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

Deep-Learning based Reconstruction of the Shower Maximum $X_{\mathrm{max}}$ using the Water-Cherenkov Detectors of the Pierre Auger Observatory