  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.
