Neural Posterior Estimation for UHECR source inference from 3D propagation simulations

The identification of ultra-high energy cosmic ray sources is one of the open challenges of high-energy astrophysics. As charged particles travel through the Universe, they are deflected by extragalactic magnetic fields and lose energy through interactions with background radiation, making source inference highly non-trivial. Existing approaches either rely on simplified propagation models or on computationally prohibitive Monte Carlo methods. Here we present a simulation-based inference framework trained on three-dimensional \texttt{CRPropa~3} propagation simulations that produces calibrated posterior distributions over source energy, distance, direction, and primary composition for individual UHECR events. The model combines a Deep Set encoder, handling the variable number of detected secondary particles, with a normalizing flow, and is trained on approximately 5 million simulated events covering a broad range of extragalactic magnetic field configurations. Validated on held-out simulations, all source parameters are recovered without systematic bias, with directional parameters best constrained and source distance most uncertain, consistent with the underlying propagation physics. Primary composition classification achieves $\geq$~98.2\% accuracy across all mass groups. This framework provides a scalable and physically interpretable interface between detailed propagation simulations and Bayesian source inference relevant for current UHECR data.