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 novel universal approximation theorems 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.

Bayesian deep learning has always garnered substantial interest in the machine learning community.

Despitethis noble goal, it has been acknowledged in the literature that statistical tests based ontheEOareoblivious totheunderlying causal mechanisms thatgenerated the disparity in the first place (Hardt et al. 2016).

Neural Information Processing Systems http://nips.cc/

Neural Information Processing Systems http://nips.cc/

Actor-critic methods, a type of model-free Reinforcement Learning, have beensuccessfully applied to challenging tasks in continuous control, often achievingstate-of-the artperformance.

Neural Information Processing Systems http://nips.cc/