Data-Driven Stochastic Model Reduction and Its Applications in Data Assimilation

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With data routinely available, many applications in science and engineering call for timely and accurate predictions that are constantly updated. In data assimilation, such predictions are made by combining data with mathematical dynamical models. To quantify the uncertainty in predictions, the dynamical models need to be repeatedly solved for many initial conditions in a timely manner. This rules out the use of first-principles dynamical models that are computationally expensive and time-consuming to solve, and prompts the need to construct effective statistical-dynamical reduced models. This project will address this issue by developing data-informed stochastic reduced models based on theories and tools in statistical inference, stochastic processes, dynamical systems, and partial differential equations. The research lies at the foundation of data-informed computational modeling and simulation, and aims at developing new mathematical and statistical theories and tools to address the challenges in data-informed predictive modeling of complex systems. The research plan is complemented by educational objectives to prepare and train students through interdisciplinary research. The goal of this project is to develop and analyze efficient algorithms to construct statistically-dynamically effective stochastic reduced models from data for complex systems, and to obtain timely predictions through data assimilation using these reduced models. The investigator plans to infer discrete-time non-Markovian reduced models from data by (i) parametrizing projections of invariant manifolds of dissipative systems, and (ii) approximating the effects of the unresolved variables on the resolved variables by nonparametric inference methods for general unknown systems. The investigator also plans to develop novel data assimilation methods for these non-Markovian reduced models and apply these methods to address the grand challenge of model reduction from noisy partial data. The work is expected to have applications in climate modeling, geophysics, fluid mechanics, and other data-informed computational modeling problems.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.