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Towards a Machine Learning Solution for Hubble Tension: Physics-Informed Neural Network (PINN) Analysis of Tsallis Holographic Dark Energy in Presence of Neutrinos

Muhammad Yarahmadi, Amin Salehi

TL;DR

This paper tackles the Hubble tension by reconstructing the expansion history $H(z)$ within a Tsallis holographic dark energy (THDE) model augmented by massive neutrinos, using a Physics-Informed Neural Network (PINN) that embeds the modified Friedmann equation into its loss. The PINN simultaneously infers $H_0$, $\Omega_\nu$, and the non-extensive parameter $\delta$, and provides uncertainty quantification via Monte Carlo dropout. Across Cosmic Chronometers data and in comparison with traditional MCMC analyses, THDE+$\nu$ with PINN demonstrates a reduction of the Hubble tension to approximately $0.5\sigma$ to $2.2\sigma$ and constrains the total neutrino mass to $\Sigma m_\nu<0.11$ eV, while revealing $\delta$ in the mildly phantom-like regime ($\delta\approx1.1$). The study highlights PINN as a robust, data-driven tool for non-parametric cosmological inference within generalized thermodynamics and demonstrates consistency with standard cosmological probes when compared to MCMC results.

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 \(z \leq 2\). The framework allows for the simultaneous estimation of the Hubble constant \(H_0\), the neutrino density parameter \(Ω_ν\), and the Tsallis non-extensivity index \(δ\). Uncertainty quantification is performed through dropout simulations, resulting in statistically consistent \(1σ\) confidence bands. Our results show that the THDE+$ν$ model, reconstructed via PINN, alleviates the statistical Hubble tension from the canonical \(\sim 5σ\) level down to a range of \(0.5σ\leq T \leq 2.2σ\), depending on the redshift sampling. Additionally, we constrain the total neutrino mass to \(Σm_ν< 0.11\,\text{eV}\). 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.

Towards a Machine Learning Solution for Hubble Tension: Physics-Informed Neural Network (PINN) Analysis of Tsallis Holographic Dark Energy in Presence of Neutrinos

TL;DR

This paper tackles the Hubble tension by reconstructing the expansion history within a Tsallis holographic dark energy (THDE) model augmented by massive neutrinos, using a Physics-Informed Neural Network (PINN) that embeds the modified Friedmann equation into its loss. The PINN simultaneously infers , , and the non-extensive parameter , and provides uncertainty quantification via Monte Carlo dropout. Across Cosmic Chronometers data and in comparison with traditional MCMC analyses, THDE+ with PINN demonstrates a reduction of the Hubble tension to approximately to and constrains the total neutrino mass to eV, while revealing in the mildly phantom-like regime (). The study highlights PINN as a robust, data-driven tool for non-parametric cosmological inference within generalized thermodynamics and demonstrates consistency with standard cosmological probes when compared to MCMC results.

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 . The framework allows for the simultaneous estimation of the Hubble constant , the neutrino density parameter , and the Tsallis non-extensivity index . Uncertainty quantification is performed through dropout simulations, resulting in statistically consistent confidence bands. Our results show that the THDE+ model, reconstructed via PINN, alleviates the statistical Hubble tension from the canonical level down to a range of , depending on the redshift sampling. Additionally, we constrain the total neutrino mass to . 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.

Paper Structure

This paper contains 41 sections, 36 equations, 9 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Motivation for Including Massive Neutrinos in the THDE Framework
  3. Motivation for Using the Physics-Informed Neural Network (PINN) Model through the Tsallis Holographic Dark Energy (THDE) Model to Alleviate the Hubble Tension
  4. Tsallis Holographic Dark Energy (THDE) with Neutrinos
  5. Friedmann Dynamics in the Presence of Massive Neutrinos and Tsallis Holographic Dark Energy
  6. Modeling the Neutrino Sector
  7. Physics-Informed Neural Network Framework
  8. Neural Network Architecture
  9. Input Layer:
  10. Hidden Layers:
  11. Multi-Head Output Layer:
  12. Physical Motivation for Architectural Choices:
  13. Implementation:
  14. Composite Loss Function and Physical Constraints
  15. Bayesian Uncertainty Quantification via Monte Carlo Dropout
  16. ...and 26 more sections

Figures (9)

  • Figure 1: Reconstruction of the Hubble constant $H_0$ and its comparison with observational constraints using the PINN framework within the Tsallis Holographic Dark Energy (THDE) model with neutrinos. Panels:(a)$H_0$ estimates for $z \leq 0.5$ across different sample sizes ($N=50,100,150,200$); (b) the same analysis for $z \leq 1.0$; (c) results for $z \leq 1.5$; (d) estimates including data up to $z \leq 2.0$. Horizontal shaded regions represent the $1\sigma$ bounds from Planck 2018 (blue) and Riess et al. (2022, orange). The PINN predictions remain within $1$--$2\sigma$ of these constraints, indicating the robustness of the THDE+neutrino scenario and the stability of $H_0 \simeq 70$--$71~\mathrm{km\,s^{-1}\,Mpc^{-1}}$ across redshift ranges.
  • Figure 2: Posterior reconstructions of the Tsallis parameter $\delta$ using the Physics-Informed Neural Network (PINN), shown for four representative redshifts in panels (a)–(d). For each redshift, $N$ independent samples were drawn from the trained network to propagate both observational and epistemic uncertainties. The resulting distributions illustrate the stability and redshift dependence of the PINN-based inference, and highlight how sampling enables a quantitative characterization of parameter uncertainties across cosmic time.
  • Figure 3: Posterior distributions of the Hubble constant $H_{0}$ obtained from different redshift ranges. Panel (a) shows the likelihood derived from cosmic chronometers at $z \leq 0.5$; (b) corresponds to $z \leq 1.0$; (c) presents the results at $z \leq 1.5$; and (d) displays the constraints up to $z \leq 2.0$. These panels illustrate how the inferred value of $H_{0}$ depends on the chosen redshift interval.
  • Figure 4: PINN-based reconstructions of the Hubble parameter $H(z)$ within the Tsallis Holographic Dark Energy (THDE) framework for fixed values of the Tsallis parameter $\delta$ ($\delta=1.0,\,1.1,\,1.2,\,1.3,\,1.4$). For each chosen $\delta$, the neural network is trained under the THDE constraints using the cosmic chronometer (CC) data and reconstructs the corresponding $H(z)$ evolution over redshift. The shaded regions denote the $68\%$ and $95\%$ confidence intervals of the CC measurements. This figure is intended to highlight the sensitivity of the expansion history to $\delta$, validate the model-consistent PINN reconstructions against observations, and indicate the redshift ranges with the greatest discriminatory power.
  • Figure 5: The MCMC results for cosmological parameters for full CC data.
  • ...and 4 more figures