This project involves the development of methodologies to estimate unknown quantities from complex, sparse, and noisy data sets by incorporating first-principles physics into statistical modeling procedures. The approach pairs new inverse problem formulations with novel probabilistic high performance computing methods to resolve crucial statistical quantities (means, correlations, confidence bands). This framework has applications in medical imaging, weather modeling, robotics, geology, and geophysics, with mechanisms for improving methods to better formulate and quantify uncertainties for real world challenges in the above domains. Furthermore, the project will incorporate the comprehensive training and the promotion of excellence in applied mathematics and statistics for a new generation of scholars who will be involved in these research activities. This project leverages an emerging Bayesian inversion formalism to estimate non-parametric physical parameters from data modeled as sparse and noisy observations of solutions to partial differential equations (PDEs). The proposed strategy blends rigorous analysis, algorithm development, numerical case studies, and modeling to build a foundational understanding of these methodologies and to expand the scope of this statistical approach to PDE inference. The project is organized around three interconnected objectives. The first objective involves the development of fluid measurement problems featuring infinite-dimensional unkno