  The correlation length-scale next to the noise variance are the most used
hyperparameters for the Gaussian processes. Typically, stationary covariance
functions are used, which are only dependent on the distances between input
points and thus invariant to the translations in the input space. The
optimization of the hyperparameters is commonly done by maximizing the log
marginal likelihood. This works quite well, if the distances are uniform
distributed. In the case of a locally adapted or even sparse input space, the
prediction of a test point can be worse dependent of its position. A possible
solution to this, is the usage of a non-stationary covariance function, where
the hyperparameters are calculated by a deep neural network. So that the
correlation length scales and possibly the noise variance are dependent on the
test point. Furthermore, different types of covariance functions are trained
simultaneously, so that the Gaussian process prediction is an additive overlay
of different covariance matrices. The right covariance functions combination
and its hyperparameters are learned by the deep neural network. Additional, the
Gaussian process will be able to be trained by batches or online and so it can
handle arbitrarily large data sets. We call this framework Deep Gaussian
Covariance Network (DGCP). There are also further extensions to this framework
possible, for example sequentially dependent problems like time series or the
local mixture of experts. The basic framework and some extension possibilities
will be presented in this work. Moreover, a comparison to some recent state of
the art surrogate model methods will be performed, also for a time dependent
problem.
