Occupation Kernel Hilbert Spaces and the Spectral Analysis of Nonlocal Operators
aa r X i v : . [ m a t h . F A ] F e b OCCUPATION KERNEL HILBERT SPACES AND THE SPECTRALANALYSIS OF NONLOCAL OPERATORS ∗ JOEL A. ROSENFELD † , BENJAMIN RUSSO ‡ , AND
XIULING LI § Abstract.
This manuscript introduces a space of functions, termed occupation kernel Hilbertspace (OKHS), that operate on collections of signals rather than real or complex functions. Tosupport this new definition, an explicit class of OKHSs is given through the consideration of a repro-ducing kernel Hilbert space (RKHS). This space enables the definition of nonlocal operators, suchas fractional order Liouville operators, as well as spectral decomposition methods for correspondingfractional order dynamical systems. In this manuscript, a fractional order DMD routine is presented,and the details of the finite rank representations are given. Significantly, despite the added theoreti-cal content through the OKHS formulation, the resultant computations only differ slightly from thatof occupation kernel DMD methods for integer order systems posed over RKHSs.
1. Introduction.
Despite the proliferation of numerical methods for and appli-cations of fractional order and nonlocal dynamical systems over the past twenty years(cf. [50, 12, 41, 31]), there are several long standing problems in the modeling ofnonlocal dynamical systems and system identification techniques for nonlinear frac-tional order systems (cf. [11, 45, 13]). Principle among these problems is the effectiverepresentation of data for modeling in nonlocal nonlinear systems. This manuscriptenables several novel data driven approaches to modeling nonlocal dynamical systems,including a nonlocal variant of Dynamic Mode Decomposition (DMD), by address-ing the problem of data representation for nonlinear time-fractional order dynamicalsystems. In particular, this work builds on work involving occupation kernels, whichembeds signal information into a function within a reproducing kernel Hilbert space(RKHS) and thus positions signal data as the fundamental unit of information of adynamical system (cf. [55, 52, 51]), by extending the idea beyond merely functionsinside of a space, but by generating Hilbert spaces of functions on collections of signalsbased on the occupation kernels themselves.Fractional order dynamical systems have been applied broadly in nearly everyscientific discipline. For example, applications include two-phase flows [66, 67], turbu-lence modeling [59], and stochastic systems (e.g. [14]). Other research has exploitedthe memory capabilities of fractional order operators in the context of materials,where classically they were used to model visco-elastic materials [33] and more re-cently biological tissues [32], and fractional order methods have also been employedin biological applications for guidance in sensor placement [62]. Extensions of controltheory have been realized through fractional order PID controllers [44], which haveseen applications to problems in aerospace and control of flexible systems [39, 58].Despite the wide application of fractional order systems, given a real world sys-tem, the development of a corresponding fractional order model is nontrivial. In theabsence of first principles through which a model may be realized, many applications ∗ This research was supported by the Air Force Office of Scientific Research (AFOSR) under con-tract numbers FA9550-20-1-0127, and the National Science Foundation (NSF) under award 2027976.Any opinions, findings and conclusions or recommendations expressed in this material are those ofthe author(s) and do not necessarily reflect the views of the sponsoring agencies. † Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620 USA [email protected] ‡ Department of Mathematics, Farmingdale State College, Farmingdale, NY 11735 USA [email protected] § Department of Mathematics, Changshu Institute of Technology,Suzhou, Jiangsu 215500, P RChina [email protected]. f fractional order dynamical systems leverage fractional operators in an unmotivatedway. This manuscript provides a new Hilbert space setting for the modeling of non-linear fractional order systems, which will provide a data driven motivation for theimplementation of fractional order operators and models.System identification and learning for fractional order systems has been largelyconstrained to linear fractional order systems. For example, [45] developed a sys-tem identification routine through the Laplace transform, while [20] gives a learningmethod for a collection linear fractional order PDEs. A notable exception is fPiNNs(cf. [35] and [41]), which uses physics informed neural network to provide a model ofa fractional order PDE, and also that of the authors which leveraged a regression ap-proach with occupation kernels [30]. The methods herein differ from that of fPiNNs inthat they are operator based (rather than neural network based) approaches to systemidentification and modeling, and constitute a contribution in a new direction for bothdata driven modeling for fractional order system and for DMD analysis techniques asa whole.One of the hurtles that have prevented the development of system identificationfor nonlocal nonlinear systems is a matter of data representation, which is resolvedin the sequel by the invocation of occupation kernels. This builds on the work in [30]in that some of the occupation kernels take the same form as in [30]. The objectiveof the work of [30] was to obtain an approximation of the dynamics themselves. Theprinciple difference is the development of nonlocal operators and a new Hilbert spaceframework, which allows for spectral decompositions of these operators for expressinga model for the system.Section 4 introduces occupation kernel Hilbert spaces (OKHSs). The construc-tion presented here is for the particular case of the Caputo fractional derivative, butis immediately generalizable to a wide range of time fractional operators where initialvalue problems may be resolved using Voltera integral equations. The developmentof OKHSs enables the generalization of a key operator used in the study of nonlineardynamical systems, namely the Liouville operator. Some Liouville operators are con-tinuous generators of semigroups of Koopman operators [9, 10, 15, 16, 17, 18], whichare the pivotal tools in the study of nonlinear dynamical systems of integer order. DMD emergedas an effective data-driven method of learning dynamical systems from trajectory datawith no prior knowledge. The DMD method aims to analyze finite dimensional non-linear dynamical systems as operators over infinite dimensional Hilbert spaces, wherethe tools of traditional linear systems may be employed to make predictions aboutnonlinear systems from captured trajectory data (or snapshots). Though based onearly work by Koopman and Von Neumann in the 1930s, DMD more recently cameto prominence as a method of identifying underlying governing principles of nonlinearfluid flows (cf. [5, 7, 26, 36, 37, 64, 65]), which compared favorably with principleorthogonal decomposition (POD) analyses.Underlying classical DMD methods are Koopman operators, which are operatorsover function spaces that represent discrete time dynamics [7, 26]. That is, given adiscrete dynamical system, x i +1 = F ( x i ), and a Hilbert space of functions, H , thecorresponding Koopman operator, K F : D ( K F ) → H is given as K F g = g ◦ F for all g ∈ D ( K F ) ⊂ H , where D ( K F ) is a given subset of H corresponding to the domain of K F . For a collection of snapshots of a dynamical system, x , x , . . . , x m , the dynamicsare then represented through observables pulled from H , as g ( x ) , g ( x ) , . . . , g ( x m ),where g ( x i +1 ) = g ( F ( x i )) = K F g ( x i ) [26, 64]. When H is a reproducing kernel Hilbert pace (RKHS) over R n with kernel function K : R n × R n → R (the definition of whichis given in the Technical Background below), then this action on the observables maybe expressed as h g, K ( · , x i +1 ) i H = g ( x i +1 ) = K F g ( x i ) = hK F g, K ( · , x i ) i H = h g, K ∗ F K ( · , x i ) i H . Hence, discrete time dynamics may be captured through the adjoint of the Koop-man operator acting on the kernel functions of a RKHS. This connection betweenKoopman operators and RKHSs is the core observation of kernel-based extended DMDpresented in [65, 24]. The kernel-based extended DMD algorithm then expands thekernel function as a column of features, collects a sequence of input-output data, andthen reduces the system through a singular value decomposition. The DMD proce-dure is completed upon determining the eigenvectors and eigenvalues of the reducedsystem, and then lifting up these eigenvalues and eigenfunctions to the Koopmanoperator to establish what are called Koopman modes for the dynamical system [65].So described is the DMD procedure for discrete time dynamical systems. Practi-tioners analyze continuous time dynamics by first fixing a time-step and then examin-ing a discrete time proxy for the continuous time system [26]. The intuition employedin this approach is that by taking small time steps, an adequate approximation of theLiouville operator may be determined by using a Koopman operator.The limiting perspective using Koopman operators has several major theoreticaldrawbacks; the most significant and fundamental is its restriction to dynamical sys-tems that admit a discretization. The symbol (or discrete dynamics) of a Koopmanoperator must be defined over all of the state space for the Koopman operator tobe well defined. The consideration of the dynamics ˙ x = 1 + x yields a discretiza-tion of x m +1 = tan(∆ t + arctan( x m )), where it can be seen that the selection of x m = tan( π/ − ∆ t ) gives an undefined value for x m +1 . Hence, many polynomialsystems do not fall under the purview of Koopman based techniques, and the par-ticular requirement that guarantees the existence of a discretization of a dynamicalsystem is forward completeness (cf. [25, Chapter 11]) which is usually established bydemonstrating that a dynamical system is globally Lipschitz continuous (cf. [6] and[23, Theorem 3.2]).To address the above limitation of the Koopman perspective, [51] introduceda combination of Liouville operators and occupation kernels to determine a finiterank representation of Liouville operators for spectral analysis. This allows for theanalysis of dynamical systems that do not admit discretizations, and as such, thisallows for the analysis of Liouville operators that were not also Koopman generators.Moreover, the introduction of occupation kernels and scaled Liouville operators in[51] over the Bargmann-Fock space yielded a compact operator for a wide range ofdynamics, and scaled Liouville operators allows for norm convergence of finite rankapproximations used in DMD procedures. The main thrust of this manuscript is toprovide a framework for the transportation of these tools to (time) fractional orderdynamical systems, where data driven modeling techniques using DMD procedurescould not be previously applied.
2. Motivation for the present approach.
Suppose that for 0 < q < D q ∗ x ( t ) = C − q R t ( t − τ ) − q ˙ x ( τ ) dτ , and let In the context of function theoretic operator theory, the function F , which defines the discretetime dynamics, is called the symbol of the Koopman operator. This terminology is common amongmany other such operators over RKHSs (cf. [49, 48, 56]) Here C q := 1 / Γ( q ) is used to avoid confusion with the occupation kernels.3 q ∗ x ( t ) = f ( x ( t )) be a nonlinear fractional order system with the initial condition x (0) = x ∈ R n . The Riemann-Liouville fractional integral is given as J q x ( t ) = C q R t ( t − τ ) q − x ( τ ) dτ, and the initial value problem may be resolved as x ( t ) = x + J q D q ∗ x ( t ) = x + J q f ( x ( t )) (cf. [12, 19]). Hence, ˙ x ( t ) = ddt J q f ( x ( t )) = D − q f ( x ( t )),where D q is the Riemann-Liouville fractional derivative operator (cf. [12, 19, 40]).Exploiting this core idea, generalizations of Liouville operators and occupation kernelsmay be defined.Specifically, two variants on the Liouville operator are explored in this manuscript;the Liouville operator of order q given as A f,q g := ∇ g ( · ) D q − f ( · ), and the frac-tional Liouville operator of order q given as A f,q g = A f,q g [ γ ]( T γ ) = q ) R T γ ( T γ − t ) − q ∇ g ( γ ( t )) D − q f ( γ ( t )) dt where g is a function on a collection of signals in a Hilbertspace that will be defined in Section 4. If the fractional Liouville operator was poisedover a RKHS consisting of functions over R n , then the image of A f,q g would be afunction of R n , which would mean that for any x ∈ R n , the quantity A f,q g ( x ) wouldbe well defined. However, as a state carries no “history,” the term D q − f ( x ) is notwell defined. This difficulty prevents the straightforward generalization of Liouvilleoperators for integer order dynamical systems to that of fractional order dynamicalsystems. Section 4 of this manuscript provides a resolution to this problem throughthe construction of a Hilbert space consisting of functions over signals. Thus, in thesequel, the domain of A f,q g is a collection of signals, and for a given continuouslydifferentiable signal θ : [0 , T ] → R n , the quantity D q − f ( θ ( t )) is well defined. Theastute reader will ask how a gradient of a functions, g , over a collection of signal isdefined and this will be quantified more precisely in Section 4.To accommodate the nonlocal requirements of the fractional Liouville operator,Section 4 constructs a Hilbert space from a RKHS, H RKHS . Specifically, for anyfunction, g , in a continuously differentiable RKHS over R n a mapping over collectionof signals arises naturally via φ g [ θ ]( t ) := g ( θ ( t )). Hence, φ g maps the signal θ tothe signal g ( θ ( · )). In combination with the canonical identification, g φ g , andthe inner product of the RKHS, h· , ·i H RKHS , an inner product on the vector space H ◦ OKHS := { φ g : g ∈ H RKHS } may be established, and the resultant Hilbert spacewill be called an occupation kernel Hilbert space (OKHS). The name for this Hilbertspace arises from occupation kernels, which will play a key role in the analysis ofnonlinear dynamical systems of fractional order, just as they have for integer ordersystems (cf. [55]). For a given signal, θ : [0 , T ] → R n , the occupation kernel oforder q > θ,q ( x ) := C q R T ( T − τ ) q − K ( x, θ ( τ )) dτ . Occupation kernelsrepresent integration after composition of the signal with a function in a RKHS, h g, Γ θ,q i = C q R T ( T − τ ) q − g ( θ ( τ )) dτ .The introduction of OKHSs and associated Liouville operators directly addressesthe problem of data integration. That is, OKHSs directly incorporate trajectory datain a kernel function contained in the Hilbert space. The definition of the Hilbert space,which consists of functions on signals , allows for the definition of nonlocal Liouvilleoperators. In turn, these nonlocal Liouville operators allow for the development ofDMD procedures on nonlinear fractional order systems.
3. Prerequisites.3.1. Reproducing Kernel Hilbert Spaces.
A (real-valued) reproducing ker-nel Hilbert space (RKHS) is a Hilbert space, H , of real valued functions over a set X such that for all x ∈ X the evaluation functional E x g := g ( x ) is bounded (cf.[42, 60, 63, 1, 3, 47, 54]). As such, the Riesz representation theorem guarantees, for ll x ∈ X , the existence of a function k x ∈ H such that h g, k x i H = g ( x ), where h· , ·i H is the inner product for H . The function k x is called the reproducing kernel functionat x , and the function K ( x, y ) = h k y , k x i H is called the kernel function correspondingto H .To establish a connection between RKHSs and nonlinear dynamical systems thefollowing operator was introduced, which was inspired by the study of occupationmeasures and related concepts (cf. [27, 2, 34, 21, 57, 22, 4, 5, 8, 28, 29, 36, 38, 61]). Definition
Let ˙ x = f ( x ) be a dynamical system with the dynamics, f : R n → R n , Lipschitz continuous, and suppose that H is a RKHS over a set X , where X ⊂ R n is compact. The Liouville operator with symbol f , A f : D ( A f ) → H , isgiven as A f g := ∇ x g · f, where D ( A f ) := { g ∈ H : ∇ x g · f ∈ H } . As a differential operator, A f is not expected to be a bounded over many RKHSs.However, as differentiation is a closed operator over RKHSs consisting of continu-ously differentiable functions (cf. [60, 46]), it can be similarly established that A f isclosed under the similar circumstances (cf. [55]). Thus, A f is a closed operator forRKHSs consisting of continuously differentiable functions. Consequently, the adjointsof densely defined Liouville operators are themselves densely defined (cf. [43]).Associated with Liouville operators in particular are a special class of functionswithin the domain of the Liouville operators’ adjoints. The following definition andproposition were given in [55], and these generalize the framework of [27] on occupationmeasures. Definition
Let X ⊂ R n be compact, H be a RKHS of continuous functionsover X , and γ : [0 , T ] → X be a continuous signal. The functional g R T g ( γ ( τ )) dτ is bounded, and may be respresented as R T g ( γ ( τ )) dτ = h g, Γ γ i H , for some Γ γ ∈ H by the Riesz representation theorem. The function Γ γ is called the occupation kernelcorresponding to γ in H . Proposition
Let H be a RKHS of continuously differentiable functions overa compact set X , and suppose that f : R n → R n is Lipschitz continuous. If γ :[0 , T ] → X is a trajectory that satisfies ˙ γ = f ( γ ) , then Γ γ ∈ D ( A ∗ f ) , and A ∗ f Γ γ = K ( · , γ ( T )) − K ( · , γ (0)) . Proposition 3.3 thus integrates nonlinear dynamical systems with RKHSs throughthe Liouville operator. In particular, the action of the adjoint of the Liouville operatoracting on an occupation kernel expressed as a difference of kernels, enables continuoustime DMD analyses that do not require an a priori discretization of the dynamicalsystem [51].
Time fractional derivatives, suchas the Riemann-Liouville fractional derivative and the Caputo fractional derivative,are defined with respect to two operators: integer order derivatives and the Riemann-Liouville fractional integral given as J q γ ( t ) := q ) R T ( t − τ ) q − γ ( τ ) dτ. The Riemann-Liouville fractional derivative of order 0 < q < γ is then given as ddt J − q γ ( t ), while the Caputo fractional derivative of order 0 < q < D q ∗ γ ( t ) := J − q γ ′ ( t ) = − q ) R t ( t − τ ) − q γ ′ ( τ ) dτ . Initial value problems for theCaputo fractional derivative may be expressed as(3.1) D q ∗ x = f ( x ) with x (0) = x , and their solution can be written in terms of a Volterra operator as x ( t ) = x + q ) R t ( t − τ ) q − f ( τ ) dτ. Riemann-Liouville based initial value problems require initial alues with respect to fractional derivatives, which are difficult to express for a physicalsystem [12]. Hence, the present manuscript will focus on dynamical systems arisingfrom the Caputo fractional derivative.
4. Occupation Kernel Hilbert Spaces.
To provide for a nonlocal Hilbertspace framework, the objective of Section 4 is to develop further the concept of oc-cupation kernels. At this point, occupation kernels have been viewed as a part of aRKHS rather than as a generator of a Hilbert space on their own. In what follows, thiswill remain true to some extent, where many OKHSs will arise from RKHSs. However,the central objects of study will be bounded functionals that come from trajectories.The focus on trajectories rather than points in R n allows for the treatment of op-erators on nonlocal information. Section 4 formalizes the necessary Hilbert spaceframework, which will enable the development of DMD analyses of (time) fractionalorder dynamical systems. Definition
Let q > . For fixed d, e ∈ N with e < d , let X = ∪ T > C d ([0 , T ] , R n ) and Y = ∪ T > C e ([0 , T ] , R ) . The OKHS of order q , H , over X is a Hilbert space thatconsists of functions g : X → Y such that for each γ ∈ X and g [ γ ] : [0 , T ] → R in Y the functional g J q g [ γ ] = C q R T ( T − t ) q − g [ γ ]( t ) dt is bounded, with g [ γ ] ∈ Y indicating the mapping of γ from X to Y by g ∈ H . For each such functional, the Riesz representation theorem yields a function, Γ γ,q ∈ H , such that h g, Γ γ,q i H = q ) R T γ ( T γ − t ) q − g [ γ ]( t ) dt, this function is called theoccupation kernel corresponding to H . The existence of OKHSs follows immediately fromthat of RKHSs.
Definition
Let X = ∪ T > C d ([0 , T ] , R n ) and Y = ∪ T > C e ([0 , T ] , R m ) and V be a vector space of infinitely differentiable functions g : R n → R m . For each g ∈ V let φ g : X → Y be the mapping that takes γ ∈ X to φ g [ γ ] := g ( γ ( · )) ∈ Y . Let W bethe vector space of maps ϕ : X → Y , under the standard operations. The operator, T : V → W given by T g := φ g , is called the canonical mapping of the scalar valuedfunction g to the signal valued function φ g . Note, since g is infinitely differentiable, φ g [ γ ] is at least as smooth as γ . Hence φ g [ γ ] ∈ Y . Theorem
Let H RKHS be an RKHS of infinitely differentiable scalar-valuedfunctions over R n with kernel function K . The space, T H RKHS := {T g : g ∈ H RKHS } is an OKHS of any order q > where the inner product is taken to be h φ, ψ i OKHS = hT − φ, T − ψ i RKHS . Proof.
From the discussion before the theorem statement, φ g is well defined as amap from X to Y for any g ∈ H RKHS . To demonstrate that H OKHS := T H RKHS is an occupation kernel Hilbert space, it must be demonstrated that the space iscomplete with respect to the norm induced by the inner product and that for any q >
0, the collection of functionals in Definition 4.1 are bounded.Note that T is linear, and hence H OKHS is a vector space. That T is injectivefollows from the consideration of constant signals. If g , g ∈ H RKHS are distinctfunctions, then there is a point x ∈ R n such that g ( x ) = g ( x ). Considering θ x ( t ) ≡ x for t ∈ [0 ,
1] which resides in X given in Definition 4.1, it follows that φ g [ θ x ]( t ) = g ( θ x ( t )) = g ( x ) = g ( x ) = φ g [ θ x ]( t ) for all t . nce injectivity is established, the inner product on H OKHS , given as h φ, ψ i OKHS := hT − φ, T − ψ i RKHS , is well defined. Moreover, T is automatically continuous with respect to the inducednorm on H OKHS .To demonstrate completeness, suppose that { φ m } ∞ m =1 ⊂ T H RKHS is Cauchywith respect to the norm induced by the inner product on H OKHS . For each m , thereis a function g m ∈ H RKHS such that φ m = T g m . Given any ǫ >
0, there is an M such that for all m, m ′ > M , the following holds, k φ m − φ m ′ k OKHS = p h φ m − φ m ′ , φ m − φ m ′ i OKHS p h g m − g m ′ , g m − g m ′ i RKHS = k g m − g m ′ k RKHS . Hence, g m is a Cauchy sequence in H RKHS , and there is a function g ∈ H RKHS such that g m → g . The continuity of the canonical identification yields φ g m → φ g ,and thus the limit of φ m resides in T H RKHS . Therefore, T H RKHS is complete and T H RKHS = H OKHS . To demonstrate the boundedness of the functional φ C q R T ( T − τ ) q − φ [ γ ]( τ ) dτ for each γ ∈ X and q >
0, let g = T − φ . It follows that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C q Z T ( T − τ ) q − φ [ γ ]( τ ) dτ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C q Z T ( T − τ ) q − g ( γ ( τ )) dτ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k g k RKHS · C q Z T ( T − τ ) q − p K ( γ ( τ ) , γ ( τ )) dτ = k φ k OKHS · C q Z T ( T − τ ) q − p K ( γ ( τ ) , γ ( τ )) dτ. As K is a continuous function on R n and γ ([0 , T ]) is compact in R n , it follows thatthe last integral is bounded for any q >
0. Note that for 0 < q <
1, the quantity( T − τ ) q − has an integrable singularity at τ = T . Remark The operator T : H RKHS → H OKHS is an isometric isomorphismfrom the base reproducing kernel Hilbert space to the constructed occupation kernelHilbert space. In fact, kT k = sup g =0 (cid:26) kT g kk g k : g ∈ H RKHS (cid:27) = sup g =0 (cid:26) k g kk g k : g ∈ H RKHS (cid:27) = 1 . Remark In general, if g is arbitrary signal valued function, then g [ γ ] and γ are not necessarily defined over the same interval. However, if H is an OKHS arisingfrom an RKHS under the canonical mapping, then given a g ∈ H and γ : [0 , T ] → R n it follows that g [ γ ] is also defined over [0 , T ] . A formula for Γ γ,q for an OKHS derived from an RKHS is obtainable in terms ofthe reproducing kernel.
Lemma
Let H RKHS be a reproducing kernel Hilbert space of infinitely differ-entiable functions then for a fixed q > and γ : [0 , T ] → R n continuous the functional g C q Z T ( T − t ) q − g ( γ ( t )) dt s bounded and thus there exists a function K γ,q ∈ H RKHS such that h g, K γ,q i RKHS = C q Z T ( T − t ) q − g ( γ ( t )) dt. Moreover, that function is given by K γ,q ( x ) = C q Z T ( T − t ) q − K ( x, γ ( t )) dt, where K is the reproducing kernel for H RKHS .Proof.
Similar to the above, let γ : [0 , T ] → R n be continuous. If g , g ∈ H RKHS then | g ( γ ( t )) − g ( γ ( t )) | ≤ k g − g k RKHS k K γ ( t ) k RKHS for all t ∈ [0 , T ] where k K γ ( t ) k RKHS = p K ( γ ( t ) , γ ( t )). Since γ is a continuousfunction on a compact set, it follows that K ( γ ( t ) , γ ( τ )) is also bounded on [0 , T ] since K is at least continuous in each variable. Therefore, k K γ ( t ) k is bounded (as afunction of t ) and the integral is bounded as well. Hence, for each γ and q > K γ,q ∈ H RKHS , such that h g, K γ,q i RKHS = C q Z T ( T − t ) q − g ( γ ( t )) dt. Note that K γ,q ( x ) = h K γ,q , K x i RKHS = h K x , K γ,q i RKHS = C q Z T ( T − t ) q − K ( γ ( t ) , x ) dt = C q Z T ( T − t ) q − K ( x, γ ( t )) dt. Theorem
Let H RKHS be a reproducing kernel Hilbert space of infinitely dif-ferentiable functions over a compact set and H OKHS be the OKHS obtained from H RKHS under the mapping T above.Proof. Let φ g ∈ H OKHS = T ( H RKHS ) be an arbitrary element in H OKHS . ByLemma 4.4, h φ g , Γ γ,q i OKHS = C q Z T ( T − t ) q − g ( γ ( t )) dt = h g, K γ,q i RKHS . Since T is an isometric isomorphism, T − is also an isometric isomorphism. Hence, h φ g , Γ γ,q i OKHS = hT ( φ g ) , T − (Γ γ,q ) i RKHS = h g, T − (Γ γ,q ) i RKHS . Since, φ g was ar-bitrary, it follows that g is also arbitrary. Moreover, h g, T − (Γ γ,q ) i RKHS = h g, K γ,q i RKHS . This is enough to show that Γ γ,q = T ( K γ,q ). . Nonlocal Liouville Operators. The objective of this manuscript is to gen-eralize existing methods for integer order dynamical systems for system identificationand DMD analysis to that of fractional order dynamical systems of Caputo type. Thissection introduces two fractional order generalizations of the Liouville operator, whichwill be employed in the sequel for DMD analysis.
Definition
Let ϕ ∈ H OKHS where H OKHS is an OKHS of order 1 arisingfrom an RKHS, H RKHS , under the canonical mapping. Since ϕ ∈ H OKHS , ϕ = φ g = T ( g ) where T is the canonical mapping and g : R n → R is a member of H RKHS .Given γ : [0 , T ] → R n , and noting ∇ g : R n → R n , define ∇ ϕ [ γ ] = ∇ g ( γ ( · )) as a signal from [0 , T ] to R n . Moreover, given an f : R n → R n and any γ : [0 , T ] → R n such that f ◦ γ : [0 , T ] → R n is differentiable, define D − q f [ γ ] = D − q ( f ( γ ( · ))) = (cid:2) D − q ( f ◦ γ ) ( · ) , . . . , D − q ( f ◦ γ ) n ( · ) (cid:3) ⊤ as a signal from [0 , T ] to R n . Here, ( f ◦ γ ) i , i = { , . . . , n } , are the componentsof f ◦ γ , and interpret D − q on the right hand side as the one dimensional Caputofractional derivative. Finally, using the standard dot product, define ( ∇ ϕ · D − q f )[ γ ] = ∇ g ( γ ( · )) · D − q ( f ( γ ( · ))) as a signal from [0 , T ] to R . Similarly, if ϕ ∈ H qOKHS then define t C q Z t ( t − τ ) − q ∇ ϕ ( γ ( τ )) · D − q f ( γ ( τ )) dτ as a signal from [0 , T ] to R . Let f : R n → R n , D q ∗ x ( t ) = f ( x ( t )) be a dynamical system of Caputo type with0 < q <
1, and recall that ˙ x ( t ) = D − q f ( x ( t )) . Definition
Let D ( A f,q ) := { ϕ ∈ H OKHS : ∇ ϕ · D − q f ∈ H OKHS } . TheLiouville operator, A f,q : D ( A f,q ) → H OKHS , corresponding to the dynamical sys-tem in (3.1) over an H OKHS arising from an RKHS is defined as A f,q ϕ [ γ ]( t ) = ∇ g ( γ ( t )) D − q f ( γ ( t )) .Alternatively, let D ( A f,q ) := { ϕ ∈ H qOKHS : J − q (cid:0) ∇ ϕ · D − q f (cid:1) ∈ H qOKHS } . The fractional Liouville operator, A f,q : D ( A f,q ) → H qOKHS , corresponding to (3.1) over H qOKHS is given as A f,q ϕ [ γ ]( t ) = q ) R t ( t − τ ) − q ∇ ϕ ( γ ( τ )) · D − q f ( γ ( τ )) dτ . The clearest way to distinguish between the two operators is through the exami-nation of the performance of their eigenfunctions on a trajectory, γ , satisfying (3.1).Specifically, suppose that φ g ∈ H OKHS is an eigenfunction of A f,q with eigenvalue λ ,then(5.1) ddt φ g [ γ ]( t ) = ∇ g ( γ ( t )) ˙ γ ( t ) = ∇ g ( γ ( t )) D − q f ( γ ( t )) = A f,q φ g [ γ ]( t ) = λφ g [ γ ]( t ) , hence φ g [ γ ]( t ) = φ g [ γ ](0) e λt . This relation on the eigenfunctions of the Liouvilleoperator agrees with that expressed for continuous time integer order systems in [51]. n contrast, suppose that φ h ∈ ˜ H OKHS is an eigenfunction for A f,q with eigenvalue λ , then D q ∗ φ h [ γ ]( t ) = C q Z t ( t − τ ) − q ∇ h ( γ ( τ )) ˙ γ ( τ ) dτ (5.2) = C q Z t ( t − τ ) − q ∇ h ( γ ( τ )) D − q f ( γ ( τ )) dτ = A f,q φ h [ γ ]( t ) = λφ h [ γ ]( t ) . Hence, φ h [ γ ]( t ) = φ h [ γ ](0) E q ( λt q ) , where E q is the Mittag-Leffler function of order q ,given as E q ( t ) := P ∞ m =0 t m Γ( qm +1) . These two different perspectives on the Liouville operator each has their ownadvantages. Specifically, the decompositions of these operators will leverage eitherexponentials or Mittag-Leffler functions. In the case of the Liouville operator, A f,q ,the decomposition for the dynamics resulting from this operator will be a linear combi-nation of exponential functions, which not only align with classical DMD analyses (cf.[51]), but are easier to compute. The decomposition of the dynamics given throughthe fractional Liouville operator, A f,q , will be a linear combination of Mittag-Lefflerfunctions, which are likely to yield better predictions for fractional order dynamicalsystems, given the intimate relationship between fractional derivatives and Mittag-Leffler functions.Interestingly, the best choice of RKHS for the foundation of an OKHS depends onthe selection of the approach to the generalization of Liouville operators. In partic-ular, the exponential dot product kernel’s native space (frequently referred to as theBargman-Fock space), aligns well with the first type of Liouville operator, A f,q , sinceits kernel functions are of the form K ( x, y ) = exp( x T y ). However, for the fractionalLiouville operator, A f,q , a multi-variable generalization of the Mittag-Leffler RKHSof the slitted plane (cf. [54]) would be more appropriate, where K ( s, t ) = E q ( s q t q ).Similar to the case of the integer order Liouville operator, each Liouville operatorinteracts nicely with occupation kernels from their respective spaces. In particular,(5.3) h A f,q φ g , Γ γ i H OKHS = g ( γ ( T )) − g ( γ (0)) = φ g [ γ ]( T ) − φ g [ γ ](0) , and(5.4) hA f,q h, ˜Γ γ i ˜ H OKHS = φ h [ γ ]( T ) − φ h [ γ ](0) , where Γ γ is the occupation kernel corresponding to H OKHS and ˜Γ γ is the occupationkernel corresponding to ˜ H OKHS . At this point, it is important to note that eachOKHS has “bounded point evaluations” through constant trajectories, α x : [0 , → R n where α x ( t ) = x . Point evaluation is then expressed as h φ g , Γ α x i H OKHS = R φ g [ α x ]( t ) dt = R g ( x ) dt = g ( x ) = φ g [ α x ](0) , and a similar result is feasible for˜ H OKHS at the cost of a constant multiple C q . Combining this result with the innerproduct relations yields(5.5) A ∗ f,q Γ γ = Γ α γ ( T ) − Γ α γ (0) and A ∗ f,q ˜Γ γ = C q (cid:16) ˜Γ α γ ( T ) − ˜Γ α γ (0) (cid:17) . These relations will be pivotal in the development of a DMD method for fractionalorder systems. . Finite Rank Representations of Liouville Operators. .For q >
0, the objective of this section is to leverage observed trajectories, M = { γ i : [0 , T i ] → R n } Mi =1 , that satisfy D q ∗ γ i = f ( γ i ) for each i = 1 , . . . , M and anunknown f : R n → R n to derive a model, G : [0 , T ] → R n , for the system for whichgiven an initial value x (0) ∈ R n the trajectory x : [0 , T ] → R n satisfying D q ∗ x = f ( x )can be estimated as x ( t ) ≈ G ( t ) . To obtain this model, a finite rank representation ofthe operators A f,q and A f,q over a given RKHS, H , will be determined through theassociated occupation kernels and the relations established in Section 5. The finiterank representation of the operator will then be leveraged as a proxy for the actualdensely defined operator, where a spectral decomposition will be determined andthe eigenfunctions will form the foundation of the data driven model. In practice, theeigenfunctions will be determined as a linear combination of the associated occupationkernels. A f,q . Let H be an OKHS determined through a RKHS, ˜ H , of continuously differentiable functionsover R n . Let β be the ordered basis of occupation kernels corresponding to M givenas β := { Γ γ i ,q } Mi =1 . The vector space span β is a finite dimensional subspace of H and hense, closed. Let P β denote the projection operator from H onto span β . Asdemonstrated in Section 5, each occupation kernel corresponding to γ i is in the domainof the adjoint of the operator A f,q . Hence, the operator P β A ∗ f,q P β is well defined over H , and this operator is of finite rank. Note that the matrix, [ P β A ∗ f,q P β ] ββ , defined interms of the ordered basis β may be expressed simply as [ P β A ∗ f,q ] ββ , since the domainof definition for the matrix is span β and P β h = h for all h ∈ span β .For a function h ∈ H , the projection of H onto span β may be written as P β h = P Mi =1 w i Γ γ i ,q , where w = ( w , . . . , w m ) T ∈ R M is obtained by solving the matrixequation(6.1) h Γ γ ,q , Γ γ ,q i H · · · h Γ γ M ,q , Γ γ ,q i H ... . . . ... h Γ γ ,q , Γ γ M ,q i H · · · h Γ γ M ,q , Γ γ M ,q i H w ... w M = h h, Γ γ ,q i H ... h h, Γ γ M ,q i H . Consequently, the finite rank representation of A ∗ f,q may be determined as[ P β A ∗ f,q ] ββ = h Γ γ ,q , Γ γ ,q i H · · · h Γ γ M ,q , Γ γ ,q i H ... . . . ... h Γ γ ,q , Γ γ M ,q i H · · · h Γ γ M ,q , Γ γ M ,q i H − × (6.2) h A ∗ f,q Γ γ ,q , Γ γ ,q i H . . . h A ∗ f,q Γ γ M ,q , Γ γ ,q i H ... h A ∗ f,q Γ γ ,q , Γ γ M ,q i H . . . h A ∗ f,q Γ γ M ,q , Γ γ M ,q i H . In (6.2), the entries of the matrix may be computed using 5.3 and the canonicalidentification φ g developed in Section 4. To wit, noting that q = 1 for the Liouvilleoperator and occupation kernel relation: h A ∗ f,q Γ γ i ,q , Γ γ j , i H = h Γ α γi ( T ) , − Γ α γi (0) , , Γ γ j ,q i H = Z T ˜ K ( γ j ( τ ) , γ i ( T )) − ˜ K ( γ j ( τ ) , γ i (0)) dτ. imilarly, for the fractional Liouville operator, A f,q , the finite rank representationis given as [ P β A ∗ f,q ] ββ = h Γ γ ,q , Γ γ ,q i H · · · h Γ γ M ,q , Γ γ ,q i H ... . . . ... h Γ γ ,q , Γ γ M ,q i H · · · h Γ γ M ,q , Γ γ M ,q i H − × (6.3) hA ∗ f,q Γ γ ,q , Γ γ ,q i H . . . hA ∗ f,q Γ γ M ,q , Γ γ ,q i H ... hA ∗ f,q Γ γ ,q , Γ γ M ,q i H . . . hA ∗ f,q Γ γ M ,q , Γ γ M ,q i H , where each entry can be computed as hA ∗ f,q Γ γ i ,q , Γ γ j ,q i H = h Γ α γi ( T ) ,q − Γ α γi (0) ,q , Γ γ j ,q i H = C q Z T ( T − τ ) q − ˜ K ( γ j ( τ ) , γ i ( T )) − ˜ K ( γ j ( τ ) , γ i (0)) dτ. The entries of the Gram matrix in (6.2) and (6.3) can be expressed as h Γ γ i ,q , Γ γ j ,q i H = h Γ α γi ( T ) ,q − Γ α γi (0) ,q , Γ γ j ,q i H = ( C q ) Z T ( T − τ ) q − ( T − t ) q − ˜ K ( γ j ( τ ) , γ i ( t )) dτ dt, with q = 1 for the Liouville operator and q matching the order of the system for thefractional Liouville operator.
7. Dynamic Mode Decompositions with Liouville Modes and Frac-tional Liouville Modes.
In Dynamic Mode Decomposition (DMD) methods forthe data driven analysis of dynamical systems, the identity function (also called the full state observable ), g id : R n → R n , given as g id ( x ) = x is individually decomposedwith respect to an eigenbasis in a RKHS corresponding to finite rank representationsof Koopman and Liouville operators similar to those determined in Section 6. Theresult is a linear combination of scalar valued eigenfunctions multiplied by a collectionof vectors obtained through the projection of the component of g id onto the eigen-basis. These vectors are called the dynamic modes of the system. When connectedwith particular operators, they are also called Koopman or Liouville modes. Subse-quently, a model for the dynamical system is determined by exploiting certain featuresof the eigenfunctions, which ultimately replace the eigenfunctions with exponentialfunctions.This section discusses the two different models that are determined through achoice of using the Liouville operator or the fractional Liouville operator over a OKHS.In place of the identity function, whose image is in R n , the DMD method presentedhere leveraged the signal valued analogue, φ g id , and exploits identities (5.1) and (6.3).From the data driven perspective, the Liouville operator and Fractional Liouvilleoperator for a particular collection of trajectories is not directly accessible. Thismotivates the use of finite rank representations such as those given in Section 6. Aneigenvector, v = ( v , . . . , v M ) T ∈ C M , with eigenvalue λ ∈ C obtained from (6.2) or(6.3) corresponds to a function in the OKHS as ϕ = √ v T Gv P Mi =1 v i Γ γ i ,q with q = 1 for(6.2) and q as the fractional order of the system in (6.3), which is an eigenfunction for he corresponding finite rank operator. To derive a model for the dynamical system,the eigenfunctions obtained from the finite rank operators will be used in place ofproper eigenfunctions of the Liouville and fractional Liouville operators.Specifically, suppose that the eigenfunctions { ϕ i } Mi =1 corresponding to the eigen-values { λ i } diagonalize the finite rank operator P β A f,q P β . The full state observablefor the OKHS, φ g id , may then be decomposed as φ g id = P Mi =1 ξ i ϕ i , where( ξ · · · ξ M ) T := (cid:16) W − (cid:0) h ( φ g id ) j , ϕ i H · · · h ( φ g id ) j , ϕ M i H (cid:1) T (cid:17) nj =1 ∈ C m × n are the Liouville modes, and W = ( h ϕ i , ϕ j i H ) Mi,j =1 . This decomposition of the fullstate observable allows the expression of an approximate model for the state, D q ∗ x ( t ) = f ( x ( t )), by leveraging (5.1) as x ( t ) = φ g id [ x ]( t ) ≈ M X i =1 ξ i ϕ i [ x ]( t ) ≈ M X i =1 ξ i ϕ i [ x ](0) e λ i t . Hence, a model for the dynamical system has been derived from the observed trajecto-ries. However, since the dynamics are governed by the Caputo fractional derivative, amodel obtained from the finite rank representation of the fractional Liouville operatormay be more suitable. In this case, suppose that { ˆ ϕ i } Mi =1 are eigenfunctions of thefinite rank operator A f,q with eigenvalues { λ i } Mi =1 . Following the same procedure asabove and using (5.2), the resultant model for the dynamical system is given as x ( t ) = φ g id [ x ]( t ) ≈ M X i =1 ˆ ξ i ˆ ϕ i [ x ]( t ) ≈ M X i =1 ˆ ξ i ˆ ϕ i [ x ](0) E q ( λ i t q ) , with ξ i being the i -th fractional Liouville mode. Heuristically, the latter model isexpected to match the state more closely, since t E q ( λ i t q ) is the solution to afractional order linear system. Additionally, if the finite sums above are replacedwith a series obtained from a collection of eigenfunctions that diagonalize the originaloperators, then the expressions are expected to result in equalities.
8. Discussion.
It should be noted that while the theoretical developments givenabove are for a scalar valued function, these methods extend naturally to vector valuedquantities by treating each dimension separately and then later stacking the dynamicmodes. This is equivalent to using vector valued kernels that are diagonal.Despite the theoretical developments in the manuscript, the actual implemen-tation of the DMD method does not differ greatly from standard occupation kernelDMD. As seen in Section 6, the computations ultimately are performed on the un-derlying RKHS rather than directly on the OKHS. Hence, the computation of Grammatrices only need a slight adjustment to accommodate the fractional integrals. Infact, these computations are exactly in line with [30], and follow from Newton-Cotesand Gaussian quadrature methods.Finally, the fractional order framework can also be posed over signal valuedRKHSs [53], which gives another approach to handling nonlocal data interpretation.OKHSs differ in that the definition of OKHSs are independent of that of RKHSs,where the RKHSs are used in this manuscript as a particular realization of OKHSs.Moreover, while signal valued spaces correspond to trajectories of a fixed length (andmay artificially adjust the lengths of trajectories using indicator functions), OKHSsallow for arbitrarily long trajectories natively.
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