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Deep ensemble graph neural networks for probabilistic cosmic-ray direction and energy reconstruction in autonomous radio arrays

Arsène Ferrière, Aurélien Benoit-Lévy, Olivier Martineau-Huynh, Matías Tueros

TL;DR

A method for reconstructing precisely the arrival direction and energy of ultra-high-energy cosmic rays from the voltage traces they induced on ground-based radio detector arrays by incorporating physical knowledge into both the GNN architecture and the input data is developed.

Abstract

Using advanced machine learning techniques, we developed a method for reconstructing precisely the arrival direction and energy of ultra-high-energy cosmic rays from the voltage traces they induced 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° 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.

Deep ensemble graph neural networks for probabilistic cosmic-ray direction and energy reconstruction in autonomous radio arrays

TL;DR

A method for reconstructing precisely the arrival direction and energy of ultra-high-energy cosmic rays from the voltage traces they induced on ground-based radio detector arrays by incorporating physical knowledge into both the GNN architecture and the input data is developed.

Abstract

Using advanced machine learning techniques, we developed a method for reconstructing precisely the arrival direction and energy of ultra-high-energy cosmic rays from the voltage traces they induced 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° 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.
Paper Structure (23 sections, 23 equations, 17 figures)

This paper contains 23 sections, 23 equations, 17 figures.

Table of Contents

  1. Introduction
  2. Simulation and Signal Conversion
  3. Efield conversion to voltage
  4. Noise modeling
  5. Graph neural network on distributed radio antenna arrays
  6. Motivations for using graph neural networks
  7. Analytical reconstruction from the wavefront
  8. Preprocessing for GNN input
  9. Architecture
  10. Training and inference procedure
  11. Performances
  12. Model convergence
  13. Direction reconstruction performances
  14. Performances on energy reconstruction
  15. Uncertainty estimation and calibration
  16. ...and 8 more sections

Figures (17)

  • Figure 1: Top: 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. Bottom: Effective lengths of the horizontal antennas at 60 MHz for a signal coming 30° in the presence of ground reflection from north as a function of the zenith angle.
  • 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 different architectures used in this study. The main gray branch represents the rGNNs, which rely solely on explicitly derived features in a data-driven approach. The inclusion of the red PWF branch describes the pGNN models, where physics-based inputs are integrated to enhance precision. In this architecture, the difference between the measured timings and the expected timings from the PWF model is added as a graph input feature which increases n$_{\rm feat}$ from 5 to 6.
  • Figure 5: Evolution of the MSE loss on the training set and validation set over epochs for the 12 ensemble members of rGNN and pGNN for energy and direction reconstruction. The introduction of the PWF information significantly decreases the training and validation mean square error.
  • Figure 6: Azimuth-angle residual ($\Delta \upphi$) versus core distance to the array center for the ensemble pGNN and the PWF estimator. The PWF estimator (orange) shows a significant bias for events landing far from the array center, while the pGNN (blue) maintains a stable resolution without bias across core distances. The shaded band indicates the central 68% interval.
  • ...and 12 more figures