Scientific Machine Learning (ML) and Artificial Intelligence (AI) now impact many diverse fields of engineering and science that include applications in sustainability, medicine and biology, and other physical sciences. These applications have spurred great interest and work on this topic and have led to development of novel algorithms and architectures. However, a more rigorous analysis of these algorithms, which is necessary for understanding their performance and pitfalls, is lacking. With this as motivation, this MS will focus on the topics that are tied to developing a more fundamental understanding of the algorithms of Scientific ML and AI. Topics of interest include analysis of the convergence of these algorithms with increasing data and complexity, and the development of new algorithms with quantifiable measures of performance. The domains of interest include, but are not limited to operator learning, optimization algorithms, transport maps, generative algorithms including transformer architectures, structure preserving frameworks, and novel function representation models.