Physics Informed Forecast of High Resolution Spatio-Temporal Data with a Functional Spatial Regression Framework with PDE Smoothing

This project addresses the critical need to improve scientific models that combine observed data with established physical laws. As researchers increasingly rely on large and complex datasets, there is growing interest in "hybrid models" that merge data-driven insights with the knowledge embedded in mathematical equations, such as those used in physics. However, current methods lack the theory necessary for reliable and interpretable results, especially in understanding spatial phenomena like fluid movement. This project will develop a new modeling framework that bridges the gap between data and physical understanding, enabling more accurate and consistent spatial predictions. The project will also foster outreach by creating open-source tools. The project develops a novel functional framework for hybrid spatial models that regularize data-driven predictions using Partial Differential Equations (PDEs). The approach formulates spatial regression as a functional optimization problem, where the solution is penalized by the governing PDE, enabling the derivation of fundamental mathematical results. The research has three main objectives: (1) to establish a theoretical foundation for these hybrid models by ensuring well-posedness; (2) to implement inference and spatial interpolation using both finite element methods and Hilbert space basis decompositions; and (3) to apply the methodology to fluid dynamics, generating physically consistent predictions. This framework is transformative in statistical methodology, integrating data science with physical modeling. It advances inferential techniques for spatial functional data and supports reproducibility through open-source software. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.