  A computational method based on the non-linear Gaussian process (GP), known
as deep Gaussian processes (deep GPs) for uncertainty quantification &
propagation in modelling of flow through heterogeneous porous media is
presented. The method is also used for reducing dimensionality of model output
and consequently emulating highly complex relationship between hydrogeological
properties and reduced order fluid velocity field in a tractable manner. Deep
GPs are multi-layer hierarchical generalisations of GPs with multiple,
infinitely wide hidden layers that are very efficient models for deep learning
and modelling of high-dimensional complex systems by tackling the complexity
through several hidden layers connected with non-linear mappings. According to
this approach, the hydrogeological data is modelled as the output of a
multivariate GP whose inputs are governed by another GP such that each single
layer is either a standard GP or the Gaussian process latent variable model. A
variational approximation framework is used so that the posterior distribution
of the model outputs associated to given inputs can be analytically
approximated. In contrast to the other dimensionality reduction, methods that
do not provide any information about the dimensionality of each hidden layer,
the proposed method automatically selects the dimensionality of each hidden
layer and it can be used to propagate uncertainty obtained in each layer across
the hierarchy. Using this, dimensionality of the full input space consists of
both geometrical parameters of modelling domain and stochastic hydrogeological
parameters can be simultaneously reduced without the need for any
simplifications generally being assumed for stochastic modelling of subsurface
flow problems. It allows estimation of the flow statistics with greatly reduced
computational efforts compared to other stochastic approaches such as Monte
Carlo method.
