orthogonal decomposition
(BayPOD) that learns the distributions of both the POD projection bases and the mapping from the system input parameters to the projected scores/coefficients so that the learned BayPOD can help predict high-dimensional systems dynamics/fields with reliable uncertainty estimates. The developed BayPOD inherits the capability of embedding physics constraints when learning the POD-based surrogate models, a desirable feature when studying complex systems with limited available data. Furthermore, the proposed BayPOD is an end-to-end solution, which unlike other surrogate-based methods, does not require separate POD and machine learning steps. The results from a real-world case study of the pressure field around an airfoil shows the potential of learnable BayPOD as a new family of reduced-order models with reliable uncertainty estimates.