  This paper deals with surrogate modelling of a computer code output in a
hierarchical multi-fidelity context, i.e., when the output can be evaluated at
different levels of accuracy and computational cost. Using observations of the
output at low- and high-fidelity levels, we propose a method that combines
Gaussian process (GP) regression and Bayesian neural network (BNN), in a method
called GPBNN. The low-fidelity output is treated as a single-fidelity code
using classical GP regression. The high-fidelity output is approximated by a
BNN that incorporates, in addition to the high-fidelity observations,
well-chosen realisations of the low-fidelity output emulator. The predictive
uncertainty of the final surrogate model is then quantified by a complete
characterisation of the uncertainties of the different models and their
interaction. GPBNN is compared with most of the multi-fidelity regression
methods allowing to quantify the prediction uncertainty.
