Abstract
Extended dynamic mode decomposition (EDMD) (Williams et al. in J Nonlinear Sci 25(6):1307–1346, 2015) is an algorithm that approximates the action of the Koopman operator on an N-dimensional subspace of the space of observables by sampling at M points in the state space. Assuming that the samples are drawn either independently or ergodically from some measure \(\mu \), it was shown in Klus et al. (J Comput Dyn 3(1):51–79, 2016) that, in the limit as \(M\rightarrow \infty \), the EDMD operator \(\mathcal {K}_{N,M}\) converges to \(\mathcal {K}_N\), where \(\mathcal {K}_N\) is the \(L_2(\mu )\)-orthogonal projection of the action of the Koopman operator on the finite-dimensional subspace of observables. We show that, as \(N \rightarrow \infty \), the operator \(\mathcal {K}_N\) converges in the strong operator topology to the Koopman operator. This in particular implies convergence of the predictions of future values of a given observable over any finite time horizon, a fact important for practical applications such as forecasting, estimation and control. In addition, we show that accumulation points of the spectra of \(\mathcal {K}_N\) correspond to the eigenvalues of the Koopman operator with the associated eigenfunctions converging weakly to an eigenfunction of the Koopman operator, provided that the weak limit of the eigenfunctions is nonzero. As a by-product, we propose an analytic version of the EDMD algorithm which, under some assumptions, allows one to construct \(\mathcal {K}_N\) directly, without the use of sampling. Finally, under additional assumptions, we analyze convergence of \(\mathcal {K}_{N,N}\) (i.e., \(M=N\)), proving convergence, along a subsequence, to weak eigenfunctions (or eigendistributions) related to the eigenmeasures of the Perron–Frobenius operator. No assumptions on the observables belonging to a finite-dimensional invariant subspace of the Koopman operator are required throughout.
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Notes
We choose to work in the general setting of dynamical systems on arbitrary topological spaces which encompass dynamical systems on finite-dimensional manifolds (in which case one can regard \(\mathcal {M}\) as a subset of \(\mathbb {R}^n\)), as well as infinite-dimensional dynamical systems, arising, for example, from the study of partial differential equations or dynamical systems with control inputs (Korda and Mezić 2016).
Since \(\mathcal {K}{:}\,\mathcal {F}\rightarrow \mathcal {F}\), the assumption of \(\mathcal {F}= L_2(\mu )\) implies that the composition relation \(\phi \circ T\), \(\phi \in L_2(\mu )\), gives rise to a well-defined operator from \(L_2(\mu )\) to \(L_2(\mu )\). In particular, this implies that \(\Vert \phi _1\circ T - \phi _2\circ T\Vert _{L_2(\mu )} = 0\) whenever \(\Vert \phi _1-\phi _2 \Vert _{L_2(\mu )} = 0\) and that \(\int _{\mathcal {M}} |\phi \circ T|^2 \, \mathrm{d}\mu < \infty \) for all \(\phi \in L_2(\mu )\).
To be more specific, if the matrix \(M_{{\hat{\mu }}_M}\) is not invertible, the solution to (10) may not be unique when viewed as a member of \( L_2(\mu )\). When viewed as a member of \(L_2({\hat{\mu }}_M)\), the solution is unique (since it is a projection onto a closed subspace of a Hilbert space). This is because in this case two functions from \(\mathcal {F}_N\) belonging to distinct \(L_2(\mu )\) equivalence classes may fall into the same \(L_2({\hat{\mu }}_M)\) equivalence class.
A sequence of operators \(\mathcal {A}_i\) converges in the operator norm to an operator \(\mathcal {A}\) if \(\lim _{i\rightarrow \infty }\Vert \mathcal {A}_i -\mathcal {A}\Vert = 0 \).
We choose to state the theorem for vector observables as this is the form of prediction typically encountered in practice. For a vector observable \(f\in \mathcal {F}^n\), the norm \(\Vert f\Vert \) is defined by \(\sum _{i=1}^n \Vert f_i \Vert _{L_2(\mu )}, \) where \(f_i\in \mathcal {F}\) is the \(i^{\mathrm {th}}\) component of f.
In Sect. 7, we show how the matrix \(A_N\) can be constructed analytically.
A sufficient condition for \(C(\mathcal {M})\) to be separable is \(\mathcal {M}\) compact and metrizable.
A sequence of functionals \(L_i\in C(\mathcal {M})^\star \) converges in the weak\(^\star \) topology if \(\lim _{i\rightarrow \infty }L_i(f) = L(f)\) for all \(f\in C(\mathcal {M})\).
A sequence of Borel measures \(\mu _i\) converges weakly to a measure \(\mu \) if \(\lim _{i\rightarrow \infty }\int f\,\mathrm{d}\mu _i = \int f\,\mathrm{d}\mu \) for all continuous bounded functions f. This convergence is also referred to as narrow convergence and it coincides with convergence in the weak\(^\star \) topology if the underlying space is compact (which is the case in our setting).
A measure \( \mu \) on \(\mathcal {M}\) is invariant if \(\mu (T^{-1}(A)) = \mu (A)\) for all Borel sets \(A\subset \mathcal {M}\) or equivalently if \(\int _{\mathcal {M}} f\circ T \, \mathrm{d}\mu = \int _{\mathcal {M}} f \, \mathrm{d}\mu \) for all continuous bounded functions f.
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Acknowledgements
The first author would like to thank Mihai Putinar for discussions on a related topic that sparked the work on this paper. We also thank Clancy Rowley for helpful comments on an earlier version of this manuscript as well as to Péter Koltai for pointing out the reference (Klus et al. 2016). This research was supported in part by the ARO-MURI Grant W911NF-14-1-0359 and the DARPA Grant HR0011-16-C-0116. The research of M. Korda was supported by the Swiss National Science Foundation under Grant P2ELP2_165166.
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Communicated by Bruno Eckhardt.
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Korda, M., Mezić, I. On Convergence of Extended Dynamic Mode Decomposition to the Koopman Operator. J Nonlinear Sci 28, 687–710 (2018). https://doi.org/10.1007/s00332-017-9423-0
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DOI: https://doi.org/10.1007/s00332-017-9423-0