astronomy/ai produced the result/The European Physical Journal C 2026 · v2
Neural networks rebuild cosmic expansion history to calibrate gamma-ray burst energies
Researchers used two neural networks to reconstruct how fast the universe has been expanding at different distances, then used that curve to work out distances to hundreds of gamma-ray bursts and fit a known relation between their energies.
spectrum · one line per step, placed by what the step does · bright lines used AI
Reconstructing gamma-ray burst energy relations with observational H(z) data in a neural network framework
The European Physical Journal C, 2026
doi:10.1140/epjc/s10052-026-16318-3 · record aix-00069 v2 · checked 2026-10-08
- AI was for
- Property prediction
- Model family
- Multilayer perceptron
- Checked by
- Held-out
- Code
- not reported
The finding the paper is about came from the AI.
What this research was about
Gamma-ray bursts are the brightest explosions astronomers see, and they can be spotted far across the universe. That makes them tempting as distance markers. The trouble is that a burst's apparent brightness only tells you its true energy output if you already know how far away it is. One well-studied pattern, the Amati relation, links a burst's total energy to the energy at which its radiation peaks. Pinning down that pattern normally means assuming a particular model of how the universe expands, which risks arguing in a circle when the bursts are then used to test such models.
A way round this is to get distances from something measured directly instead. Measurements of the Hubble parameter, which describes the expansion rate at a given redshift, exist at scattered points. The authors set out to turn those scattered points into a smooth curve without committing to a cosmological model, and then use that curve to calibrate the bursts.
Where AI came in
The reconstruction of the expansion curve was done by neural networks, standing in for the parametric fits or kernel-based smoothing usually used for this step. Two were built. One was a conventional network with a single hidden layer of 4096 units, trained from scratch on the 32 Hubble measurements; because the dataset was too small to split, the researchers trained it on 1000 resampled versions of the data and took the average as the answer and the spread as the uncertainty. The second was a Bayesian version, where probability distributions rather than fixed values are assigned to the network's internal numbers, sampled with a Markov chain method.
Everything downstream rested on that output. The reconstructed curves were integrated to give distances to 115 bursts from one catalogue and 129 from another, converted into energies, and then fitted for the Amati relation's slope, intercept and scatter by a separate statistical sampler. The reported slopes, 1.231 and 1.218 for one sample and 1.381 for the other, therefore depend on the networks' reconstruction. A leave-one-out check, holding back each Hubble point in turn, was used to inspect how well the network reproduced it.
Written by AIxSci from the checked record below, to give context for readers outside the field. It is not part of the record.
The work
Technical · from the record
The authors reconstruct the Hubble parameter as a function of redshift from 32 observational Hubble data points using two neural networks: a single-hidden-layer artificial neural network trained on bootstrap resamples, and a Bayesian neural network whose weight posterior is sampled with the No-U-Turn Sampler. The reconstructed H(z) is integrated to obtain luminosity distances for 115 GRBs from the A220 compilation and 129 from the J220 compilation below z = 1.965, which are then used to fit the Amati relation by MCMC. For A220 the reported slope is b = 1.231 (+0.093/-0.093) from the ANN and b = 1.218 (+0.096/-0.094) from the BNN; for J220 both approaches give b = 1.381. The paper reports that the two neural-network slopes agree with each other and, within 1 sigma, with earlier model-independent low-redshift calibrations.
How AI was used
Two learned regressors stood in for parametric or kernel-based reconstruction of the cosmic expansion history. For the deterministic network, a grid search over width (powers of two from 2^7 to 2^14) and depth (up to four hidden layers) was scored with a chi-square risk statistic computed on the full Hubble dataset, selecting one hidden layer of 4096 neurons with ELU activations, Adam, an initial learning rate of 0.01 decaying with iteration, and 5750 epochs; because the dataset was too small to split, the full data were used for training and 1000 bootstrap resamples were each trained independently, the ensemble mean taken as the reconstruction and its variance as the uncertainty, with leave-one-out cross-validation used to inspect residuals. For the Bayesian network, independent zero-mean Gaussian priors were placed on all weights and biases, a Gaussian likelihood was assumed for the Hubble measurements, and the posterior was sampled by NUTS, with network width and prior variance chosen by WAIC; predictions were obtained by marginalising over the posterior. Both reconstructions were then integrated to luminosity distances, converted to isotropic energies and rest-frame peak energies with analytic error propagation, and fitted for the Amati slope, intercept and intrinsic scatter using the emcee sampler with 32 walkers for 5000 steps, 1000 discarded as burn-in and thinning by 10. Implementations used PyTorch and Pyro.
The shape of the work
Structural · the record, drawn
no AI
Assemble Hubble parameter and GRB samples
Obtaining raw data, whether by measurement, download or retrieval.
We utilise 115 GRBs from A220 dataset and 129 GRBs from J220 dataset.where the paper describes this · verbatim
AI
Select ANN architecture and train on bootstrap resamples
Fitting model parameters, including fine-tuning an existing model. The AI stood in for statistical model.
we utilise the full dataset during training and employ bootstrap resampling to assess the stability of the modelwhere the paper describes this · verbatim
AI
Reconstruct H(z) as ANN ensemble average
Running a trained model over new data to predict, classify or score. The AI stood in for statistical model.
We generate 1000 bootstrap samples, and the final reconstruction is obtained by averaging over these realizationswhere the paper describes this · verbatim
AI
Leave-one-out cross-validation of the reconstruction
Testing outputs against ground truth.
For further validation, we perform Leave One Out Cross Validation.where the paper describes this · verbatim
AI
Infer BNN weight posterior with NUTS
Fitting model parameters, including fine-tuning an existing model. The AI stood in for statistical model.
we adopt MCMC methods to train our BNNswhere the paper describes this · verbatim
AI
Reconstruct H(z) by marginalising the BNN posterior
Running a trained model over new data to predict, classify or score. The AI stood in for statistical model.
Model predictions are obtained by marginalising over the posterior,where the paper describes this · verbatim
no AI
Derive GRB distances and isotropic energies
Cleaning, filtering, normalising or labelling data already obtained.
the luminosity distance of the GRBs are calculatedwhere the paper describes this · verbatim
no AI
Fit Amati relation parameters by MCMC
Fitting model parameters, including fine-tuning an existing model.
We perform Markov Chain Monte Carlo (MCMC) sampling using the emcee affine-invariant ensemble samplerwhere the paper describes this · verbatim
What the record says
Technical · every part carries its own basis
+ in the paper~ our reading− not reported
How to read the quotations. A quotation shows where the paper describes something. It does not quote every value beside it: one passage locates a part of the work, and values without their own quotation are our reading of that passage.
The calibrated luminosity distances, and hence the reported Amati relation parameters, are derived from the neural-network reconstruction of H(z); no non-learned reconstruction of their own is carried through to the reported constraints
We employ an Artificial Neural Network to reconstruct the calibration relation directly from the data.where the paper describes this · verbatim
For further validation, we perform Leave One Out Cross Validation.where the paper describes this · verbatim
What this paper did not report
Technical · absence is published deliberately
- CodeWhether the code is available is not stated.
- Trained model weightsWhether the trained model is available is not stated.
- DataWhether the data are available is not stated.
- ComputeThe hardware or time used is not stated.
- How many were testedThe paper gives no count of what was tested.
- Version of Artificial Neural Network (one hidden layer, 4096 neurons, ELU, Adam)Which version of the model was used is not stated.
- Version of Bayesian Neural Network (NUTS/HMC posterior sampling)Which version of the model was used is not stated.
- What step 4 replacedThe paper gives no basis for what the AI stood in for.
About this article
Record aix-00069, version 2, checked by a person on 2026-10-08. The record describes the paper; it does not assess whether the paper's findings are right. The paper is published under CC-BY; quotations are at most 25 words. How we work · Report an error