Modern Koopman Theory for Dynamical Systems

Learning Dynamical Systems via Koopman Operator Regression in Reproducing Kernel Hilbert Spaces

“…Inequality ( 25) is a direct consequence of the definition of the separation. On the other hand, (26) follows immediately from (7) and the fact that…”

“…Then, the dynamical system can be decomposed into superposition of simpler signals that can be used in different tasks such as system identification and control, see e.g. [7]. More precisely, given an observable f ∈ span{ψ i | i ∈ N} there exist corresponding scalars γ f i ∈ C known as Koopman modes of f , such that…”

“…Further, the error degrades as t increases. This implies that the prediction error (t = 1) is only controlled by the risk but in general forecasting (t ≥ 1) will gets increasingly harder for larger t. Second, Inequality (7) shows that (λ i , S π ψ i ) is approximately an eigenpair of the Koopman operator A π . Indeed, it guarantees that each eigenfunction of an estimate G, considered as an equivalence class in L 2 π (X ), approximately satisfy the Koopman eigenvalue equation.…”

“…In the context of dynamical systems, this estimator is known as kernel Dynamic Mode Decomposition, and is of utter importance in a variety of applications [7,23]. Note that, however, PCR does not minimize the empirical risk under the low-rank constraint.…”

“…In light of recent machine learning progress, it is appealing to ask if the properties of dynamical systems can be estimated (learned) from empirical data. Beyond machine learning this question has a long history in dynamical systems [7]. The go-to reference for data-driven algorithms to reconstruct dynamical systems is [23], where numerous methods based on so-called dynamic mode decomposition (DMD) are discussed along with interesting applications.…”