  This work introduces Jacobian-scaled K-means (JSK-means) clustering, which is
a physics-informed clustering strategy centered on the K-means framework. The
method allows for the injection of underlying physical knowledge into the
clustering procedure through a distance function modification: instead of
leveraging conventional Euclidean distance vectors, the JSK-means procedure
operates on distance vectors scaled by matrices obtained from dynamical system
Jacobians evaluated at the cluster centroids. The goal of this work is to show
how the JSK-means algorithm -- without modifying the input dataset -- produces
clusters that capture regions of dynamical similarity, in that the clusters are
redistributed towards high-sensitivity regions in phase space and are described
by similarity in the source terms of samples instead of the samples themselves.
The algorithm is demonstrated on a complex reacting flow simulation dataset (a
channel detonation configuration), where the dynamics in the thermochemical
composition space are known through the highly nonlinear and stiff
Arrhenius-based chemical source terms. Interpretations of cluster partitions in
both physical space and composition space reveal how JSK-means shifts clusters
produced by standard K-means towards regions of high chemical sensitivity
(e.g., towards regions of peak heat release rate near the detonation reaction
zone). The findings presented here illustrate the benefits of utilizing
Jacobian-scaled distances in clustering techniques, and the JSK-means method in
particular displays promising potential for improving former partition-based
modeling strategies in reacting flow (and other multi-physics) applications.
