Phylogenetic networks provide a means of describing the evolutionary history of sets of species believed to have undergone hybridization or horizontal gene flow during the course of their evolution. The mutation process for a set of such species can be modeled as a Markov process on a phylogenetic network. Previous work has shown that a site-pattern probability distributions from a Jukes-Cantor phylogenetic network model must satisfy certain algebraic invariants, i.e. polynomial relationships. As a corollary, aspects of the phylogenetic network are theoretically identifiable from site-pattern frequencies. In practice, because of the probabilistic nature of sequence evolution, the phylogenetic network invariants will rarely be satisfied, even for data generated under the model. Thus, using network invariants for inferring phylogenetic networks requires some means of interpreting the residuals when observed site-pattern frequencies are substituted into the invariants. In this work, we propose an approach that combines statistical learning and phylogenetic invariants to infer small, level-one phylogenetic networks, and we discuss how the approach can be extended to infer larger networks. This is joint work with Travis Barton, Colby Long, and Joseph Rusinko.