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