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Reconstructing f(T) Gravity from Observational Hubble Data: A PINN Approach

This study employs a Physics-Informed Neural Network (PINN) to non-parametrically reconstruct the Hubble parameter from observational data and subsequently derive the corresponding f(T) gravity function, demonstrating a robust, data-driven framework that aligns with the standard ΛCDM cosmological model.

Original authors: RAVI KANT MISHRA, Priya Awasthi

Published 2026-09-09

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CC BY 4.0

Original authors: RAVI KANT MISHRA, Priya Awasthi

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The universe is expanding, and it is doing so at an accelerating pace. This discovery, confirmed by decades of observations ranging from the light of distant exploding stars to the faint afterglow of the Big Bang, stands as one of the most profound facts of modern cosmology. To explain this acceleration, scientists have traditionally relied on a standard model that includes a mysterious force called dark energy, which pushes galaxies apart. However, this explanation comes with significant theoretical headaches, leading many researchers to ask if gravity itself might behave differently on the largest scales than we currently understand. Instead of adding a new, invisible substance to the cosmic inventory, some physicists propose that the rules of gravity need a slight adjustment. One such adjustment involves a theory called teleparallel gravity, which describes the force not as the curvature of space and time, but as a twisting or torsion of the fabric of the universe. If this twisting is the true driver of cosmic acceleration, then the mathematical function describing it should be deducible directly from how fast the universe is expanding at different points in its history.

A team of researchers at the Sant Longowal Institute of Engineering and Technology in India has taken a novel approach to this problem, using a type of artificial intelligence to read the history of the universe directly from observational data. Rather than guessing a specific mathematical formula for how gravity might change, they employed a system known as a Physics-Informed Neural Network. This is a computer program designed to learn patterns from real-world measurements while simultaneously obeying the fundamental laws of physics. The researchers fed the network data on the expansion rate of the universe at various distances, known as redshifts, and asked the system to reconstruct the entire expansion history without assuming a pre-set model. The network was trained to ensure its predictions matched the actual measurements while also satisfying the physical equations that govern how the universe should behave. The result was a smooth, reliable map of cosmic expansion that did not rely on any preconceived notions of what the universe should look like.

From this reconstructed expansion history, the team was able to work backward to determine the behavior of the torsion scalar, the specific quantity that measures the twisting of spacetime in this alternative gravity theory. By analyzing how this twisting changed over time, they were able to derive the specific mathematical function that describes the modified gravity, effectively reverse-engineering the laws of the universe from the data itself. The study found that the expansion history derived by the artificial intelligence aligns remarkably well with the standard cosmological model, particularly at lower distances where data is most abundant. The reconstructed model suggests that the universe is currently expanding at a rate of approximately 70.1 kilometers per second per megaparsec, with matter making up about 26 percent of the total cosmic energy budget. These values fit comfortably within the ranges expected by the standard model, suggesting that while the universe may be governed by a slightly different geometric rule, it still looks very much like the one we already know.

The researchers also examined the stability of their findings by testing how the model reacted to small changes in its internal settings and by analyzing the errors between their predictions and the actual data. The system proved to be highly stable, with the errors clustering tightly around zero, indicating that the artificial intelligence did not systematically overestimate or underestimate the expansion rate. The derived function describing the modified gravity could be accurately represented by a simple cubic polynomial, a type of curve that fits the data with an extremely high degree of precision. This suggests that the complex behavior of the universe's expansion can be captured by a relatively straightforward mathematical relationship, even without assuming that relationship in advance. The study demonstrates that machine learning, when guided by physical laws, can serve as a powerful tool for uncovering the underlying mechanics of the cosmos directly from observation, offering a new way to test whether our current understanding of gravity is complete or if a subtle twist in the geometry of space is the key to the universe's accelerating fate.

Technical Summary: Reconstructing f(T) Gravity from Observational Hubble Data via PINN

Problem Statement
The accelerating expansion of the universe, confirmed by Type Ia Supernovae, CMB, BAO, and Observational Hubble Data (OHD), presents a fundamental challenge within General Relativity, often attributed to dark energy or the cosmological constant ( $\Lambda$ ). While the standard $\Lambda$ CDM model fits observations well, it faces theoretical issues such as the coincidence problem and the Hubble tension. Modified gravity theories, specifically $f(T)$ gravity (an extension of Teleparallel Gravity where the torsion scalar $T$ is generalized to an arbitrary function $f(T)$ ), offer an alternative geometric explanation for cosmic acceleration without invoking dark energy.

A critical gap exists in reconstructing the specific functional form of $f(T)$ directly from data without assuming a pre-defined parametric model (e.g., linear or power-law). Traditional non-parametric methods like Gaussian Process Regression often rely on kernel assumptions that may introduce bias, while standard Artificial Neural Networks (ANNs) may produce solutions that violate physical laws. This study addresses the need for a model-independent, physically consistent reconstruction of the cosmic expansion history and the subsequent derivation of the $f(T)$ function.

Methodology
The authors employ a Physics-Informed Neural Network (PINN) framework to reconstruct the Hubble parameter $H(z)$ directly from OHD. The methodology integrates observational data with the governing physical equations of teleparallel gravity.

Data Source: The network is trained on a dataset of 31 observational Hubble data points spanning redshifts $z \in [0.07, 2.36]$ (Table 1).
Network Architecture: A feed-forward neural network with one input neuron (redshift $z$ ), two hidden layers (8 neurons each) using hyperbolic tangent activation functions, and one output neuron ( $H(z)$ ).
Loss Function Construction: The training minimizes a composite loss function ( $L_{total}$ $L_{t o t a l}$ ) comprising:
- Data Loss ( $L_{data}$ ): The $\chi^2$ difference between the predicted $H(z)$ and observational data.
- Physics Loss ( $L_{phy}$ ): Enforces the relationship between the deceleration parameter $q(z)$ and $H(z)$ , specifically penalizing deviations from the physical definition $q(z) = -1 + (1+z)\frac{H'(z)}{H(z)}$ .
- Monotonicity Loss ( $L_{mono}$ ): Ensures $H(z)$ behaves monotonically (non-increasing with redshift) to prevent non-physical oscillations.
- Boundary Loss ( $L_0$ ): Constrains the present-day Hubble parameter $H_0$ .
- Clipping Loss ( $L_{clip}$ ): Maintains numerical stability by confining predictions to physically acceptable ranges.
Optimization: The network is trained using the Adam optimizer with adaptive learning rates and momentum.
Reconstruction Chain:
- The trained PINN yields a smooth, non-parametric $H(z)$ .
- Using the FLRW metric relation in teleparallel gravity, $T = -6H(z)^2$ , the torsion scalar $T(z)$ is derived.
- The field equations for $f(T)$ gravity are utilized to numerically integrate and reconstruct the function $f(T)$ without assuming an analytical form.
- The resulting numerical $f(T)$ is approximated by a cubic polynomial for analytical comparison.

Key Results

Hubble Parameter Reconstruction: The PINN successfully reconstructs $H(z)$ across the observed redshift range. The reconstruction aligns closely with the standard $\Lambda$ CDM model, particularly at low and intermediate redshifts. The majority of observational data points fall within the $1\sigma$ and $2\sigma$ confidence bands of the PINN prediction.
Cosmic Expansion History: The reconstructed deceleration parameter $q(z)$ indicates a transition from deceleration to acceleration at $z_t \approx 0.6$ , consistent with current cosmological understanding.
f(T) Function: The model-independent reconstruction of $f(T)$ demonstrates a smooth, stable dependency on the torsion scalar $T$ . The numerical reconstruction is well-approximated by a cubic polynomial ( $f(T) = aT^3 + bT^2 + cT + d$ ) with a coefficient of determination $R^2 = 0.999996$ and an RMSE of $1.7488 \times 10^2$ .
Parameter Constraints: A $\chi^2$ -based likelihood analysis in the $(\Omega_m, H_0)$ plane yields best-fit values of $H_0 \approx 70.1 \text{ km s}^{-1} \text{ Mpc}^{-1}$ and $\Omega_m \approx 0.26$ . The confidence contours show a negative correlation between these parameters, consistent with standard cosmological degeneracies.
Convergence and Stability: The loss landscape analysis reveals a smooth, basin-like structure with a single well-defined minimum, indicating the trained solution is stable and robust against weight perturbations. Residual analysis confirms the model is globally unbiased, with errors centered around zero.

Significance and Claims
The paper claims to establish a direct, data-driven link between observational cosmology and modified teleparallel gravity. By utilizing PINNs, the study demonstrates that it is possible to:

Reconstruct the cosmic expansion history without assuming a specific cosmological parametrization.
Derive the underlying $f(T)$ gravity function directly from observational constraints, ensuring the solution satisfies both data fidelity and physical laws (teleparallel field equations).
Provide a robust framework for investigating the late-time evolution of the universe in teleparallel gravity, offering an alternative to parametric reconstruction methods that may suffer from kernel selection bias or physical inconsistency.

The authors conclude that the PINN approach serves as an effective, model-independent tool capable of recovering both expansion history and modified gravity functional forms from observational datasets, validating the potential of physics-informed machine learning in cosmological reconstruction.

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