Information Shapes Koopman Representation
Authors: Xiaoyuan Cheng
Wenxuan Yuan
Yiming Yang
Yuanzhao Zhang
Sibo Cheng
Yi He
Zhuo Sun
Venue: ICLR 2026 Oral (2026)
Keywords: Koopman Operator
Latent subspace reconstruction
representation for physical systems
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)
TL;DR: Because the Koopman operator is infinite-dimensional, identifying tractable finite-dimensional subspaces is challenging. We aim to construct these subspaces through information theory.

Abstract: The Koopman operator provides a powerful framework for modeling dynamical systems and has attracted growing interest from the machine learning community. However, its infinite-dimensional nature makes identifying suitable finite-dimensional subspaces challenging, especially for deep architectures. We argue that these difficulties come from suboptimal representation learning, where latent variables fail to balance expressivity and simplicity. This tension is closely related to the information bottleneck (IB) dilemma: constructing compressed representations that are both compact and predictive. Rethinking Koopman learning through this lens, we demonstrate that latent mutual information promotes simplicity, yet an overemphasis on simplicity may cause latent space to collapse onto a few dominant modes. In contrast, expressiveness is sustained by the von Neumann entropy, which prevents such collapse and encourages mode diversity. This insight leads us to propose an information-theoretic Lagrangian formulation that explicitly balances this tradeoff. Furthermore, we propose a new algorithm based on the Lagrangian formulation that encourages both simplicity and expressiveness, leading to a stable and interpretable Koopman representation. Beyond quantitative evaluations, we further visualize the learned manifolds under our representations, observing empirical results consistent with our theoretical predictions. Finally, we validate our approach across a diverse range of dynamical systems, demonstrating improved performance over existing Koopman learning methods.

Published as a conference paper at ICLR 2026
INFORMATION SHAPES KOOPMAN REPRESENTATION
Xiaoyuan Chengl* Wenxuan Yuan
Yiming Yangb Yuanzhao Zhang*
Sibo Cheng" Yi He
Zhuo S,n
'Dynamic Systems Lab, University College London
*Department of Earth Science and Engineering, Imperial College London
Santa Fe Institute
*CEREA, ENPC and EDF R&D, Institut Polytechnique de Paris
"School of Statisties and Data Science, Shanghai University of Finance and Economics
"Institute of Big Data Research, Shanghai University of Finance and Economics
Xiaoyuan Cheng and Wenxuan Yuan contributed equally to this work.
ABSTRACT
The Koopman operator provides a powerful framework for modeling dynamical systems and has attracted growing interest from the machine learning commu- nity. However, its infnite-dimensionall nature makes identifying suitable finite- dimensional subspaces challenging, especially for deep architectures. We argue that these difficulties come from suboptimal representation learning, where la- tent variables fail to balance expressivity and simplicity. This tension is closely related to the information bottleneck (IB) dilemma: constructing compressed rep- resentations that are both compact and predictive. Rethinking Koopman learn- ing through this lens, we demonstrate that latent mutual information promotes simplicity, yet an overemphasis lapse onto few dominant modes. In contrast, expressiveness is sustained by the Neumann entropy, which prevents such collapse and encourages mode diversity. This insight leads us to propose an information-thheoratee Lagrangian formulation that explicitly balances this tradeoff. Furthermore, we propose a new algorithm based on the Lagrangian formulation that encourages both sim- plicity and expressiveness, leading stable and interpretable Koopman rep- resentation. Beyond quantitative evaluations, we further visualize the learned manifolds under our representations. observing empirical results consistent with our theoretical predictions. Finally, we validate our approach across diverse range of dynamical systems, demonstrating improved performance over exist- ing Koopman learning methods. The implementation is publicly available at attps://gitan com/ woom///eeuan52222 Intoooatt
simplicity may cause latent space to col-
INTRODUCTION
Modeling and predicting the behavior of nonlinear dynamical systems are fundamental problems science and engineering (Brunton al., 2020; Kovachki et al., 2023; Mezic, 2020). Classical approaches typically rely nonlinear differential equations contrast, the Koopman operator framework offers a compelling alternative: represents nonlinear evolution linear transformation in appropriate function space (Koopman, 1931; Fritz, 1995).
black-box learning methods. In
Motivation. This linearization principle has recently attracted significant attention the deep learn- ing community, it enables complex nonlinear dynamics to be modeled and predicted using linear representations. However, integrating this framework into deep architectures poses fundamental challenge: the Koopman operator is inherently infiniteddimensional necessitating the identification or learning of a suitable finite-dimensional subspace for practical implementation. Deep learning models, most notably variational autoencoders (VAEs). have been employed to approximate such
"Email Xiaoyuan Cheng and Wenxuan Yuan to ucesxo4 Bucl. ac uk and wenxuan, sacaaanuuunggggg yuanfqq.com. 1 "Correspnndigg Author. Corresponding Zhuo Sun: sunzhuo@mai s.shufe edu. en.

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subspaces in a purely data-driven manner (Otto & Rowley, 2019; Pan & Duraisamy, 2020; Liu et al., 2023). Yet in practice, the resulting representations often suffer from instability, mode col- lapse fail to produce reliable dynamics. To address these challenges. some studies incorporate domain-specific priors-such as symmetry, conservation laws, dissipation, or ergodicity ( (Vaidya & Mehta, 2008; Weissenbacher et al., 2022; Azencot al., 2020; Cheng et al., 2025) into the Koopman representation. While effective in restricted settings, such approaches lack general prin- ciples for guiding Koopman representation. This calls for more general and principled approach to constructing diieee-dmensional representations, one that balances simplicity, in the form of latent linearity, with sufficient expressiveness (more literature review in Appendix C).
Information Bottleneck View. A natural way to achieve the tradeoff between simplicity and expres- siveness is through the lens of the Information Bottleneck (IB). The classic IB framework formalizes the idea that good representation should compress the input as much as possible while preserv- ing information relevant to a downstream task (Tishby et al., 2000; Tishby & Zaslavsky, 2015). In the context of representation learning (see Table 1), this typically means finding latent vari- able z that minimizes Complexity (x, 2)' from input x, while retaining expressiveness by improving Relevance(2, 2, y) (Vera et al., 2018). Instead of static latent representation, the goal of Koopman representation is to predict the future state In, given the current state hn-1 via a latent variable 2n-1- This gives rise to a dynamical information bottleneck formulation: we aim to learn Koopman rep- resentation 2n-1 with maximal linear predictability of future state Ta, while remaining as compact as possible.
Table 1: Information- ceemaation-thooeticc comparison between standard and Koopman representations. Here, 8 controls the trade-off' between simplicity and future-state expressiveness.
Latent Representation
Koopman Representation
Goal
Disentangled
Predictive
Info. Flow
Koepmar operiier
Sp
Zn
Lagrangian
Complesityy
x)
Relevance(2, v)
B Complexity 1, Fm Relevance (zn 1, 2nn
Why Is Finding a Good Koopman Representation Challenging? Learning Koopman represen- tation imposes stricter constraints than conventional latent representation models (see Table 1). In VAE or B-VAE (Kingma et al., 2013; Burgess et al., 2018), the focus is on reconstructing the input sampling from its distribution, which only requires the latent representation z to contain enough information about y. However, in Koopman learning, the latent space needs to support linear forward spaces. This constraint implies that the latent representa-
from 2, 2n+1 1 in some tion must not only capture information about the current state but also conform to a linear predictive structure (structural consistency) (Mardt et al., 2018; Kostic et al., 2023a;b; 2024), which imposes a stronger restriction. Prior work has shown that simply increasing the dimensions of the latent space does not necessarily improve performance (Li et al., 2020; Brunton et al., 2021), underscoring the importance of maintaining temporal coherence, i.c., ensuring that latent trajectories evolve con- sistently over time to prevent instability and error accumulation. Moreover, predictive sufficiency requires that the latent representation retains enough Koopman modes to faithfully reconstruct the system's 'ss future trajectories, such that multi-step prediction accuracy is preserved (Wang et al., 2022; 2025). Unlike standard VAEs and their variants, which emphasize flexible latent representations to support reconstruction, Koopman models demand dynamically consistent latent representations: small deviations can propagate and amplify over time. In summary, while conventional represen- tation learning emphasizes disentanglement and reconstruction, Koopman representation learning requires three key properties: temporal coherence, predictive sufficiency, and structural consisteney. The IB framework provides a meta view to navigate these trade-offs. It enables us to ask the central question:
Is it possible to learn Koopman representations that are both structurally simple and expressive, under the guidance of information-theereer principles?
Motivated by this question, we approach the problem from a fresh IB perspective, leading to core contributions: Theoretical Insight. We develop an coammttionthhoreete framework for Koopman representation, proving that mutual information controls error bounds while von Neumann entropy
'y Complexity(2, =) mean the mutual information [(:;; 2) in the IB framework, quantifying how much of the input information x is retained in 2. Relevance(z,) y) denotes T(2; v), the information carries about y.
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determines the effective dimension. By disentangling the information content of Koopman repre- sentations, we reveal how temporal coherence, predictive sufficiency, and structural consistency are governed by latent information and how these components are intrinsically connected to the spec- tral properties of the Koopman operator. This yields a novel information-theorate Lagrangian that extends the elassical IB principle by explicitly incorporating dynamical constraints, thereby making the fundamental trade-off between simplicity and expressivity in Koopman representation mathe- matically explicit. Principled Framework. Building on our anfformation-theoree derive a tractable, architecture-agneen rithm. Each term of the loss corresponds directly to one of the three desiderata- temporal coherence, predictive sufficiency, and structural consistency. This yields a general algorithm that is broadly ap- plicable: it extends naturally from physical dynamical systems to high-dimensional visual inputs and graph-structured dynamics, and our empirical results validate the theoretical predictions.
Lagrangian, loss function that translates our theory into practical algo-
tempooaa
PRELIMINARIES
Notation. Let MM C IR" be a finiteddimensional manifold equipped with a measure Consider discrete-time nonlinear map T .M- .M, so that the state v; M evolves according 2t
T(z:-1). We denote by H. = L jt, the Hilbert space of real-valued observables ø M + R.
Definition 2.1 (Koopman Operator (Koopman, 1931)) The Koopman operator K. H H is
linear operator acting on observables as
(x) <(T(x)), for & E H. x EEM.
(1)
Despite the appeal of lifting nonlinear dynamiis into linear forward via Koopman representation, practical approximations require projecting the ceinite-dimensione function space onto a finite- dimensional subspace. In the Koopman learning framework, this restriction manifests learning finite set of effective latent features {Φ1, 22. Φa} that map the state x as latent representation == Φ(x) E Z $ H. where is the latent space spanned by the selected latent features. The center of this paper is on discussion how to find good representation z. To ground the principles of information theory, we introduce some essential technical definitions.
Definition 2.2 (Mutual Information (MacKay, 2003)) Given two random variables x and y with joint probability distribution p(x,y) y) and marginal distributions p(x) and p(y), the mutual informa- tion 1 (x; y) quantifies the amount of information shared between x and and is defined as
I(x; v) Elloog p(x,y) 1 Ellog p(y)z)) ply)
p(z)p())
Definition 2.3 (Von Neumann Entropy (Witten, 2020)) Let p Radd be symmetric, positive semidefinite matrix with trace 1. The von Neumann entropy of p is defined as
S(p) -tr(p p).
If ( are the eigenvalues of p. then = >; log Ae This value reflects latent effec- tive dimensions: close if p concentrated on a single direction, and close log p spread uniformly. More connection with effective dimension is given in Appendix E.
Intuitively, mutual information and the von Neumann entropy provide principled way measure the predictability and the intrinsic effective dimension of Koopman representation. Building on these preliminaries, we can quantify the preserved information under Koopman representation.
3 METHOD
Our approach proceeds as: (1) probabilistic analysis in Koopman representation how information iitommation-thcorttiic to Koopman spectral properties; (3) general Lagrangian formulation guide better representation.
loss drives error accumulation; (2) an information theoretic characterization linking lost information
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3.1 INFORMATION FLOW IN KOOPMAN REPRESENTATION
A probabilistic view of Koopman representation. Firstly, we denote Ti: and 21:t as the states and their corresponding autoregressively generated latent variables from time step I, respec- tively. According to the direct information flow in Table I, the Koopman representation induces the following trajectory distribution given To-
p pk (21./10) - f p(20l10))
II (((zmlzn-1)((;;;) ddzddz ... dzg. mm
(2)
Here, p(zp (o) acts as the encoder, mapping the initial state into latent variable. The latent forward
is modeled by a linear Gaussian transition, where M(zn/zn-1 (2,, [Ken-1, Σ) is probabilistic representation of equation with variance Σ. This directly reflects Definition 2.1, as the latent evolution is constrained to be linear. Finally, each state In is reconstructed from its corresponding latent variable 2n via a decoder p(z, (zn). typically instantiated as a Gaussian. We now turn to the fundamental question of whether information is inevitably lost during latent propagation.
Proposition (Information Loss in Latent Evolution) Let In-1 + Zn-1 K Zn + Ja rep-
resent the information propagation in Koopman representation as shown in equation 2. Then, by the property of mutual information, the following holds:
( 2 [(2n-1: x..) 2 (2n=11;2n)
The detailed proof and its multi-step extension provided in Appendix F.I. The first inequal- ity reflects that the mapping Tx-1 + 2n-1 is compressed representation, which may discard predictive information about Zn. The second inequality indicates that the latent forward propaga- tion 2n-1 + 2n is governed by Koopman operator, inherently limits the information that can be preserved in the latent space. As result, 1(=n-1; =n,) is larger than [2, 2n), since the future state Fn generally carries more dependencies on 2n-1 than the latent evolution alone. In this sense, (-x-1i sets the information limit of Koopman representation by the operator K.
While Proposition 1 shows the degradation of information along latent propagation, it remains an abstract statement that is not directly tractable under the complex trajectory distributions in equa- tion 2. To obtain tractable measure, turn to the Kullback-Leiblee (KL) divergence as natural way to quantify the discrepancy between true and Koopman-inducas trajectories:
DKL (p(# (2)) || KR (z1: < DKL (p(#1al=0) |p
KR
(I1=e|=0)) (
+ Eenc + Eua Eres
(4)
Here, is the true distribution and p is the ideal Koopman model distribution in equation without any approximations. kk is the variational approximation, Eencs Ena and E are approxi- mation errors induced by the latent representation, Koopman operator and reconstruction (see details in Appendix F2). This motivates the following result, which formalizes how the information gap translates into autoregressive error bound for Koopman representations.
Proposition (Autoregressive Error Bound of Koopman Representation) The distribu- tion discrepancy between the true and moopman-inducce trajectories is bounded by the information gap as
lpp(ria 20) - KR (xia xo)]lrv <
(pa(ap( 20)
b1 kR z0)) +
(5)
(1 ((n E.
Here, || Ilrv is the total variation distance. The upper error bound is obtained
Ep(xia || (B,Ka[aaa æo]
xuoll
< C,
2 Σ
(I(x--1322) - 1----I(aaa [(2n-15 + E, (6)
where O is positive constant and & is related to the approximation error in equation 4.
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The proof is in Appendix F.3. The KL divergence between the true and Koopman-induced induced trajectory distributions reflects how much temporal coherence is lost during representation. Here, I(xn-1i Zn quantifies the intrinsic dynamical coupling T in the original system, while (2n zn) characterizes the information of that coupling that exists under Koopman representation. Since T(2n-15 2n) acts the information limit (see Proposition 1), the gap - ) 7(2n-11 1; 2n) measures the infor- mation that is lost when nonlinear dynamics are approximated by Koopman representation. Also, link the upper/lower error bounds and distribution discrepancy in equations (lower bound see equation 25). It reflects the prediction error is bounded by the step- wise information limit.
3.2 INFORMATION COMPONENTS IN KOOPMAN REPRESENTATION
The latent mutual information quantifies the magnitude of error, but does not uncover how this loss relates to Koopman spectral properties. To sharpen the insight from Propositions I and 2, we consider the aggregated quantity [2x; (e), which measures the total information available to the decoder p(2;) 2.). Our focus is on how much of this information can be stably propagated from past latent variables Zt-n-
(n):
Overall Architecture
(c):
Spectral Water-Filling Effect
MI
VNE
MI+
Input 1t-1
Latont space
Output or,
Encoddr
Decoder
Koooouii K Operater
Koyyaa Opertor
alog ((
logp(suls))
Spectral Mode
Allocated Information
(b):
Spectral Information Disentanglement
I(R 4i)
Encodlng
M(ai I Rs-x)
I(anz mt-1)
Original Space
Decoding
Space
Figure 1: Information theoretic Koopman framework. (a) Structure overview, (b) Information dis- entanglement with spectral interpretations, and (c) Water-filling effect of Mutual Information (MI) and von Neumann entropy (VNE) on spectral information allocation.
Proposition (Information Disentanglement and Spectral Property) The mutual infor- mation 22) can be disentangled into summation of three distinct components, each with a spectral interpretation (see proof in Appendix F.4, see Figure 1(b)):
Component
coherent
Fast Fast-dissipating
Residual no counterpart
Spectral property Mutual info term
A
T(ze-n;
I [2200111 =(-n)
I(2;: xx-1)
The decomposition shows that Koopman representations preserve temporal-coheree informa- tion associated with spectral modes of the Koopman operator whose eigenvalues lie near the unit circle, while information linked dissipating modes (\>) < 1) decays rapidly and noise-
like components have no spectral support, hence compressible.
(1). Temporal-coheree information I (zt-n 24) (see closed form equation 29). This term represents information that persists along the latent evolution 2t-n - ... + It corresponds to conserved or slowly dissipating information that remains stable during latent evolution. From spectral per- spective, these are associated with Koopman modes whose eigenvalues are near to the complex unit circle (i.e., [X] z 1), implying that the corresponding information is propagated almost losslessly across time and hence remains mutually informative between past and present latent variables.
(2). Fast-dissipating information [2x; x 1(t-)))))))) (see closed form equation 35). This term reflects short-term dependencies that arise from the most recent state _t- beyond what is already encoded in the past latent state 2--n- Such information provides transient predictive power but quickly leaks out, since the autoregressive latent evolution 2t-n inputs -. In contrast, these contributions are associated with Koopman modes whose eigenvalues satisfy [^] < 1, indicating exponential information dissipation with forward steps. Consequently,
2t cannot continually access external
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the mutual information they contribute vanishes rapidly as the time step increases, making those modes inherently cannot be captured by temporal-coherent information.
(3). Residual information I (2; I' ((-1) (see closed form equation 36). This term measures unpre- dictable information in the current state that cannot be explained from the past state. It corresponds to information injected the present step-sued such noise anomalies that interferes with con- structing a coherent latent state. Unlike temporal-coheren or fast-dissipating modes, these residuals have no spectral counterpart in the Koopman operator: they are not tied any eigenvalue structure. From the IB perspective, such non-predictive component is compressible. Having the disentangled information, the next question is how latent mutual information shapes Koopman representations.
explained
Proposition (The Role of Latent Mutual Information) Maximizing the lutent mutual in- formation [(z 24) allocates spectral weights to temporally coherent modes in the latent space, thereby enhancing the relevance of the Koopman representation. However, excessive emphasis this objective can lead to mode collapse, where the representation concentrates on only a few dominant modes and loses effective dimension (see Figure (())).
In Koopman representation, the latent mutual information admits a closed form
(x 2
log det(I + Mn & ( Mn
where det denotes the determinant, I is the identity matrix, C == Cou(zt-n) is the latent covari- ance matrix and M, == Ei=o K'I(K()T is the n-step linear forward covariance (see detailed explanation and proof in Appendix F.5). We find that, from a Lagrangian perspective, maximiz- ing [zz-n: ze) of Koopman representation under the finite variance constraint tr(C) < zo leads to a water-filling allocation: variance is distributed along the directions corresponding to the largest
eigenvalues of Mn K""C(K") M, These directions correspond to temporally coherent modes, which explains why higher latent mutual information enhances relevance. However, when the spec- trum this matrix is highly skewed, the water-filling solution degenerates into low-rank allocation, squeezing information into only a few dominant directions. This effect reduces the effective dimen- sion of the latent space Z. causing some spectral weights to vanish (cf. equation 43), To address the collapse induced by skewed spectral allocation, next analyze how effective dimension can be preserved through entropy regularization.
Proposition 5 (Effective Dimension and Anti-Collapse) Low effective dimension (see Proposition 4) in Koopman representation indicates information collapse to few dominant modes and limits the model's ability to represent rich modes. Penalizing the von Neumann entropy S (((ey)
encourages more expressive and spectrally diverse representations.
Connecting to Proposition 4, Appendix F.6 contains a detailed proof via a water-filling and La- grangian view. The normalized operator tr(C) can be regarded as a density matrix in Hilbert space, and the effective dimension can be measured as exp(S) (see Definition E.2). When penalized with large the von Neumann entropy, the water-filling solution attains non-zero allocation across all modes, preventing variance from collapsing entirely onto a few dominant directions (cf. equa- tion 46). This ensures positive distribution of spectral weights across all modes, thereby avoiding degenerate spectra and increasing the effective dimension of the latent space Z (see Figure (())..
C
3.3 INFORMATION FORMULATION FOR PRACTICAL IMPLEMENTATION
The preceding analysis (Propositions 3, 4 and 5) reveals a fundamental trade-off in Koopman repre- sentation learning: maximizing latent mutual information enhances temporal coherence and predic- tive ability but risks mode collapse, whereas entropy regularization promotes spectral diversity for predictive sufficiency. Based on this principle, we formulate the following unified Lagrangian:
max log [= 2x) - [I =..). + 7.5 ir(C)
log ((x))==))
where a, 39 and y Lagrangian multipliers. In equation 8, the first term in equation 7 preserves temporal-coheree information, the second term penalizes dist-dissipating or confounding compo- nents (z - T(2 Ft -l==-))) [(2150/000 see proof in equation 31), the third term

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rewards larger von Neumann entropy of the normalized covariance to promote spectral diversity in the latent space Z. Lastly, p(zz 2t) is the reconstruction terms from predicted latent variable &
While the Lagrangian in equation 8 captures the desired nnformation-thcoree trade-offs, it is not directly computable. To make practical, we derive a tractable loss function for satisfying temporal coherence, predictive sufficiency and structural consistency
max Σ
(= Pn)
+
B1E 3l (2n (u..) [lg Th (Zn (n-1)) 1)]
+
B H aa)
Temporal coherence
Structural consistency
Encoder entropy
+
log Pa (
C SS CF(C)
+ LELBO-
Reconstraction
Predictive sufficiency
In VAE structure (shown in Figure (a)), each component of the loss plays a distinct role in balanc- objectives: (1) The mutual information [ P..) captures temporal
ing the information-theow coherence by linking 2n to its temporal neighborhood Pn, = { cludes immediate past and future latent states; in practice, this can be computed either via the closed form in equation 7 for low-dimensional latents, or approximated by InfoNCE (Wu et al., 2020) for high-dimensional settings. (2) The term -Epo(.. [log go. (=
S S k}, which in-
Hy (2nl serves as an equivalent representation of the conditional mutual information I (1 with linear Gaussian transition (2( 2n-1) = [2 Ev)) and entropy of encoder Hp (zn (2,)) (see Appendix G.1). Minimizing this KI. not only encourages the latent representation to capture infor- mation from the state Tn, but also compresses fast-dissipating and residual components, ensuring that the representation remains expressive yet simple. Here, (t.. [log go (zn|zn-1)) enforces structural consistency in latent space. (3) The term log Pa (Ir)) zn. is the decoder loss from predicted
latent variable 2n, (4) Neumann entropy term S
e is computed from the normalized co- [+[C]]
variance matrix C = EZ () T)T of the latent codes within a minibatch of size B. This promotes spectral diversity and guards against mode collapse, ensuring that the learned Koopman representation retains predictive sufficiency. (5) LELBO is the Evidence Lower Bound (ELBO) for training stability and reconstruction (see more analysis and implementation details in Appendix G. For AE structure, Epo (#n (a,)) [log qu (Zn (an-))))) degenerates into I,² loss enforcing the structural consistency || || Kn and ELBO becomes AE reconstruction term (see G.2).
4 EXPERIMENTS
Tasks. We evaluate our approach across three types of dynamical data: (1) Physical simulations, including Lorenz 63, Kármán vortex street, Dam flow, and weather forecasting task (ERA5), which test the ability to capture nonlinear, stochastic and high-dimensional physical dynamics; (2) Visual- input control, including image-based Planar, Pendulum, Cartpole, and 3-Link manipulator, which evaluate the ability extract latent dynamics from high-dimensional visual inputs while control- lable in latent spaces; and (3) Graph-structured dynamics prediction, including Rope and Soft Robotics, which tests generalized abilities of latent representation on dynamics with graph struc- tures (see experimental details in Appendix G.3).
Metrics. We assess performance on both forecasting and control. For forecasting, we report (i) normalized root mean square егrоr (NRMSE) for short- and long-term predictions (for physical simulation and
dynamics), (ii) physical consistency metrics based on spectral dis- tribution enrors based on 1000- step sequences (SDEs), (iii) distributions of state measured by the Lelbeack-Leibler divergence (KID), and (iv) structural similarity index (SSIM) for physical sim- ulations. (v) the quality of ciw-dimensional manifold construction from high-dimensional visual inputs. For control, we measure the success rate of latent-space control of visual inputs following the setting in (Levine et al., 2020).
Baseline Algorithms. We compare against competitive baselines for each type of task. For physi- cal simulations, include VAE (Burgess et al., 2018), Koopman Autoencoder (KAE) (Pan et al., 2023), Koopman Kernel Regression (KKR) (Bevanda et al., 2023), and a SOTA Koopman variant for chaos Poincaré Flow Neural Network (PFNN) (Cheng et al., 2025). For visual-input control,
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consider VAE-based representation learning methods, including Embed to Control (E2C) (Ban- ijamali et al., 2019), as well as Prediction, Consistency and Curvature (PCC) (Levine et al., 2020), together with KAE. For graph-structured dynamies, compare with Compositional Koopman Operator (CKO) (Li et al., 2020), the current SOTA method for graph-structured dynamics.
Table 2: Performance comparison of five algorithms on physical simulation tasks. PENN is designed for chaotic dynamics and is thus not evaluated on Dam Flow. Here, N-NRMSE and N-SSIM denote errors at N prediction steps, values in parentheses indicate variance, and SDE is the spectral distribution error. Best results are highlighted in bold with green, second best are shaded in blue.
Task
Metric
VAE
KAB
KKR
PFNN
Ours
5- N-NRMSE
0.005 (0.002)
0.006 (0.003)
0.004 (0.002)
0.005 (0.003)
0.003 (0.002)
Lorenz 63 (n = 3)
20-NRMSE
0.011 (0.007)
0.014 (0.009)
0.009 (0.008)
0.011 (0.007)
0.007 (0.004)
50-NRMSE
0.019(0.011) (0.011)
0,023 (0.013)
0,017 (0.009)
0.017 (0.007)
0.013 (0,008)
KLD
1.047
0.464
0.342
0.293
0.285
5-NRMSE
0,127 (0.005)
0.149 (0.01 I)
0.114 (0.065)
0.075 (0.007)
0.068 (0.006)
20-NRMSE
0.134 (0.003)
0.195 (0.015)
0.157 (0.057)
0.125 (0.012)
0.114 (0.015)
Kármán Vortex
50-NRMSE
0.211 (0.018)
0.233 (0.027)
0,209 (0.028)
0.137 (0.015)
0.138 (0.018)
(n 64 x 64 x 2)
5-SSIM
0.743 (0.100)
0.719 (0.030)
0.868 (0.087)
0,920 (0.030)
0,936 (0.025)
20-SSIM
0.720 (0.079)
0.586 (0.039)
0.732 (0.086)
0,800 (0.0 050)
0.823 (0.047)
50-SSIM
0.539 (0.045)
0,571 (0.037)
0,581 (0.061)
0.710 (0.030)
0.688 (0.020)
SDE
0.538
0.620
0.799
0.278
0.256
5-NRMSE
0,030 (0.001)
0.037 (0.000)
0.019 (0.003)
0.018 (0.001)
20-NRMSE
0.033 (0.000)
0,042 (0.000)
0.028 (0.002)
0.024 (0.001)
50-NRMSE
0.034 (0.000)
0.046 (0.001)
0.031 (0.002)
0.026 (0.003)
Dam Flow
5-SSIM
0.522 (0.021)
0.419 (0.031)
0.720 (0.034)
0.760 (0.012)
(n 64 x 64 x
20-SSIM
0.443 (0.007)
0,282 (0.024)
0,584 (0.025)
0.627 (0.010)
50-SSIM
0.404 (0.005)
0,176 (0.008)
0.502 (0.010)
0.577 (0.006)
SDE
0.563
0.488
0.373
0.244
N-MMMSE
0.055
0.058
0.049
0.028
10-NRMSE
0.063
0.068
0.060
0.035
ERAS Weather (channel avg)
50- NRMME
0.118
0.074
0.079
0.068
5-SSIM
0.666
0.664
0.697
0.867
10-SSIM
0.606
0.635
0.808
50-SSIM
0.481
0.707
0.695
0.781
Result Analysis. Our analysis is organized around the contributions established in proposi- tions(Section 3), and we structure the discussion by addressing the following key questions. (1) Does the latent mutual information determine the predictive limit of the Koopman representation? (Proposition 2) Yes. This is verified by the quantitative results of physical simulations in Table 2. Consistent with proposition, the prediction error under Koopman representation inevitably accumu- lates and is bounded by the latent mutual information. By regularizing with latent mutual informa- tion, both short- and long-term predictions are improved. Notably, PFNN (Cheng et al., 2025) is a SOTA model specifically designed with domain knowledge, while method, grounded in general information theory, achieves comparable performance on chaotic tasks (Lorenz 63 and Kármán vortex). Compared with other Koopman- baase methods, our approach yields substantial improvements in both physical consistency and predictive accuracy.
(2) How is the preserved information parttcclllyy that associated with Koopman eigenmodes near the unit circle-shaped by latent mutual information and von Neumann entropy in constructing a dynamics-relevam manifold? (Proposition 4 and 5) The preserved information manifests in Koop- man modes with eigenvalues lying close to the unit circle, capturing the recurrent structure of the Kármán vortex limit cycle, as shown in Figure 2 (left). However, KAE suffers from some eigenval- ues collapse toward zero, reducing the effective latent dimension. This collapse explains the drift observed in its autoregressive prediction. In contrast, our model captures the limit-cycle structure and produces stable autoregressive trajectories, consistently revolving around the true orbit (Fig- ure 2, right). Baselines such as KKR and PFNN also capture limit-cycle structure (via one-step reconstruction) but gradually deviate from the correct trajectory over long horizons. By incorporat- ing latent mutual information, we ensure that temporal-coherent information retained, while von Neumann entropy prevents eigenvalue degeneration and preserves sufficient modes. Consequently, the information behind those modes can be preserved over long horizons, which directly translates into improved long-term prediction accuracy and statistical consistency, as also reported in Table 2.
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BatA3
(e(())
Re(A)
Onestop Prodiction
Autorogressive Prediction
Figure 2: Eigenvalue comparison and manifold visualization of the Kármán vortex street. Left: Eigenvalue distributions of Koopman operators. Right: t-SNE visualization of learned latent mani- folds for five methods on the Kármán vortex. The underlying dynamics is abstracted as a limit cycle.
(3) How does explicit information-theorele regularization sufficiently capture essential dynamics, compared with VAFs and Koopman autoencoders? (Proposition and 5) As shown in the recon- structed manifolds Figure 3, our method produces latent manifold that aligns most closely with the ground truth. For E2C, which is directly built on a VAE architecture, the latent geometry is heavily distorted (the loss of coherence). The manifold learned by KAE collapses into nearly one- dimensional structure, reflecting the lack of effective dimensions in its latent space. PCC, modified VAE-based method designed to improve manifold construction, demonstrates partial improvement but still exhibits gap compared with our approach. By preserving both effective dimensionality and temporal coherence, our Koopman representation achieves the best average control performance in both noiseless and noisy environments (Table and in Appendix G.5.2).
Ground Iruhh
E2C manifold
POD menifold
KAE menifold
Our manifodd
Figure 3: Latent manifolds of Planar visualized using locally linear embedding. The first subfigure shows the ground truth, while the second to fifth depict manifolds learned by different algorithms.
Ground Truth
KAE
KKR
PFNN
Ours
2018-01-01
08:00
Error
LEGOR
GRWRL
(2000
BELIZ
2018-01-00
68:00
tal
Error
Figure 4: Comparison of continuous predictions for the global humidity starting from 2018 01 00 00 to 2018 01 08 08 00. Error maps in the lower panels demonstrate that, compared with other models, showing with more stable and accurate results of our model (see more demonstration in Appendix G.5.1).

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(4) How robust are the findings under noise, extended prediction horizons, and large-scale settings? (Proposition and 2) Our method remains robust under both noisy observations and extended pre- diction horizons. As shown in Table 2 and Figure 4, it maintains stable performance in long-tem men-teemm rollouts and physical statistics in large scale weather forecasting. Moreover, our approach supports control under noisy environments, achieving competitive performance. These quantitative results are consistent with our probabilistic propositions.
(5) To what extent can our Lagrangian formulation be generalized diverse architectures and adapted to support downstream lasks? (Proposition 1-5) Our formulation demonstrates broad appli- cability: it consistently improves performance across physical simulations (see Table 2), visual per- ception tasks for manifold construction and control (see Figure 3, Tables 8 and in Appendix G.5.2), and graph-structuree dynamics prediction (see Figure 5). These gains indicate that the proposed La- grangian principle is architecture-agnoste and can be readily incorporated into different settings to enhance both predictive accuracy and task effectiveness (more results are referred Appendix G.5).
Prodictior
Prediction
Ropa
Prodiction Seft
Prodiction
26-
20
100
310
20
40
100
32
20
100
steps
steps
Figure 5: Comparison of prediction over 100 rollout steps. The left two figures show results for the Rope environment (n E [40, 56) with and without noise; the right two subfigures are the results for the Soft environment (n [160, 224)).
0,8 2,
0,y as
--
2-18-2r-1
2-
Porckin Alple
Figure 6: Ablation study on the pendulum task. Latent manifolds are learned from high -ish-dimensioaal pendulum images, where the ground-truth phase space is isomorphic to St x R. Color represents the pendulum angle. Each subplot corresponds to removing or adjusting one regularization term: latent mutual information (a), KL divergence (3), and von Neumann entropy (7).
Ablation Studies. We analyze the effect of varying each Lagrangian multiplier to understand its role in shaping Koopman representation. the pendulum task, the ground-truth phase space is S IR consisting of periodic angle and an angular velocity. The ablation study in Figure 6 illustrates how each regularization term contributes to recovering this manifold from high-dimensional visual inputs. Without mutual information regularization (a 0), temporal coherence is lost and the latent space degenerates into scattered points without geometric structure. Without structural consistency (3 = 0), the latent manifold collapses, highlighting its role in enforcing the dynamics of Koopman representation. Removing the von Neumann entropy term (7 0) retains the circular S compo- nent but suppresses the R dimension, indicating the necessity of preserving effective dimensions. Increasing mutual information alone concentrates the representation on the S' component (reflect- ing Proposition 4), while regularizing with von Neumann entropy yields manifold that closely approximates the full S x IR structure. These observations align with the theoretical roles of the three penalties: temporal coherence, structural consistency and predictive sufficiency.
5 CONCLUSION
We presented new perspective on Koopman representation by formulating it through an information-theorae lens, leading to a general Lagrangian formulation that balances simplicity and expressiveness. Our analysis reveals the relationship between Koopman spectral properties and information in deep architectures. The proposed algorithm based on the Lagrangian formulation consistently improves the performance in wide range of dynamical system tasks.
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ACKNOWLEDGMENTS
Zhuo Sun is supported by Fundamental Research Funds for the Central Universities 20251 10590 of Shanghai University of Finance and Economies.
ETHICS STATEMENT AND REPRODUCIBILITY STATEMENT
This work raises specific ethical concerns beyond standard practices in machine learning research. All methods, datasets, and hyperparameters are described in detail, and core code is released in supplementary materials.
REFERENCES
Daniel Alpay. Reproducing kernel spaces and applications, volume 143. Birkhäuser, 2012.
Hassan Arbabi and Igor Mezic. Ergodic theory, dynamic mode decomposition, and computation of spectral properties of the koopman operator. SIAM Journal Applied Dynamical Systems, 16 (4):209--2126, 2017.
Omri Azencot, N Benjamin Erichson, Vanessa Lin, and Michael Mahoney. Forecasting sequential data using consistent koopman autoencoders. In International Conference on Machine Learning, pp. 475-485. PMLR, 2020.
Francis Bach. Information theory with kernel methods. IEEE Transactions on Information Theory, 69(2)752-775, 2022.
Mark R Baker and Rajendra Patil. Universal approximation theorem for interval neural networks. Reliable Computing, 4(3):235-239) 1998.
Ershad Banijamali, Rui Shu, Hung Bui, Ali Ghodsi, et al. Robust locally-linees controllable embed- ding. In International Conference on Artificial Intelligence and Statistics, pp. 1751-1759, PMLR, 2019.
Alain Berlinet and Christine Thomas-Agnan Reproducing kernel Hilbert spaces probability and statistics. Springer Science & Business Media, 2011.
Petar Bevanda, Max Beier, Armin Lederer, Stefan Sosnowski, Eyke Hüllermeier, and Sandra Hirche. Koopman kernel regression. Advances in Neural Information Processing Systems, 36:16207- 16221, 2023.
Petar Bevanda, Max Beier, Armin Lederer, Alexandre Capone, Stefan Sosnowski, and Sandra Hirche. Koopman-equuvvarian gaussian processes. arXiv preprint arXiv:2502.0666 06645, 2025.
Stephen P Boyd and Lieven Vandenberghe. Convex optimization. Cambridge university press, 2004.
Morten Breivik and Thor I Fossen. Guidance laws for planar motion control. In 2008 47th IEEE Conference on Decision and Control, 570-577. IEEE, 2008.
Steven L Brunton, Bernd R Noack, and Petros Koumoutsakos, Machine learning for fluid mechanics. Annual review of fluid mechanics, 52(1):477-508 2020.
Steven L Brunton, Marko Budišić, Eurika Kaiser, and Nathan Kutz. Modern koopman theory for dynamical systems. arXiv preprint
2021.
Christopher P Burgess, Irina Higgins, Arka Pal, Loie Matthey, Nick Watters, Guillaume Des- jardins, and Alexander Lerchner. Understanding disentangling in beta-vae. arXiv preprint arXiv:1804.0333 03599. 2018.
Xiaoyuan Cheng. Yi He, Yiming Yang, Xiao Xue, Sibo Cheng, Daniel Giles, Xiaohang Tang, and Yukun Hu. Learning chaos in linear way. arXiv preprint 2rXiv:2503.14702 2025.
Thomas M Cover. Elements of information theory. John Wiley & Sons, 1999.
11

---

Published as a conference paper at ICLR 2026
Imre Csiszár, Paul C Shields, et al. Information theory and statisties: A tutorial. Foundations and Trends@ in Communications and Information Theory, 1(4):417-528, 2004.
Suddhasattwa Das and Dimitrios Giannakis. Koopman spectra in reproducing kernel hilbert spaces. Applied and Computational Harmonic Analysis, 49(2):533660 607, 2020.
Suddhasattwa Das, Dimitrios Giannakis, and Joanna Slawinska. Reproducing kernel hilbert space compactification of unitary evolution groups. Applied and Computational Harmonic Analysis, 54:75-136, 2021.
Marco Federici, Patrick Forré, Ryota Tomioka, and Bastiaan S Veeling. Latent representation and simulation of markov processes via time-lagged information bottleneck. arXiv preprint wwXiv:2309.07200 2023.
J Fritz. John von neumann and ergodic theory. The Neumann Compendium (F. Br ody and T. amos, eds.), World Scientific, Singapore, pp. 127-131, 1995.
Katsuhisa Furuta, Masaki Yamakita, and Seiichi Kobayashi. Swing up control of inverted pendulum. In Proceedings of IECON'91: 1991 International Conference on Industrial Electronics, Control and Instrumentation, pp. 2193-2198, 1991.
Shlomo Geva and J. Sitte. A cartpole experiment benchmark for trainable controllers. Unpublished benchmark description (often cited in RL literature), 1993. often referenced as the visual cart- pole, see e.g. in RL benchmarks.
Hans Hersbach, Bill Bell, Paul Berrisford, Shoji Hirahara, András Horányi, Joaquín Muñoz-Sabater, Julien Nicolas, Carole Peubey, Raluca Radu, Dinand Schepers, Adrian Simmons, Cornel Soci, Saleh Abdalla, Xavier Abellan, Gianpaolo Balsamo, Peter Bechtold, Gionata Biavati, Jean Bid- lot, Massimo Bonavita, Giovanna De Chiara, Per Dahlgren, Dick Dee, Michail Diamantakis, Rossana Dragani, Johannes Flemming, Richard Forbes, Manuel Fuentes, Alan Geer, Leo Haim- berger, Sean Healy, Robin J. Hogan, Elías Hólm, Marta Janisková, Sarah Keeley, Patrick Laloy- aux, Philippe Lopez, Cristina Lupu, Gabor Radnoti, Patricia Rosnay, Iryna Rozum, Freja Vam- borg, Sebastien Villaume, and Jean-Noël Thépaut. The era5 global reanalysis. Quarterly Journal of the Royal Meteorological Society, 146(730): 1999)2)):99999)099 2049, 2020.
Patrick Kidger and Terry Lyons. Universal approximation with deep narrow networks. Conference on learning theory, pp. 2306-2327, PMLR, 2020.
Diederik P Kingma, Max Welling, al. Auto-encoding variational bayes, 2013.
Bernard O Koopman. Hamiltonian systems and transformation in hilbert space. Proceedings of the National Academy of Sciences, 17(5):315-318, 1931.
Bernard O Koopman and v Neumann. Dynamical systems of continuous spectra. Proceedings of the National Academy of Sciences, 1833:2555-263, 1932.
Milan Korda and Igor Mezić. Optimal construction of koopman eigenfunctions for prediction and control. IEEE Transactions on Automatic Control, 65(12):5114455 2020.
Vladimir Kostic, Pictro Novelli, Andreas Maurer, Carlo Ciliberto, Lorenzo Rosasco, and Massimil- iano Pontil. Learning dynamical systems via koopman operator regression in reproducing kernel hilbert spaces. Advances in Neural Information Processing Systems, 35:4017-4031, 2022.
Vladimir Kostic, Karim Lounici, Pietro Novelli, and Massimiliano Pontil. Sharp spectral rates for koopman operator learning. Advances Neural Information Processing Systems, 36:32328- 32339, 2023a.
Vladimir R Kostic, Pietro Novelli, Riccardo Grazzi, Karim Lounici, and Massimiliano Pontil. Learn- ing invariant representations of homooooggessuss stochastic dynamical systems. arXiv preprint (.Xiv:2307.09912 2023b.
Vladimir R Kostic. Karim Lounici, Prune Inzerilli, Pietro Novelli, and Massimiliano Pontil. Consis- tent long-term forecasting of ergodic dynamical systems. In Forty-first International Conference Machine Learning, 2024.
12

---

Published as a conference paper at ICLR 2026
Nikola Kovachki, Zongyi Li, Burigede Liu, Kamyar Azizzadenesheli. Kaushik Bhattacharya, An- drew Stuart, and Anima Anandkumar. Neural operator: Learning maps between function spaces with applications to pdes. Journal of Machine Learning Research, 24(89):1-97, 2023.
Nir Levine, Yinlam Chow, Rui Shu, Ang Li, Mohammad Ghavamzadeh, and Hung Bui. Predic-
tion, consistency, curvature: Representation learning for locally-linear control. arXiv preprint arXiv:1909.01506, 2020.
Yunzhu Li. Hao He, Jiajun Wu, Dina Katabi, and Antonio Torralba. Learning compositional koop- man operators for model-based control. arXiv preprint arXiv:1910.0226 2020.
Yong Liu, Chenyu Li, Jianmin Wang, and Mingsheng Long. Koopa: Learning non-stationary time series dynamics with koopman predictors. Advances in neural information processing systems, 36:1 1es11271-12290 2023.
Jan CA Lubbe. Information theory. Cambridge university press, 1997.
David JC MacKay. Information theory, inference and learning algorithms. Cambridge university press, 2003.
Andreas Mardt, Luea Pasquali, Hao Wu, and Frank Noé. Vampnets for deep learning of molecular kinetics. Nature communications, 911):,, 2018.
Alexandre Mauroy, Y Susuki, and Igor Mezic. Koopman operator in systems and control, volume 7. Springer, 2020.
Igor Mezic, Koopman operator, geometry, and learning. arXiv preprint arXiv:2010.05377, 2020.
Samuel Otto and Clarence W Rowley. Linearly recurrent autoencoder networks for learning dynamics. SIAM Journal Applied Dynamical Systems, 18(1):558-592 2019.
Shaowu Pan and Karthik Duraisamy. Physics-informed probabilistic learning of linear embeddings of nonlinear dynamics with guaranteed stability. SIAM Journal on Applied Dynamical Systems, 19(1):480-500 2020.
Shaowu Pan, Eurika Kaiser, Brian M de Silva, Nathan Kutz, and Steven L Brunton. Pykoop- man: A python package for data-driven approximation of the koopman operator. arXiv preprint ar0iv:2306. 12962, 2023.
Stephan Rasp, Stephan Hoyer, Alexander Merose, Ian Langmore, Peter Battaglia, Tyler Russell, Alvaro Sanzhez-Gonzalez, Vivian Yang, Rob Carver, Shreya Agrawal, Matthew Chantry, Zied Ben Bouallegue, Peter Dueben, Carla Bromberg, Jared Sisk, Luke Barrington, Aaron Bell, and Fei Sha. Weatherbench 2: A benchmark for the next generation of data-driven global weather models. Journal of Advances in Modeling Earth Systems, 1666)=2023MS000001,, 2024.
Olivier Roy and Martin Vetterli. The effective rank: A measure of effective dimensionality. In 2007 15th European signal processing conference, pp. 606-610. IEEE, 2007.
James V Stone. Information theory: A tutorial introduction to the principles and applications of information theory. 2024.
Naoya Takeishi, Yoshinobu Kawahara, and Takehisa Yairi. Learning koopman invariant subspaces for dynamic mode decomposition. Advances in neural information processing systems, 30, 2017.
Naftali Tishby and Noga Zaslavsky. Deep learning and the information bottleneck principle. In 2015 ieee information theory workshop (itw), pp. 1-5. Ieee, 2015.
Naftali Tishby, Fernando Percira, and William Bialek. The information bottleneck method. arXiv preprint physics/0004057, 2000.
Umesh Vaidya and Prashant G Mehta. Lyapunov measure for almost everywhere stability. IEEE Transactions on Automatic Control, 53(1):307-333, 2008
Darko Veberič. Lambert w function for applications in physics. Computer Physics Communications, (3(12):2622-2628 2012.
13

---

Published as a conference paper at ICLR 2026
Matias Vera, Pablo Piantanida, and Leonardo Rey Vega. The role of the information bottleneck in representation learning. In 2018 IEEE international symposium on information theory (ISIT), pp- 1580-1584. IEEE, 2018.
Haoqing Wang, Xun Guo, Zhi-Hong Deng, and Yan Lu. Rethinking minimal sufficient representa- tion in contrastive learning. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp. 16041-16050, 2022.
Yiyi Wang, Jian'an an Zhang, Hongyi Duan, Haoyang Liu, and Qingyang Li. Rethinking selectivity in state space models: A minimal predictive sufficiency approach. arXiv preprint arXiv:2508.0315. 2025.
Matthias Weissenbacher, Samarth Sinha, Animesh Garg, and Kawahara Yoshinobu. Koopman q- learning: Offline reinforcement learning via symmetries of dynamics. In International conference on machine learning, pp. 23645-23667. PMLR, 2022.
Matthew O Williams, Ioannis G Kevrekidis, and Clarence W Rowley. A data-driven approximation of the koopman operator: Extending dynamic mode decomposition. Journal of Nonlinear Science, 2506):130711346, 2015.
Edward Witten. A mini- introuuttooitnonnn to information theory. La Rivista del Nuovo Cimento, 43(4):
187-227, 2020.
Mike Wu, Chengxu Zhuang, Milan Mosse, Daniel Yamins, and Noah Goodman. On mutual in- formation in contrastive learning for visual representations. arXiv preprint arXiv:2005. 2020.
Xingjian Wu, Xiangfei Qiu, Hongfan Gao, Jilin Hu, Bin Yang, and Chenjuan Guo. k2 vae: koopman-kalman enhanced variational autoencoder for probabilistic time series forecasting. arXiv preprint arXiv:2505.23:: 2025.
Yuanchao Xu, Jing Liu, Zhongwei Shen, and Isao Ishikawa. Reinforced data-driven estimation for spectral properties of koopman semigroup in stochastic dynamical systems. arXiv preprint arXiv:2500..m 042.65559.4465 2025a.
Yuanchao Xu, Kaidi Shao, Isao Ishikawa, Yuka Hashimoto, Nikos Logothetis, and Zhongwei Shen. A data-drives framework for koopman semigroup estimation in stochastic dynamical systems. arXiv preprint 1331012501.13301, 2025b.
Yuanchao Xu, Kaidi Shao, Nikos Logothetis, and Zhongwei Shen. Reskoopnet: Learning koopman representations for complex dynamics with spectral residuals. arXiv preprint arXiv:2501.00701, 2025c.
Yiming Yang, Xiaoyuan Cheng, Daniel Giles, Sibo Cheng, Yi He, Xiao Xue, Boli Chen, and Yukun Hu. Tensor-var: Efficient four-dimensional variational data assimilation. In Forty-second Inter- national Conference on Machine Learning, 2025.
Raymond W Yeung. Information theory and network coding. Springer Science & Business Media, 2008.
Luo Yining, Chen Yingfa, and Zhang Zhen. Cfdbench: A large-scale benchmark for machine learn- ing methods in fluid dynamics. arXiv preprint arXiv:2310.05963 2023.
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THE USE OF LARGE LANGUAGE MODELS (LLMS)
During the preparation of this manuscript, the authors used ChatGPT to polish the writing (e.g.. improving grammar, readability, and clarity), The content, technical contributions, and conclusions of the paper were developed entirely by the authors, who take full responsibility for all ideas and results presented.
A NOTATION
Table 3: Notations in the Main Text
Notations
Meaning
d
latent dimension
det
determinant of matrix
n
state dimension probability distribution
p KR
probability distribution of the Koopman-induced trajectory
KR
variational approximation of the Koopman-induced trajectory
t
time step
tr
trace of matrix
state of dynamical systems
2
latent variable of dynamical systems
C H I I K. DEL
latent covariance matrix
Hilbert space
mutual information
identity matrix
Koopman operator
KL divergence
p))
Lebesgue space equipped with inner product
Ma M N S
step linear forward covariance
dintee-amensional manifold
Gaussian distribution
von Neumann entropy
T Z
discrete-time nonlinear map of dynamics
latent space spanned by ( Φa}
a, 3,7 7
Lagrangian multipliers
X
eigenvalues density matrix observable/feature
P
1, 0.w
parameters for neural networks in Koopman representation
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B A GENTLE INTRODUCTION TO APPENDIX
The appendix is organized to complement the main paper with precise definitions, detailed theoret- ical analysis, and additional experimental material. Given the information density of the appendix, we provide here a short roadmap to guide the reader:
Appendix More Literature review. We provide the classic to modern Koopman learn- ing methods.
Appendix D Limitations and Future Directions. We state the limitations of framework and propose some future directions for Koopman representation.
Appendix E. Technical definitions. We collect the technical background, notations, and formal definitions used throughout the paper for case of reference.
Appendix F. Theoretical analysis. This section develops our theory step by step, where each proposition answers a natural "next question" the following chain:
Q1. Will information be lost? (Proposition
Q2. If so, how much is lost? (Proposition 2)
Q3. What kind of information is lost? (Proposition 3)
Q4. How can we optimize information retention? (Proposition 4)
Q5. How can avoid negative side effects such mode collapse? (Proposition 5)
In this way, the proofs form a coherent progression: each proposition is the answer to the next natural question raised by the previous one.
Appendix G. Experimental setup and additional results. We provide implementation details, dataset descriptions, and supplementary results to support and validate the proposi- tions made in the main text.
This structure ensures that readers can navigate the appendix according to their interests: consult Appendix A for notation, Appendix B for the full theoretical journey.
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C MORE LITERATURE REVIEW RELATED TO KOOPMAN REPRESENTATION
The Koopman operator was originally introduced by Koopman and von Neumann as a linear embed- ding of Hamiltonian dynamical systems (Koopman, 1931; Koopman Neumann, 1932). However, its infiniteddimensional nature makes it difficult to identify suitable handcrafted basis functions us- ing conventional methods (Brunton et al., 2021). To address this, kemnel techniques from functional analysis have been employed as bases for learning the Koopman operator (Das & Giannakis, 2020; Das et al., 2021; Kostic al., 2022; Bevanda al., 2025). Owing the well-posed properties of kernel functions in reproducing kernel Hilbert spaces (e.g., linearity, existence, and convergence guarantees), the Koopman operator can directly approximated via (extended) dynamic mode de- composition (DMD or EDMD) (Williams et al., 2015; Takeishi et al., 2017; Arbabi & Mezic, 2017; Xu et al., 2025a). Despite these thcoretical advantages, fixed kernel functions are often too restrictive to capture general function space (Berlinet & Thomas-Agnan 2011; Alpay, 2012).
In contrast, deep learning frameworks provide a more flexible alternative: leveraging the universal approximation property of neural networks (Baker & Patil, 1998; Kidger & Lyons, 2020), they al- low learning general Koopman representation without relying on predefined kernels. Following this principle, (variational) autoencoder (AE/VAE) architectures have been widely adopted to tract features spanning the Koopman subspace (Liu et al., 2023; Wu et al., 2025; Xu et al., 2025b;c). The resulting latent representations are flexible and support downstream tasks such as prediction and control (Li et al., 2020; Mauroy et al., 2020; Korda & Mezic, 2020; Weissenbacher et al., 2022). However, these representations are typically learned in a purely supe-supe manner, lacking ex- plicit grounding in dynamical systems theory. To improve their reliability, recent studies incorporate domain-specific priors-such as symmetry, conservation laws, dissipation, or ergodicity-into the Koopman representation (Vaidya & Mehta, 2008; Weissenbacher et al., 2022; Azencot et al., 2020; Cheng et al., 2025). Within the VAE setting, recent studies (Federici et al., 2023) have started to link Markovian dynamics and information theory, demonstrating that time-lagged tricks exploit mutual information to obtain better latent representations. While existing approaches are effective for specific dynamical systems, a formal theoretical foundation for guiding the learning of Koopman representations remains insufficient. In this work, we investigate how general coormation-theorete principles can be employed to fill this gap.
D LIMITATIONS AND FUTURE DIRECTIONS
A current limitation of our framework is that it does not address the sample complexity or non- asymptotic convergence of the Koopman representation; future work could explore more rigor- ous theoretical analyses in this direction. In addition, recent studies have highlighted connections between kernel methods (Kostic et al., 2022: 2023a; 2024) and information theory (Bach, 2022), suggesting an interesting avenue for extending conventional kernel techniques Koopman theory through an information-theoree perspective.
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E KEY TECHNICAL DEFINITIONS AND RELATED PROPERTIES
Definition E.1 (Density Matrix (Bach, 2022)) density matrix E [Rdxd a real symmetric ma- trix satisfying:
p positive semi-defintte: p
*The trace of p is I: tr(p) =1
Such matrix can be interpreted as probubillty-weighteed combination of orthonormal directions R. admits a spectral decomposition:
d
d
Σ P.u. where p. 2 0, Σ P; 1, and 02 E Rd with ||ul| 1.
2=1
Definition E.2 (Effective Dimension (Roy Vetterli, 2007)) Given a density matrix on Hilbert space, the effective dimension is defined as
(p)(p) == cxp(S(p)) (S(p)),
where S(p) -Trpp p) is the von Neumann entropy of
The effective dimension measures how many directions in a representation space are substantially used. Given symmetric, positive semi-definite matrix p with unit trace, its Neumann entropy
S(p) -tr(p log,))
quantifies the spectral diversity of p. The effective dimension is then defined as
dest (p) exp(S(p)).
so that (p) can be interpreted the number of dimensions effectively occupied by the latent variable. In particular, den(p) = when p is concentrated on a single direction (pure state in quantum mechanics), while deat (p) d when p is maximally mixed and spreads uniformly all directions. higher effective feature dimension often required to ensure predictive sufficiency.
(2)
H H(x)
I(z;z)
H(*|²)
Figure 7: A Venn diagram illustrating entropy, conditional entropy, and mutual information between the true state a and latent variable z. Hxx) and H²2) denote the Shannon entropy (total information) of x and 2, respectively. Their symmetric overlap. [(2; æ), represents the mutual information that quantifies how much information about the true dynamics is preserved in the Koopman representa- tion. The non-overlapping regions, (xx22) and (=]æ),, correspond to the residual uncertainty not captured by T(2; ())
Definition E.3 (Entropy, Mutual Information and Conditional Mutual Information) Let I, y.y. be random variables. Beyond the Definition 2.2 in the main text, we give a standard definition (conditional) mutual information based Shannon entropy (shown in Figure 7, (Csiszár et al., 2004)).
DI Entropy of random variable is defined as
(1x) - / p(x) p(r)dd.
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D2 Mutual Information is defined as
xx2;)) = H() - (2,
equivalently,
[(=;)) - H()) ((=)-HHi -H()))(())
It can also be expressed in terms of the KullbackLeiibler (KL) divergence:
1(x;)) = DKu(p(æ,, 8) p(x))p()))
where
DRL(D())l((t))= = / p(xx) log p(x) dr.
q(x)
D3. Conditional Mutual Information is defined
T | 2) H(x 2) H(y =) - H 12),
equivalently,
[( 2) Hx | =) H(x v)
In terms of KL divergence.
[[ | 2) E DKL(p(>; y =z) || p( =))(() =))
-E, |KL[p(<> 12,2) || =))]
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F THEORETICAL FRAMEWORK
To ground our theoretical analysis, we first formalize the autoregressive structure of Koopman rep- resentations illustrated in Figure 8. The original dynamics evolve nonlinear transformation
Zt-n > Ft-n+1
Zz-
In parallel, states are encoded into latent variables žt-n- which propagate linearly under the Koop- man operator K and are subsequently decoded back to approximate the original states. This two- layer structure makes clear where information may dissipate: (i) during encoding from r to 2, (ii) along the linear latent evolution govemed by K., and (iii) during reconstruction from x. An- alyzing this flow of information is therefore essential for understanding the fundamental limits of Koopman representations, and the proofs of the following propositions will be developed around this structure.
Et
Itt-n+1
encoding
decoding
decoding
2t-n
21 2t-n+1
21
Figure 8: The autoregressive structure of Koopman representation. Top row (solid arrows): the original states Σt-n evolve under the nonlinear map T. Bottom row (dashed arrows): the states are first encoded into latent variables 2,-.. which then evolve linearly under the Koopman operator K. The latent variables subsequently decoded back to approximate the original states. Thus, the latent evolution under Koopman representation captures essential structure but do not directly contain the full state information. Note: the dashed arrows represent the approximated Koopman representation in latent space, whereas the solid arrows denote the true underlying dynamics.
Fact F.1 As shown in Figure 8, we set that the latent variable 2t obtained via a probabilistic encoder that depends only on the current true state It, i.e., p(2; I²). Consequently, the information content of 2 cannot exceed that of It, so (2x) S H(x))) (due the data processing inequality (Stone, 2024)). Moreover, under this setting, 2t is conditionally independent of any other variable the dynamical system given Nt, i.e.,
1(2 O (x)) =0 O, or equivalently
2z Itz
where 0 denotes any variable in the dynamical system.
F.1 PROOF OF PROPOSITION 1
Beyond establishing Proposition F.1, it crucial to highlight the phenomenon of information dissi- pation in the autoregressive Koopman representation (see Figure 8) and to clarify how the Koopman operator connects to the information limit. By a fundamental principle of information theory, compressed representation never increase the available information. While this observation yields one-step inequality 3, our analysis extends it to the multi-step analysis, where the cumulative effect of encoding, autoregressive latent evolution via K., and decoding be rigorously tracked. This extension makes explicit how information gradually dissipates at each stage of the Koopman representation, ultimately leading the error accumulation.
Lemma F2 (Chain Rule of Information) For variables x, y, the mutual information with their joint variable satisfies
I(x; I(x; (v. 2)) I(x; v). where (y. 2) is treated as the joint random variable with distribution ply, 2). According to the non-negativity of conditional mutual information, it is obvious that
I(x; (y, 2)) 2 [(x; v)-
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Proof We prove this proposition steps.
Step I. According to the autoregressive structure of the Koopman representation, the first in- equality 3 follows directly from the data processing inequality (Stone, 2024). Since the latent variable zt-1 is compressed representation of It-1, the mutual information between succes- sive states cannot exceed that induced by the original dynamics T. Formally,
(1x-1;
Z 1(ze-1;zz).
(10)
This can be derived based on the Fact F.1, it can be factorized as 1 (x Øx) = 1( 2(-);; - (z 11(1-11)) (Lemma F.2)
([(=)-1;2-1)) - - ((2 (Fact F.I)
[=(-1; + 1000-1:z0(20-1
(Lemma F.2)
2 [((-1;;11
(non-negativity of mutual information)
(11)
Thus, the first inequality is derived.
Step 2. In terms of seconding inequality, derive it follow
1(2-11:11) = I( (x1,21) - (2)1-122(22g) (Lemma F.2)
[(=-1; 2() [(2 (Fact F.I)
(()));;=)) + [ (21111;2e/2z) 2 (()-1:2))
(Lemma F:2)
(non-negativity of mutual information)
(12)
Here, the proof for the proposition ends.
Beyond the proposition, we would like disentangle the what information is lost during the Koop- man representation. Combining equations 11 and 12, we obtain
[(I(-1;2() I[ + 7(
(13)
=1(2-1i4) + (2 x1 + 1i
then,
I(x1-1;21) 122-112)) = 12 21) [1 22 21-1)
(14)
step-wise information gap in Koopman representation
Furthermore, extend from one-step mutual information multi-step ones (with 1), such that
[(xz-nj ) = 1(IIt-n 2tn);; )); -
(Lemma F.2)
[((It-n) 2 xx()
(Fact F.1)
[(zt-n X.) + (It-nj te
(Lemma F.2)
[(ztnj 21) (Zz-ni Te (2) (e
(repeating previous procedure)
(15)
Also, I(xt-ni 2y 2)-)) can be disentangled as follows:
Step 1 - Introduce St-n+1 Apply Lemma F22
I(It-niZe 2)-n)= = I(21-ni 4 (2-n) + I Ixx 22-ns 2x-n+1).
Here, the graphical structure ensures that:
I( 2tte-n Lt, 2t-n) = 0,
so there no correction term.
Step 2 - Introduce 2t-n42. Expand the second term via Lemma F.2:
I(It-niZz Zt-n 2t-n11) = I(xt-ni 2t-n42 Ze-n; 2t-)+1)
[ Dt Zt-ns 2t-u41 2 )
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Again, the graphical structure implies:
1 [ 4-12 Ft: 2t-ni 24- 2t-++1) = 0.
Step 3 Repeat recursively. For each = n2 + 1, - n + 2, ..., t:
[ft-ni F4 2t-nci-1) (ate-n 2i (t-nxi-1) + (rt-ni Tt 2t-ma);
where
nii = ........
Equivalently, in compact notation:
(c 12--
[(re-ni (ite-niz//z-aa 2-22)- (Zt-ni
(16)
Thus, by combining equations 15 and 16, the multi-step lost information becomes
(xx=nnix)) (2t-ni
(17)
Σ / ( Fe( + (z
Here, we interpret the physical meaning the three parts Koopman representation. Each term [2: 2/ =)(=-)1 measures how much new information about past state zt-n revealed by the la- tent variable the latent variable 24, given all previous latent states. Physically, this corresponds to the fast-dissipating modes that decay quickly; more latent steps added, this residual information diminishes rapidly to zero.
The second term [2 measures the residual dependency between the past and the current state that cannot be fully represented the latent sequence {zt-nit} due to the compressed representation.
The quantity 7(=:-n; W2 (x) measures the information about the state Iy that remains in the past la- tent variable zı- but not preserved in the current latent 2t. positive value therefore indicates information loss during latent evolution. This phenomenon arises from Koopman modes with eigen- values |\| < whose contributions decay over time and thus dissipate predictive information the Koopman representation.
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F.2 DERIVATION OF VARIATIONAL DISTRIBUTION
The derivation of discrepancy between true and Koopman-induced trajectories is listed as follow:
DEL (p(#1: (o) |l KR (21:(<0))
p( p K
(x vo)
=E[log [lg
pK 200 9 (x [T)
p(x1
R (1)) (2111
-I3 log p (
log
p(11:1/20))
kR
<E| log
+ log
k (2
RR
20 I1st 100
p(x1:0/<00)
xK p(z0 220)
I p( |nn ()p(kn kmn)
-IE log
+ |8| log
(
(ro)
n= 1 K (2n, nn Izn )r (5 2n
(18)
p(x11:1(20)
p(2a)aro)
=E log
+ log
p (211:1
g (< kro)
pl 2m 22-1)n
+ E log p (2n)
E log
q (zn k2-1))
qk H (I Izn
DKL (p(x1:1/0)) K ( (zo)
+ DEL (p(zo (ro) |I K (zolzz)))) (z))
discrepancy between Koopman-induce induced distributions
representation
DKL (p(zn 2n 1) t (z (mllz-11) + DKL (p(2nlzz) || RK (I (l))
1==1
Koopman operator error.
error, S.r.
Here, gKR denotes the variational approximation. For notational convenience, denote the last three terms in equation 18 by & Ema and Eree, respectively. Also, the logic from third line to fourth line holds since the inequality follows from the fact that marginalization cannot increase KL divergence.
F.3 PROOF OF PROPOSITION 2
Before proving Proposition we first introduce a technical lemma.
Lemma F.3 (Pinsker's Inequality (Yeung, 2008)) For any two probability distributions and the same space X, the total variation distance
1 4||rv == sup p(X) x(X)
x
is equal one half of their L distance:
lp I gllrv= 1llvllrv= = /Io(e) -ID(x)-q(x)) q(x)) dx.
Moreover, it is bounded by the Kullback Leibler divergence:
- - = lm(x) g(x)) do &
/ 1 (ell))
Proof The proof proceeds in four steps: (1) establish the connection between KL divergence and total variation distance, (2) relate KL divergence latent mutual information, (3) derive the upper error bound via information-ther
limits, and (4) show that the lower bound decays exponentially with increasing latent matual information.
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Step I. By applying Pinsker's inequality (Lemma F.3) and equation 18, directly bound the distributional discrepancy as
[1((
(on S V Dia. (p(a)a))0)) (x11())())) Eenc + Eura Erce
(19)
Step 2. The connection between mutual information and KL. divergence is given as follows:
21) (22-1;:24)
p(rre p(11)211111 Ellog p(2)12e-1) p(z))
=B|log
(Definition 2.2)
p(z+)
p(1r|21111) + [El1og p(x+))
p(())
=F[log
((2r122-1)
p(aa (152)24-1))100 ((1)(())-))((z)) =Ellog p(z1)p((2 20)2))))
(20)
B(arr=]]
=E(log p(ar x1-1
p(a
(Bayes' rule)
p(z) 21-1)
p(2](z1)
=F[log (1-1) 1)
p(ze 11-1)
p( 24)
By recursively using the result in equation 20, we can summation the results
(T - 12 zn))
Bloog p(xn|22n-1) (n- plzn Rnn)
(21)
p( 2n-1) pl (zn)
p(z0 wo)) I p(I (n
II p(zal
=Ellog
p(za rr)) Im==1 pl (2)))) Ih p(
Here, p(Ix(=n-1) and p( zm 1n- are governed the original nonlinear dynamics T and Koopman operator (Zz M. respectively. Based on this fact, we can further to develop equation 21
p( mp)
I
n= p(rn In-1)
I
Bllog
p (ro)
IT pl 1n-1)
I pl En)
p(20,,
-Ellog p (Kltt,,
(22)
p(œ1a)
ZEllog pk p
=DKL (p(Z)alm0)) KR KR (o))
Plugging equation 22 into equation in Step have
p(a):1200) KR (
DRL. (p(2):)) ((m + Eonc + Em + Emc)
2
(23)
Σ (I(n-11;4m) [22 kn)) + Eenc + Eim + Exc.
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Step 3. Based on the distributional discrepancy in equation 23, we have the following inequality
| [T æo] - Eplx1= t æool|2
- /1 aia dq (x11e) x0) - f 21 dp(x1: x
(Lebesgue measure)
<||x1e11b (x1a (o) p( mo) dæ
(triangle inequality)
distance
520 [|o(010 xo) KR ( 2o)llv
(Lemma F.3)
<0 (P((1 xo) ( æ0))
(Pinsker's inequality)
2 (((n-112m)) 1(2n-1; 2n)) + Eone + Era + Eyers (via equation 19). <τ
where the state x lies a compact space M with a complete metric, ensuring 1x S SO co. The proof ends.
Step 4. The classical rate-distortion theorem (Cover, 1999) states that Tiat RRnxt under L error distortion via the ideal Koopman model p KR, the minimal achievable distortion D is bounded by
2 exp( (z1=)) exp( nt Since the entropy ((ra) can be wiaally measured by the mutual information Ex-1 1(zn-1:00 In) of original dynamics T. Then the accumulative mean-squared error after steps given zu is bounded below as
nt
2 E((nn-zzza)
D 2
(24)
2Te nt
Ep [|| (anaaaaa 2 Cexpl
Σ n=1
I(2n-1iZn) Eene + Ena + Erec)), (25)
where the constant C = trajectory.
nt Z
exp(
2 Et
absorbs the marginal entropy of the ((In-15xm))
Remark F.4 A special case of Proposition is ergodic system, conditioning on xo and then taking the snngttmmed average
lim t-00 t
DKL(p(ELa) 2o) q KR RR(((11 l mo)) S lim t-000
Σ (I(Zn-1, - (22--1,2)) 2n)), ==1
which follows from Proposition 2. The left-hand side is the relative entropy rate, and the inequality shows that the dynamic discrepancy between p and gt K.R. can be controlled by the per-step informa- tion difference.
F.4 PROOF OF PROPOSITION 3
Beyond establishing Proposition 3, it is even more important to clarify the connection between spectral theory and the information components of the Koopman representation. Before proceeding to the detailed proof, we first derive the closed-form expression of latent mutual information under the Koopman representation. This will allow to interpret the spectral properties of the Koopman operator from an information-thhorrtic perspective.
For one-step forward under Koopman representation, have
21 K2t-1 + 6--1. 6--1 ~ ~N0, E).
For multi- step forward, it can be recursively derived as
2t K(K22-2 6-2) 4-11,
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cirle
Unidiee
10
Expesvalion
1.0
Exgervaluss (aw7)
Epensshans C'n
0.5
0.5
1m03
0.0
(mtA)
Ro(A)
Figure 9: Spectral behavior of the Koopman operator under different regimes. Left: Figenvalues of K. (orange dots) lie the complex unit circle (D>l 1), and those of K with 7 (blue crosses) remain on the unit circle, indicating temporal coherence and preservation of information. Eigenvalues K. lie strictly inside the complex unit circle (l>l 1), and the spectrum of KC" contracts toward the origin as n increases, reflecting fast mixing and information dissipation.
Right:
then,
21 = K 21
Σ Ka--
Gaussian distribution for all i. Therefore, 2, follows the distribution NNCC"* 2t-n EnD (C' For convenience, denote the covariance matrix as
Here, cx- (0, 2) is wetatiiiineeeedeeeeee Gaussia
M, = C'
with 2 N(K" stn; Mn)
(26)
2=0
Without loss of generality, we can set žt-n ~ A0, C) and the covariance matrix is denoted as C = Cov(z Given the latent variable 2t n. the conditional entropy (zzezzen)) is calculated
as
H(z122ea)) log (2 det Mn).
(27)
On the other hand, the entropy H (z+) is calculated as
H (z+)= = 2
log log (2TTe* det ( + Mn)].
(28)
Proof The proof of this proposition proceeds in three steps: (1) we first interpret latent mu- tual information in relation to the spectral properties of the Koopman representation; (2) we disentangle the mutual information (22; clarify the role of each component and asso- ciated spectral behavior; and (3) we derive the closed-form expression of muiual information under the Koopman representation, which highlights how the Koopman operator governs the information flow.
Step 1. Spectral Properties in Latent Mutual Information. Based on the Definition E.3, mutual information I (zt-nz 2x) is calculated via equations 26, 27 and 28
122 21) = H (2,) - H(2ekee-a)
1
log( (2Te)) det (K"C((K) + M.)) - log((2 det Mn]
det
2
1 (2mett del (KC"C(KO)) + Mn)
(29)
log
2
(2 det Mn
2 log det(I + MM
M,
We now examine how the behavior of the Koopman representation depends on its spectral prop- erties, by analyzing the cases where the eigenvalues of K. satisfy [k] > |X] z 1, and [X] <
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[X] 1: The Koopman representation is explosive: || grows exponentially, the term K²"((I")) dominates. As a result, the mutual information (z)-ni diverges with n., reflecting amplification of initial uncertainty.
XX] 1: The Koopman representation is temporally coherent: K="C(K") remains bounded, and when noise is small the mutual information is approximately conserved at a con- stant level. This corresponds to the temporal-coherent information component.
XX < 1: The Koopman represent is fast mixing: K - 0 as + OC, so the additional term vanishes relative to Mn. Thus the mutual information I[=--ni 24) - 0, indicating that information from the remote past information is asymptotically lost due to contraction and noise accumulation.
An illustration given in Figure where the left panel shows the [^] case with eigenvalues lying on the unit circle, and the right panel shows the [>] < case with eigenvalues contracting toward the origin as n increases.
Step 2. Disentanglement of mutual information (Z)5
According the chain rule of mutual information, we have
((z) Hx 2) + I(2;ar)
(30)
where H(x 2) measures the irreducible uncertainty of x given the latent variable (i.e., the information lost in the latent space), and 122;x) quantifies the total amount of information about x preserved in the latent representation.
this sense, I (2;x) can be regarded as the maximum information that the decoder can retain about the data through the latent variables. To better understand the role of structural con- sistency, we next decompose [(2; x) into components, in order to examine how much of this retained information is altributable to the latent forward via Koopman representation.
T(21; 2x)
=I(2t, (Zt-n; Ut)) [( (Fact F.l)
=1(z-ni 2x) + 1 [(t) We (2t)))))
=I(z-ni 21) + I(2;; (z4-1, 21 =t-n) ) - - (Fact F.I)
(31)
=T( 21) + [(2; (12;:1111220 (2))))))))))))) + T (2+ 2e 2 It-1)
=1(z--i *)) + ([,;;1-1| + (zz; (2) 2t (Tt-1) w
=I(2-ni 2)) + [(2::(1-]<<-))) (2 2e (1-11 - I
-I( 21) + [(2 (22 + [ (22:: 11)
Step 3. According to Step I, the latent mutual information has been thoroughly explained; we now turn to the second and third terms in equation 31.
For linear conditional Gaussian (Lubbe, 1997), the mutual information for
1 det En det Ebje
I(a, 2 log det Da,bie
(32)
where Fagle and E.G denote the conditional covariance matrices of a and b given c, respectively, and Da denotes the joint conditional covariance matrix (a, b) given c, i.e..
Dale
Eable
Ea.Me
Ebale
Ebje
with Eable and Enole being the conditional cross-covariances.
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The closed-form expression for the mutual information [22 2t-n) is derived as follows. Conditioned on 2t- we have the Snner--Gaussiae relations
2t-1 Rt-n N(K n- z-n; Mn-1);
(33)
21 Z4- ~ N(K 2t-n:
2nn: Ma),
where Ms. = " Assume It-l Dār-1 + - with 6x-1 ~ N(0, It), inde- pendent of the process noise. Then
Mn
KM DD
E Daco 1 Ilc
(34)
DM,-1D' DM
RR]
Plugging equation 33 equation 34 into the Gaussian closed-form identity yields
de((M,)
R)
12 [2-nn)= 2 log
(35)
Mn K.,,-1D' DMn-1D' R
det
(DMn-IKT
=t- scales with n,, suffices to study the growth of M, Eizo KC (CT If the spectral radius p(K.) < 1, then M,, converges to the unique solu- tion 2 of the discrete Lyapunov equation Mx = E K. hence I(2;; zr_n) remains bounded (" "ccoppresiblle", Conversely, if p(C) 1, then Mn. diverges and the infor- mation grows unbounded along the unstable directions. When p(K.) zs 1, the growth is slow and reflects long-term temporal coherence (then contradicts with equation 29, and can be captured by the latent mutual information). Therefore, this compressible information corresponds to the spectral radius p(K.) <1
To analyze how [(2;; Xt-
As for the residual term (21;31 2t-1), it can be expanded as
12gg0 (t-1) H (x--)) 1) - H(x 24, 2t-1)
(36)
Hz,, xx-1) - log ((tze())))
where the second equality follows from the observation model 2; D2, - with e N(O, R), which implies H | 24, 2;-1) = 2 log ( (ooe)))R. Therefore, this residual mutual infor- mation depends only the noise covariance R and original dynamics T, but it has no spectral counterpart in the Koopman operator.
Summary. The mutual information T(2;; x.) naturally decomposes into three parts: (i) temporal-coherent information I(2t-n 21), which captures temporal coherence when eigen- values of Koopman operator X 1; (ii) fast-dissipating information [22 zt-n), which remains bounded only in the stable regime p(K.) < 1; and (iii) residual information [22 xe(xt))))))) which reflects observation noise and has no spectral counterpart. These three components and their spectral interpretations are summarized in Table 4.
Table 4: Spectral interpretation of information components in Koopman representation
Information
Spectral property
Temporal behavior
Information mean-
ing Predictable, low en- tropy Transient, com- pressible under IB Noise, anomalies, non-predictive left- step overs
Temporal-coheren
1=1
Long-lived, persis- Rapidly decaying. short-lived
Fast-dissipating
A x<
Residual Confounding
(no spectral coun- terpart)
Unpredictable, injected at present
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F.5 PROOF OF PROPOSITION 4
To establish this proposition, we analyze the problem from Lagrangian perspective. Specifi- cally, we investigate how the Koopman representation behaves when the latent mutual information T(=e-nj²)) =q) is maximized under a finite variance constraint. This perspective reveals a water-filling allocation principle that governs how variance is distributed across spectral modes, thereby clarify- ing the connection between latent mutual information and the latent variable 2.
Proof 4 Consider latent variable under Koopman representation showing equation 26
tt KC Zt-n +, s e~ NO, Mn), C == E.
(37)
where denotes the covariance matrix of 2t-n- The matrix C characterizes the spectral dis- tribution of the latent variable, and our goal is to investigate how maximizing latent mutual information influences Koopman representation.
According to the previous proof in equation 29, we have
log det +1+ Mn (I"")C(K") M, b).
(zt-ni 2e)
Denote the singular value decomposition Mn (K.) = V with gi 2 0. Then
I(zeni 21) = 2 log det (I+ Uddiag( CV
(38)
As tr(C) measures the total second moment, we impose tr(C) S C' for some finite constant C. This assumption ensures that the Koopman representation has bounded total variance, preventing degenerate solutions where the variance grows without bound.
Under the constraint tr(C) S C. maximizing the latent mutual information under Koopman representation becomes an optimization problem as
log det (I Udiag( (ociaa((//iVV VV (..)UT)
max C 2
(39)
s.t. tr(C)= S C.
equation 38, the matrices U and V are orthogonal, and C a symmetric positive semidefinite matrix with eigenvalues (pi....,Pa} with 0 for all i. We interpret these eigenvalues as spectral weights of the Koopman observablesfeatures, indicating how variance allocated across the observableffeature directions (Øi,... Φa defined in equation 1. Then, optimization problem in equation 39 becomes a water-filling problem as
max
log(1 + g.P.).
(40)
Psi DEn SC 2 1=1
The Lagrangian formullaiooo becomes
d
1 Σ log(1+ + g.p.)) pl Σ p. 0) - Σ C C
T.P:
(41)
2
t=1
1=1
According to
(KKT) condition (Boyd Vandenberghe, 2004), we obtain
94
4 17, 0, T)
pi
(42)
2(1 + g.P.))
2μ
9i
Based the non-negativity of the eigenvalues p, 0 for all i, the optimal allocation is
p; max {0, 2u 91
(43)
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where i is the Lagrange multiplier determined by the variance budget constraint. This solution characterizes the spectral weights the Koopman representation along each observableffeature direction, and reveals two key phenomena:
Concentration on temporally coherent modes. Since gi depends on the Koopman eigenvalues through K"', larger ps equation 43 are assigned eigen-directions with [X] = 1, corresponding to temporal temporal-coherent coheerent modes.
Mode collapse. Because Ei= P; S C, variance is preferentially allocated to direc- tions with larger gain 9s. while less informative directions receive zero weight. This leads to low-rank allocation (low effective dimension) where only subset of modes are retained.
In summary, this proof demonstrates that maximizing latent mutual information is equivalent to water-filling allocation of spectral weights, which naturally explains why emphasizing this objective can lead mode collapse in the Koopman representation.
F.6 PROOF OF PROPOSITION 5
Connecting to Proposition 4, we continue to prove Proposition 5 via Lagrangian formulation. With- out entropy regularization, the solution degenerates to low-rank (mode collapse); with entropy regularization, the solution assigns non-zero weights to directions, improving effective dimen- sion.
Proof According to Definition 2.3, we can normalize C into density matrix C then tr(C),
S S( =
Σ i=1
Pt tr(C)
log
Pi tr(C)
with pı is the eigenvalue C.
Under a given regularization coefficient this normalization allows us to improve the effecitve dimension. Based on equation 41, the modified Lagrangian formulation under the regularized Von Neumann entropy becomes
C log(1 g.P.) + +
Ps tr(C)
log
tr(C)
Pt
1
P
C)
Σ [=1
TRP.
(44)
1=1
Based the KKT condition, we have
OL
9:
Pi + tc(C)
11
(log
(45)
Öp:
2(1 + g.P.)
tr(C)
where n under the KKT condition. The solution of equation 45 is
Ps + 1).
(log
2(1 g:pa)
ir(C)
tr(C)
Since a closed-form solution is not directly available, we proceed with further algebraic trans- formation.
By reorganization,
Pi tr(C)
tr(C) Ft
log
-11
2(1 + g.P:)
Exponential both sides:
Pi
tr(C)
gi
11 -1)
exp
(C(C)
?
2(1 + g.P:)
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Then,
Pr = (r(C) exp
(r(()
Fi
-1 -1)
2(1 + gips)
We can transform the above form as,
tr(C)
9:
ir(C) -1 tr(c)
Pt exp
11
(46)
2(1 + 9:pi)
>0
C100
-1- tr(C)
Here C1 = :r(CC)exp
a positive constant Introducing y 9ePr. we can
write equation 46 via algebraic transform:
(y-1) exp ( (- tr(()gr = 9:C1.
(47)
27y
The above equation is related to the form a exp(x) rz = constant, we can solve it via the generalized Lambert W function (also known as r-Lambert W see (Veberië, 2012))
tr(C)
y
2
(48)
W
(((
g:Cr
Here, W1 W1/lg,,;, is the r-Lambert W function. Since
2-1
the clomed of pt becomes
41
tr(C)
Pi
27
1
> 0.
(49)
9i
W1/(s.C.)
B.C.
Then, the solution of equation 44 under the regularized von Neumann entropy assigns non-zero spectral weight to all observableffeature directions ( &d), since p: holds according to equation 46, Consequently, the effective dimension is provably improved.
"W, denotes the generalized Lambert W function, defined solution of rexp(x) + ræ constant.
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G PRACTICAL DETAILS, IMPLEMENTATION AND EXPERIMENTAL DETAILS
G.1 IMPLEMENTATION DETAILS FOR VAE
Algorithm Information InformationnTheorcii Theoretic Koopman Representation (VAE, probabilistic)
Require:
network parameters (a, B, 7): learning rate ny number of epochs K; batch size B; neighbor window k; temperature т.
Dataset 2> (z,,
1: Initialize encoder po (=|ar,, decoder P= and latent dynamics network go (zn|22zn1))
2: for epoch to K do
3: for each minibatch (.1,..., Fn) from D do
4: Sample latents 2; po(=(2;)-
5:
Temporal coherence (InfoNCE): For each Zno treat its temporal neighbors P, {2n=l
1 I i < k} as positive samples, compute
1
exp(2 2p/T) log
I(2ni Pn)
Σ PEP
Pn
exp(2 2//T)
j==1
6:
Structural consistency: compute latent likelihood
E Epo(z,, [log qu (Zn (2n22n11)
7: Predictive sufficiency: compute covariance C B E. - 2))2z T)T, where
Σ, 24 normalize P C/tr(C), then compute
= - X log Aj where xy denotes j th eigenvalue of P.
8:
Standard Evidence Lower Bound (ELBO) ferm (for training stability and reconstruc- tion):
CELRO log P. (w 1/2n-1) DKL. (Po2(2-11111-11 [I Z)).
9:
Total Loss:
L= ---na[2nnPP)) + Bpo(s.. flo go + Hpo xnn)
+ (unl + xS(( + LHLO
10:
Update 0, w, 1h using Adam step
11: end for
12:
end for
Connection to Structural Consistency. By Definition E.3, when given 2n-1 the conditional mu- tual information is
1(zn; Fn Zn-1) = H (zn zn-1) H (Zn Ina Zn-1).
Since the encoder is independent of 2x-1 in our setting (according to Figure 8), this simplifies to (Znn5nn 2n-1) H (zn zn-1) H xx)..
(50)
Consider the encoder distribution
(= xn), which maps observations to latent variables, and the Koopman prior
(zn kn-1) = N(zn K2p2n-1, Exp)
which models latent evolution as linear Gaussian transition governed by the Koopman operator
K.p.
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Then the conditional mutual information can be equivalently written as the following form according to Definition E.3 or equation 50:
1 [Zn Fn in-1)) = po lann)
log
po Po(2n 2
Fn
(51)
(0o | Zn-1) 1)
Expanding the term in equation 51, we
1( (Znj Fn
2n-1) po (2m 2.)) log go Zn-1)] Hr (Zn In..
has two effects:
1. Alignment with Koopman dynamics.
The expectation term ppp(ss [n-] log qu (2n Zn-1)) requires samples drawn from the encoder to lie in gions of high likelihood under the Koopman prior. Since the prior is parameterized as linear Gaussian transition, minimizing the KL forces the encoder outputs to be predictable under a linear structure.
2. Entropy regularization. The entropy term Hpg (zn( encourages that the encoder not to be deterministic.
Together, these effects ensure that the latent variables produced by the encoder not only encode information about the current state but also evolve consistently with the linear Gaussian dynamics imposed by the Koopman operator. Formally,
(2 2n)) F (2n 2n-1)
evolves approximately linearly under K.p,
which enforces structural consistency in the latent space.
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G.2 IMPLEMENTATION DETAILS FOR AE
Algorithm 2 Information-Theerrii Koopman Representation (AE, deterministic)
Require:
Dataset D = { M hyperparameters (, 7); learning rate number of epochs K' batch size B; neighbor window temperature т.
1:
Initialize deterministic encoder 2n fe().), decoder 2, [wx),, and Koopman operator Kg.
2: for epoch = 1 to K do
3: for each minibatch {a1,.... FB) from D do
4: Encode latents fo(x;) for i 1, I..... B.
5:
Temporal coherence (InfoNCE):
exp(2 zp/T)
I(zn: Pn) =
1 [Pal
Σ log pEPn
Σド=1 exp(2, 2//T)
B
6:
Structural consistency (deterministic):
Kaoup || Kyp 11
7: Predictive sufficiency: compute S(P) from normalized covariance P
tr(C)' C
T E(, 2)( -))==)
8:
Reconstruction:
Cice = || (zn)))
9: Total Loss:
L= Lre (zni + BL 7S(P).
10: Update 0, w, va with Adam step 7).
11: end for
12:
end for
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G.3
EXPERIMENT SETTINGS AND ADDITIONAL RESULTS
G.3.1
PHYSICAL SIMULATION
Lorenz. The Lorenz dataset is generated from the classical Lorenz system of ordinary differential equations (ODEs), which model simplified atmospheric convection. The governing equations are:
i o x): y x(p y: 1 my _2,
(52)
where a 10, 28, and 8 = 8/3 are the standard chaotic parameters. The system is integrated using a fixed time step At 0.1 with fourth-order Runge- Kutta method. The resulting trajecto- ries exhibit chaotic behavior and are commonly used as benchmarks for nonlinear dynamical system identification.
Kármán Vortex. The Kármán vortex street dataset is generated from the two-dimensional in- compressible Navier-Stokes equations, which describe the velocity field (u, u) and pressure of a viscous fluid:
Ou
du
Du
Op
8²u u
8uu
+ u
+
01
dr
θy
dr
Re
Ox
0y²
dv
du
dv
Op
1
0²v
020
+
u
+
(53)
Ōt
Ox
dy
dy
Re
Oz
0y²
du
Ov
O,
By
where Re = UULV is the Reynolds number, defined with characteristic velocity U, length L, and kinematic viscosity v. The training dataset covers flows with Re E [40, 1000), while the test dataset focuses on Re = 1000. The flow is simulated around cylinder, producing the characteristic al- ternating vortex shedding pattern. The domain is discretized on a x 64 grid, with time step At = 0.001 and both and velocity components are recorded at each grid point (data from (Yining et al., 2023)).
Dam Flow. The dam flow dataset is also generated from the two-dimensional incompressible Navier-Stokes equations, using the same formulation as in the Kármán vortex case. The train- ing dataset spans Re [40, 1000] and the test dataset uses Re 1000. The flow is initialized in rectangular channel with a fixed dam obstacle, where an imposed inlet velocity drives the fluid past the dam-like structure, generating simple wake pattern downstream. The domain discretized on a 64 x 64 spatial grid, with temporal resolution At = 0.1 (data from (Yining et al., 2023)).
The ERA5 dataset a global atmospheric reanalysis produced by the European Centre for Medium- Range Weather Forecasts (ECMWF), providing physically consistent estimate of the large-scale circulation from 1940 to the present (Hersbach et al., 2020). The physic state consists five chan- nels: 500,hPa geopotential, 850,hPa temperature, 700,hPa specific humidity, and 850,hPa wind com- ponents the zonal and meridional directions. We train all baselines from 1979-01-01 to 2016-01- 01 and test after 2018-01-01 (data from (Rasp al., 2024)). Illustrations are provided in Figure 10.
G.3.2 VISUAL INPUTS
Planar System In this task the main goal is to navigate an agent in a surrounded area on plane (Breivik & Fossen, 2008), whose goal is to navigate from corner to the opposite one, while avoiding the six obstacles in this area. The system is observed through a set of 40 x 40 pixel images taken from the top view, which specifies the agent's location in the area. Actions are two- dimensional and specify the x y direction the agent's 's movement, and given these actions the next positional state of the agent is generated by deterministic underlying (unobservable) state evolution function. Start State: one of three corners (excluding bottom-right). Goal State: bottom- right cornrr Agent's Objective: agent is within Fuclidean distance of from the goal state.
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Lorenz63
Kármán Vortex (e).
Dam Flow
(d). Geopotential
Temperature
Humidity
Wind
Figure 10: Examples of physical simulation: (ar.nz 63, (b). Kármáá Vortex, (c).Dam Flow, (d).ERA5
Inverted Pendulum Sp & Balance This is the classic problem of controlling an inverted pendulum (Furuta et al., 1991) from 48 x 48 pixel images. The goal of this task is to swing up an under-actuated pendulum from the downward resting position (pendulum hanging down) the top position and to balance it. The underlying state of the system has two dimensions: angle and angular velocity, which is unobservable. The control (action) I-dimensional, which the torque applied to the joint of the pendulum. To keep the Markovian property in the observation (image) space, similar to the setting in E2C, each observation Ia contains two images generated from consecutive time-frames (from current time and previous time). This is because each image only shows the position of the pendulum and does not contain any information about the velocity. Start State: Pole is resting down (SwingUp), randomly sampled in + /6 (Balance). Agent's Objective: pole's angle is within = from an upright position.
CartPole This is the visual version of the classic task of controlling a cart-pole system (Geva & Sitte, 1993). The goal in this task is to balance pole on moving cart, while the cart avoids hitting the left and right boundaries. The control (action) is I-dimensional, which is the force applied to the cart. The underlying state of the system s; 4-dimensional, which indicates the angle and angular velocity of the pole, as well as the position and velocity of the cart. Similar to the inverted pendulum, in order to maintain the Markovian property the observation 2; is stack of two 80 80 pixel images generated from consecutive time-frames Start State: Pole is randomly sampled in t /6. Agent's Objective: pole's angle is within + 10 from an upright position.
3-link Manipulator - SwingUp & Balance The goal in this task to move a 3-link manipulator from the initial position (which is the downward resting position) final position (which is the top position) and balance it. In the 1-link case, this experiment is reduced to inverted pendulum. that of acrobot, except that have torques applied to
In the 2-link case the setup is similar all intermediate joints, and in the 3-link case the setup is similar to that of the 3-link planar robot arm domain that used in the E2C paper, except that the robotic arms modeled by simple rectangular rods (instead of real images of robot arms), and our task success criterion requires both the swing-up (manipulate final position) and balance. The underlying (unobservable) state 8, of the system is 6-dimensional, which indicates the relative angular velocities and relative angles of the 3 links. Start State: Pole is resting down. Agent's Objective: pole's angle is within + from an upright position.
The control algorithm is linear quadratic control in the latent space and the corresponding control horizon follows the setting in (Levine et al., 2020).
G.3.3 GRAPH-STRUCTURES DYNAMICS FOR SIMULATION
In the numerical experiments, we adopt the graph environments introduced (Li et al., 2020), where interactions among objects modeled differently according to their connection types and physical
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Figure 11: The examples of visual inputs: Planar (left), Pendulum (middle), Cartpole (right).
properties. These environments are designed capture diverse interaction dynamics through Koopman representation (see illustrative examples in Figure 12), as detailed below:
In the Rope environment, the top mass is fixed height and is treated differently from the other masses, resulting in two distinct types of self-interactions one for the top mass and one for non-top masses. Additionally, there are eight types of interactions between different objects. Each mass is represented by four dimensions, encoding its state and velocity. Objects in relation can be either the top mass or a non-top mass, yielding four possible combinations. Interactions may occur between adjacent masses or between masses that two hops apart. In total, this gives 4 2 types of interactions between different objects. Training is performed on environments with 5-9 objects, while testing uses 10-14 objects. The overall dimensionality ranges from 40 56.
In the Soft environments, quadrilaterals are categorized into four types: rigid, soft, actuated, and fixed, each with its own form of self-interaction fined between two quadrilaterals only if they are connected at point or along an edge. Connections from different directions treated as distinct relations, with eight possible directions: up, down, left, right, up-left, down-left, up-right, and down-right. Relation types also encode the category of the receiving object, resulting total of (8 ++ 1) x 36 possible relation types between objects. Training is conducted on environments with 5-9 quadrilaterals, while testing uses 10-14 quadrilat- erals. Each quadrilateral is represented by a 16-dimensional vector, giving a total dimensionality ranging from 160 to 224.
For interactions between objects, edge is de-
noisy environment, the additive noise is zero-mean Gaussian with standard deviation equal to 10% of the standard deviation of the observation data.
Rope
Rope
Soft
Soft 2
Graph of Soft 2
Figure 12: Examples of ropes and soft robots. Left: Blue nodes denote the initial states of Rope 1 and Rope 2, while orange nodes show their states after 40 time steps. Right: Interconnected quadrilaterals indicate the initial states, and the boxes represent the states of the soft robots after 40 time steps. The second soft robot (Soft can be abstracted as graph structure shown the right.
G.4 IMPLEMENTATION ALGORITIM THREE TASKS
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Table 5: Model structures across experimental environments. Here, K2; + Bog denotes a con- trolled latent transition with linear control input a, (Visual Inputs case). KC(A) denotes an adjacency- conditioned Koopman operator, corresponding to a shared Koopman composition modulated by the adjacency matrix A (i.e., K(AA) == A K. in graph environments; see Li al. (2020, Page 4) for details).
Environment
Structure
Key Features
Physical Simulation
AE (G.2)
Kay; reconstruction; Koopman linear forward; InfoNCE; von Neumann entropy
Zt+1
Visual Inputs (Con-
VAE (G.1)
2++1 = K2, + Ba, + (Linear Gaussian); VAE ELBO; reconstruction; InfoNCE; von Neumann entropy
trol Tasks)
Graph-structured Dynamics
AE (G.2)
2++1 K((A) (udjacency-conddiiionee) reconstruction; InfoNCE; von Neumann entropy
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G.5
MORE EXPERIMENTAL RESULTS
G.5.1
PHYSICAL SIMULATIONS
20
0
20
25
25
40
20
-25
0
25
25
0
25
0
25
50
y
Ground Truth
Ours
Figure 13: Comparison of sampled spatial distributions based 100000-step step data for Lorenz 63. Green denotes the ground-truth distribution from the physical solver, and purple denotes samples generated by our method. Across both marginal and joint projections, the distributions exhibit close agreement, demonstrating that our empirical results capture the underlying dynamics.
Kármán vortex street: Continuous Prediction from t=0.7s
Error
Error 0.85
Comparison of continuous predictions for the Kármán vortex street starting from 0.7s. Ground truth contrasted with predictions from VAE, KAE, KKR, PFNN, and our method. Error maps in the lower panels demonstrate that, compared with other models, our method achieves the closest agreement and effectively prevents collapse in the predicted fields.
Figure 14:
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CFDBench Dam Flow: Continuous Prediction from t=5.0s
Ground Truth
5.5s
=
33
14
Error
t= 7.5s
13
Error
21
Figure 15: Comparison of continuous predictions for the dam flow starting from t 5.0s. Ground truth are contrasted with predictions from VAE, KAE, KKR, and our method. Error maps in the lower panels demonstrate that, compared with other models, our method more effectively prevents collapse in the predicted fields for dam flow.
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Table 6: Per-channel performance comparison on ERA5 weather forecasting, N-NRMSE and N- SSIM denote errors at N prediction steps; values in parentheses indicate the standard deviation across test samples. Even under the highly stochastic and high-dimensional ERA5 weather dy- namics, our approach outperforms all baselines across both short-term and long-term prediction horizons.
Channel
Metric
KAE
KKR
PFNN
Ours
5-NRMSE
0.058 (0.020)
0.061 (0.027)
0.046 (0.012)
0.023 (0.005)
10-NRMSE
0.068 (0.018)
0.074 (0.026)
0.062 (0.018)
0.032 (0.010)
50-NRMSE
0.157 (0.071)
0.082 (0.025)
0.082 (0.016)
0.075 (0.017)
Goopotential
S-SSIM
0.860 (0.054)
0.852 (0.064)
0,882 (0.036)
0.964 (0.011)
10-SSIM
0.836 (0.051)
0.820 (0.065)
0.848 (0.052)
0.943 (0.028)
50-SSIM
0.665 (0.195)
0.790 (0.049)
0.765 (0.055)
0.806 (0.039)
5-NRMSE
0.049 (0.019)
0.052 (0.032)
0.040 (0.009)
0.022 (0.003)
10-NRMSE
0.056 (0.017)
0.064 (0.028)
0.051 (0.015)
0.026 (0.006)
50-NRMSE
0.114 (0.054)
0.074 (0.030)
0.067 (0.018)
0.063 (0.019)
Temperature
5-SSIM
0.866 (0.046)
0.862 (0.058)
0.888 (0.026)
0.956 (0.008)
10-SSIM
0.844 (0.044)
0.829 (0.063)
0.859 (0.042)
0.942 (0.019)
50-SSIM
0.671 (0.216)
0.802 (0.059)
0.803 (0.051)
0.825 (0.042)
5-NRMSE
0.064 (0.029)
0.069 (0.043)
0.055 (0.011)
0.031 (0.003)
10-NRMSE
0.077 (0.027)
0.085 (0.041)
0.070 (0.020)
0.038 (0.006)
50-NRMSE
0.165 (0.086)
0.093 (0.040)
0.088 (0.021)
0.084 (0.028)
Humidity
5-SSIM
0.859 (0.056)
0.856 (0.076)
0,880 (0.026)
0.954 (0.008)
10-SSIM
0.818 (0.064)
0.805 (0.095)
0.835 (0.053)
0.933 (0.021)
50-SSIM
0.663 (0.220)
0.781 (0.087)
0.776 (0.058)
0.799 (0.057)
5-NRMSE
0.055 (0.009)
0.056 (0,011)
0.053 (0.006)
0.032 (0.006)
10-NRMSE
0.059 (0.008)
0.061 (0.009)
0.060 (0.008)
0.041 (0.010)
50-NRMSE
0.096 (0.030)
0.063 (0.006)
0.070 (0.007)
0.062 (0.005)
Wind direction
S-SIIM
0.505 (0.1
0.502 (0.141)
0.537 (0.084)
0.814 (0.051)
10-SSIM
0.433 (0.115)
0.415 (0.134)
0.424 (0.120)
0.721 (0.107)
50-SSIM
0.300 (0.251)
0.361 (0.085)
0.267 (0.100)
0.382 (0.068)
5-NRMSE
0.051 (0.008)
0.051 (0.009)
0.050 (0.006)
0.031 (0.005)
10-NRMSE
0.054 (0.007)
0.054 (0.008)
0.055 (0.007)
0.040 (0.010)
50-NRMSE
0.060 (0.006)
0.057 (0.005)
0.087 (0.025)
0.057 (0.005)
Wind direction
5-SSIM
0.240 (0.159)
0.247 (0. 183)
0.300 (0.091)
0.649 (0.098)
10-SSIM
0.163 (0.133)
0.163 (50)
0.208 (0.101)
0.499 (0.163)
50-SSIM
0.105 (0.077)
0.093 (0.082)
0.165 (0.193)
0.094 (0.074)
Table 7: Training time statistics for different models and tasks. Epoch times are reported as mean + std (in seconds). For Ours, InfoNCE and entropy (von Neumann entropy) rows correspond to the total per-epoch computation. Notably. the overhead introduced by InfoNCE and von Neumann entropy is marginal, accounting for only a small percentage of the total training time.
Task
Metric
VAB
KAE
KKR
Ours
Epoch time (s)
172.91
4.11
195.26 + 3.17
186. 1.86
182.52 = 2.53
201.23 + 1.07
Kármán vortex
InfoNCE time Entropy time (5)
13.78 + 0.93
0.97 + 0.13
Epoch time (8)
16.21
0.29
16.95 0.33
17.48 + 0.29
18.92 0.36
Dam Flow
InfoNCE time
0.76 # 0.04
Entropy time (5)
0.56 0.08
Epoch time (s)
240.20 + 0.52
224.95 0.54
242.33 + 0.71
253.24 + 2.05
ERAS
InfoNCE time
15.09 + 0.80
Eatropy time (s)
7.43 0.64
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Ground Truth
Ours
2019 2018-01-01
08:00
Error
2016-01 -88
08:00
1900
Error
Comparison of continuous predictions the global geopotential starting from 2018 01 - 01 00 00 to 2018 01 08 08 00. Ground truth are contrasted with predictions from KAE, KKR, PFNN, and our method. Error maps the lower panels demonstrate that, compared with other models, showing with more stable and accurate results of our model.
Figure 16:
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Ground Truth
Ours
2018 2019-01-01
08:00
Error
2031
35
2016-01 -01-88
08:00
2358
19700
225
20L0
201
201
23100
201
Error
Comparison of continuous predictions for the global temperature starting from 2018 01 01 00 00 to 2018 01 08 08 00. Ground truth are contrasted with predictions from KAE, KKR, PFNN, and our method. Error maps the lower panels demonstrate that, compared with other models, showing with more stable and accurate results of our model.
Figure 17:
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Ground Truth
Ours
2018 2019-01-01
08:00
Error
2018-01 -08
08:00
1744
1231
12
Error
Figure 18: Comparison of continuous predictions for the global u-direction wind starting from 1018-01-011 -00 00 to 2018-01 -08 08 00. Ground truth are contrasted with predictions from KAE, KKR, PFNN, and our method. Error maps the lower panels demonstrate that, compared with other models, showing with more stable and accurate results of our model.
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Ground Truth
Ours
2018 2019-01-01
08:00
Error
T1AL
2016-01-00 -08
08:00
Error
Figure 19: Comparison of continuous predictions for the global v-direction wind starting from 0018-01-01 -00 00 to 2018 011 -08 08 00. Ground truth are contrasted with predictions from KAE, KKR, PFNN, and our method. Error maps the lower panels demonstrate that, compared with other models, showing with more stable and accurate results of our model.
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Epoch 1
Epoch 10
Epoch 20
Epoch 30
Epoch 40
24
68
0.8
0.
0.5
54
0.4
0.4
0.4
0.2
0.2
1mm Values
(a)
Epoch
Epoch
Epoch
Epoch 80
Epoch
2.
0.3
0.5
24
0.5
3.5
0.5
0.5
52
0.2
9.2
0.2
0.2
Egen
Eggan Valuss
Egen
Eion Values
Egon
With Y
Without
Epoch
Epoch 10
Epoch
Epoch 30
Epoch 40
(b)
Epoch 50
Epoch 60
Epoch
Epoch 80
Epoch 90
Epoch 1
Epoch 10
Epoch 20
Epoch 30
Epoch 40
(c)
Epoch 50
Epoch 60
Epoch 70
Epoch BD
Epoch 90
33
Figure 20: Comparison between y 0.0 and y 0.5 on the physical simulation task with latent dimension 32. (a) Evolution of the eigenvalue spectrum over training epochs. (b) Heatmap of the latent covariance matrix for 7 0.0 across epochs. (c) Heatmap the latent covariance matrix for y 0.5 across epochs. With the addition of the von Neumann entropy regularizer, two clear effects emerge during training. First, the latent covariance matrix no longer collapses to a few dominant modes: the eigenvalue distribution becomes more uniform and remains close to full rank throughout optimization (see (a)), indicating that the model learns a richer set of modes rather than compressing them into low-dimensional subspace. Second, the covariance structure transitions from highly sparse (when 0) to dense and full-rank under entropy regularization (from (b) to (c)), verifying theoretical prediction in Proposition 5.
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Kármán Vortex
Dam
Entrosy
Entrosy
Entrosy
Entrapy Loes
Entrup/ Locs
Total Loss
(a)
Epozh
Epoch
Epoch
Figure 21: Visualization of the von Neumann entropy regularization loss and the total training loss over epochs for the physical simulation tasks. The stable behavior of both the total loss and the von Neumann entropy loss indicates that our training procedure is numerically stable and robust across different systems.
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G.5.2 VISUAL PERCEPTION
Table 8:
Percentage Goal (%) for different algorithms under noisy rollout. A higher value indi- cates that the system reaches closer to the goal within a fixed number of control steps.
Domain
E2C
PCC
KAE
Ours
Planar (n 40 40)
6.2 (1.5)
34.8 (3.6)
5.1 (1.2)
39.6 (2.8)
Pendulum (n = 48 x 40 x 2)
45.5 (3.9)
59.8 (3.2)
26.3 (2.8)
62.7 (2.9)
Cartpole (n 80 80 x 2)
8.1 (1.6)
(3.5)
57.2 (3.8)
61.9 (3.0)
3-link (n = 80 x 80 2)
(1.0)
21.3 (1.9)
2.1 (0.6)
19.5 (2.0)
Table 9; Percentage to Goal (%) for different algorithms under noiseless rollouts.
Domain
E2C
PCC
KAE
Ours
Planar (n 40 40)
37.8 (3.5)
71.4 (0.6)
12.2 (1.4)
73.8 (0.5)
Pendulum (n = 48 x 40 x 2)
88.5 (0.6)
90.1 (0.5)
68.2 (1.9)
91.5 (0.4)
Cartpole (n 80 80 x 2)
39.5 (3.3)
94.1 (1.6)
98.5 (0.3)
97.6 (1.2)
3-link (n = 80 x 80 2)
21.5 (0.9)
(1.6)
11.8 (0.8)
47.9 (1.4)
G.5.3 GRAPH-STRUCTURED DYNAMICS
Time Direction
Rope
Soft
Figure 22: Comparison of ground truth and predictions time for rope and soft-body dynam- ics. Top row (Rope): red dots indicate ground truth positions, while blue dots show our predicted trajectories. Middle and bottom rows (Soft): translucent shapes represent ground truth deforma- tions, and solid colored blocks denote our predictions. The time axis progresses from left to right.
48

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Published as a conference paper at ICLR 2026
G.5.4 MODEL ARCHITECTURE
Table 10: Model architecture for Kármán Vortex and Dam Flow task with input dimension (C, H, W), where C' denotes the number of velocity components and H x W is the spatial res-
olution.
Components
Layer
Layer number
C, W)
Activation
Convolution Block
C 8C, (2 )
ReLU
Convolution Block
3
8C + 64C, TG. 16
W
ReLU
Encoder
Convolution2d
64C 128C,
T6 C.W
ReLU
Flatten
128C;
18 16
H W.
Fully Connected
CHW + d,
Koopman Operator
Linear
d, d,
Fully Connected
d, - CHW
ReLU
Transpose
CHW (128C, H TO:
W
Conv ConyTranspose Block
3
128C + 8C, (H,WW
ReLU
Decoder
Conv TranToospose
8C' + (H, (H,
Conv2d Refinement
ReLU
Table 11: Model architecture for ERA5 task with input dimension (C, H, W), where C denotes the number of channels and x W is the spatial resolution, using a factorized-atteniin encoder.
Components
Layer
Layer number
C, (H,
Activation
Conv2d
C'
GAC
(H, W)
5
Encoder
Conv2d
64C
64C
( T
H W.
FactorizedBlock
(4 W
GELU
Flatten
64C
H W 4CHHW 5
5
4
Fully Connected
4CHW
_> d.
Koopman Operator
Linear
d, de
Fully Connected
d. + 4CHW 3
ReLU
Transpose
4CHW 64C H w)
T
Conv Transsosod
64C 4 5 (H,W)
ReLU
Decoder
Conv2d
64C
5 + C, (H.W)
Conv2d Refinement
C (H,
ReLU
For fair comparison, the architectures for visual input and graph-structured dynamics follow the settings in Levine et al. (2020) and Li et al. (2020).
Table 12: Hyperparameter settings across physical simulations, visual inputs, and graph-structured dynamics
Parameter
Symbol
Physical Sim.
Visual Inputs
Graph Dyn.
Temporal coherence Structural consistency von Neumann entropy
2.00
3.00
2.00
8
2.00
2
0.10
0.50
0.10
InfoNCE neighborhood
k
5
5
49

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Published as a conference paper at ICLR 2026
G.6 BASELINE ALGORITHMS
Physical Simulation Tasks.
VAE (Kingma al., 2013): Baseline implemented using standard variational autoencoder with nonlinear forward map in latent space. Code available at htttps://g$tthi com/bveeilic/Varitt -ana-aattennoodeee
KAE (Pan et al., 2023): Koopman learning with an autoencoder architecture. Code avail- able at https: /tt////ttttt com/ dynami cs abb atb/pykoopmam.
KKR (Bevanda et al., 2023): For the low-dimensional Lorenz-63 system, we adopt fixed kernel functions as basis following the implementation in https: huu.t com/TUM-IIR/Koon hwt.///////..ee...ooo For high-dimensional systems, we use deep kernel fea- tures following Yang al. (2025), with code available at https: / thub. com/ wydm/nggucl/eensovvaa http://ennoovvr/bbloaoa htt.///aaa//baaaaaaaaaaaaaaaaaaa htto//aan//nnn monde L/KS_modell py..
thub.
PFNN (Cheng et al., 2025): A state-of-the-art Koopman variant for learning and predicting chaotic dynamies. Code available https: //t:://gitttt com Coom/Hy23333/PMM
Visual Inputs.
E2C (Banijamali et al., 2019): A latent embedding approach based on the VAE framework.
Code available at https:/ / thub. com/ er jangg//n////
KAE (Pan et al., 2023): Koopman learning with autoencoder architecture. Code avail- able at https: /tteps://gitan //gttub.. com/
PCC (Banijamali al., 2019): state-of-the-art latent embedding algorithm also based on the VAE framework. Code available at
com/ nA wttm//iinARReeeeeeeeeeeeeee /
Graph-Structured Dynamics.
CKO (Li et al., 2020): A Koopman based framework for learning and predicting general graph-structured dynamics. Code available at https: //g/ thub. com/ Yunzhuti// Compos t ona KoopmanOperators.
50


Reviews:
[{"review_id": "F5hNWN9vSH", "reviewer": "Reviewer_Ho1y", "summary": "The key idea of this paper is that learning a good Koopman latent space can be seen as an information bottleneck problem: the goal is a latent representation that is both simple (compact, linear) and expressive (able to predict the future).\nTo formalize that, the authors derive an information-theoretic Lagrangian with three main terms:\n- Latent mutual information -- to keep temporal coherence and predictive power.\n\n- Von Neumann entropy -- to prevent mode collapse and keep latent diversity.\n\n- Conditional mutual information -- to suppress irrelevant or fast-dissipating information.\n\nThey build a new Koopman learning algorithm that implements this Lagrangian, combining VAEs with information-theoretic regularization. Experiments on chaotic systems (Lorenz, vortex flows), visual control tasks, and graph-structured dynamics show better long-term prediction and more stable latent manifolds than existing Koopman Autoencoders or PFNNs.\nConceptually, they link Koopman spectral modes to information flow -- coherent modes near |λ|=1 correspond to persistent information, while dissipative modes correspond to information loss.", "questions": "1. Can the authors formally prove that maximizing I(z_{t−n}; z_t) indeed selects eigenfunctions corresponding to Koopman modes with |λ|≈1? Or is this just a heuristic interpretation?\n\n2. How is the von Neumann entropy computed robustly from noisy minibatch covariances? What happens if the covariance matrix is rank-deficient?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 3, "strengths": "- The connection between spectral theory and the information components of the Koopman representation is very interesting. This can bring good information-theoretic insight into dynamical representation learning.", "weaknesses": "1. The paper often claims that it learns a Koopman operator or a Koopman representation, but the actual results only show a **latent linear predictor** -- not a true Koopman operator. The authors present a latent-linear model and refers to the learned matrix $K_{\\psi}$ as a “Koopman operator”. However, unlike DMD or EDMD, which explicitly enforce the Koopman relation φ(T(x)) = K φ(x) over sampled trajectories, this work only trains $K_{\\psi}$ to minimize predictive error ($z_{t+1} ≈ K_{\\psi}  z_t$) or a KL divergence with a linear-Gaussian prior. This ensures linear predictability in the learned coordinates, but not Koopman invariance of the feature map φ(x). The paper provides no invariance or reconstruction tests. Consequently, $K_{\\psi}$ may simply act as a best-fit linear regressor in latent space rather than an approximation of the true Koopman operator. More clear, the paper trains a neural encoder $φ_θ(x)$ that maps x_t → z_t,\nand a linear map $K_{\\psi}$ that predicts: $z_{t+1} ≈ K_{\\psi}  z_t$.  This looks like DMD in the latent space. However, the key difference is how $φ_θ$ is chosen and what it represents. In DMD/EDMD, φ is fixed (a known basis) and directly tied to the system's state, so the learned K approximates the true Koopman operator on that basis. In this paper, $φ_θ$ is learned jointly with $K_{\\psi}$ to minimize reconstruction and prediction losses. There is no constraint ensuring $φ_θ$ produces Koopman-invariant coordinates. As a result, $K_{\\psi}$ is just the best linear predictor in whatever feature space $φ_θ$ learns, not necessarily the Koopman projection of the true dynamics.\n\n2. Some propositions depend on unproven assumptions as the proofs and closed-form identities (in the Appendix) repeatedly assume linear–Gaussian dynamics, full-rank covariances, and ergodicity to move from general nonlinear dynamics to tractable formulas. These assumptions are used without careful justification (or statements of when they hold). for instance,  Appendix F (derivations for conditional mutual information): the Gaussian closed-form identity and linear–Gaussian relations are introduced and used to produce equations (32)–(35). It writes z_{t} | z_{t−n} ∼ Normal(Kⁿ z_{t−n}, M_n) and uses linear Gaussian formulas to get closed-form mutual information (e.g., eq. (33)–(35)). Those closed forms are valid if the latent process is linear Gaussian. But the encoder is a neural network mapping high dimensional x to z; there is no general reason the induced z dynamics are exactly linear Gaussian. The paper uses the Gaussian identities to make spectral claims about Koopman eigenvalues and MI, and then presents those claims as general. Moreover\n- Ergodicity: Remark F.4 after Proposition 2 uses an ergodic limit for long-time averages.\n- Full-rank covariance / density matrix manipulations: von Neumann entropy derivations and water-filling solution in Appendix F.5 (equations around 44–49).\n\n3. Calculating mutual information and von Neumann entropy for neural networks is challenging, but the paper does not adequately explain the methods used for their estimation in high-dimensional spaces.\n\n4. The “von Neumann entropy” regularization uses S(C/tr(C)), but C may not be always positive semidefinite.\n\n5. The approximation I(z_t;P_t) ≈ InfoNCE loss is questionable for temporal sequences; no justification for mutual information estimation validity.\n6. The KL-based error bound (Proposition 2) assumes independence that may not hold for nonlinear latent dynamics.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "FexsBUnK8s", "reviewer": "Reviewer_Ga5Z", "summary": "The authors explore a new, information-theoretic approach to Koopman operator learning, which is used to model dynamical systems. They argue that existing methods struggle because latent variables often fail to balance expressivity (capturing rich dynamics) and simplicity (remaining compact and interpretable). Drawing parallels to the information bottleneck principle, they introduce a Lagrangian formulation that explicitly trades off these two goals. In this setup, latent mutual information promotes simplicity, while von Neumann entropy encourages diversity and avoids mode collapse. They propose a corresponding algorithm that stabilizes learning, improves interpretability, and yields better empirical performance on various dynamical system benchmarks.", "questions": "Q0. Does the proposed approach still model a true Koopman operator, or has it effectively become an information-theoretic Lagrangian framework—based on mutual information and entropy—that only mimics Koopman behavior?\n\nQ1. How sensitive is the proposed Lagrangian formulation to the relative weighting between mutual information and von Neumann entropy? In other words, does performance degrade noticeably if this balance is not carefully tuned?\n\nQ2. Can the authors demonstrate the scalability of their method by applying it to higher-dimensional or real-world dynamical systems to validate its robustness beyond synthetic benchmarks?\n\nQ3. Are there other approximate Koopman operator methods that could be compared against the proposed approach? It would be helpful to see how this method performs relative to alternatives that may more closely capture the underlying manifold, even if the proposed one achieves better overall results.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "S1. The paper presents a fresh conceptual perspective by linking Koopman learning with the information bottleneck framework, which deepens theoretical understanding and connects two previously distinct research areas.\n\nS2. The proposed Lagrangian and algorithmic formulation is both principled and practical, demonstrating consistent empirical improvements and more interpretable latent structures.\n\nS3. The paper provides a well-structured and thorough analysis of the information components underlying the Koopman operator, offering a clear theoretical justification for the proposed Lagrangian formulation.\n\nS4. The visualizations of the learned manifolds are insightful and effectively illustrate how the proposed method improves representation quality and interpretability.\n\nS5. The quantitative results show that the proposed approach consistently outperforms existing methods, supporting its practical advantages and robustness.", "weaknesses": "W1. The framework, while elegant, is conceptually dense and abstract, making it difficult to assess how easily it can be implemented or scaled to complex, high-dimensional real-world systems.\n\nW2. The empirical validation, though broad, seems to focus mainly on illustrative examples—more rigorous or comparative testing on large-scale benchmarks would strengthen the claims of generality and robustness.\n\nW3. While the information-theoretic framing is creative, the use of mutual information and entropy as tradeoff terms resembles existing regularization and variational methods, making the contribution partly incremental rather than entirely groundbreaking.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "lej0azJ9Oy", "reviewer": "Reviewer_KxuT", "summary": "The paper proposes a new representation learning scheme based on the Information Bottleneck framework for learning Koopman operators. It addresses a common flaw in current methods, mode collapse, by introducing a novel information-theoretic Lagrangian that balances latent mutual information (for simplicity) against von Neumann entropy (to ensure expressiveness). The paper experimentally evaluates the proposed algorithm across a range of dynamical systems, demonstrating improved performance against baseline methods.", "questions": "1. How sensitive the algorithm is with respect to hyperparameter optimization? I haven't found any discussion on hyperparameter selection; is there any intuition or heuristic for choosing (k, alpha, beta, gamma)? Especially since the ablation study in Figure 5 clearly demonstrates that the balance between these terms is critical to success.\n2. In the abstract, the Authors noted that KO's infinite-dimensional nature makes finding a good finite-dimensional subspace challenging, especially for deeper architectures. Could you elaborate on why deeper architectures tend to suffer more?\n3. At the beginning of section 3.2, the authors said, “The latent mutual information quantifies the magnitude of error, but not the nature of the information lost in the Koopman representation.”, which sounds peculiar to me. Could you elaborate on what you mean by “nature of the information lost”?\n4. I don’t understand the meaning of the red and blue arrows in Proposition 3. Does that mean we want to maximize the blue and minimize the red?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "The paper addresses an important challenge in data-driven dynamical systems: learning stable, interpretable, and expressive finite-dimensional representations of the Koopman operator. The authors see the problem through a general, principled, and information-theoretic approach to manage the simplicity-expressiveness trade-off. The latent mutual information ($I(z_{t-n}; z_t)$) promotes temporal coherence but can lead to mode collapse, while von Neumann entropy ($S(\\mathcal{C})$) promotes expressiveness and spectral diversity, actively preventing such collapse. This leads to a novel information-theoretic Lagrangian objective for Koopman learning. The paper is well-written and logically organized, making it easy to follow and comprehend. Its clarity is further enhanced by the inclusion of informative diagrams.", "weaknesses": "1. The paper's core premise is that standard representation learning (like VAEs) is insufficient for Koopman models because it prioritizes reconstruction over the more restrictive constraint of a linear predictive structure (structural consistency). While this claim is valid, the paper frames this as a relatively novel insight, overlooking a significant body of existing literature that has already identified and addressed this exact problem. For instance, works on VAMPNets (arXiv:1710.06012)  and, more recently, DPNets (\tarXiv:2307.09912) are explicitly designed to learn representations that linearize the dynamics, moving far beyond simple reconstruction. By not citing or benchmarking against these more relevant baselines (as well as other related works, such as arxiv:2309.07200), the paper overstates the novelty of its problem formulation, making it difficult to assess how this new solution compares to other state-of-the-art methods targeting the same fundamental challenge.\n\n2. There is a gap between the \"pure\" information-theoretic Lagrangian (Eq. 8) and the tractable loss function used for optimization (Eq. 9). The practical implementation relies on several approximations:\n    \n- The latent mutual information $I(z_{t-n}; z_t)$ is approximated using the InfoNCE contrastive loss, which is a lower bound, not the true MI.\n- The von Neumann entropy $S(\\frac{\\mathcal{C}}{tr(\\mathcal{C})})$ is calculated using a stochastic approximation of the covariance matrix $\\mathcal{C}$ derived from a single minibatch (Algorithm 1, line 7). This estimate is likely to be very noisy and may introduce high variance into the gradients, potentially affecting training stability and the quality of the final manifold.\n\n3. As the authors state in their appendix (Appendix D), the framework does not yet provide formal guarantees on sample complexity or the non-asymptotic convergence of the representation. It is unclear how many data samples are needed to reliably estimate the information-theoretic quantities (especially the minibatch-based entropy) and converge to a meaningful Koopman subspace.\n\n\n_Minor_:\n\n- In Eq. 14, line 1003, I believe that it should be $I(x_{t−1}; x_t) − I(z_{t−1}; z_t)$.\n- Figure 7 is a bit misleading, in the decoding phase, as it is mentioned in the caption of the figure, “ The latent variables are subsequently decoded back to approximate the original states,” but in the diagram, it goes back to the original state. Therefore, one could wonder why $I(z_{t-1}; x_t | z_t)$ is not zero.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "pdk5mFmAcf", "reviewer": "Reviewer_hgxb", "summary": "This paper presents an information-theoretic framework for learning Koopman representations of nonlinear dynamical systems.In this paper, authors proposed a unified framework that balances simplicity and expressiveness in learning finite-dimensional Koopman representation.", "questions": "(1) How sensitive are the results to the choice of von Neumann entropy regularization coefficient $\\gamma$? What happens if you set $\\gamma = 0$ (no entropy regularization) on all tasks quantitatively?\n(2) The current formulation involves covariance and entropy computations, then how does the computational cost scale for high-dimensional latent spaces?\n(3) How does your approach perform on stochastic dynamical systems, where the Koopman operator is no longer deterministic setting?\n(4) The InfoNCE computation treats temporal neighbors as positives. But if the system has $T$-periodic dynamics, shouldn't $z_{T}$ also be a positive? How do you handle periodicity?", "limitations": "", "rating": 6, "confidence": 2, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "(1) The proposed information-theoretic Lagrangian provides a clear mathematical formulation for balancing simplicity and expressiveness of predictive sufficiency;\n(2) The information decomposition in Proposition 3 connecting mutual information components to Koopman spectral properties is insightful. This bridges dynamical systems theory with information theory smoothly;\n(3) Experiments across three diverse domains demonstrates broad practical application.", "weaknesses": "(1) While the paper provides asymptotic and information-theoretic insights, it lacks finite-sample guarantees or non-asymptotic convergence results;\n(2) The practical loss in Eq. 9 is a little bit heavy, with many tunable hyperparameters $\\alpha, \\beta, \\gamma$. It is unclear how sensitive performance is to these values;\n(3) Missing comparisons or discusstion of the following recent deep Koopman methods:\n  (a) https://www.nature.com/articles/s41467-018-07210-0;\n  (b) https://openreview.net/pdf?id=Svk7jjhlSu;\n  (c) https://pubs.aip.org/aip/cha/article/27/10/103111/151485;\n  (d) https://www.sciencedirect.com/science/article/abs/pii/S0021999124004431;\n  (e) https://link.springer.com/article/10.1007/s00332-019-09567-y.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Oral)