Operator Learning for Smoothing and Forecasting

TL;DR

This paper develops a theoretical framework for data-driven smoothing and forecasting in dynamical systems using neural operators, with universal approximation theorems and experiments on classical systems.

Contribution

It introduces a novel theory underpinning neural operator methods for data assimilation and forecasting, establishing existence and approximation properties.

Findings

Universal approximation theorems for data-driven smoothing and forecasting.

Theoretical analysis applied to Lorenz `63, Lorenz `96, and Kuramoto-Sivashinsky systems.

Validation of the theory through numerical experiments.

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 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.