Model-independently calibrating the luminosity correlations of gamma-ray
  bursts using deep learning
  Gamma-ray bursts (GRBs) detected at high redshift can be used to trace the
Hubble diagram of the Universe. However, the distance calibration of GRBs is
not as easily as that of type Ia supernovae (SNe Ia). For the calibrating
method based on the empirical luminosity correlations, there is an underlying
assumption that the correlations should be universal over the whole redshift
range. In this paper, we investigate the possible redshift dependence of six
luminosity correlations with a completely model-independent deep learning
method. We construct a network combining the Recurrent Neural Networks (RNN)
and the Bayesian Neural Networks (BNN), where RNN is used to reconstruct the
distance-redshift relation by training the network with the Pantheon
compilation, and BNN is used to calculate the uncertainty of the
reconstruction. Using the reconstructed distance-redshift relation of Pantheon,
we test the redshift dependence of six luminosity correlations by dividing the
full GRB sample into two subsamples (low-$z$ and high-$z$ subsamples), and find
that only the $E_p-E_{\gamma}$ relation has no evidence for redshift
dependence. We use the $E_p-E_{\gamma}$ relation to calibrate GRBs, and the
calibrated GRBs give tight constraint on the flat $\Lambda$CDM model, with the
best-fitting parameter $\Omega_{\rm M}$=0.307$^{+0.065}_{-0.073}$.
