A co-kurtosis based dimensionality reduction method for combustion datasets

TL;DR

This work addresses PCA’s insufficiency for capturing extreme chemical events in combustion data by introducing CoK-PCA, which uses the fourth-order co-kurtosis tensor $T_{ijkl}$ and its cumulant $K_{ijkl}$, factorized via matricization and SVD to obtain orthonormal co-kurtosis principal vectors. Reduced representations project data onto the top $n_q$ vectors and reconstruct linearly as $\mathbf{X}_q=\mathbf{Z}_q\mathbf{A}_q^T$, offering a contrast to PCA’s covariance-based eigenvectors. The method is validated on a synthetically representative dataset, a zero-D homogeneous reactor ignition case, and a 2D DNS of HCCI, showing that CoK-PCA more accurately captures stiff dynamics and improves the accuracy of species production rates and heat release rate in reacting zones, with robust performance under nearby test conditions. These results indicate that higher-order moments provide a promising path for more faithful reduced-order models in combustion, and motivate exploring non-linear reconstructions and hybrid PCA/CoK-PCA strategies in future work.

Abstract

Principal Component Analysis (PCA) is a dimensionality reduction technique widely used to reduce the computational cost associated with numerical simulations of combustion phenomena. However, PCA, which transforms the thermo-chemical state space based on eigenvectors of co-variance of the data, could fail to capture information regarding important localized chemical dynamics, such as the formation of ignition kernels, appearing as \rev{extreme-valued} samples in a dataset. In this paper, we propose an alternate dimensionality reduction procedure, co-kurtosis PCA (CoK-PCA), wherein the required principal vectors are computed from a high-order joint statistical moment, namely the co-kurtosis tensor, which may better identify directions in the state space that represent stiff dynamics. We first demonstrate the potential of the proposed CoK-PCA method using a synthetically generated dataset that is representative of typical combustion simulations. Thereafter, we characterize and contrast the accuracy of CoK-PCA against PCA for datasets representing spontaneous ignition of premixed ethylene-air in a simple homogeneous reactor and ethanol-fueled homogeneous charged compression ignition (HCCI) engine. Specifically, we compare the low-dimensional manifolds in terms of reconstruction errors of the original thermo-chemical state, and species production and heat release rates computed from the reconstructed state. \rev{The latter -- a comparison of species production and heat release rates -- is a more rigorous assessment of the accuracy of dimensionality reduction.} We find that, even using a simplistic linear reconstruction, the co-kurtosis based reduced manifold represents the original thermo-chemical state more accurately than PCA, especially in the regions where chemical reactions are important.

Figures (16)

• Figure 1: A bi-variate synthetic dataset with a few extreme-valued samples representing an anomalous event. Red and blue lines are the principal vectors obtained from PCA and CoK-PCA, respectively. Solid and dashed lines indicate the first and the second principal vectors, respectively.
• Figure 2: Error ratio ($r_i$) for variables $v_1$ and $v_2$ in synthetic data. Darker blue and brown colors: error ratio based on maximum error, lighter blue and brown colors: error ratio based on average error.
• Figure 3: Alignment between PCA and CoK-PCA principal vectors from homogeneous reactor dataset represented as a dot product for each mode.
• Figure 4: Plots of error ratio $r_i$ ($i\in \{M,A\}$, see Eq. \ref{['eq:error-ratio']}) for reconstructed data, $n_q=5$, from homogeneous reactor dataset. Error ratio $r_i$ based on maximum (a) and average (b) errors of species mass fractions and temperature. Error ratio $r_i$ based on maximum (c) and average (d) errors of species production rates and heat release rate.
• Figure 5: Plots of error ratio $r_i$ ($i\in \{M,A\}$, see Eq. \ref{['eq:error-ratio']}) for reconstructed data, $n_q=15$, from homogeneous reactor dataset. Error ratio $r_i$ based on maximum (a) and average (b) errors of species mass fractions and temperature. Error ratio $r_i$ based on maximum (c) and average (d) errors of species production rates and heat release rate.