Neural networks and standard cosmography with newly calibrated high
  redshift GRB observations
  Gamma-ray bursts (GRBs) detected at high redshift can be used to trace the
cosmic expansion history. However, the calibration of their luminosity
distances is not an easy task in comparison to Type Ia Supernovae (SNeIa). To
calibrate these data, correlations between their luminosity and other observed
properties of GRBs need to be identified, and we must consider the validity of
our assumptions about these correlations over their entire observed redshift
range. In this work, we propose a new method to calibrate GRBs as cosmological
distance indicators using SNeIa observations with a machine learning
architecture. As well we include a new data GRB calibrated sample using
extended cosmography in a redshift range above $z>3.6$. An overview of this
machine learning technique was developed in [1] to study the evolution of dark
energy models at high redshift. The aim of the method developed in this work is
to combine two networks: a Recurrent Neural Network (RNN) and a Bayesian Neural
Network (BNN). Using this computational approach, denoted RNN+BNN, we extend
the network's efficacy by adding the computation of covariance matrices to the
Bayesian process. Once this is done, the SNeIa distance-redshift relation can
be tested on the full GRB sample and therefore used to implement a cosmographic
reconstruction of the distance-redshift relation in different regimes. Thus,
our newly-trained neural network is used to constrain the parameters describing
the kinematical state of the Universe via a cosmographic approach at high
redshifts (up to $z\approx 10$), wherein we require a very minimal set of
assumptions on the deep learning architecture itself that do not rely on
dynamical equations for any specific theory of gravity.
