# Koopman Operator Theory
for Infinite-Dimensional Systems:
Extended Dynamic Mode Decomposition and Identification of Nonlinear PDEs

A. Mauroy  Department of Mathematics and Namur Institute for Complex Systems
(naXys), University of Namur, Belgium
(e-mail: alexandre.mauroy@unamur.be).

###### Abstract

We consider Koopman operator theory in the context of nonlinear infinite-
dimensional systems, where the operator is defined over a space of nonlinear
functionals. The properties of the Koopman semigroup are described and a
finite-dimensional projection of the semigroup is proposed, which provides a
linear finite-dimensional approximation of the underlying infinite-dimensional
dynamics. This approximation is used to obtain spectral properties from data,
a method which can be seen as a generalization of Extended Dynamic Mode
Decomposition for infinite-dimensional systems. Finally, we exploit the
proposed framework to identify (a finite-dimensional approximation of) the Lie
generator associated with the Koopman semigroup. This approach yields a linear
method for nonlinear PDE identification, which is complemented with
theoretical convergence results.

###### keywords:

Koopman operator, infinite-dimensional systems, partial differential
equations, spectral analysis, nonlinear identification

##  1 Introduction

Koopman operator theory is a powerful framework which provides an alternative
approach to dynamical systems. Through the so-called Koopman (or composition)
operator (Koopman (1931)), nonlinear dynamical systems can be interpreted as
linear, higher-dimensional systems that are amenable to systematic analysis.
Under the impulse of the seminal work Mezić (2005), the Koopman operator
framework has grown in popularity over the last decade in dynamical systems
theory, and more recently has attracted attention in control theory (see
Mauroy et al. (2020) and references therein). However, to the author’s
knowledge, the research effort has focused on finite-dimensional systems, and
little work has been devoted to infinite-dimensional dynamical systems, such
as nonlinear partial differential equations (PDEs). In this context, one can
mention the early work by Banks (1985), which defined (an equivalent of) the
Koopman operator on a separable $L^{2}$ space of functionals (themselves
defined on a Hilbert space). Later on, Dorroh and Neuberger (1996) followed a
different path, defining the composition operator in the space of continuous
functionals and investigating the properties of the associated Lie generator.
In the very recent work by Farkas and Kreidler (2020), this approach was
pushed further and leveraged in the framework of strongly continuous
semigroups. Finally, it is also recently that the Koopman operator framework
has been considered to study the spectral properties of nonlinear PDEs (Mezić
(2020); Nakao and Mezić (2020)).

In this paper, we further exploit and develop the Koopman operator framework
for systems described by infinite-dimensional differential equations. We do
not rely on strongly continuous semigroup theory, but rather adopt the
approach proposed by Dorroh and Neuberger (1996). In this context, the Lie
generator is shown to be related to a Gâteaux derivative and a finite-
dimensional approximation of the Koopman operator is presented, which allows
to approximate a nonlinear, infinite-dimensional system by a linear, finite-
dimensional one. Based on this framework, our main contributions are twofold.
First, we complement the spectral analysis in Nakao and Mezić (2020) by
developing a data-driven method to compute the spectral properties of the
Koopman operator. This method is a generalization of the Extended Dynamic Mode
Decomposition (EDMD) method (Williams et al. (2015)) for infinite-dimensional
systems and is more general than the mere application of standard EDMD to
discretized PDEs. Second, combining our previous work (Mauroy and Goncalves
(2020)) with the developed framework, we propose a novel method for nonlinear
identification of infinite-dimensional systems. This method relies on linear
techniques, does not require to evaluate time derivatives, and is complemented
with theoretical convergence results. It is an indirect alternative method to
recent direct methods for data-driven discovery of nonlinear PDEs (Rudy et al.
(2017); Long et al. (2018); Li et al. (2019); Gurevich et al. (2019)).

The rest of the paper is organized as follows. In Section 2, we introduce the
Koopman operator framework for infinite-dimensional systems, with a focus on
the Lie generator and finite-dimensional approximation. Section 3 is devoted
to spectral analysis and presents the generalized Extended Dynamic Mode
Decomposition (EDMD) method. The identification method for infinite-
dimensional systems is presented in Section 4 along with theoretical
convergence results and numerical examples. Finally, concluding remarks are
given in Section 5.

##  2 Koopman operator theory for infinite-dimensional systems

###  2.1 Koopman semigroup

We consider (infinite-dimensional) dynamical systems of the form

$\dot{u}=W(u)\,,\qquad u\in\mathcal{U}$ (1)

where $\mathcal{U}$ is a (separable) Hilbert space and
$W:\mathcal{D}(W)\to\mathcal{U}$ is a nonlinear operator, with
$\mathcal{D}(W)$ the domain of $W$. If the system is described by a PDE, then
$W$ is typically a differential operator. Moreover, we assume that $W$
generates a (nonlinear) semiflow $(\varphi_{t})_{t\geq
0}:\mathcal{U}\to\mathcal{U}$, i.e. $u(t)=\varphi^{t}(u)$ is a classical
solution to the abstract differential equation (1) associated with the initial
condition $u_{0}\in\mathcal{D}(W)$. We also make the standing assumption that
each map $\varphi^{t}:\mathcal{U}\to\mathcal{U}$, $t\geq 0$, is continuous and
that the mapping $t\mapsto\varphi^{t}(u)$ is continuous from $\mathbb{R}^{+}$
into $\mathcal{U}$ (strong continuity).

The semigroup of Koopman operators (or Koopman semigroup in short) associated
with (1) is defined on a space of “observable-functionals.”

######  Definition 1 (Koopman semigroup)

Consider the space $\mathcal{E}$ of complex-valued functionals
$\xi:\overline{\mathcal{U}}\to\mathbb{C}$, where the definition domain
$\overline{\mathcal{U}}\subset\mathcal{D}(W)$ is invariant under
$\varphi^{t}$. The semigroup of Koopman operators $(U^{t})_{t\geq 0}$
associated with the semiflow $(\varphi_{t})_{t\geq 0}$ is defined by
$U^{t}\xi=\xi\circ\varphi^{t}$, $\xi\in\mathcal{E}$. $\diamond$

In the rest of the paper, $\mathcal{E}$ is the space of continuous bounded
functionals, endowed with the supremum norm
$\|\xi\|=\sup_{u\in\overline{\mathcal{U}}}|f(u)|$, i.e.
$\mathcal{E}=C(\overline{\mathcal{U}})$.

###  2.2 Lie generator

Following the work by Dorroh and Neuberger (1996), we define the Lie generator
of the Koopman semigroup.

######  Definition 2 (Lie generator)

The Lie generator of the semigroup $(U^{t})_{t\geq 0}$ is the linear operator
$L:\mathcal{D}(L)\to\mathcal{E}$ that satisfies

$L\xi(u)=\lim_{t\downarrow
0}\frac{U^{t}\xi(u)-\xi(u)}{t}\qquad\xi\in\mathcal{D}(L)$ (2)

for all $u\in\overline{\mathcal{U}}$. $\diamond$

######  Remark 1 (Infinitesimal generator)

The Lie generator is reminiscent of the infinitesimal generator of strongly
continuous semigroups (Engel and Nagel (1999)). However, the limit in (2) is
defined pointwise, while it is defined in the strong sense in the case of the
infinitesimal generator. In fact, the infinitesimal generator of the Koopman
semigroup cannot be defined unless the Koopman semigroup is strongly
continuous (i.e. $\lim_{t\downarrow 0}\|U^{t}\xi-\xi\|=0$
$\forall\xi\in\mathcal{E}$). This property does not hold in our setting, but
could be valid with the mixed topology on $C(\overline{\mathcal{U}})$, as
shown in Farkas and Kreidler (2020). $\diamond$

The Lie generator enjoys a few properties (e.g. dense domain, bounded
resolvent) and we refer to Dorroh and Neuberger (1996) for more details. In
the case of a Koopman semigroup associated with a semiflow generated by the
abstract differential equation (1), it follows from the chain rule property
that the Lie generator is given by

$L\xi(u)=\lim_{t\downarrow
0}\frac{\xi(\varphi^{t}(u))-\xi(u)}{t}=D_{D_{t}\varphi^{t}(u)}\xi(u)=D_{W(u)}\xi(u)$

where $D_{W(u)}\xi(u)$ denotes the Gâteaux derivative of $\xi$ at $u$ in the
direction $W(u)$:

$D_{W(u)}\xi(u)=\lim_{\lambda\rightarrow 0}\frac{\xi(u+\lambda
W(u))-\xi(u)}{\lambda}\,.$

Note that this can be interpreted as the Lie derivative associated with the
infinite-dimensional vector field $W(\cdot)$.

###  2.3 Finite-dimensional representation

It is convenient to approximate the Koopman semigroup $U^{t}$ or the Lie
generator $L$ in a finite-dimensional subspace of $\mathcal{E}$. Toward that
end, we can consider the compressions
$U_{n}^{t}=P_{n}U^{t}|_{\mathcal{E}_{n}}$ and
$L_{n}=P_{n}L|_{\mathcal{E}_{n}}$, where
$\mathcal{E}_{n}\subset\mathcal{D}(L)$ is a $n$-dimensional subspace of
$\mathcal{E}$ and $P_{n}:\mathcal{E}\to\mathcal{E}_{n}$ is a projection
operator. Suppose that $\mathcal{E}_{n}$ is spanned by the basis of
functionals $\\{\xi_{i}\\}_{i=1}^{n}$. In this basis, the finite-dimensional
operators $U^{t}_{n}$ and $L_{n}^{t}$ can be represented by the matrices
$\mathbf{U}\in\mathbb{R}^{n\times n}$ and $\mathbf{L}\in\mathbb{R}^{n\times
n}$, respectively, which are defined so that

$U^{t}_{n}\xi_{j}=\sum_{i=1}^{n}\mathbf{U}_{ij}\xi_{i}\quad\textrm{and}\quad
L_{n}\xi_{j}=\sum_{i=1}^{n}\mathbf{L}_{ij}\xi_{i}\,.$ (3)

The choice of the basis functions is crucial, as it affects the accuracy of
the approximation and the performance of the methods based on this
approximation (see below). However, finding the optimal set of basis functions
is not trivial, and may require a priori knowledge on the system.

##  3 Spectral analysis and Extended Dynamic Mode Decomposition

The spectral properties of the Koopman operator reveal important geometric
properties of the underlying dynamics (see e.g. Mezić (2005), and Mauroy and
Mezić (2012); Nakao and Mezić (2018) in the context of phase reduction). In
this section, we exploit the proposed framework for infinite-dimensional
systems and compute the spectrum of the Koopman operator from data. This
yields a generalization of the Extended Dynamical Mode Decomposition method
for infinite-dimensional systems.

###  3.1 Spectrum of the Koopman operator

We consider the spectrum of the Lie generator (2), i.e. the set of (Koopman)
eigenvalues $\lambda$ such that $L\xi_{\lambda}=\lambda\,\xi_{\lambda}$ for
some (Koopman) eigenfunctional $\xi_{\lambda}\in\mathcal{E}$.

#### Linear systems.

It is well-known that the spectrum of linear finite-dimensional systems is
contained in the spectrum of the related Koopman operator. As shown in Mezić
(2020) and Nakao and Mezić (2020), this result also holds for infinite-
dimensional systems. Consider a linear system $\dot{u}=Au$, $u\in\mathcal{U}$,
and suppose that $\lambda$ is an eigenvalue of $A$, so that there exists an
eigenfunction $w_{\lambda}\in\mathcal{D}(A)$ with
$A^{*}w_{\lambda}=\bar{\lambda}w_{\lambda}$, where $A^{*}$ denotes the adjoint
operator of $A$ and $\bar{\lambda}$ is the complex conjugate of $\lambda$.
Then the functional $\xi_{\lambda}(\cdot)=\langle\cdot,w_{\lambda}\rangle$
satisfies

$\begin{split}L\xi_{\lambda}(u)=D_{Au}\langle u,w_{\lambda}\rangle=\langle
Au,w_{\lambda}\rangle=\langle u,A^{*}w_{\lambda}\rangle&=\langle
u,\bar{\lambda}w_{\lambda}\rangle\\\
&=\lambda\,\xi_{\lambda}(u)\,,\end{split}$

so that $\lambda$ is a Koopman eigenvalue. Moreover, it is easy to verify that
$(\xi_{\lambda})^{\alpha}$ is a Koopman eigenfunctional associated with the
Koopman eigenvalue $\alpha\lambda$, provided that it belongs to the space
$\mathcal{E}$. If $A$ generates a strongly continuous (linear) semigroup
$(\varphi^{t})_{t\geq 0}=(T^{t})_{t\geq 0}$, then the spectral mapping theorem
implies that $e^{\lambda t}$ is an eigenvalue of the operator $T^{t}$ (see
e.g. Engel and Nagel (1999)). It follows that

$U^{t}\xi_{\lambda}=\langle T^{t}u,w_{\lambda}\rangle=\langle
u,(T^{t})^{*}w_{\lambda}\rangle=\langle
u,e^{\bar{\lambda}t}w_{\lambda}\rangle=e^{\lambda t}\,\xi_{\lambda}$

and $e^{\lambda t}$ is also an eigenvalue of $U^{t}$.

###  3.2 Extended Dynamic Mode Decomposition for
infinite-dimensional systems

Extended Dynamic Mode Decomposition (EDMD) is a data-driven method that builds
a finite-dimensional approximation of the Koopman semigroup and computes the
approximate spectral properties of the operator (Williams et al. (2015)). It
can be easily extended to the infinite-dimensional framework that we consider
here.

Suppose we are given a set of $m$ pairs
$(u_{k},\varphi^{t_{s}}(u_{k}))\in\overline{\mathcal{U}}^{2}$, where $t_{s}$
is a sampling time, in such a way that we can measure the values of $n$
functionals
$(\xi_{i}(u_{k}),\xi_{i}(\varphi^{t_{s}}(u_{k})))\in\mathbb{R}^{2}$ for all
$i\in\\{1,\dots,n\\}$. The generalized EDMD method proceeds with the following
steps.

  1. 1.

Compute the data matrices

$\mathbf{\Xi}_{1}=\left(\begin{array}[]{ccc}\xi_{1}(u_{1})&\cdots&\xi_{n}(u_{1})\\\
\vdots&&\vdots\\\ \xi_{1}(u_{m})&\cdots&\xi_{n}(u_{m})\end{array}\right)$ (4)

and

$\mathbf{\Xi}_{2}=\left(\begin{array}[]{ccc}\xi_{1}(\varphi^{t_{s}}(u_{1}))&\cdots&\xi_{n}(\varphi^{t_{s}}(u_{1}))\\\
\vdots&&\vdots\\\
\xi_{1}(\varphi^{t_{s}}(u_{m}))&\cdots&\xi_{n}(\varphi^{t_{s}}(u_{m}))\end{array}\right)\,.$
(5)

  2. 2.

Provided that $m\geq n$, a matrix approximation of
$U^{t_{s}}_{n}=P_{n}U^{t_{s}}|_{\mathcal{E}_{n}}$ is given by the least
squares solution $\mathbf{U}=\mathbf{\Xi}_{1}^{\dagger}\mathbf{\Xi}_{2}$,
where $\mathbf{\Xi}^{\dagger}$ denotes the Moore-Penrose pseudoinverse of
$\mathbf{\Xi}$. Note that, in this case, $P_{n}$ is the discrete orthogonal
projection

$P_{n}\xi=\underset{\tilde{\xi}\in\mathcal{E}_{n}}{\textrm{argmin}}\sum_{k=1}^{m}|\xi(u_{k})-\tilde{\xi}(u_{k})|^{2}\,.$
(6)

  3. 3.

The eigenvalues $\lambda_{U}$ of $U^{t_{s}}$ are approximated by the
eigenvalues of $\mathbf{U}$ and estimates of the eigenvalues $\lambda_{L}$ of
the Lie generator are given by $\lambda_{L}=\log(\lambda_{U})/t_{s}$.
Moreover, Koopman eigenfunctionals are approximated in the basis of
functionals by the components of the corresponding (right) eigenvectors of
$\mathbf{U}$.

The method only requires to know the samples $u_{k}$ in a weak sense, i.e.
through the values $\xi_{j}(u_{k})$ of a finite number of functionals. For
instance, these values could be the weighted averages of
$u_{k}:X\to\mathbb{R}$ over the definition domain $X$. For the specific choice
of evaluation functionals $\xi_{j}(u)=u(x_{j})$ with $x_{j}\in X$, we recover
the classical DMD method applied to a discretized version of the infinite-
dimensional system (1).

######  Remark 2 (Convergence properties)

The EDMD method must be used with some care since it might yield spurious
eigenvalues. Similarly to Korda and Mezić (2018), convergence properties
should be characterized as $m,n\rightarrow\infty$ and the general validity of
the spectral mapping theorem should be investigated. We leave these questions
for future research. $\diamond$

###  3.3 Numerical example

We illustrate the generalized EDMD method with the Burgers equation

$\dot{u}=-u\frac{\partial u}{\partial x}+\frac{\partial^{2}u}{\partial
x^{2}}\qquad u\in L^{2}([-1,1])$ (7)

associated with homogeneous Dirichlet boundary conditions $u(-1)=u(1)=0$. The
dynamics (7) is conjugated to the linear diffusion dynamics
$\dot{v}=\partial^{2}v/\partial t^{2}$ through the so-called Cole-Hopf
transformation (Hopf (1950)). As explained in Nakao and Mezić (2020); Page and
Kerswell (2018), this implies that the Koopman spectrum associated with the
Burgers dynamics coincides with the Koopman spectrum associated with the
diffusion dynamics and it follows that Koopman eigenfunctionals are of the
form $\xi(v)=\langle sin(k\pi x/2),v(x)\rangle^{\alpha}$ (in the new variable
$v$) and associated with the Koopman eigenvalues $\lambda=\alpha(k\pi/2)^{2}$.

We use the dynamics (7) to generate $m=50$ data-pairs taken from $10$
trajectories (with random, arbitrarily chosen initial conditions of the form
$(x^{2}-1)\cos(a\pi x+b\pi)$, $a,b\in[0,1]$). The sampling time is
$t_{s}=0.2$. Estimates of the Koopman eigenvalues are computed through the
EDMD method, with $n=27$ basis functionals $\xi=\langle\cos(a_{j}(\pi
x/2)+b_{j}\pi/2),(u(x))^{k}\rangle^{l}$, with $(j,k,l)\in\\{1,2,3\\}^{3}$ and
where $a_{j},b_{j}$ are randomly chosen over the interval $[0,1]$. As shown in
Fig. 1, dominant eigenvalues $-\alpha(\pi/2)^{2}$, $\alpha\in\mathbb{N}$, are
captured. This is consistent with the results presented in Page and Kerswell
(2018) and shows the importance of selecting an augmented basis of nonlinear
functionals to capture other eigenvalues than the principal ones of the form
$(k\pi/2)^{2}$, $k\in\mathbb{N}$. Note that other (complex) eigenvalues may
appear over different numerical tests. They should be considered in light of
future theoretical analysis (see Remark 2), and more advanced methods should
be proposed to distinguish true eigenvalues from spurious ones.

Figure 1: The generalized EDMD method is used to compute the Koopman
eigenvalues associated with the Burgers dynamics (7). Dominant eigenvalues
$-\alpha(\pi/2)^{2}$, $\alpha\in\mathbb{N}$ (blue circles) are correctly
estimated (red crosses).

##  4 Identification of infinite-dimensional systems

In this section, we use the Koopman operator framework for infinite-
dimensional systems in the context of identification. Our goal is to identify
the coefficients $c_{i}\in\mathbb{R}$ of the infinite-dimensional dynamics

$\dot{u}=W(u)=\sum_{i}c_{i}W_{i}(u)\,,$ (8)

with $u\in\mathcal{U}=L^{2}(X)$ (where $X\subset\mathbb{R}^{p}$ is a compact
set), using $m$ data pairs $(u_{k},\varphi^{t_{s}}(u_{k}))$ generated by the
dynamics (8). We assume that the operators
$W_{i}:\mathcal{D}(W_{i})\to\mathcal{U}$ are known a priori, so that this can
be seen as a parameter estimation problem. Note that (8) may be described by a
partial differential equation, but the proposed method is not limited to that
case (see Section 4.3).

###  4.1 Lifting identification method

The lifting identification method proposed in our previous work (Mauroy and
Goncalves (2020)) is generalized to the case of infinite-dimensional systems.
This method consists in three steps.

  1. 1.

_Lifting of the data._ We compute the data matrices (4) and (5) with the basis
of functionals

$\xi_{i}(u)=\langle W_{i}(u),w\rangle$ (9)

where $\langle\cdot,\cdot\rangle$ denotes the inner product in $L^{2}(X)$ and
$w\in L^{2}(X)$ is a weighting function. We suppose without loss of generality
that $W_{1}(u)=u$ (possibly with the coefficient $c_{1}=0$ in (8)), so that
the linear functional $\xi_{1}(u)=\langle u,w\rangle$ belongs to
$\mathcal{E}_{n}$.

  2. 2.

_Identification of the Lie generator._ We compute the matrix representation
$\mathbf{U}=\mathbf{\Xi}_{1}^{\dagger}\mathbf{\Xi}_{2}$ of the compression
$U_{n}^{t_{s}}$ in the subspace
$\mathcal{E}_{n}=\textrm{span}(\xi_{1},\dots,\xi_{n})$ (step 2 in Section
3.2). Then we obtain the finite approximation $\widetilde{\vec{L}}^{(t_{s})}$
of the Lie generator by taking the matrix logarithm

$\widetilde{\vec{L}}^{(t_{s})}=\frac{1}{t_{s}}\log(\mathbf{\Xi}_{1}^{\dagger}\mathbf{\Xi}_{2})\,.$
(10)

Note that this approximation is not equal to the matrix representation
$\mathbf{L}$ of the $L_{n}$.

  3. 3.

_Identification of the coefficients._ Estimates $\hat{c}_{i}$ of the
coefficients $c_{i}$ are given by the entries of the first column of
$\widetilde{\vec{L}}^{(t_{s})}$, i.e.
$\hat{c}_{i}=\widetilde{\vec{L}}^{(t_{s})}_{i1}$.

The proposed method does not require to compute time derivatives and is
therefore an alternative to the direct methods proposed in Rudy et al. (2017);
Li et al. (2019). It is also noticeable that the use of basis functionals of
the form (9) bears similarity to the method proposed in Gurevich et al.
(2019), which makes a clever use of a weak formulation of the data in space
and time. We note that this method requires sufficiently long time-series to
allow accurate time integration, while our method can deal with independent
data pairs.

###  4.2 Convergence results

Now we show that the estimated coefficients converge to the true coefficients
as $t_{s}\rightarrow 0$. This is summarized in the following proposition.

###### Proposition 1

Let $\varphi^{t}$ be the continuous flow generated by (8) and let
$\mathcal{E}_{n}\subset\mathcal{D}(L)$ be the subspace spanned by the basis
functionals $\xi_{i}(\cdot)=\langle W_{i}(\cdot),w\rangle\in
C(\overline{\mathcal{U}})$, with $\xi_{1}(\cdot)=\langle\cdot,w\rangle$.
Assume that the pairs
$(u_{k},\varphi^{t_{s}}(u_{k}))\in\overline{\mathcal{U}}^{2}$ are such that
the $n\leq m$ vectors $(\xi_{i}(u_{1}),\dots,\xi_{i}(u_{m}))$, $i=1,\dots,n$,
are linearly independent. Then,

$\lim_{t_{s}\rightarrow 0}\widetilde{\vec{L}}^{(t_{s})}_{i1}=c_{i}$

where $\widetilde{\vec{L}}^{(t_{s})}$ is given by (10).

The discrete orthogonal projection $P_{n}$ (6) is well-defined since the
vectors $(\xi_{i}(u_{1}),\dots,\xi_{i}(u_{m}))$, $i=1,\dots,n$, are linearly
independent.

It is clear that

$\frac{1}{t_{s}}\log
U_{n}^{t_{s}}\xi_{1}(u)=\sum_{i=1}^{n}\widetilde{\mathbf{L}}_{i1}^{(t_{s})}\xi_{i}(u)$

and we also have

$L_{n}\xi_{1}(u)=P_{n}D_{W(u)}\langle u,w\rangle=P_{n}\langle
W(u),w\rangle=\sum_{i=1}^{n}c_{i}\xi_{i}(u)\,.$ (11)

Since $P_{n}\xi_{1}=\xi_{1}$, it follows that

$\begin{split}\left|\sum_{i=1}^{n}\left(\widetilde{\vec{L}}^{(t_{s})}_{i1}-c_{i}\right)\xi_{i}(u)\right|&=\left|\left(\frac{1}{t_{s}}\log
U^{t_{s}}_{n}-P_{n}LP_{n}\right)\xi_{1}(u)\right|\\\
&=\frac{1}{t_{s}}\left|\left(\log U^{t_{s}}_{n}-\log
A_{n}^{t_{s}}\right)\xi_{1}(u)\right|\end{split}$ (12)

where $A^{t_{s}}_{n}=\mathcal{E}_{n}\to\mathcal{E}_{n}$ is the finite-
dimensional operator $A^{t_{s}}_{n}=e^{t_{s}P_{n}LP_{n}}$. Since the basis
functionals $\xi_{i}$ are linearly independent,

$\lim_{t_{s}\rightarrow 0}\frac{1}{t_{s}}|(\log A_{n}^{t_{s}}-\log
U_{n}^{t_{s}})\xi_{1}(u)|=0\qquad\forall u$ (13)

implies that

$\lim_{t_{s}\rightarrow
0}\left|\widetilde{\vec{L}}^{(t_{s})}_{i1}-c_{i}\right|=0\,.$

Thus it remains to show that (13) holds.
Since $\lim_{t_{s}\rightarrow 0}\|A^{t_{s}}_{n}-I\|=0$, it is easy to check
that

$\lim_{t_{s}\rightarrow 0}\frac{1}{t_{s}}\left\|\log
A_{n}^{t_{s}}-(A_{n}^{t_{s}}-I)\right\|=0$ (14)

[note that $\lim_{t\rightarrow 0}(\log f(t)-(f(t)-1))/t=0$ for all $f\in
C^{1}$ such that $\lim_{t\rightarrow 0}f(t)=1$] and it follows similarly that

$\lim_{t_{s}\rightarrow 0}\frac{1}{t_{s}}\left\|\log
U_{n}^{t_{s}}-(U_{n}^{t_{s}}-I)\right\|=0\,.$ (15)

Since the mapping $t\mapsto(A_{n}^{t_{s}}-U_{n}^{t_{s}})\xi(u)$ is
differentiable, the mean value theorem implies that

$\frac{1}{t_{s}}\left|(A_{n}^{t_{s}}-U_{n}^{t_{s}})\xi(u)\right|=\left|\frac{d}{dt}\left(A^{t}_{n}U^{t_{s}-t}\xi(u)\right)|_{t=\tau}\right|$

for all $\xi\in\mathcal{E}_{n}$ and $u\in\overline{\mathcal{U}}$, and for some
$\tau\in[0,t_{s}]$. Then we can write

$\begin{split}\frac{1}{t_{s}}\left|(A_{n}^{t_{s}}-U_{n}^{t_{s}})\xi(u)\right|&=\left|A_{n}^{\tau}(P_{n}LP_{n}-P_{n}L)U^{t_{s}-\tau}\xi(u)\right|\\\
&\leq|A_{n}^{\tau}P_{n}LP_{n}(U^{t_{s}-\tau}\xi(u)-\xi(u))|\\\
&\quad+|A_{n}^{\tau}P_{n}L(P_{n}\xi(u)-\xi(u))|\\\
&\quad+|A_{n}^{\tau}P_{n}L(\xi(u)-U^{t_{s}-\tau}\xi(u))|\\\
&\leq|A_{n}^{\tau}P_{n}LP_{n}(\xi(\varphi^{t_{s}-\tau}(u))-\xi(u))|\\\
&\quad+|A_{n}^{\tau}P_{n}L(\xi(u)-\xi(\varphi^{t_{s}-\tau}(u)))|\end{split}$

where we used the fact that $P_{n}\xi=\xi$ and that $U^{t}$ and $L$ commute.
Since the functionals $A_{n}^{\tau}P_{n}LP_{n}\xi$ and $A_{n}^{\tau}P_{n}L\xi$
are continuous and the flow $\varphi^{t}$ is continuous in $t$, we have

$\begin{split}\lim_{t_{s}\rightarrow
0}|A_{n}^{\tau}P_{n}LP_{n}(\xi(\varphi^{t_{s}-\tau}(u))-\xi(u))|=0\\\
\lim_{t_{s}\rightarrow
0}|A_{n}^{\tau}P_{n}L(\xi(u)-\xi(\varphi^{t_{s}-\tau}(u)))|=0\end{split}$

so that

$\lim_{t_{s}\rightarrow
0}\frac{1}{t_{s}}\left|(A_{n}^{t_{s}}-U_{n}^{t_{s}})\xi(u)\right|=0\,.$ (16)

Finally (14), (15) and (16) yield

$\begin{split}&\lim_{t_{s}\rightarrow 0}\frac{1}{t_{s}}|(\log
A_{n}^{t_{s}}-\log U_{n}^{t_{s}})\xi(u)|\\\ &\leq\lim_{t_{s}\rightarrow
0}\frac{1}{t_{s}}\left(|(\log
A_{n}^{t_{s}}-(A_{n}^{t_{s}}-I))\xi(u)|\right.\\\ &\left.\quad+|(\log
U_{n}^{t_{s}}-(U_{n}^{t_{s}}-I))\xi(u)|+|(A_{n}^{t_{s}}-U_{n}^{t_{s}})\xi(u)|\right)=0\,.\end{split}$

This concludes the proof. $\square$ Although the result requires that the
sampling time $t_{s}$ tend to zero, accurate results can be obtained even if
$t_{s}$ is not so small (see Section 4.3). However, large sampling times might
affect the method performance. A more detailed study of the error convergence
rate as $t_{s}$ tends to zero is left for future research.

When an arbitrary basis of functionals is chosen so that $\langle
W(\cdot),w\rangle\notin\textrm{span}\\{\xi_{1},\dots,\xi_{n}\\}$, we have the
following corollary.

###### Corollary 2

Under the assumptions of Proposition 1, but with $\xi_{i}(\cdot)=\langle
V_{i}(\cdot),w\rangle$, $V_{i}:\mathcal{D}(V_{i})\to\mathcal{U}$, the
estimated nonlinear operator
$\hat{W}^{(t_{s})}=\sum_{i=1}^{n}\widetilde{\vec{L}}^{(t_{s})}_{i1}V_{i}$
satisfies

$\lim_{t_{s}\rightarrow
0}\hat{W}^{(t_{s})}=\underset{\tilde{W}\in\textrm{span}(V_{1},\dots,V_{n})}{\textrm{argmin}}\sum_{k=1}^{m}\left|\left\langle\tilde{W}(u_{k})-W(u_{k}),w\right\rangle\right|^{2}\,.$

It follows from (11), (12) and (13) that

$\lim_{t_{s}\rightarrow
0}\sum_{i=1}^{n}\widetilde{\vec{L}}^{(t_{s})}_{i1}\xi_{i}(\cdot)=P_{n}\langle
W(\cdot),w\rangle\,,$

which implies that

$\lim_{t_{s}\rightarrow
0}\left\langle\hat{W}^{(t_{s})}(\cdot),w\right\rangle=P_{n}\langle
W(\cdot),w\rangle\,.$

The result follows from the definition of the discrete orthogonal projection
(6). $\square$

###### Remark 4.1

The proof of Proposition 1 is adapted from a proof in Mauroy and Goncalves
(2020), but does not rely on semigroup theory. In particular, only weak (i.e.
pointwise) convergence properties are used in the proof of Proposition 1,
while strong convergence is considered in the proof in Mauroy and Goncalves
(2020). For this reason, strong convergence results can be obtained in Mauroy
and Goncalves (2020) in the case of arbitrary bases, but a weaker result is
obtained here (see Corollary 2). $\diamond$

###  4.3 Numerical examples

We can now use the lifting identification method with two illustrating
examples: a nonlinear partial differential equation and a nonlinear diffusive
dynamics on a graphon.

####  4.3.1 Nonlinear partial differential equation.

We aim at identifying the coefficients of the dynamics

$\dot{u}=-2u-0.5(1+u)\frac{\partial u}{\partial
x}+(1-0.2u)\frac{\partial^{2}u}{\partial
x^{2}}+0.1\frac{\partial^{3}u}{\partial x^{3}},$ (17)

$u\in L^{2}[0,5]$, with homogeneous Dirichlet boundary conditions
$u(0)=u(5)=0$. The PDE is used to generate $m=50$ data pairs, taken from $25$
trajectories (with random, arbitrarily chosen initial conditions of the form
$x(x-5)\cos(a\pi x/5+b\pi)$, $a,b\in[0,1]$). The sampling time is $t_{s}=0.3$.
The lifting identification method is used with $n=12$ basis functionals (9),
with the nonlinear operators

$W_{i}(u)\in\left\\{u^{j}\frac{\partial^{k}u}{\partial
x^{k}},\,j\in\\{0,1,2\\},\,k\in\\{0,1,2,3\\}\right\\}$

and the weighting function $w(x)=\exp(-1/(1-(x/L)^{2}))$. Fig. 2 shows that
the coefficients $c_{i}$ are estimated with small error. The results are
compared with a direct identification method where time derivatives $\dot{u}$
are estimated through (forward) finite differences and the coefficients are
obtained through least squares regression of $\langle\dot{u},w\rangle$ over
basis functionals of the form (9).

Figure 2: The lifting identification method is used to recover the PDE (17).
The coefficients $c_{i}$ (blue circles) are estimated with small error (red
crosses). The lifting method outperforms a straightforward direct method based
on finite-difference estimation of time derivatives (green crosses). Nonzero
coefficients correspond to $W_{1}(u)=u$, $W_{4}(u)=\partial u/\partial x$,
$W_{5}(u)=u\,\partial u/\partial x$, $W_{7}(u)=\partial^{2}u/\partial x^{2}$,
$W_{8}(u)=u\,\partial^{2}u/\partial x^{2}$, and
$W_{10}(u)=\partial^{3}u/\partial x^{3}$.

####  4.3.2 Nonlinear diffusive dynamics on a graphon.

The graphon $G:[0,1]^{2}\to[0,1]$ is used to describe the limit of a sequence
of dense graphs, and can be interpreted as the infinite-dimensional version of
an adjacency matrix. Here, we consider a nonlinear diffusive dynamics on the
graphon $G(x,y)=-1+0.7x+0.3y$, which is described by the integro-differential
equation

$\begin{split}\dot{u}=&-0.5u+1.5u^{2}-u^{3}\\\
&+\int_{0}^{1}(-1+0.7x+0.3y)(u(t,y)-u(t,x))\,dy.\end{split}$ (18)

We generate $m=50$ pairs of data, taken from $25$ trajectories (with random
initial conditions of the form $0.1a\cos(b\pi x+b\pi)$, $a,b\in[0,1]$). The
sampling time is $t_{s}=0.5$. The lifting identification method is used with
$n=7$ basis functionals (9), with the operators

$\begin{split}&W_{1}(u)=1,\quad W_{2}(u)=u,\quad W_{3}(u)=u^{2},\quad
W_{4}(u)=u^{3},\\\
&W_{j}(u)(x)=\int_{0}^{1}f_{j}(x,y)\,(u(t,y)-u(t,x))\,dy\quad
j\in\\{5,6,7\\}\\\ \end{split}$

where $f_{5}(x,y)=1$, $f_{6}(x,y)=x$, and $f_{7}(x,y)=y$. The weighting
function is $w(x)=x^{2}$. As shown in Fig. 3, the coefficients are correctly
estimated and, in particular, the graphon is identified. We note however that
some limitations could arise if more complicated graphons were used, due to
the fact that a larger set of basis functionals might not be linearly
independent.

Figure 3: The lifting identification method is used to recover the dynamics
(18) on a graphon. The coefficients $c_{i}$ (blue circles) are correctly
estimated (red crosses).

##  5 Conclusion

In this paper, the Koopman operator framework has been leveraged in the case
of infinite-dimensional nonlinear systems. Building on previous theoretical
works, we have considered a semigroup of composition operators and the
associated Lie generator on a space of continuous functionals. A finite-
dimensional representation of these operators has been proposed and used in
the context of spectral analysis. This approach yields a generalization of the
data-driven EDMD method for infinite-dimensional systems. We have also
developed a novel identification method, which allows to identify nonlinear
PDEs although it relies solely on linear techniques. This method has been
complemented with convergence results.

The Koopman operator framework for infinite-dimensional systems is still in
its infancy. Convergence properties of the finite-dimensional approximation of
the Koopman operator should be thoroughly studied, in particular in light of
the recent results by Farkas and Kreidler (2020) in semigroup theory. This
could provide some insight into the results obtained with the generalized EDMD
method. Moreover, the proposed lifting identification method does not seem to
be robust to noise. In this context, the weak formulation of the method could
be further exploited to enhance noise robustness (see e.g. Gurevich et al.
(2019)) and some guidelines to carefully select the weight functions could be
provided. In future work, the lifting method should also be compared with
state-of-the-art PDE identification methods.

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