Trajectory-optimized cluster-based network model for the sphere wake

TL;DR

The paper introduces a trajectory-optimized cluster-based network model (tCNM) that improves nonlinear model reduction accuracy for fluid flow data, especially in complex wake dynamics, by refining centroid placement and trajectory representation.

Contribution

It presents a novel trajectory-optimized clustering method that enhances the accuracy of cluster-based network models for nonlinear flow dynamics, outperforming previous methods like k-means++ and POD.

Findings

tCNM reduces representation error by 5 times compared to closest centroid approximation.

tCNM achieves error levels comparable to POD of the same order.

The model is interpretable, stable, and fully automatable for complex flow regimes.

Abstract

We propose a novel trajectory-optimized Cluster-based Network Model (tCNM) for nonlinear model order reduction from time-resolved data following Li et al. ["Cluster-based network model, " J. Fluid Mech. 906, A21 (2021)] and improving the accuracy for a given number of centroids. The starting point is k-means++ clustering which minimizes the representation error of the snapshots by their closest centroids. The dynamics is presented by 'flights' between the centroids. The proposed trajectory-optimized clustering aims to reduce the kinematic representation error further by shifting the centroids closer to the snapshot trajectory and refining state propagation with trajectory support points. Thus, curved trajectories are better resolved. The resulting tCNM is demonstrated for the sphere wake for three flow regimes, including the periodic, quasi-periodic, and chaotic dynamics. The representation error of tCNM is 5 times smaller as compared to the approximation by the closest centroid. Thus, the error at the same level as Proper Orthogonal Decomposition (POD) of same order. Yet, tCNM has distinct advantages over POD modeling: it is human interpretable by representing dynamics by a handful of coherent structures and their transitions; it shows robust dynamics by design, i.e., stable long-time behavior; and its development is fully automatable, i.e., it does not require tuneable auxiliary closure and other models.