Simulation-based inference for direction reconstruction of ultrahigh-energy cosmic rays with radio arrays

Ultrahigh-energy cosmic-ray (UHECR) observatories require unbiased direction reconstruction to enable multimessenger astronomy with sparse, nanosecond-scale radio pulses. Explicit likelihood methods often rely on simplified models, which may bias results and understate uncertainties. We introduce a simulation-based inference pipeline that couples a physics-informed graph neural network (GNN) to a normalizing-flow posterior within the Learning the Universe Implicit Likelihood Inference framework. Each event is seeded by an analytic plane-wavefront fit; the GNN refines this estimate by learning spatiotemporal correlations among antenna signals, and its frozen embedding conditions an eight-block autoregressive flow that returns the full Bayesian posterior. Trained on about 8,000 realistic UHECR air-shower simulations generated with the ZHAIRES code, the posteriors are temperature-calibrated to meet empirical coverage targets. We demonstrate a subdegree median angular resolution on test UHECR events, and find that the nominal 68% highest-posterior-density contours capture mml:mathmml:mn71</mml:mn>mml:mo%</mml:mo>mml:mo±</mml:mo>mml:mn2</mml:mn>mml:mo%</mml:mo></mml:math> of true arrival directions, indicating a mildly conservative uncertainty calibration. This approach provides physically interpretable reconstructions, well-calibrated uncertainties, and rapid inference, making it ideally suited for upcoming experiments targeting highly inclined events, such as GRAND, AugerPrime Radio, and BEACON.