Structure-preserving deep learning has recently received high attention, e.g., the development of the symplecticity-preserving neural networks \(\textrm{SympNets}\) for learning the flow of a Hamiltonian system. The incorporation of structural properties in neural networks has been shown to produce qualitatively better long-time predictions. It is still a great challenge to obtain computationally efficient learning algorithms for high-dimensional problems. In this work, we propose dimensionality reduction with the proper symplectic decomposition (PSD) of the time-series training datasets in conjunction with \(\textrm{SympNets}\) . PSD was originally proposed to obtain symplectic reduced-order models of Hamiltonian systems. We demonstrate the proposed purely data-driven approach by learning nonlinear localized discrete breather (DB) solutions in a one-dimensional crystal lattice model. We find that learning the SPD-reduced Hamiltonian dynamics is not only more computationally efficient compared to learning the whole high-dimensional model, but we also recover accurate spectral results from the neural network predictions in contrast to predictions by learned non-symplectic proper orthogonal decomposition (POD) dimensionality-reduced dynamics.