Explore advanced concepts in sampling, inference, and data-driven physical modeling through this 37-minute conference presentation delivered at UCLA's Institute for Pure & Applied Mathematics (IPAM) workshop on Scientific Machine Learning. Delve into cutting-edge research methodologies presented by a University of Massachusetts, Amherst faculty member, focusing on the intersection of statistical sampling techniques, probabilistic inference, and physics-informed machine learning approaches. Gain insights into how these mathematical frameworks are being applied to solve complex scientific problems through data-driven modeling techniques. The presentation was recorded as part of IPAM's specialized workshop series dedicated to advancing the field of scientific machine learning through rigorous mathematical foundations.