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

Source: arXiv - 2602.23321v1

Overview

A team of astrophysicists and machine‑learning researchers has built a deep‑ensemble graph neural network (GNN) that can infer the arrival direction and energy of ultra‑high‑energy cosmic rays from the raw voltage traces recorded by autonomous radio antenna arrays. By treating each triggered antenna as a node in a graph and injecting physical priors into the network, they achieve sub‑degree angular precision and a ∼ 16 % energy resolution—far better than previous data‑driven methods—while also providing calibrated uncertainty estimates.

Key Contributions

• Graph‑based representation of radio arrays – each antenna becomes a node, with edges encoding spatial relationships, enabling the GNN to naturally exploit the irregular geometry of real‑world detector layouts.
• Physics‑aware GNN architecture – custom layers embed known propagation and antenna response physics, reducing the amount of simulated training data needed.
• Deep ensemble for probabilistic output – multiple independently trained GNNs are combined to produce both point estimates and well‑calibrated confidence intervals for direction and energy.
• State‑of‑the‑art reconstruction performance – angular resolution of 0.092° and electromagnetic energy resolution of 16.4 % on realistic simulated data with noise and background.
• Robustness & domain‑shift analysis – systematic tests (noise level variation, antenna failures, simulation‑real gaps) demonstrate how to detect when the model’s predictions may become unreliable.

Methodology

1. Data preparation – Simulated extensive‑air‑shower events generate voltage waveforms at each antenna. After a trigger, only the antennas that recorded a signal are kept.
2. Graph construction –
   • Nodes: per‑antenna feature vectors (e.g., peak voltage, pulse width, time‑of‑arrival, local noise level).
   • Edges: distances and relative orientations between antenna pairs, optionally weighted by expected signal correlation.
3. Physics‑informed GNN – The network stacks message‑passing layers that respect causal signal propagation (e.g., time‑delay constraints) and includes learnable kernels that mimic antenna gain patterns.
4. Deep ensemble – Five GNNs are trained from different random seeds and data shuffles. Their outputs are aggregated to produce a mean prediction and a variance that serves as an uncertainty estimate.
5. Training & loss – A multi‑task loss combines a directional loss (angular distance) and an energy loss (relative error), plus a regularization term that penalizes over‑confident uncertainty predictions (negative log‑likelihood).
6. Evaluation & robustness checks – Performance is measured on a held‑out simulated set with realistic noise, and stress‑tests are run by altering noise levels, removing antennas, or injecting systematic biases to mimic real‑world domain shifts.

Results & Findings

• Physics priors matter: Adding propagation constraints reduced the required training set size by a factor of three while preserving accuracy.
• Ensembling improves reliability: The variance across ensemble members correlates strongly with actual prediction error, enabling automatic flagging of low‑confidence events.
• Robustness: When up to 20 % of antennas are randomly disabled, angular error degrades gracefully (< 0.15°), and the ensemble’s uncertainty grows accordingly, signaling the loss of confidence.

Practical Implications

• Next‑generation radio cosmic‑ray observatories (e.g., GRAND, IceCube‑Gen2 radio extensions) can deploy fewer antennas or operate with higher noise environments while still achieving precise reconstruction, lowering hardware and maintenance costs.
• Real‑time event filtering – The calibrated uncertainties allow an autonomous array to decide on‑the‑fly whether to trigger a deeper read‑out or discard an event, saving bandwidth and storage.
• Transferable workflow – The graph‑based pipeline is agnostic to the specific detector geometry, making it applicable to other sparse sensor networks (e.g., seismic arrays, distributed IoT monitoring) where the physical layout is irregular.
• Open‑source tooling – The authors release the GNN modules and training scripts, giving developers a ready‑made template for building physics‑aware graph models on custom datasets.

Limitations & Future Work

• Simulation‑only validation – Results are demonstrated on high‑fidelity Monte‑Carlo data; real‑world deployment will need extensive domain‑adaptation studies.
• Computational load – While inference is fast on a modern GPU, the ensemble (five models) still requires ~ 10 ms per event, which may be a bottleneck for ultra‑high‑throughput arrays.
• Energy range – The study focuses on ultra‑high‑energy (> 10¹⁸ eV) showers; extending to lower energies may demand richer feature engineering or larger ensembles.
• Future directions proposed by the authors include:
  • Incorporating self‑supervised pre‑training on unlabeled real data to bridge the simulation‑real gap.
  • Exploring Bayesian GNNs for a more principled uncertainty treatment.
  • Scaling the approach to multi‑modal data (e.g., combining radio with surface‑detector or fluorescence measurements).

Bottom line: By marrying graph neural networks with domain knowledge and deep ensembles, the authors deliver a practical, high‑precision solution for reconstructing cosmic‑ray properties from noisy, sparsely sampled radio data—opening the door for smarter, cheaper, and more autonomous astrophysical observatories.*

Authors

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

Paper Information

• arXiv ID: 2602.23321v1
• Categories: astro-ph.IM, cs.LG
• Published: February 26, 2026
• PDF: Download PDF