We calibrate the distance and reconstruct the Hubble diagram of gamma-ray bursts (GRBs) using deep learning. We construct an artificial neural network, which combines the recurrent neural network and Bayesian neural network, and train the network using the Pantheon compilation of type-Ia supernovae. The trained network is used to calibrate the distance of 174 GRBs based on the Combo-relation. We verify that there is no evident redshift evolution of Combo-relation, and obtain the slope and intercept parameters, γ = 0.856 −0.078 and logA = 49.661 +0.199 −0.217, with an intrinsic scatter σint = 0.228 +0.041 −0.040. Our calibrating method is independent of cosmological model, thus the calibrated GRBs can be directly used to constrain cosmological parameters. It is shown that GRBs alone can tightly constrain the ΛCDM model, with ΩM = 0.280 +0.049 −0.057. However, the constraint on the ωCDM model is relatively looser, with ΩM = 0.345 +0.059 −0.060 and ω < −1.414. The combination of GRBs and Pantheon can tightly constrain the ωCDM model, with ΩM = 0.336 +0.055 −0.050 and ω = −1.141 +0.156 −0.135.