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

TL;DR

This work develops an online framework for learning Koopman operators from streaming data by coupling recursive subspace identification with Grassmannian-subspace novelty detection. Data from new dynamical regimes are selectively archived and used to update a lifted linear model, whose observables are learned via Gaussian processes, enabling adaptive model order and reduced basis complexity. The approach demonstrates accurate online tracking of regime changes and outperforms fixed-dictionary EDMD in nonlinear settings, with practical benefits for streaming sensing and control. Overall, the method offers a computationally efficient and adaptive means to maintain high-fidelity operator-based models in the presence of regime-switching dynamics.

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.

Figures (8)

• Figure 1: Commutative diagram illustrating the propagation of the observable $g(x_t)$, projected observable $\Psi(x_t)$ and its realization $z_t$ using the operators $\mathcal{K}, K_{\Psi}$ and $K$, respectively.
• Figure 2: (a) Grassmannain distance $(G)$ between the subspace spanned by the extended observability matrix of the incoming dataset $\mathcal{D}_{i+1}$ and the recursively updated subspace $\mathbf{\Gamma_i}$. (b) The red points represent the $\epsilon$-redundant datasets, while the blue points represent the useful ones.
• Figure 3: (a) Grassmannian distance $d_{Gr}(\mathbf{\Gamma}_i, \hat{\Gamma}_{i+1})$ at each dataset between the subspace spanned by extended observability matrix of the system in eq.\ref{['eq:ex1_lin']} and the identified system ($K,C$). (b) Identified System order ($r$). (c) Eigenvalues of $K_i$, updated with each streaming dataset. (d) Singularvalues of $\Xi_i$, updated with each streaming dataset.
• Figure 4: Grassmannain distance $(G)$ between the subspace $\Gamma_{i+1}$ spanned by the extended observability matrix of the incoming dataset and the recursively updated subspace $\mathbf{\Gamma_i}$. The dotted black line represents the threshold value of $\epsilon=0.01$. The red line represents the cumulative average of $G$ computed over the number of datasets.
• Figure 5: Trajectories sampled in the first 600 datasets are shown in blue (single-well oscillations), and the following datasets from 601 to 900 are shown in red (double-well oscillations).