Learning Koopman eigenfunctions and invariant subspaces from data: symmetric subspace

M. Haseli, J. Cortés
IEEE Transactions on Automatic Control 67 (7) (2022), 3442-3457

Abstract

This paper develops data-driven methods to identify eigenfunctions of the Koopman operator associated to a dynamical system and subspaces that are invariant under the operator. We build on Extended Dynamic Mode Decomposition (EDMD), a data-driven method that finds a finite-dimensional approximation of the Koopman operator on the span of a predefined dictionary of functions. We propose a necessary and sufficient condition to identify Koopman eigenfunctions based on the application of EDMD forward and backward in time. Checking this condition requires the comparison of the eigendecomposition of matrices whose size grows with the size of the dictionary. To address this, we propose the Symmetric Subspace Decomposition (SSD) algorithm which provably identifies the maximal Koopman-invariant subspace and the Koopman eigenfunctions in the span of the dictionary. We also introduce the Streaming Symmetric Subspace Decomposition (SSSD) algorithm, an online method that only requires a small, fixed memory and updates its estimate of the invariant subspace as new data is received. We prove that, given a data set, SSSD and SSD find the same solution.

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