Learning Koopman Eigenfunctions and Invariant Subspaces From Data: Symmetric Subspace Decomposition

This article develops data-driven methods to identify eigenfunctions of the Koopman operator associated with 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. Moreover, we propose the Symmetric Subspace Decomposition (SSD) algorithm, an iterative method that provably identifies the maximal Koopman-invariant subspace and the Koopman eigenfunctions in the span of the dictionary. We also introduce the Streaming SSD algorithm, an online extension of SSD that only requires a small fixed memory and incorporates new data as is received. Finally, we propose an extension of SSD that approximates Koopman eigenfunctions and invariant subspaces when the dictionary does not contain sufficient informative eigenfunctions.