Biological networks constructed from varied data, including protein-protein interactions, gene expression data, and genetic interactions can be used to map cellular function, but each data type has individual limitations such as bias and incompleteness. Unsupervised network integration promises to address these limitations by combining and automatically weighting input information to obtain a more accurate and comprehensive result. However, existing unsupervised network integration methods fail to adequately scale to the number of nodes and networks present in genome-scale data and do not handle partial network overlap. To address these issues, we developed an unsupervised deep learning-based network integration algorithm that incorporates recent advances in reasoning over unstructured data, namely the graph convolutional network (GCN), and can effectively learn dependencies between any input network, such as those composed of protein-protein interactions, gene co-expression, or genetic interactions. Our method, BIONIC (Biological Network Integration using Convolutions), learns features which contain substantially more functional information compared to existing approaches, linking genes that share diverse functional relationships, including co-complex and shared bioprocess annotation. BIONIC is scalable in both size and quantity of the input networks, making it feasible to integrate numerous networks on the scale of the human genome.