The present paper addresses the study of non-arbitrariness in language within a deep learning framework. We present a set of experiments aimed at assessing the pervasiveness of different forms of non-arbitrary phonological patterns across a set of typologically distant languages. Different sequence-processing neural networks are trained in a set of languages to associate the phonetic vectorization of a set of words to their sensory (Experiment 1), semantic (Experiment 2), and word-class representations (Experiment 3). The models are then tested, without further training, in a set of novel instances in a language belonging to a different language family, and their performance is compared with a randomized baseline. We show that the three cross-domain mappings can be successfully transferred across languages and language families, suggesting that the phonological structure of the lexicon is pervaded with language-invariant cues about the words' meaning and their syntactic classes.