The obstacle is known in the trade as the circularity problem, and it has plagued gamma-ray burst cosmology for nearly two decades. The most widely used tool for taming these explosions is the Amati relation, an empirical correlation between the rest-frame spectral peak energy of a burst and its isotropic-equivalent radiated energy. The catch is that the isotropic energy cannot be measured directly. It must be derived from the burst’s bolometric fluence and its luminosity distance, and luminosity distance depends on the cosmological model you assume. Historically, researchers calibrated the Amati relation by assuming a standard Lambda-CDM cosmology, then used the calibrated relation to constrain cosmological models. The reasoning goes in a circle: the answer you get out is quietly shaped by the assumption you put in.

Cosmologists have proposed a variety of escape routes over the years. Some teams interpolated distances from low-redshift supernovae without invoking any particular background cosmology, while others fitted the burst relations and cosmological parameters simultaneously in a single statistical framework. A different strategy, adopted by Amati and collaborators, exploited observational Hubble data gathered from cosmic chronometers, galaxies whose stellar populations act as natural clocks that tick out the age of the Universe at different redshifts. Because these Hubble measurements are essentially model-independent, they can anchor burst calibrations without smuggling in a preferred cosmology. The Delhi team builds on this idea but replaces the older reconstruction techniques, such as Bezier polynomials and Gaussian process regression, with something more modern and more flexible: neural networks.

The choice of machine learning is deliberate. Gaussian process regression, long the standard non-parametric tool in cosmology, implicitly assumes Gaussian statistics and is notoriously sensitive to the choice of kernel function, which can quietly shape the reconstructed curve. Artificial neural networks, by contrast, impose far fewer assumptions on the data and learn the underlying function directly from observations. The researchers trained a network on 32 observational Hubble parameter measurements spanning redshifts from 0.07 to 1.965, drawn from an updated compilation by Ratra and colleagues. With so few data points, architecture choices matter enormously, so the team ran a systematic grid search, scoring candidate networks with a RISK function drawn from statistical decision theory, which measures the expected squared prediction error while accounting for observational uncertainties.

The search converged on a strikingly simple answer: a single hidden layer of 4096 neurons, using the exponential linear unit activation function and the Adam optimizer with a decaying learning rate over 5750 epochs. Wider networks offered no meaningful improvement, and deeper architectures with two, three or four hidden layers showed clear signs of overfitting, producing spurious oscillations rather than smooth reconstructions. Smaller networks underfit the data, yielding higher risk values. The chosen configuration sat in a sweet spot balancing flexibility against generalization, and leave-one-out cross-validation confirmed that normalized residuals scattered around zero with no systematic trend in redshift, indicating the reconstruction was unbiased and not driven by any individual data point.

Standard neural networks, however, have a well-known weakness: they produce point predictions without honest error bars, and can even confidently report inaccurate values. For sparse, noisy cosmological datasets this is a serious liability. The team addressed it in two ways. First, they supplemented the network with bootstrap resampling, generating 1000 resampled datasets by drawing the 32 measurements with replacement, training the network independently on each, and averaging the resulting ensemble of reconstructions. The variance across the ensemble provides an estimate of reconstruction uncertainty, though the authors caution this primarily captures sampling variability rather than full model uncertainty. Second, and more ambitiously, they deployed a Bayesian neural network, which treats every weight and bias in the network as a random variable with its own probability distribution.

The Bayesian formulation, originally developed by Bishop and by MacKay in the 1990s, transforms network training into probabilistic inference. Starting from zero-mean Gaussian priors on all parameters, the network updates its beliefs through the likelihood of the observed Hubble data, yielding a full posterior distribution rather than a single best-fit set of weights. Because the posterior is high-dimensional and nonlinear, the researchers sampled it with the no-u-turn sampler, an adaptive variant of Hamiltonian Monte Carlo implemented in the Pyro probabilistic programming library. Model selection was guided by the widely applicable information criterion, which balances predictive accuracy against model complexity and naturally encodes Occam’s razor. Notably, the analysis found that predictive performance was largely insensitive to network width once sufficient capacity was reached, while the prior variance played the dominant role in regulating effective model complexity.

With the Hubble parameter reconstructed by both methods, the team converted the expansion history into luminosity distances by integrating the inverse expansion rate, then calibrated the Amati relation for two independent burst catalogues: the A220 sample of 220 long gamma-ray bursts assembled by Khadka and Ratra, and the J220 sample from Jia and colleagues, drawn from the Swift and Fermi missions. Because the calibration data extend only to redshift 1.965, the researchers restricted the fitting to 115 bursts from A220 and 129 from J220 below that cutoff, using Markov chain Monte Carlo with the emcee sampler to constrain the relation’s slope, intercept and intrinsic scatter simultaneously.

The results are reassuringly consistent across methods and datasets. For the A220 sample, the artificial network yielded a slope of 1.231 with uncertainties of about 0.09, while the Bayesian network gave 1.218, agreeing within one sigma. Both match previous model-independent calibrations based on Gaussian processes and interpolation techniques, which found slopes near 1.29 to 1.30. For the J220 sample, both networks delivered an identical slope of 1.381 with tighter uncertainties, consistent with simultaneous-fitting and redshift-binning analyses. The J220 sample also showed reduced intrinsic scatter, suggesting a more homogeneous burst population or fewer observational systematics, a difference the authors attribute to variations in sample selection, redshift coverage and measurement uncertainties between the catalogues.

The broader message is twofold. First, the close agreement between the two neural network approaches demonstrates that machine-learning calibration of gamma-ray bursts is robust against the choice of algorithm, strengthening confidence that these explosions can eventually serve as genuine cosmological probes rather than model-dependent curiosities. Second, the Bayesian framework emerges as the preferred tool for the future: it propagates uncertainty from the data through the network parameters to the final prediction in a principled way, captures the epistemic uncertainty that deterministic networks miss, and guards against overfitting when data are sparse. As upcoming missions swell the high-redshift burst sample, and as researchers extend the technique to other burst correlations such as the Ghirlanda, Yonetoku and Dainotti relations, Bayesian networks may become the standard machinery for turning the Universe’s most violent explosions into its most far-reaching measuring sticks.

Subject of Research: Model-independent calibration of gamma-ray burst Amati relation using neural network reconstruction of observational Hubble parameter data

Article Title: Reconstructing gamma-ray burst energy relations with observational H(z) data in a neural network framework

Article References: Aurpa, N. B., Habib, A. D., & Rani, N. (2026). Reconstructing gamma-ray burst energy relations with observational H(z) data in a neural network framework. The European Physical Journal C, 86(9), Article 1133. https://doi.org/10.1140/epjc/s10052-026-16318-3

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16318-3

Keywords: gamma-ray bursts, cosmology, Amati relation, neural networks, Bayesian neural networks, Hubble parameter, circularity problem, luminosity distance, machine learning, observational Hubble data, cosmic expansion, Markov chain Monte Carlo