Operator Learning for Smoothing and Forecasting

• URL: http://arxiv.org/abs/2603.20359v1
• Date: Fri, 20 Mar 2026 16:04:55 GMT
• Title: Operator Learning for Smoothing and Forecasting
• Abstract summary: We develop a theory to underpin data-driven methods to solve smoothing problems.<n>We establish the first universal approximation theorem for purely data-driven algorithms for both smoothing and forecasting of dynamical systems.<n>Results are illustrated with experiments studying the Lorenz 63, Lorenz 96 and Kuramoto-Sivashinsky dynamical systems.
• Score: 1.261395649216268
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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