The U.S. Department of Energy is considering implementing the direct feed approach for the vitrification of low-activity waste (LAW) and high-level waste (HLW) at the Hanford site in Washington state. If implemented, the nuclear waste with a higher concentration of alkali/alkaline-earth sulfates (than expected under the previously proposed vitrification scheme) will be sent to the vitrification facility. It will be difficult for the existing empirical models to predict sulfate solubility in these glasses or design glass formulations with enhanced sulfate loadings in such a scenario. Further, the existing models are unable to produce reliable predictions when applied to HLW glasses whose composition falls outside of the range encompassed by the database used to develop/calibrate the models. Accordingly, this study harnesses the power of artificial intelligence (machine learning, ML) with a goal to address the limitations of the existing models. Toward this, three ML models have been trained using a large database; comprising >1000 LAW and HLW glasses and encompassing a wide range of glass compositions and processing variables. Next, the ML model with the best prediction performance has been used to quantitatively assess and rank the influence (i.e., importance) of glasses' compositional/processing variables on the SO3 solubility in the glasses. Finally, on the premise of such understanding of influential and inconsequential variables, two closed-form analytical models - with disparate degrees of complexity (one highly parametrized and one with fewer input variables) - have been developed. Results show that both analytical models produce predictions of SO3 solubility in LAW and HLW glasses with an accuracy analogous to ML models and substantially higher than the analytical models that represent the current state-of-the-art. Overall, this study's outcomes present a roadmap - informed by data and channeled by artificial intelligence - that can be leveraged in the future to design nuclear waste glasses with unprecedented sulfur loadings.