| Abstract: |
| Machine learning has opened new frontiers in purely data-driven algorithms for data assimilation, 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 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 machine 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 data assimilation and forecasting of dynamical systems. We work in 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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