  Gaussian process regression is a widely-applied method for function
approximation and uncertainty quantification. The technique has gained
popularity recently in the machine learning community due to its robustness and
interpretability. The mathematical methods we discuss in this paper are an
extension of the Gaussian-process framework. We are proposing advanced kernel
designs that only allow for functions with certain desirable characteristics to
be elements of the reproducing kernel Hilbert space (RKHS) that underlies all
kernel methods and serves as the sample space for Gaussian process regression.
These desirable characteristics reflect the underlying physics; two obvious
examples are symmetry and periodicity constraints. In addition, non-stationary
kernel designs can be defined in the same framework to yield flexible
multi-task Gaussian processes. We will show the impact of advanced kernel
designs on Gaussian processes using several synthetic and two scientific data
sets. The results show that including domain knowledge, communicated through
advanced kernel designs, has a significant impact on the accuracy and relevance
of the function approximation.
