Physics-based Penalization for Hyperparameter Estimation in Gaussian Process Regression

In Gaussian Process Regression (GPR), hyperparameters are often estimated by maximizing the marginal likelihood function. However, this data-dominant hyperparameter estimation process can lead to poor extrapolation performance and often violates known physics, especially in sparse data scenarios. In this paper, we embed physics-based knowledge through penalization of the marginal likelihood objective function and study the effect of this new objective on consistency of optimal hyperparameters and quality of GPR fit. Three case studies are presented, where physics-based knowledge is available in the form of linear Partial Differential Equations (PDEs), while initial or boundary conditions are not known so direct forward simulation of the model is challenging. The results reveal that the new hyperparameter set obtained from the augmented marginal likelihood function can improve the prediction performance of GPR, reduce the violation of the underlying physics, and mitigate overfitting problems.