A Bayesian PINN Framework for Barrow-Tsallis Holographic Dark Energy with Neutrinos: Toward a Resolution of the Hubble Tension

We investigate the Barrow-Tsallis Holographic Dark Energy (BTHDE) model using both traditional Markov Chain Monte Carlo (MCMC) methods and a Bayesian Physics-Informed Neural Network (PINN) framework, employing a range of cosmological observations. Our analysis incorporates data from Cosmic Microwave Background (CMB), Baryon Acoustic Oscillations (BAO), CMB lensing, Cosmic Chronometers (CC), and the Pantheon+ Type Ia supernova compilation. We focus on constraining the Hubble constant $ H_0 $, the nonextensive entropy index $ q $, the Barrow exponent $ \Delta $, and the Granda-Oliveros parameters $ \alpha $ and $ \beta $, along with the total neutrino mass $ \Sigma m_\nu $. The Bayesian PINN approach yields more precise constraints than MCMC, particularly for $ \beta $, and tighter upper bounds on $ \Sigma m_\nu $. The inferred values of $ H_0 $ from both methods lie between those from Planck 2018 and SH$_0$ES (R22), alleviating the Hubble tension to within $ 1.3\sigma $-$2.1\sigma $ depending on the dataset combination. Notably, the Bayesian PINN achieves consistent results across CC and Pantheon+ datasets, while maintaining physical consistency via embedded differential constraints. The combination of CMB and late-time probes leads to the most stringent constraints, with $ \Sigma m_\nu < 0.114 $ eV and $ H_0 = 70.6 \pm 1.35 $ km/s/Mpc. These findings suggest that the BTHDE model provides a viable framework for addressing cosmological tensions and probing modified entropy scenarios, while highlighting the complementary strengths of machine learning and traditional Bayesian inference in cosmological modeling.