数据驱动的Koopman谱分析：选择关键观测器与kernel方法
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本文主要探讨了Koopman理论在非线性偏微分方程（PDE）中的应用，特别是在动态模式分解（DMD）算法中的角色。Koopman理论的核心在于通过构建Koopman算子来捕捉系统复杂的非线性动态行为。在这个过程中，选择用于构造Koopman算子的可观测函数（observables）至关重要，因为它们直接影响到逼近非线性动力学的准确性。
选取适当的可观测函数是关键挑战，因为它们不仅影响DMD算法的计算效率，还决定着得到的Koopman特征值、特征函数（即Koopman模）的质量。文章提出了一种基于核函数的数据驱动方法，这种方法避免了因需要大量基函数来填充足够丰富的可观测空间而产生的计算问题。通过选择核函数，可以隐式地定义一组可观测函数，从而简化计算并提高Koopman算子的近似精度。
作者Matthew O. Williams, Clarence W. Rowley和Ioannis G. Kevrekidis着重展示了如何通过选择核函数以及构建Koopman算子来处理几个经典非线性PDE的例子，包括 Burger's方程、非线性薛定谔方程、Cubic-Quintic Ginzburg-Landau方程和反应扩散系统。这些示例展示了通过适当观测函数选择和Koopman算子构建，能够揭示复杂系统在时间和空间上的物理可解释特征，并且与流形学习方法建立了联系。
该研究的价值在于它提供了一个有用的中间步骤，即既具有可解释性又相对易于计算的Koopman算子近似，它介于原始的DMD方法和计算密集型的扩展DMD（EDMD）之间。通过这种方法，研究人员能够在高维状态空间中有效地进行Koopman谱分析，为理解和控制复杂的动态系统提供了新的工具。然而，文中也强调了当前面临的挑战：如何找到一种原则性的方法来选择最合适的可观测函数，这是Koopman理论能否在实际问题中发挥其潜力的关键所在。
4 M.O. WILLIAMS, C.W. ROWLEY, AND I.G. KEVREKIDIS
2.2.
Extended Dynamic Mode Decomposition.
Extended Dynamic Mode De-
composition (Extended DMD) [
7
] is a regression procedure whose solution produces
a finite-dimensional approximation of the Koopman operator. To obtain this ap-
proximation, we define a basis set that consists of
K
scalar observables, which we
denote as
ψ
k
for
k
= 1
, . . . , K
, that span
F
K
⊂ F
. We also define the vector valued
observable, ψ : M → R
K
, where
ψ(x) =
ψ
1
(x)
ψ
2
(x)
.
.
.
ψ
K
(x)
. (3)
In this application,
ψ
is the mapping from physical space to feature space. Any
φ,
ˆ
φ ∈ F
K
can be written as
φ =
K
X
k=1
a
k
ψ
k
= ψ
T
a,
ˆ
φ =
K
X
k=1
ˆa
k
ψ
k
= ψ
T
ˆ
a, (4)
for some set of coefficients
a,
ˆ
a ∈ C
K
. Although
F
is unknown, we assume access to
a data set of snapshot pairs:
{(x
m
, y
m
)}
M
m=1
, y
m
= F (x
m
), (5)
where
x
m
, y
m
∈ M
. One important special case of such a data set is a single
time-series of data, which can be written in the above form by “grouping” sequential
pairs of snapshots that were obtained with a fixed sampling interval, say ∆t.
Let
ˆ
φ
=
Kφ
+
r
, where
r ∈ F
is a residual function that appears because
F
K
is not necessarily invariant to the action of the Koopman operator. Using the
notation in
(4)
, the objective of the Extended DMD procedure [
7
] is to define a
mapping from some given
a
to a new vector
ˆ
a
that minimizes this residual. Because
the Koopman operator is linear, this mapping can be represented by a matrix
K ∈ R
K×K
. To determine the entries of
K
, the Extended DMD approach takes
ideas from collocation methods typically used to solve PDEs, but uses the
x
m
as
the collocation points rather than a pre-determined grid [
24
,
25
]. As a result, the
finite dimensional approximation is
K , Ψ
+
x
Ψ
y
, (6a)
where
Ψ
x
,
ψ(x
1
)
T
ψ(x
2
)
T
.
.
.
ψ(x
M
)
T
, Ψ
y
,
ψ(y
1
)
T
ψ(y
2
)
T
.
.
.
ψ(y
M
)
T
, (6b)
are in R
M×K
, and + denotes the pseudoinverse.
In Williams et al.
[7]
, the relationship
Ψ
+
x
= (
Ψ
T
x
Ψ
x
)
+
Ψ
T
x
was used to rewrite
(6) as
K , G
+
A, (7a)
G = Ψ
T
x
Ψ
x
=
M
X
m=1
ψ(x
m
)ψ(x
m
)
T
, A = Ψ
T
x
Ψ
y
=
M
X
m=1
ψ(x
m
)ψ(y
m
)
T
, (7b)
which is advantageous when the number of snapshots is much larger than the number
of basis functions, i.e.,
M K
, because each term in the sum can be evaluated
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