The Auger Engineering Radio Array (AERA) is part of the Pierre Auger Observatory and is
designed to investigate cosmic-ray induced air showers using radio measurements. The ultimate
goal of AERA is to recover the three-dimensional electromagnetic field originating from the air
shower with the measured voltage time traces of the antennas, which is a challenging task. The
electric field measurements are modified in the detection process by the frequency- and direction-
dependent antenna response, and is superimposed by noise. We use conditional Invertible Neural
Networks (cINNs) to learn posterior distributions, from which the most likely electromagnetic field
given a measured voltage time trace can be inferred. We extend the method with an autoencoder to
further enhance robustness, reduce the parameter space, and decouple the cINN from data shape.
We will present an overview of the method and its application to simplified simulation data with
typical properties of AERA and evaluate the methods reconstruction quality.