KoopSTD: Reliable Similarity Analysis between Dynamical Systems via Approximating Koopman Spectrum with Timescale Decoupling
Authors: Shimin Zhang
Ziyuan Ye
Yinsong Yan
Zeyang Song
Yujie Wu
Jibin Wu
Venue: ICML 2025 poster (2025)
Keywords: Neural Network Similarity
Neural Data Analysis
Dynamical Systems
Primary area: general_machine_learning->evaluation

Abstract: Determining the similarity between dynamical systems remains a long-standing challenge in both machine learning and neuroscience. Recent works based on Koopman operator theory have proven effective in analyzing dynamical similarity by examining discrepancies in the Koopman spectrum. Nevertheless, existing similarity metrics can be severely constrained when systems exhibit complex nonlinear behaviors across multiple temporal scales. In this work, we propose **KoopSTD**, a dynamical similarity measurement framework that precisely characterizes the underlying dynamics by approximating the Koopman spectrum with explicit timescale decoupling and spectral residual control. We show that KoopSTD maintains invariance under several common representation-space transformations, which ensures robust measurements across different coordinate systems. Our extensive experiments on physical and neural systems validate the effectiveness, scalability, and robustness of KoopSTD compared to existing similarity metrics. We also apply KoopSTD to explore two open-ended research questions in neuroscience and large language models, highlighting its potential to facilitate future scientific and engineering discoveries. Code is available at [link](https://github.com/ZhangShimin1/KoopSTD).

KoopSTD:
Reliable Similarity Analysis between Dynamical Systems via Approximating Koopman Spectrum with Timescale Decoupling
Shimin Zhang" 1
Ziyuan Ye"
Yinsong Yan
Zeyang Song?! 21 Yujie Wu Jibin Wu 134
Abstract
siderable attention within the machine learning commu- nity (Hotelling, 1992; Kriegeskorte et al., 2008; Hardoon et al., 2004; Korbllihh et al., 2019; Williams et al., 2021; Klabunde et al., 2025; Huh et al., 2024). Recent develop- ments in representation space analysis have shed light on how internal representations of deep neural networks are influenced by network architectures (Hermann & Lampinen, 2020; Nguyen et al., 2021), training methods (Wang et al., 2018; Mehrer et al., 2020; Csiszárik et al., 2021), and data distributions (Ding et al., 2021; D' Amour et al., 2022). One fundamental assumption behind most existing similarity analyses is that the representation a network can closely correspond to its functional characteristics. This assumption holds for static network architectures, such as convolutional neural networks (CNNs) and feed-forwart networks (FFNs), which do not involve complex internal dynamics.
Determining the similarity between dynamical systems remains a long- standing challenge in both machine learning and neuroscience. Recent works based on Koopman operator theory have proven effective in analyzing dynamical similarity by ex- amining discrepancies in the Koopman spectrum. Nevertheless, existing similarity metrics can be severely constrained when systems exhibit com- plex nonlinear behaviors across multiple temporal scales. In this work, we propose KoopSTD, dy- namical similarity measurement framework that precisely characterizes the underlying dynamics by approximating the Koopman spectrum with ex- plicit timescale decoupling and spectral residual control. We show that KoopSTD maintains invari- ance under several common representation- space transformations, which ensures robust measure- ments across different coordinate systems. Our extensive experiments physical and neural sys- tems validate the effectiveness, scalability, and robustness of KoopSTD compared to existing sim- ilarity metrics. We also apply KoopSTD to ex- plore two open-ended research questions in neuro- science and large language models, highlighting its potential to facilitate future scientific and engi- neering discoveries. Code is available at link.
However, for non-linear dynamical networks such as re- current neural networks (RNNs), large language models (L.LMs), and brain networks, two significant challenges arise concerning this assumption. Firstly, two dynamical systems can exhibit distinct representational structures while preserving computational equivalence in their underlying dynamics (Maheswaranathan et al., 2019; Dubreuil et al., 2022). Secondly, two dynamical systems may demonstrate different dynamical behaviours while maintaining similar representational structures (Russo et al., 2020; Driscoll et al., 2024). To facilitate the comparison of temporal dynam- ics, various temporal similarity measures have been pro- posed, such as dynamic time warping (DTW) (Sakoe & Chiba, 1978) and cross-correlation (Rabiner, 1978), which conduct comparisons after addressing temporal misalign- ments between representations. However, these measures still struggle to tackle the two challenges mentioned above, particularly when the temporal correspondences between sequences cannot be well established.
1. Introduction
Investigating and comparing non-linear neural networks through the lens of representation space has attracted con-
"Equal contribution 'Department of Data Science and Artificial Intelligence, The Hong Kong Polytechnic University, Hong Kong SAR, China Depparment Electrical and Computer Engincering, National University of Singapore, Singapore 'Department of Com- puting, The Hong Kong Polytechnic University. Hong Kong SAR, China 'Research Center of Data Science and Artificial Intelligence, The Hong Kong Polytechnic University, Hong Kong SAR, China. Correspondence to: Jibin Wu <jibin. w polyu.edu.hk>.
Recent advances in Koopman operator theory (Koopman, 1931; Kutz et al., 2016; Brunton et al., 2022) offer promis- ing solution by transforming the analysis of non-linear dy- namical systems into a linear, infinite-dimensine of observables. This transformation enables us to analyze non-linear systems using the extensive array of powerful techniques available in linear systems theory. Several efforts
space
Proceedings of the 4nnd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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KoopSTD: Dynamical similarity mcasurement via approximating Koopman Spectrum with Timescale Decoupling
have since started to push the boundaries of dynamical simi- larity metrics via linear operator representations, such as the Koopman spectral kernel-based metric (Fujii et al., 2017), operator- breeator-based metric (Ishikawa et al., 2018) and dynamical similarity analysis based on Hankel Al- ternative View of Koopman (HAVOK-bard -baee DSA) (Ostrow et al., 2024). However, existing approaches are not sensitive to cross-scale coupling in temporal dynamics, which limits their ability to capture multi-scale dynamical patterns in- trinsic to many physical and biological systems. Moreover, these methods are prone to spectral pollution, where spuri- ous cigenvalues arise in the numerical approximation of the Koopman operator (Lewin& & Séré, 2010; Klus et al., 2018; Colbrook et al., 2019; Kostic et al., 2023; Philipp et al., 2023), potentially undermining the reliability of similarity measurements.
across five brain regions (detailed in Sec. 4.4). The results show strong correspondence with the estab- lished structural organization of auditory pathways, suggesting a structural functtionall coupling in the hu- man auditory cortex.
We apply KoopSTD to analyze LLMs, specifically GPT-2 and LLaMA, of varying sizes (detailed in Sec. 4.4). Our findings demonstrate that larger-scale models exhibit significantly higher coherence in their internal dynamics compared to smaller variants, pro- viding a fresh perspective of the scaling laws governing LLMs.
The rest of this paper is organized as follows. Section 2 provides a comprehensive review of similarity metrics for both static systems and non-linear dynamic systems. Sec- tion 3 introduces our proposed metric and establishes its theoretical groundedness through rigorous analysis. In Sec- tion 4, we present our experimental results both synthetic and real-world datasets, followed by discussion of our key findings. Section 5 concludes the paper and outlines potential directions for future research.
To address these limitations, we introduce KoopSTD, framework to measure similarity between dynamical sys- tems by approximating the Koopman Spectrum of dynam- ical systems with Timescal Decoupling Specifically, the transition in the original state space is first decomposed into different frequency components using the short-time Fourier transform (STFT) to capture temporal dynamics across mul- tiple timescales. KoopSTD then extracts spatiotemporal pat- terns by performing Koopman mode decomposition (KMD) on the obtained fime-frequency representation, revealing the intricate interplay among decoupled timescales. Finally, the resulting Koopman spectrum is refined by filtering out spurious modes based on the spectral residuals, ensuring verified and more accurate representation of the underlying dynamics.
2. Related Work
There two main research lines in measuring the similar- ity between dynamical systems: representation-batt meth- ods and dynamics- based methods. In the representation- based methods, the linear analysis approaches, particularly canonical correlation analysis (CCA) (Hotelling, 1992), represent one of the carliest attempts. To capture non- linear relationships, one set of approaches draws from in- formation theory, utilizing metries such as mutual infor- mation (Kraskov et al., 2004; Belghazi et al., 2018) and divergence (Lin, 1991). Another line of solutions analyzes geometric relationships in representa- tional spaces by comparing the patterns of neural responses across different conditions, such as representational similar- ity analysis (RSA) (Kriegeskorte et al., 2008) and centered kernel alignment (CKA) (Kornblith al., 2019). Although these representation similarity metrics enable mooal system comparison, they are sensitive to simple transforma- tions of the representations, such as data perturbations (Ding et al., 2021) or subset shifts (Davari et al., 2023). Similar to (Ding et al., 2021), generalized shape metrics (Williams et al., 2021) provide a distance measurement by defining a rigorous mathematical framework between representations through optimal linear transformations.
Our contributions can be summarized as follows:
We propose a dynamical similarity metric, KoopSTD, to effectively analyze the similarities between high- dimensional multi-scale nonlinear dynamical systems (detailed in Sec. 3).
We theoretically demonstrate that KoopSTD remains invariant under a broad class of invertible linear trans- formations, including isotropic scaling, rotation, and permutation (see Sec.3.4), which ensure the robust- ness of the proposed metric under common transfor- mations in representation space (Kornblith et al., 2019; Williams et al., 2021; Klabunde et al., 2025).
We conduct comprehensive experiments on both physi- cal and neural systems to validate the effectiveness (de- tailed in Sec. 4.2) and robustess (detailed in Sec. 4.3) of KoopSTD, demonstrating significant improvements over existing similarity metrics.
However, these representation- -aaedd approaches face crit- ical limitations when comparing dynamical systems, even after applying temporal alignment techniques like cross- correlation (Rabiner, 1978) or DTW (Sakoe & Chiba, 1978). These limitations motivate recent advances to shift the com-
We apply KoopSTD to analyze fMRI data from the hu- man auditory cortex, revealing a functional hierarchy
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KoopSTD: Dynamical similarity mcasurement via approximating Koopman Spectrum with Timescale Decoupling
parison from static representations to the underlying dynam- ics of the non-linear systems themselves (i.e,, dynamics- based methods). Early efforts to compare dynamical sys- tems focused on model-specific approaches. The fxxett topology analysis was proposed by (Maheswaranathan et al., 2019) to extract universal dynamical properties across RNN architectures via constructing directed graphs from fixed points and their transition probabilities. Similarly. (Chen et al., 2023) introduced inner product-based similarity mea- sures for comparing filter subspace similarity, though this method was limited to systems with identical architectures. To achieve model-agnostic dynamical similarity metrics, researchers developed methods that directly analyze sys- tem dynamics. (Fujii et al., 2017) pioneered this direction by proposing Koopman spectral kernels, which general- ized Binet-Cauchy kernels nonlinear dynamical systems through Koopman operator spectral analysis. Similarly, (Ishikawa et al., 2018) constructed a metric using the ratio of empirical Ferron-Frobenius operator norms over infi- nite time, eliminating the need for exponential discoumting. More recently, (Ostrow et al., 2024) advanced the field with DSA based on HAVOK, which evaluates similarity by ana- lyzing how vector fields transform under orthogonal maps to identify topologically conjugate systems.
scalar amplitudes, &; denotes eigenfunctions, and v repre- sents Koopman modes,
s(x(1)) = (x()) = (ox(((0))o.
This decomposition describes the complex flow patterns by a hierarchy of simpler processes, with their linear su- perposition reconstructing the full system dynamics. The resulting representation allows comparison of any two dy- namical systems according to the principle of topological conjugacy (Ostrow et al., 2024; Redman et al., 2024).
While KMD provides theoretical framework that decom- poses a dynamical system into set of modes associated with eigenvalues of the Koopman operator, its exact compu- tation is often intractable for real-world systems. Dynamic mode decomposition (DMD) (Williams et al., 2015; Schmid, 2022) addresses this limitation by offering a data-driven ap- proach to approximate this decomposition. To approximate the Koopman operator, DMD constructs a finite-dimensional linear matrix A through least-squares optimization with two snapshots W₂ {[g(=(0)), (I.. (00-))) 1 and W = [
g (x(2). ..... g(>(t)) in observable space:
Despite these advances, these -adel-agnostic operator- theoretic metrics face inherent spectral pollution is- sues (Lewin & Séré, 2010: Colbrook et al., 2019) when approximating infiniteddimensional operators with finite- dimensional matrices, and operating primarily in the tempo- ral domain, limiting both their practical utility and ability to capture multi-scale dynamics.
= arg min [W - AW = WWw1, x+ (3)
where |1-11p- is the Frobenius norm and i denotes the pseudo- inverse. Thus, the spectral components can be obtained by solving A$ &A, where A diag(AA, .2. Av) is the diagonal matrix containing the Koopman eigenvalues, and T is the corresponding eigenvectors.
3. Method
3.2. KoopSTD Captures Multi-Scale Dynamics in the Coordinate
3.1. Preliminary on Koopman Operator Theory
Nonlinear dynamical systems are inherently complex and challenging to analyze directly in the time domain due to their intricate interactions across multiple timescales, or equivalently, their diverse frequency components. As a prominent method in signal processing, the STFT pro- vides time-frequency representation through windowed Fourier analysis of the time series. Given a time series X T Na with sampling time T and feature dimension Na. the STFT can be regarded as a mapping & that produces time-frequency representation Z,
The Koopman operator theory provides a powerful frame- work for representing nonlinear dynamical systems in infinite-dimensional Hilbert function space. This allows us to leverage powerful linear analysis techniques such as spectral analysis, mode decomposition and linear predic- tion, while still capturing the full nonlinear dynamics of the original system. For a time-discrete system described by arit + 1] = [((tt) that evolves over a state-space Q2 C IRd under the mapping& F Ω + Ω the Koopman operator K corresponding to observables g Ω + C is defined as follows:
h : X + Z,
(4)
(1) (go= x ZEΩ, g ge (
where Z E (( with / and denoting STFT win- dow length and hop size, respectively. N, = ( + 1) .Na is the total number of frequency components obtained by con- catenating frequency bins across all dimensions. Note that KoopSTD also supports other -requuery techniques (e.g., wavelet transform).
A key aspect of modern Koopman operator theory is Koop- man Mode Decomposition (KMD). It breaks apart a spa- tiotemporal signal into infinite triplets {(),, 0r, m where X; represents Koopman eigenvalues that reflects
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KoopSTD: Dynamical similarity mcasurement via approximating Koopman Spectrum with Timescale Decoupling
To effectively capture multi-scale temporal dynamics, Koop- STD leverages Koopman analysis approximate the evolu- tion of the system within the digen-iime-frqueency coordi- nates defined by the right singular vectors V of Z.
We give systematic discussion about the Galerkin and alternative methods for estimating the spectral error bound of linear operators in Appendix B.I.
To systematically demonstrate KoopSTD, let us consider two dynamical systems X₃ (t + 1]=
(i = 1] = (x2))) where X, E IR IR, and X2 E R re- spectively. The data sampled from two systems are first transformed into the eigen-time- feeeuneer coordinate via STFT and SVD, resulting in two embeddings. We subse- quently apply Eq. (7) to both embeddings for extracting the top digenvalue-eigenveeee pairs with the minimum spectral residuals. These pairs represent the evolution of r accurate modes, which capture the system's 's overall dy- namical behaviors. The eigenvalues explicitly reflect the evolving amplitudes of these modes, making it reasonable to eigenvalues { and {Aa,} to measure the discrepancy d(F1,,F2) between two dynamical systems via symmetric permutation invariant distance metrics. Here, we adopt F2) as the Wasserstein distance:
[t)) and X2[t +
Z = UEV"
(5)
where denotes the conjugate transpose. Then, charac- terize the finiteddimensional operator A outlined in Eq. (3) based on the temporal snapshots Wv, W from singular vectors V, denoted as Aif.
Apf = WyWL.
(6)
Note that V forms a basis in the ggeenttim-frequenne coor- dinates, extracting dominant temporal modes from the origi- nal time-frequency space. Based on these modes, the Koop- man operator approximation Atf in Eq. (6) explicitly quan- tifies the underlying dynamics across multiple timescales.
3.3. KoopSTD Measures the Dynamics Dissimilarity with Reliable Koopmanism
d(F, inf =EP(r)
Σ
|uos - Se
(8)
By applying DMD eigen-time-frequee the approximated operator A, captures the evolution on different timescales. To assess the dynamical similarity be- tween two systems, one could employ similarity metrics such as Wasserstein distance or Procrustes analysis on the operators Atj. However, when performing discrete spectral analysis, the occurrence of spurious eigenvalues, which are not associated with the real dynamics of the operator, can lead to spectral pollution. This phenomenon can signifi- cantly distort the comparison, weakening the reliability of the similarity measure and potentially leading to inaccurate conclusions about the dynamical behavior of the systems.
observations,
where P(r) denotes the set consisting of all permutations of r elements. The pseudocode of KoopSTD is summarized in Algorithm 1.
Algorithm KoopSTD Pseudocode
Input: two time series, X1 RTix and X₂ TTxxNa2; STFT window size, E Z+; STFT hop size,
8 E = number of preserved modes, r E Z+
Output: Dynamics dissimilarity di hetween X, and X₂ Procedure DMD STEDDTTT(X.)
Here, we incorporate the advances in spectral error control via Galerkin method (Fletcher & Fletcher, 1984) for linear operators (Williams et al., 2015; Colbrook & Townsend, 2024) to identify and exclude Koopman modes with unreli- able spectral residuals. Specifically, given Ny eigenvalue- eigenvector pairs {}), و for the approximated Koop- man operator Auf, we explicitly measure the accuracy of each candidate eigenpair by computing its squared relative residual as follows:
STFT(X, Solve Z == UEV*
Approximate Aif by Eq. (6)
Return Auf
End Procedure
Procedure RESCONTROL(A
Solve A. ΦA for eigenpairs (51 Nr
for j 1 to N, do
Compute the residual of (d ú3}} by Eq. (7)
end for Top accurate eigenvalues A = diag(S: Se )2,...,
0), it [M- - |MPO]
(7)
Return A
End Procedure
where = WW N Wt WwWw. and Ả denotes the complex conjugate of eigenvalues. A detailed derivation of Eq. (7) is provided in Appendix B.2. This residual quantifies the deviation from the expected spectral properties, thus serving as a standard for accurate mode selection and reliable dynamical similarity metric.
Atr.1 + DMD X1, (,
A + DMD X2, L,
A1 < RESCONTROI.
Ag + DESCONTROL(AAf21
Compute the dynamics dissimilarity d by Eq. (8)
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KoopSTD: Dynamical similarity mcasurement via approximating Koopman Spectrum with Timescale Decoupling
3.4. cuinsformation-Invaiinnn Property
where TR = =(X + XQ: : Q O(Na),, and O(Na) == (RRNaxNa, Na, Q"Q INal denote the orthogonal group.
In this section, we prove that KoopSTD exhibits funda- mental invariance properties, ensuring its robustness against common transformations in the representation space. To facilitate a more concise and unified theoretical analysis, we reformulate the KoopSTD in Eq. (8) into a more general form as shown in Definition 1.
Property 3. (Permutation invariance). KoopSTD remains invariant for any permulation transformations Tp on the representation space,
=2)) = =d( F2).
(13)
Definition 1. (KoopSTD). Let Fi; R R be two dynamical systems with their finite- minteedimeessonal approximations of Koopman operators A1, A2. The dy- namics dissimilarity between two systems can be defined as:
where Tp = (X XQ, ;T E PNat. PNa denote the set of all permutations on (1....NaJ), and for π E PN Q2 E NaxNa denote the permutation matrix.
d(F 2 d(M(A))
(9)
The proof of Proposition I and Property 1-3 are provided in Appendix A.
where X() denotes the eigenvalue spectrum and d(:,-)
can be symmetric permulation invariant metrics (e.g., Wasserstein distance and Jensen-Shannon divergence).
4. Experimental Results
In this section, we first introduce the experimental setup in Sec. 4.1. We then conduct a comprehensive comparison between the proposed KoopSTD and existing metries in Sec. 4.2, utilizing classical Lorenz systems, followed by ablation study of KoopSTD. To verify the robustness of KoopSTD, we further conduct experiments under two challenging scenarios in Sec. 4.3: a) different system dy- namics yield similar sampled representations, or b) similar system dynamics manifested in different sampling represen- tations. Finally, in Sec. 4.4, we apply KoopSTD to address open-ended research questions in neuroscience and LLMs.
Building upon the KoopSTD formalized in Definition 1, a natural follow-up question is whether this similarity metric possesses any theoretical groundedness? Our answer, in Proposition 1, is yes: the similarity measurement of Koop- STD remains invariant under invertible linear transforma- tions on the representational space.
Proposition Let [t + +] = (x) and X2lt + ((2()) be two time-discrete dynamical systems, governed by mappings F2. Na + NN, Then, the dissimilarity di F2) measured by KoopSTD is invari- ant under broad class of invertible linear transforma- tions T on the representational space, such that:
4.1. Experimental Setup
(10)
To evaluate the performance of KoopSTD, we com- pare it with four existing similarity metries, including two representation--asee methods, CKA (Korbblith et al., 2019) and Procrustes analysis (Williams et al., 2021), as well as two dynamics- methods, (CC) (Rabiner, 1978) and HAVOK- -bseed DSA (Ostrow et al., 2024). Details on their implementation can be found in Appendix C. Each metric maps the dynamical pat- terns into low-dimensional metric space through pairwise dissimilarity measures, where effective metric would result in samples of the same dynamical class forming well- separated clusters. The quality of these clusters, and thus the effectiveness of the metrics, is evaluated using the Sil- houette coefficient. For a given similarity metric MI applied to a N -ssmpllss dataset with & distinct dynamical behavior clusters, the coefficient can be expressed as:
Note that T includes, but is not limited to, some common transformations, such as isotropic scaling, rotations, and permutations (Kornblith et al., 2019; Williams et al., 2021; Klabunde et al., 2025), which frequently appear in real- world neural data or network representation spaces. As result, KoopSTD demonstrates several notable properties: Property 1. (Isotroiic scaling invariance). KoopSTD remains invariant for any isotropic scaling Tis on the representation space,
d( Fa))
(11)
where Tis = + + XQ : Q = gINab and q E R denotes the scaling factor.
(Rotation invariance). KoopSTD remains invariant for any rotation transformations Tn on the rep- resentation space,
Property 2.
1 b(1) a(i) I(M = N max (a(i).
(14)
where a(i) is the average distance between sample i and other samples in the same cluster (intra-cluster distance), and b(i) is the minimum average distance between sample
d())(Fi,FF)) = F(),
(12)
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KoopSTD: Dynamical similarity mcasurement via approxinating Koopman Spectrum with Timescale Decoupling
Centered Kernel Alignment
Procrustes Analysts
Crozs Corretation
HAVOK
KoopSTD
U Ie-- le Rine like
Perisatio22 Periodic-1
Chaotic
Axte-1
Axie-1
Axte-1
Aste-1
Axile- 1
Figure 1. Comparison different metries across five classes Lorenz systems via MDS projection. The more separable the patterns are. the better the performance of the metric.
i and all samples in the nearest neighboring cluster (inter- cluster distance). The Silhouette coefficient ranges from -1 to I, where values close to I indicate well-defined clusters, values close to O suggest overlapping clusters, and values close to 1 imply misclassification.
their shared recurrent patterns in broader temporal scales.
To further understand the contribution of each component in KoopSTD, we conduct ablation studies, which are de- tailed in Appendix E. The results confirm that both timescale decoupling and spectral residual control in KoopSTD are crucial for distinguishing different dynamical behaviors. Specifically, the üime-frequency transformation enables ef- fective separation between chaotic dynamics and other types by decoupling complex temporal patterns across multiple scales. Moreover, the spectral residual control for Koopman operator rank reduction improves the distinction between globally similar but locally different dynamics, particularly for periodic orbits, by ensuring the selection of accurate and relevant dynamical modes.
Metries Representational
Dynamical
Systems
CKA Procrustes
CC HAVOK KoopSTD
Lorenz Systems
-0.05
-0.04
-0.27
0.47
0.94
PDM
-0.04
-0.02
-0.30
0.90
0.99
Attractors
Flip-Flop
0.20
0.98
-0.16
0.10
-0.04
RNNs
Table 1. Quantitative analysis of existing similarity metrics across thrce systems using the Silhouette coefficient. For the experiments of Lorenz system and PDM attractors, higher values indicate het- ter performance, while in the 3-bit Flip-Flop RNNs experiments, values approach to 0 are preferred. A detailed description of these metrics is provided Appendix C.I.
4.3. Robustness Analysis
In this section, we compare KoopSTD with other metrics under challenging conditions where either a) different sys- tem dynamics yield similar sampled representations, b) similar system dynamics manifested in different sampling representations.
4.2. Effectiveness Analysis
Geometric Homogeneity. The first experiment aims to assess whether the proposed metric can accurately capture the underlying dynamical disparity despite relatively similar trajectories. Following (Ostrow al., 2024), we examine three noisy attractors: bistable attractors (BA), line attrac- tors (LA), and point attractors (PA), representing unstable, stable, and leaky integration in perceptual decision-making, respectively. Through adversarial optimization (Galgali et al., 2023), the anndition-averaged trajectories of LA and PA become almost identical to BA's, leaving only noise- induced deviations during recurrence to reveal their distinct dynamics. In this experiment, we generated 50 systems for each attractor class with 100 sampled trajectories per system.
The Lorenz system is defined by three coupled nonlinear differential equations: da = ( - x), dy dē
2 = x((p - =) - y, and dz zy- - B2. By setting Ø = 10, B -/. and varying p. we obtain five classes of trajectories: line-like (p- 10), ring-like (p- = 20), two periodic orbits (p = 152, 220), and chaotic (p- = 75). These trajectories exhibit behaviors ranging from stable to highly nonlinear with scale dynamics, making them ideal for testing various similarity metrics.
As shown in Table 1, KoopSTD achieves the highest Sil- houette coefficient (0.94) among all compared similarity metrics, indicating its superiority in analyzing multi-scale dynamics. Figure illustrates that KoopSTD, when visu- alized through multidimensional scaling (MDS) (Kruskal, 1964), maintains low intra-cluster distances and high inter- cluster distances with the lowest variance. Contrastingly, representation-bast based metrics (i.e., CKA A and Procrustes anal- ysis) fail to differentiate between distinct dynamics. The two dynamics- gynamics-based metrics (i.e., CC and HAVOK- -based DSA) can separate line-like like and ring-like dynamics but struggle to differentiate between periodic and chaotic dynamics due to
According to the intra- and inter-cluster distances shown in Figure 2, Procrustes analysis identifies all samples to be the same due to its emphasis on geometric aspects, where all samples exhibit similar trajectories in the 2D space. While HAVOK-basee DSA and KoopSTD are both capable of separating different dynamical patterns, KoopSTD exhibits stronger consistency on the inter-cluster distance across dif-
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KoopSTD: Dynamical similarity mcasurement via approximating Koopman Spectrum with Timescale Decoupling
Certered Kernel Alignment
(b)
Procrustes Analysis
Intra-class Dissimilgrity Siriiiirrtyyy (b)
Cross Correlation
Intra Intra-class class Dissimilarity (4))
Inter-class Dissimila milllanty aiarrtyyy (+)
Inter-clase Dissimilrryy (?)
Inter-class Dissimilarity (1)
1.5x10
2x102
1x10
1x10
2=10
2x10
Intra-class Dissimilarity
1.5x10"
8x10
5x10
15x10*
1.5x10
6x10
Ex10
1=10*
1=10-
11-10*
4x10
4x10
Sx10
2x10
5-10*
5+10
8A-BA LAIPPAPP BA CAPA PA-BA
BA-BA LA PA PA.PRA
BA
PA.BA
BA- CA-A PAlPA.P PAPP LABBA 2 PAP
(d)
HAVOK
(e)
KoopSTD (Ours)
Intra-class Dissimilarityy (-))tt((())
Intra-class pire-class Dissimilarity [46]
Inter- class Dissimilarityy (P)
8-10
0x10 +
0.8
0.8
Inter-class Dissimllarity
6-10
-6x10
0.6-
-0.6
4-10 4-
-4x10*
0.4-
0.4
2-10 4-
2+10
0.2-
0.2
0.0
0.0
BA-8 LA-L Pa. PA.PAA
BA-BA unl La LA
LN PA-BA
Figure 2. Comparison of different metrics between attractor systems. Each class (BA: bistahle attractor, LAA: line attractor, PA: point
attractor) contains 50 systems with 100 sampled trajectories per system. The bars show both intra-class distances within each attractor type and inter-class distances between attractor types. Better metries are characterized by lower intra-class distances (left part of each subfigure) and higher ilter-class distances (right part of cach subfigure).
Centered Kernel Alignment
Procrustes Analysts
Cross Correlation
HAVOK
KoopSTD
Varillo RNN
LSTM
Ario-2
Aste-2
Gue to
ADD2
GRU
Auts-1
Axte-1
Axte-1
Axti-11
Figure 3. Comparison of metrics across three types of RNN via MDS projection. The more inseparable the patterns are, the better.
ferent samples. As result, KoopSTD achieves an almost perfect clustering performance with Silhouette coefficient of 0.99. In contrast to KoopSTD, representation based met- rics fail to effectively distinguish both inter-class and intra- class differences in neural dynamics.
in capturing dynamic similarity amidst geometric variations. On the other hand, KoopSTD yields a Silhouette coefficient of -0.04, indicating minimal separation their underlying dynamics, further demonstrating its capacity to effectively uncover the shared intrinsic dynamics across different net- work architectures.
Geometric Disparity. The second experiment investigates whether the metric can correctly identify neural network sys- tems with the same recurrent dynamic despite differences in geometric network architectures. To explore this, we trained 60 one-layer RNNs with varying architectures on the widely recognized b-b Flip-Flop task (Sussillo Barak, 2013). More details for the task and training can be found in Appendix D. Notably, Figure 10 shows that when pro- jecting network hidden states onto their first three principal components, trajectories across different RNN architectures consistently trace the vertices of a three-dimensional cube, revealing a shared computational structure.
4.4. Empower Scientific Discoveries through KoopSTD
KoopSTD is a powerful tool that can facilitate complex dynamical system analysis across various domains. We next demonstrate its potential for scientific research in both com- putational neuroscience and LLMs by answering two funda- mental questions: RQ1. To what extent can the KoopSTD reveal functional correspondences among distinct regions of the human auditory cortex? RQ2. What novel perspec- tives can KoopSTD offer regarding the relationship between model size and the functional behavior of LLMs?
As shown Figure all existing metrics identify the dispar- ity between networks in solving the 3-bit Flip-Flop task to varying degrees, with CKA, CC, and HAVOK-basee DSA detecting partial differences, and Procrustes Analysis detect- ing the full extent. This points to their reduced robustness
Discover Structural-function Relationship between Cortical Regions
We investigated the functional dissimilarity patterns across human auditory cortical regions using KoopSTD on the

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KoopSTD: Dynamical similarity measurement via approximating Koopman Spectrum with Timescale Decoupling
"Narratives" dataset (Nastase et al., 2021). This dataset com- prises cortical fMRI recordings of 68 individuals listening to the same audio story. Experimental details are provided in Appendix C.5.
LLAMA322 18
16
LLAMA 3.2 38
LLAMA 86
As illustrated in Figure 4, our analysis reveals a distinct hier- archical organization functional relationships among dif- ferent auditory areas, including primary core area AI, three surrounding belt regions (i.e., LBelt, MBelt, and PBelt) and RI in the anterior insular cortex. Specifically, the diagonal block of AI, LBelt, and MBelt regions exhibit high func- tional similarity, suggesting shared computational properties in early auditory processing stages. In contrast, the PBelt and RI regions show substantially higher functional dis- similarity compared to other auditory areas, implying their distinct functional role in the auditory processing hierarchy. We leave more comparisons in Appendix D.
Axds-1
Figure 6. Dynamics dissimilarity hetween different scales (IB, 3B, and 8B) of LLaMA measured by KoopSTD.
2019) models with increasing sizes (1 (120M, 340M, 760M, 1.5B parameters). We performed 5 independent inference runs with different random seeds for each instruction, col- lecting 375 samples from the last layer of each model (75 instructions x 5 runs), resulting in a total of 1500 samples.
Intriguingly, these functional dissimilarities analyzed by KoopSTD correspond closely to the known myelination patterns in the human auditory cortex (Glasser et al., 2016; Baker et al., 2018), where regions exhibit similar myclin concentrations also display similar functional properties (see Appendix Figure 11). This alignment between structural and functional organization provides fresh insights into the relationship between cortical myelination and functional specialization in auditory processing.
As shown in Figure 5, KoopSTD effectively captures sig- nificant dynamical discrepancies between different scales of GPT-2 models. Notably, the dynamics of representations produced by larger models (i.e., GPT-2 760M and 1.5B) show higher consistency across different instructions. This is reflected in both the KoopST based dissimilarity matrix and the low- ww-dimensional visualization, where larger models exhibit more concentrated clusters, suggesting potentially more stable and consistent dynamical patterns in their repre- sentation spaces. Moreover, in the low- -indeinnsioonl space, representations from larger models exhibit substantial over- lap. These observations demonstrate a hypothesis: 'Repre- sentation dynamics evolve towards greater consistency and stability as the model scale increases.'
A1
A1
Liett
LBolt
NBelt
PSet
Auto-2
R)
MBelt
0.1
To validate this hypothesis beyond GPT-2, we conducted additional experiments on the LLaMA family of mod- els (Dubey et al., 2024) with IB, 3B, and 8B param- eters using 50 instructions randomly sampled from the Massive Multitask Language Understanding (MMLU) dataset (Hendrycks et al., 2021). The similarity results of LLaMA in Figure share the same phenomenon as GPT-2 to some extent, as shown by the convergent dynamics in representation space when scaling up model parameters. These findings shed light the nuanced differences be- tween LLMs of varying sizes, offering new insights into the scaling laws of LLM (Kaplan et al., 2020).
RV
A1 LBelt MBelt PBclt
Axis-1
Figure 4. Functional dissimilarity between different auditory re- gions (A1, LBelt, MBelt, PBelt, and RI) measured by KoopSTD.
Validating LLM Scaling Laws and Beyond
To make an effective similarity comparison, we randomly selected 75 instructions from the Databricks Dolly 15K dataset (Conover et al., 2023). For each instruction, we conducted experiments using four GPT-2 (Radford al.,
5. Conclusion
120M
0.
In this work, we introduce KoopSTD, a similarity measure- ment framework for analyzing dynamical similarity between nonlinear systems across multiple timescales via Koopman operator theory. Through extensive experiments on physical systems, neural systems, brain networks, and language mod- els, we demonstrate the effectiveness and broad applicability of KoopSTD. Looking forward, KoopSTD shows promise as a quantitative metric for knowledge distillation in large
340
760
GPT2-120M
GPT2 340M
GPT2-76DM
1.50
GP12-1.58
120M
347M
3.58
Axio-1
Figure 5. Dynamies dissimilarity between different scales (120M, 340M, 760M, and I.5B) of GPT-2 measured by KoopSTD.
8

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KoopSTD: Dynamical similarity mcasurement via approximating Koopman Spectrum with Timescale Decoupling
language models, potentially guiding the development of more efficient architectures while preserving dynamic be- haviors. The framework could be broadly applied to analyze complex dynamical systems across disciplines, from charac- terizing conformational changes in molecular dynamics and protein folding to investigating neural dynamics in different cognitive states and pathological conditions. These appli- cations may provide valuable insights into the fundamental principles governing both biological and artificial dynamical systems.
framework's ' success will ultimately depend on the research community's 's commitment to addressing these challenges through continued development and responsible application.
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This work was supported by the National Natural Sci- ence Foundation of China (Grant No. 62306259), Re- search Grants Council of the Hong Kong SAR (Grant No. C5052-23G, PolyU25216423, and PolyU15217424), and The Hong Kong Polytechnic University (Project IDs: P0043563, PO046094)
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The proposed KoopSTD framework represents significant advance in analyzing complex dynamical systems, with im- plications that extend beyond its technical contributions. This tool has the potential to accelerate scientific discover- ies across multiple disciplines by enabling more accurate analysis of complex systems in neuroscience, physics, and artificial intelligence. neuroscience, KoopSTD's ability to analyze brain dynamics could assist in the early detec- tion of neurological disorders through the identification of atypical dynamical patterns, potentially improving clinical outcomes. Within Al research, the framework could guide the development of more efficient neural architectures by providing deeper insights into internal dynamics, potentially reducing computational costs and the environmental impact of AI systems.
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KoopSTD: Dynamical similarity mcasurement via approximating Koopman Spectrum with Timescale Decoupling
A. Proofs of Theoretical Properties
To give better proof to the theoretical analysis, we move similarity metric provided in the main manuscript here.
Restatement of Definition (KoopSTD). Let F2, R + Na be two dynamical systems with their diiieedimmesional approximations of Koopman operators A1; A2. The dynamics dissimilarity between the two systems defined as:
d(F - d (M(A)), X(A2))
(15)
where X(-) denotes the eigenvalue spectrum and d(.,)) can be any symmetric, permutation-invann -nnvriiinn metric (e.g., Wasserstein distance or Jensen-Shannon divergence).
A.1. Notations
Table 2. Summary of Notations
Symbol
Description
Symbol
Description
I.
Koopman operator
JF
Original nonlinear mapping functions
probability measure of L H Observable function
Ω
Original state-space
g
Composition function
(A, V)
KMD triplet
Approximated operator
W, W
Snapshot data
A, 4
Eigenvalue and eigenvector of A Time-frequency embedding
X
Sampled data original space
Na
The original feature dimension
T
The time dimension Approximated operator by KoopSTF
N₃
The frequency dimension
Aif
(4 (A,,0,) 03
Ny
Eigenvalue-eigenvectoo pair of Arf
M,N,O
See Eq. (7)
B
Size of the dataset
((F,,F2)
The dissimilarity between two systems
p
Power parameter in Wasserstein distance STFT window length
The number preserved eigenpairs
STFT hop length
A.2. Proof the Transfornation-IIvvaiinl Property
Let Xi + = (X)) and X2(1 + 1] = Fx(((()) be two time-discrete dynamical systems with state variables x, X2 N., and they are governed by mappings F2. R,,, + R Now we prove that the distance d(Fi. between two systems calculated by KoopSTD remains invariant under invertible linear transformations T, such that:
d(T((i)F))) = d(Fi,F2).
(16)
where T= (X + XQ : Q E R). GL(Na, R) denotes the general linear group of all invertible matrices Q E NaNNa
Proof. Consider a dynamical system governed by the discrete-time evolution equation X|I + 1] = F(X((), where X E Na represents the state the system at time The KoopSTD seeks to approximate the Koopman operator by firstly mapping X E Na into time-freyuency representation Z E =l Specifically, the entry Zm.k.. corresponding to time frame m, frequency index & and feature index is given by:
[-1
Zm k. Σ X(I,n]wlf (
(17)
where I, respectively denote the length and step size of window function w[-] in STFT. By the linearity of the STFT, this transformation can be expressed in matrix form as
Z STFT(X) X2
(18)
12

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KoopSTD: Dynamical similarity mcasurement via approximating Koopman Spectrum with Timescale Decoupling
where Les is the transformation matrix representing the STFT operation. Next, KoopSTD computes the right singular vectors V of Z by solving SVD:
zzZ (VEU") -''V".
(19)
The Koopman operator A can then be approximated based on temporal snapshots (iee. Wv and Wv') of the right singular vectors V Z:
WwW'v'
(20)
For the transformed representation XQ, can also derive its time-frequency embedding Zy by substituting X XQ into Eq. (18):
Zf = STFT(XQ) = L, =QQ.
(21)
Similarly, to obtain VT, we substitute = ZT into Eq. (19):
ZiZy = (ZQ)= = Q'""Q=
(22)
Then, we derive the transformed right singular vectors V and the temporal snapshots WV.T. V,T as:
Vy = Q'V Wv.7 Q Wv. W Q'
(23)
Finally, we express the transformed approximated Koopman operator in terms of Aç:
A-= = Wv,7 W Q'
(24)
We observe that applying invertible linear transformations to the representational space within the KoopSTD framework corresponds to a similarity transformation of the approximated Koopman operator. Consequently, the transformed operator AT is similar to the original A, implying that they share the same eigenvalues:
X(AT) = A(A).
(25)
According to Definition 1, we have the following equivalence:
= (2),
(26)
Thus, we have proven that KoopSTD is invariant under a broad class of invertible linear transformations.
Since the following transformations on the feature dimension are also invertible linear transformations, we can deduce that KoopSTD remains invariant to the following transformations,
Isotropic scaling: Tis (X XQ = 4INNN1 E R.
Rotations: Ta = {X + XQ Q E O(N)), where O(Na) == (IR NaxNa QTQ IN,} denotes the orthogonal group.
Permutations:
Tp fX XQ, : P(N.)), where P(Na) is the set of permutations {1...., Na).
B. Koopman Spectral Residual Control
B.1. Related Works for Spectral Error Estimation
Spectral error estimation is critical for evaluating the fidelity Koopman operator approximations, as spectral pollution can distort interpretations of system dynamics. Such artifacts commonly result from finite- miiee-dimensionll truncation and suboptimal basis selection in approximation schemes (Klus et al., 2018; 2020; Philipp et al., 2023).
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KoopSTD: Dynamical similarity mcasurement via approximating Koopman Spectrum with Timescale Decoupling
In KoopSTD, we adopt the Galerkin approximation (Williams et al., 2015), projection-based technique that has demon- strated effectiveness in Residual Dynamic Mode Decomposition (ResDMD) (Colbrook & Townsend, 2024), to suppress spectral pollution and eliminate spurious modes in the Koopman spectrum. By restricting the approximation to well-chosen subspace, this method enables finer control over spectral content and yields more robust representations of the system's 's dynamics. Nonetheless, the performance of Galerkin methods is inherently constrained by the expressiveness and scalability of the basis functions (Davies & Stephens, 1983). A limited basis can potentially fail to capture intricate dynamics, while increasing the basis size significantly raises computational costs, undermining efficiency.
Beyond Galerkin methods, several alternative approaches have been proposed to estimate spectral error bounds of Koopman operators. For instance, sparse operator learning frameworks (Hou et al., 2023) target mixing stochastic processes and leverage sparsity in reproducing kernel Hilbert spaces (RKHS) to balance interpretability and sample efficiency. On the other hand, kernel-based methods (Klus et al., 2018; 2020; Kostic et al., 2022; Philipp et al., 2023) offer rigorous convergence guarantees for the spectral approximation of Koopman or transfer operators, and are particularly suited for a variety of stochastic settings, including Markov chains and stochastic differential equations. These methods offer complementary trade- offs terms of approximation accuracy, theoretical guarantees, computational scalability, and applicability to stochastic dynamics. In future work, we plan to investigate the integration of these techniques into the KoopSTD framework to enhance its robustness and precision in quantifying dynamical similarity.
B.2. Detailed Derivation of Eq. (7)
Here, we provide step-by-step derivation of Eq. (7). ResDMD (Colbrook Townsend, 2024) aims discard modes associated with large spectral residuals, determined by their inconsistency with the spectral convergence properties predicted by DMD's quadrature-based approximation. Specifically, for candidate eigenvalue-eigenvecoor pair (A, 6) of K, the corresponding eigenfunction
((x) = Em (z))), where N, represents the number of observables g()), the accuracy of (A, 0) can be estimated by the squared residual:
res(A, 6) = 1 [KŌ(x) - xx(x)) dw())
((IC- -)) (K- - k)6)
(27)
IKd,Ko (Ao, KZ) (Kd, (d) + (A0, Td)
(Kg0, Kgo) - Mgg0 Kgo) - A(Kg0, g0) + (g0
In the framework of KoopSTD, the terms Kg and g respectively denote snapshots W/y and Wv. Then, by substituting
M = WWWW, -ww,w/ and O = WyWv. Wv. we can derive:
res(S, = [Kg0,Kgo)
'igo, Kgo) (CCg + (913
= 8" (i(WWW W/ >WEW: - SwTWr + 1&/3W;
(28)
[' âV ÂN+ +1/O0
Finally, the squared relative residual can be expressed as:
res (2, 0) JAC - 2
S + IÅPOJO
(29)
||62112
N0
C. Detailed Experimental Settings
C.1. Alternative Similarity Metrics for Comparison
To demonstrate the versatility of the proposed KoopSTD in measuring dynamical similarity, we compare its clustering performance against four existing similarity metries based on their calculated dissimilarity matrices from datasets. Below, we provide details for each metric:
Dynamica similarity analysis based on Hankel Alternative View of Koopman (HAVOK- DSA) Although research on Koopman- based similarity analysis began a decade ago, HAVOK-base DSA proposed by (Ostrow et al.,
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KoopSTD: Dynamical similarity measurement via approximating Koopman Spectrum with Timescale Decoupling
Line-like, p=10
Ring-like, p=20
Periodic[1-1], p=220 Periodic(1-1-2-21, p=152
Periodic[1-1], p=75
Figure 7. Our dataset contains five distinct dynamical behaviors generated by the Lorenz system, each represented by its corresponding trajectory in the phase space. The "Periodic [1-1-2-2]" trajectory refers periodic orhit where the first equivalent point is visited twice, followed by the second equivalent point being visited twice. In Figure I, Periodic [1-1] and Periodic [1-1-2-2] are respectively denoted by Periodic-1 and Periodic-2.
2024) forwards a step leveraging the theoretically powerful HAVOK framework (Brunton al., 2017) to capture the intrinsic dynamics measured systems, To achieve linear representation of the dynamics, HAVOK first constructs Hankel representation by applying time-delay embedding the original data. It then fits truncated-rank regression models the singular vectors of the Hankel matrix. The chosen truncation rank is claimed to provide optimal linear fit, forming a -epprrn-nnvarintt subspace. For more details of HAVOK, please refer to (Brunton et al., 2017). For the implementation of HAVOK-badd -based DSA, we utilize the official GitHub repository provided by (Ostrow et al., 2024). The results presented in our paper reflect the best performance achieved after careful hyperparameter tuning, including the number of delays, delay interval, and truncation rank.
Cross-correlation (CC) Cross- Ctoss-correlation is defined as the normalized dot product between time series and it can be efficiently computed using numpy.correlate. In our study, we define the dissimilarity between two dynamics as the reciprocal of the maximum correlation value.
Procrustes Analysis Procrustes Analysis is specialized form of generalized shape metrics (Williams al., 2021), which quantifies the dissimilarity between two representations by optimally aligning them through orthogonal trans- formation. By minimizing the sum of squared differences between corresponding elements, Procrustes Analysis serves as a powerful tool for comparing geometric structures while preserving their relative spatial relationships. The optimization in our study is implemented by Adam optimizer with a learning rate of 0.01.
Centered Kernel Alignment (CKA) CKA quantifies the similarity between representations by evaluating the alignment of their kernel matrices after centering. Since empirical evidence suggests that using nonlinear kernel in CKA does not offer significant advantages over the linear kernel (Davari al., 2023), opt for the linear kernel in our study to maintain simplicity and efficiency.
C.2. Lorenz System
We use the Lorenz system to generate a dataset consisting of time series with various dynamic behaviors in the phase space. Specifically, we set = 10 and B = S while varying p to produce five distinct types of trajectories: line-like (p 10). ring-like (p- = 20), periodic orbit [1-1] (p = 220), periodic orbit [1-1-2-2] (p 152), and chaotic (p = 75).
For each A. we run simulations starting from the same initial point (-8, 27) for total of 800 seconds, with a time step of dt 3. We then retain the last 300 seconds, assuming that the dynamic behavior has stabilized. Their trajectories visualized in Figure 7. Next, construct dataset by extracting observations from these trajectories. Specifically, randomly clip 20-second segments and generate 30 time series instances, resulting in total of 150 samples. Each sample represents the system's is evolution over 20,000 timesteps in 3
We apply KoopSTD to first transform the data by STFT with a window length of 500 and a hop length 1. Next, we fit the DMD model and preserve only the 10 eigenpairs of the approximated Koopman operator with the minimum spectral residuals for the subsequent pairwise dissimilarity computation.
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KoopSTD: Dynamical similarity mcasurement via approximating Koopman Spectrum with Timescale Decoupling
Bistable Attractor
Line Attractor
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Figure The trajectories of single-trial samples (faint) and their condition-averagee counterparts (bold) three Perceptual Decision- making attractors (reproduced from (Ostrow et al., 2024)). The background black arrows indicate the vector field of each attractor system.
C.3. Perceptual Decision-making Attractors
We visualize the single-trial samples' trajectories on the 2D space alongside their condition-average. red and blue respectively denote different processes for making different decisions. While the condition-averagee trajectories of the three attractor systems are optimized to appear nearly indistinguishable (detailed implementation refer to (Galgali et al., 2023; Ostrow et al., 2024)), their underlying dynamics are markedly different, as highlighted by the vector fields (background arrows). Specifically, the vector field of a bistable attractor has two stable fixed points (representing two decisions), with trajectories converging toward one of these points based on initial conditions. In contrast, a line attractor's vector field directs the dynamics along a specific trajectory, with no tendency to diverge. point attractor, on the other hand, has a vector field that points inward toward single fixed point, indicating that all trajectories globally converge to this point.
The dataset comprises 1.0 attractor systems, each with 100 trials, where each trial consists of two conditions evolving for 10,000 timesteps in 2 The proposed KoopSTD effectively separates three distinct attractor systems using the STFT with a window size of 1,0 1,024, and a hop length of 128. Additionally, we retain the top 5 eigenpairs from the fitted operator matrix obtained via DMD.
C.4. 3-bit Flip-Flop Recurrent Neural Networks
In the 3-bit Flip-Flop task, as shown by Figure 9, three separate input streams (grey) deliver occasional pulses with values of -1 or at random intervals. The network is required to retain the most recent pulse value from each channel and continuously output until a new pulse arrives (dark green solid line), triggering a state switch. The task tests the network's ability to maintain memory over time while adapting unpredictable and asynchronous changes, pushing its capacity for dynamic memory retention and updating.
We train population of RNNs with three network architectures: vanilla RNN with tanh activation, LSTM, and GRU, Each network consists of single recurrent layer with 256 hidden units, followed by a linear readout layer. Following the experimental setup in (Schuessler et al., 2024), we inject zero-mean, isotropic white noise into the hidden states during training to promote robust and stable recurrent dynamics. The initial recurrent and output weights are drawn from centered normal distributions. Each network is trained with batch size 32 and learning rate of 0.01, with different random seeds. To ensure task performance, we only save checkpoints where the MSE falls below 0.001.
According to (Maheswaranathan al., 2019), while the geometry of RNN activations can be highly sensitive to variations in network architectures, the underlying computational dynamics-sun such as the topological structure of fixed points points-often remain consistent. To explore this, we visualize the hidden states of well-trained RNNs with different architectures in Figure 10. By reducing the dimensionality to three using PCA, we observe that the resulting trajectories exhibit qualitatively similar structures across architectures, tracing the corners a three- dineessinnn cube. This cube further characterized by eight stable fixed points (Golub & Sussillo, 2018), highlighted in black.
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KoopSTD: Dynamical similarity mcasurement via approximating Koopman Spectrum with Timescale Decoupling
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Figure Illustration of the onput-oulput scheme 3-bit Flip-Flop task. The input pulse sequence is shown in dark gray, the ground truth is represented by dark green solid line, and the output of well-trained model is depicted by yellow dashed line.
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Figure PC3 hidden states trajectories of Vanilla RNN, LSTM, GRU when solving 3-bit Flip-Flop task.
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KoopSTD: Dynamical similarity mcasurement via approximating Koopman Spectrum with Timescale Decoupling
For each trained network, we extract hidden layer outputs using the same batch of inputs. The hidden representation for each trial is denoted as h E R 12552256 where 125 corresponds to the time dimension. We present the result of KoopSTD> with parameters | 10, 8 1, 10; however, our method remains robust to hyperparameter selection.
C.5. Auditory Cortex
Our experiments utilize the Narratives dataset (Nastase et al., 2021), comprchensive collection of auditory story-listening [MRI data comprising recordings from 345 unique participants across 28 naturalistic spoken stories, each lasting approx- imately 3 to 56 minutes. For details on data acquisition and preprocessing please refer to the original paper. To analyze auditory cortical regions, we extract fMRI recordings from five areas-A1, LBelt, MBelt, PBelt, and RI-defined on the cortical surface in ísaverage6 space using a multimodal cortical parcellation. For simplicity, we use a subset of the data, specifically from 68 participants listening to the "Pieman" story, which has duration of 450 seconds. With an fMRI sampling rate of 0.67 Hz, each recording consists of 300 time points.
With 1 10, 1,r r 50, the result illustrated in Figure 4 by KoopSTD can be reproduced.
C.6. Large Language Models
GPT-2. GPT-2 (Generative Pre-trained Transformer 2) (Radford et al., 2019) is large-scale language model developed by OpenAI, designed for natural language understanding and text generation. It is based on the Transformer architecture and trained on a diverse corpus of internet text using unsupervised learning. GPT-2 features multiple model sizes, ranging from 124M to 1.5B parameters, and generates coherent, contextually relevant text given a prompt. Unlike traditional task-specific models, GPT-2 excels in zero-shot and few-shot learning, demonstrating strong performance across various NI.P tasks without fine-tuning. To measure the dynamic discrepancy between models, we apply KoopSTD on the hidden representation of their last layer, using the hyperparameters 150, s = 1,r r 10.
LLaMA. LLaMA (Large Language Model Meta AI) (Dubey al., 2024) is a family of open-weight language models developed by Meta, designed for efficient and scalable natural language processing. It offers strong performance across various NLP tasks while being more accessible and resource-efficient than many proprietary models. In our experiments, we use LLaMA 3.2 for the IB and 3B models and LLaMA 3.1 for the 8B model. The measured results by KoopSTD presented in Figure 6 can be reproduced by the hyperparameters = 50, 1, 10.
D. Detailed Experimental Results
Structural functional relationship between cortical regions. To better illustrate the functional-structue coupling patterns in the auditory cortex identified by KoopSTD in Sec. 4.4, we present the anatomical organization and myelin distribution of early auditory regions in Figure 11. This structural homogeneity aligns with their shared functional properties observed in our analysis, where A1, MBelt and LBelt areas are characterized by comparable high myclin content (shown in red-yellow). In contrast, RI and PBelt regions exhibit distinct myelination patterns, suggesting their relative independence in both structural and functional aspects. This anatomical evidence, particularly the myelin distribution pattern, provides additional support for our KoopSTD-based findings regarding the hierarchical organization and functional coupling within the auditory cortical network.
E. Ablation Study
We demonstrate that KoopSTD better captures system dynamics compared to the recent HLAVOK-basen DSA (Ostrow et al., 2024), which relies on truncated-ramt regression of time-delay embeddings (detailed in C.1). KoopSTD's s advantage stems from two key aspects: decomposing system transitions across multiple timescales using time-freyeency transformation, and filtering out spurious modes based on their spectral residuals. To validate, we conduct ablation studies on comparing different trajectories generated by the Lorenz system using HAVOK-based DSA, time-delay embedding with residual-controlled Koopman spectrum (an ablation), time- free-freueency transformation with truncated Koopman spectrum (another ablation), and the proposed KTDD
Figure 12(a) demonstrates that both time-nryy transformation and residual-controlles Koopman spectrum contribute to KoopSTD's performance on the Lorenz system. By filtering spurious dynamical modes, KoopSTD successfully distinguishes
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KoopSTD: Dynamical similarity measurement via approximating Koopman Spectrum with Timescale Decoupling
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Figure The early auditory regions are displayed on a folding map in Panel A, while Panel presents myelin mapping scaled from 4% 96% to emphasize variations in myelin density within the heavily myelinated auditory cortical regions (reproduced from (Glasser al., 2016)).
between line-like and ring-like trajectories. This step results in a more concentrated eigenvalue distribution on the complex plane (i.e., ring-like trajectory denoted by green triangles, Figure 12(b)), preserving only the modes that capture essential dynamics. Meanwhile, the freeufeyy transformation effectively reveals the disparity between complex dynamical patterns through the multi-scale temporal analysis (i.e., chaotic behavior denoted by red circles, periodic orbits denoted by yellow triangles and purple squares, Figure 12(c)).
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KoopSTD: Dynamical similarity measurement via approximating Koopman Spectrum with Timescale Decoupling
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Figure 12, Ablation study on time- aee-freuynncy transformation and cositual-controlledd Koopman spectrum, each columns from left to right corresponds to one of the ablations.
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Reviews:
[{"review_id": "V7bAIKlH7h", "reviewer": "Reviewer_ttZN", "summary": "This paper is part of the broad hot topic of comparing non-linear neural networks through the lens of representation space and focuses more specifically on the measurement of dissimilarity between non-linear dynamical systems.  These systems, due to their non-linearity, can be analyzed using the Koopman operator theory, which transforms them into a linear, infinite-dimensional space of observables. The authors introduce KoopSTD, a novel metric for assessing dynamical similarity by comparing their Koopman spectra. Their method involves 1) decomposing transitions in state space using the short-time Fourier transform (STFT), 2) extracting spatiotemporal patterns via Koopman mode decomposition (KMD), and 3) filtering out spurious modes by selecting the top $r$ eigenpairs of the Koopman operator with minimal spectral residuals.\n\nThe main contributions of the new metric are:\n\n- Experiments: Applied to physical and neural systems, comparing it to existing similarity metrics like Centered kernel\nalignment (CKA) (, Procrustes analysis, Cross-Correlation, Hankel\nAlternative View of Koopman (HAVOK-based DSA).\n\n- Theory: KoopSTD satisfies three invariance properties: isotropic scaling, rotation, and permutation invariance.\n\n- Applications: Used for analyzing fMRI data and Large Language Models (LLMs).", "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": "Thanks for your careful review and comments!\n\nGlad to see that you deem our work interesting and have a promising impact on many fields. Please kindly find our point-to-point responses below.\n\n## Concerns\nQ1: Presenting the invariance property as 'theorem'.\n\n>We revise the invariance property from \"Theorem” to \"Proposition”.\n\nQ2: Using \"theoretical guarantees\" to describe the properties of KoopSTD.\n\n> We revise the \"theoretical guarantees\" to \"theoretical groundedness\" accordingly.\n\nQ3: Improper usage of \"Koopman matrix\" and \"$𝐗_{t+1}=𝐀_{1}𝐗_{t}$\".\n\n> We correct the terminology to \"approximated Koopman operator\" and \"$g(𝐗_{t+1})=𝐀_{1}(g(𝐗_{t}))$\", respectively.\n\nQ4: Incorrect equality $d(cλ(𝐀_1),cλ(𝐀_2))=d(λ(𝐀_1),λ(𝐀_2))$.\n\n> We fix this error by removing line 250 (left) in the main manuscript to let $d(c𝓕_1,c𝓕_2)=d(λ(𝐀_1),λ(𝐀_2))$ directly.\n\nQ5: Unclear proof of three invariance properties.\n\n> We refine our proof by unifying three invariance properties into a single proposition below.\n\n**Proposition.** Let $𝐗_1[t+1]= 𝓕_1({𝐗_1[t]})$ and $𝐗_2[t+1]= 𝓕_2({𝐗_2[t]})$ be two time-discrete dynamical systems, where $𝓕_1, 𝓕_2 : ℝ^{N_d} \\to ℝ^{N_d}$. The dissimilarity $d(𝓕_1, 𝓕_2)$, as measured by KoopSTD, remains invariant under invertible linear transformations 𝓣,\n$$\n    d(𝓣(𝓕_1, 𝓕_2)) = d(𝓕_1, 𝓕_2),\n$$\nwhere $𝓣=${$𝐗 ↦ 𝐗𝐐: 𝐐 \\in GL(N_d, ℝ)$}.\n\n**Proof.** For a dynamical system $𝐗[t+1]= 𝓕({𝐗[t]})$. The KoopSTD seeks to obtain the approximate Koopman operator $𝐀$ by firstly mapping $𝐗\\inℝ^{T\\times N_d}$ into a time-frequency representation $𝐙\\inℝ^{(\\frac{T-l}{s}+1)\\times(\\frac{l}{2}+1)\\times N_d}$ using STFT.\n\nBy the linearity of the STFT, we can express the process of STFT as:\n$$\n    𝐙=\\text{STFT}(𝐗)=𝓛_{tf}𝐗,\n$$\nwhere $𝓛_{tf}$ refers to the linear operator of the STFT.\n\nThen, KoopSTD computes the right singular vectors $𝑽$ of $𝐙$ by solving SVD:\n$$\n   𝐙^* 𝐙=(𝑽\\Sigma 𝑼^*)(𝑼\\Sigma 𝑽^*)=𝑽\\Sigma^2𝑽^*.\n$$\n\nTherefore, the Koopman operator can be approximated based on temporal snapshots $𝐖_{𝑽}$ of $𝑽$:\n$$\n    𝐀 = 𝐖_𝑽'𝐖_𝑽^{†}.\n$$\n\nGiven any transformed input $𝐗𝐐$, we can also derive its time-frequency representation $𝐙_{𝓣}$:\n$$\n    𝐙_{𝓣}=\\text{STFT}(𝐗𝐐)=𝓛_{tf}𝐗𝐐=𝐙𝐐.\n$$\n\nThen, the transformed singular vectors $𝑽_{𝓣}$ of $𝐙_{𝓣}$ can be derived as follows:\n$$\n    𝐙_{𝓣}^* 𝐙_{𝓣}=(𝐙𝐐)^* 𝐙𝐐=𝐐^* 𝐙^* 𝐙𝐐 =(𝐐^* 𝑽)\\Sigma^2(𝑽^* 𝐐),\n$$\n$$\n    𝑽_{𝓣}=𝐐^*𝑽.\n$$\n\nThe transformed snapshots $𝐖_{𝑽,𝓣}$ of $𝑽_{𝓣}$ can also be derived:\n$$\n    𝐖_{𝑽,𝓣}=𝐐^*𝐖_{𝑽},\n$$\n\n$$\n    𝐖'_{𝑽,𝓣}=𝐐^*𝐖'_𝑽.\n$$\n\nThus, the transformed approximated Koopman operator $𝐀_{𝓣}$ can be denoted as follows:\n$$\n    𝐀_{𝓣}=𝐖_{𝑽,𝓣}'𝐖_{𝑽,𝓣}^{†}=𝐐^* 𝐀(𝐐^*)^{-1}.\n$$\n\nBy definition of similarity transformation, the matrices $𝐀_{𝓣}$ and $𝐀$ are similar, implying that they share the same eigenvalues:\n$$\n    λ(𝐀_{𝓣})=λ(𝐀).\n$$\nBased on Definition (1) and the above derivation, we have:\n$$\n    d(𝓣(𝓕_{1},𝓕_{2}))= d(λ(𝐀_{1,𝓣}),λ(𝐀_{2,𝓣}))=d(λ(𝐀_{1}), λ(𝐀_{2})) = d(𝓕_1,𝓕_2).\n$$\nSince the following transformations on the feature dimension are also invertible linear transformations, we can deduce that KoopSTD remains invariant to the following transformations,\n- **Isotropic scaling**$:𝓣_{IS}=${$𝐗↦ 𝐗𝐐:𝐐=qI_{N_d},q\\in ℝ^+$}.\n- **Rotations**$:𝓣_{R}=${$𝐗 ↦ 𝐗𝐐:𝐐 \\in O(N_d)$}, where $O(N_d) :=$ {$ℝ^{N_d \\times N_d}, 𝐐^T𝐐=I_{N_d}$} denote the orthogonal group.\n- **Permutations**$:𝓣_{P}=${$𝐗 ↦ 𝐗𝐐_{\\pi}: \\pi \\in P(N_d)$}, where $P(N_d)$ is the set of permutations on {$1,...,N_d$}.\n\nQ6: The properties of KoopSTD stem from the invariance of the characteristic polynomial under similarity transformations. Does KoopSTD remain invariant to all similarity transformations?\n\n> As shown in our response to Q5, applying any invertible linear transformation to $𝐗$ is equivalent to applying a similarity transformation on the approximated Koopman operator $𝐀$, thereby ensuring KoopSTD's measurement remains invariant to all invertible linear transformations.\n\n> We emphasized the invariance properties in terms of *isotropic scaling, rotations and permutations* in the manuscript because they are essential for developing a reliable metric that ensures **robust** similarity measurements across different systems.\n\n## Related work of spectral error estimation provided by the reviewer\n\n> Thank you for providing insightful related work!\n\n> We agree that the performance of spectral error estimation using the Galerkin method can be influenced by the choice of basis. We have carefully read these relevant studies. The `Introduction` section will cover them briefly. Additionally, we will cite some of them in the `Conclusion` section as future work in effectively extracting the reliable modes.\n\n\n## Typos pointed out by the reviewer\n> Thanks for your careful review! We have proofread the manuscript, the typos will be fixed accordingly.", "claims_and_evidence": "The main claims regarding the new KoopSTD metric are as follows:\n\n- Theoretical: The KoopSTD metric satisfies three key invariance properties: isotropic scaling (Theorem 1), rotation (Theorem 2), and permutation invariance (Theorem 3).\n\n-  Experimental: The KoopSTD procedure outperforms existing similarity metrics (CKA, CC, Procrustes, HAVOK-DSA) in distinguishing five Lorenz systems with varying Rayleigh numbers $\\rho$.  It is robust in capturing the underlying dynamical differences, even in the presence of similar or noisy attractors. Additionally, KoopSTD can correctly identify neural network systems with identical recurrent dynamics despite differences in their geometric network architectures.\n\n- Prospects: The paper examines KoopSTD's ability 1) to uncover functional correspondences in the human auditory cortex and 2) to provide insights into the relationship between model size and the functional behavior of LLMs.\n\n\nI believe the experimental section is quite convincing in demonstrating the effectiveness and capability of KoopSTD to distinguish (or identify) systems in situations where the dynamics show trajectory similarities (or differences in geometric architectures).\n\nThe work on using KoopSTD in computational neuroscience and LLMs, although exploratory, is interesting and could open up new perspectives.\n\nHowever, the theoretical part seems less developed: the statements about invariance properties (isotropic scaling, rotation, and permutation), presented as 'theorems,' are on the level of textbook exercises, and their proofs need revision.", "methods_and_evaluation_criteria": "The chosen benchmark datasets effectively showcase the procedure’s properties. Its effectiveness is demonstrated through the study of Lorenz systems with different Rayleigh numbers, leading to five distinct trajectory classes. KoopSTD’s ability to differentiate systems, even when their trajectories appear similar, is tested using three noisy attractors, where distinctions arise solely from noise-induced deviations. Additionally, experiments with three 3-bit Flip-Flop RNNs of varying architectures show that KoopSTD outperforms representation-based methods (CKA, Procrustes) and dynamical approaches (CC, HAVOK) in identifying neural network systems with the same recurrent dynamics, as measured by the Silhouette coefficient.", "theoretical_claims": "There are several issues with the theoretical claims and their proofs. The concept of __Koopman matrix__  is used in improper manner. In author's notation, for system $\\mathcal F_1$ the Koopman matrix $A_1$ is an __approximation of an operator__ and equation  $X_{t+1} = \\mathbf A_1 X_t$ makes no sense at all. The systems are not linear autonomous systems (which is the meaning of the equation). The __embeddings/representations  are crucial part__ of Koopman based approaches, and there they are not treated properly.\n\nAssuming one correctly writes whole operator formalism, and assuming that one proves the metric induced by the operators and not approximations (which is not in my opinion sufficient to imply guarantee theoretical guarantees of the method, but only suggest its groundedness), still theorems would need rewriting. Namely:\n\n__Theorem 1__: The equality $d(c\\lambda(\\mathbf A_1),c\\lambda(\\mathbf A_2))=d(\\lambda(\\mathbf A_1),\\lambda(\\mathbf A_2))$ seams to be false. Using it, it follows immediately from definition (8) that $d(c\\mathcal F_1,c\\mathcal F_2)=|c|d(\\mathcal F_1,\\mathcal F_2)$. Moreover, the notation $d(c\\mathcal F_1,c\\mathcal F_2)$ is clumsy and the proof is unclear as it stands. Similarly, Equations (17) (proof of Theorem 2) and (18) (proof of Theorem 3) should be rewritten in a clearer and more understandable way.\n\n\n__Theorems 1-3__: All results are in fact a direct consequence of the well-known fact that the characteristic polynomial is invariant under similarity transformations. In particular, similar matrices have the same eigenvalues and, therefore, the same characteristic polynomial.Similarity transformations include scaling (Theorem 1: $A_1^c=DAD^{-1}$ where $D=cI$), rotation (Theorem 2), \\textit{shearing} (if $\\mathbf {A}_1^{\\text{sheared}}=P\\mathbf {A}_1P^{-1}$ for some invertible matrix $P$), and orthogonal transformations (Theorem 3).", "experimental_designs_or_analyses": "I haven't verified the validity of the experimental designs in detail.", "relation_to_broader_scientific_literature": "I believe there are recent works based on the related aera of \\textit{learning Koopman operators} that provide precise spectral bounds,\nIndeed, to effectively capture multi-scale temporal dynamics, the proposed KoopSTD first transforms the data into the time-frequency representation space via STFT, resulting in two time-frequency embeddings that extract the top $r$ eigenvalue-eigenvector pairs by minimizing spectral residuals (Equation (7)). The following papers provide bounds for the estimation errors of Koopman operators in different frameworks. I believe some of them (at least) should be cited; perhaps the estimation procedures developed in these works could allow for extracting the top $r$ eigenvalue-eigenvector pairs more efficiently than using a Galerkin approximation (which can be computationally expensive and can fail if the modes of the system are not well-separated in the frequency domain) followed by minimizing the residual error, as is done in the paper. \n\n- Hou et al., Sparse learning of dynamical systems in RKHS: An operator-theoretic approach. In International Conference on Machine Learning, 2023. ($\\beta$-mixing stochastic processes). \n- Klus et al., Data-driven model reduction and transfer operator approximation Journal of Nonlinear Science, 2018. (Time-homogeneous stochastic processes).\n- Klus et al., Eigendecompositions of transfer operators in reproducing kernel Hilbert\nspaces. Journal of Nonlinear Science, 30(1):283–315,\n2020c\n- Kostic et al., Learning dynamical systems via Koopman operator regression in reproducing kernel Hilbert spaces, NeurIPS, 2022. (Markov chains)\n- Kostic et al., Sharp spectral rates for Koopman operator learning, NeurIPS, 2023. (Time-reversal-invariant stochastic dynamical systems).\n- Philipp et al., Error bounds for kernel-based approximations of the Koopman operator}. Applied and Computational Harmonic Analysis, 2024. (Stochastic differential equations).\n- Bevanda et al. Koopman Kernel Regression In Advances in Neural Information Processing Systems 36 (deterministic dynamical systems)\n- Drmac and Mezic, A data driven Koopman-Schur decomposition for computational analysis of nonlinear dynamics, arXiv preprint arXiv:2312.15837", "essential_references_not_discussed": "This comment is the same as the one in the previous section \"Relation to broader scientific literature\".", "other_strengths_and_weaknesses": "The main body is well-written and enjoyable to read. The application of KoopSTD to cortical fMRI data to highlight functional dissimilarities in auditory processing, aligning with myelination patterns, is particularly promising for its potential use in other areas.\n\nThe main weaknesses stated in other fields are the cause of my score. If the authors thoroughly revise the statements of the theorems and their proofs to address this question, I would be happy to reconsider my evaluation.", "other_comments_or_suggestions": "L161: I think it should be $g(x^t)=\\sum_{i=1}^\\infty\\lambda_i^{\\textcolor{red}{t}}\\phi_i(x^0)v_i$.\n\nL130 (second column): Frobenius.\n\nL208: as $\\mathbf{A_{tf}}=W_V'W_V^\\dag$ (Equation (6)), it should be $\\mathcal M=W_V^{'*}W_V'$, $\\mathcal N=W_V^{*}W_V'$ and $\\mathcal M=W_V^{*}W_V$. \n\nL175(second column): to use eigenvalue\\textbf{s}.\n\nL250, Equation (10): The sequence of equalities is incorrect. It should be: $d(\\mathcal C(\\mathcal F_1,\\mathcal F_2))=d(\\mathcal F_1,\\mathcal F_2)$.\n\n L250 (second column, table): PDM Attractors \\textbf{Flip-Flop} RNNs.\n\nL688: The Koopman matri\\textbf{ces} $\\mathbf A_1$ and $\\mathbf A_2$.\n\n L705: The \\textbf{system} $\\mathcal F_1$ [...] transform\\textbf{s} as follows.\n\n L776: I think it should be $\\mathcal K$ instead of $K$.\n\n L790: Following Equation (30).\n\n L1120: five \\textbf{different} dynamics.", "questions_for_authors": "The KoopSTD procedure is interesting and effective for studying similarity between dynamical systems, resistant to spurious eigenvalues, and has promising applications in neuroscience and LLMs. However, the presentation of the theoretical part is unconvincing in its current form. Is KoopSTD invariant to all similarity transformations (see section 'Theoretical claims')?", "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": "BBL9ml0Zua", "reviewer": "Reviewer_WLZa", "summary": "A method for computing dissimilarity between dynamical systems is proposed. It is based on the Wasserstein distance between sets of eigenvalues of the Koopman operators of the dynamics. Here two major heuristics are adopted. One is the STFT-based observable, which the authors motivate as a way to deal with multiple timescales of dynamics. The other is spectral residual as proposed in resDMD, with which the method focuses only on important eigenvalues and becomes more robust. The proposed method's utility is validated with different datasets.\n\n### Update after rebuttal\n\nThank you for the detailed response. It would strengthen the paper to include the discussion about the phase information. I would keep my original positive rating.", "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": "We sincerely appreciate the reviewer's valuable feedback!\n\nPlease kindly find the point-to-point response to the questions and suggestions.\n\nQ1: How about applying DMD on the complex part of STFT results instead of the real part?\n\n> We appreciate the reviewer’s insightful question. We agree that discarding phase information will enhance robustness, as phase is highly sensitive to time shifts and minor perturbations. This sensitivity is particularly problematic in fMRI and other biological signals, where recording conditions and preprocessing can introduce unavoidable phase shifts.\n\n> From the perspective of the Koopman operator theory, the magnitude, which represents the signal's energy distribution, generally evolves in a smoother manner and is more amenable to linear approximation, making it more suitable for DMD. On the other hand, phase information is often linked to transient and localized changes, which are more challenging to capture within the linear framework of the Koopman operator theory. Therefore, in the KoopSTD framework, the real part is more suitable.\n\n> We acknowledge that incorporating phase information in a controlled manner to enrich the description of system dynamics could be an interesting future direction. We will add a discussion on this point in the `Conclusion` section in the revised manuscript.\n\n> Once again, we sincerely thank the reviewer for bringing this to our attention.\n\nQ2: How can we determine the hyperparameters of KoopSTD in practice, including the number of modes?\n\n> Thanks for pointing out this question! We agree that the choice of hyperparameters — particularly the STFT window length $l$ and the number of preserved modes $r$ — deserves more discussion. We will add a brief description in the `Experimental Results` section in the revised main manuscript and a detailed discussion in the revised appendix.\n\n> In most DMD-related work, hyperparameters are typically chosen heuristically, relying on prior knowledge and experience with the specific system’s characteristics[1,2,3]. In contrast, by conducting an additional sensitivity analysis, we demonstrate that our KoopSTD remains robust across a wide range of these parameters, reducing the reliance on manual tuning. Specifically, we conducted the experiments on the classical Lorenz system, measuring the silhouette coefficients of the distance matrices obtained by KoopSTD with various $<l,r>$ combinations. In this setting, the silhouette coefficient$\\in [-1,1]$ is the larger, the better.\n\n> The results are summarized in the table below. It is noteworthy that the results in most of the configurations are greater than 0.9. This indicates that KoopSTD is able to consistently distinguish different dynamical behaviors, even with diverse hyperparameter settings, highlighting its robustness in practice without requiring extensive hyperparameter tuning.\n\n| $l \\, \\backslash \\, r$ |   5    |  10   |  15   |  20   |  25   |\n|------------------------|--------|-------|-------|-------|-------|\n| **500**                | 0.898  | 0.931 | 0.959 | 0.969 | 0.962 |\n| **600**                | 0.918  | 0.965 | 0.973 | 0.968 | 0.946 |\n| **700**                | 0.947  | 0.961 | 0.971 | 0.979 | 0.985 |\n| **800**                | 0.904  | 0.931 | 0.945 | 0.943 | 0.956 |\n| **900**                | 0.827  | 0.942 | 0.939 | 0.920 | 0.929 |\n\n> Note that increasing $r$ within a reasonable range preserves more meaningful modes, potentially enhancing the representation of the system's dynamic patterns. However, continuously increasing $r$ presents two key drawbacks:\n1. Firstly, an excessively large number of modes can introduce uncontrollable spectral error, reducing the accuracy of the Koopman approximation.\n2. Secondly, the computational cost can significantly rise. While the computational cost between a single pair comparison remains manageable, comparing all pairs in a large dataset becomes substantially more time-consuming.\n\n[1] Kutz et al., Multiresolution dynamic mode decomposition. In SIAM Journal on Applied Dynamical Systems, 2016.\n\n[2] Noack et al., Recursive dynamic mode decomposition of transient and post-transient wake flows. In Journal of Fluid Mechanics, 2016.\n\n[3] Brunto et al., Chaos as an intermittently forced linear system. In Nature communications, 2017.", "claims_and_evidence": "The experiments nicely support the claim, i.e., ability to distinguish different dynamics and not to distinguish the same dynamics.\n\nA bit tricky point seems to lie in the use of STFT. If I understood correctly, the method uses STFT's spectral magnitudes only and dispose the phase information (because in Algorithm 1, the space of $\\mathcal{Z}$ is $\\mathbb{R}$, not $\\mathbb{C}$). The authors motivate the use of STFT with multiscale dynamics, but I think disposing the phase information is also an important feature of the method mainly for robustness. An ablation study dropping the STFT and instead taking time-dealy only is presented in appendix, which is nice, but I am curious about what happens if you do STFT but not dispose the phase, i.e., apply resDMD on the complex values outcome of STFT.", "methods_and_evaluation_criteria": "The method is simple yet reasonable. The empirical evaluation makes sense and is clearly presented.", "theoretical_claims": "The basic properties of the proposed dissimilarity are analyzed, which seem correct.", "experimental_designs_or_analyses": "(see Methods And Evaluation Criteria section)", "relation_to_broader_scientific_literature": "Dynamics comparison appears in various field and is indeed an important problem. The paper should have decent impact in many fields.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "The proposed method is simple, which I think is a good feature as a method generally applicable in various domains. Probably the choice of the number of modes to use is somewhat tricky in practice, so some comments if any on it would be appreciated.", "other_comments_or_suggestions": "N/A", "questions_for_authors": "N/A", "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": "fgU5Kf2WLc", "reviewer": "Reviewer_YjRQ", "summary": "This paper proposes KoopSTD, a new framework for comparing the similarity of non-linear dynamical systems by analyzing their Koopman spectra across multiple timescales. The main idea is to first transform time-series data into a time-frequency representation via short-time Fourier transforms, which extracts multi-scale dynamics. From there, the authors approximate the Koopman operator—a linear operator used to describe nonlinear system evolution—and apply a residual-based filtering step (Residual DMD) to remove “spurious” eigenvalues. The final step computes a similarity measure between the remaining, most accurate spectral components of any two dynamical systems. The paper demonstrates that KoopSTD (1) effectively separates different classes of Lorenz-type and other simulated chaotic dynamics, (2) robustly handles cases where geometric trajectories look similar but have different underlying dynamics (or vice versa), and (3) extends to real scenarios such as analyzing auditory cortex fMRI data and internal representations of large language models. Overall, the contributions center on using multi-scale Koopman analysis and spurious-mode filtering to yield a reliable, permutation-invariant metric of dynamical similarity.", "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": "We sincerely appreciate your thorough assessment of our work. We are delighted that you recognize KoopSTD's contributions to providing a robust and multi-scale similarity metric for nonlinear dynamical systems. In particular, we deeply appreciate your time and careful attention in examining our experimental setup, as well as the discussion of its theoretical foundations and empirical validation. Your recognition that our results convincingly support our claims is especially encouraging.\n\nThank you again for your time and evaluation!", "claims_and_evidence": "Broadly, yes. The authors provide synthetic (Lorenz-like systems, attractors) and empirical (fMRI data, LLMs) evidence showing that KoopSTD better distinguishes or unifies multi-scale dynamical systems than existing metrics. They also demonstrate how Residual DMD helps filter spurious eigenvalues. While more large-scale or higher-dimensional results could further confirm generality, the current experiments convincingly support the paper’s main claims on multi-scale robustness and spectral reliability.", "methods_and_evaluation_criteria": "Yes. The paper’s Koopman-based metric aligns naturally with multi-scale dynamical systems, and the authors employ Lorenz/attractor setups plus fMRI and LLM datasets, each demonstrating realistic temporal patterns. While relatively controlled, these examples effectively validate the method’s ability to handle different types of temporal complexity, which suits the paper’s goal of broadly characterizing non-linear dynamics across diverse settings.", "theoretical_claims": "The proofs are correct. The proofs of the invariance of dissimilarity are detailed.", "experimental_designs_or_analyses": "I looked in detail at the setup of the Lorenz/attractor experiments, the 3-bit Flip-Flop RNN tasks, and the fMRI and LLM tests. Each uses a clear procedure: collecting time-series data, transforming with STFT, approximating the Koopman operator, filtering eigenvalues, then comparing metrics. The sampling sizes and time lengths appear sufficient to capture relevant dynamics, and the train/test splits or multiple runs (e.g., different initial seeds) help confirm robustness. Overall, the paper’s experiments are reasonably well-structured and consistent with their stated aims.", "relation_to_broader_scientific_literature": "1. it builds on classical Koopman operator theory (Koopman, 1931; Kutz et al., 2016) and its practical approximation via dynamic mode decomposition (Williams et al., 2015), extending these ideas by incorporating short-time Fourier transforms to decouple dynamics across multiple timescales. This multi-scale analysis addresses the challenge of cross-scale coupling, which has been a known limitation in earlier Koopman-based metrics (e.g., Fujii et al., 2017; Ostrow et al., 2024).\n\n2. By integrating Residual DMD techniques (Colbrook & Townsend, 2024), the paper tackles spectral pollution issues that have been highlighted in prior work (Lewin & Sére, 2010; Drmac et al., 2018). \n\n3. This paper also relates to the broader literature on representation similarity analysis (e.g., Kornblith et al., 2019; Williams et al., 2021) by providing a robust, invariant measure for comparing the dynamics of complex systems such as neural and language models.", "essential_references_not_discussed": "No", "other_strengths_and_weaknesses": "NA", "other_comments_or_suggestions": "NA", "questions_for_authors": "NA", "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 KoopSTD, a novel method for comparing nonlinear dynamical systems by approximating their Koopman spectra with a timescale-decoupling approach. The method leverages STFT-based observables to address cross-scale coupling and applies residual-based filtering to isolate meaningful spectral components. The paper demonstrates the utility of KoopSTD through well-structured experiments on physical systems, neural networks, fMRI data, and large language models, showing it to be both effective and robust across settings. While the theoretical exposition—particularly regarding invariance properties—was initially underdeveloped, the authors responded thoroughly by refining proofs, clarifying terminology, and addressing reviewer concerns in detail. The method’s simplicity, general applicability, and potential for impact across neuroscience, machine learning, and system analysis make it a valuable contribution. I recommend acceptance.
Decision: Accept (poster)