At the highest energies, cosmic rays can be detected only indirectly by the extensive air showers they create upon interaction with the Earth's atmosphere. While high-statistics measurements of the energy and arrival directions of cosmic rays can be performed with large surface detector arrays like the Pierre Auger Observatory, the determination of the cosmic-ray mass on an event-by-event basis is challenging. Meaningful physical observables in this regard include the depth of maximum of air-shower profiles, which is related to the mean free path of the cosmic ray in the atmosphere and the shower development, as well as the number of muons that rises with the number of nucleons in a cosmic-ray particle. In this contribution, we present an approach to determine both of these observables from combined measurements of water-Cherenkov detectors and scintillation detectors, which are part of the AugerPrime upgrade of the Observatory. To characterize the time-dependent signals of the two detectors both separately as well as in correlation to each other, we apply deep learning techniques. Transformer networks employing the attention mechanism are especially well-suited for this task. We present the utilized network concepts and apply them to simulations to determine the precision of the event-by-event mass reconstruction that can be achieved by the combined measurements of the depth of shower maximum and the number of muons.