The scintillation light distribution produced by photodetectors in positron emission tomography (PET) provides the depth of interaction (DOI) information required for high-resolution imaging. The goal of positioning techniques is to reverse the photodetector signal’s pattern map to the coordinates of the incident photon energy position. By considering the DOI information, monolithic crystals offer good spatial, energy, and timing resolution along with high sensitivity. In this work, a supervised deep neural network was used for the approximation of DOI and to assess through Monte Carlo (MC) simulations the performance on a small-animal PET scanner consisting of ten 50 × 50 × 10 mm3 continuous Lutetium-Yttrium Oxyorthosilicate doped with Cerium (LYSO: Ce) crystals and 12 × 12 silicon photomultiplier (SiPM) arrays. The scintillation position was predicted by a multilayer perceptron neural network with 256 units and 4 layers whose inputs were the number of fired pixels on the SiPM plane and the total deposited energy. A GEANT4 MC code was used to generate training and test datasets by altering the photons’ incident position, energy, and direction, as well as readout of the photodetector output. The calculated spatial resolutions in the X-Y plane and along the Z-axis were 0.96 and 1.02 mm, respectively. Our results demonstrated that using a multilayer perceptron (MLP)-based positioning algorithm in the detector modules, constituting the PET scanner, enhances the spatial resolution by approximately 18% while the absolute sensitivity remains constant. The proposed algorithm proved its ability to predict the DOI for depth under 7 mm with an error below 8.7%.