Gamma-Ray Bursts Calibrated by Using Artificial Neural Networks from the Pantheon+ Sample

In this paper, we calibrate the luminosity relation of gamma-ray bursts (GRBs) by Artificial Neural Networks (ANN) which is employed to analyze the Pantheon+ sample of type Ia supernovae (SNe Ia) in a manner independent of cosmological assumptions. The A219 GRB dataset are used to calibrate the Amati relation ($E_{\rm p}$-$E_{\rm iso}$) at low redshift with the ANN framework, facilitating the construction of the Hubble diagram at higher redshifts. Cosmological models are constrained with GRBs at high-redshift and the latest observational Hubble data (OHD) via a Markov Chain Monte Carlo numerical approach. For the Chevallier-Polarski-Linder (CPL) model within a flat universe, we obtain $\Omega_{\rm m} = 0.321^{+0.078}_{-0.069}$, $h = 0.654^{+0.053}_{-0.071}$, $w_0 = -1.02^{+0.67}_{-0.50}$, and $w_a = -0.98^{+0.58}_{-0.58}$ at the 1-$\sigma$ confidence level, which indicating a preference for dark energy with potential redshift evolution ($w_a \neq 0$). These findings by using ANN align closely with those derived from GRBs calibrated by using Gaussian Processes.