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Artificial neural networks (ANNs) have proven to be a valuable tool for the data-driven construction of empirical models. The predictive capabilities of the ANN approximations are often validated within a statistical framework. For many applications, one can rely on a first-principles background that allows for the construction of hybrid models. By constrasting the predictions of the hybrid models with the theory, valuable insight can be obtained about the predictive capabilities of the approximation and the first-principles formulation itself. In this contribution, we illustrate the use of ANNs for the construction of hybrid models for the determination of mixing rules for the BACK thermodynamic equation of state (EOS). The hybrid approximations are constructed in three levels, with the progressive addition of more information to the empirical formulation. Using an ANN within the framework of an EOS substantially increases the potential for applications allowing for the estimation of thermodynamic properties other than the ones used for network training.