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.

These methods can be broadly categorized into two types: function learning and operator learning approaches. In function learning, the goal is to directly learn the solution.

Fourier Neural Operators (FNOs) have shown promise for solving partial differential equations (PDEs).

Here, we present novel adaptations for convolutional neural networks to demonstrate that they are indeed able to process functions as inputs and outputs.

We consider the problem of discretization of neural operators between Hilbert spaces in a general framework including skip connections. We focus on bijec-tive neural operators through the lens of diffeomorphisms in infinite dimensions.

This research offers a fresh take on neural operators with a framework Representation equivalent Neural Operators (ReNO) designed to address these issues.