Title
Data Assimilation: From the Ensemble Kalman Filter to Operator Learning
Abstract
Ensemble Kalman methods constitute a set of tools for model-based forecasting which can be used for state estimation, with potential for uncertainty quantification, in dynamical systems. The methodology is widely used in the geophysical sciences and is a key component of the best weather prediction systems worldwide. Despite this empirical success, analysis of the accuracy of ensemble Kalman methods, especially in terms of their ability to accurately quantify uncertainty, is lagging. In this talk we consider a unifying approach to algorithms that rests upon transport of probability measures and mean field stochastic dynamical systems. We demonstrate how this mean field framework enables analysis quantifying the error of the ensemble Kalman filter when viewed as an approximation of the true filtering distribution. The ensemble Kalman filter, serving as a training data preprocessor, is also an essential component of emergent data-driven forecasting methods. In this talk, we show how operator learning may be applied at various stages of the data assimilation pipeline to effect data-driven forecasting. We introduce an operator formulation of transformers, the architecture behind the success of large language models, and discuss how its application to data assimilation offers a promising new approach.