Operator Learning for Smoothing and Forecasting
It addresses the lack of analysis for data-driven methods in data assimilation and forecasting, providing foundational theory for researchers in machine learning and dynamical systems.
The paper developed a theoretical framework to underpin purely data-driven methods for smoothing and forecasting in dynamical systems, establishing the first universal approximation theorem for such algorithms and illustrating results with experiments on Lorenz and Kuramoto-Sivashinsky systems.
Machine learning has opened new frontiers in purely data-driven algorithms for data assimilation in, and for forecasting of, dynamical systems; the resulting methods are showing some promise. However, in contrast to model-driven algorithms, analysis of these data-driven methods is poorly developed. In this paper we address this issue, developing a theory to underpin data-driven methods to solve smoothing problems arising in data assimilation and forecasting problems. The theoretical framework relies on two key components: (i) establishing the existence of the mapping to be learned; (ii) the properties of the operator learning architecture used to approximate this mapping. By studying these two components in conjunction, we establish the first universal approximation theorem for purely data-driven algorithms for both smoothing and forecasting of dynamical systems. We work in the continuous time setting, hence deploying neural operator architectures. The theoretical results are illustrated with experiments studying the Lorenz `63, Lorenz `96 and Kuramoto-Sivashinsky dynamical systems.