Deep neural networks are a powerful technique that have found ample applications in several branches of physics. In this work, we apply deep neural networks to a specific problem of cosmic ray physics: the estimation of the muon content of extensive air showers when measured at the ground. As a working case, we explore the performance of a deep neural network applied to large sets of simulated signals recorded for the water-Cherenkov detectors of the Surface Detector of the Pierre Auger Observatory. The inner structure of the neural network is optimized through the use of genetic algorithms. To obtain a prediction of the recorded muon signal in each individual detector, we train neural networks with a mixed sample of simulated events that contain light, intermediate and heavy nuclei. When true and predicted signals are compared at detector level, the primary values of the Pearson correlation coefficients are above 95%. The relative errors of the predicted muon signals are below 10% and do not depend on the event energy, zenith angle, total signal size, distance range or the hadronic model used to generate the events.