Deep Koopman Learning using Noisy Data
Authors: Wenjian Hao
Devesh Upadhyay
Shaoshuai Mou
Venue: Accepted by TMLR ()

Abstract: This paper proposes a data-driven framework to learn a finite-dimensional approximation
of a Koopman operator for approximating the state evolution of a dynamical system under
noisy observations. To this end, our proposed solution has two main advantages. First, the
proposed method only requires the measurement noise to be bounded. Second, the proposed
method modifies the existing deep Koopman operator formulations by characterizing the
effect of the measurement noise on the Koopman operator learning and then mitigating it
by updating the tunable parameter of the observable functions of the Koopman operator,
making it easy to implement. The performance of the proposed method is demonstrated on
several standard benchmarks. We then compare the presented method with similar methods
proposed in the latest literature on Koopman learning.

Published in Transactions on Machine Learning Research (04/2025)
Deep Koopman Learning using Noisy Data
Wenjian Hao
hao93@purduc.cmmmmmmm edu
School of Aeronautics and Astronautics Purdue University
Devesh Upadhyay
devesh. wwwehh..ppddyyy@@sabbbe.cccm
Saab, Inc.
Shaoshuai Mou
mous@pundue.com
School of Aeronautics and Astronautics Purdue University
Reviewed on OpenReview: https: //
// opwnreveew.weww.oom net/ forum? id- 16Rm6T2
Abstract
This paper proposes a data-driven framework to learn a finite-dimensional approximation of a Koopman operator for approximating the state evolution of dynamical system under noisy observations. To this end, our proposed solution has two main advantages. First, the proposed method only requires the measurement noise to be bounded. Second, the proposed method modifies the existing deep Koopman operator formulations by characterizing the effect of the measurement noise on the Koopman operator learning and then mitigating it by updating the tunable parameter of the observable functions of the Koopman operator, making it easy to implement. The performance of the proposed method is demonstrated on several standard benchmarks. We then compare the presented method with similar methods proposed in the latest. literature on Koopman learning.
1 Introduction
Directly dealing with complex nonlinear dynamical systems for model-basee control design has remained a challenge for the control community. One long-standing solution to this problem has been to use linearized models and the associated vast body of knowledge for lincar analysis. Lincar control theory is a very rich and well-developed field that provides rigorous control development with methods for stability and robustness guarantees. Lyapunov showed that for linearized system that is stable around an equilibrium point, there exists region of stability around this cquilibrium point for which the original nonlinear system is also stable A.Lyapunov (1992). Recent advances in data-driven methods have spurred new and increased research interest in machine learning (MI.) based methods for deriving reduced-order models (ROM) as surrogates for complex nonlinear systems. This has also led to the adoption of these methods for developing control and autonomy/ antonomy/auttmaaioon automation solutions for robotic and unmanned systems. Examples include learning dynamics using deep neural networks (DNNs) MMrppyy 2002); Gillespie et al. (2018), physics-informed neural networks
(PINNs) Raissi et al. (2019), and lifting lincarization mothods such as Koopman opcrator methods Koopman
1931 Koopman & Neumann
1932); Mezid [2015); Proctor al. 2018 Mauroy & Goncalves (2016).
Lifting linearization allows one to represent nonlinear system with equivalent linear system in lifted, nigher-dincnsional space. One widely adopted lifting method is the extending dynamic mode decomposition (EDMD) Williams et al. (2015), which lifts the state space to a highee-ddimensional space, for which the temporal evolution approximately linear Korda & Mezić (2018). It is, however, typically difficult to find an exact finite-dimensional lincar representation for most nonlinear systems. Further, the Koopman operator focuses on non-autonomous systems: de f(z, u. u,t). This poses challenge for the control design of dynamical systems in the choice of sufficient basis function necessary for the lifted system to be linear

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Published in Transactions on Machine Learning Research (04/2025)
and exact. Extensions to such systems require truncation-based approximations, and the finite-dimensional representation is no longer exact. To this end, various sigen-deromposition-bssss truncations are proposed.
In Lusch et al. (2017) the authors proposed using deep learning methods to discover the eigenfunctions of the
approximated Koopman operator, and Young ct al. (2019); Lusch et al. (2018); Han et al. (2020); Bevanda et al. p employed DNNs as observable functions of the Koopman operator, which are tuned based on collected state-control pairs by minimizing an appropriately defined loss function which is also referred as the deep Koopman operator method (DKO). Recent work, such as Hao et al. (2024), has extended the DKO method to approximate nonlinear time-varying systems. These methods rely
set of measured
output variables that collectively define some nonlinear representation of the independent state variables. Establishing a sufficient set of these observable functions remains an active area of research. Further, real-world noisy measurements impose additional challenges. Additionally, for most practical systems, it is also critical to find computationally feasible approximation methods for extracting finiteddimensional representations.
Related work. While Koopman-based methods have been proven to be effective in learning dynamics from system's input-output data pairs. However, in practical real-world applications, the output measurements are noisy and can result in biased estimates of the lincar system. Even if the noise of the state variables is assumed to be uncorrelated, the nonlinear transformations in the observables may lead to complex noise- influence correlations between the noise-free states and the transformed observables. Several methods are proposed to solve the measurement noise issue. One solution [Sotiropoulos, (2021) is to directly measure the states and the observables, this, however, may not always be possible. Noisy measurements are also shown to further complicate the anti-causal observable problem when dealing with the lifting of controlled systems Sclby (2021). In other approaches, authors in Dawson ct al. (2016); Hemati al. (2017) introduce total least square (TLS) methods in DMD, in Haseli & Cortés (2019) the authors propose a combination of the EDMD and TLS methods to account for the measurement noise, and in Sinha et al. (2020); Wanner & Mezic (2022): Sinha et al. (2023) the authors propose to solve the EDMD with measurement noise as a robust Koopman
operator problem which a min-max optimization problem.
This paper extends the DKO method to the scenario where the system state data is corrupted by unknown but bounded measurement noise. As already discussed, this creates the challenge of generating additional noise transformations impacted by the DNN-derived basis functions of the deep Koopman operator. This leads to distortion of the noise, and the properties of the measurement noise and associated correlations may not remain the same after lifting. The contributions of this work are that we first propose a data-driven framework to Iearn the deep Koopman operator from the system states-inputs data pairs under unknown and bounded measurement. noise, and then we provide numerical evidence that our proposed method can approximate the system dynamics with reasonable accuracy adequate for control applications.
This paper is organized as follows. Section 2 states the problem. Section B presents the proposed algorithm and its theoretical development. The numerical simulations and comparison of the algorithms are shown in Section 4 Finally, Section 5 concludes the paper.
We denote || || as the Euclidean norm. For a matrix A E mx* || A ||r denotes its Frobenius norm, A' denotes its transpose, and At denotes its Moore-Penrose pseudoinverse. Additionally, L, represents the n x n identity matrix.
Notations.
2 Problem
Consider the following time-invariant system:
a=1 + 1) = f(x((1) u()), a00) given,
(1)
y(1) = x1)) + w()),
(2)
where t = 0, 2... denotes the time index, (xt) E Ra and w(t) E RT denote the system state and control input, respectively, y(t) E R denotes the measured state, wr()) E R corresponds to the unknown measurement noise, which is assumed to be bounded (i.c., w(t) I| Wmaz), and f R" x Rm + R" denotes the dynamics mapping which is assumed to be unknown.
2

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Published in Transactions on Machine Learning Research (04/2025)
Suppose an observed system states-impure trajectory from time to T is denoted as:
(yy,, u,) t 0, 2, .T).
(3)
One approach to approximating the unknown dynamics in Eq. ① using & is through the deep Koopman operator (DKO) method. We begin with an approximate representation of the original system dynamics +( (((()), ((), 0) with (()) ==0), where =(t) IR" represents the system state. The function f is constructed based on the Koopman operator theory-based process, described by:
g + 1).0) 0) = Ag (z()). Bu(t),
(4)
a(t) = Cg (t). 0),
(5)
where g(: 0) RD + IR" is represented by Lipschitz continuous DNN with known architecture and tunable parameters 0 E IR, and A E R B E IR** C' !R are variable matrices to be determined
later. Here, Eq. 44) with r 2 n represents the dynamics evolution in the lifted space R", and Eq. (5) defines
the mapping between the lifted space I' and the original space R", By integrating Eq.
the deep
Koopman operator dynamics can be expressed as follows:
@t + 1) = f(i(t). u(t), 0) C()g(E(t) 0) + Bu(t)), 2(0) = x0).
(6)
The problem of interest is to determine the constant matrices A', B", C* and the optimal parameter l using the noisy trajectory & in Eq. such that, for any 0 S T 1, the following approximation holds:
g 0*) A (y:( 0*) B"u
(7)
32 C* C"g(y 0").
For notational brevity, we define the of A', B", C³ or satisfying Eq. 7-1 the Deep Koopman Representation (DKR), which will be referenced throughout this paper.
K 1= B" B', 0"
(9)
3
Main Results
This section first outlines the main challenges and key ideas underlying the proposed approach, followed by the presentation of an algorithm to achieve the DKR in Eq.
3.1 Challenges and Key Ideas
To achieve the DKR, natural approach is minimize the following estimation errors using & in Eq. 3
B" C 0 arg min A.B.C.
1
Σ | f(y., u, 0)1
(10)
2T
A fundamental challenge in solving Eq. 10 arises from the fact that the true system states æ, are unknown
in our problem setting.
To address this challenge, propose alternative minimization problem to Eq. 10
To proceed, for any
0 Os S we first introduce the notation:
Ee+1 f(z,, u, Ō 0¹) + B u.) + E.
and
Di+1 fly., u. 0') = O'' B' +
= where f and f are introduced DKO dynamics achieved using the noise-free and noisy trajectories, respec- tively, based on the same function g(., 0). The terms & and ? represent the estimation errors that arise from solving the following optimization problems:
T
Á" B' Ō' 0" = arg min
æ++1 - f(æ,, uc, 0)l
(11)
A.B.C.O 27

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Published in Transactions on Machine Learning Research (04/2025)
and
1
A", O", arg min
Yot1 f(yes ue 0)1
(12)
A, 2T
Then, by expanding the error function in Eq. 10 using the introduced f and f, and applying the triangle inequality, we obtain:
| f(ar, ux 0*) + f( uc 0*) f(y.. u.
u. 0¹) + f(V. u4 0*) - ye+1 + Vi+1 f(ve, un u,, 0)1 (13) ,,, 0)1 + Il (zt, un 07) fly.. uc u. 0*) 1|a +| ||2.1|
S Mc+1 - fly., ue
Note that E and E are typically small positive constants. For detailed analysis of these estimation errors, we refer to the existing work Hao et al. (2024). Removing the constant terms from the upper bound derived
in Eq. 13,, we formulate the following loss function to achieve the DKR in Eq. 9
L,(A,,,C,)) = 2T Σ (ll Ve+1 f(y,, ur, 0)1 [I ur 0*) flyes ur 0*) 11²).
(14)
minimize the upper bound of Eq. 110) as in Eq. 14). This minimization problem is split into two components. First, given a noisy trajectory &. the goal is to determine dynamical model that approximates the relationship between yr, uy and Vt+1 as stated in Eq. 112. The second component focuses on minimizing the norm
Using this development, we propose that instead of minimizing the residual in Eq. 10
difference between the model f(a; ur 0*) from Eq. (11) and f(y,, ur 0*) from Eq. (12). The key challenge in solving Eq. [44 lies in quantifying the difference between the two models, f(z,, ue 0*) and f(y:, u, O*).
Remark 1 Note that, in contrast to the conventional error function of || - f(v,, us 0112 | Wi+1 - ffye, uc 0)1 |1 f(y, un 0)12 + || the proposed loss function in Eq. {14) substitutes the term ||W++al| with the discrepancy between the system dynamics, llf 112. This formulation allows for further minimization by tuning 0, thereby enhancing robustness and slability in estimation.
3.2 Algorithm
We now introduce an algorithm to solve Eq. 14 utilizing the noisy data from Eq. addressing the first term in Eq. (1.. for which we define the following loss function:
3
We start by
1 2T lg Ag(y,, 0) Buell 1ly. (0(3).0)11))
L,C.A.
(15)
where the first and second parts of L5.1 represent the estimation errors in the lifted space, as described in Eq. 4 and the original space, as outlined in Eq. ( respectively.
If the matrices A, B, C are known, the optimal 0 that minimizes L₁,1 can be directly obtained using the gradient descent method. However, when these matrices are unknown, an alternative iterative approach can be employed. Specifically, let & 0, 1, be the iteration index, and let 0₂ represent the estimation of 0* at the k-th iteration. Initially, the relationship between the constant matrices and 0g is established based on the trajectory g. Once this relationship is identified, the gradient VÆL can be computed only using 6g, enabling the application of the gradient descent method to minimize L by iteratively updating 0..
To this end, we first introduce the following data matrices formed from E:
Y = you Vı. yr-Il E R
Y
= Vy1. N2. .yrl E R U [uo, u ur ur E RMxT
G = (g 0z).g(() g(v,, 0z), -9 R.. G = (g( 0z).g(ue 0z), glyr, 0x)] E RiXT
(16)
It leads to the following compact form of Eq. 15 using given 0g and the definition of the Frobenius norm:
G+ U
(A,(A,
G -
[A B
1 + Y
CG
(17)
2T

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Published in Transactions on Machine Learning Research (04/2025)
GR U
If the matrices G2 E Rext and then the parameter 0g can be updated using the following rule:
(R(+))x Eq. (17) have full row ranks (i.e., are right-invertible),
[ Bgl, Oj arg min f, = Gk [A.B).c
YG
(18)
U
0x+1 = 02 - a& VoL,,,1 (Ai, Big Cj 0s),
Og given,
(19)
where the step size ar satisfies the standard conditions Σ ap and Σx= of oc, and
VoLg, Bi, Cj 02)
1 -
Σ
(Vog(ye+1,0) 0x) ( Aig(v,, 0x) Bi
t=
-(CCVeg(y: 0n)) (y. Cig(vr, 0k))
Note that the matrices A Bi, and Cj remain constant while computing VoL,,, B C2 0z).
To minimize the second part of Eq- 14h. which quantifies the discrepancy between the two models, far,, un Or) and f(y,, u, )*), observe that this difference arises due to the measurement noise wt- To systematically characterize this discrepancy under ws, we define the following loss function:
T-1
L,,
1
Σ
max ang
f(xx un 0²) f(y,, un 0*) 1|
2T
(20)
1
max(
1g
g(v., 0²) - - + I||A,, B B'',, + +I[Ō
-x13).
2T
C. and A B", C' are obtained following the same procedure outlined in Eq. (18). using the same function g(: 0), but derived from noise-free data and noisy data, respectively. Let G and G denote the data matrices computed with an arbitrary given 0, analogous to G and G₁ in Eq. (16), respectively. Given that the system state æ; in Eq. (20) unknown within our problem setting, we now introduce the following theorem to determine the optimal 0" that minimizes (0,)
Here, we assume that the matrices
Theorem If the unknown measurement noise ||we|| is bounded by wmar and g(. 0) is a Lipschitz con- tinuous function, then the optimal 0 that minimizes the following loss function will also minimüze Eq.
200:
( = 2T (11
G
12
((1
G
P + G
G U
1
G
+ G
(21)
+ (GG')
YG 1
G
1|
Proof of Theorem is given in the Appendix. Based on Lr in Eq. following loss function:
17 and ?, in Eq.
21
one have the
1 - G ( [A,
G U
(A,
1 Y - cGI] + (0)
(22)
2T
In summary, rather than directly minimizing the residual Eq. 10), this paper proposes a method determinc A", B", C and 0* that minimize Eq. (22), which serves as an upper bound for Eq. (10 The proposed algorithm follows iterative process. In each iteration k, the algorithm first. computes the matrices [A;, Bū] and C using Eq. (18). Then, the gradient descent method is employed to update 0, to find the optimal 0* that minimizes L, in Eq. (22p. This iterative procedure is summarized in Algorithm referred to as deep Kooyman learning with the noisy data (DKND) in the rest of this paper.
5

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Published in Transactions on Machine Learning Research (04/2025)
Algorithm 1: Deep Koopman learning with the noisy data (DKND)
Input: Y, Y. U in Eq. (16).
Output: A" B* C", 0"
Initialization: Set the learning rate sequences {ax | and terminal accuracy 2 0, build DNN 9(-,0) (): R" > R" with given Og E IR".
for k- = 2, do
Compute [A [B] and Cg by solving Eq. (18 and construct the loss function Ey in Eq. (22).
replacing its [A, B). C, and 0 with [A Bx C₁ and 0x. respectively.
Update the 02 using the gradient descent: 0k+1 Oz ag VoL, (A] B C5 0x) Stop if L, < c and save the resulting A B Cj: and 0₂ as A', B", ,",
C. and 0
end
Remark 2 Note that by following the definition of Moore-Pennood inverse (i.e., for any D E R with full
row rank; = = D'(DD')=t), the inverse terms {
G
G
) and (GG')-1 may pose challenges in com-
U
puting the gradient of Ly. One way to address this issue is utilize the relation 0 - K--KK-100 KKeKKKKKKK where K E Ruxn This approach enables gradient computation involving matric inverses in more manage- able form.
4 Experiments
In this subsection, first demonstrate the performance of the proposed algorithm by analyzing the es- timation errors between the predicted system states and the true noise-free states across four benchmark dynamics: one 2D simple linear discrete time-invariant dynamics:
0.9
-0.1
0 [
1 0
T++1
0
0.8
+
Ue 2O
cartpole (x, E IR,, ee al. (2016), and one real-world example of unmanned surface vehicles (æ,, E R6, E R²), of which the details can be found in Li et al. (024). Then, we compare the proposed algorithm with related methods.
: ur E R) and lunar lander (x, E R E IR²) examples from the Openai gym Brockman
Experiment setup. this experiment, we first gather noise-free state-input pairs D {(x,, uT) from the aforementioned four examples, where u, represents randomly generated control inputs drawn from uniform distribution bounded between -1 and 1. Subsequently, we introduce three types of bounded mea- surement noise: Gaussian noise (wx²) with mcan = 0 and standard deviation = 2, Poisson distribution (w(") with an expected separation À = 3, and uniform distribution (w generated from the open interval (1,,)). To ensure bounded noise, we apply a clipping procedure to the measurement noise. These noise types are added to the system states to yield noisy measurements. Specifically, we denote the noisy measurements under Gaussian noise g = 2t + w.. The corresponding dataset, denoted D {(yf uu) used for the experiments. To facilitate training and testing, we allocate 80% of DG to train DKND (denoted DG reserving the remaining 20% for testing (denoted D For performance evaluation, compute the rool mean square deviation (RMSD) over the lest dataset DG
RMSD(D =
Σ
| - f(y., un 0+)
DG test
(DE test
where IDC denote the number of data pairs (yr, uz) in DC and f represents the estimated dynamics obtained from the proposed DKND method. Additionally, we compare the performance DKND against three bascline algorithms: DK, which solves Eq. (12) using noisy measurements y., DMDTLS from Dawson et al. (2016), and the multilayer perceptron (MLP) approach. To fairly evaluate the algorithms, we assign the above methods with the same DNN structure, training parameters (e.g., learning rate, training epochs, etc.).

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and training and testing datasets. To mitigate the influence of random initialization of DNN parameters, each gradient-based method is run for 10 experimental trials. The average RMSD and their standard deviations are reported in the tables over these 10 trials.
RMSD
Methods
2D example
Cartpole
Lunar lander
Surface vehicle
proposed
0.1963++0.000
0.3705++0.000
0.4007 0000
0.3059++00091
Gaussian
DK
0.214910.0106
0.560410.003.
0.4730 .0051
0.306810.0027
noise
MLP
0.2118=0.0016
0.3987=0.000 000006
0.4650=0.002
0.2960=:000000
DMDTLS
0.2483
29.9163
0.6836
2.1779
Wman = 0.8
Wmar = 1.0
Wmer 1.0
Wmaa = 1.0
proposed
0.4431#0.0015
0.6975=00.002
1926990000029
0.7642:#0.0311
Poisson
DK
0.4707+-0.020
0.7996++0.007
0.8565++0.006
0.7768++0.0013
noise
MLP
0.464441000011
0.6808=t0.0017
0.8329110:0119
0.7727
DMDTLS
0.4709
3.7518
0.9268
2.0631
Wmer 1.2
Wmar 1.3
Wmer 1.5
Wman 1.5
proposed DK MLP DMDTLS
1.47111000089
2.1534+:..292 2912
2.122410:5110
1.754110.0117
Uniform noise
1.7493=:0.188 1877
2.3839+00.000 1021
2.6577=0.0422
1.5712
2.3287=000036
4.0224+0.0000
0$612-000034
1.7127
27 77.2998
39. 2297
286. #86.4929
24.2376
Wman 5.3
Wmar 7.2
Wmaa 8.2
Wmaa 1.2
Table 1: Averaged RSMD over training data.
RSMD
Methods proposed
2D example
Cartpole
Lunar lander
Surface vehicle
0.2074+0.000S
0.3974+0.0077
0.4877++000185
0.4539++00061
Gaussian
DK
0.2124+:0.0072
0.6190=1.0000
0.8433=10.078
0.56421:0.100
noise
MLP
0.3021+0.0311
0.8757:0.0172
1.9348:10.1598
0.7048:1.0088
DMDTLS
0.2514
28.3258
0.6551
2.4107
Wmar = 0.8
Wmar = 1.0
Wmar = 1.0
Wmar = 1.0
proposed
0.4551+0.0014
0.7118#0.0003
0.8268+0.00255
0.9456:+0.0485
Poisson noise
DK
0.4084+4000211
0.8281 +0000888
1. 1.0857+0.1158
1.0846 +00 1584
MLP
0.4888+0.0003
1.059611.00299
1.831640.1631
0.922910.0612
DMDTLS
0.4709
4.4250
0.8958
3.3943
Wmar 1.2
Wmar 1.3
Wmar 1.5
Wmar 1.5
proposed
L4832-40.0117
2. 1.162-10.270
-332331066963
2.2739+101100
Uniform noise
DK
2.0752++0.277
2.3234+0.100 1033
3. 3.0809++0.060 06339
3.9145+00.400
MLP DMDTLS
2.6835
4.8319
6. 6.289440.1411
3.1910 +0.4081
26.7608
40.9191
288.291 298.
24.6145
Wmar 5.3
Wman 7.2
Wmar = 8.2
Wman 1.2
Table 2: Averaged RSMD over lesting data.
Results analysis. As presented in Tables. 2 the proposed DKND method achieves smaller average RSMD and standard deviation on testing data when compared to other methods, even as the complexity of the dynamics is increasing. Specifically, when the noise follows uniform distribution and w, grows larger, the gap in RSMD between the proposed DKND and DK methods becomes more pronounced. Note that the RSMD for all gradient-based comparison methods over the training data does not show significant differences. This can be altributed to the fact that their training processes are terminated al the same terminal accuracy. Figs. 4 display detailed estimation error plots across the testing data, with shaded regions indicating the variability across 10 trials. Due space limitations, additional experimental details, such the generation of measurement noise, the structure of the DNNs, and the training parameters, are provided in the Appendix.

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DMOTLS
DMOTTS MLP
2.5
OK DKND
DK
Eatiattoon eroos
Erumation enars
Exlimation eroors
1.0
0.5
6.2
0.0
100
20
40
100
1.0
Data Indixx 1K0
index
(a) Gaussian noise.
(b) Poisson noise.
(c) Uniform noise.
Figure 1: Prediction errors over lesting data for the linear dynamics example.
MLP DK
MLP
MIP
DK
OK
Exteraation errors
4-
Exiimation ertos
10.0
2.5
0.0
40
600
1.00
120
20
400
60
80
100
120
20
40
60
80
100
120
Data indec (4)
incex
Data Indlex (k)
(a) Gaussian noise.
(b) Poisson noise.
(c) Uniform noise.
Figure Prediction errors over testing data for the cartpole example.
DMDTLS DKND
DMDTLS
MLP
MLP
Estimation erces
35:
Esiimation
30
0
50
100
150
200
250
300
100
15.0
200
230
300
50
100
150
200
250
300
Data index
Data index [k]
Data index [k]
(a) Gaussian noise.
(b) Poisson noise.
(c) Uniform noise.
Figure Prediction errors over testing data for the hmar lander example.

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Published in Transactions on Machine Learning Research (04/2025)
2.0
10
MLP
DK DKND
DIOND
DKND
1.5
Estimation eners
Edimationnn
2.0
1.5
Extimation
0.5
0.0-
0.5
20
50
60
30
40
70
80
30
600
80
(k)
indea
(a) Gaussian noise.
(b) Poisson noise.
(c) Uniform noise.
Figure 4: Prediction errors over testing data for the surface vehicle example.
5 Concluding Remarks
In this paper, have introduced data-driven framework called Deep Koopman Learning with Noisy Data (DKND) to address the challenge of learning system dynamics from data affected by measurement noise. By learning dynamics, we refer to estimating dynamics where, given yt, Ur, the output of the estimated dynamics, L.+1, approximates the true system state 2++1 with reasonable accuracy. The key contribution of this work lies in modifying the existing deep Koopman framework by explicitly characterizing the noise effect on the learned representation in Eq. (9) and mitigating the impact of noise on the DKR through tuning the
DNN parameters to minimize Eq. (21) requiring only that the measurement noise be bounded. We evaluated the proposed DKND framework on datasets with three different types of measurement noise, using examples including simple 2D dynamics, cartpole, lunar lander, and surface vehicle systems. Our results demonstrate the robustness of DKND under different types of measurement noise compared to related methods.
Limitations. Since the formulation presented in this paper only addresses the scenario where the measure- ment noise is bounded, the effect of this bound on the performance of the proposed approach is not formally investigated and remains an open question. Due to the non-convex nature of DNN optimization, the DKND framework is inherently limited to achieving local minima. Future research could explore several aspects, including the design of optimal control strategies based on the learned dynamics using the measured noisy system states.
Acknowledgments
This material is based upon work supported by the Defense Advanced Research Projects Agency (DARPA) via Contract No. Any opinions, findings and conclusions, or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the DARPA, the U.S. Government, or Saab, Inc.
and under subcontract to Saab, Inc. as part of the RefleXAI project.
References
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The general problem of the stability of motion. International Journal of Control, 55(3), 1992.
Petar Bevanda, Max Beier, Sebastian Kerz, Armin Lederer, Stefan Sosnowski, and Sandra. Hirche. Koop- manizingflows: Diffeomorphically learning stable koopman operators. orXiv preprint arX0v:2112.04055 2021.
Greg Brockman, Vicki Cheung, Ludwig Pettersson, Jonas Schneider, John Schulman, Jie Tang, and Wojciech Zaremba. Openai gym. arXiv preprint arXiv:1606.011 01540, 2016.
Scott TM Dawson, Maziar $ Hemati, Matthew O Williams, and Clarence W Rowley. Characterizing and correcting for the effect of sensor noise in the dynamic mode decomposition. Experiments in Fluids, 57: 1-19, 2016.
9

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Morgan T Gillespie, Charles M Best, Eric C Townsend, David Wingate, and Marc D Killpack. Learning nonlinear dynamic models of soft robots for model predictive control with neural networks. In 2018 IREE International Conference on Soft Robotics (RoboSoft), pp. 39 45. IEEE, 2018.
Yiqiang Han, Wenjian Hao, and Umesh Vaidya. Deep learning of koopman representation for control. 2020 59th IEEE Conference on Decision and Control (CDC), pp. 1890 1895. IEEE, 2020.
Wenjian Hao, Bowen Huang, Wei Pan, Di Wu, and Shaoshuai Mou. Deep koopman learning of nonlinear time-varying systems. Automatica, 159:111372, 2024.
Masih Haseli and Jorge Cortés. Approximating the koopman operator using noisy data: noise-resilient extended dynamic mode decompasition. In 2019 American Control Conference (ACC), pp. 5499-5504. IEEE, 2019.
Maziar Hemati, Clarence W Rowley, Eric A Deem, and Louis Cattafesta. De-biasing the dynamic mode decomposition for applied koopman spectral analysis of noisy datasets. Theoretical and Computational Fluid Dynamics, 31:349-368, 2017.
Bernard O Koopman. Hamiltonian systems and transformation in hilbert space. Proceedings of the national acodemy of sciences of the united states of america, 17(5):315, 1931.
Bernard O Koopman and J Neumann. Dynamical systems of continuous spectra. Proceedings of the
National Academy of Sciences, 18(3):255 263, 1932.
Milan Korda and Igor Mczić. Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control. Aulomatica, 93:149 160, 2018.
Jianwen Li, Hyunsang Park, Wenjian Hao, Lei Xin, Jalil Bloss, Kyle Pattison, Christopher Vo, Devesh Upadhyay, et al. C3d: Cascade control with change point detection and deep koopman learning for autonomous surface vehicles. arXiv preprint 003:02203.00592. 2024.
Ajinkya Chaudhary, Meredith
Bethany Lusch, Steven L Brunton, and J Nathan Kutz. Data-driven discovery of koopman eigenfunctions using deep learning. Bulletin of the American Physical Society, 2017.
Bethany Lusch, J Nathan Kutz, and Steven L Brunton. Deep learning for universal linear embeddings of nonlinear dynamics. Nature communications, 9(1):1-10, 2018
Alexandre Mauroy and Jorge Goncalves. Linear identtficcttin of nonlinear systems: A lifting technique based the koopman operator. In 2016 IEEE 55th Conference Decision and Control (CDC), pp. IEEE, 2016.
Igor Mezić. On applications of the spectral theory of the koopman operator in dynamical systems and control theory. In 2015 54th IEEE Conference on Decision and Control (CDC), pp. 7034-7041. IEEE, 2015.
Kevin Patrick Murphy. Dynamic bayesian nelworks: representation, inference and learning. University of California, Berkeley, 2002.
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10

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Published in Transactions on Machine Learning Research (04/2025)
Subhrajit Sinha, Bowen Huang, and Umesh Vaidya. On robust computation of koopman operator and prediction in random dynamical systems. Journal of Nonlinear Science, 30(5):2057-2000) 2020.
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Mathias Wanner and Igor Mezic. Robust approximation of the stochastic koopman operator. SIAM Journal on Applied Dynamical Systems, 21(3):1930-198) 2022.
Matthew O Williams, Joannis G Kevrekidis, and Clarence W Rowley. A data driven approximation of the koopman operator: Extending dynamic mode decomposition. Journal of Nonlineor Science, 25(6)) 1307-1346, 2015.
Enoch Young, Soumya Kundu, and Nathan Hodas. Learning deep neural network representations for koop- man operators of nonlinear dynamical systems. In 2019 American Control Conference (ACC), pp. 4832 4839. IEEE, 2019.
A
Appendix
In this section, we first provide the proof Theorem 1 followed by an overview of the detailed simulation setup used in the experiments presented throughout this paper.
A.1 Proof of Theorem
Recall that G and G denote the data matrices constructed using arbitrary given parameter vector 0, in the same structure as Gy and G2 in Eq. (16), respectively. Before presenting the proof, we introduce the following notions. Define ōg: = gly,, 0) g c, 0) as the difference between the DNN g(., 0) evaluated at the observed noisy system state yt = + and the true system state It. Next, we introduce the following data matrices:
R-XT AG = [8g1. 8g2. ògrl E RTXT AG jōgo 8g1 8gr-1l
G [g 0), g 11, (( 0)] E RTXT
G [g(x) 0).00))))) g 22
g xT 0) RT
(23)
pays FT, 37 E x X [1i 02 ... Ørl R
W [wo, wT WT wT-1 RoxT W [1], W2 wrl E BaxT
For brevity, we omit the iteration index k the constant matrices of f and f are assumed to be computed using the same function g(, 0). Utilizing Eq. (16) and Eq.
23 we obtain:
Y X Y X W, G G +AG G-G.+AG
(24)
We now proceed by minimizing Eq. 11 (over the noise-free trajectory) with respect to the dynamics matri- ces. The solution to this problem is analogous to the one derived in Eq. (18) (over the noisy trajectory). By utilizing the notations introduced in Eq. 23 the following results can be obtained through a reformulation
of Eq. 18
[
A = G
(25)
(26)
11

---

Published in Transactions on Machine Learning Research (04/2025)
We then expand Eq. in the following expression:
- 26
using Eq.
24
and the definition of the ChoorePPenrose inverse. This results
[ [A, B"]
G2
G AG
= G
-G AG)
U
U
G - AG
G (G-AC AG
G - AG
= (G AG)
(
U
(G [
G
G
(AG - G)AG' - AGG'
-AGU"
G - AG)AG AGG' AGU")
I
Ü
U
UAG
0
M.
(27)
and
C XG = (Y- W)(G-AG))
= =(Y -Y - WY-W)(G- - AG)((G AG) -G)((G))G)GGAG))))
(28)
= -YGG- - - (Y+W)AG - WG ( GG' GG' -G-G+AG)AG''
M.
G 0
[
[
Note here that B = G
and C = YG'(GG')-1 Applying Ghorman-Morrisse
formula to Eq.
- 28 that is, for given invertible matrix P E R2 and column vectors m, E R", if
+ 1'p-mm * 0, the following holds:
(P me p-1 P-1mu'p-
1 v'P-l lmm
Using this formula, we derive the following results:
A A" A
[A B'JM P
(I,4 Pm -
(29)
and
P-1
(30)
Here, recall the dynamics difference (0*) defined in Eq. 20 where replace Ō+ with 0, as 0 represents the optimal solution of L Ly., which is found using the same variable 0 in the overall loss function Ly = L, + L as defined in Eq. [4]. given by:
T-1
(0)
max
Σ
[g θ*) g3r 0) + A' (
B + -
27
By following Eq.
29
30
L (0) becomes
1
Ly2(0)
max (N 10 P-P.P_ [Ā", B'J)M,,
(I+m + P-- N P-1 -
| - -
2T
wi
M.P (I, +1.PP- M - - N.P-1 Σ
[1g( θ¹) - -g( 0)113).
1=0
(31)
To proceed and for clarity, we define dg.
= gxz, + w Wman 0) g(xx, 0) and
AGmaz [ōgman
8g maz
Tar R AG Gmmax jgg
max
8g 8g
maz
8g
max
E grxT
W maa [Waar, Wmaz." ... W. Wmar E R Y Ymax + W
12

---

Published in Transactions on Machine Learning Research (04/2025)
Accordingly, we define the following matrices under the maximum measurement noise:
N max
[G AG AG + AGmas G', max U'},
N mar
(Y mar + W mas) AG' W max G',
Mmos (A (AGaz G) G) maz
AGmaz G'
A Gmaz U
(32)
-UAG max
Mmaz= = (-G + AGmaz) A max
AGman G'.
Then, by applying the triangle inequality and utilizing the fact that g(x, 0) is Lipschitz continuous with Lipschitz constants Le and Lg, where Ls denotes the Lipschitz constant of with respect to æ and Le denotes the Lipschitz constant of g with respect to 0, the function ( can be expressed as:
1
(0) 27 (Il (N (ama P - [A", B 1" P (Ir4m +P- PP- MMaan) N maz P-1 13
+ (N (NmanD - (+)MMar P (I, +P "M-") Ñ Nmar P-1 |
(1g 0²) - -g(x: + Wmann 0)113)
t=
1
(1l N max P-1
|1 + [A", (1?) |I Mmon P-1 11 (I+4m + P P-PPPMMaann Mmax)) 111
113 + N max P-1
|1?
27
+ (1l N Nman P
1 +
C 11:) M Mias P-1 | (I, P P-
P-1
11
-TL,g)
(33)
where Lg = /2maxx Liz, Lo}. Observing that the upper bound in Eq. 33 contains complex terms
P and || (I, P-1PMm D-MMaaz) 13, which complicate the minimization of the upper bound by tuning 0, we propose an alternative approach. Instead of attempting to find a solution that makes (I++m - - =|3-=
[I++m + P 1p-IMmaz
and (I,++I P..+--11p M 113= O, we aim to achieve P p-'Mmas | 0 and P P-1M maa = 0 such that (I++m PP--Mmaar= 1 m and || (I, + P-1
r, and
then proceed minimize the remaining terms. Thus, define the upper bound of Eq. (33 as:
L L,, = 27 (II [ MaN P-1
|1 + [A", "AA,""] 1 (12) |I Mmar P-1 1 P P Myaa 112 + N maz P-1 ||
+ (Il Ñ Nman
P
1
C 11:) M MaadP P-1 1 P
P Mman || + N Nmaz P-1 || +TI,
(34)
< 27 ||Nmax
1|2 P-P1p | + [ B*] I12)) || M Mya | p-1 11 + N Naa |1:11 P-1 1|
+ (Il Nmax
||:11
PP
|| C* 1 Maax 1 P-1 11 + N Nmax || P-1
|| +TL2)
Here, using the the Lipschitz continuity of g and Eq.
32 one obtains:
N maz
1 < (Il G | + G
1| + +(T
L. Wman) 2+ (TL)
|| Mmaz
1/ < (2 ||
G
1/ 2
U 1/ +TT Wmaz Lex )²
(35)
N max
| < (Il Y
1 ((TTmaxx))))
(TL, )² + (T (axm) )² G 1|:
M Mmaz
| (2 || G || (T (T
Finally, using Eq.
35 and removing the constant terms while merging the repeated terms from Eq. 34
the resulting loss function expressed as:
È. (0)
1
P
1 ((I & 1 +
+
[
B
1 G +
I
G l1))
27
(36)
G U
where ( B+ T
and غh YG
13

---

Published in Transactions on Machine Learning Research (04/2025)
A.2
Simulation details
In this subsection, provide the simulation details regarding the experiment in Section
A.2.1
Computation resource and training parameters
2D dynamics
Cartpole Adam
Lunar lander
Surface vehicle
Optimizer
Accuracy (e)
le -
Training epochs (S)
le4
Learning rate (ax)
1e -
The number of data pairs (T)
500
600
1600
600
Compute device
Apple M2, 16GB RAM
Table 3: Training parameters.
A.2.2 DNNs architecture
The DNN architectures of method DKND and DK used in this paper are presented in Table 4 We refer to http://tw.....ogr.ooo////aaaaa/a
html for the definition functions Linear(), ReLU() and we denote layer' as the i-th layer of the DNN and Linear([n, ml) denotes a linear function with a weight matrix of shape n x m. Since for the DKND and DK methods, the input of its DNN observable function is the
2D dynamics
Cartpole
Lunar lander
Surface vehicle
layer lype
Linear( [2, 512))
Linear ([4, 512)
Linear( 6, 512)
Linear( (6, 512)
layer2 type
Re (e)
ReLUU)
Re ReTU()
layer" type
Linear Linear([12, 128))
Linear ([12, ([near([51,,)0 128))
Linear( ([11,, 128])
Linear (512, 128))
layer type
Le
Re
ReLUU)
ReLU()
layer" type Linear( (128, 4)
Lincor([128) 6)
Linear( [288, (12n 4))
Linear((122 10))
Table 4: DNN structures of DKND and DK.
mcasured state Ys and the imput of the MLP method is stacked vector of [3/. ufl', show the DNN structure of the MLP method in the following table.
2D dynamics
Cartpole
Lunar lander
Surface vehicle Linear( ((8, 512))
layer' type
Linear( (0o,arr(3,55 512))
Linear(55, (1,))))
Linear( (8,5111 512))
layer" type
ReLU()
ReLU()
ReLU()
ReLU()
layer" type
Linear (51err(51,,00 128))
Linear ([12,, (inear[512, 128))
Linear (512, 128)
Linear (512, 128))
layer* type
ReLU()
ReLU()
ReLU()
ReLU()
layer" type
Linear [128, 4)
Linear([128, 6))
Linear [128, 4))
Linear ([12,,1118) 10))
Table 5: DNN structures of MLP.
14


Reviews:
[{"review_id": "EfDnLeCEBa", "reviewer": "Reviewer_Ch1T", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "The strengths of the paper are as follows:\n\n1) The authors build on the idea of the deep Koopman operator, which relies on finding a suitable “lifted” space of observables (via a deep neural network) in which the system behaves linearly.\n\n2) Unlike standard deep Koopman approaches that directly fit noisy input-output data, the paper introduces an additional term in the loss function to account for bounded noise and mitigate its impact in the learned Koopman operator.\n\n3) One of the key sights of their approach is that after states are transformed by nonlinear observables, the noise is no longer simply additive or uncorrelated.\n\n4) Three different noise distributions (Gaussian, Poisson, Uniform) were tested with various magnitudes of noise, demonstrating that their approach (which they termed as DKND) consistently achieves more accurate predictions (in terms of lower RMS error) compared to baseline methods (DMDTLS, DKL, MLP).\n\nThe weakness of the paper are as follows:\n\n1) As seen from (3), they assume a full state feedback which is much of a simplification to the standard measurement mode y=Hx+w. This weakens the contribution and applicability of their to partial observable systems.\n\n2) Why is it assumed that the relation between the lifted state g and the original state xhat is linear?\n\n3) The advantage of their approach is only when they assume that the measurement noise is bounded (Theorem 1). If that is the case, how is your approach different from linear systems with bounded additive external disturbances acting on the systems. For these type of system, the research literature is quite mature.\n\n4) There is a lot of previous work in literature that have considered stochastic systems with different noise characteristics [1-5]. How is our approach different from these, especially in the problem they are considering?\n\n\n\n[1] Wanner, Mathias, and Igor Mezic. \"Robust approximation of the stochastic Koopman operator.\" SIAM Journal on Applied Dynamical Systems 21, no. 3 (2022): 1930-1951.\n\n[2] Sinha, Subhrajit, Bowen Huang, and Umesh Vaidya. \"On robust computation of koopman operator and prediction in random dynamical systems.\" Journal of Nonlinear Science 30, no. 5 (2020): 2057-2090.\n\n[3] Mauroy, Alexandre, and Jorge Goncalves. \"Linear identification of nonlinear systems: A lifting technique based on the Koopman operator.\" In 2016 IEEE 55th Conference on Decision and Control (CDC), pp. 6500-6505. IEEE, 2016.\n\n[4] Sinha, Subhrajit, Bowen Huang, and Umesh Vaidya. \"Robust approximation of koopman operator and prediction in random dynamical systems.\" In 2018 Annual American Control Conference (ACC), pp. 5491-5496. IEEE, 2018.\n\n[5] Haseli, Masih, and Jorge Cortés. \"Approximating the Koopman operator using noisy data: noise-resilient extended dynamic mode decomposition.\" In 2019 American Control Conference (ACC), pp. 5499-5504. IEEE, 2019.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper addresses the problem of learning a finite-dimensional approximation of the Koopman operator from system data that are corrupted by unknown but bounded measurement noise.\n\nThe main motivation of their work is that traditional data-driven Koopman methods typically assume either noise-free measurements or require strong noise assumptions, limiting their applicability in real-world settings. In this paper, these authors aim to relax these assumptions by only requiring that the noise be bounded.\n\nIn addition to learning the governing matrices A,B of the linear system, they also learn an additional matrix C which maps the lifted state and the original state.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "1) As seen from (3), they assume a full state feedback which is much of a simplification to the standard measurement mode y=Hx+w. This weakens the contribution and applicability of their to partial observable systems.\n\n2) Why is it assumed that the relation between the lifted state g and the original state xhat is linear?\n\n3) The advantage of their approach is only when they assume that the measurement noise is bounded (Theorem 1). If that is the case, how is your approach different from linear systems with bounded additive external disturbances acting on the systems. For these type of system, the research literature is quite mature.\n\n4) There is a lot of previous work in literature that have considered stochastic systems with different noise characteristics [1-5]. How is our approach different from these, especially in the problem they are considering?", "broader_impact_concerns": "NA", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "8t3Duddq31", "reviewer": "Reviewer_MLZK", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n   * The paper studies an important problem where noises are presented. The derivations presented in the paper are rigorous and the statements are mathematically correct.\n\nWeaknesses:\n   * While the paper presents some theorems and derivations to motivate the final algorithm, the algorithm itself lacks theoretical guarantees. It would be interesting to analyze the algorithm more and understand how it compares to other existing methods. Convergence results, if any, would also be helpful.\n   * The presentation of the paper can be improved. See comments below.\n   * The empirical advantages of the proposed DKND algorithm are not clear from the numerical simulations. It seems that the gain is only marginal.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper studies the learning of a nonlinear dynamical system under noisy measurements of the states. Driven by the Koopman operator theory, it proposes to linearize the dynamical system by nonlinearly embedding the state vector into a higher-dimensional observation space. Building upon the linear system, the paper derives a DKND algorithm to handle the noises and implements it by alternatively optimizing the measurement embedding and the linear system. Numerical experiments are shown.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "There are a few comments on the presentation of the work:\n1. The most crucial comment is that I would like to see more content on how the proposed method handles the noises and why it is better than other methods, as this is the core contribution of the paper. So far, the majority of the content in section 3 is spent on deriving the algorithm. While this is important, perhaps the derivation can be compressed for more discussion about the clear properties of the proposed algorithm.\n\n2. The figures on page 8-10 certainly need reorganization. They should be made more compact so that a page can fit at least two of them. Captures should be expanded to explain what they are. If possible, the boxes of the four subfigures (a)-(d) should be aligned.\n\n3. On page 3, you wrote \"$g(\\cdot, \\boldsymbol{\\theta})$ is Lipschitz continuous with Lipschitz constant $L_g$.\" The authors should make it clear which variable is changing. If you mean that $g$ is Lipschitz continuous in the first variable fixed a fixed $\\boldsymbol{\\theta}$, does $L_g$ depend on $\\boldsymbol{\\theta}$? Finally, I don't see $L_g$ pop up again later in the manuscript. Did I miss something, or why is $L_g$ important?\n\n4. There are also many minor typos. I suggest that the authors proofread the manuscript again before resubmitting:\n\n   1. In the last paragraph of page 2, $\\xi$ should be made bold. Also, what is $\\hat{\\mathbf{u}}\\_t$?\n   2. In (5) and (6), it doesn't make sense that $\\hat{\\mathbf{x}}\\_{t+1} = \\mathbf{C} g(\\hat{\\mathbf{x}}\\_{k+1}, \\boldsymbol{\\theta})$.\n   3. After (6), the expression $g(\\cdot, \\boldsymbol{\\theta}): \\mathbb{R}^n \\times \\mathbb{R}^p \\rightarrow \\mathbb{R}^r$ should be changed to either $g(\\cdot, \\cdot): \\mathbb{R}^n \\times \\mathbb{R}^p \\rightarrow \\mathbb{R}^r$ or $g(\\cdot, \\boldsymbol{\\theta}): \\mathbb{R}^n \\rightarrow \\mathbb{R}^r$.\n   4. In the second equation of the \"Key ideas\" subsection, $\\tilde{\\mathbf{C}}$ should be changed into $\\overline{\\mathbf{C}}$.\n   5. At the end of page 4, (20) is an objective instead of a problem. You should not say \"there exists a solution to (20).\"\n   6. I don't know the precise formatting rules of TMLR, but I recommend using parentheses for equations (e.g., (2) instead of 2) unless the formatting rules suggest otherwise.", "broader_impact_concerns": "NA", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "wDRPLWNLRI", "reviewer": "Reviewer_SPcz", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "### (0)\nThe problem, observable learning on noisy data, is challenging but important. The paper concisely motivates it and discuss relevant previous studies.\n\n### (1)\nI have two major concerns. One is about the theoretical construction of the method. In my understanding, the core of the method is the transition from Eq. (33) to Eq. (35), which claims that the minimizer of (35) coincides with that of (33). I cannot follow the reasoning here. Let us think of the following two minimization problems:\n- $\\operatorname{minimize}_\\theta ~ f(\\theta) + || ( h(\\theta) - g(\\theta) ) c(\\theta) - h(\\theta)  ||^2$ ... (A)\n- $\\operatorname{minimize}_\\theta ~ f(\\theta) + || g(\\theta) ||^2 + || h(\\theta) - g(\\theta) ||^2 $ ... (B)\n\nI think they are analogous to (33) and (35) (before the inequality); with $f(\\theta) = || \\delta g_k ||^2$, $g(\\theta) = [ \\bar{A}^*, \\bar{B}^* ]$, and $h(\\theta) = n v' p^{-1}$, $c(\\theta) = u v' p^{-1} (I + v' p^{-1} u)^{-1}$.\nHowever, in general, (A) and (B) do not have the same minimizers, a minimal counterexample being $f(\\theta)=\\theta^2$, $h(\\theta)=\\theta$, $g(\\theta)=c(\\theta)=1$.\nMoreover, taking the upper bound of the first-second lines of (35), the minimizer is not necessarily preserved, either.\nThus the reasoning around (33) and (35) seems to need further clarification.\n\n### (2)\nAnother concern is about the empirical evaluation. Although the method is motivated by noisy measurements, no experiments are conducted on any real-world data. This may not necessarily be a fatal drawback resulting in an immediate rejection but certainly diminishes the convincingness of the paper.\n\n### (3)\nHere are other minor points:\n- Page 1: \"the Koopman operator fails ...\" --> what do you mean by \"fails\"?\n- Page 1-2: \"dynamic mode decomposition (DMD) ... lift the state space to a higher-dimensional space\" --> the standard DMD does not do the lifting.\n- Page 5, in Remark 1: \"invertible\" --> right-invertible\n- Page 5, just after (24): \"$w_1 \\geq 0$, ... $w_6 \\geq 0$\" --> from where these weights appeared? No explanation found.\n- The Gaussian and Poisson noises are not bounded.\n\n2025-02-14: a small typo corrected", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "A method for learning observables of Koopman operators with observation noise is proposed. The idea is based on the upper-bounding of noise effects, resulting in a kind of regularization terms in the final objective function. The performance of the method is examined with four types of simulated data.", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "(1) Most importantly, please elaborate on the proof of the main theorem, in particular around the transition from Eq. (33) to Eq. (35).\n\n(2) Additionally but still importantly, please consider adding experiments on real-world data.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: 2 of the 3 reviewers recommended acceptance. The third believed that the paper was weak in its impact, and therefore recommended reject (although this is not grounds for rejection in the TMLR case). I agree that this work could be stronger, but I believe it is of sufficient interest as is to the broader community. Therefore, I am recommending accept as is.
Decision: Accept (poster)