Jorge Cortés
Professor
Cymer Corporation Endowed Chair
Learning Koopman eigenfunctions and invariant subspaces from data:
Symmetric Subspace Decomposition
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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Mechanical and Aerospace Engineering,
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