Nonlinear Data-Driven Approximation of the Koopman Operator Dan Wilson1 1Department of Electrical Engineering and Computer Science University of Tennessee

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Nonlinear Data-Driven Approximation of the Koopman Operator

Dan Wilson ∗1

1Department of Electrical Engineering and Computer Science, University of Tennessee,

Knoxville, TN 37996, USA

October 11, 2022

Abstract

Koopman analysis provides a general framework from which to analyze a nonlinear dynami-

cal system in terms of a linear operator acting on an inﬁnite-dimensional observable space. This

theoretical framework provides a rigorous underpinning for widely used dynamic mode decom-

position algorithms. While such methods have proven to be remarkably useful in the analysis

of time-series data, the resulting linear models must generally be of high order to accurately

approximate fundamentally nonlinear behaviors. This issue poses an inherent risk of overﬁt-

ting to training data thereby limiting predictive capabilities. By contrast, this work explores

strategies for nonlinear data-driven estimation of the action of the Koopman operator. General

strategies that yield nonlinear models are presented for systems both with and without con-

trol. Subsequent projection of the resulting nonlinear equations onto a low-rank basis yields

a low order representation for the underlying dynamical system. In both computational and

experimental examples considered in this work, linear estimators of the Koopman operator are

generally only able to provide short-term predictions for the observable dynamics while compa-

rable nonlinear estimators provide accurate predictions on substantially longer timescales and

replicate inﬁnite-time behaviors that linear predictors cannot.

1 Introduction

Model identiﬁcation is a necessary ﬁrst step in the design, optimization, control, and estimation of

complex dynamical systems. When the mechanisms that underlie the dynamics are well understood,

models can often be derived using ﬁrst principles approaches and subsequently ﬁt to available data.

However, in applications where an underlying system is too complicated to write down the under-

lying equations, data-driven model identiﬁcation can be a powerful alternative [1], [2]. Substantial

progress has been made in recent years in the development of algorithms for inferring dynamical

models strictly from time-series data. Dynamic mode decomposition (DMD) [1], [3], [4] is one such

algorithm, with the ability to represent the evolution of snapshot data in terms of a collection of

linear modes with associated eigenvalues that determine the growth/decay/oscillation rates. This

general framework has inspired numerous variations that can, for instance, incorporate the inﬂuence

of an exogenous control input [5], account for noise and uncertainty [6], and continuously adjust

when the system parameters are time-varying [7].

While DMD has been used in a wide variety of applications to explicate the underlying behavior

of snapshot data in terms of eigenmode and eigenvalue pairs, without additional modiﬁcations it

∗corresponding author: dwilso81@utk.edu

arXiv:2210.04792v1 [math.DS] 10 Oct 2022

yields a linear estimator for the underlying dynamics. Alternative approaches have been developed

to identify fully nonlinear representations for the underlying equations from data [8], [9], [10],

[11]. These methods typically consider a large nonlinear function library and subsequently use

machine learning algorithms to choose a sparse subset that best matches the training data. While

such methods can be readily applied to identify sparse representations of highly nonlinear (and

even chaotic systems), their eﬃcacy is dependent on the choice an appropriate nonlinear function

library. Related machine learning approaches using neural networks have also achieved success for

prediction in highly nonlinear dynamical systems [12], [13], [14].

Many of the data-driven model identiﬁcation strategies described above have a close connection

to Koopman analysis [15], [16], [17]. Koopman-based approaches generally allow for the repre-

sentation of a nonlinear dynamical system as a linear operator acting on an inﬁnite-dimensional

observable space. Such approaches are distinct from standard linearization techniques that consider

the dynamics in a close neighborhood of some nominal solution. Rather, the goal of Koopman anal-

ysis is to identify a linear operator that can accurately capture fundamentally nonlinear dynamics –

the key challenge is in the identiﬁcation of a suitable ﬁnite basis to represent the action of the gen-

erally inﬁnite dimensional Koopman operator. The connection between DMD and spectral analysis

of the Koopman operator is well established [4], and in applications where high-dimensional data

is readily available, the DMD algorithm can indeed be used to provide a ﬁnite-dimensional approx-

imation of the Koopman operator. Extensions of the DMD algorithm have illustrated that more

accurate approximations of the Koopman operator can be obtained using DMD in conjunction with

a set of lifting functions [18] and/or time-delayed embeddings of snapshot data [19]. Additional

accuracy can also be obtained using an adaptive strategy that uses DMD to obtain a continuous

family of linear models and actively chooses the one that provides the best representation at every

instant [20].

How best to approximate the action of the Koopman operator from snapshot data remains an

open question. DMD separates data into snapshot pairs and subsequently ﬁnds a linear operator

that provides a least squares ﬁt for the mapping from one snapshot to the next. This is currently the

most widely used approach. An obvious advantage of linear estimators of the Koopman operator

is that they allow for subsequent analysis using a wide variety of linear techniques. Nonetheless,

such linear estimators are not always suitable for highly nonlinear systems since ﬁnite dimensional

linear operators cannot be used, for instance, to replicate the inﬁnite-time behavior of systems with

multiple hyperbolic ﬁxed points or systems with stable limit cycles. Further limitations of linear

estimators can also be seen in [25] which established the diﬃculty of representing even the relatively

simple Burgers’ equation in terms of a linear operator owing to the existence highly degenerate

Koopman eigenvalues. Alternatively, nonlinear models obtained from data-driven techniques are

often more diﬃcult to analyze, but can admit lower dimensional realizations and can often provide

accurate representations of chaotic behavior [9], [12], [13]. Recent works have considered nonlinear

estimation strategies. For instance, [21] approximates separate Koopman operators that result

for diﬀerent values of an applied control input and uses this information to formulate a switching

time optimization problem. Related approaches consider bilinear approximations of the Koopman

operator for control systems [22], [23], [24].

This work explores strategies for nonlinear data-driven estimation of the action of the Koopman

operator. General strategies that yield nonlinear models are presented for systems both with and

without control. In the various examples considered in this work, only short term predictions of

the dynamical behavior can be obtained using linear estimators for the Koopman operator. By

contrast, nonlinear estimators are able to provide accurate long-term estimates for the dynamics

of model observables and yield accurate information about limit cycling behaviors and basin of

attraction estimates. The organization of this paper is as follows: Section 2 provides necessary

background on Koopman operator theory along with a brief description of associated data-driven

model identiﬁcation techniques including DMD [1], extended DMD [18], and Koopman model

predictive control [26]. Section 3 proposes algorithms for obtaining a nonlinear approximation

for the Koopman operator from snapshot data in both autonomous and controlled systems. The

proposed approach is related to the extended DMD algorithm in that it considers a dictionary of

functions of the observables, however, instead of estimating the action of the Koopman operator

on each of the elements of the dictionary, the explicit nonlinear dependence of the dictionary

elements on the observables is retained. A variety of examples are presented in Section 4. Here,

linear estimators for the Koopman operator are generally able to provide short-term predictions

for the dynamics of observables; comparable nonlinear estimators provide accurate predictions on

substantially longer timescales and accurately identify inﬁnite-time behaviors. Concluding remarks

and suggestions for extension are provided in Section 5.

2 Background

2.1 Koopman Operator Theory

Consider a discrete-time dynamical system

x+=F(x),(1)

where x∈Rnis the state and Fgives the potentially nonlinear dynamics of the mapping x7→ x+.

The Koopman operator K:F → F acts on the vector space of observables so that

Kψ(x)≡ψ(F(x)),(2)

for every ψ:Rn→Rbelonging to the space of observables F. This operator is linear (owing to the

linearity of the composition operator). As such, it can be used to represent the dynamics associated

with a fully nonlinear system. Approaches that use Koopman analysis are distinct from standard

linearization techniques that are only valid in a close neighborhood of some nominal solution.

Note that while the Koopman operator is linear, it is generally inﬁnite-dimensional [15], [16], [17].

In practical applications, the critical challenge of Koopman analysis is in the identiﬁcation of a

ﬁnite-dimensional approximation of the Koopman operator.

2.2 Finite Dimensional Approximation of the Koopman Operator

Dynamic mode decomposition (DMD) [1], [3], [27] is one standard approach for identifying a ﬁ-

nite dimensional approximation of the Koopman operator. To summarize this algorithm, one can

consider a series of data snapshots

si= (g(xi), g(x+

i)),(3)

for i= 1, . . . , d where g(x)∈Rm=ψ1(x), . . . , ψm(x)is a set of observables obtained from the

data and x+

i=F(xi). The goal of DMD is to identify a linear dynamical system of the form

i=Agi,(4)

where gi=g(xi), g+

i=g(x+

i), and A∈Rm×mmaps the observables from one time step to the

next. Such an estimate can be found according to a least-squares optimization,

A=X+X†,(5)

where X≡[g1. . . gd], X+≡[g+

1. . . g+

d], and †denotes the pseudoinverse. As a slight modiﬁcation,

instead of taking the pseudoinverse of Xas in Equation (5), it is often desirable to obtain a lower

rank representation by ﬁrst taking the singular value decomposition of Xand truncating terms

associated with low magnitude singular values [5], [28]. Notice that the DMD algorithm as described

above does not require knowledge of the underlying state and as such, can be implemented in a

purely data-driven setting. DMD often struggles in applications where few observables are available,

i.e., when mis small. In such cases, extended DMD (EDMD) can be used [18], which considers a

lifted observable space

h(x) = g(x)

flift(g(x))∈Rm+b,(6)

where flift(g(x)) ∈Rbis a possibly nonlinear function of the observables called a ‘dictionary’. As

before, letting hi=h(xi) and h+

i=h(x+

i) comprise snapshot pairs with

H≡h1. . . hd,

H+≡h+

1. . . h+

d,(7)

an estimate for the Koopman operator using the lifted coordinates can be obtained according

to Alift =H+H†. The EDMD approach can provide more accurate estimates of the Koopman

operator than the standard DMD approach. Indeed, in some cases the estimated Koopman oper-

ator converges to the true Koopman operator in the limit as both the lifted state and number of

measurements approach inﬁnity [29], [30]. Possible choices of lifted coordinates include polynomi-

als, radial basis functions, and Fourier modes [18]. Additionally, delay embeddings of time series

measurements of observables [28], [19] have also yielded useful results in a variety of applications.

2.3 Koopman-Based Model Identiﬁcation With Control

Koopman-based approaches can readily be generalized to actuated systems [26, 31, 32]. Following

the approach suggested in [26], consider a controlled dynamical system

x+=F(x, u),(8)

with output also given by Equation (2). The above equation is identical to (1) with the incorporation

of a control input u∈Rq⊂ U. Following the approach from [26], one can deﬁne an extended state

space that is the product of the original state space Rnand the space of all input sequences

l(U) = {(ui)∞

i=0|ui∈ U}. Deﬁning an observable φ:Rn×l(U)→Rbelonging to a space of

observables H, the nonautonomous Koopman operator K:H → H can be deﬁned according to

Kφ(x, (ui)∞

i=0) = φ(F(x, u0),(ui)∞

i=1).(9)

Leveraging the EDMD algorithm, an estimate for the nonautonomous Koopman operator can be

obtained by deﬁning a vector of lifted coordinates

p(xi) = 



g(xi)

flift(g(xi))

ui

,(10)

and determining an estimate for the linear dynamical system p(x+

i) = Acp(xi), where Ac∈

R(m+b+q)×(m+b+q). As noted in [26], one is generally not interested in predicting the last qcompo-

nents of p(x+

i), i.e., those associated with the control input. As such, the estimation of the ﬁnal

qrows of Accan be neglected. Letting ¯

Acorrespond the ﬁrst m+brows of Ac. Partitioning

A=A Bwith A∈R(m+b)×(m+b)and B∈R(m+b)×q, a linear, ﬁnite dimensional approximation

of the Koopman operator can be obtained using a series of snapshot triples

wi= (hi, h+

i, ui),(11)

for i= 1, . . . , d. Recall that hiand h+

iwere deﬁned below Equation (6). Once again, deﬁning H

and H+as in (7) and letting Υ = u1. . . ud, an estimate for ¯

Acan be obtained according to

A=A B=H+H

Υ†

,(12)

ultimately yielding the state space representation

i=Ahi+Bui.(13)

Using the above equation, the evolution of the observables can be recovered from the ﬁrst mentries

of h(x).

3 Nonlinear Approximations of the Koopman Operator

3.1 Nonlinear Predictors For Autonomous Systems

The estimation strategies summarized in Sections 2.2 and 2.3 yield linear models, for instance, of

the form (4) and (13). The strategy detailed below allows for additional nonlinear terms in the

prediction of the dynamics. To begin, consider an unperturbed, discrete time dynamical system of

the form (1) with observables g(x)∈Rm. Leveraging the delayed embedding approaches considered

in [28] and [19], one can deﬁne a lifted state

γi=





h(xi)

h(xi−1)

h(xi−z)







,(14)

where z∈Ndetermines the length of the delayed embedding and h(x) was deﬁned in Equation

(6). Here, γi∈RMwith M= (z+ 1)(m+b). Next, a secondary lifting is deﬁned

σi=γi

fn(γi),(15)

where fn(γi)∈RLis an additional, generally nonlinear function of the lifted state γi. The term

fnrepresents an additional user speciﬁed lifting of the data. For example, these terms can be

comprised of polynomials, radial basis functions, and Fourier modes [18]. Letting σiand σ+

ibe

the lifted coordinates on successive iterations, a direct implementation of the EDMD algorithm

detailed in Section 2.2 would seek a matrix Athat solves

min

A"d

i=1 ||σ+

i−Aσi||F#,(16)

for a collection of data (σi, σ+

i) for i= 1, . . . , d where || ·||Fdenotes the Frobenius norm. Alterna-

tively, one can instead neglect the prediction of the ﬁnal Lstates because they are direct functions

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NonlinearData-DrivenApproximationoftheKoopmanOperatorDanWilson*11DepartmentofElectricalEngineeringandComputerScience,UniversityofTennessee,Knoxville,TN37996,USAOctober11,2022AbstractKoopmananalysisprovidesageneralframeworkfromwhichtoanalyzeanonlineardynami-calsystemintermsofalinearoperatoractingonan...

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Nonlinear Data-Driven Approximation of the Koopman Operator Dan Wilson1 1Department of Electrical Engineering and Computer Science University of Tennessee

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