Jacobian-Scaled K-means Clustering for Physics-Informed Segmentation of Reacting Flows

TL;DR

This paper introduces Jacobian-scaled K-means (JSK-means), a physics-informed clustering method that replaces the standard Euclidean distance with a centroid-dependent, Jacobian-weighted distance to capture dynamical similarity in thermochemical composition space. By deriving scaling matrices from the chemical Jacobian evaluated at centroids, JSK-means biases cluster centroids toward regions of high chemical sensitivity while preserving the input data. Demonstrations on a high-fidelity hydrogen–air detonation dataset show that JSK-means concentrates clusters near the wavefront and triple-point zones, and POD analyses confirm distinct dynamical separation in composition and source-term spaces. The approach promises improved partition-based modeling in reacting flows and multi-physics problems, with potential extensions to other clustering methods and applications.

Abstract

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.

Figures (15)

• Figure 1: Computational domain for channel detonation simulation.
• Figure 2: Cropping procedure for an instantaneous detonation flowfield at $t=50$$\mu$s used to generate the clustering dataset. Colorbar ranges for $T$, $\rho Y_{H2}$, and $\rho Y_{H2O}$ are $[300,3200]$ K, $[0.001,0.006]$$\text{kg}/\text{m}^3\text{s}$, and $[0.1,0.2]$$\text{kg}/\text{m}^3\text{s}$ respectively.
• Figure 3: Singular value distribution for un-normalized (top) and normalized (bottom) chemical Jacobians obtained from dataset $\Phi$. The x-axis denotes the singular value index. For each index, spread in corresponding singular value for all $N$ sample points in $\Phi$ is plotted and colored by temperature. Points correspond to samples above 500 K. Bands are alternately shaded for ease of visualization.
• Figure 4: (Left) Plot of source term (black) and Jacobian (blue) versus phase space variable $\phi$ for the 1D toy problem in Eq. \ref{['eq:1d_toy_problem']}. Variables in the plot have been normalized as per the procedure outlined in Sec. \ref{['sec:scaling']}. (Right) Plot of normalized objective functions for standard (solid) and physics-guided (dashed) K-means clustering approaches versus number of iterations of the K-means algorithm. The first 300 iterations is a burn-in phase that utilizes the standard K-means algorithm in Alg. \ref{['alg:standard_kmeans']}. The next 300 iterations utilize the modified physics-guided K-means algorithm in Alg. \ref{['alg:jacobian_kmeans']}.
• Figure 5: Cluster visualizations provided by standard K-means (top row) and JSK-means algorithm (bottom row) for $K=5$, $10$, and $15$. Centroid locations are provided as the filled markers in each plot.