Eur. Phys. J. C (2025) 85: 1301
https://doi.org/10.1140/epjc/s10052-025-14993-2

Regular Article - Theoretical Physics

Towards a machine learning solution for hubble tension: Physics-Informed Neural Network (PINN) analysis of Tsallis Holographic Dark Energy in presence of neutrinos

Department of Physics, Lorestan University, Khoramabad, Iran

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Received: 26 June 2025
Accepted: 26 October 2025
Published online: 14 November 2025

Abstract

We present a Physics-Informed Neural Network (PINN) framework for reconstructing the redshift-dependent Hubble parameter $H(z)$ within the Tsallis Holographic Dark Energy (THDE) model extended by massive neutrinos. In this approach, the modified Friedmann equation is incorporated into the neural network loss function, enabling training on Cosmic Chronometers data up to $z\le 2$. The framework allows for the simultaneous estimation of the Hubble constant $H_0$, the neutrino density parameter ${\mathrm{\Omega }}_{\nu }$, and the Tsallis non-extensivity index $\delta$. Uncertainty quantification is performed through dropout simulations, resulting in statistically consistent $1\sigma$ confidence bands. Our results show that the THDE+$\nu$ model, reconstructed via PINN, alleviates the statistical Hubble tension from the canonical $\sim 5\sigma$ level down to a range of $0.5\sigma \le T\le 2.2\sigma$, depending on the redshift sampling. Additionally, we constrain the total neutrino mass to $\mathrm{\Sigma }{m}_{\nu }<0.11\text{eV}$. A detailed comparison with the traditional Markov Chain Monte Carlo (MCMC) analysis demonstrates the consistency of both methods, while highlighting the competitiveness of the PINN-based THDE framework as a robust, data-driven approach for non-parametric cosmological inference within generalized thermodynamics.