Compression of the Koopman matrix for nonlinear physical models via hierarchical clustering

TL;DR

This work tackles the scalability of Koopman-operator methods when using EDMD with large dictionaries by introducing a hierarchical-clustering-based compression of the Koopman matrix. The authors cluster rows and columns to form a reduced matrix $K'$, then construct compressed dictionaries before and after the action and recover compatibility with time evolution through a recovery matrix $R$. Empirical results on a cart-pole dataset show that the proposed clustering-based compression maintains predictive accuracy while achieving substantial speedups and memory savings, outperforming SVD-based low-rank approximations at similar budgets. The method offers a practical, non-neural alternative for efficient linear-analysis of nonlinear dynamics with potential broad applicability in prediction and control of complex systems.

Abstract

Machine learning methods allow the prediction of nonlinear dynamical systems from data alone. The Koopman operator is one of them, which enables us to employ linear analysis for nonlinear dynamical systems. The linear characteristics of the Koopman operator are hopeful to understand the nonlinear dynamics and perform rapid predictions. The extended dynamic mode decomposition (EDMD) is one of the methods to approximate the Koopman operator as a finite-dimensional matrix. In this work, we propose a method to compress the Koopman matrix using hierarchical clustering. Numerical demonstrations for the cart-pole model and comparisons with the conventional singular value decomposition (SVD) are shown; the results indicate that the hierarchical clustering performs better than the naive SVD compressions.

Figures (10)

• Figure 1: Cart-pole model. The external force $u$ only acts on the cart horizontally.
• Figure 2: Linear computation via the Koopman operator. Instead of the original nonlinear dynamics, we employ the linear operator $\mathcal{K}$ in the observable space. In practice, we estimate a finite-dimensional matrix to approximate the linear operator.
• Figure 3: An example of a dendrogram constructed from hierarchical clustering. Here, the five elements are clustered hierarchically according to distance.
• Figure 4: An example of the original matrix for the clustering. Since the Koopman matrix is square, we start from this square matrix in the following explanation.
• Figure 5: Hierarchical clustering for the matrix. Rows and columns are hierarchically clustered separately.