A hybrid data-driven-physics-constrained Gaussian process regression framework with deep kernel for uncertainty quantification

Abstract

Gaussian process regression (GPR) is a well-known machine learning method employed for various applications such as uncertainty quantifications. However, GPR is an inherently data-driven method that requires a sufficiently large dataset. If appropriate physics constraints (e.g. those expressed in partial differential equations) can be incorporated, the amount of data can be greatly reduced, and the accuracy can be further improved. In this study, we propose a hybrid data-driven-physics-constrained Gaussian process regression framework. We encode the physics knowledge with a Boltzmann-Gibbs distribution and derive our model using the maximum likelihood (ML) approach. We apply the deep kernel learning method. Through the training of a deep neural network, the proposed model learns an adaptive covariance function from both data and physics constraints. The proposed model achieves good results in high-dimensional problems, and correctly propagates the uncertainty while requiring very limited labelled data.