The atmospheric depth of the air shower maximum $X_{mathrm{max}}$ is an observable commonly used for the determination of the nuclear mass composition of ultra-high energy cosmic rays. Direct measurements of $X_{mathrm{max}}$ are performed using observations of the longitudinal shower development with fluorescence telescopes. At the same time, several methods have been proposed for an indirect estimation of $X_{mathrm{max}}$ from the characteristics of the shower particles registered with surface detector arrays. In this paper, we present a deep neural network (DNN) for the estimation of $X_{mathrm{max}}$. The reconstruction relies on the signals induced by shower particles in the ground based water-Cherenkov detectors of the Pierre Auger Observatory. The network architecture features recurrent long short-term memory layers to process the temporal structure of signals and hexagonal convolutions to exploit the symmetry of the surface detector array. We evaluate the performance of the network using air showers simulated with three different hadronic interaction models. Thereafter, we account for long-term detector effects and calibrate the reconstructed $X_{mathrm{max}}$ using fluorescence measurements. Finally, we show that the event-by-event resolution in the reconstruction of the shower maximum improves with increasing shower energy and reaches less than $25~mathrm{g/cm^{2}}$ at energies above $2times 10^{19}~mathrm{eV}$.
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