The Kernel Perspective on Dynamic Mode Decomposition
Authors: Efrain Gonzalez
Moad Abudia
Michael Jury
Rushikesh Kamalapurkar
Joel A Rosenfeld
Venue: Accepted by TMLR ()

Abstract: This manuscript takes a critical look at the interactions between Koopman theory and reproducing kernel Hilbert spaces with an eye towards giving a tighter theoretical foundation for
Koopman based dynamic mode decomposition (DMD), a data driven method for modeling
a nonlinear dynamical system from snapshots. In particular, this paper explores the various
necessary conditions imposed on the dynamics when a Koopman operator is bounded or
compact over a reproducing kernel Hilbert space.
Ultimately, it is determined that for many RKHSs, the imposition of compactness or boundedness on a Koopman operator forces the dynamics to be affine.
However, a numerical method is still recovered in more general cases through the consideration of the Koopman operator as a closed and densely defined operator, which requires
a closer examination of the connection between the Koopman operator and a RKHS. By
abandoning the feature representation of RKHSs, the tools of function theory are brought to
bear, and a simpler algorithm is obtained for DMD than what was introduced in Williams
et al (2016). This algorithm is also generalized to utilize vector valued RKHSs.

Published in Transactions on Machine Learning Research (09/2024)
The Kernel Perspective on Dynamic Mode Decomposition
Efrain H Gonzalez"
ehgonza@sandiae
Deportment of Mathematics ond Statistics
University of South Florida
Moad Abudia
abudia @okstate. edu
School of Mechanical and Aerospace Engineering
Oklahoma State University
Michael Jury
mjurySSuft.c edu
Department of Mathematics
University of Florido
Rushikesh Kamalapurkar
rkamalapurkar edu
Department of Mechanical and Acrospace Engineering
Universily of Florida
Joel A. Rosenfeld
rosenfeldjj Trsff. edu
Department of Mathematics and Statistics
University of South Florida
Reviewed on OpenReview: https: // openreveewwww net/ form? id- sI88aV7hG
Abstract
This manuscript takes a critical look at the interactions between Koopman theory and repro- ducing kernel Hilbert spaces with an eye towards giving a tighter thcoretical foundation for Koopman based dynamic mode decomposition (DMD), a data driven method for modeling nonlinear dynamical system from snapshots. In particular, this paper explores the various necossary conditions imposed the dynamics when Koopman operator is bounded or compact over reproducing kernel Hilbert space.
Ultimately, it is determined that for many RKHSs, the imposition of compactness or bound- edness on Koopman operator forces the dynamics be affine.
However, numerical method is still recovered in more general cases through the consid- eration of the Koopman operator as closed and densely defined operator, which requires a closer examination of the connection between the Koopman operator and a RKHS. By abandoning the feature representation of RKHSs, the tools of function theory are brought to bear, and simpler algorithm is obtained for DMD than what was introduced in Williams et al (2016). This algorithm is also generalized to utilize vector valued RKHSs.
Introduction
Dynamic mode decomposition (DMD) has been gaining traction as model-free method of making short-run predictions for nonlinear dynamical systems using data obtained as snapshots trajectories. DMD aims to obtain finite rank representation of the Koopman operator by studying its action on the full state observable (i.e. the identity function) (Rowley al. 2009). Koopman operators over reproducing kernel Hilbert spaces (RKHSs) were studied to take advantage of infinite dimensional feature spaces to extract more information from the snapshots of system in (Williams et al. 2015a). This perspective also enacts dimensionality reduction by formulating the DMD method in reproducing kernel Hilbert space (RKHS) framework and implicitly using the kernel trick to compute inner products in the higl--dimensional space of observables. In
NNo at the Department of Cognitive & Emerging Computing, Sandia National Laboratories 'The following YouTube playlist discusses the content of this manuscript: https///oottue..cooaaaalylleseee

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Published in Transactions on Machine Learning Research (09/2024)
(Williams et al. 2015b), it is shown that kernel-based DMD produces collection of Koopman modes that agrees with other DMD results in the literature.
The introduction of kernel based techniques for Koopman analysis and DMD has catalyzed a new direction for Koopmanism. The new perspective added by kernel methods is that of approximations and is at the center of the work presented this manuscript. Universal RKHSs, such as those corresponding to Gaussian RBFs and exponential dot product kernel functions, have the ability approximate any continuous function over compact subset of IR" to any desired accuracy up to computational precision. Morcover, when the kernel function is continuous or bounded, convergence in RKHS norm yields point wise everywhere convergence and uniform convergence over compact subsets of IR" (Steinwart & Christmann, 2008). This everywhere convergence stands in contrast to the perspective through the lens of ergodic methods, where convergence results only hold almost everywhere. For additional insight into the point wise everywhere convergence property described above see example 1 in Appendix A.I
The study of dynamical systems through the Koopman formalism over RKHSs manifests as a search for functions that are close to being eigenfunctions of the Koopman operator, rather than the actual eigenfune- tions. Since only finite amount of data can be available for the study of an infinite dimensional operator, actual cigenfunctions typically cannot be computed. In fact, there is no requirement that K₂ even has any eigenfunctions. The universality property arises in the search for Koopman "eigenfunctions, which, given a particular Koopman operator and kernel space, might not exist, since the existence of an eigenfunction depends on the Hilbert space. However, the formal equation, (FF(x)) - Ay(xx) = may still hold for some continuous function ip. If one is working over a RKHS corresponding to universal kernel, then for any given compact set and 0, there is function & H such that lp(x) {(x)] e for all in that compact set, which provides for an "approximate" eigenfunction. Here we define approximate eigenfunctions follows:
Definition For E an approximate cigenfunction & is a function in the Hilbert space H together with E C such that [Kr((x)) Xp(x)) < for all E X.
This is particularly important for DMD methods, which attempt to construct a finite rank approximation of a Koopman operator from a finite collection of observed snapshots. Note that obtaining approximate eigenfunctions as in Example D is not dissimilar to the objective of ergodic methods, where approximation of system invariants and eigenfunctions using time averages is sought. The existence of eigenfunctions depends on the selection of the Hilbert space, will be shown in Section and eigenfunctions may not be present even in the L ergodic setting (Budišić et al. 2012).
The objective of this manuscript is to present the kernel perspective of Koopmanism as a distinct study from the ergodic perspective. With that goal in mind, the paper is structured in the following way:
Section 2: Introduces reproducing kernel Hilbert spaces (RKHSs) which will be necessary for the remainder of the paper.
Section 3: Introduces Koopman operators, their relationship DMD, and the differences between the ergodic and kernel perspectives. Here the pointwise convergence properties in the RKHS are presented.
Section 4: Discusses properties of Koopman operators over RKHSs. It is demonstrated that assump- tions of boundedness and compactness hold for very small collection of Koopman operators.
Section 5: A numerical algorithm for a Koopman based DMD method is presented, which relaxes the assumptions of bounded and compact Koopman operators to densely defined Koopman operators. This yields a new theoretical foundation for the study of Koopman DMD over RKHSs. The new algorithm still relies on some of the same matrices presented in (Williams et al. 2015b).
Scction 6: Extends the developed DMD algorithm to vector valued RKHSs.
Section 7: Provides a numerical example that compares the developed DMD method that pre- sented in WWiliiams et al. 2015b).
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Appendix: Includes examples and proofs referenced in the text as well as few numerical experiments which show that the developed DMD algorithm has nearly identical results to that developed by [Williams et al. 2015b).
2 Reproducing Kernel Hilbert Spaces
Definition 2. A Reproducing Kernel Hilbert Space (RKHS) H over a set X is Hilbert space composed of functions from X to C such that for all a E X the evaluation functional Ex. == [(x is bounded.
Therefore, for all x E X there exists function K₂ E H such that f(x) = Kz) for all f E H. The function K2 is the reproducing kernel centered at x and the function K X x + C defined by [(x,y) y) = (Ky, Kz) is the unique kernel function corresponding to H [Aronszajn] 1950 Throughout most of this manuscript, the methods used will be restricted RKHSs of real valued functions. However, for some specific examples in Section will be more convenient employ RKHSs of complex valued functions of real variable.
Reproducing kernels can be equivalently expressed as realizations of inner products of feature space mappings in (NN (Steinwart & Christmann 2008).
Proposition 1. Given orthonormal basis for RKHS, {em()))m=1 H, the kernel function may be expressed as K(x, v) = Em=I em (a))m)mm)) where V (x) == (e1(z), (n(2))e2(2)) (2(2).....)) .) (N) is called a feature map. Equivalently, given a feature mapping V : - (N), there is RKHS whose kernel function is given as K(x, v) = (x)), (y))e.
We will make repeated use of projections onto finite dimensional vector spaces arising from spans of collections of kernels centered at snapshots from a dynamical system.
Proposition 2. For a collection of centers {.i,..., ,.m) the projection of a function g E H ondo a = span gg....., K.. given as arg min I/h - H. The projection can be expressed in terms of a as Pag = Ei=1 w; W, where the weights wi satisfy
(111
K(x1,Tm)
Wi
((x1)
:
(1)
((xm, 21)
K(Im,
Wm
g(xm)
Proof. The projection of function a Hilbert space onto a closed subspace is determined by finding the closest member of that subspace to the function being projected. The matrix equation may be established by expressing the projection as Pag E WyK,, expanding E w,K2, 1 al via inner products, and setting the derivative with respect to = (W1, wm)) TT E R.m to zero, resulting in the system of linear equations in equation !
If the functions (Kzj...., K2 } are linearly independent then the Gram matrix in equation 1 is non-singular. In that case, the weights {wr) are unique.
Definition 3. A RKHS of real valued functions, H, over Ω C R", is said to be universal if for any compact V VC e > 0, and h E C(V), there a function h E H such that || Tillo < e, where C(V) denotes the set of continuous functions defined on V.
Many commonly used kernel functions satisfy this universality property, including the Gaussian RBF kernel functions and the exponential dot product kernel functions (Steinwart & Christ 2008
3 Koopman Operators over RKHSs
The theory of Koopman operators has been long intertwined with ergodic theory, where ergodic theoretic methods justify almost everywhere convergence claims of time averaging methods invariants of the Koop- man operator. The Birkhoff and Von Neumann ergodic theorems are posed over L! (R) and LP(R) for p > 1, respectively (Walters 2000). However, the invariants for Koopman operators not always analytic even
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smooth. Hence, ergodic theorems do not give guarantees of convergence within most RKHSs, which are frequently composed of real analytic functions. Furthermore, even though ergodic theorems guarantee the existence of invariants over (IR), the invariant itself is hidden behind a limiting operation via time averages (Walters. 2000). Hence, there is an expected error in the constructed invariant that stems from finitencss of data.
The objective of DMD methods is to find functions within the function space that nearly achieve eigenfunction behavior. Specifically, for Koopman operator, Kp, and c <> 0, the objective is to find H and À C
for which |Kr((zz) Xp(z)) for all in given workspace. When such function is discovered, the eigenfunction witnesses the snapshots, Fi+1 F(zi), as an exponential function as ((1++1) = X)((z)) + 1-1#+ 1-A
E. If P is a proper eigenfunction for Kp, then may be taken to be zero, and ((z++1) X (2((1))
only for almost all within the domain of interest. For < e may be relaxed to || X < since (Kr((x) - < K x < Ce, where C > 0 depends on the kernel function and the point z. If the kernel function is continuous and the domain is compact, a finito C' may be selected uniformly for that domain. In the special case of the Gaussian RBF kernel function and the domain being R", C may be taken to be 1.
Ergodic methods generally yield [Kp}(x)->((())
RKHS, the condition (Krā(x)-
Proposition 3. If KF is compact and the finâte rank approzimation of Kp, which we will denote as RE, is within of Kr with respect to the operator norm, and if & is a normalized eigenfunction for Kp with eigenvolue X, then ||Cp@p x S ||rr Rr S [| R S..
Hence, if we obtain a finite rank approximation of K2 that is within with respect to the operator norm, then an eigenfunction of the finite rank approximation will be an approximate eigenfunction of Kpp. This approximation is important in DMD, where the eigenfunctions are utilized to generate an approximation of the full state observable, (x) == x, one dimension at time, as outlined Section 5
If an accurate finite rank approximation of the Koopman operator can be obtained in a RKHS, then the approximation of the overall model is accurate point- everywhere via the same proof given in (Rosenfeld) et al. 2022). and uniformly over compact sets when the RKHS consists of continuous functions. In contrast, the approximation is only accurate almost everywhere when considering Koopman operators posed over (RR. Point-wise everywhere approximation is a distinctive advantage of kernel based methods. This convergence result is less clear from an invocation of the kernel trick machine learning, and exemplifies the advantage of the operator thcoretic considerations introduced this manuscript.
The approximation of the Koopman operator in the operator norm topology naturally leads to the question of when Koopman operator, or more generally, composition operator, can be compact. Compactness is a central issue for DMD as every approximation is indeed of finite rank, stemming from the observed data. In 1979, it was established in (Singh & Kumar 1979h that composition operators over I,2 (1) cannot be compact when j is non-atomic. Indeed, (TR) has no compact composition operators.
Koopman operators over most frequently considered RKHSs are compact for very narrow range of dynamics. In fact, most Koopman operators over these spaces are not even bounded, as will be expanded upon in Section 4.2.3 Unboundedness is added complication for the valid implementation of DMD, addressed in Section using densely defined and potentially unbounded Koopman operators. Alternatively, other classes of compact operators over RKHSs connected to dynamical systems can be leveraged for DMD procedures to give convergence guarantees (see, e.g., Rosenfeld et al. 2022
The remainder of this section introduces densely defined Koopman operators over RKHSs, where Lemina enables the DMD algorithm introduced in Section 5
Definition 4. Let H be a RKHS over R" For function F R + R we define the Koopman Operator (sometimes called a composition operator), KF D(KF)) + H, as Krg go F where D(KF) {g E H go F E H}. When D(Kr) is dense in H, Kr is said to be densely defined.
While not all densely defined Koopman operators over RKHSs are bounded, they are all closed operators
Pedersen 2012)
Lemma Let F X + X be the symbol for a Koopman operator over RKHS H over a set X. KF D(CF) - H is a closed operator.
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The proof of the above lemma is provided in Appendix B.1
Proposition 4. If Koopman operalor is densely defined, ils adjoinl is densely defined and closed.
Proof. Given the kernel function centered at 24,
=
(g. Kr(x))
Therefore, the linear functional g + (Krg, K₂) is bounded over H. Thus, K E D(K).) for all z E X, and Kr = F(a))- Hence, cach kernel function is in the domain of the adjoint of a densely defined Koopman operator, and as the span of kernel functions is dense their RKHS, the adjoint is densely defined.
4 A Different Landscape for Koopman Operators
This section examines properties of Koopman operators over RKHSs. The selection of space fundamen- tally changes the behavior of Koopman operators over that space, where properties such as the lattice of eigenfunctions, common cigenfunctions for different discretizations, and boundedness of the operators may not hold. In the succeeding subsections we discuss each of these properties and provide counter examples for each of these propertics in Appendix A.3 for Koopman operators corresponding the continuous time dynamics i = (x2 -z1)". We discuss that many RKHSs only support bounded Koopman operators when the discretized dynamics are affine, and in the case of the Gaussian RBF's native space, we provide a novel proof of this fact in Appendix [3.2]
Subsoquently, the notation FAt will denote the discrctized dynamics corresponding to à = (x2 -m1) T with fixed time step A > 0.
Recently, kernel methods have been adapted for the study of DMD and Koopman operators, largely through the guisc of extended DMD, where kernels are leveraged to simplify computations via the kernel trick. However, the adjustment from the classical study of Koopman operators through ergodic theory to that of reproducing kernel Hilbert spaces leads to significant. differences in the Koopman operators and their properties. In most cases, the ergodic theorem cannot be directly applied to recover invariants of the operation 9 + 9 o F, for given F, since those invariants may be nonsmooth. This section exemplifies some of the distinguishing properties of Koopman operators over RKHSs, and in some cases, illustrates their limitations.
Much of the classical properties of Koopman operators is established in a variety of specific contexts, such T.2 spaces of invariant measures and I,1 spaces, can be seen in (Budišić et al. 2012 Kawahara 2016 Kutz
ct al. 2016b Brunton & Kutz 2019 Brunton et al. 2021). Properties of the Koopman operator strongly depend on the selection of underlying vector space, and boundedness,, compactness, eigenvalues, elc. change based on this selection. While Koopman operators were introduced by Koopman in 1931 in (Koopman, 1931 and then later picked up by the data science community in the carly 2000s (c.g. (Mezić) 2005 Kutz et al. 2016b)), the study of such operators and their properties continued in earnest throughout, the 20th century as composition operators (e.g. (Shapiro 2012 This is particularly important. for RKHSs, where the specification of a bounded or densely defined Koopman operator over a particular space yields strong restrictions on the available dynamics.
4.1
Concerning Sampling and Discretizations
4.1.1
Forward Complete Dynamics
In applications, Koopman operators enter the theory of continuous time dynamics through discretization
of the continuous time dynamical system (Bittracher et al. 2015 Mauroy & Mezić 2016). That is, given the dynamical system & = f(x), the system is discretized through the selection of fixed time-stop, At > O, as fm + [x(())d where the right hand side plays the role of the discrete dynamics. However, Tm+ Fmm J.m
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for such discretization exist for arbitrarily large values of m, it is necessary that the dynamics be forward complete. The forward completeness assumption restricts the class of continuous dynamics on which Koopman based methods may be applied. For example, the continuous time dynamics, == does not admit a discrctization, since it is not forward complete. Example 2 in Appendix A.2 demonstrates where discrelization fails lor 3 + DMD method that circumvents this requirement, by utilizing Liouville operators and occupation kernels, may be found in (Rosenfeld al. 2022)
4.1.2 Sampling and Data Science
Ergodic based methods as employed in (Budišić et al. 2012 Mezić 2005 Kutz et al. 2016b b Takeishi et al. 2017) provide a methodology for obtaining invariants and eigenfunctions for Koopman operator almost everywhere. That is, by selecting continuous representative of an equivalence class in an L space for the invariant measure, at almost every point within the domain, time averaging against that representative will converge an invariant of the operator. However, this is "probability 1 result, and the number of points where it may fail can potentially be uncountable. Without any external information concerning the convergence, there is no true guarantee that at a particular selected point, the time-averaged approximation will be close to the evaluation an actual invariant at that point. Such computational issues are precisely where the strength of kernel methods manifest. To illustrate the kernel method consider the following theorem:
Thcorem 1. Let F R + R" be discretization of dynamical system with the corresponding Koopman operalor, KF H H, and let H be RKHS over R" consisting of continuous functions. Suppose further that 0 and R H + H is an approrimation of KF such that the norm difference is bounded as [[F - Krll < c. Suppose that & E H and is a normalized eigenfunction of Re with eigenvalue X. Then & is an upprozimate eigenfunciton of Kp.
Proof.
(Kr(()) - xp(=)== (Kr(()) (x(x)((2)) IK(rr Kr)p, K(,x))nll & ||Cr Kr]ll SC. C
and C' O is constant that depends on the kernel function and prespecified compact domain. The compact domain may be extended to all of R in some cases, such as when the kernel function is the Gaussian RBF kernel function.
Thus it can be seen that kernel spaces and approximations that are close to the Koopman operator, in operator norm, can provide functions that behave similar to eigenfunctions the Koopman operator. More- over, the difference behavior from proper eigenfunction governed pointwise by how close the operalor approximation is in the first place.
4.2 Properties of the Operators
This section will consider the classical Fock space consisting of entire functions as a function space over which the Koopman operator is defined. The Fock space is used extensively in Quantum Mechanics (Hall 2013) and it is space where operators have been well studied (Zhu, 2012).
Definition 5. The Fock space is a RKHS, with kernel function K(2, w) = ="= The kernel function for the Fock space over C may be obtained through a product of single variable kernels as K(2, w) = e
21 2n The Fock space is given as
F²(C) (10)=110=) - Σ amz
8
8
Σ
am Pmm < 00
Closely related to the Fock space is the exponential dot product kernel, e" where for a single variable, the exponential dot product kernel's native space may be obtained by restricting the Fock space to the reals, and

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then taking the real part of the restricted functions. Through conjugation of the exponential dot product kernel, the Gaussian RBF may be obtained as
1211122 - ""V -1llll11///2 exp ( 2 2
Ka(a,
=e =
2
and performing the same operation on the Fock space kernel over C" yields
(2 w)²
Kc(2, w) = = =2 = n²
exp
2
which is the kernel corresponding to the complexified native space for the Gaussian radial basis function
over C (cf. (Steinwart & Christmann 2008 ). This space may be expressed as
(C( - (0 F*(C))
and the native space corresponding to the Gaussian RBF can be obtained by taking the real parts of functions from H& and restricting to R".
4.2.1 Lattice of Eigenfunctions
As presented in (Budišić et al. 2012; Klus et al. 2015), the eigenfunctions of Koopman operators over - (I) form a lattice. That is if ffu and f22 are two eigenfunctions for the Koopman operator, then is p1 P2- For the lattice to occur more generally, it is necessary for the product of the eigenfunctions to be member of the underlying vector space. This closure property holds, for example, in the space of continuous functions and other Banach algebras. Hilbert spaces are not generally Banach algebras, and since it is desirable to work over Hilbert spaces for properties such as best approximations, projections, and orthonormal bases (cf. (Folland, 1999)), it is important to point out that the closure property of eigenfunctions of Koopman operators does not hold in general. For example the eigenfunctions of K,,, do not form lattice. Fundamentally, powers of p(²) = /4 E (C( are not all contained in the Fock space. This ultimately a consequence of growth conditions imposed by the RKHS norm. For more details consult Appendix A.3.1
4.2.2 Common Eigenfunctions
The intuition behind the use of Koopman operators in the study of continuous time dynamical systems is that eigenfunctions for the Koopman operators should be "close" to that of the Koopman generator for small timesteps. However, semi-groups of Кoopman operators do not always share a common collection of eigenfunctions. Since each Koopman operator obtained through a fixed time-step may produce a different collcction of cigenfunctions, there no way to distinguish which, any, should correspond to cigenfunctions of the Koopman generator. In Appendix A.3.2 we show that an eigenfunction for the Koopman operator corresponding to Fz/2 and is not an eigenfunction for the Koopman operator corresponding to Fr/3-
4.2.3 Boundedness of Koopman Operators
Throughout the literature, it is frequently assumed that Koopman operators are bounded. This assumption manifests as an unrestricted selection of observables in the study of the Koopman operator. When a Koopman operator is a densely defined operator whose domain the entire Hilbert space, it is also closed (Pedersen 2012). Hence, by the closed graph theorem (cf. (Folland) 1999 Theorem 5.12)), such an operator must be bounded. Furthermore, the collection of finite rank operators is dense in the collection of bounded operators over Hilbert space in the strong operator topology (SOT) (cf. (Pedersen 2012
Paragraph
4.6.2)). Convergence in SOT was independently studied in the work (Korda & Mezic 2018 where the
DMD routine was demonstrated to converge to a bounded Koopman operator in SOT.
As mentioned in Korda & Mezić 2018), SOT convergence does not in general lead to convergence of the eigenvalues. To preserve spectral convergence, the finite rank approximations produced by DMD algorithms need to converge to Koopman operators in the operator norm topology. The most direct approach, and
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one that leads to good pointwise estimates of eigenfunctions, is through the use of compact Koopman operators. However, it isn't immediately clear when one can expect continuous dynamical system to yield compact Koopman operator through discretization. For example, the Koopman operator corresponding to discretization of the continuous time system = 0 is the identity operator, I, for any fixed time step, and I is not compact over any infinite dimensional Hilbert space.
In addition, for any given RKHS, the collection of bounded Кoopman operators is very small. It was demonstrated in (Carswell et al., 2003) that Koopman operator over the Fock space is bounded only when the corresponding discrete dynamics are affine. It follows that the same result holds over the exponential dot product kernel's native space.
It may perhaps be less obvious that this result extends to the Gaussian RBF's native space. A proof of this fact was first published in (Ikeda et al. 2022). We provide a novel proof this result that leverages the Gaussian RBF kernel's connection to the Fock space in Appendix B.2 The thcorem and corollary relatod to our proof is shown below. These proofs demonstrate that even for popular selections of RKHSs, the collection of bounded Koopman operators is small.
Theorem 2. If FF; CC + C" is entire function, and Kr is bounded on HG, then F(z) As + b for matria A C and vector b e C". Here, HG represents the Gaussian RBF's native space.
Corollary 1. If F is real entire vector valued function, and K5 is bounded the Gaussian RBF's native space over R", then F is affine.
Hence, for the most commonly used kernel function in machine learning, the collection of bounded (and hence compact) Koopman operators over its native space is restricted to only those Koopman operators corresponding to affine dynamics. Each selection of RKHS and kernel function will yield a correspondingly small collection of bounded Koopman operators. It should be noted that Koopman operators were completely classified over the classical sampling space, the Paley-Wiener space (Chacón & Giménez, 2007), as also being those that correspond to affine dynamics, and it is simple exercise to show that the native space for the polynomial kernel also only admits bounded Koopman operator when the dynamics are affine.
In summary, thus far it has been established that Koopman operators are only bounded in the case where the dynamics are affinc over the following spaces:
The polynomial kernel space, K(x,v) v) = (1 x))"
The Fock Space, ((2, w) = 20
The exponential dot product kernel's native space, K (zz y) cax"v
The Gaussian RBF's native space, K K(x, u) exp (- m11)
Paley- Wiener spaces, K ((x, y) sinc(a(x - y)), and
Other spaces discussed in [Ikeda et al. 2022).
Consequently, in most practical respects Koopman operators over RKHSs should not be assumedd to be bounded, and certainly not compact.
5 Dynamic Mode Decomposition with Koopman Operators over RKHSs
As product of its genesis in the machine learning community, many DMD procedures appeal to feature space, and this holds in implementations of kernel-based extended DMD (Williams et al.) 2015b), which casts the snapshots from a finite dimensional noulinear system inlo an infinite feature space. The direct involvement of the feature space in the estimation of the Koopman operator leads to rather complicated numerical machinery. To avoid directly computing the infinite dimensional vectors that result, an involved collection of linear algebra techniques are leveraged extract the Koopman modes. Here is shown that this process may be simplified and that a procedure that directly involves the kernel functions centered
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the snapshots simplifies the design of DMD algorithms. This approach keeps with the spirit of the "kernel trick," " where feature vectors are never directly evaluated and only accessed through evaluations of the kernel function itself.
The algorithm presented in this section is designed around scalar valued RKHSs, which necessitates the de- composition of the (vector valued) full state observable component-wise. complete vector valued algorithm is discussed in Section 0 where it is demonstrated that the present algorithm is computationally equivalent to the vector valued algorithm for certain selections of kernel operators.
Recall from Proposition 2 in a real valued Hilbert space, the projection of function onto collection of linearly independent basis functions, ",,..., UM : is a linear combination of those functions, Pg EØ= Wyy where the weights wi may be determined by solving
)
(u) uard :
W1
(g. u1)
=
(uM, w1) (um, u was
We will use Definition B in order to aid us in creating and denoting the finite rank representation of Kr-
fai,..., a.. is an ordered basis for V, and suppose that B (3,,..., .....,,,) is an ordered
and 3 is denoted as [E] where the j-th column of this matrix contains the weights of the vector Eag corresponding to the ordered basis B.
Throughout this algorithm, a Koopman operator will be assumed to be densely defined, as Section g demon- strated that most Koopman operators cannot be expected to be bounded or compact. An additional as- sumption will be made that the kernel functions themselves reside in the domain of the Koopman operator. It should be noted that since the kernels are always in the domain of the adjoint the Koopman operator (sce Section 3). a finite rank representation of the adjoint of the Koopman operator may thus be derived without assuming that the kernels are in the domain of the Koopman operator.
For the sake of the derivation, it is also assumed that the Koopman operator is diagonalizable, which is not generally expected to be true. However, the finite rank representations leveraged in this manuscript are almost al always diagonalizable,, since the set of non-diagonalizable matrices are of measure zero in the collection of all matrices. Moreover, for periodic data sets, the adjoint of the Koopman operator will be invariant on the span of the collection of kernel functions centered at the snapshots, and thus, the finite rank representations will be explicitly the adjoint of the Koopman operator on that subspace, which supports the assumption of the availability of eigendecompositions for the Koopman operator in the periodic or quasiperiodic settings.
For given collection of snapshots {I1, .2.,.., 22.,,, H the goal is to determine finite rank representation of K that is derived from the kernel functions centered at the snapshots. To express finite rank representation, the ordered basis a {km....., kzm-1} } is leveraged. In particular, if P. is the projection onto span(a), the operator PaKy maps span(a) to itself, which enables the discussion of cigenfunctions and eigenvalues of PaKF using only functions in span(a).
Proposition 5. Given a collection of snopshots {Ii, T2,,, C X and the corresponding ordered basis a = (ke, 1 .... Rem-r> } of kernel functions, g -m a,ke, and PaKpg= = Em-l bike, then
Gb = Ia.
While availability of time series of snapshots (21, developed method does not require such time series. It can also implemented using arbitrary snapshots (21, and (y1, y2,.., ym provided yı F(r,).
(21,22,-00 22, ... am) such that 0i+1 F(x;) more typical case, the
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where (a1 (01....,.)))) a...,amm1)) b (b1, ..,,..bee given by
1) (m1)) and the Gram matrix G and the interaction matria I are
((11,4m-1)
and I ((22, K(22, Fm-1)
21
K 1.mm-13mm-l
K ((Im)ti)
K (Imm fm-1))
Proof- Using Proposition ((F()),
2 the reproducing property, and the fact that Kphiz,
kin; (F(x))
[K 1= apken kZDH
EE= I (KKpken kzi)
Ee= m (x((
Gb
Ia.
KF Ees ackix kem-1)
Ems a (Kpkss kmm-1)
Em-l (GK(mmm
If .ry....,k. Ka are linearly independent, then G is invertible and = G- i.e., the operator PaKp, restricted to span(a) is uniquely represented by the matrix [PaKFla = G-'z. If is singular, then the projected function PoKyg admits multiple sets of coefficients b, that satisly PaK,g = Σin beka, In this case, the finite rank representation is not unique. In this paper, we study two specific approaches to compute finite rank representation for the case where G singular or numerically singular.
The first approach, summarized in Algorithm where is user-selected regularization coefficient and 4 is an 1x x identity matrix.
regularized regression, i.e., [P,Kr]o (G + elm-1) -
The second approach is a truncated pscudoinverse approach, i.c., PaKCels == G where G is the trun- cated pseudoinverse of G, computed by zeroing the singular values of G that are smaller than 2 0.
It should be noted that and are precisely the matrices examined [Williams et al. 2015b after the use of truncated SVD. If the truncated pseudoinverse approach used to implement the method developed in this paper, then the generated results are identical to those generated by the KDMD method in (Williams et al. 2015b). The results obtained by using the regularized regression approach differ from those obtained using the KDMD method in (Williams et al. 2015b), but not substantially
In Appendix we provide empirical evidence for the similarity of the results obtained from our method and the method employed in (Williams et al. 2015b).
The objective of DMD is to use the finite rank representation determined above to create data driven model of the dynamical system. This makes use of fundamental property of eigenfunctions of the Koopman operator.
Lemma 2. Suppose that is an eigenfunction of KF with eigenvalue X. Evaluoting the eigenfunction at snapshot reveals p(x1+1) = (x(()))
Proof. Let tp be an eigenfunction of Kpp with eigenvalue X, where F represents the discrete time dynamics such that T++1 F(x)). Furthermore, recall the definition of the Koopman operator presented in Definition Then
>xp(as) = Kry((1) = &0 (F(x:) (#++1)
Therefore, p(x+41) = X)((x1).
Proposition 6. Suppose that 1 is a complete set of eigenvectors of the Koopman operator, KFF, corresponding to the eigenvalues { For a slate x R", let (x)a be the d-th component of for d= l......,n n. If it assumed that the mapping I + (I)s in the RKHS (as it is when the native space for the eoponential dot product space (Steinwart & Christmann 2008) ) then each snapshot may be reconstructed as
I++1
lim
Σ (eMp)pp((
(3)
+000
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Proof. Since {pp)ī= is cigenbasis for Kp corresponding to the cigenvalues {Ay}p=n and the mapping + (x)a is in the RKHS,
M
(x)a
lim
Σ
(E
M
+00
$=1
for some coelficients ((S),M)))==1 By stacking each (æ)aa, the full state observable gia, given by gia(x) = x, is expressed as
M
Dia
lim Σ {)MD)(x).
(4)
+00
Therefore, equation 3 can be derived by using equation and Lemma 2
Note that since the Koopman operator is not generally a normal operator, { is not expected to be an orthonormal basis, and hence, there may be nonzero influences between the coefficients obtained by projection and this is expressed by the additional index M in &,M- This means that series representation of the docomposition as expressed (Kawahara, 2016 Brunton & Kutz, 2019) is not always possible. Hence, Koopman modes are not fixed quanlilies unless there is an orthonormal basis of eigenfunclions for the Koopman operator.
Since the Koopman operator is approximated here by finite rank representation, perfect reproduction of Dia through series of eigenfunctions is not possible. Instead, eigenfunctions ions determined through the finite rank representation are used to construct the approximation of gia. In particular, the matrix [PaKp]j is the matrix representation of P.KFF
Proposition 7. vj is eigenvector for the matrix [PaKr]a with eigenvalue Xg, then Exe (1y (0)))((a,))) is an cigenfunction of PaK.y
Proof. By the definition of cigenvector, [PaKrlau = Azuz. Therefore,
K(I,z1)
PoKF
Σ
(o)))(((.0)))
[PoKFlauj= (()))
K(xxfm--1)
Using Proposition 7 the corresponding normalized eigenfunction is denoted by
m
(x)x
Vo,Cuu, ((r,)))))
(5)
where G= = (K x2)) m is the Gram matrix associated with the snapshots and the kernel function and (t)t denotes the conjugate transpose.
Using a finite rank representation of equation it easy to see that the d-th row of the matrix & == (& Êm-1) of Koopman modes is comprised of the components of (æ)a along the (non-orthogonal) directions pj. That is,
m-1
gia(;;) = x; Σ (z,)
(6)
jml
which yields
G X,
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where X: == (x) ... Fm-1) is the data matrix and
V : (Vitoms
Pm a Gum
is the matrix of normalized eigenvectors of [PoAFlla
Similar to the finite rank representation above, G is non-singular, then the matrix of Koopman modes is unique, given by { X(V"G)-' Tf is singular and if regularized regression used, then modes are com- puted using { (G+ of the finile rank representation with absolute value smaller than are removed from the computation, along with the corresponding eigenvectors, and the modes are computed using X(VTG))
If the truncated pscudoinverse approach selected, then eigenvalues
Using the approximate cigenfunctions, Lj equation and Lemma 2 data driven model for the system is obtained
I++1 RJ
(x1))
(7)
Furthermore, the discrete-time dynamics may be approximated (by setting = 1) as
m-1 =())))(())-- = E (x))
(8)
Consider a contimuous system - = f(x) with f E C² and let a compact set X be forward invariant under the flow of the system. If the discreto-time system Th+1 = F(x)) obtainod via discretization of the continuous-time system with time step T, and if (e (p) is eigenpair of resulting Koopman operator K for all T > 0, then along a solution xx-)) of I = f(x), we have dp(x(()) = (p(((t)) Mauroy & Mezić, 2016 Section III-A). Motivated by this relationship, the continuous- time dynamics may be approximated as
dt
Σ log(>;) (x1
f(x) R0 f(x)
Note that in the above construction, the cigenfunctions 9j are assumed to be common across all time-steps T. In view of Example 5 not all eigenfunctions of the Koopman operator may be common eigenfunctions. As a result, the continuous-time model in equation g is a heuristic, albeit a useful one (see Figure in the appendix).
6 Vector Valued Considerations
The attentive reader will notice that the ultimate objective of DMD is to achieve an approximation or decomposition of the function (ia(x) := x, the full state observable. However, the full state observable is R" valued, whereas the RKHSs in question consist of scalar valued functions. Consequently, the coefficients that are determined to approximate the full state observable have simply separated the individual components of lıa and established an approximation for each of them separately. The weights for these approximations are stacked to make the vector valued coefficients.
vector valued. Up to now, the methods
This section shows how the vector valued coefficienss arise naturally from a projection onto vector valued functions in vector valued RKHS. With the right selection of kernel operator, this projection operation reduces to the setting of Section 5
If the goal is to decompose the full state observable all at once, then one must appeal to vector valued RKHSs, which produce an operator valued kernel function.
Proposition Given vector valued RKHS, H, consisting of functions that map set to Hilbert space y, there is for each X and y a function Ka E H such that (g, Kz,) (g(x),
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Algorithm 1: Pscudocode for the kernel perspective based DMD algorithm. Upon obtaining the Koop- man modes, the approximate cigenfunctions, and the cigenvalues, equation
is used to compute 24+1-
Input: Snapshots, X = [T1 22, {m] Input: Kernel function K R x R + of an RKHS
Compute the Gram mairix, G = ( if G is ill-conditioned then if regularized regression is used then Set G = G + elm-1 else truncated psoudoinverse is used then Set G-1 -
end if end if
Compute the interaction matrix, I= (K( ar) (22
Compute the finite rank representation of the Koopman operator, (Pakp] G-11 Compute cigenvalues, Aj, and cigenvectors, Bj, of [PaKr]a if truncated pseudoinverse is used then
For all j. if [Ayl < then set Xy 0 and 2/j Set (VTG)-'=(VTO)
end if
Compute the matrix of Koopman modes, & = x (VTG)-
Compute approximate eigenfunctions, equation
Output: Koopman modes, E, for j L.....
Output: Approximate eigenfunctions, (x(x)) for = 1.
Output: Eigenvalues Xj for = 1...., -
The mapping v + Kz.v is linear, and hence, the kernels are expressed as K2 where K2 : - H is a bounded operator. In this case, for each g E H, Kig = g(x). The operator valued kernel associated with H is given as
K(x,y) = K,
- H,
and note that when y = 12", this operator valued kernel a matrix whose entries are scalar valued functions.
When a vector valued RKHS, H, is obtained from scalar valued RKHS, Å with kernel k(æ, y), as nm where the inner product of two elements, 9 (91,..--9n)) and h (hı,..., hT) from H is given (g. h) (g), ir then the operator valued kernel is given k(x, v) which has many convenient properties for computation. More details concerning vector valued RKHSs may be found in (Camnll el. al.
2010
Agler & McCarthy 2023).
Here the Koopman operator is defined just as it would be for a scalar valued space: KFg = g o F, and the difference here is that is vector valued. Similar relationships between the Koopman operator and the kernel functions still hold, where
(KFg, (g(F(z)), = (g. KFG)))
hence KjKK, KF(z)-
Now given collection of snapshots, {.1...., FM). the associated operator valued kernels are Kz, Kexr For decomposition of the full state observable, Fia(x) = x, one needs to interface with the Hilbert space and construct a finite rank approximation of Kpp as was done in the prior section. This means that a subspace within H needs to be constructed, say spanned by a collection of basis functions o, so one can write PaKPa.
The selection of the basis is less constrained than in the scalar valued case, where only the kernels centered at the snapshots were considered. In this case, not only must the snapshots be considered, but also complete
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freedom of the choice of R" to select each K KgV. A natural choice is to select v from the standard basis of R This leads to
(Kzo.e11-.... K KIM-1,011 K Kzo,22 K2M- Kzo.eur.... KIM-11aa) }:
which a large basis for problems such as the cylinder flow example, where n is of the order 80, 000. Consequently, the gram matrix corresponding to this basis square matrix of dimension M n.
So for arbitrary operator valued kernels, the DMD algorithm presented in this paper would have to invert M dimensional matrix, and the computation time would scale with the dimension of the space ((M n) using standard inversion algorithms. This scaling would make benchmarks such as the cylinder flow example completely infeasible.
Computational leasibility can be achieved through the judicious selection of the kernel operator. Here we leveraged kernel operator of the form
K(x, v) = k( I
with k a scalar valued kernel function. In this case,
(Kzu, Kyw)) k( v)(u, w)
and the two functions Ky and Kyw are orthogonal when v and are orthogonal in R". This means that the Gram matrix composed of the given above is going to be block diagonal matrix with each block being the Gram matrix corresponding to the scalar valued kernel. Thus only matrix of size M must be inverted, and each dimension treated individually. Significantly, the inversion problem no longer scales with the dimension.
Alternatively, if we instead use a different scalar valued space for each dimension, we would get n different Gram matrices to invert. The complexity would scale with the dimension, but linearly rather than cubicly.
Proposition 9. Suppose that k(æ, y) corresponds to a scalar valued RKHS, H, and let H be an R valued RKHS such that if g E H, then = product (g, h)H = 2i (g,, h Then H is vector valued RKHS.
9T)" where 9s E for each i ,,..., equipped with the inner
Proof. Let E RR and X then
(g(x), u)R=1 S [[g(z)]l
(gi, k(, x)) P|ul|R=
S vk(a,
1 |u|| <k(z,
hence the functional g + (g(x), u) bounded. In this setting, K(x y) diag(k(x, y)... .................... k(a, 8)), and (Kzei Kyez)Rn 0 i x That H is vector valued RKHS follows from the above.
If { ØN is a collection of snapshots such that F(x,) = æ1+1, then finite rank approximation of KCF may be constructed by examining the image of the operator on Kgney for i N and j = 1,..., and then projecting back onto the span of these kernels. The projection operation requires the computation of the Gram matrix for the basis (K which a block diagonal matrix, where each block corresponds a selection of dimension through ej. Thus, if & = (k(x,, xr)) the gram matrix manifests as
G=
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Similarly, the interaction matrix is block diagonal. Consequently, the vector of weights corresponding to each kernel function is composed of smaller vectors of length N - 1, each one corresponding to different dimension. Hence, each dimension may be treated independently. 'The eigenfunctions for this finite rank representation the Koopman operator are then composed n identical collections of functions, differing only in the dimension they are supported in. Let .1.....99 N H be these scalar valued functions, and write sa,, == e.e, be the corresponding vector valued cigenfunction.
The full state observable is projected onto this collection of eigenfunctions
N
(ia(a) (x) ~ Pa.i))
Σ (a))
Σ Wassei
i=
Here, Ei=1 Wa = Ess where & is the Koopman mode from the previous section.
7 Numerical Example
In this section compare the results obtained from the developed DMD method and the method developed in (Williams et al. 2015b on a benchmark dataset. Two additional numerical examples are included in Appendix C
7.1 Periodic flow around cylinder
The website (Kutz et al. 2016a) accompanying the textbook (Kutz et al. 2016b} provides several data sets that serve as benchmarks for spectral decomposition approaches to nonlinear modeling, which have been released to the public through their website. This section utilizes the cylinder flow data set to demonstrate the results the developed DMD method. The cylinder flow example is numerically generated, and the data provided corresponds a planar steady state flow of the system. The data consists of 151 snapshots, containing values of the vorticity of the flow at several mesh points in a plane, recorded every 0.02 seconds. In this demonstration, snapshots 1 through 30 are used as the input data, and snapshots 2 through 31 are used as output data, assuming that the ith and (i + 1)th snapshots satisfy Zi+1 = F(z;) for some unknown nonlinear function F. The snapshots are normalized so that the largest 2-norm among all snapshots is 1.
(a) This figure shows reconstruction of the vorticily field the same time points using two differrnt ker- nels. The left column shows reconstruction of the initial state, Z1. The middle column shows recon- structions of the state, 215, and the right column corresponds to reconstruction the state Zag.
(b) This figure presents prediction of the vorticity field the same time points using two different nels. The left column presents prediction of the state mss- The middle column shows prediction of the state æ101, and the right column corresponds to prediction of the state T151
Figure Reconstruction and prediction of the vorticity a fluid flow past a cylinder using two different kernel functions. The first rows contains the ground truth, the second rows leverages the Gaussian RBF kernel, and the third row uses the exponential dot product kernel.
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-10
.10-
[|xx(t) - 2(
1
[1x0() +
0.5
0.5
0
0.5
1
1.5
2.5
3
0.5
2
2.5
[s]
&
(a) Method developed in this paper
(b) Kernel DMD method in
Williams et al.
2015b
Figure 2: For the periodic flow example, this figure shows the 2-norm of the error between the true snapshots and the snapshots generated using DMD. The results for the Gaussian RBF kernel and the exponential dot product kernel identical.
The Koopman modes, cigenvalues, and eigenfunctions are then computed using the developed technique and snapshots 2 through 31 are reconstructed using equation 7 DMD is implemented using the ex- ponential dot product kernel, ((x xxp(,==>) (with μ 1), and the Gaussian RBF kernel,
K(x,y) = exp (- vll3) (with = 1).
To further demonstrate the accuracy of the obtained fmile-dimennsionll representation of the Koopman operator, snapshots 32 through 151 are predicted from snapshot using equation 7 The code used to generate these results is publicly available, see Gonzalez et al. (0023
7.2 Discussion
Figures la and 2 show the ability of the developed DMD technique to reconstruct the training data. The relative reconstruction errors associated with this system are of the order of le - 8. Additionally, figure la shows that for the periodic flow example, similar reconstruction results can obtained when using the Gaussian RBF kernel or the exponential dot product kernel.
As shown in figures abdand 2 given the first 31 snapshots, the developed DMD technique is able to predict of order 1e 4, without the knowledge of the underlying physies, F. Furthermore, expected, the performance of the developed method is nearly identical to the baseline Kernel DMD method developed in (Williams et al. 2015b).
the remaining 120 snapshots, with relative prediction
8 Conclusion
This manuscript revisits theoretical assumptions concerning DMD of Koopman operators, including the existence of lattices of eigenfunctions, common eigenfunctions between Koopman operators, and boundedness and compactness of Koopman operators. Coutteccxamples that illustrate restrictiveness of the assumptions are provided for each of the assumptions. In particular, a major theoretical result. is established to show that the native RKHS of the Gaussian RBF kernel function only supports bounded Koopman operators the dynamics affine. Moreover, kernel-based DMD algorithu that simplifies the algorithm in (Williams et al.) 2015a) and presents it in operator theoretic context is developed and then validated through simulations The developed DMD algorithm is also extended to vector valued RKHSs.
Acknowledgments
This research was supported by the Air Force Office of Scientific Research (AFOSR) under contract numbers and FA9550-21-1-0134, and the National Science Foundation (NSF) under award numbers
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2027976, 2027999, and 1900364. Any opinions, findings and conclusions recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the sponsoring agencies.
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Steven L Brunton and J Nathan Kutz. Dala-driven science and engineering: Machine learning, dynamical systems, and control. Cambridge University Press, 2019.
Steven L Brunton, Marko Budišić, Eurika Kaiser, and Nathan Kutz. Modern koopman theory for dynamical systems. arXiv preprint arXiv:2102. 12086, 2021.
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A Examples
In this section we provide detailed examples that support some of the claims made throughout this paper. The title of each subsection is used as a way of linking the section in the paper with the examples that are relovant to that section.
A.1 Introduction
Example 1. Given a Koopman operator KCp corresponding to the discrete dynamics, F, an > O, and a compact subset Ω that is invariant for the dynamics, it follows that if ó satisfies
KK A S
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then
(o(F"(x)) X²)((x) & Ce Am+1 1 A
for x E Ω where the accuracy of the resultant models depend on c and X. Significantly, as c tends to zero, so does the difference (0(ET (z)) Xm ((x) at every point in a Q.
A.2 Forward Complete Dynamics
Example 2. Consider the one dimensional dynamics:
i ==1+2²
For fixed 0 < At < T/2 the corresponding discrete time dynamics are given as:
mm+1 (am(arctan(mm)) + Al).
Setting
3'm tan(n/2 Att,
it is clear that Fm+1 undefined. Consequently, the composition symbol,
F(x) tan(arctan(x)) + At)
for the hypothetical Koopman operator, not well defined over R for any selection of At.
A.3 Properties of the Operators
The following simple example will be leveraged throughout the discussions in the ensuing subsections. Example 3. Consider the dynamical system
3 (22 -21)²
which corresponds to circular dynamics in the plane, For any fixed 0 = At, the discretization of this system yields the linear discrete dynamics
(cos(0) -in(()) (O) cos(0) Fm-
Fm++1
That is, the discretization corresponding to a fixed time-step results in fixed rotation of R?, To simplify the presentation, we use C a model for R,,,, where rotation of the complex plane reduces to multiplication by a unimodular constant,
2m+1 c Zm
(10)
The corresponding discrete time dynamics will be written as
Fo(z) = ' z.
A.3.1 Lattice of Eigenfunctions
Example 4. Consider the dynamical system mentioned in example 3 If 0 in equation 10 the discrete dynamics corresponding to rotation by in the complex plane become
2m+1 e 2m
2m
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The corresponding Koopman operator, CF, (C) (C), is given
Kr,9(=) :- 9(-z).
Hence, every even function is an eigenfunction for this Koopman operator with eigenvalue 1. Any function g E (C) exhibits strict bound its growth rate (cf. Zhu 2012
To wit,
[l le/2
Io(=)]= (g. x)) ()) (o())) s [
That is, if a function is in the Fock space then the function is of order at most 2, and if the function is of order 2 it has type at most 1/2 (cf. (Boas) 2011)). Conversely, if an entire function is of order less than 2, is of order 2 and type less than 1/2, then it also in the Fock space. While lunctions of order 2 and lype 1/2 can be in the Foek space, it does not hold for every such function. For example, e= is of order and type 1/2, but is not in the Fock space.
it is the Fock space, and
Thus, (0z) e is cigenfunction for KF, in the Fock space. However, 9 space, and cannot be an eigenfunction for Kr, (C) - (C).. Hence, the eigenfunctions for Kr, do not form lattice.
e 9 is not in the Fock
A.3.2 Common Eigenfunctions
Example 5. Consider again the dynamical system in example 3 but instead set 0 = T T/2, then
F Fx/2(z)= = iz.
In this case, the polynomial 24 + 2 E (C) is an eigenfunction for the Koopman operator corresponding to 7/2 with eigenvalue 1, However, 2t 8 is not an eigenfunction for KF,,as as
(2) + (3) = 24 + 22//3_8
and the constants cannot be factored out of the polynomial as an eigenvalue.
B Proofs
B.1 Proof of Lemma
Lemma 1. Let F X + X be the symbol for Koopman operator over RKHS H over set X. Kr
D(Kr) + H is closed operator.
Proof. Suppose that {9m Jm C D(KF)) such that Im - g H and KF9m + H. To show that K is closed, we must show that g E D(KF) and KFg = h. This amounts to showing that h = 9 F, by the definition of D(Cp). Fix E X, then
h(xx)= = (h, Kz)
lim (Krgm, K2) h lim gm (F(x)) m=+00
m=+00
lim (gm, KF(w)) = (g, K = g(F(()))
m-=00
As was an arbitrary point X, h = g( (F(()) and the proof complete.
B.2 Proof Regarding Boundedness of the Koopman Operator
As stated in Section 1.2.3 here we provide a novel proof that shows that a Koopman operator over the Gaussian RBF's native space bounded only when the corresponding discrete dynamics are affine. In order achieve this we begin by introducing and proving a few lemmas.
Lemma If KF a bounded operator over the Gaussian RBF's native space, then F real analytic vector valued function over R
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Proof. If Kp is bounded, then (x) (y(F(x)) exp((I1F(()) -1))) is in the RBF's native space for each y E IR" Since every function in the RBF's native space is real analytic, so is (F()))) and thus, the logarithm, (x) - vll2 = - [1) + 2y F(z) - llvl is real analytic. This holds if y 0, and hence [IF(x)) l2 is real analytic. Thus, for every y, the quantity y² F(x) is real analytic. That cach component of F(x) is real analytic follows from the selection of y as the cardinal basis elements Ik", and this completes the proof.
Lemma 5. F F is real analytic vector valued function that yields bounded Koopman operator, KF, over the Gaussian RBF's native space, then its extension to an entire function, F C - C induces a bounded operator over Hc(C").
Proof. Since an entire function on C" is uniquely determined by its restriction to R", it follows that the span of the complex valued Gaussian RBFs with centers in I= is dense in HG. Moreover, to demonstrate that Kr is bounded, it suffices to show that there is a constant C such that
(= Ko(F(s,,
(11)
is posilive kernel. By the above remark, it sulfices to show this for real x, y E R", but then this is equivalent to the statement that KrF is bounded over the Gaussian RBF's native space over R".
Theorem 2 F C" + C" is an entire function, and K5 is bounded on HG, then F(z) = Az b for matrix A Cuxn and vector b CH.
Proof. If KF is bounded, then it has a bounded adjoint, Kj,, which acts on the complex Gaussian as K))(().)) = Ko(,.(zz)) In particular, there a constant C such that [| 1
C²IIK((; P) Noting the identity (Ka((,))l = exp (2 (82))) and taking the logarithm, it
follows that
Σ (og((() + Σ (32,) ) < log(C²) + ||
(12)
From this inequality, it follows that for each coordinate j = 1.... n, the harmonic function has linear growth. That is, there is constant Ô so that [uy(2)] s ([1+11=112) for all 2 E C". It follows (e.g. from the standard Cauchy estimates) that (2(2) u2(x iy) must be an affine lincar function of x and y: and therefore, so must its harmonic conjugate uz(z), and we conclude that F(z) Az + b.
(F)
C Further Numerical Examples
C.1 The Duffing oscillator
In this numerical example, sixteen 10-seconds long trajectories of the Duffing oscillator, given by d; 222 and a -0... - 24, are generated starting from initial states on uniform grid on the set 1-1,1] [- 1,11. 1]. The trajectories are sampled every 0.5 seconds to generate set of snapshots. The snapshots are used to build a predictive model of the oscillator using the method developed in this paper (implemented using regularized regression with c <= 10-6 and the exponential dot product kernel with / = 20) and using 10-3 for zeroing singular values and the exponential dot product kernel with fa = 20). To evaluate the predictive models, 15-seconds long trajectories of the oscillator starting from 100 initial conditions randomly selected from the set 1-1.5,155) [-1.5, 1.5] are generated by integrating the contimuus--imme model in equation using variable time-step fourth order Hunge-Kutta integrator. These trajectories are compared against. the true trajectories starting from the same initial conditions, generated by integrating the true dynamics using the [e(t])((e)]] where zx-) same integrator. The metric used for comparison is the pointwise relative error e(t) max; ([1) is the true trajectory and &(.) is the predicted trajectory. Figure 3 shows the pointwise mcan of the relative error across the 100 trajectories along with the spread of the relative error from the pointwise minimum the pointwise maximum.
the baseline method from (Williams et al.] 2015b) (implemented using a threshold
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(a) Method developed in this paper, reconstruction using the contimuous- time model in equation g
(b) Kernel DMD from (Williams et al. 2015b). recon-
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(c) Method developed in this paper, reconstruction using the discrete-time model in equation g
(d) Kernel DMD method in (Williams et al. 2015b), re-
construction using the discrete-tima model in equation
Figure For the Duffing oscillator example, this figure shows the pointwise mean of the relative error between the true trajectories and the trajectories generated using DMD, The shaded area shows the pointwise maximum and the pointwise minimum over 100 randomly initialized trajectories.
To demonstrate the effectiveness of the heuristic contimuous-time model in equation 9 the true trajectories, sampled every 0.5 seconds, are also compared against trajectories generated using the discrete-time model in equation [8] Figure 3 also shows the pointwise mean of the relative error across the 100 trajectories along with error bars that show the pointwise maximum and the pointwise minimum relative error.
C.2 Turbulent flow
Another Numerical example uses a data set that accompanics (Jakob et al., 2020), which consist of flow simulations that range from laminar flow configurations to turbulent flow configurations. The data can be accessed from their website (Jakob et al.) 2021). The flow data used in this paper (id number 7999) corresponds to 2-dimensional turbulent flow with Reynolds number of 4092.216 and Kinematic viscosity of 5.45 exp -05. The data consist of 1001 snapshots, each snapshot contains flow velocities in the horizontal and vertical directions measured on regular grid of size 512x512 in two dimensions with domain [0, 1] [0, 1]. The snapshots are separated by 0.01 seconds. In this demonstration, snapshots 1 through 300 are used as the imput data, and snapshots through 301 are used as output data, assuming that the ith and (2 + 1)th snapshots satisfy z++1 F(x,) for some unknown nonlinear function F. The snapshots are normalized so that the largest 2-norm among all snapshots 1.
The Koopman modes, eigenvalues, and eigenfunctions are then computed using the developed technique and snapshots 2 through 301 are reconstructed from snapshot 1 using equation DMD is implemented using the Gaussian RBF kernel, K K(x,1) y) = exp (- | =v13) (with & 0.75 and 0.0001). The baseline method
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(a) Difference between prediction generated using the method in [Wilianns et al. 20155) and the truncated pseudoinverse implementation of the method developed in this paper.
(b) Difference between trajectories generated using the method in (Williams et al. (2055b) and the regularized regression implementation of the method developed in this paper.
Figure 4: For the Duffing oscillator example, this figure shows the pointwise mean of the between the trajectories predicted using the continuous-time model in equation 9 generated by the method in (Williams et al. 2015b) and the method developed in this paper. The shaded area shows the pointwise maximum and the pointwise minimum over 100 randomly initialized trajectories.
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Figure 5: For the turbulent flow example, this figure shows the 2-norm of the error between the true snapshots and the snapshots generated using DMD.
from Williams et al. 2015b) is implemented using the same kernel with μ 0.000518. The predictive model in equation is also used lo generate 700 additional snapshots.
C.3 Additional Discussion
The ability of the developed DMD technique reconstruct the training data from the same initial condition is apparent from figures 3 and 5 In the duffing oscillator example the pointwise mean of the relative error is shown to be nearly zero. The relative reconstruction errors for the turbulent flow of the order of le 2.
The duffing oscillator example, is also used to highlight the similarity in predictions generated by the DMD method developed in this manuscript and that proposed by WWiliaam et al. 2015b). Figure shows that the difference between the predictions generated by a truncaled pseudoinverse implementation of the method developed in this paper and the predictions generated by the method in (Williams et al.) 2015b) are of the order of 1c As expected, figure g further highlights that choosing to use the truncated pseudoinverse implementation as opposed to the regularized regression implementation results in predictions that are more similar to those obtained under the method in (Williams et al. 2015b). Furthermore, figure 3 also shows
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that for the duffing oscillator example the contimuous-timg model in equation 9 results in smaller pointwise mean relative error than the discrete-time model in equation g
In the case of the turbulent flow, as shown in the relative prediction errors for the method developed in this paper and the baseline kernel DMD method developed in (Williams et al. 2015b) quickly increase to 1. Since the data are normalized so that the largest 2-norm among all snapshots 1, a relative prediction error of signifies a very poor match between the predicted state and the true state. Such performance degradation is expected for turbulent flows since the underlying assumption that the ith and + 1)th snapshots satisfy 21+1 = Fx)) for some unknown nonlinear function F is unlikely to generalize outside the training data when the flow is turbulent. From figure 5 the method developed in this paper appears to perform slightly better than the baseline. This performance improvement. can be attributed to the explicit regularization of the Gram matrix, where the regularization parameter can be tuned in addition to the kernel width parameter. (Williams et al. 2015b), the regularization done implicitly through the pseudoinverse operation. We postulate that by manipulating the threshold used to decide which singular values are zero in the pseudoinverse computation, the performance of the kernel DMD method can be improved.
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Reviews:
[{"review_id": "nGlaGkj9TI", "reviewer": "Reviewer_ot35", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strenghts\n\n- The paper is of interest for the TMLR community. The theoretical contribution is sound and readers might be interested in the characterization of bounded Koopman operators over Gaussian RKHSs.\n\n- The numerical experiments complement nicely the theoretical derivations.\n\nWeaknesses:\n- The paper questions many assumptions regarding Koopman operators but leaves some out of questionning. While I understand the interest of looking at a Koopman operator as an operator from $H \\to H$, it introduces a main question that needs addressing in my opinion: \"how big is $\\{g \\in H | g \\circ F \\in H \\}$\" ? Is the densely defined assumption really reasonnable for say the Gaussian RBF kernel ? What type of $F$ are needed to have $\\forall x \\in X, y \\mapsto k(F(y), x) \\in H $ ?\n\n- The assumption \"$x \\mapsto x$ is in $H$\" is a heavy one and must be better highlighted. What happens with the gaussian kernel which violates this assumption ?\n\n- The relation to [Williams 2015a] is not well explicited in my opinion. From what I understand, both methods yield the same matrix whose eigendecomposition is used to approximate the dynamics. Is the difference only regarding the assumptions made on the Koopman operator ? Or is it in the regularization performed when the matrix is not invertible ? Or further down the line when computing the modes ?\n\nAdditional remarks:\n- In the text before equation (5), when writing the decomposition of $(x)_d$, there is a typo on the index: $\\phi_i$ should be $\\phi_j$.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper proposes a theoretical contribution related to the study of Koopman operators over RKHSs. It proves that if the Koopman operator of a real entire vector-valued function $F$ is bounded over the Gaussian RBF space, then $F$ is affine.\n\nFurthermore, the paper discusses the feasibility of the assumptions that are often made about the Koopman operator and that may not be relevant when they are considered on RKHSs, arriving at the conclusion that they cannnot be assumed to be bounded and compact.\n\nThe paper then proposes to work in the context of densely defined operators and establishes a DMD method in this case that closely mimics previous work about kernel DMD. Numerical experiments highlight the benefit of the approach.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "- Improve the presentation of the differences with previous work [critical]\n- Better discuss the assumptions (densely defined + identity in the RKHs) [critical]", "broader_impact_concerns": "None", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "IhWvSXYie7", "reviewer": "Reviewer_64sb", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n1. To my knowledge, a somewhat novel, kernel based perspective on DMD/Koopmanism has been promulgated in this work, along with a more impressive set of counterexamples to suggest why Koopmanism isn't necessarily well suited for anything beyond affine problems, before further analysis provides a strong counter-counter-argument as to why the situation is not critically deficient after all, if the KOs are considered to be densely defined. If the reader can slog through the text (see weakness 1), the proposed theory is an interesting and fruitful way to consider the existing empirical literature in these directions and a stark departure from the ergodic perspective that heavily dominates how Koopmanism is formalized usually. \n\n2. To further bolster the value of their case, the authors investigate some naturally implied algorithms within the perspective and somewhat validate that their results are equivalent to existing methods (see weakness 2).\n\nWeaknesses:\n1. I don't mean to be harsh, but the work is badly-written - I would consider myself well versed with most of the involved subject matter and it still took me an unreasonable amount of time to simply figure out when known facts were being repeated and when something novel was being added. It is even more confusing why this should be the case since in my opinion, the novel contributions are truly exciting and remarkable. A lot of the mathematical arguments, results, and proofs have been presented in an almost stream of consciousness style that is not amenable to reading about such dense technical matters.\n\n2. Numerical analysis is not insufficient, but also far from conclusive. It is also presented in an sub-optimal manner. For ex, in both Fig. 7.1 and 7.3, the relevant quantity of interest is not the outputs of the two methods, but the difference between the two method outputs (I am not suggesting the removal of those figures, but explicitly the quantity of interest is the difference between those two sub-figures in each case - that comparison should be present). I would also be more comfortable if at least a couple of more experimental verifications were included, rather than just 2.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The work is focused on characterizing the properties of Koopman operators (KO) over reproducing kernel Hilbert spaces (RKHS), as a way to bring forth a somewhat novel Kernel perspective to $Koopmanism$ - contrasting it with the usual ergodic picture. A central result for example showcases how the class of bounded Koopman operators over RKHS is miniscule even when the considered dynamics is being governed by analytic maps - an additional assumption that the map be affine is critical to impart enough structure to support bounded KOs (and even then in a restricted Guassian RBF setting). \n\nHowever, while seemingly a very strong restriction on the viability of $Koopmanism$ (the lack of compactness and/or boundedness in operators often makes functional analytic problems completely intractable), within an assumption of densely defined KOs (amongst other smaller ones), we find that enough structure is still preserved to make it numerically viable, making this an impressive attempt at providing a novel, kernel based theoretical perspective on DMD.\n\nThe proposed theory leads to some natural algorithms, which are tested for their advantages and then compared with existing methods to demonstrate equivalence.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "On a technical/mathematical level, I have no issues and have found no substantial errors.\n\nOn a reading front, I think the authors have struggled with the intersectional nature of this work - writing for an ML audience while needing to leverage fairly standard but still bulky and involved mathematical machinery. My personal advice would be to write every novel result/deduction/argument out in a standard mathematical format. \n\nWhere arguments are being developed, put them into the service of proving or conjecturing a precise statement. Where claims are being made, always state them as precise statements/lemmas/theorems. Everytime a definition is stated, it should be as an indexed object that can be referred to instantly by simply looking at the right definition number, rather than as a sentence in some hard to find paragraph. Etc. Etc. \n\nI don't mean to suggest that the above approach is completely absent: I am strongly recommending that this approach be applied to every mathematical portion of this paper. To further simplify reading, the authors can then pick and choose the most important statements and lemmas to build a storyline of precise statements and implications in the \"main text\" and then put everything else (including all the proofs) in a well-referenced appendix. This way, people who care can easily access full technical details while the rest can follow the important portions of the story without needing to untangle the entire technical machinery.\n\nIn summary, the work is accurate and convincing but highly unclear: I can't recommend acceptance in good faith until that last point is resolved. However, I would again strongly acknowledge for the editors that I have no technical changes requested or errors to point out here - if the struggles in reading were simply mine, please feel free to completely discount the above.", "broader_impact_concerns": "Not Applicable", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "ih3VyP5JVc", "reviewer": "Reviewer_fD6Z", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "### Strengths\n\n+ The derivation of \"kernel DMD\" from the perspective of finite rank representation of the Koopman operator sounds indeed interesting.\n\n+ The writing is not bad, I could follow the main reasoning though I am not well equipped with such mathematical discussions.\n\n### Weaknesses\n\n- That being said, I feel that the relationship between the claims scattered in Sections 4-6 could have been clearer. A short paragraph on how they relate to each other might help.\n\n- The numerical examples do not seem wrong, but it is unclear how they are valuable. I am not following the discussions that happened in the previous submission, but my first impression is that they were added just to make the paper slightly more ML-ish. A possibility I would raise (which I am not recommending but just showing) is to defer the numerical examples to an appendix to fit the paper within the nominal 12-page length.\n\n- How the discussions in Sections 4.1, 4.2.1, and 4.2.2 relate to the discussion afterward in Sections 5 and 6 is unclear.\n\n- The novelty of the result in Section 4.2.3 is unclear. For example, could you please clarify the connection to Ikeda et al. (2022), if any?\n\n[Ikeda et al., 2022] Ikeda et al., Boundedness of composition operators on reproducing kernel Hilbert spaces with analytic positive definite functions, Journal of Mathematical Analysis and Applications 511:126048, 2022.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "An algorithm similar (or almost identical) to the kernel DMD is derived in a perspective different from previous studies. Specifically, the authors start at the projection of the Koopman operator onto a set of basis functions comprising kernel functions centered at the snapshots. Although the resulting algorithm seems to be basically identical to what is known as kernel DMD (eg WIlliams+ 2015), the contribution of this work is claimed to be its derivation from the given perspective. Moreover, some analyses of the Koopman operator in RKHS and its eigenfunctions are presented.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "(1) As mentioned above, please elaborate, if any, on the connection between the result in Section 4.2.3 and that in Ikeda et al. (2022).\n\n(2) The difference between the two algorithms used in Section 7.2 could be clearer.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: All reviewers recommend acceptance and I believe the merits of the paper (the novel theoretical perspective) is worth being published in TMLR. 

The authors spent considerable amount of time revising their manuscript and the improvement in clarity was noted by reviewers. None of the reviewers provided additional (specific) points that the authors could address in another round of revision, so I am recommending acceptance as is.
Decision: Accept (poster)