astro-ph.IMFeb 26, 2026
SaveReconstruction of cosmic-ray direction and energy in radio arrays using deep ensemble graph neural networks
Organizations: Université Paris-Saclay, CEA, List, F-91120 Palaiseau, France · Sorbonne Université, CNRS, Laboratoire de Physique Nucléaire et des Hautes Énergies (LPNHE), 4 Pl. Jussieu, 75005 Paris, France · National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100101, China · Sorbonne Université, UPMC Univ. Paris 6 et CNRS, UMR 7095, Institut d’Astrophysique de Paris, 98 bis bd Arago, 75014 Paris, France · Instituto de Física La Plata, CONICET - UNLP, Boulevard 120 y 63 (1900), La Plata - Buenos Aires, Argentina
Abstract
Using advanced machine learning techniques, we developed a method to reconstruct the arrival direction and energy of ultra-high-energy cosmic rays from the voltage traces they induce on ground-based radio detector arrays. In our approach, triggered antennas are represented as a graph structure, which serves as input for a graph neural network (GNN). By incorporating physical knowledge into both the GNN architecture and the input data, we improve the precision and reduce the required size of the training set with respect to a fully data-driven approach. This method achieves an angular resolution of 0.092 degrees and an electromagnetic energy reconstruction resolution of 16.4% on simulated data with realistic noise conditions. We also employ uncertainty estimation methods to enhance the reliability of our predictions, quantifying the confidence of the GNN's outputs and providing confidence intervals for both direction and energy reconstruction. Finally, we investigate strategies to verify the model's consistency and robustness under real-life variations, with the goal of identifying scenarios in which predictions remain reliable despite domain shifts between simulation and reality.
Figures & tables
Figure 1: Left: Magnitude of the modeled RF chain transfer function for the X and Y (solid blue) and Z (dashed orange) antenna arms. The magnitude response shows frequency-dependent gain variations. Right: Effective lengths of the horizontal antennas at 60 MHz for a signal coming 30° from north in the presence of ground reflection as a function of the zenith angle.
Figure 2: Spectral radiance map derived from the temperature map (top) and effective area map ( ) (bottom) for arm X of the detector at 60 MHz as a function of right ascension (RA) and latitude. The red star indicates the zenith at detector’s location. Both plots correspond to a local sidereal time of 18:00.
Figure 3: Example of a graph constructed from a simulated event. Triggered antennas are represented as nodes (blue dots), with edges (gray lines) connecting each antenna to its eight nearest neighbors. The color of each node corresponds to the signal amplitude recorded at that antenna, illustrating the spatial distribution of the detected signals across the array.
Figure 4: Illustration of the architecture used in this study. The model consists of a scaling layer, 4 EdgeConv layer where each node aggregates information from its 8 nearest neighbors into a vector of 256 features, 2 pooling layers that aggregate the node features into a single vector, and a final MLP that outputs the reconstructed quantities and their associated uncertainties. The number of input features .
Figure 5: Evolution of the mean squared error on the training and validation sets over epochs for the 12 ensemble members GNN for direction (top) and energy (bottom) reconstruction. The validation error stabilizes early in both cases. The transparent envelopes represent the standard deviation over all 12 models
Figure 6: Distribution of angular errors for single models (gray), the ensemble mean prediction (blue), and the PWF estimator (orange). The ensemble model achieves a mean angular error of 0.092°, significantly outperforming both single models (0.11°) and the PWF baseline (0.16°). The table summarizes the performances of all models considered. The quantities shown are the mean error on zenith angle, the standard deviation of the error on zenith angle, the mean angular error, and the 95th percentile of the angular error distribution.
Figure 7: Zenith-angle residual ( ) versus event parameters for the ensemble model. Top-left: zenith; top-right: primary energy; bottom-left: triggered-antenna multiplicity; bottom-right: distance to the array center. Performance depends mainly on multiplicity; no significant bias is observed. The shaded band indicates the central 68% interval.
Figure 8: Conditional distribution of versus lateral core distance for the PWF (left) and ensemble GNN (right), with each column normalised to 100%. The lateral core distance corresponds to the distance between the shower core and the shower axis projected onto the ground. The vertical line correspond to the edge of the infill. The bi-modality introduced by the PWF for events on the edge of the infill is completely corrected by the GNNs
Figure 9: Distribution of relative energy errors . Blue and orange filled histograms: ensemble predictions for proton and iron primaries, respectively. The ensemble model achieves an energy resolution of 16.4% (68% quantile of the absolute relative residuals), compared with 19.5% for single models. A small bias is visible in the ensemble: iron events are slightly underestimated while proton events are slightly overestimated, but resolution is the same.
Figure 10: Relative energy error as a function of event parameters for the ensemble model. Top-left: zenith angle; top-right: electromagnetic energy; bottom-left: number of triggered antennas; bottom-right: distance from to the array. Blue points denote individual events; the black curve shows the bin-wise mean with a shaded 68% envelope, representing the energy resolution as a function of the horizontal axis. Bias is observed at high zenith angles corresponding to large distances of the emission point.
Figure 11: Standardized residuals for the two direction components (top: , ) and for (bottom). The red curve is a standard Gaussian distribution . The close match supports the Gaussian assumption.
Figure 12: Coverage plots for the zenith angle (orange), azimuth angle (green), and logarithmic electromagnetic energy (blue). The red dotted lines indicate the 68%, 95%, and 99.7% confidence levels expected for a standard normal distribution. The black dashed diagonal corresponds to a perfectly calibrated model, while the gray shaded region represents the expected statistical fluctuations. Overall, the reconstructed quantities follow the expected calibration behavior with only mild deviations.
Figure 13: Direction reconstruction with uncertainty estimation. The individual models (green dots) predict a correction to the PWF direction (blue cross), there mean defines the deep ensemble prediction (green cross). The predicted covariance matrix gives the predicted uncertainty distribution which allows to create confidence intervals around the predicted value. Here the true value is inside the "1 " interval (68% isoline).
Figure 14: Top: Angular resolution (median values) and zenith angle bias (mean values) as a function of trigger threshold for the ensemble model (solid line), the PWF estimator (crosses) and single models (transparent dots). The angular resolution and error on stay stable. Bottom: Energy resolution as a function of the trigger threshold values for ensemble (solid line) and single models (transparent points). TThe energy resolution stays stable but biases with increasing threshold from 0.1% at to -16% at 9 .
Figure 15: Angular resolution (median values) and zenith angle bias (mean values) as a function of the fraction of deactivated antennas for the ensemble model (solid line) and the PWF estimator (crosses) and single models (transparent dots). The angular resolution degrades from 0.09° to 0.11° with 40% antennas deactivated. Bottom: Energy resolution as a function of the fraction of deactivated antennas for the ensemble model. The energy resolution worsens from 16.4% to 23.0% with 40% antennas deactivated.
Figure 16: Angular resolution (median values) and zenith angle bias (mean values) as a function of miscalibration for the ensemble model (solid line), the PWF estimator (crosses) and single models (transparent dots). The angular performances don’t degrade. Bottom: Energy resolution as a function of the fraction of miscalibration for the ensemble model and single model. The energy resolution worsens from 15.9% to 24.0% with 20% smearing on antenna amplitudes. A bias also appears above 10% miscalibration
Appendix figures & tables2 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 17: Energy resolution as a function of training set size. Ensemble predictions (circles) and individual seed predictions (squares) are shown for models trained with (orange) and without (blue) PWF input. Energy resolution is defined as the 68% quantile on energy relative error .
Figure 18: Angular resolution as a function of training set size for direction reconstruction. Ensemble predictions (circles) and individual seed predictions (squares) are shown for models trained with (orange) and without (blue) point spread function (PWF) corrections. Angular resolution is defined as the median angular error in degrees.