Collaborative Research: Augmenting Continuous Data Assimilation to Perform Equation Discovery with Applications in Geophysics

Forecasting complex societally important systems such as weather, ocean currents, and groundwater flow remains a grand challenge, especially when models must account for noisy or incomplete data. Traditional physics-based models offer scientific interpretability but often rely on idealized assumptions that limit accuracy. Conversely, recent advances in machine learning approaches have led to more accurate predictive models, but these are inherently "black boxes," lacking scientific insight into the underlying physical mechanisms. This project will use observable data to systematically adapt and modify existing physically derived models, thus staying true to both the traditional and data-driven approaches. The methodology is adaptable to a wide range of goals, such as optimizing predictions, matching observed statistics, or identifying unknown model parameters. Applications of this work will include problems of great interest to society and industry by identifying more efficient models for turbulence, which has ramifications from weather prediction to the development of engines and design of aircraft. The investigators will also apply this method to develop reliably predictive models for groundwater flow, a key issue for national water and food security. Further, the project will advance education and workforce development by mentoring undergraduate and graduate students and facilitating interactions with National Laboratories and private-sector stakeholders. The project builds on recent combined efforts of the investigators demonstrating an algorithm capable of "on-the-fly" parameter and model discovery. The investigators will mathematically rigorously justify this algorithm (and similar renditions of it), quantify its limitations, and lay the mathematical foundation for further improvement. The investigators will apply this method to identify large eddy simulation (LES) models for turbulent flows, and to correct reduced-dimensional models (e.g., from three dimensions to two dimensions) can be modified to accurately capture important statistics. The data assimilation technique will be extended to the Richards equation for groundwater flow, and the new parameter identification algorithm will be used to identify the precise form of spatially varying diffusion tensors which is critical for porous media and groundwater flows. The project will also develop algorithms for optimal sensor placement (where optimality is defined as accurately representing key characteristics of the system from partial observations available from limited mobile and/or static sensors), enhancing the ability to accurately reconstruct and predict full-system behavior from sparse measurements. 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.