Reconstructing Gamma Ray Burst Energy Relations with Observational H(z) data in Neural Network Framework

Gamma-ray bursts (GRBs) offer a powerful probe of the cosmic expansion history far beyond the redshift range accessible to Type Ia supernovae. However, the study of cosmological models using GRBs is hindered by the circularity problem, which arises from assuming a fiducial cosmological model during GRB luminosity distance calibration. In this work, we perform a model-independent calibration of GRB luminosity relations using observational measurements of the Hubble parameter from the A220 and J220 compilations, thereby avoiding explicit cosmological assumptions. We employ an Artificial Neural Network to reconstruct the calibration relation directly from the data. In addition, we implement a Bayesian Neural Network framework as an alternative approach, enabling a data-driven treatment of both statistical and systematic uncertainties. The calibrated GRB sample is used to constrain the Amati relation, and we systematically compare the outcomes obtained from different calibration techniques and datasets. We find that the Amati relation slopes derived from the two neural network approaches are consistent with each other and with previous low-redshift calibrations obtained using model-independent methods. The Bayesian Neural Network approach provides a more robust framework for propagating uncertainties in the calibration procedure.