ResKoopNet: Learning Koopman Representations for Complex Dynamics with Spectral Residuals
Authors: Yuanchao Xu
Kaidi Shao
Nikos K. Logothetis
Zhongwei Shen
Venue: ICML 2025 poster (2025)
Keywords: Koopman operator
data driven dynamical system
dictionary learning
spectral analysis
Primary area: general_machine_learning->representation_learning

Abstract: Analyzing the long-term behavior of high-dimensional nonlinear dynamical systems remains a significant challenge. While the Koopman operator framework provides a powerful global linearization tool, current methods for approximating its spectral components often face theoretical limitations and depend on predefined dictionaries. Residual Dynamic Mode Decomposition (ResDMD) advanced the field by introducing the \emph{spectral residual} to assess Koopman operator approximation accuracy; however, its approach of only filtering precomputed spectra prevents the discovery of the operator's complete spectral information, a limitation known as the `spectral inclusion' problem. We introduce ResKoopNet (Residual-based Koopman-learning Network), a novel method that directly addresses this by explicitly minimizing the \emph{spectral residual} to compute Koopman eigenpairs. This enables the identification of a more precise and complete Koopman operator spectrum. Using neural networks, our approach provides theoretical guarantees while maintaining computational adaptability. Experiments on a variety of physical and biological systems show that ResKoopNet achieves more accurate spectral approximations than existing methods, particularly for high-dimensional systems and those with continuous spectra, which demonstrates its effectiveness as a tool for analyzing complex dynamical systems.

ResKoopNet:
Learning Koopman Representations for Complex Dynamics with Spectral Residuals
Yuanchao Xu"
Kaidi Shao 2
Nikos Logothetis Zhongwei Shen'
Abstract
1. Introduction
In the analysis of complex dynamical systems, fundamen- tal challenge is accurately capturing and understanding the long-term behavior of highly nonlinear systems, particu- larly those with high dimensionality. Various data-driven methods (Brunton & Kutz, 2019; Schetzen, 2006; Wiggins, 2003; Slotine & Li, 1991: : Tu et al., 2014; Lan & Mezić, 2013; Mezić, 2005; Xu et al., 2025; Ishikawa et al., 2024) have been developed these approaches, the Koopman operator framework (Koop- man, 1931; Koopman & Neumann, 1932) has emerged as powerful tool due to its ability to globally linearize nonlin- ear systems. Unlike local linearization methods (Hartman, 1960; Grobman, 1959). which only approximate dynamics near fixed points, the Koopman operator transforms the en- tire system into a linear form within an infiniteddimension function space, enabling the application of spectral analysis techniques to study complex dynamics.
Analyzing the long-term behavior of high- dimensional nonlinear dynamical systems re- mains significant challenge. While the Koop- man operator framework provides powerful global linearization tool, current methods for ap- proximating its spectral components often face theoretical limitations and depend on predefined dictionaries. Residual Dynamic Mode Decom- position (ResDMD) advanced the field by in- troducing the spectral residual to assess Koop- man operator approximation accuracy; however, its approach of only filtering precomputed spec- tra prevents the discovery of the operator's s com- plete spectral information, limitation known as the 'spectral inclusion' problem. We introduce ResKoopNet (Residual- Koopman-leaming Network), novel method that directly addresses this by explicitly minimizing the spectral residual to compute Koopman eigenpairs. This enables the identification of more precise and complete Koopman operator spectrum. Using neural net- works, our approach provides theoretical guaran- tees while maintaining computational adaptabil- ity. Experiments on variety of physical and bio- logical systems show that ResKoopNet achieves more accurate spectral approximations than ex- isting methods, particularly for dighediiensional systems and those with continuous spectra, which demonstrates its effectiveness as a tool for analyz- ing complex dynamical systems.
address this challenge. Among
Despite its promise, practical computational challenges arise from the infinite-dimensionaal nature of the Koopman opera- tor. Numerical methods such as Extended Dynamic Mode Decomposition (EDMD) have been developed to approxi- mate the Koopman operator using a finite set of observables, which makes it possible extract dynamic modes from data. Important convergence results for EDMD were estab- lished by Korda & Mezić (2018), which proved that as the number of samples increases, EDMD converges to the L,2- orthogonal projection of the Koopman operator on a finite- dimensional subspace. They further showed convergence in the strong operator topology as the dimension of the sub- space increases, and that accumulation points of the spectra of EDMD correspond to eigenvalues of the Koopman op- erator with associated eigenfunctions converging weakly, provided the weak limit is nonzero. However, despite these theoretical advances, EDMD faces notable limitation in spectral approximation. It suffers from spectral pollution, where discretization introduces spurious eigenvalues unre- lated the true operator. Moreover, EDMD struggles to accurately capture continuous spectra, which are crucial for characterizing chaotic and complex systems. As a result, EDMD can miss critical information about the underlying dynamics, particularly in systems exhibiting rich, complex
'Equal contribution 'Department of Mathematical and Sta- tistical Science, University of Alberta, Edmonton, Canada "International Center for Primate Brain Research, Chinese Academy of Sciences, Shanghai, China. Correspondence to: Zhongwei Shen <zhongwei @eteeeeiuullbba ualberta.ca>.
Proceedings of the 4nnd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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ResKoopNNet
behavior.
continuous spectrum. [-T,, denotes a periodic interval. The discrete spectrum is particularly important for insights into long-term behavior, such periodicity and stability, while the continuous spectrum characterizes complex be- haviors such as mixing, chaos, and transport phenomena (Arbabi & Mezic, 2017: Yang et al., 2022; Brunton et al., 2017). Our analysis emphasizes finding these spectral com- ponents including both discrete and continuous part from KMD.
To address these limitations, Residual Dynamic Mode De- composition (ResDMD) (Colbrook & Townsend, 2024) was introduced to offer convergence guarantees through spec- tral residual that measures how well the estimated Koopman spectra are. By assessing convergence of spectral residual, ResDMD eliminates spurious spectral components that do not represent the system's true dynamics, which enhances the reliability of spectral estimation. ResDMD also intro- duces the concept of pseudospectrum, regions where the spectral content is sigmificant even without discrete eigen- values, which is particularly valuable for analyzing systems with continuous spectra. However, ResDMD only filters precomputed polluted spectra, which limits its ability to independently refine spectral estimates.
Extended Dynamic Mode Decomposition (EDMD) (Williams et al., 2015) is a widely used data-driven method for approximating the Koopman operator and its spectral components in KMD. It selects a set of observables { ( to form a subspace VNNx span {cb [ The method constructs a dimensiime approximation of the Koopman operator by solving least-square problem based on system snapshots and selected dictionary, allowing the computation of eigenvalues, eigenfunctions, and Koopman modes. While common dictionaries include polynomials, Fourier basis, and RBF functions, selecting an optimal dictionary remains system-dependent and challenging. Given data snapshots {(x;, y with y/s = F(x,) and { i.i.d. by lu, two data matrices Vx and Vy are formed by evaluating the dictionary on these data points:
In this paper, we propose Residual- desidual-based Koopman-learing Network (ResKoopNet), which overcomes these limitations by minimizing spectral residuals over eigenpairs while ex- ploring the optimal dictionary space. ResKoopNet em- ploys neural networks to optimize dictionary functions for the Koopman invariant subspace, which enables both dis- crete eigenvalue computation and pseudospectrum analysis. Through experiments both simulation models and high- dimensional real-world systems, we demonstrate its superior accuracy and scalability over other methods.
(1())) :
VNN (x1)
2. Preliminary on Koopman Operator
V ==
4/1 (2m)
ww (#m)
Consider discrete-time dynamical system (9, 14) governed by map F Ω - Q where & _ IEd is the state space, and is a probability measure. The evolution of the system is described by: Th+1 F(xx), h E Z+
4/ (91)
UNx (v1)
yy =
w/ (vm)
uh (um)
EDMD computes the approximated Koopman matrix repre- sentation by
The Koopman operator IC acts on observable g E F as:
Kg - go F.
K / Vy.
where the inner product and corresponding norm for F is defined as (f.g), = Jn fg du and [|fí] = ff,f) respectively. Notice that K. is linear even though F is non- linear. Another key aspect of modern Koopman operator theory is Koopman Mode Decomposition (KMD) (Mezić, 2005), which represents system dynamics through its spec- tral components. In particular, for given observable g. KMD provides the expansion following
where V is the pseudoinverse of V The eigenvalues of K provide approximation of the Koopman operator's discrete spectrum, and the Koopman eigenfunctions Φ; are approximated as Φ; = VV,,, where v, E CN is the i-th eigenvector of K and
V =
is (row) vector of observables from the dictionary.
g(I,,) (xx))
Σ aA""x((Fa)++ + | | (K)
ind $0.9 (20) d0,
3. Koopman Operator Learning
EDMD directly approximates the Koopman operator and its spectral components from data but suffers from spectral pol- lution, where discretization the infinite-dimensionne oper- ator introduces spurious eigenvalues that have no relation
continuous spectrum
discrete spectrum
where 4 are the eigenfunctions of K, cx are expansion coefficients, and Φo.g represents the contribution from the
2

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ResKoopNeet
to the true spectrum. Residual Dynamic Mode Decompo- sition (ResDMDD) (Colbrook & Townsend, 2024) improves spectral accuracy by filtering out these polluted eigenvalues via spectral residual measurement. However, it only filters precomputed spectra and cannot fully discover the Koop- man operator's complete spectral information, a limitation known as the 'spectral inclusion' problem where parts of the true spectrum may be missed entirely.
Eq. (1) can be approximated (see A.2 for more details):
1
res A, 0)²
V wy - y 3
X W
m
(3)
Notice that Eq. (3) is calculated only for precomputed eigen- pairs. However, the spectral residual serving as a threshold only filters polluted eigenpairs: it does not explore addi- tional accurate eigenpairs or enhance the approximation. For an extreme case, the spectrum of the approximated Koopman operator could be trivial (Colbrook & Townsend, 2024, Example 3.1). This is precisely the spectral inclu- sion problem we mentioned earlier, where the computed speetrum is constrained to a subset of the true spectrum, but could potentially omit critical spectral components. Ad- dressing this fundamental limitation is a core objective of our work.
To address these issues, propose Residual-based Koopman-learning Network (ResKoopNet), which mini- mizes spectral residual over eigenpairs while exploring the optimal dictionary space. This approach identifies a more precise and complete spectrum of the Koopman operator while ensuring all learned dictionary functions satisfy small spectral residual criteria. We note that while we employ simple neural network as optimizer in this work, the specific choice of optimization tool can be adapted based on the data structure and properties of the dynamical system being studied.
Continuous Spectra and Pseudospectrum The concept of pseudospectrum is particularly useful when studying the continuous spectrum of the Koopman operator, as it captures regions where the resolvent norm (i.c., I[(KC - > is large, even in the absence of discrete eigenvalues. order to compute the pseudospectrum, a set of grid points as candidate spectrum values {= 2n. is scanned in the complex plane using the spectral residual; specifically, for each grid point 2j E C, we compute the following SVD problem
3.1. ResDMD Review
Now, suppose we have an estimated eigenvalue- eigenfunction pair (A, 6) of IC where À E C and & = Vv = Em NK thu, E VNK is expressed in the dictionary functions. Assuming & is normalized, i.e., | = 1, the spectral residual that measures the accuracy of this eigenpair is defined as:
TJ = min res VV.) v,ECN,
(4)
0) = So [K((z) - {b(x))² du(x)
which uses the dictionary & computed from our proposed ResKoopNet method. The approximated c-pseudospectrum containing the continuous spectra is then given by {z, Tj < c}. More details on proof of convergence can be found in Colbrook & Townsend (2024, Appendix B).
Sn
Nx Σ
(KCiA, ((KCi,,KKb))) Nabhe Kapg)a
wJ=1
-X(Co 33) + (r) 13)
3.2. Residual-based Koopman-learning Network (ResKoopNet)
where U; A denote the complex conjugates of Ug X.
This spectral residual in Eq. (1) quantifies how far the esti- mated eigenpair deviates from the true spectrum. In practice, we can approximate this residual using data samples through Galerkin methods (Boyd, 2013; Colbrook & Townsend, 2024); more specifically, the following convergence results hold as the number of data points m + oc:
General framework In this section, we present the Residual-based -eapmann Network (ResKoopNet), which minimizes spectral residual to address the spectral inclusion problem and yield more accurate and complete Koopman spectrum. The overall pipeline is illustrated in Figure 1. ResKoopNet optimizes the dictionary functions by minimizing the total spectral residual J as following
lim 1 m-
VV V
(Wh, Wy)
N K res (Xi, dh:
m
=
1
lim m= m
= (Wi, Kib)))
(2)
over all computed eigenpairs (()i, (AA,,)))) (1), This loss di- rectly influences the quality of the finieedimennional Koop- man approximation and ensures the learned dictionary cap- tures essential spectral features without relying on precom- puted spectra or post-processing.
lim
. Wy = (vi, 10* (C
m +00 m
where * denotes conjugate transpose of matrix or adjoint operator. Using these relationships, the spectral residual in
3

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ResKoopNeet
In this work, we parameterize the dictionary functions (1;0) using simple feedforward neural network, shown on the right side of Figure 1, where 0 denotes the network parameters. Unlike EDMD, which minimizes pre- diction error, our method explicitly targets spectral accuracy. Remark 3.1. Notice that the neural network architecture can be replaced by more advanced models or other optimizers depending on the system and data structure.
different performance in capturing the spectral properties of the Koopman operator.
Remark 3.2. To ensure the numerical stability when comput- ing Gt, we add a small perturbation, i.e., K = (G+oT)-1A for some small number 0.
As shown in Eq. (6), ResKoopNet provides an explicit ex- pression for the optimal Koopman matrix K given a fixed dictionary function L. Although the formulation is sim- ilar to EDMD, the key distinction between ResKoopNet and EDMD lies in their loss functions; more specifically, while EDMD minimizes the overall prediction error || || V 1| ResKoopNet minimizes the residual projected onto the estimated eigenspace, i.e., (I V W This eigenspace projection fundamentally changes the op- timization landscape by weighting errors along spectral di- rections, ensuring that the learned Koopman matrix better captures the underlying eigenstructure rather than merely minimizing average prediction error. Therefore, they arise from distinct theoretical foundation.
Optimizer
Initiaize @1 for v (z; 0)
Input: data snapchots {(x, y.) ) 1--1
Construnt R (0)
Look for optimal dictionary (Va, stv, }
Campute: residual ((O)
Update / (rg
Output Koopman matrix X, poeudospedrum, elgentunctions
J
From Residual to Machine Learning This section explains how a simple neural network is integrated into ResKoop- Net. In the optimizer as shown in Figure 1, the neural network parameterizes the dictionary V (a;; 0) and minimize the parametrized total spectral residual (() given in the following:
Retum (2(2;)) ()
The optimizer on the right searches for the optimal dictionary by minimizing Eq. (3).
Figure 1:
1 [ y (0) Vx (x)))
In ResDMD's framework, the spectral residual in Eq. (3) measures how well a computed eigenpair from the dataset, in other words, ideally we want the total spectral residual J == >i that we can treat J as loss function. Moreover, minimizing J is also equivalent to the following minimization problem:
(()
(7)
m
where (()) and V()) depend on the parameters A. The op- timal Koopman matrix K(0) is computed following Eq. (6):
i res( (Ai, 6²)² approaches to zero, which implies
Nx
k())= G(8))A())
(8)
The algorithm alternates between updating ((0) via Eq. (8) and optimizing J(0) w.r.t. 0 by stochastic gradient descent via Eq. (7). While it is possible to optimize both ((0) and (()) simultaneously, as done in Takeishi et al. (2017) and Otto & Rowley (2019), our separate procedure ensures computational efficiency and numerical stability (Li et al., 2017).
min J = min I1 ([([yy-WxKKVVV V K
(5)
where each column of the matrix V is (right) eigenvector v; of the matrix K. Thus, with dictionary & fixed, we can obtain a closed form for the optimal Koopman matrix K as following
K = GAA,
(6)
Computing Algorithm In our neural network implementa- tion, we include some non-trainable functions to enhance the dictionary computation. Specifically, we add a vector of constant one and the coordinates of the state space as non-trainable basis in the output layer, which helps avoid trivial solutions, i.c., J = for some initial 0. For the network architecture, we build simple feedforward neural network with three hidden layers, where each hidden layer size can be specified during training. We use the hyperbolic tangent (tanh) function as the activation function for the hidden layers and employ the Adam optimizer for updating the network parameters 9. Adam is particularly well-suited for this task due to its ability to adapt the learning rate for
where G - 4 V A = 1 Vy By (See A.3 for more details on deriving the Eq. (5) and Eq. (6)).
Although Eq. (6) resembles the EDMD formulation for com- puting the Koopman matrix representation K, ResKoopNet and ResDMD fundamentally differ from EDMD in their theoretical foundation. While they share a similar matrix expression, this expression is derived from minimizing the spectral residual in Eq. (3) rather than the prediction error that EDMD minimizes, as shown in Appendix A.3. Specifi- cally, our approach explicitly incorporates IC* as shown in Eq. (2), focusing on spectral accuracy rather than ob- servable prediction. This distinction leads to significantly
4

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ResKoopNett
Algorithm 1 ResKoopNet
by the Davies-Bouldin index. Note that we include Hankel- DMD (Arbabi & Mezic, 2017) consistent benchmark across all experiments due to its theoretical guarnntees and applicability to both low and high- -igh-dimensional systems with- out requiring predefined dictionaries. It uses time-delayed state measurements by Takens Embedding Theorem (Tak- ens, 2006) instead of using a pre-defined dictionary or ap- plying Galerkin approximation as in EDMD and ResDMD methods (see Appendix 1. for more details).
Input: Dataset X, Y. number of observables Nk. learn ing step 6, regularization parameter loss function threshold c > 0, grid points (21,... .n,
1: Initialize 0, thus initializing (0)
2: Compute (0) and its cigenvector matrix V(0) repeat
Update 0- 0 (0)
Compute (0))) L Vx A(0) in L Vy
m
Update (0)- = (G(0) + +al) and (())
until J(0) <
4.1. Pendulum
Output: (0). eigenpairs {()b, di & and c-pseudospectrum {zj: Ty < <) (See Eq. (4)).
The pendulum system a measure- Hamiltonian system, which theoretically constrains its spectrum to the unit circle. The system is governed by:
Ø = sin(
(0, 0) E [- T x [-15, 15),
each parameter, which can lead to faster convergence in the alternating optimization process between the network parameters and the Koopman matrix K(#) formulation. The computing steps are illustrated the following Algo- rithm 1. ResKoopNet's practical advantages come with computational demands due its iterative optimization. Each gradient update scales linearly with system dimen- sionality and network parameters. While individual steps are lightweight, repeated least-squares optimizations make convergence slower than standard numerical methods, with stochastic gradient descent achieving an (1/n)) rate.
where 0 represents the angular displacement from equilib- rium and 0 the angular velocity. The dynamics exhibit two distinct regimes: oscillatory motion for low-energy initial conditions, and rotational motion when the initial energy exceeds the separatrix threshold (Lusch et al., 2018).
For our analysis, we sample 90 and 240 initial conditions uniformly distributed in the domain, each evolving for 1000 time steps with At 0.5. This gives datasets of approxi- mately 9 x 10 and 2.4 x 10 snapshots, respectively. We choose 3 hidden layers and use 300 neurons and 350 neuron in each hidden layer in both cases, respectively.
We also give a brief discussion on the convergence analysis of ResKoopNet in Appendix A.4, which leverages exist- ing results from approximation theory in the Barron space (Haykin, 2009; Weinan et al., 2019).
15
1.5
4.5
4. Application in physical and biological systems
-05
In this section, we present three examples to demonstrate ResKoopNet's effectiveness in estimating key components of Koopman Mode Decomposition: spectrum, eigenfune- tions, and Koopman modes. In the pendulum example, we compare with EDMD, EDMD-DL. Hankel-DMD, and Res- DMD, and show that ResKoopNet captures the full Koop- man spectrum with significantly fewer dictionary observ- ables than required by ResDMD, while other methods either produce spectral pollution or miss the continuous spectrum. Our turbulence analysis demonstrates ResKoopNet's ability to detect not only acoustic vibration and turbulent fluctua- tion (as in ResDMD (Colbrook& & Townsend, 2024, Section 4.3.1, Section 6.3)) but also to recover the fundamental pres- sure field structure through the lowest-residual Koopman mode, which is feature not captured by Hankel-DMD us- ing the same dataset. In the neural dynamics example, we compare against three methods across five mice datasets, with ResKoopNet achieving superior clustering of neural states corresponding to different visual stimuli, as quantified
-1.0
-1.
-1.
.18
06
10
N.S 100
N K 300
1.-
10-
05
0.-
-55
-05
-15
Figure 2: The four plots depict the spectrum of the Koopman operator, constructed using varying dictionary sizes Ng of 25, 50, 100, and 300. Each plot utilizes 90 initial points to illustrate the impact of increasing the dictionary size on approximating the spectrum of the Koopman operator.
As shown in Figure 2, ResKoopNet requires only Nx = 300 observables to approximate the full spectrum. While the
5

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ResKoopNett
of data points (Figure 3), where NK 350 observables remain sufficient. Figure 4 compares ResKoopNet with four classical methods using the 90-initial-point dataset. Both EDMD and ResDMD employ Hermite functions (order 15) for 0 and Fourier functions (order 20) for 0, while Hankel- DMD uses a time delay of 150. Though Hankel-DMD produces eigenvalues near the unit circle, it only captures the discrete point spectrum and cannot reconstruct the con- tinuous spectral components. We include Hankel- DDDD here for consistency in benchmarking across all experiments, de- spite its limitations in higher-dimensional systems examined later. The shaded region in the ResDMD plot represents the e-pseudospectrum, which approximates regions where the spectral residual is smaller than a threshold value; however, with 300 basis functions, ResDMD fails fully recover the unit circle. Overall, this pendulum example demonstrates that ResKoopNet achieves accurate and complete spectral recovery using much fewer observables than other classical methods.
K3
K 150
1.0
10
035
082
-05
0.5
-3.3
-15
13
25
144
05
08
-10
-10
-10 -05
00
10
15
-0% 50
1.
Figure 3: Same example as Figure 2 but with larger data size, using 240 initial points to show the effect of increasing dictionary size on the Koopman spectrum approximation.
4.2. Turbulence
Recovering spatial patterns is a typical goal of DMD- based methods, especially in fluid dynamics (Schmid, 2022; Mezić, 2013). Figure 5(a) shows the ground truth pressure distribution, in which a clear asymmetry exists between the upper and lower airfoil surfaces. This fundamental spatial pattern is critical for analyzing aerodynamic performance and structural integrity. While the original method has successfully detected acoustic vibration and tur- bulent fluctuation shown in Colbrook & Townsend (2024, Section 6.3), it fails to recover the global spatial structure of the pressure field. In contrast, ResKoopNet achieves a significant improvement by accurately reconstructing this dominant spatial pressure field structure itself from its lead- ing Koopman mode shown in Figure 5(b), which contains essential information about the flow characteristics around the airfoil.
2240-LUU 228
-32
EDMD: Nx 300
EDMD-DL: Nx = 300
The turbulent flow dataset from Colbrook & Townsend (2024, Section 6.3) models a two-dimensional airfoil system with Reynolds number 3.88 x 10" and Mach number 0.07. The data captures a pressure field at 295, 122 spatial points across 798 time steps, sampled every 2 x 10-5 seconds. Given the extreme dimensionality of this X E IR 197882955,122 matrix, we apply truncated SVD on the data matrix to re- tain only the top 150 components and project the data via XV US IR 798x150 After computing Koopman modes in this reduced state space using ResKoopNet, we map them back to the original dimensions through multiplication with V. To maintain consistency with the baseline, same number of observables NK 250 in Colbrook & Townsend (2024, Section 6.3), including 99 trainable neural- network-based functions, one constant function, and 150 principal components obtained from the truncated SVD.
-00
-25 1030
-00
62 30
180
Hankel-DMD: NK 300
ResDMD: NK 300
Figure 4: Comparison with classical methods. Eigenvalue spectra computed from a 300 300 Koopman matrix using EDMD, EDMD-DL, Hankel-DMD, and ResDMD.
original ResDMD work reported using 964 observables (Colbrook & Townsend, 2024, Section 4.3.1) 1), our additional tests indicate that approximately 460 basis functions can pro- duce reasonable circular approximations with ResDMD (see Figure 7 in Appendix A.7.1). Nevertheless, ResKoopNet still achieves comparable accuracy with significantly fewer observables. This efficiency persists even with larger sizes
the
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ResKoopNNet
Figure 5(b) shows the first Koopman mode computed by ResKoopNet, which corresponds to the mode whose eigen- value has the largest absolute real part, and it also has the smallest spectral residual among all computed modes. This mode not only captures the most dominant dynamics in the system but also closely resembles the original pressure field. This correspondence demonstrates that ResKoopNet is able to recover the principal spatial feature of the turbulent flow, something that Gernel-ResDMii and Hankel-DMD could not achieve, which also justifies that the proposed method ResKoopNet can effectively address the spectral inclusion problem where traditional methods often fail to recover the complete spectrum of the Koopman operator. This mode likely corresponds to the largest singular value observed in the SVD, as shown in Figure 5(e) and (f). The dominant singular value highlights the strongest spatial mode in the
(True) Pressure Field
Mode 1 (Residual: 8.36e-07)
Z
05
92
41
Data
(a)
Mode 3 (Residual: 1.38e-02)
Mode 7 (Residual: 1.59e-02)
fuctuttinn
N
00-
05
05
02
02
X
(c)
(d)
data, which aligns with the physical air pressure field and its global structure.
All Singular Values (Log)
First 150 Singular Values (Log).
102
Singuler Value
10*
Singuler Value
In addition to the first mode, Figure 5(c) and (d) show the third and seventh Koopman modes computed by ResKoop- Net, associated with the eigenvalues having the third and seventh largest absolute values of their real parts. These two modes have the second and third smallest spectral residuals and correspond acoustic vibration and turbulent fluctua- tion, as also observed in the Kernel-ResDMD study (Col- brook& & Townsend, 2024, Section 6.3). The ordering of the Koopman modes in Figure (db is based on the ascend- ing values of spectral residual, indicating how well each mode aligns with the spectrum of the underlying Koopman operator.
304-
30*
10a
204
250
400
800
25
15
200
125
150
Index
Index
(e)
(f)
Figure 5: Turbulence detection using Koopman modes from 250 observables. (a) shows the original 2D pressure field. (b) shows Koopman mode which has the smallest residual and closely matches the original pressure field. (c) and (d) show the acoustic vibration and turbulent fluctuation by the Koopman modes with the 2nd and 3rd smallest spectral residual. (e) and (f) show the plot of all singular values and first 150 singular values when applying truncated SVD method, respectively.
For further comparison, we include in Appendix A.8.2 four Koopman modes computed using Hankel-DMD with a time delay of 5, selected similarly based on the small- est spectral residual. These modes, however, do not re- veal the global pressure pattern observed in ResKoopNet's first mode, which further validates the unique capability of ResKoopNet recover large-scale coherent structures in turbulent systems.
the experiments, mice viewed natural videos while their neural signals were recorded via calcium imaging in the primary visual cortex, reflecting the activity of thousands of neurons. Here, we focus on the state partitioning of neu- ral signals. Specifically, in each mouse, six video stimuli were repeatedly shown, creating ideal conditions define brain states. Neural activity during repeated trials with the same stimuli is assumed to reflect the same underlying dy- namic system, enabling Koopman decomposition methods to uncover and separate these brain states.
4.3. Neural dynamics identification in mice visual cortex
Since ResKoopNet directly minimizes the spectral residuals based eigenfunctions, its estimated evolution of eigen- functions over time should ideally capture latent dynamics. To evaluate how effectively ResKoopNet reveals latent tem- poral dynamics real data, we apply it to a dataset of high-dimensional neural signals and demonstrate its advan- tages over a series of classical methods: the Hankel-DMD, EDMD (combined with RBF basis) and DMDD..
The dataset consists of neural recordings from five mice, each exposed video stimuli, repeated 9-10 times for a total of around 60 trials. Each recording captures the activity of over 7,000 neurons, with each 10-second video sampled at 50 Hz, resulting in 300 data points per trial.
These methods are selected as representative approaches for handling high-dimensional data.
We applied ResKoopNet and three classical Koopman de- composition methods (Hankel-DMD, EDMD with RBF ba- sis, and Kernel- ResDDD) to this dataset, using different
The dataset is part of the open dataset on mice from the com- petition "Sensorium 2023" (Turishcheva et al., 2024b;a). In

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Elgenfunctions State
20
50
Eigentunctors
10
0.25
0.15
30
0.1
0
State
State
State
Slate
State 5
State
B
Mouse-1
Mouse-2 ResKoopNet
Mouse-3 ResKoopNet
Mouse 44
Mouse 5 ResKoopNet
ResKoopNet
ResKoopNet
Dim:
Dim
Dir:
0.5
Dim.
Dim.
Dim.
Dim.
C
Hankel DMD
Hankel DMD
Hankel DMD
Hankel DMD
Hankel DMD
0.1
0.1
0.1
0.05
0.05
Dime2
0.02
5im.
0
Dim
-0.05
-0.00
.0.04
-0.1
-0.1
0.1
0.1
Dim. 1
Dim.
Dim.
Dim.
Dim.
D
EDMD+RBF
KeDD
Multi-dimensional Scallng
0.2
30
RecKoopNei
0.1
Davies- Boul-Bn
20
EDMD++RE RBI
Dim.
Dim,
Ker ResDMD
0+
10
-0.05
..2
-011
0.5
-0.1
0.1
4
Mouse Index
Figure 6: ResKoopNet outperforms Hankel-DMD in identifying latent dynamic structures in neural signals with dictionary size of 50. (A) (Top) 50 Koopman eigenfunctions estimated by ResKoopNet across 6 states characterized by different video stimuli in an example mouse. Each trial contains 300 data points (10s at 50Hz). (Bottom) 50 Koopman eigenfunctions approximated by Hankel-DMID, each 50 points long, reflecting the dimension of the Hankel matrix. (B) 2D representation of Koopman eigenfunctions for all tested mice, computed by ResKoopNet and reduced via Multidimenssional Scaling (MDS). Trials of the same state cluster well. (C) Same as (B) but computed with Hankel-DMD, showing no clear state separation. (D) 2D representation of Koopman eigenfunctions for the first mouse, computed by EDMD with an RBF basis. See Appendix Figure 14 for full results. (E) Same (D) but computed with
See Appendix Figure 15 for full results. (F) -Buudin Indices (DBIs) evaluating clustering quality across four methods (ResKoopNet, Hankel-DMD, EDMD+RBE, and Kernel-ResDMin for five mice. Lower DBI values for ResKoopNet indicate better clustering.
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ResKoopNeet
implementations and Koopman subspace dimensions. For ResKoopNet, trained dictionaries on all snapshots from each mouse to avoid overfitting, We reduced the data to 24 dimension via truncated SVD, together with the constant and 25 trainable bases, resulting in 50 eigenfunctions. The neural network we use here consists of 3 hidden layers, each with 200 neurons. The hyperparameter scanning results are included in Appendix Figure 17. which demonstrates that our parameter choice is within the robust range. The decomposed eigenfunctions are shown in Figure 6A(t))), with markers indicating ground truth state separations. For Hankel-DMD, we built a Hankel matrix with a delay of 50, producing 50 eigenfunctions per trial. In EDMD with RBF basis, we used the SVD-truncated 300 basis and 1000 RBF functions, resulting in 1301 eigenfunctions. For Kernel- ResDMD, we used normalized Gaussians as kernel func- tions, setting the Koopman subspace dimension to 299 eigenfunctions based on Colbrook et al. (2023). See Ap- pendix A.9.4 for method details and Appendix A.10 for dictionary size justification. These eigenfunctions, shown in Figure 6A(bottom), Appendix Figure 14A, and Appendix Figure 15A, are compared to ground truth trial identities.
The 2D MDS visualization reveals a clear separation of fea- tures for all 5 mice using ResKoopNet (Figure 6B), whereas other method shows clear clustering (Figure 6C-E, Ap- pendix Figure 14B, Appendix Figure 15B). To quantify this clustering, we calculate the Davies-Bouldin index (DBI), a measure of clustering quality that assesses how compact and well-separated the clusters are. A lower DBI indicates more compact clusters that are farther apart from each other, which corresponds to better clustering. The DBI is signifi- cantly lower for ResKoopNet (Figure 6F), suggesting that it captures the latent dynamic structure more effectively than all three other methods. Similar clustering patterns are confirmed with UMAP and t-SNE (Appendix Figure 16).
5. Conclusion and future work
In this paper, we introduced ResKoopNet, novel approach that minimizes spectral residuals to accurately approximate Koopman operators for complex dynamical systems. This method successfully resolves the spectral inclusion prob- lem while capturing both discrete and continuous spectra with high accuracy. Our experiments across physical and biological systems demonstrate that ResKoopNet signifi- cantly outperforms existing methods in identifying coherent structures and uncovering latent dynamics, particularly in high- figh-dimensional applications.
In this study, Koopman eigenfunctions are interpreted as representing dynamical features corresponding to the video stimuli. Consequently, their effectiveness in capturing these key, stimulus-related dynamics is evaluated by assessing how well cigenfunctions from trials with the same video stimulus cluster together and remain distinguishable from those associated with different video stimuli. This effec- tively transforms the problem into clustering task based the separability of eigenfunctions across différent stimuli. It is noteworthy that for the ResKoopNet case, averaged trial differences are visibly clear. It is also worth clarifying that clustering these processes is not biologically necessary, as simple clustering of mean firing rates is sufficient (see Ap- pendix Figures 10, 11. 12). However, this dataset provides an ideal benchmark with clear ground truth labels, enabling accurate comparison of Koopman eigenfunction estimation methods. Effective estimation of Koopman eigenfunctions can be valuable for more complex tasks, such as unsuper- vised latent state identification or decoding object dynamics from video stimuli, which offers promising directions for future research.
Despite these advantages, ResKoopNet has several limita- tions. First, the use of neural networks increases compu- tational cost compared classical numerical algorithms (see Appendix A.6). Second, unlike stochastic approaches such as VAMP (Mardt al., 2018) and SDMD (Xu et al., 2025), ResKoopNet currently does not account for stochas- ticity (see Appendix A.5 for comparison discussion). In addition, its performance can vary depending on the neural architecture and training configuration.
There are several promising directions to address these limi- tations and extend ResKoopNet's 's capabilities. Many physi- cal laws, such as conservation principles symmetry con- straints, are inherently encoded in the spectral properties of dynamical systems. Incorporating such domain knowl- edge directly into the neural network architecture could significantly enhance the learning of Koopman spectral in- formation. Future work could explore physics-informed neural network designs, develop efficient training strategies to reduce computational costs, and extend the framework to handle stochastic dynamics. Ultimately, ResKoopNet and its future variants have the potential to become power- ful tools for understanding and predicting the behavior of complex dynamical systems across diverse scientific and engineering applications, and bridge the gap between data- driven methods and physical principles.
We use Multi-dimennional Scaling (MDS) visualize how these eigenfunction- features cluster according to ground truth states. MDS reduces data dimensionality based on similarities, making it ideal for visualizing clustering performance. While Uniform Manifold Approximation and Projection (UMAP) and t-distributed Stochastic Neighbor Embedding (t-SNE) are alternative methods, we show MDS results in 2D space (Figure 6B-B), with similar results for UMAP and t-SNE in Appendix Figure 13, Appendix Fig- ure 14C, D and Appendix Figure 15C, D.
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Acknowledgements
Colbrook, M. J. and Townsend, A. Rigorous data-driven computation of spectral properties of koopman operators for dynamical systems. Communications Pure and Applied Mathematics, 27(1):221-283,22
Authors are grateful to Matthew Colbrook for valuable con- sultation on theoretical aspects of ResDMD and providing fluid data, to Bin Han for discussion on computational is- sues, to Rouslan Krechetnikov for discussion turbulence analysis, to Michel Beserve and Rory Bufacchi for proof- reading and to Turishcheva Polina for providing permission of using the Sensorium 2023 dataset.
Colbrook, M. J., Ayton, L. J., and Szōke, M. Residual dynamic mode decomposition: robust and verified koop- manism. Journal of Fluid Mechanics, 955:A21, 2023.
Cybenko, G. Approximation by superpositions of sig- moidal function. Mathematics of Control, Signals, and Systems, 2(4):303-314, December 1989. doi: 10. 1007/BF02551274. URL /tttpps://ha..co science/ hal-03753170
Impact Statement
This work contributes advancing Koopman operator learn- ing through neural network-based spectral residual mini- mization. ResKoopNet enhances data-driven analysis in fields such as physics, engineering, and neuroscience by capturing continuous spectral structures and enabling au- tomatic dictionary exploration. Future work could explore implementation with physical constraints and extensions stochastic systems. No direct ethical concerns ticipated, this research primarily focuses on data-driven methods for dynamical systems analysis.
Grobman, D. M. Homeomorphisms of systems of differ- ential equations. Doklady Akademii Nauk SSSR, 128(5): 880 000-881,40 1959.
Hartman, P. A lemma in the theory of structural stability of differential equations. Proceedings of the American Mathematical Society, 11(4):610-620, 1960.
Haykin, S. Neural Networks and Learning Machines. Pearson International Edition. Pearson, 2009, ISBN 9780131293762. URL https: /bbokks///ooo.ooo google. ca ca Ba//Books@id-KCCWAAACCC
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Xu, Y., Shao, K., Ishikawa, I., Hashimoto, Y., Logothetis, N., and Shen, Z. A data-driven framework for koopman semi- group estimation in stochastic dynamical systems, 2025. URL https: /
WRRLhttss://axiie aaxiv...vvv. org/ abs/ 250.. 13301.
Yang, Zhao, J. Song, J., Wu, J., Zhao, C., and Leng, H. A hybrid method using havok analysis and machine learning for predicting chaotic time series. Entropy, 24 (3), 2022. ISSN 1099-4300. doi: 10.3390/c240300408. URL https:tt///// wwww, mápi com 1099-4300/24
3/408.
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A. Appendix
A.I. Source Code
For reproducibility, the source code is available at this link.
A.2. Calculation steps for Eq. (3)
Here we are going to show how spectral residual implies (3).
Consider @ = V/ = Enn viv, with || = 1, then
So (K((()) - X0(x))
In (dp(x)
/ [Kd(x)= xx(x)
(Ko KO - AO)
(Ko, (KC, - (A0,KØ); KO) - (KO, A0) + (16, AO)
X(VV,,AVVV - A(KMVV VV),, + (VV. VV)
NK
Nx
Nx
Nk
Kedyv)) X Nx
NK
NK
NK
KNps v
Σ
Kbbyv)l 1( _
Vivi
Σ j=1
Kibivi
Σ
dyy + 12/2
ViVii Σ
,)
1=1
j=1
j=1
Nk
Σ
VK Kabs)ay - A Ve Kabp) > Σ
(K 1lC 1l))
1212
w-=
Nk
Σ
vi
(KC)), Aebi,,Kas)' Kubs)n X(Cis, 2b)n + (1r Vj
+J=1
Nx
Mr Vrlis - - Vy Wy + +1112 1 1 1 vs
Vi
i,==
v"
V By X - + 11 v
= Eq. (3))
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A.3. Details for deriving (5)
J
res 02)2
NK 1
v
- A + 11 Vi
m
Nx
[v ( -V (W )vv
NK
1
(Vyve, Vyv,) - Uxyv,
m
-V Vyvi) + (Vx WxKvi)
Nx
- VxKv,1
( 1 V
m
Next, by matrix calculus with denominator layout convention, we try to find minimal of J:
dtr((J) dk
dJ
(since J is a scalar)
dK
Nw
d
v
dK
m i=1
Nx
d
L A - K A KK
tr
v;
dk
Nx
d
d
d
tr (vv + d Ok (v
ti [v.
to (v]
dK
dK
dK
Nk
Σ
-vvvv - +G+
Nk 222 + 2GK)vvv (G is symmetric)
=
where tr() is trace of a matrix and == U 2x x A = Wy L Vi Vy.
Since eigenvector 1/ is not a zero vector, ट. is not zero matrix. So
22 + 2GK
+ K
Remark A.1. To solve d ak tr (v Kv GKvi). we simply rewrite it as
ak tr (V, = * tr ((Kv)))((Kv))
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A.4. Discussion on convergence analysis
To understand how neural networks enhance ResKoopNet and build up the theoretical framework of convergence, it is important first introduce Barron space (Pinkus, 1999; Cybenko, 1989; Haykin, 2009; Barron, 1993). Barron space characterizes functions efficiently approximated by two-layer neural networks, which is central to ResKoopNet. By leveraging networks that approximate functions within this space, ResKoopNet can flexibly optimize the dictionary functions for Koopman operator approximation, making it highly effective for complex, high-dimensional systems.
function f belongs to Barron space B if it can be represented as:
f(x) = J ao x))(da dw),
where is the activation function, a weight vector, a is coefficient, and p probability distribution. The complexity of f is measured by the Barron norm f|l:
(%s [allwelippdee du))
|/ inf PEP,
where py is the set of distributions for which f can be represented. This framework provides foundation for analyzing approximation errors in neural networks.
The following theorem (Weinan al., 2020) discusses the approximation capabilities of two-layer neural networks within this context, establishing foundation for the subsequent analysis.
Theorem A.2 (Direct Approximation Theorem, L versoon), For any f E B and N, there exists two-layer neural network fr with neurons ((a;, w.)) such that
||A| VF
- frlll S
This implies an approximation error decreasing at rate of O(1/ (F). where r the number of neurons. In ResKoopNet, the dictionary (x; 0) (1; ") is parameterized by a neural network with parameters 0. Assuming the true dictionary functions v/2 B, Theorem A.2 ensures that Y (z; 8) can approximate the optimal dictionary spanning the Koopman invariant subspace BNx F with error >> 0, provided is sufficiently large.
Remark A.3. Notice that, the "two-layer neural network" in the Theorem A.2 statement refers a hidder layer + an output layer, which is the most standard and general setting. Our implementation uses three hidden layers. This does invalidate the result, deeper networks achieve least the same approximation power (Pinkus, 1999).
We want to show two convergence results here: (1) (z; f) approaches the true invariant subspace of KC; (2) The eigenpairs (Xe, Φ.) and pseudospectrum approximate K's true spectrum as J(0) -
Assumption A.4. To formalize convergence, we make the following assumptions:
(a) The optimal dictionary functions {w; NK spanning IC's invariant subspace lie in B.
(b) The loss J(0) is strongly convex and Lipschitz continuous in 0, and the neural network (x; has bounded gradients, facilitating gradient-based optimization.
Now, consider Barron space B which is dense in F. Given NR fixed, let BNN span [xb?] NK be the true invariant Ji=l
subspace. By Theorem A.2, for each vy, there exists a neural network approximation (,; 0,) with r neurons such that:
1| - te( 0.)1l S 1 ll
Vf
The total dictionary error is:
Nx Σ
Nk
CNK
- (1:0,)13
11 - (.(:0)))11
1
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where CNK = En | il is finite under Assumption A.4(a). As co, V(r;; 0p) - Np in the Frobenius norm, which ensures the dictionary approximated by neural network can represent BNK-
Algorithm I updates 0 via stochastic gradient descent (SGD) with step size ó and computes K( iteratively until [(,,) < e where 8% represents n-th iteration of parameters 6. For a Lipschitz continuous J()) with constant L (by Assumption A.4(b)) and a strongly convex region near the optimum 0" (assumed locally for simplicity), SGD converges at a rate of ((1/n) in expectation (Bottou et al., 2018):
- -
S 2mp
where 7) is the strong convexity constant and is the iteration number. In practice, J(0) is non-convex due the neural network, and the alternating update with K (0) stabilizes the process. As iteration step - and amount of data points m - oc, J (0,) + 0, which implies res (A, Φ.) - 0 for all i.
With (() + 0, we now assess the spectral error: (As, 6z) < c, then |/CØ; /ro:lla//lodill. < vc, indicating A and d; are approximate eigenpairs of converges to KC's true spectrum as NK + oo and - O, as pointed out in (Colbrook Townsend, 2024, Theorem B.1). Uniform convergence on compact subsets of C follows from the density of BN in F and Dini's theorem, pointed out in Colbrook & Townsend (2024, Lemma B.
The non-convexity and system complexity may slow the spectral approximation in practice. High-order convergence (e.g.. polynomial) could arise for smooth dynamics (See Colbrook & Townsend (2024, Theorem 3.1) for more details), which is useful for further study.
Remark A.5. The computational cost (Appendix A.6) scales with r, t, and m, which trades off with accuracy. Adaptive selection of r and early stopping when J(O) < e can optimize this balance.
A.5. Highlights of ResKoopNet compared with typical existing neural network- -baed Koopman framework
Our ResKoopNet method takes a fundamentally different approach from existing deep learning methods by building upon the residual- based framework of ResDMD rather than the different variational principles of VAMPnets (Tian & Wu, 2021; Wu & Noé, 2020: Mardt et al., 2018) or the deep autoencoder structure in (Lusch et al., 2018). By incorporating spectral residual measures into deep learning and introducing structured representation that captures dependencies among eigenvalues, we achieve more compact and interpretable models for nonlinear systems with continuous spectra. This approach enables us to directly minimize Koopman spectral approximation errors while avoiding the high-dimensional representations or point-spectrum limitations of previous methods.
loss functions following the
If we take the VAMP framework as an example, here are the connections and differences. The proposed loss function and the VAMP score share the goal of optimizing approximations of the Koopman operator's spectral properties, establishing a connection in their ultimate purpose. However, although they both depend on the covariance matrices (in our manuscript Equation 3.2), their methodologies differ significantly. Our residual-based method directly minimizes the spectral approx- imation error of the Koopman operator and accommodates both point and continuous spectra, while the VAMP score follows a variational framework, maximizing the sum of singular values to approximate the point spectrum, primarily for stochastic systems (though see an exception in Tian & Wu (2021)). Moreover, while VAMP is specifically designed for Markov processes and requires the Koopman operator to be Hilbert- Schhmdd,, our approach focuses on deterministic systems and enables a more comprehensive spectral analysis that incorporates continuous spectra. This distinction in scope and methodology highlights how the two frameworks complement each other in addressing different aspects of spectral estimation.
A.6. Discussion of computation costs
Despite the various advantages of the ResKoopNet framework, one significant limitation is its higher computational cost compared to the original ResDMD and other classical methods.
Theoretical Perspective: The ResKoopNet algorithm's computational demands stem primarily from its iterative optimization process. Each iteration involves a gradient descent update with complexity scaling linearly with both system dimensionality and neural network parameters. Though individual gradient steps are computationally lightweight for standard network architectures, the algorithm's 's efficiency issue lies in its repeated ceast-squares optimizations. Below we provide a detailed comparison between computation costs of several methods we used in this paper.
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The computational costs of EDMD, ResDMD, EDMD-DL, Hankel-DMD, and Nes vary significantly based on their core computational steps and specific configurations. For dataset with 10 data points and Nx- = 300 dictionary functions (for EDMD-based methods), the theoretical complexity and runtime differ across methods. EDMD involves least squares and eigenvalue decomposition, with complexity of O + +Nk), making it the fastest method, and typically requiring only seconds to minutes for computation. ResDMD extends HDMD by adding residual evaluation and pseudospectrum computation. The residual evaluation introduces an additional O(N and pseudospectrum computation across n: grid points incurs O(r. N ). resulting in total complexity of O(N m + NxNa) ). This leads to runtimes ranging from minutes to hours, depending on the resolution of the pseudospectrum grid. EDMD- DL incorporates dictionary learning through stochastic gradient descent (SGD), where each iteration involves matrix construction (O (O())m)), Koopman matrix computation (O(N ))), and neural network forward/backward propagation (0(10)), with [0] representing the total network parameter size). With l SGD iterations, the total complexity becomes k m + N + (01)). leading to runtimes also in the range of minutes to hours depending on k. ResKoopNet, which builds on EDMD-DD shares the same complexity, O m + NN + [0))). but includes the explicit use of Koopman matrix eigenvectors and optional pseudospectrum computation, making its runtime slightly longer than EDMD-DL for high-resolution spectral analysis. Hankel-DMD, using time delay embedding dimension T, constructs Hankel matrix (O(Tm)), performs singular value decomposition (SVD) (O(T'm)), and computes the eigenvalues of a reduced T x T matrix (). The total complexity is O(Tm + T'm + I"), and the runtime is heavily influenced by T, typically ranging from minutes to hours. While EDMD is computationally the most efficient, ResDMD and Hankel-DMD provide higher precision and robustness in spectral analysis at the expense of increased runtime, and EDMD-DL and ResKoopNet offer flexibility and accuracy through dictionary learning, with additional SGD iterations and optional pseudospectrum computation contributing to their computational burden.
Comparison over methods:
O
Empirical Perspective: In our experiments, without computing the pseudospectrum, the computational cost of ResDMD typically ranges from seconds to minutes. ResKoopNet, on the other hand, can require tens of minutes several hours, depending on factors such as data dimensionality, the number of snapshots, hidden layer configurations, dictionary sizes, and training convergence criteria.
Trade-off Between Cost and Accuracy: While ResKoopNet's additional computational steps introduce higher costs, they enhance the accuracy and robustness of Koopman eigenpair estimation by allowing automatic dictionary learning and minimizing spurious spectral components. This trade-off makes ResKoopNet particularly valuable in applications where precision critical. However, its computational demands render it less suitable for real-time or online Koopman model learning tasks.
Detailed Analysis of Computational Bottlenecks: The complexity O(k(N&m + N + (0)) arises from ResKoopNet's iterative optimization, alternating between updating the neural network parameters 0 and computing the Koopman matrix (0). The dominant term OKNN m)) reflects the cost of constructing data matrices V x (0) and y (0) at each iteration, followed by gradient computation. For high-dimensional systems (large m) or complex architectures (large (0]), the number of iterations required for convergence increases, particularly if the loss J(O) 1 ( xx (
is non-convex or ill-conditioned. The term O(kN2) ) stems from solving (()) = (G(0) + a))-)A(B). where matrix inversion scales cubically with Nx. The optional pseudospectrum computation, with complexity (m becomes significant for systems with continuous spectra, as n, grid points are scanned to evaluate residuals = miny, (e5 Vv;)). Comparatively, ResDMD's s lower cost (O(N3 K m + N2N4k) avoids iteration, but sacrifices dictionary adaptability.
To mitigate these costs, we will consider some methods: (1) We consider Mini-Batch SGD since it uses batch size b < m that reduces per-iteration complexity to O(N + N + (l). potentially lowering overall cost despite increased lt as validated by Barron space convergence properties. (2) We can use preconditioning to improve the condition number of G()) via diagonal scaling or incomplete Cholesky factorization could accelerate convergence, reducing k:
Theoretical Extensions: The O(1 (n) SGD rate assumes convexity, yet ((8)'s 's non-convexity suggests sublinear convergence. A bound like ||C - I (Appendix A.4) could quantify accuracy, while determining minimal NR and k for spectral completeness remains a future work. Extending ResKoopNet to stochastic systems (e.g., adding noise terms to spectral residual) embedding physical constraints (e.g., unitary spectra for Hamiltonian systems) could enhance its scope, respectively, which also aligns with future work outlined in the S.ction
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A.7. Extra tests for ResDMD and ResKoopNet
A.7.1. RESDMD WITH LESS DICTIONARY FUNCTIONS
While the original ResDMD implementation used 964 basis functions to capture the full spectrum of the nonlinear pendulum. additional tests demonstrate that the unit circle spectrum can be reasonably approximated by ResDMD with fewer observables. Figure 7 shows the progressive improvement in spectral approximation the dictionary size NK increases. With nearly 500 basis functions, ResDMD begins to adequately reconstruct the unit circle structure, though still requiring substantially more observables than the 300 basis functions needed by ResKoopNet.
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A.7.2. RESKOOPNET WITH DIFFERENT HYPERPARAMETERS
We test ResKoopNet on the dataset which has 90 initial points, used in the Section 4.1. The Figure 8 shows that when tunning the number of hidden layers from to 4 and the number of neurons in each layer from 250 to 350, the results change significantly. We can also tell that a suitable NN structure would be about 3 hidden layers with 300 neurons in each hidden layer. More neurons and hidden layers would be fore help, but cost more computational resource. This result also shows that our method is robust to hyperparameters tuning.
A.8. Hankel-DMD
1. JUSTIFICATION OF USING LANKEL-DMD AS COMPARISON IN ALL EXPERIMENTS
Hankel-DMD operates by constructing a Hankel matrix from time-delayed measurements of the system state, based on Takens' embedding theorem, which states that time-delayed coordinates can reconstruct the state space dynamical systems. Hankel-DMD also falls within the framework of Extended Dynamic Mode Decomposition (EDMD), as it effectively uses time-delayed states dictionary functions. This connection introduces convergence conditions specific to time-delay embeddings, differing from those associated with standard EDMD implementations. This makes Hankel-DMD a natural choice for comparison in the pendulum system. Specifically, the method enables a more detailed extraction of the system's 's modes and dynamics, with theoretical guarantees established in works like (Arbabi & Mezic, 2017), which proved its convergence for ergodic systems.
Practically, the approach involves constructing large matrix of time-shifted copies of measured data, where the number of delays determines how many past states are considered. This theoretically grounded framework is particularly effective when the system states have good temporal resolution and has shown strong performance in analyzing high- nigh-dimensional dynamical systems. Consequently, we also apply Hankel-DMD to the turbulence and neural dynamics experiments to evaluate its effectiveness in these representative high-dimensional settings.
As results, although its performance rivals ResKoopNet in the simple pendulum system by showing eigenvalues points near the unit circle and containing some polluted eigenvalues, which are close to the ground truth unit circle, we would like to emphasize that it capture the point spectrum and miss the full spectral information. When it comes high-dimensional systems, fails to capture key dynamics in higher-dimensional systems, as seen in the later experiment (Section 4.2 and Section 4.3).
A.8.2. APPLICATION IN TURBULENCE
Here we present the Koopman modes computed by Hankel-DMD for comparison with the ResKoopNet results. As shown in Figure 9, despite having small residuals, these modes fail to clearly capture the fundamental pressure field structure that was successfully identified by ResKoopNet's first Koopman mode (see Figure 5). The discrepancy observed for
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and each column shows the model with a different neuron count (250, 275, 300, 325, 350).
Hankel-DMD (Arbabi & Mezic, 2017) is due its fundamentally different theoretical framework. While ResKoopNet (and other EDMD-based methods) use Galerkin framework with convergence guarantees established via minimizing the spectral residual, the Hankel-DMD approach is based on Takens' embedding theorem (Takens, 2006). Therefore, the spectral residual metric, which is derived and justified within the Galerkin projection framework, does not fully extend to the Hankel-DMD setting. Even if Hankel-DMD DMD yields small residual, this does not ensure that its modes accurately capture the underlying dynamics (such as the fundamental pressure field structure).
A.9. Practical details for neural data analysis
1.9.1. DATASET DETAILS AND EXPERIMENTAL SETUP
The dataset utilized in this study is part of the open dataset provided for the 'Sensorium 2023' competition (Turishcheva et al., 2024b). The dataset consists of calcium imaging recordings from the primary visual cortex of mice. During the experiments, the mice were presented with natural video stimuli while the activity of thousands of neurons was recorded. The objective of the competition is to predict large-scale neuronal population activity in response to different frames of the stimulus videos, based the hypothesis that population dynamics in the primary visual cortex, driven by visual stimuli, encode significant information about the dynamics of the videos (Basole et al., 2003; Onat et al., 2011; Hénaff et al., 2021).
A.9.2. TASK DEFINITION AND RATIONALE
In contrast to the competition's 's prediction objective, our study focuses on the task of state partitioning of neural signals. While prediction remains feasible, we aim to demonstrate that state partitioning is sufficient to highlight the superiority of ResKoopNet over series of other methods in uncovering the latent dynamies of the system. Specifically, each experiment,
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Figure 9: The plots illustrate turbulence detection using the four Koopman modes computed by Hankel-DMD, which are ranked with their corresponding spectral residual from the smallest, similar as in ResKoopNet case.
set of six video stimuli was repeatedly presented to each mouse, creating ideal conditions for defining brain states. The recording setup remained consistent for each mouse, ensuring that the neural activities could be interpreted as originating from the same dynamical system, with the primary variable being the input stimulus.
We hypothesize that during repeated trials with identical visual stimuli, the underlying dynamics of the neural system remain consistent. Consequently, the recurrence of the same brain state expected during these trials. This provides reliable basis for testing the efficacy of Koopman decomposition methods in uncovering latent dynamics and distinguishing these states.
A.9.3. DATASET STRUCTURE AND DIMENSIONALITY
The dataset includes neural recordings from five mice, with each mouse responding to six distinct video stimuli, presented in 9-10 repeated trials (resulting in approximately 60 trials in total). Each trial involves recordings of over 7000 neurons. The duration each video stimulus is 10 seconds, with a sampling rate of 50 Hz, yielding 300 data points (299 snapshots) per trial. Thus, the data be analyzed consists of a high-ding dinensiinnii time series with 7000+ observables per snapshot.
A.9.4. IMPLEMENTATIONS OF RESKOOPNET AND OTHER CLASSICAL METHODS
We compare here four methods: the proposed ResKoopNet and three classical Koopman decomposition methods for high-dimensional systems: the Hankel-DMD, the EDMD with RBF basis, and the Kernel ResDMD. We applied them to the 5 datasets, although with slightly different implementations and different dimensions of approximated Koopman invariance subspace.
For ResKoopNet, we train the dictionaries with all the snapshots recorded in each mouse such that the total snapshot
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number is the product of the snapshot number in one trial and the number of all trials. This is to avoid overfitting with the small snapshot numbers within trial. The high-dimensional data is first reduced to 300 dimensions with Singular Value Decomposition. The dimension of the Koopman subspace is chosen to be 50 (to be compared with Hankel DMD fairly), consisting of 25 trained bases and 25 pre-chosen ones (constant and the first-degree polynomials of the SVD-ed 24 dimensions). One can find the decomposed eigenfunctions in Figure (A(t))), with a marker of the ground truth state separations based on stimulus identity.
For Hankel-DMD. the Koopman eigenfunctions were approximated using the eigenvectors of the Hankel matrix. Specifically, the Hankel matrix was formed as in Equation 53 from Arbabi & Mezic (2017), using all the observables from one trial of each mouse with delay of 50. Consequently, the snapshot size became 249 times the observable number, and the resulting number of eigenfunctions was 50, each with length of 50. The Hankel-DMD eigenfunctions for each trial of data are shown in Figure 6A (bottom), alongside the ground truth trial identities for comparison.
For EDMD with RBF basis, the nigh-dimensional dataset is first reduced to 300 dimensions with SVD. Then RBF basis is calculated with 1000 RBF functions. The choice of the basis number is decided based on classical experiments of using RBF basis to estimate the Koopman operator of Duffing systems (Li et al., 2017).
For Kernel ResDMD, it is variant of Kernel EDMD (Kevrekidis et al., 2016), the dimension of the Koopman invariant subspace should corresponds to the sample number (in time). Given the data size to be 300, we have 299 snapshots, resulting in 299 Koopman bases. The detailed calculated is performed for each trial with the program provided in the original ResDMD paper (Colbrook et al., 2023; Colbrook & Townsend, 2024). We chose the kernel function as the commonly-used normalized Gaussian function in the calculation.
The Koopman eigenfunctions from both ResKoopNet and other methods represent dynamical features corresponding to one of the six video stimuli. To evaluate how well the eigenfunctions capture the latent dynamics, we assess the similarity of the features for trials with the same stimulus and their dissimilarity from those corresponding to different stimuli. Effectively, this makes the problem a clustering task, where the separability of the Koopman eigenfunctions reflects how well they capture the key dynamic components related to the stimuli.
A.10. Choice justification of dictionary sizes
In this section, we provide justifications for the use of different dictionary sizes (i.e., the number of Koopman eigenfunctions) in the aforementioned four methods for the neural dynamics experiment.
First, the high-dimensional data was pre-pree proppeoeeeeed using SVD to reduce its dimensionality to 300. Then, for the four methods:
1. For ResKoopNet, we selected 25 trained basis functions and 24 first-order monomial basis functions the dictionary for the 24 reduced observables. This choice ensures the dictionary size is the same as Hankel DMD, making it fair comparison between the performance of these two methods. The last two methods uses higher dictionary sizes but still remain ineffective in capturing temporal dynamics, thus not affecting the fairness of choosing 50 dietionaries here. The size of the trained dictionary was set to be at least equal to the original observable size to ensure a sufficiently rich dictionary for the Koopman invariant subspace.
2. For Hankel DMD, the number of delays (as dictionary size/number of eigenfunctions) is first constrained by the temporal sample size (i.e., snapshot size) because it cannot exceed the maximum snapshot size. Therefore, it is impossible to choose the same dictionary size as the ResKoopNet example. Choosing the delay too small will result in an insufficient dictionary size to span the Koopman invariant subspace, and too large will reduce the actual snapshot size to estimate the covariance matrices in the estimation of the Koopman matrix. Therefore, we chose a compromise delay number of 50 that satisfies both needs.
3. For RBF basis, in principle, we can use the same dictionary size. However, our previous experience with a similar dataset and the results of using the RBF basis for the EDMD method all suggest that the performance will be better with more dictionary functions. Therefore, we chose 1000 RBF basis and the original 300 first-order monomial basis a better condition compared to the same dictionary size with ResKoopNet.
4. For Kernel ResDMD, the dictionary size is theoretically determined to be the number of snapshots. Therefore, we cannot make the dictionary size consistent with the ResKoopNet example.
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Based the above justifications, we believe our choices of dictionary sizes reasonable and ensure fair comparison across the methods.
A. 1... VISUALIZATION AND CLUSTERING PERFORMANCE
To visualize the clustering of high-dimension Koopman eigenfunctions, we perform dimensionality reduction using Multi-dimensional Scaling (MDS). MDS particularly useful for visualizing high-dimensional data by preserving pairwise similarities (Kruskal, 1964) (here we use correlation as a measure of similarities). While UMAP (McInnes et al., 2018) and t-SNE (Van der Maaten & Hinton, 2008) are alternative visualization methods, with different emphasis on global- relationships, we primarily use MDS in this study and provide UMAP and t-SNE results in the supplementary materials (see Appendix Figure 13A, B. Appendix Figure 14C, D and Appendix Figure 15C, D). UMAP in implementation is still correlation-based For t-SNE estimation use the perplexity of 15, value for optimal separation.
By applying MDS, the high-dimensional eigenfunction- baased features are reduced to a low-dimensional space. For illustration, we present the results of reducing the feature space to two dimensions (Figure 6B-E). The ResKoopNet reduced features for the six types trials (corresponding to the six video stimuli) are well-separated for all five mice (Figure 6B). In contrast, the Hankel-DMD features show no clear clustering structure (Figure 6C). Similarly, the features produced by EDMD with RBF basis and Kernel ResDMD do show clear separability (Figure 6D-E, Appendix Figure 14B-D, Appendix Figure 15B-D).
A. 10... CLUSTERING QUALITY METRICS
We further quantified the clustering quality by calculating the Davies-Boulde Index (DBI) for both Koopman decomposition methods across all mice (Figure 6F). The DBI designed to assess the compactness of clusters and the separability between them. lower DBI indicates better clustering performance. ResKoopNet features yield significantly lower DBI scores compared to other methods, confirming that ResKoopNet produces more clearly defined clusters corresponding to the ground truth trials. Similar clustering results are observed with UMAP and t-SNE (see Appendix Figure 16), further supporting the superior performance of ResKoopNet in capturing the latent dynamic structure compared to the other classical methods.
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Figure 12: Comparison of clustering quality based Davies-Bouldis Indices for clustering with mean firing rates and
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Reviews:
[{"review_id": "sxQC35xrUj", "reviewer": "Reviewer_1CwD", "summary": "The paper introduces ResKoopNet, a neural network-based method for learning Koopman operator representations of high-dimensional nonlinear dynamical systems. \nResKoopNet aims to address limitations of previous methods of learning Koopman operators from data such as Extended Dynamic Mode Decomposition that discover spurious eigenpairs from data, and the spectral inclusion problem related to the difficulty in capturing the entire true spectrum of the Koopman operator, especially for systems with continuous spectra.\n\nThe method extends Residual Dynamic Mode Decomposition (ResDMD) by explicitly minimizing a spectral residual loss function. The minimisation of the spectral residual contributes in avoiding spectral pollution, where the discretisation of the infinite dimensional operator to a finite matrix results in the discovery of spurious eigenvalues that are numerical artifacts.\n\nThe main contributions  of  this submission is using a feedforward neural network to automatically select the dictionary functions, overcoming the limitations imposed by predefined basis dictionaries used in the approximation of the operator, and using the spectral residual loss in the optimisation.\n Through minimising the spectral residual, ResKoopNet aims to approximate both discrete and continuous spectra.\nNumerical results demonstrate ResKoopNet's accuracy on several example systems: on a classical pendulum system, on a turbulent flow, and on neural dynamics from mouse visual cortex.\n\nFor the pendulum system the proposed approach is shown to outperform existing methods in approximating the Koopman operator in terms of the number of basis functions (observables) required for accurate spectrum approximation (proposed method seems to require much fewer basis functions compared to ResDMD for same amount of data ) .", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "1. We would like to thank the reviewer for the helpful comments. The updated figures and proof can be seen here: https://anonymous.4open.science/r/rebuttal_materials-14918/\n\n2. Regarding the benchmarking question:\n \n(1) In the 2nd experiment, the 300 basis functions we have chosen are indeed different from the 250 basis functions used in the original ResDMD paper (Colbrook & Townsend, 2024). We have re-done the experiment and the results are shown in **Fig5_new.png**; \n\n(2) In the neural experiment, to fairly compare with the 50 bases used in Hankel-DMD, we re-estimated the Koopman eigenfunctions using 50 dictionaries (24 SVD-truncated bases, one constant, and 25 trainable bases). The performance remained comparable to the case with 501 bases. See **Fig_6_new.png and Fig_13.png, Fig_16.png** for the averaged eigenfunctions and clustering results.\nWe would like to clarify that the results are robust to hyperparameter choice (hidden layer number and neuron size in the hidden layer) when they reach a threshold for a given dataset. To illustrate this, we have added a hyperparameter scan result in **Fig_8.png** by scanning the hidden layer size from 1 to 4 and each layer’s neuron size in the range of [250, 275, 300, 325, 350]. The figure shows that the performance is robust with the increase of both hyperparameters, suggesting that varying the layer size will not trigger sensitive changes in the approximation results after a certain threshold. Similar robustness results are shown for the neural experiment in **Fig_17.png**: with smaller layer size and layer number, the clustering performance is not stable but becomes robust after the two hyperparameters reach a threshold. This justifies our network structure of 3 layers of 200 neurons.\n\n3. We agree that the topic of “pseudospectrum” should be separated. We can put it at the bottom of Section 3.1 with the title “Continuous Spectra and Pseudospectrum”. \n\n4. We will add some background details for turbulence example: ‘The turbulent flow dataset from Colbrook & Townsend (2024, Section 6.3) models a two-dimensional airfoil system with Reynolds number $3.88 \\times 10^5$ and Mach number $0.07$. The data captures a pressure field at 295,122 spatial points across 798 time steps, sampled every $2 \\times 10^{-5}$ seconds’.\n\n5. We will extend the computational cost analysis and incorporate computational bottlenecks and theoretical extensions. \n\n6. The colormap values of all Koopman modes in **Fig5_new.png** now have the same scale.  \n\n7. In the turbulence experiment, we will add further explanation of how truncated SVD is applied before using ResKoopNet. We applied a change of basis method here; specifically, consider the data matrix decomposed by truncated SVD $X=USV^\\top$ with truncation $k=150$ singular values, then multiplying $V$ on both sides from the right to get $XV=US$, which is the lower dimensional data matrix projected by matrix $V$; then we apply ResKoopNet and compute its (low-dimensional) Koopman modes; then multiply the matrix $V^\\top$ from the left to recover the original Koopman modes. Koopman modes and eigenfunctions are ranked by ascending spectral residuals, where smaller residuals indicate better approximations. \n\n8. We added singular value plots in Figures 5(e) and (f) (see **Fig5_new.png**), revealing a significantly large singular value corresponding to the dominant spatial pattern captured by Koopman mode 1 in Figure 5(b). \n\n9. Yes, we used SVD for both high-dimensional experiments(2nd and 3rd). In the turbulence example, we chose the 150 reduced dimension following Colbrook & Townsend, 2024. In the neural example, we reduce the data into 300 dimensions as it is a manageable dimension compared to the original dimension (>7000), but thanks to the reviewer we have demonstrated that with a smaller truncated dimension (24) we also get meaningful eigenfunctions (see previous reply). We also use the same method (filtered by spectral residual) in selecting eigenvalues in 1st pendulum experiment, selecting Koopman modes in 2nd turbulence experiment, and selecting eigenfunctions in 3rd neural experiment.\n\n10. In **Fig5_new.png (c)(d)**, we have also illustrated the acoustic vibration and turbulent fluctuation characteristics. The original Hankel-DMD results in Appendix Figure 7 are similar to Figure 5(c)(d) and those in the ResDMD paper, which has no new characteristic to specify.\n\n11. Regarding the SVD procedure: we first reduced spatial dimension to 150 using truncated SVD. Then, we applied ResKoopNet to obtain 1+150+149=300 Koopman modes ranked by ascending spectral residual. Here, '1' denotes a constant non-trainable basis, '150' are SVD-reduced spatial coordinates, and '149' are trainable bases (later adjusted to '99' for a total of 250 to match the original ResDMD setup). We then selected the less \"polluted\" (low-dimensional) Koopman modes and mapped them back to the original high-dimensional space using the SVD matrix $V$.", "claims_and_evidence": "The authors claim that ResKoopNet addresses the spectral inclusion and spectral pollution issues of previous Koopman approximation methods. For the pendulum system the authors demonstrate that the method effectively tackles the spectral pollution issue and the spectral inclusion by increasing the number of observables.", "methods_and_evaluation_criteria": "- The use of spectral residual as a loss function is conceptually valid and justified given existing literature, making it suitable for evaluating the accuracy of spectral approximation of the operator.\n\n\n\n- The choice of the pendulum system as a benchmark makes absolutely sense, because the method can be compared against ground truth. However the same is not true for the other two benchmarks. For the turbulent flow experiment I am not entirely convinced that the method outperforms competing methods (see my questions below), while regarding the neural data experiment I am not sure about the motivation of applying the method in this setting.\n\n- However, more explicit justification for choosing specific hyperparameters (e.g. the size and structure of neural networks, number of eigenfunctions, and dimensionality reductions) would be beneficial. See also my questions below.", "theoretical_claims": "I checked the derivations in A.2 and A.3", "experimental_designs_or_analyses": "Yes I checked the presented experiences and I have included some of my issues and comments also in the other comments\n\nRegarding the experiment with the application on neural data:\n- I am not sure why the authors choose to compare their method and the competing method in Figure 6A using a different number of eigenfunction (500 vs 50). In A9 the authors justify partly their choice, but I would expect to see how their method performs with 50 basis functions to compare it to Hankel-DMD, and to see in the main text a comparison with a competing methods with the same number of functions. Since the constraint for the Hankel-DMD and the kernel ResDMD is the number of snapshots I would expect first to see a comparison of all methods with this number of bases, and then additionally present results with more bases where possible. Thus, I am not entirely convinced about the validity of the remaining analyses performed on this dataset, but I remain open to being persuaded otherwise by the authors.", "relation_to_broader_scientific_literature": "The method builds upon extensive literature on data-driven approximation of dynamical systems, and in particular on data-driven approximation of the Koopman operator. Specifically it extends recent work that uses the spectral residual to clear the identified spectral eigenpairs from spurious eigenpairs introduced through the numerical approximation of the infinite operator. It extends the existing literature by first using a neural network as a dictionary of observables/basis functions instead of using pre-selected basis, and uses the spectral residual as a loss function. Existing DMD methods optimise the dictionary of active basis function in the approximation by including a L1 cost on the basis coefficients.", "essential_references_not_discussed": "I am not aware of any, but I do not word directly in this sub-field.", "other_strengths_and_weaknesses": "**Strengths:**\n- Identifies Koopman operator by directly optimising the error of the approximation of the operator in the eigenspace\n- Can approximate the operator for systems with continuous spectra, unlike EDMD-like approaches\n\n\n\n**Weaknesses:**\n\n- As mentioned by the authors the method is sensitive to hyperparameter selection.\n\n- The writing and organising of the text could benefit from a bit of restructuring. For instance, the reference to the pseudospectrum in page 4 should be in my view under a separate subsection. Also when describing the experiments, the reader needs sometimes more background details to understand the setting. E.g. when describing the turbulent example the reader has to go to the paper of  Colbrook &\nTownsend (2024) to understand the experimental setting.\n- High computational demands discussed in A.5, but it is acceptable as first implementation of the approach that can be improved on in follow-up work", "other_comments_or_suggestions": "- Page 5: the authors write \" Even with a much larger amount of dataset\": I think they want to write amount of datapoints\"\n- the fonts in most figures are too small to be readable.\n- In Fig. 5, since you are trying to relate the 2D pressure field of the original system to the first eigenfunction of the learned Koopman operator, I would suggest to rescale the colormaps so that the associated regions in the two upper plots have same/similar colouring. (minor)", "questions_for_authors": "- For the high dimensional turbulent system, do you reduce first the dimensionality of the state with SVD, and then compute the Koopman approximation? If yes, how do you select the reduced dimensionality? Can you show a plot with the singular values? Do you follow the same procedure for all methods?\n\n- For the turbulent flow experiment, in the paper of Colbrook &Townsend (2024), the authors show both for DMD and ResDMD that the eigenfunctions have some characteristic details indicative of the acoustic sources. In the plots you provide in the main text from the proposed method, and in the appendix from the Hankel-DMD these details are absent. As a non-specialist, I wonder why this is the case, and whether you can still claim successful approximation of the operator, when such details are not captured by the proposed method.\n- Related to the previous question, throughout the high dimensional experiments the authors seem to select to first reduce the dimensionality of the data to 300 through SVD and then apply the method. How do you select that value?", "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": "", "extra_scores": {}}, {"review_id": "M5i1b3x0Vl", "reviewer": "Reviewer_QgoQ", "summary": "The paper focuses on Koopman operator analysis and builds on the Residual Dynamic Mode Decomposition (ResDMD), which uses the spectral residual to evaluat the accuracy of a Koopman operator approximation and to perform filtering of a computed spectrum. Here, the proposal is to use the spectral residual iteratively during the spectral estimation process, iterating between optimizing a neural network that parameterizes a dictionary and minimizing the residual to identify the Koopman eigenpairs. The paper reports numerical experiments for multiple systems/datasets including a pendulum system, a turbulence system, and neural recordings from the visual cortex of mice.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "1. We would like to thank the reviewer for the helpful comments and suggestions. The updated figures and proof can be seen here: https://anonymous.4open.science/r/rebuttal_materials-14918/\n\n2. Indeed, we didn’t notice the paper [R1]: \"Another Look at Residual Dynamic Mode Decomposition in the Regime of Fewer Snapshots than Dictionary Size\" and we would like to thank the reviewer for pointing it out. As the reviewer mentions, we indeed conducted our work before the emergence of [R1]. However, after reading this paper, we confirm that Algorithms 7 and 8 in [R1] are different from our proposed method. We first briefly go over both algorithms here: Algorithm 7 is almost the same as the kernel-ResDMD algorithm in the original ResDMD paper(Colbrook & Townsend (2024)) except for the condition of fewer snapshots than dictionary size. It still computes the $\\epsilon$-pseudospectrum and corresponding approximate eigenfunctions as in ResDMD. The grid points $z_j$ serve as a candidate of eigenvalue to be tested on in the 3rd step of Algorithm 7, and will be kept only if its ‘’residual” is smaller than a threshold. However, the dictionary used in ‘’residual” is important. In our ResKoopNet method, the dictionary is obtained by minimizing ‘’spectral residual”, which is theoretically guaranteed by Appendix B.2 in the original ResDMD paper(Colbrook & Townsend (2024)). As for Algorithm 8, it is just a variant of Algorithm 7 where a specific candidate of eigenvalue $\\lambda$ replaces the grid points $z_j$. So, we would like to point out that our contribution has nothing to do with any modification of improvement on any specific part in ResDMD such as pseudospectrum, rather we proposed a computing method that exploits the theoretical framework of spectral residual and addresses the  **spectral inclusion** problem, and validate it in a few examples including a real-world problem.\n\n3. Based on the reviewer’s suggestion, we have improved the convergence analysis, which is mentioned in the file **convergence_proof.pdf**.  Since the function space $\\mathcal{F}$ we considered here is $L^2$, we will be able to apply Theorem A.3. Under a few assumptions as in Assumption A.4, we have shown the convergence of the trained dictionary to the optimal dictionary and the convergence of estimated eigenpairs to the true spectrum of the Koopman operator $\\mathcal{K}$. Now it not only shows the existence of the dictionary but also shows the convergence of the dictionary and we would like to thank the reviewer for the suggestion.\n\n4. The last part of Section 3.2 named ‘’Computing Algorithm” on page 4 briefly introduces the architectures and layers, etc. We have investigated other parameter settings and tried different layer numbers. Specifically, in the pendulum example, we have scanned the layer number in the range [1,2,3,4] and layer size in the range of [250, 275, 300, 325, 350] and demonstrated the robustness of our parameter choices. This result is shown in **Fig_8.png**. The hyperparameter scanning results for the neural experiment are included in **Fig_17.png**: with smaller layer sizes and layer numbers, the clustering performance is not stable but becomes robust after the two hyperparameters reach a threshold. However, integrating the spectral residual-based Koopman operator approximation method with more complicated neural networks like PINO or KAN is beyond the scope of current work and will be addressed in future extensions.", "claims_and_evidence": "Please see \"Other Strengths and Weaknesses\"", "methods_and_evaluation_criteria": "Please see \"Other Strengths and Weaknesses\"", "theoretical_claims": "I checked the provided calculation steps in Appendices A.1 and A.2.", "experimental_designs_or_analyses": "Please see \"Other Strengths and Weaknesses\"", "relation_to_broader_scientific_literature": "Please see \"Other Strengths and Weaknesses\"", "essential_references_not_discussed": "[R1] Colbrook, Matthew J. \"Another look at Residual Dynamic Mode Decomposition in the regime of fewer Snapshots than Dictionary Size.\" Physica D: Nonlinear Phenomena 469 (2024): 134341.", "other_strengths_and_weaknesses": "Strengths\nS1.\tThe paper introduces a novel Koopman analysis framework.\n\nS2.\tThe reported experiments are thorough and the results demonstrate a noticeable improvement over existing techniques. \n\nS3.\tSection 4.3 is particularly interesting and highlights the strengths of the proposed methods in terms of its ability to capture key dynamics and perform effective clustering. \n\nS4. The paper provides a good discussion of relationships to other neural-network based Koopman operator frameworks (Appendix A.4). \n\nS5. The discussion of the computational costs (Appendix A.5) is helpful.\n\nS6. There is good detail and justification for experimental design choices (Appendix A.9) \n\nS7. The visualizations and clustering quality analysis, and related discussion, provide valuable insight into the behaviour of the proposed method (Appendix A.9).\n\nWeaknesses\n\nW1.\tThe paper’s novel technical contribution is limited. The paper does not cite [R1] (available from 30. Aug 2024), which outlines an algorithm (Algorithms 7 and 8) to compute the pseudospectra by minimizing the residual. This appears to be identical to the minimization proposed here for a fixed dictionary. Given that this component of the algorithm has already been published, the remaining innovation of this paper consists of incorporating a neural network to parameterize the basis functions and iterating between the minimization steps. As the authors acknowledge, this approach has been widely used in the neural Koopman operator literature, albeit with the focus on the prediction-based loss function (e.g., Li et al., 2017). \n\nW2.\tDespite its key innovation being the introduction of a neural network to jointly learn the dictionary while minimizing the spectral residuals to determine the Koopman eigenpairs, the experimental section of the paper does not explore any design choices related to the neural network (architecture, layers, convergence, etc.) \n\n[R1] Colbrook, Matthew J. \"Another look at Residual Dynamic Mode Decomposition in the regime of fewer Snapshots than Dictionary Size.\" Physica D: Nonlinear Phenomena 469 (2024): 134341.\n\n\nAFTER REBUTTAL:\n(1) The authors have clarified the differences between their work and [R1].\n(2) There has been further experimental exploration of the architectural choices. \n(3) The authors have provided further theoretical results concerning convergence. While these are welcome and strengthen the paper, I would observe that the inclusion of a proof in the rebuttal appears to violate the review process policies, which impose a strict character limit and require that links only contain figures and tables. This inclusion potentially provides the authors of this paper with an unfair advantage. I would like the Area Chair to comment on this. \n\nWith the above factors in mind, I have increased my overall score from 2 (weak reject) to 3 (weak accept).", "other_comments_or_suggestions": "The paper’s contribution would be considerably stronger if it:\n(a)\tExpanded on the convergence analysis in Appendix A.3. The current analysis is very brief and does not provide a clearly-stated convergence result. The final discussion essentially focuses on the existence of the dictionary rather than the ability of the algorithm to converge to it. \n\n(b)\tConducted a more thorough investigation of the potential ways to incorporate the neural network (architectural choices, learning behaviour, convergence). Since this is the major novel contribution of the work, seems neglected in the experimental analysis.  I suspect that the authors were unaware of [R1] (or even conducted some of the research associated with this work prior to the emergence of that paper). If this paper were the first to propose minimizing the spectral residual (as well as incorporating a neural network) then the contribution would be more substantial and it would be more reasonable to focus entirely on other non-neural experimental aspects.\n\n{R1] Colbrook, Matthew J. \"Another look at Residual Dynamic Mode Decomposition in the regime of fewer Snapshots than Dictionary Size.\" Physica D: Nonlinear Phenomena 469 (2024): 134341.", "questions_for_authors": "Q1. It is possible that there is a more meaningful distinction between Algorithms 7 & 8 in [R1] and the procedure proposed here, and that I have misinterpreted one of the algorithms due to notational differences. If so, I would appreciate if the authors could clarify what they perceive as the critical difference.", "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": "", "extra_scores": {}}, {"review_id": "bYWxm1SepF", "reviewer": "Reviewer_K3yo", "summary": "This research presents a novel method for approximating the spectral components of the Koopman operator for discrete-time deterministic dynamical systems by minimizing spectral residuals. Unlike traditional methods that rely on predefined dictionaries, this approach utilizes a neural network to optimize dictionary functions to discover a more precise and complete spectrum of the Koopman operator for complex dynamical systems. The method employs an alternating optimization procedure, where a neural network learns the suitable observables while the Koopman matrix is iteratively updated via least-squares estimation. The method's effectiveness is validated on a pendulum system, a turbulence model, and neural recordings, demonstrating superior accuracy over classical methods.1", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "1. We would like to thank the reviewer for the helpful comments. The updated figures and proof can be seen here: https://anonymous.4open.science/r/rebuttal_materials-14918/\n\n2. Regarding “Hankel-DMD works almost as well as the proposed method”: While Hankel-DMD performs well in Experiment 1's lower-dimensional case, it underperforms in Experiments 2 and 3's higher-dimensional settings. We included it across all experiments for consistency and fair comparison. In Experiment 2, ResKoopNet successfully captured a dominant Koopman mode showing the main air pressure structure near the airfoil that both ResDMD and Hankel-DMD missed (see ResDMD Fig.17 and our Appendix Fig.7) In Experiment 3, ResKoopNet succeeded whereas Hankel-DMD and others failed. These results show that our method addresses the spectral inclusion by capturing important spectral components that other methods miss.\n\n3. We acknowledge that our original turbulence example used 300 basis functions, while ResDMD used 250. We re-did the experiment and the updated results are shown in **Fig_5_new.png**.\n\n4. In the neural experiment, to ensure a fair comparison with the 50 bases in Hankel-DMD, we re-estimated Koopman eigenfunctions from ResKoopNet using 50 dictionary elements (24 SVD-truncated bases, one constant, and 25 trainable bases). The resulting performance is comparably strong to the original 501-basis case. Detailed approximations of eigenfunctions and clustering outcomes are provided in the updated **Fig_6_new.png, Fig_13.png and Fig_16.png**.\n\n5. Regarding the optimality comments:\n\n(1) Regarding the comment that ‘in which sense the neural network's learned representation is optimal’: the ‘optimal’ representation in ResKoopNet is not optimal in the classical sense (e.g. a unique minimizer of an L2 error as in EDMD); rather, it means that the learned dictionary minimizes a very specific loss: ‘spectral residual’ so that the finite-dimensional Koopman operator best captures the true spectral properties (both discrete and continuous as explained in ResDMD of the underlying Koopman operator given the data. In other words, by minimizing the spectral residual, the network is trained to produce a dictionary (or representation) that ‘optimally’ balances the ability to approximate the Koopman eigenpairs against the error introduced by discretization and finite sampling. Minimizing this loss(Eq.7 in the paper) pushes the network toward a representation where the computed eigenpairs have a very low spectral residual. \n\n(2) Regarding the comment that 'with a finite number of samples, one cannot be sure that if it is small, it means that the real object is also small', the authors justify the 'optimality' by proving convergence results in Appendix A.4(see **convergence_proof.pdf**): as the number of samples increases and as spectral residual loss function $J(\\theta)$ below a threshold, the approximation of the dictionary converges (and hence the `optimality’ hold). So while practically the cost function is an estimate, the theory from ResDMD assures that, in the limit, a near-zero cost implies a good approximation of the true Koopman spectrum.\n\n(3) Additionally, the convergence analysis in Appendix A.4 (pages 14–15, in red) is expanded to demonstrate the convergence of both the dictionary and the spectrum, see **convergence_proof.pdf**.\n\n(4) The data may contain noise, but our work uses a deterministic framework without accounting for stochastic effects. We plan to address noise and stochasticity in future research (see the Conclusion Section (second paragraph)).\n\n6. Regarding other DMD methods with neural networks: Since ResDMD is EDMD-based, we compared it with EDMD-DL(Li et al., 2017). ResDMD and kernel-ResDMD have full spectrum theoretical guarantees but do not address spectral inclusion issues, while EDMD and EDMD-DL are effective in both low and high dimensions, they lack similar guarantees as ResDMD. Future work will explore advanced neural network structures, e.g., PINO or KAN, and include further comparison with other neural network-based DMD methods.\n\n7. We performed hyperparameter tuning and included them in **Fig_8.png**, demonstrating the robustness of our parameter choices. The hyperparameter scanning results for the neural experiment are included in **Fig_17.png**: with smaller layer size and layer number, the clustering performance is not stable but becomes robust after the two hyperparameters reach a threshold. Therefore we have chosen 3 layers of 200 neurons.\n\n8. We have now used the same normalized color map value for each Koopman mode figure (**Fig_5_new.png**). \nHankel-DMD does not perform as well as ResKoopNet in Fig. 7 of Appendix A7.2 because its foundation(Takens’ embedding) is different from the Galerkin framework used in EDMD and ResDMD, from which the spectral residual metric is derived. Thus, even a small residual for Hankel-DMD may not capture key dynamics (like the main pressure field structure and clustering brain state).", "claims_and_evidence": "1. The authors claim that their method addresses spectral inclusion and reduces spectral pollution for complex dynamical systems of high dimensions, which is not well-supported. In none of the experiments do they show that there exists a subset of the true spectrum that is not attainable from ResDMD or other methods, but instead that their method can do that. In Experiment 1, which is a low-dimensional system, they only show that for finding the whole spectrum, ResDMD needs a bigger size dictionary. Also, it is noteworthy that Hankel-DMD works almost as well as the proposed method. In Experiment 2, the authors do not directly compare their method with ResDMD using the same size dictionary. While they are using Nk = 300, the figures in the ResDMD are when Nk = 250. This problem also exists in the last experiments, and the size of the dictionaries is very different. So in a nutshell, for the first experiment (low-dim) they did not show that they are able to find some part of the spectrum that the other methods cannot, and for the others (high-dim) the comparison is not fair in my opinion. Thus, I would suggest the authors provide more evidence to support their argument.\n\n2. The authors use the term “optimal” representation several times in the paper, but they do not specify in which sense the neural network's learned representation is optimal. How do they know that? Since the cost function they are using is an estimate of spectral residual, with a finite number of samples, one cannot be sure that if it is small, it means that the real object is also small. Furthermore, there might be noise in the data.\n\n3. In addition, the claim for convergence analysis is a bit shaky, which I will elaborate on in the theoretical claims section.", "methods_and_evaluation_criteria": "Yes, the proposed method is sensible in general. Using neural networks to learn representations for the Koopman operator is a well-known technique in the field. Also, the datasets used are aligned with the proposed method and the problem. However, I am not sure that the evaluation criteria for showing the superiority of the methods make sense if there is not a fair comparison between different methods.", "theoretical_claims": "Yes, I checked all the theoretical claims in the appendix. The authors demonstrate equivalences between different equations presented in the main body of the paper, which appear correct. However, the convergence analysis is, in my opinion, somewhat shaky. First, the proposed method uses a three-layer neural network, whereas the theory provided is for two-layer networks. Second, the norm of the functions derived from density arguments is unknown. Thus, as we considering sum of the norm of these functions for each mode it is not clear how one might ensure the epsilon guarantee.", "experimental_designs_or_analyses": "I checked the experiments and analysis. As I mentioned before, the experiment design could be improved to make the comparison more fair.", "relation_to_broader_scientific_literature": "The key contributions of the paper build upon prior advancements in Koopman operator theory, particularly addressing limitations in existing spectral approximation methods such as Extended Dynamic Mode Decomposition (EDMD) (Williams et al., 2015) and Residual Dynamic Mode Decomposition (ResDMD) (Colbrook & Townsend, 2024). While EDMD provides a finite-dimensional approximation of the Koopman operator, it suffers from spectral pollution and lacks guarantees of capturing the full spectrum, especially in high-dimensional systems. ResDMD improves upon this by introducing spectral residuals to filter inaccurate eigenvalues, but it still relies on predefined dictionaries and does not refine spectral estimates. ResKoopNet advances this line of research by optimizing dictionary functions through a neural network, thus addressing the spectral inclusion problem and improving spectral accuracy. This aligns with recent trends in machine learning-assisted dynamical systems analysis, such as deep autoencoders for Koopman embeddings (Lusch et al., 2018) and kernel-based Koopman methods (Kevrekidis et al., 2016), but differs in its explicit spectral residual minimization approach.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "The paper is highly original in its integration of spectral residual minimization with neural network-based dictionary learning, addressing key limitations of existing Koopman approximation methods. Its significance lies in improving spectral accuracy for complex dynamical systems, with strong implications for physics, engineering, and neuroscience. In terms of clarity, the paper is generally well-written and the concepts are explained in a logical flow. Overall, the paper presents an important contribution but would benefit from broader benchmarking and a clearer discussion of trade-offs.", "other_comments_or_suggestions": "Typos:\n“ … its only filters precomputed spectra …” → “... it only filters …”\n\"... These results are presented in Appendix Figure 7 and Appenxic A.7.2.\" → appendix A.7.2\n“In this chapter ..” → “In this section…”", "questions_for_authors": "1. The authors only compared their method with classical methods, which use kernel functions. I wonder if there are any DMD methods leveraging neural networks that could be included for comparison?\n2. It is unclear how the authors determined the hyperparameters for the proposed method and the other methods. I wonder if hyperparameter tuning was performed, especially since some of the comparison methods are kernel methods, where using different kernel functions might yield different results. If yes, I think it will be great to mention to it in the main body.\n3. In the turbulence experiment, the color maps vary across different Koopman modes and methods. Would it be possible to use normalized Koopman modes to better facilitate comparison and verification of the results?\n4. The authors reported that despite having small residuals, the Hankel-DMD modes fail to clearly capture the fundamental pressure field structure. Does this not suggest that the residual metric may not be a reliable indicator of representational quality, and thus the final representation may not be truly “optimal”?", "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": "", "extra_scores": {}}, {"review_id": "URByAeaux4", "reviewer": "Reviewer_2cVC", "summary": "This paper introduces a new deep learning based approach for approximating Koopman eigenvalues based on Residual DMD. By learning a dictionary representation that minimizes a spectral residual loss function, the authors demonstrate their approach is able to perform well on several data sets.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "1. Thank you for your insightful suggestions. The updated figures and proof can be seen here: https://anonymous.4open.science/r/rebuttal_materials-14918/\n\n2. Thank you for raising this point about Barron spaces and the network architecture. To clarify, Barron spaces (Appendix A.3) provide a theoretical framework for analyzing functions efficiently approximated by neural networks with certain properties. Their mathematical properties, notably being dense in  $L^2$ spaces under certain conditions, ensure that functions in  $L^2$  can be approximated arbitrarily well by Barron functions. This justifies applying Theorem A.3 and supports our extended analysis (see the file **convergence_proof.pdf**).\n\n3. Regarding the network architecture notation: the “2-layer” network mentioned in the theorem refers to a single hidden layer plus an output layer, which is the standard notion in approximation theory. Practically, we used three hidden layers (Section 3.2) for improved empirical performance, but insights from the 2-layer theory still inform deeper network convergence behavior (see **convergence_proof.pdf**).\n\n4. The convergence analysis in Section A.4 relies on this framework because it establishes that as the number of neurons increases, the approximation error decreases at a specific rate (proportional to $1/\\sqrt{r}$ where $r$ is the number of neurons). This gives us confidence that with sufficiently large networks, ResKoopNet can theoretically approximate the true Koopman eigenfunctions with quantifiable error bounds, which is essential for the spectral analysis guarantees we discuss (see **convergence_proof.pdf**).\n\n5. You are correct that the original ResDMD requires fewer basis functions. We discuss this further and include a new experiment (**Fig_7.png**). Still, results show that ResDMD needs around 500 manually selected dictionary elements, whereas ResKoopNet achieves better performance with only 300 basis functions without manual selection.\n\n\n6. Regarding Fig. 5(c) and (d): You are correct; these were identical due to complex conjugate eigenvalues. In the updated version (**Fig_5_new.png**), we've repeated the experiment using 250 dictionary elements (matching ResDMD from Colbrook & Townsend, 2024), resulting in distinct Figures 5(c) and (d). We have also added singular value plots in the new Figures 5(e) and (f).\n\n7. Regarding the neural experiment, as suggested by the reviewer, we have included new figures:\n\n(1) To ensure a fair comparison with the 50 bases in Hankel-DMD, we re-estimated the Koopman eigenfunction from ResKoopNet using 50 dictionaries (24 SVD truncated bases, one constant, and 25 trainable bases). As a result, the performance is similarly good to the performance with 501 bases. For details of approximated averaged eigenfunctions and clustering results please refer to **Fig_6_new.png, Fig_13.png and Fig_16.png**. A hyperparameter scanning is also included in **Fig_17.png**.\n\n(2) The reviewer is right to point out that both the averaged response across trials and the mean firing rates are sufficiently distinct for each video stimulus. **Fig_10.png** shows trial-averaged neural activities for an example mouse, and **Fig_11.png(A)** presents mean firing rates across all mice for the six stimuli.**Fig_11.png(B–D)** demonstrates clustering directly based on mean firing rates, confirming that mean firing rate information alone is sufficient, with performance comparable to ResKoopNet (**Fig_12.png**).\n\n(3) We agree clustering via Koopman eigenfunctions is biologically not strictly necessary here given the clear stimulus-driven differences. However, this dataset provides an ideal benchmark for Koopman eigenfunction estimation methods. Notably, standard approaches like Hankel DMD, kernel ResDMD, and EDMD (RBF basis) fail to differentiate these distinct processes, likely due to limitations in handling high-dimensional data or eigenfunction estimation.\n\n(4)The effectiveness of ResKoopNet in accurately estimating Koopman eigenfunctions demonstrates its strength and broader applicability, particularly for more challenging tasks such as unsupervised latent brain state identification or decoding/reconstructing object movements from the videos, forming promising directions for future research.\n\n8. Regarding 'Why not use, for instance, the dictionary learning EDMD to find a good spectral approximation': The Koopman matrix updating formula $\\tilde{K}$ (Eq.6) matches EDMD, but it is derived from a distinct loss function (Eq.5), as shown in Appendix A.2. This loss function difference is fundamental; ResDMD (via Galerkin approximation, Eq.(2)) approximates both $\\mathcal{K}$ and $\\mathcal{K}^*\\mathcal{K}$, whereas EDMD approximates only $\\mathcal{K}$. Thus, during optimization, ResKoopNet yields a $\\tilde{K}$ that better captures Koopman's spectral properties.", "claims_and_evidence": "All major claims are supported by clear evidence.", "methods_and_evaluation_criteria": "The authors evaluate their computational framework on two benchmark data sets (pendulum and turbulence) and one data set that has not been frequently studied (mouse neural activity). The authors compare their approach to several different methods that are commonly used, and appear to make the comparisons fairly.", "theoretical_claims": "I did not follow the discussion of Barron spaces and convergence in Appendix A.3. This is not my expertise, but I did feel that the introduction to Barron spaces was a little terse and it was not clear to me why one would expect them to be appropriate spaces to consider for ResKoopNet, since the networks used were three layers (Sec. 3.2 -- not two layers, as mentioned in Appendix A.3).", "experimental_designs_or_analyses": "The experiments were largely well done, but I have several specific comments/questions: \n\n1. The authors claim that ResDMD requires 964 dictionary elements to compute the pseudospectra of the nonlinear pendulum (Sec. 4.1). However, Fig. 1 of Colbrook and Townsend (2024) only shows the spectra computed for 150 and  964 dictionary elements. The authors show that 300 dictionary elements does not lead to the complete pseudospectra (Fig. 4), but they do not show that at least 964 are needed. It could be that significantly fewer dictionary elements than 964 are needed (but greater than 300). I think therefore this claim should be modified. \n\n2. The authors show that the higher Koopman modes computed by ResKoopNet approximate acoustics in the turbulence example. Figs. 5c and d look identical to me. Could this be an error and the same figure plotted twice? If not, is there a reason the 6th and 7th Koopman modes are so similar? \n\n3. The results on using Koopman mode decomposition for clustering neural activity across stimuli is interesting. I think there are three basic analyses that could strengthen these results. First, could the authors plot the neural population recordings (averaged across trials) for each stimuli. Is there a clear difference in the activity across stimuli (in which case, maybe such a dynamics based clustering is not necessary). Second, what happens if you cluster the trials just based on the mean activity (for each neuron) across the trial? I could imagine different visual stimuli driving different average activity in the neural population, that could be picked up by the Koopman eigenfunctions associated with eigenvalue 1. And third, what happens if you use fewer Koopman modes, computed from ResKoopNet, in your clustering? It seems like comparing having 500 modes vs. having 50 (in the case of Hankel-DMD) could affect the comparison.", "relation_to_broader_scientific_literature": "This work could be improved by motivating more clearly why ResDMD is such a powerful method and why using the spectral residual loss function is so important. Why not use, for instance, the dictionary learning EDMD to find a good spectral approximation. I think the authors have reasons for this sprinkled throughout the paper, but making it more clear would be helpful. Additionally, the authors discuss dictionary learning EDMD and work showing convergence bounds of EDMD (Korda and Mezic, 2018), but then they also say things like \"ResKoopNet employs neural networks to optimize dictionary functions\" (page 1) and \"EDMD lacks theoretical gaurantees of convergence\" (page 1), without discussing this highly relevant work. Finally, the authors say that \"as shown in (6), ResKoopNet provides an explicit expression for $\\tilde{K}$\" (page 3). But this expression is just from EDMD, and not specific to ResKoopNet. These last two points give slightly misleading representations on what this works contributions are.", "essential_references_not_discussed": "The authors cited everything that I think is essential (but could use with more referencing of those works throughout the text, as noted above).", "other_strengths_and_weaknesses": "**Strengths** \n1. I thought the idea of using the residual spectral loss was innovate and appeared to lead to strong results. \n2. The application of ResKoopNet to neural data (while I have some reservations about it, see above) is novel and interesting. \n3. Developing better tools for approximating the continuous part of the Koopman spectra is important and I think a contribution of this paper (that could even be discussed more in the intro). \n\n**Weaknesses**\n1. The experimental results could be improved, as discussed above. \n2. The authors could make their contributions more clear (and not make claims that could be misleading about their contributions), as discussed above.\n3. The authors could make the paper easier to follow by including more discussion on ResDMD, particularly on Kernel ResDMD, since they compare to this method several times. Additionally, writing out the Koopman mode decomposition could allow the authors to introduce Koopman methods better. Finally, explaining why the true pseudospectra covers the entire unit circle in Sec. 4.1 would be helpful. \n4. Finally, there are a number of typos and grammatical errors throughout the paper. While I very much understand the presence of these things, the amount was bordering on making it harder to read.", "other_comments_or_suggestions": "The authors rely a number of times on referencing results from the original ResDMD paper. This makes it harder for the reader who is not very familiar with ResDMD to follow at places, and to compare the results. I think one last way for the authors to improve their paper is to include more results/discussion to make the paper as self contained as possible, even if this all this additional information is added to the Appendices.", "questions_for_authors": "1. Are Fig. 5c and d the same figure? If not, why are the modes so similar? \n2. How much do the clustering results in Fig. 6 rely on the learned dynamics (as opposed to the trial average responses of the neurons/mean firing rate for each stimuli)? How dependent are they on the exact number of Koopman eigenfunctions used for ResKoopNet?", "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": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This paper introduces ResKoopNet, a neural network-based framework for approximating the Koopman operator using a spectral residual loss. The method aims to address spectral pollution and inclusion issues prevalent in prior approaches such as EDMD and ResDMD. While the core idea of integrating neural dictionary learning with residual-based spectral refinement is incremental, the paper provides a thoughtful and well-motivated extension to existing Koopman operator methods. The experimental validation across diverse systems—including a pendulum, turbulent flow, and neural recordings—demonstrates the potential of the approach, although some concerns remain regarding fair comparisons (e.g., dictionary sizes across methods) and clarity in defining optimality and convergence. The authors were responsive during the rebuttal, conducting additional experiments and clarifying technical points, including convergence arguments and hyperparameter robustness. While the methodological novelty is modest, the empirical value, clarity of exposition, and relevance to ongoing work in Koopman learning justify a weak accept.
Decision: Accept (poster)