Online learning of Koopman operator using streaming data from different dynamical regimes

TL;DR

This paper introduces an online method for learning the Koopman operator from streaming data, capable of identifying different dynamical regimes, reducing data requirements, and adaptively learning basis functions for improved modeling.

Contribution

It proposes a novel approach that uses Grassmannian distance to select relevant data segments and adaptively learns basis functions, enhancing efficiency and accuracy in Koopman operator estimation.

Findings

The method effectively identifies data from different dynamical regimes.

It reduces the amount of data needed for updating the Koopman model.

It adaptively learns basis functions, optimizing model complexity.

Abstract

The paper presents a framework for online learning of the Koopman operator using streaming data. Many complex systems for which data-driven modeling and control are sought provide streaming sensor data, the abundance of which can present computational challenges but cannot be ignored. Streaming data can intermittently sample dynamically different regimes or rare events which could be critical to model and control. Using ideas from subspace identification, we present a method where the Grassmannian distance between the subspace of an extended observability matrix and the streaming segment of data is used to assess the `novelty' of the data. If this distance is above a threshold, it is added to an archive and the Koopman operator is updated if not it is discarded. Therefore, our method identifies data from segments of trajectories of a dynamical system that are from different dynamical regimes, prioritizes minimizing the amount of data needed in updating the Koopman model and furthermore reduces the number of basis functions by learning them adaptively. Therefore, by dynamically adjusting the amount of data used and learning basis functions, our method optimizes the model's accuracy and the system order.