The term data-driven describes computational methods for numerical problem solving which have been developed by the field of data science; these are at the intersection of computer science, mathematics, and statistics. When applied to a domain science like nuclear physics, especially with the goal of deepening scientific insight, data-driven methods form a core pillar of the computational science endeavor. In this dissertation I explore two problems related to theoretical nuclear physics: one in the framework of numerical statistics, and the other in the framework of machine learning. I) Historically our understanding of the structure of the atomic nucleus, the quantum many-body problem, has been built upon many layers of approximation, since the computational complexity of the problems is so large. One of the most flexible and enduring models, the configuration-interaction shell model, allows for detailed calculations of arbitrary scope. I lay out a simple framework for uncertainty quantification in empirical shell model calculations, thus providing not only error bars for large-scale calculations, but also insight for theory optimization and experimental design. II) Nuclear cross sections are an integral component in many different applications including astrophysics and nuclear medicine, but descriptions of cross sections are often very “data-heavy”. Huge libraries consisting of cross section evaluations, a combination of experimental measurements and theoretical results, are dense with information and thus ripe for data-driven methods. I have developed a deep learning system to learn trends in cross sections across the nuclear landscape. This system can predict cross sections for any nuclide and also can be used as an ensemble predictor. This is to my knowledge the first generative adversarial model developed for analyzing trends in nuclear data libraries.