Efficient Identification of Linear Evolutions in Nonlinear Vector Fields: Koopman Invariant Subspaces

Abstract

This paper presents a data-driven approach to identify finite-dimensional Koopman invariant subspaces and eigenfunctions of the Koopman operator. Given a dictionary of functions and a collection of data snapshots of the dynamical system, we rely on the Extended Dynamic Mode Decomposition (EDMD) method to approximate the Koopman operator. We start by establishing that, if a function in the space generated by the dictionary evolves linearly according to the dynamics, then it must correspond to an eigenvector of the matrix obtained by EDMD. A counterexample shows that this necessary condition is however not sufficient. We then propose a necessary and sufficient condition for the identification of linear evolutions according to the dynamics based on the application of EDMD forward and backward in time. Due to the complexity of checking this condition, we propose an alternative characterization based on the identification of the largest Koopman invariant subspace in the span of the dictionary. This leads us to introduce the Symmetric Subspace Decomposition strategy to identify linear evolutions using efficient linear algebraic methods. Various simulations illustrate our results.

Sparsity-promoting algorithms for the discovery of informative Koopman-invariant subspaces

“…When the number of features is larger than the number of data snapshots, EDMD eigenvalues can be misleading (Otto & Rowley 2019) and often plagued with spurious eigenfunctions that do not evolve linearly even when the number of data snapshots is sufficient. Analytically, it is clear that a Koopman eigenfunction in the span of the dictionary will be associated with one of the eigenvectors obtained from EDMD, given Ψ x is full rank, and contains sufficient snapshots M (Haseli & Cortés 2019). Indeed, the EDMD is an L 2 projection of the Koopman operator under the empirical measure (Korda & Mezić 2018b).…”

“…This is effectively a recursive implementation of EDMD. Recently, Haseli & Cortés (2019) showed that given a sufficient amount of data, if there is any accurate Koopman eigenfunction spanned by the dictionary, it must correspond to one of the obtained eigenvectors. Moreover, they proposed the idea of mode selection by checking if the reciprocal of identified eigenvalue also appears when the temporal sequence of data is reversed, which is similar to the idea of comparing eigenvalues on the complex plane from different trajectories, as proposed by Hua et al (2017).…”

Learning Stable Models for Prediction and Control

“…Besides simplicity, using the Koopman linear representation for control can in certain cases also outperform feedback policies that are based on the underlying nonlinear dynamics [28], [29]. However, unless finite-dimensional Koopman invariant subspaces exist [28], [30], [31], the operator is infinitedimensional and renders practical use challenging. Studies seek finite-dimensional approximations using methods such as the Dynamic Mode Decomposition (DMD) [32] extended DMD (EDMD) [33], [34], Hankel-DMD [35], or closedform solutions [36], [37] which use state measurements to approximate Koopman operators.…”

Parallel Learning of Koopman Eigenfunctions and Invariant Subspaces For Accurate Long-Term Prediction

“…Here, we gather some important results from [22], [31] regarding the SSD algorithm and its output. (A.…”

“…The next result shows that the eigendecomposition of K SSD captures all the functions in the span of the original dictionary that evolve linearly in time according to the available data. Theorem A.3: (Identification of Linear Evolutions using the SSD Algorithm [31]): Under Assumption 3.1, K SSD w = λw for some λ ∈ C and w ∈ C cols(KSSD)\{0} if and only if there…”