We demonstrate the application of deep neural networks as a machine-learning tool for the analysis of a large collection of crystallographic data contained in the crystal structure repositories. Using input data in the form of multiperspective atomic fingerprints, which describe coordination topology around unique crystallographic sites, we show that the neural-network model can be trained to effectively distinguish chemical elements based on the topology of their crystallographic environment. The model also identifies structurally similar atomic sites in the entire data set of similar to 50000 crystal structures, essentially uncovering trends that reflect the periodic table of elements. The trained model was used to analyze templates derived from the known crystal structures in order to predict the likelihood of forming new compounds that could be generated by placing elements into these structural templates in a combinatorial fashion. Statistical analysis of predictive performance of the neural-network model, which was applied to a test set of structures never seen by the model during training, indicates its ability to predict known elemental compositions with a high likelihood of success. In similar to 30% of cases, the known compositions were found among the top 10 most likely candidates proposed by the model. These results suggest that the approach developed in this work can be used to effectively guide the synthetic efforts in the discovery of new materials, especially in the case of systems composed of three or more chemical elements.