Conformal Online Learning of Deep Koopman Linear Embeddings
Authors: Ben Gao
Jordan Patracone
Stephane Chretien
Olivier Alata
Venue: NeurIPS 2025 poster ()
Keywords: Conformal Online learning; Koopman representation learning; Streaming dynamical systems modeling
Primary area: general_machine_learning
TL;DR: We introduce Conformal Online Learning (COL), a novel framework for adaptively updating Koopman-invariant representations of nonlinear dynamical systems from streaming data.

Abstract: We introduce Conformal Online Learning of Koopman embeddings (COLoKe), a novel framework for adaptively updating Koopman-invariant representations of nonlinear dynamical systems from streaming data. Our modeling approach combines deep feature learning with multistep prediction consistency in the lifted space, where the dynamics evolve linearly. To prevent overfitting, COLoKe employs a conformal-style mechanism that shifts the focus from evaluating the conformity of new states to assessing the consistency of the current Koopman model. Updates are triggered only when the current model’s prediction error exceeds a dynamically calibrated threshold, allowing selective refinement of the Koopman operator and embedding. Empirical results on benchmark dynamical systems demonstrate the effectiveness of COLoKe in maintaining long-term predictive accuracy while significantly reducing unnecessary updates and avoiding overfitting.

Conformal Online Learning of Deep Koopman Linear Embeddings
Ben Gao
Jordan Patracone"
Université Jean Monnet Ssinn-Etienne, CNRS, Institut d' Optiquu Graduate School, Inria, Laboratoire Hubert Curien UMR 5516, F-42023, SAINT- ETIENNE, France fr
Université Jean Monnet Saien-Etienne, CNRS, Institut d'Optique Graduate School, Inria, Laboratoire Hubert Curien UMR 5516,
F-42023, SAINT- ETIENNE, France jordan. watrracoe@univ-st-eee.....
fr
Stéphane Chrétien Université Lyon 2 Laboratoire ERIC Bron, France
Olivier Alata
Université Jean Monnet Saint-Etienne, CNRS, Institut d' d'Optique Graduate School, Laboratoire Hubert Curien UMR 5516, F-42023, SAINT- ETIENNE, France olivier, al wlwaaaaniisssseeeeeeeea eeinene......ee. fr
stephane s www.et.iinnnnii--yyy.......
Abstract
We introduce Conformal Online Learning of Koopman embeddings (COLoKe), a novel framework for adaptively updating Koopman-invariin representations of nonlinear dynamical systems from streaming data. Our modeling approach com- bines deep feature learning with multistep prediction consistency in the lifted space, where the dynamics evolve linearly. To prevent overfitting, COLoKe employs a conformal-style mechanism that shifts the focus from evaluating the conformity of new states assessing the consistency of the current Koopman model. Updates are triggered only when the current model's prediction error exceeds dynami- cally calibrated threshold, allowing selective refinement of the Koopman operator and embedding. Empirical results on benchmark dynamical systems demonstrate the effectiveness of COLoKe maintaining long-tern predictive accuracy while significantly reducing unnecessary updates and avoiding overfitting.
Introduction
Understanding the evolution of complex systems over time is essential in numerous disciplines, including robotics [Bruder al. 202T finance [Mann and Kutz, 2016), physics (Kaptanoglu et al.) 2020). chemistry [Klus et al. 2020 and neuroscience [Chrétien et al. [023]] A particularly powerful framework for this analysis is provided by aperator-theorette approaches, which recast nonlinear dynamics into linear evolution in function space. Among these, the Koopman operator [Budisić et al., 2012 Mauroy et al. 2020) plays a central role: describes progression of measurement functions (or observables) defined over the state space, yielding infiniteddimensional linear representation of nonlinear systems. Its spectral decomposition offers principled way to characterize the long-term behavior and underlying structure of the dynamics [Brunton et al.) [2022] Although the Koopman operator is infinite- ceinite-dimensioneee veloped to approximate it in mnite-dimensional setting. Notable among these are Dynamic Mode Decomposition (DMD) Rowley et al., 2009) and its nonlinear extensions, such as Extended DMD (EDMD) [Williams et al. 2015 and its variants (see [Jin et al. 2024 and references therein). Recent
several numerical methods have been de-
Research conducted within the context the Inria IIT Associate Team.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

---

Table 1: Positioning of COLoKe with respect to the state-the arr...
No need
Online
Adaptative embedding
Built-in reconstruction
for history update
ODMD
Zhang et al.
2019
R-EDMD Sinha el al. 2019
2023
x
DKLT Hao et al. 2024
V
batch only batch only batch only
/
X
BatchOnline Mazouchi et al.
2023
V
R-SSID [oyya and lallapraadda 2024
X
Online [Liang et al. 2022
COLoKe (ours)
work has also proposed truncated Signature features rooted in path theory (Chrétien al. 2025 Another line of research consists of kernel learning formulations in reproducing kernel Hilbert spaces [Kostic et al. 2022 Hou et al. 2023 In addition, deep learning techniques have been lever-
aged to learn expressive Koopman-invariant representations, using neural networks and autoencoder
architectures estimate significant observables [Li et al. 2017 Wehmeyer and Noč, 2018 Lusch
et al. 2018 Yeung et al., 2019 Otto and Rowley 2019]. For example, Xu et al. (022bb] addresses
this by explicitly minimizing the spectral residual. The stochastic Koopman operator extends the
classical Koopman framework to random dynamical systems. capturing the evolution of observables
under stochastic influences [Crmjaric- et al. 2020 Kostic et al. 2023 Xu et al. 2025a]
Most existing methods, however, operate in the offline setting, assuming access the entire dataset in advance. Yet, in many applications-sucl as online monitoring, adaptive control, or real-time forecasting- data arrive sequentially, and the system may evolve in non-stationary fashion (Korda) and Mezić 2018). Several online methods have been proposed for this task. For instance, Online DMD Zhhang et al., 2019] and Online EDMD (Sinha al.] 2019 2023] incrementally update Koopman operator estimates as new data arrives. However, these methods typically rely on linear observables or fixed dictionaries, limiting their expressiveness. More recent methods using neural networks (e.g. [Liang et al. 2022 Hao et al.] 2024)) lack principled learning strategies and often rely on retraining the model using a fixed number of steps, regardless of whether the update is necessary. These limitations highlight the need for online learning strategies that are not only memory-efficien but also adaptive in order to update models only when required by the incoming data.
We address this need by introducing Conformal Online Learning of Koopman embeddings (COLoKe), whose high-level principle is sketched in Figure [ Our approach combines deep Koopman repre- sentation learning with novel repurposing of conformal prediction principles to decide when to adapt the model as data arrive in a streaming fashion. Updates are triggered only when necessary, reducing both computational burden and overfitting. This contributes to the growing body of work on online Koopman learning (Sinha et al. 2023] Loya and Tallapragada| [2024] Mazouchi et al. [003)] Our method differs by enabling adaptive embeddings, built-in reconstruction, and real-time updates while remaining memory- eeffire-efficient (see Table To the best of our knowledge, COLoKe is the first principled approach for online learning driven by cnfformal-based updates.
Contributions. In summary, our contributions are: (i) We propose a novel online learning framework that leverages conformal prediction to guide adaptive model updates; (ii) We instantiate this framework in the context of Koopman operator learning, enabling expressive online regression of nonlinear dynamical systems through deep data-adaptive embeddings; (iii) We provide a theoretical analysis establishing a dynamic regret bound under mild assumptions.
Outline.
The rest of the paper is organized as follows. In Section 2 we present the mathematical background on Koopman operator theory and conformal prediction, which form the foundation of our approach. Section 3 introduces our main method and details how a conformal-basee
update strategy permits to update model parameters adaptively in an online fashion. Section 4 provides empirical validation on range of benchmark dynamical systems, comparing COLoKe to existing online Koopman learning methods. Proofs and implementation details are deferred to the appendix.
2 Mathematical background
In this section, we present the two key components of our method, namely the Koopman operator and the conformal prediction framework.
2

---

2.1 Data-driven learning of the Koopman operator
Let measurable space (X, Ex), with X' Rd and Σx Borel olaleerra on X, and let T: X - X' be measurable, time-invarian deterministic map. Hereafter, we consider a discrete- time autonomous dynamical system governed by the iteration rule 2(+1 T(2)) for all E N, which describes the evolution of the system as sequence of states {#4 entirely determined by the initial condition To E X and the update rule T. A classical approach to analyze such kind of nonlinear dynamical systems is through the Koopman operator formalism. Rather than studying the trajectories in state space directly, the Koopman approach lifts the dynamics to an infinite- anfnite-dimmnsionall space of observables FF (e.g., (2 pi) for some Borel measure (d). The Koopman operator K:: F F defined by
(Kf))z) - ((T())), E
(I)
which describes the evolution of observables along trajectories of the system. Importantly, is a linear operator, even when T is nonlinear, making it powerful tool for the spectral analysis of nonlinear dynamics [Brunton et al. 2022]. In particular, if o E F is an eigenfunction of IC with eigenvalue À E i. Kip Xip then along trajectory (Iz), the observable evolves linearly: (p(x)) = (zo)
defines an injective embedding of the state space, the system can be linearized via the coordinate transformation - (yp1 (z)), . P0
follows that, when a set of eigenfunctions { .Pr)
+ (r.()) (x))..
In this lifted space, the nonlinear dynamics evolve linearly, providing a compelling framework for Koopman-bane based analysis and control [Korda and Mezić 2018, Mauroy et al. 2020).
In many real-world scenarios, the transition map T governing the dynamics unknown or inaccessi- ble, and we must instead rely on observed trajectories of the system { This shift has led to the development of data-driven approximations of the Koopman operator K. Motivated by the fact that Koopman cigenfunctions evolve linearly along trajectories, one typically seeks a set of observables {f1...., fm] C FF that spans subspace that is approximately invariant under the action of K., In the ideal setting where S = (pan(fi....
fm) is invariant under K. i.c. Kff E S for all f E S, the restriction of K. to S admits an exact representation by finite-dimensional matrix K E Cm and the evolution of observables in this subspace is governed by the linear relation
( = (x(x,,,
denotes the lifted representation of the state. This insight motivates the search for dindeemnsiioall Koopman -ivaariann subspaces that admit such linear representations. Methods like EDMD (Williams et al.) 2015] approximate this setting by fixing dictionary {.fe but their performance is limited by the expressiveness and suitability of the chosen observables. To overcome this limitation, recent approaches [Takeishi et al., 2017 Lusch et al., 2018 Yeung al. 2019 Otto and Rowley 2019] propose learn both the feature map + and the linear operator K jointly using neural networks, leading to deep Koopman embeddings.
where ((x) =
2.2 Conformal prediction for online data
Conformal prediction (Vok et al., 1999 2005 Romano et al. 2019 Angelopoulos and Bates 2021 is a distribution-Fn -ree framework for uncertainty quantification that constructs valid prediction sets with finite-sample guarantees.
Given past assigns a conformity score s y) each candidate output y. The conformal prediction set is defined as C. = {y E Y s(Iz, y) S ). where q: is a quantile calibrated to ensure P(y: E C,) 2 1 - a. The set C, is thus interpreted are deemed sufficiently "conformal" (i.e., not too surprising) with respect to the current model and past observations. While this framework is powerful and requires no distributional assumptions beyond exchangeability, it is not directly applicable to time series or online learning, where data are typically non-exchangeable. In such settings, standard conformal methods may yield miscalibrated or overly conservative intervals. Addressing this challenge has motivated the development of adaptive conformal approaches that can track distribution shifts over time Gibbs and Candes) [021] I] Xu et al.) 2021 Angelopoulos et al. 2023]. In particular, Conformal PID Control was recently introduced in Angelopoulos et al. [2023] as dynamically calibrated version of conformal prediction. Here, "PID" refers to the use of Proportional, Integral, and (optionally) Derivative feedback carms-standard components in control theory-usee used to adaptively adjust the prediction threshold. Rather than fixing the quantile threshold qe in advance, the method updates it online in response to conformity violations.
pairs {<,,, ye) W=) a model produces a prediction ÿe for a new input T2, and
set of plausible outputs: it contains all candidate values of y that
3

---

KOOPMAN EMBEDDING
CONFORMITY-BASED UPDATE
[2 10- -er
Po. (az)
Score ss(se Eq.
B)
NN
window
Set @++11 (scc Eq.
YES
Do (21) & Ki Do, (ste-r)
S Q+
Update parameters (0c, Kc)
NO
Wait for new datum at time <- 1+1
Figure 1: Schematic representation of COLoKe. The model receives rolling window of observations, lifts them via a partially-learned feature map, computes a conformity score based multi-step prediction error, and updates parameters only the score exceeds a conformal threshold.
Among its variants, we consider the conformal PI control scheme: after observing whether y Ct. a binary error signal e = 11,e & C.} is computed, and the threshold adjusted via:
c +re
a++1 q + 7 a)
Σ
(e.
a)
EEN,
(3)
Proportional term (P)
Integral (1)
where 7 >> is learning rate and Fr is nonlinear saturation function [Angelopoulos et 2023 acting as an integral correction term. This Pl-style control loop allows the prediction region to adaptively expand or contract maintain the desired coverage. Under mild regularity conditions, ensures that the empirical coverage converges, i.e., + ET E - a as T co.
In this paper, we revisit the conformal mechanism from a novel perspective. Rather than using conformal PI control to build uncertainty sets for 1/,, we reinterpret the conformity score diagnostic tool for the model itself, in particular for deciding whether a Koopman embedding remains consistent over time if needs to updated. This reinterpretation will be developed in Section 3.2
3
Online conformal learning of deep Koopman embedding
3.1 From online learning...
We consider an online learning setting with bounded memory, where the goal is to incrementally learn Koopman-invann invariant subspace from sequentially observed dynamics. Let N denote the current time, and let E N' be fixed window size. each time step we assume access finite buffer of recent observations D, { ..,,,, J. which serves to incrementally update the current Koopman approximation.
At each time step f, we denote by t X C the current feature map and by K, E Cm the corresponding finite-dimensional Koopman operator. Following (Li al. [017]. also advocate to enforce interpretability and preserve part of the original state, by designing the feature map to include both the identity and learnable nonlinear component, i.e.,
48,
(x)
z., Lo, (a:)
Vo E X
(4)
where d.. X Cm- is neural network with parameters 0,. This structure ensures that the lifted representation retains access to the state while learning an additional embedding & (ar) from data. In order to capture temporal consistency within the buffer, we seek to minimize multi-step prediction error recent observations on D,. This leads the following loss function used for online updates
Definition 3.1 (Online Koopman training loss), At each time step, the parameters (0,, K2) are updated by minimizing the multi-step prediction loss, i.c.,

---

minimize (9,,Kc)
Le(O,, Ki) = Σ (a, Ki)
(5)
(e,r)EI,
where the index I, = {(s, N2 t- w S < S f) collects all valid mtlti-step
prediction pairs within the buffer D,, and each loss term is defined as
(0,, K() =
Lo. (xx+7) - K² (xstT
(6)
which accumulates the discrepancies between the lifted state at time + and all intermediate predictions obtained by successively applying Kg to earlier lifted states at times {s...., 8 + T 1}.
This formulation is motivated by two design principles. First, shown by Otto and Rowley 2019)) multi-step prediction promotes the identification of persistent spectral modes and approximate Koopman eigenfunctions, thereby improving long-term prediction. Second. by explicitly including the state x in the lifted representation. (4). the model embeds a reconstruction constraint directly into the consistency loss. This coupling climinates the need for a decoder, as prediction errors in the lifted space naturally reflect discrepancies in the original coordinates.
Remark 3.1 (Prediction conformity score). In particular, have that
4- (0,, Ki) = Σ (xz) - KT Do, (12-r)12
which quantifies the discrepancy between the current lifted state Do, (Ix) and its multi-step predictions from all previous states in the buffer. This term corresponds to predicting x; from each of the past w observations using the learned Koopman operator K,, and thus provides a direct measure of temporal consistency toward the present.
In principle, one could perform multiple optimization steps to minimize Cr(O,, K,) at each time t, thereby reducing the residual prediction error as much possible within the local buffer. However, such an approach may lead to overfitting to recent data and degrade generalization. This behavior is typical of baseline online learning schemes that rely fixed optimization schedules without accounting for model confidence or adaptation needs. To mitigate this issue, we propose a data-driven stopping rule inspired by conformal PID control, introduced in the next section, which adaptively decides whether additional updates are beneficial.
3.2 .o To conformal online learning
Assuming that the Koopman embedding, parametrized by (0--1, K,-1), has been adequately trained on the previous buffer D;-1 = { 1 we now aim to determine whether it remains consistent with the newly observed state x;, Rather than retraining unconditionally at each step,
initialize (0,, K,) == (0--1, K,-1) and update the model only when the incoming data indicates significant deviation from previously learned dynamics.
To assess this deviation, we rely on the prediction conformity score introduced in Remark 3.1 which measures the discrepancy between the current lifted state oo, (2:) and its multi-step predictions from the past window. This score acts as a proxy for temporal alignment and forms the basis of our adaptive update rule. To formalize this idea, we define score function s, by treating the prediction conformity score as a function of the test point x E X, with model parameters (0,, K2) fixed:
st(2, (0t,
Σ 10. (x) - KJ De. (II-+))1
(8)
Note the difference with (7. in the sense that s(x, (0,, K.)) is function of the additional variable
X. The score s(x, (Or, K,)) is instrumental in defining the prediction set introduced now. Inspired by the conformal prediction interval (PI) framework described in Section 2.2 prediction set as
we define a
C. {o E X st (0., K()) g.
(9)
5

---

Algorithm 1 Conformal Online Learning of Koopman embeddings (COLoKe)
Require: Buffer size w, initial parameters (0m-1, K Kw-1). step-size v) >> 0
1: Initialize conformity threshold Gu
2: for L = w, + ,.... do
3: Observe new state za and update buffer D; = fzi-w)...,
4: Set (0,, K2) (0--1,Ki1)
5: Compute prediction conformity score 3; 6- w-- (0,, Kt)
See Eq.
6: Update threshold: 4++1 < ConformalPI(g))
See Eq. with C2
1(s > 4t
7: while $ > qt do
8: Perform gradient-based step: (0,, Ki) t- (81, V Ki)
See Eq. (5)
9: Recompute 8, +- e (0;; K₂)
10: end while
11: end for
where qi > is a calibration threshold that controls the conformity level. In the standard conformal prediction setting, the set C₁ serves as prediction region for the next state Tr. This interpretation treats C, as a (1 c--confidence region which the next observation expected to fall, based on past conformal scores.
In our setting, however, we use this prediction set in a novel way: not for uncertainty quantificationn but as decision rule for model adaptation. Specifically, if the newly observed state 2; lies outside Cr the prediction error is considered too large relative to past conformity, and an update of (0,, K2) is triggered. Otherwise, the model is retained without further training. This repurposing of conformal principles provides lightweight, data-driven mechanism for online learning.
Crucially, while classical conformal prediction evaluates the conformity of new states, our approach shifts perspective evaluate the conformity of parameter configurations. That is, rather than asking whether a new observation aligns with a fixed model, we ask whether there exists any parameter pair (0, K) under which the current observed state I, is conformal.
In this spirit, and since we are not primarily interested in constructing a prediction interval for 2t, we introduce the novel notion of prediction score set.
Definition 3.2 (Prediction score set), Given a newly observed state xy and a conformity threshold qe > the prediction score set time is defined
S. = s Param.),
(10)
where Param, = {00, K) such that s = (1z, (0, K)) = f- (0, K) S gr). This set contains all prediction scores attainable at H, by Koopman models that satisfy the current calibration constraint.
(0)
In this view, if the current model satisfies f- (0,, K2) E St (or equivalently. 4.-w, (0,, K,) S q). it deemed temporally consistent and it is retained. Conversely, if e,-w, (f,, K2) & S, (i.e.. l (f,. K2) > qt). the current model is no longer consistent with past dynamics, and an update of (0,, K2) triggered. Note that, by design, there no need to construct Param, explicitly. We refer to the resulting adaptive online learning scheme, driven by conformity -baed update decisions, as Conformal Online Learning of Koopman embeddings (COLoKe). The full procedure implementing online learning is summarized in Algorithm aand sketched in Fig. We provide below a preliminary bound for the dynamic regret of COLoKe.
the principles of conformal-basee
Theorem 3.3 (Dynamic regret of COLoKe). Let (0t, K.) be the parameters produced by Algorithm 7 and let (0t, K;) E argmin Lo, K) denote any depeneen optimal model minimizing the loss at step Further assume:
(A1) Each La is T-smooth with || L. K)ll < B; (A2) The oracle path has bounded total variation and squared variation:
V1 = 11100 Kin) (07, KTIl < co, := Σ 1; K;1) K?)K < so;
(A3) The conformity thresholds satisfy Σte ae S O (ah(T)) for some sublinear, nonnegative, nondecreasing function h;

---

x10
16
Offine
q:
8
800
COLoKe
10"3
test error
400
q:
imto
30
90 30
60
90
0
50
100
train time (seconds)
(a) Left: Adaptive update behavior with threshold gt. score st, and update counts; Right: sublinear growth supporting (A3) in Theorem 3.3
(b) COLoKe vs. offline Koop- man learning
Figure 2: Illustration and empirical support for COLoKe's adaptive learning strategy.
Then the dynamic regret satisfies: Σts (c, k(), (ah(T)) + VT Sr).
Proof. The proof is deferred to Appendix
Assumptions (A1) are standard in online learning and naturally satisfied in our setting. The loss L. is smooth in both / and K, provided that the neural network Φ employs smooth activations. The bounded variation of the dynamic oracle (A2) reflects slowly evolving or piecewise-stationar dynamics, which commonly arise in practice. The key nonstandard assumption is (A3), which bounds the cumulative conformity thresholds gr. and which follows from our use of conformal PI. More specifically, the function h comes from the saturation function in the update rule (3), as designed in Angelopoulos et al. 2023]. While we state (A3) as an assumption, can be viewed as an empirical hypothesis: in regimes with stable distributions and smoothly adapting models, conformity thresholds decrease rapidly, leading to sublinear accumulation. This behavior consistently observed experiments (see Fig. 2a and deriving it remains important direction for future work. For intuition, imagine a hypothetical scenario where (A3) does not hold, for instance where the cumulative grows linearly. This would imply that the scores stay roughly constant From a conformal prediction standpoint, this means that predictive uncertainty does not shrink as more data arrives, indicating the model fails to learn better representation. Such behavior may arise in non-autonomous systems under frequent abrupt changes, or when the model lacks the expressivity learn the dynamics. While the former case is out of the scope of the present paper, the second case can be mitigated by augmenting the dimension of the lifted space.
time.
Our proposed conformal online learning approach can be applied to general online non-convex learning settings beyond Koopman framework. An important contribution to online non-convex learning is due to (Suggala and Netrapalli 2020), who achieves optimal regret even under non-convex and adversarial losses by assuming access to an offline oracle. Subsequent work on dynamic regret explores variants handle changing environments [Xu and Zhang 2024 Although these approaches specify how to update model parameters at each iteration, they do not consider when such updates should take place or what extent. Our approach provides statistically principled answer these questions.
4 Numerical experiments
We now conduct experiments on multiple datasets derived either from solving differential equations as- sociated with canonical dynamical systems, from real-world sequential measurements. For the sake of reproducibility. all datasets and methods can be found at and we report full implementation details well as complementary numerical studies in the supple- mentary material.
4.1 Illustration and validation
To build intuition about our conformity-based update mechanism, we begin by illustrating how COLoKe behaves in controlled setting. Then, we assess whether it (i) accurately recovers the
7

---

DA M1
10
12a Ag
-2
10-1
10-33
D4
0-
x 10
5x 10
50
100
(a) Eigenvalues estimation error as functions of the dynamic t.
(b) Eigenfunctions {y11, (P2, (Ps} (from left to right) correspond-
ing to the oracle (top) and estimated COLoKe (bottom).
Figure 3: Convergence of the Koopman eigenvalue and eigenfunction estimates in the online setting.
underlying Koopman model, and (ii) achieves predictive performance comparable to offline learning approaches. Beyond overall accuracy, we are particularly interested in evaluating the quality of the estimated spectral properties. To this end, we consider the following analytically tractable system with known Koopman eigenvalues and cigenfunctions.
Setting. To validate the spectral accuracy of our online algorithm, we consider a benchmark system with analytically known Koopman eigenvalues and eigenfunctions [Brunton et al. 2016 2022
(((t) = au((),
E R
(11)
b(t)) = b(u(t) - u (t)).,
where á denotes the time derivative u. For a = -0.05 and = 1, the system admits a single attracting manifold
u² The Koopman spectrum contains the eigenvalues {); -1, XE -0.05, = - with known analytical eigenfunctions. To generate data, we simulate trajectories of the state vector x, = [u(t), v(t)] using a fixed integration step 0.01. We produce 1000 training trajectories of length 100 by sampling initial conditions uniformly from [_,, and similarly generate 1000 additional trajectories for testing.
Illustration and role of conformity- updates. Fig. (left) depicts the evolution of the calibration threshold de (blue line) and the associated prediction score set S₁ (shaded blue region). When the model yields nonconformant score 8; > de (top orange), updates the score becomes conformant s; S Ge (bottom orange). The number of updates (green crosses) varies across time, reflecting when and how much the model must adjust maintain consistency. As training progresses, the steady decay of q: reflects growing confidence and temporal alignment the model with the data. This increased accuracy tightens the score set St, making conformity harder to achieve- yet this is a desirable outcome, as ensures high precision adaptation. Importantly, the be maintained without overfitting
triggered until
number of updates remains controlled, showing that conformity or instability. To complement these observations, Fig. 2a (right) reports the cumulative thresholds, which exhibit sublinear growth of order (VVT), thereby supporting Assumption (A.3) in Theorem 3.3 Further analysis about how the conformal apdate-triggs mechanism adapts to different dynamical systems can be found in Appendix C
Spectral properties. We monitor the spectral behavior of the learned Koopman operator by diagonal- izing K, at each time step and tracking its cigenvalues. Although cigenvalues may be complex in general, the true values in this setting real. Fig. aag..aale repoots the estimation emrors over time, showing that COLoKe recovers the correct spectrum, the estimated eigenvalues stabilize around their true values. At the end of training, obtain real-valued eigenvalues (-1.0091, -0.04996, -0.1001} }. which closely match the ground-truth, The estimated eigenfunctions, shown in Fig. 3b align with the oracle up scalar factor.
Comparison with offline Koopman learning. To assess the computational efficiency of COLoKe, we compare it with an offline deep Koopman model trained on full trajectories using the same neural architecture. Figure 2b reports the test error as function of training time (in seconds), measured NVIDIA RTX 2000 ADA GPU. Note that, the test error is evaluated on separate

---

Table Numerical evaluation synthetic and real (indicated by x) datasets. For synthetic datasets, the first and second line corresponds to generalization error and online enror respectively.
ODMD
OEDMD
OnlineAE
OLoKe
COLoKe
1.1- 1.1-10-3
2.5-10"
1.0-10 10
2.1-10-b
2.4- 20 10
Single
(+3.6-10
(#2.8 10288)107")
(#7.7- 10-")
-t6.6-10-1
(#3.6 10
attractor
4.6 10-6
1.5 -.-10-2
7.4. 10-6
7.5-10-6
7.6 7.6-10-1
(17.3 (L7.3-107))
(14.7.10 107))
(12.8 10
(12.5 10
(19.6 10.6.1001 10"*)
2.5 10
6.8-10-3
8.7 10-3
10-1
3.1.10-6
Duffing
(+7.8-10
(+4.5.10-3)
(+2.5 10-3)
(+1.0.10-9)
(+2331007
oscillator
1.9 10
1.8-00-3
2.0- 10-3
2.3 10
7.
(#1.5 10"
(#3.3-107))
(#6.2 10
(#4.0-10 1075)
(#1.9 10-100
2.1-10-a
2.1-10-3
10-2
6.6-10%
3.8.10
VdP
(13.6-10
(13.2.10-9)
(13.0 10
(11.5-10")
(11.2.10
oscillator
1.1-10-3
1.1-10-3
8.8-10-3
9.2-10
100
107
(07.8.10-))
(#1.0- 10-3)
(+3.0-10")
(+1.410
2.7 207-10-1
5.5-10-1
5.9-10-1
7.6 7,6-10-3
6.--10-3
Lorenz
(+1.3-10-9)
(+2.2.10-3)
(48.4-1033)
(+1.8-10-9
(+1.0.10-))
system
1.0 100-10-1
2.7-10-1
3.8-10
1.7-10-
3.3-10
(45.8 10
(+3.1.10"3)
(42.6 10
(43.0 10
(+1.1.10
1.2 102-10-1
1.5-10-1
7.9-10-*
9.7 10-2
7.3 7
ETD
(12.9 10-1)
(12.6.107))
(17.3 10-3)
(18.5 10-3)
(16.3 LL6.3-10-3 10-3)
8.3 10-3
8.0 8.0-10-3
1.24 10
8.8 10-3
18-10-3
EEG
(+1.13 10-3)
(t1.11-10-)
(+1.81 10-3)
(49.7-10 10-
(+8.5 (18.5-10-3)
50 (+7.5-10
7.3-10-4
6.4 10-4
4.8- 10-10-4-0
4
Turbulence
(+8.9-
(#9.0 10-1)
(+6.3 10-1)
(#5.9-10 10-1)
set of full trajectories, making it a reliable measure of generalization rather than step-wise prediction accuracy. While the offline method requires optimizing over the entire dataset, COLoKe incrementally adapts its model and reaches lower test error in significantly less time. The gap widens as training progresses, highlighting the advantage of online updates in terms of both speed and generalization. This experiment confirms that COLoKe delivers competitive predictive performance while being substantially more frugal computationally.
4.2 Comparison with online baselines
We now evaluate the benefits of COLoKe against state-of-thh-art online approaches suite of benchmark dynamical systems, commonly used Koopman and machine learning studies, and spanning wide range of complexity.
Datasets. We consider dynamical systems with single attractor (Single Attractor) [Brunton et al.) 2016), two stable spirals and saddle point (Duffing oscillator) (Williams al., 2015). limit cycle (Van der Pol oscillator) [Sinha et al. 2019), and a chaotic regime with a strange attractor
(Lorenz system) [Kostic et al. 2022 We complement them with three real-world datasets: the Electricity Transformer Dataset (ETD) [Zhou al. 2021], the EEG Motor Movement/Imagery Dataset [Schalk al. 2004 Goldberger et al. 2000 and a real turbulence dataset CASES-99 [Earth]
Observing Laboratory [999]. These datasets introduce additional challenges such as noise and
potential distribution shilts. Full details are provided in Appendix B.I
Baselines. We compare COLoKe against the online learning strategies listed in Table While the table includes both online and batch-based methods, our evaluation focuses only those compatible with fully streaming, one-sample-at Saample-at---tinn updates, excluding methods that require access to data batches. For R-EDMD [Sinha et al.] 2019 2023), since the radial basis function dictionary requires the full data to estimate the centers (see Table l). we use polynomial dictionary. And the reconstruction matrix is estimated with the current buffer by regularized pseudo-inverse. This purely online variant

---

is coined Online EDMD (OEDMD), For completeness, we also include variant of COLoKe, referred as OLoKe, in which the conformal update rule is dropped and replaced by a standard strategy; at each time step, the parameters are updated using a fixed number of gradient steps upon receiving new sample. Implementation details are reported Appendix B.2
Metrics. We report two complementary metrics to evaluate model performance. The generalization error measures one-step prediction error on unseen trajectories. The online prediction error, on the other hand, quantifies performance during streaming inference by averaging the training prediction loss over time as new data arrives. For real datasets, models are evaluated on single trajectory and only online prediction eπror are computed. We also report the computation time of each method across all datasets Appendix D
Results. Table reports the performance averaged over 5 random splits of 2000 trajectories for simu- lated datasets, along with standard deviations of the means. Several important observations emerge. First, OEDMI underperforms ODMD, primarily due to two factors: the reconstruction matrix is estimated online, which limits accuracy; the appropriate choice of dictionary (i.e., polynomial degree) is unknown and not adaptively selected. Second, COLoKe, combining flexible neural architecture with a principled update scheme, consistently outperforms all baselines across both synthetic and real-world datasets. In particular, it systematically outperforms its fixed-step counterpart OLoKe, demonstrating the benefit of adaptive model refinement through conformity-baie based updates. On the chaotic Lorenz system. (C)OLoKe achieves an improvement of nearly two orders of magnitude over the baselines, highlighting its effecttivnness capturing highly complex and sensitive nonlinear dynamics. Third, on con-autonomous real-world datasets, COLoKe achieves the best online perfor- mance, highlighting the capacity of conformal control to dynamically adjust to distribution shift. Altogether, these results show that COLoKe combines expressive modeling with principled online adaptation, making it a strong and reliable choice for real-time learning in complex, ever-evolving dynamical environments.
Efficiency of adaptive updates. To further highlight the benefits of conformal- updates, we compared COLoKe to OLoKe trained with fixed update budgets of I, 5, 10, 50, 100, and 150 iterations on the high-dimensionas real EEG dataset. Unlike these
9.5c-03
Online error
O-5
9.0c-03-
baselines,
8.5e-03
O-10
COLoKe does not require manual tuning: its updates are triggered automatically based on conformity. As shown in Figure it achieves both lower online error and substan- tially lower execution time than all fixed-budget variants of OLoKe. This demonstrates that adaptive conformal triggering not only prevents unnecessary computations, but also outperforms any fixed-step strategy without the need to search for an optimal iteration.
O-150
8.0c-03
COLaKe
O-50
0-100
100
150
200
Execution time (s)
Figure 4: Pareto front comparing COLoKe to fixed-step OLoKe variants.
5 Conclusion
We have introduced COLoKe, principled framework for online learning of Koopman-invariant representations, where conformal prediction is repurposed to adaptively trigger updates only when the model becomes temporally inconsistent. By moving beyond fixed-step updates, COLoKe achieves accurate and efficient learning of linear embeddings for nonlinear dynamics when data arrive sequen- tially. This work opens several promising directions for future research. On the theoretical front, our results highlight the need for more comprehensive understanding of conformity- based online learning, with potential relevance well beyond Koopman operator estimation. The main limitation of our analysis is assumption (A3), whose validity remains open although being empirically supported; future work could aim to derive it from first principles for autonomous systems. On the methodologi- cal front, extending COLoKe to non-autonomous systems [Gao et al. 2025) could further broaden its applicability and help unlock practical Koopman learning in real-time, resource-constee resource-constrained and dynamically evolving environments.
10

---

Acknowledgements This work was sponsored by public grant overseen by Auvergne- Raaeee-RRoneAAAeee Alpes region, Grenoble Alpes Metropole and BPIFrance, as part of project I-Démo Région "Green AI".
References
Anastasios Angelopoulos, Emmanuel Candes, and Ryan Tibshirani. Conformal pid control for time series prediction. In A. Oh, T. Naumann, A. Globerson, K. Saenko, M. Hardt, and S. Levine, editors, Advances in Neural Information Processing Systems, volume 36, pages 23047-23074. Curran Associates, Inc., 2023.
Anastasios N Angelopoulos and Stephen Bates. A gentle introduction to conformal prediction and distribution-free uncertainty quantification. arXiv preprint arXiv:2107.0551,, 2021.
Aadyot Bhatnagar, Huan Wang, Caiming Xiong, and Yu Bai. Improved online conformal prediction via strongly adaptive online learning. In International Conference on Machine Learning, pages 2337-2363. PMLR, 2023.
Daniel Bruder, Xun Fu, R. Brent Gillespie, C. David Remy, and Ram Vasudevan. Data-driven control of soft robots using koopman operator theory. IEEE Transactions Robotics, 37(3);948-961, 2021.
Steven L. Brunton, Bingni W. Brunton, Joshua L. Proctor, and J. Nathan Kutz. Koopman invariant subspaces and finite linear representations of nonlinear dynamical systems for control. PLOS ONE, 11(2):e0150171, February 2016.
Steven L. Brunton, Marko Budišié, Eurika Kaiser, and J. Nathan Kutz. Modern koopman theory for dynamical systems. SIAM Review, 64(2):229-340, 2022.
Marko Budisić, Ryan Mohr, and Igor Mezić. Applied koopmanism. Chaos: An Interdisciplinary Journal of Nonlinear Science, 22(4), 2012.
Stéphane Chrétien, Ben Gao, Astrid Thébault Guiochon, and Rémi Vaucher. Leveraging the power of signatures for the construction of topological complexes for the analysis of multivariate complex dynamics. In International Conference on Complex Networks and Their Applications, pages 283-294. Springer, 2023.
Stéphane Chrétien, Ben Gao, Jordan Patracone, and Olivier Alata. Using signatures and koopman operator to learn non-lincar dynamics. In International Conference on Geometric Science of Information, pages 83-92. Springer, 2025.
Nelida Črnjarić-Zic, Senka Maćešić, and Igor Mezić. Koopman operator spectrum for random dynamical systems. Journal of Nonlinear Science, 30(5):0007-2566, 2020.
UCAR Earth Observing Laboratory. Cooperative atmospheric surface exchange study (cases-99), 1999. Dataset.
Ben Gao, Jordan Patracone, Olivier Alata, and Stéphane Chrétien. Apprentissage incrémental de l'opérateur de koopman pour systèmes non-autonomes via prédiction conforme. In GRETS/ XXXe Colloque Francophone de Traitement du Signal et des Images, 2025.
Isaac Gibbs and Emmanuel Candes. Adaptive conformal inference under distribution shift. In
M. Ranzato, Beygelzimer, Y. Dauphin, P.S. Liang, and J. Wortman Vaughan, editors, Advances in Neural Information Processing Systems, volume 34, pages 1660-1672. Curran Associates, Inc., 2021.
Ary L. Goldberger, Luis A. N. Amaral, Leon Glass, Jeffrey M. Hausdorff, Plamen Ch. Ivanov, Roger G. Mark, Joseph F. Mietus, George B. Moody, Chung-Kang Peng, and H. Eugene Stanley. Physiobank, physiotoolkit, and physionet. Circulation, (0123))e215-e200 2000.
Wenjian Hao, Bowen Huang, Wei Pan, Di Wu, and Shaoshuai Mou. Deep koopman learning of nonlinear time-varying systems, Automatica, 159:1 2024.

---

Boya Hou, Sina Sanjari, Nathan Dahlin, Subhonmesh Bose, and Umesh Vaidya. Sparse learning of dynamical systems in RKHS: An operator-theorett approach. In Proceedings of the 40th International Conference on Machine Learning, volume 202 of Proceedings of Machine Learning Research, pages 13325-13352. PMIR, 23 29 Jul 2023.
Yuhong Jin, Lei Hou, and Shun Zhong. Extended dynamic mode decomposition with invertible dictionary learning. Neural Networks, 173:106177, 2024.
Alan A Kaptanoglu, Kyle D Morgan, Chris J Hansen, and Steven L Brunton. Characterizing magnetized plasmas with dynamic mode decomposition. Physics of Plasmas, 27(3), 2020.
Stefan Klus, Feliks Nüske, and Boumediene Hamzi. Kernel-based approximation of the koopman
generator and schrödinger operator. Entropy, 22(7):722, 2020.
Milan Korda and Igor Mezić. Linear predictors for nonlinear dynamical systems: Koopman operator
meets model predictive control. Automaticu, 93: 13:149-160, 2018.
Vladimir Kostic, Pietro Novelli, Andreas Maurer, Carlo Ciliberto, Lorenzo Rosasco, and Massimiliano 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 in Neural Information Processing Systems, 01.32328-323333 2023.
Qianxiao Li, Felix Dietrich, Erik M. Bollt, and Ioannis G. Kevrekidis. Extended dynamic mode decomposition with dictionary learning: A data-driven adaptive spectral decomposition of the koopman operator. Chaos: An Interdisciplinary Journal of Nonlinear Science, 27(10), 2017.
Zhichao Liang, Zixiang Luo, Keyin Liu, Jingwei Qiu, and Quanying Liu. Online learning koopman operator for closed-loop electrical neurostimulation in epilepsy. IEEE Journal of Biomedical and Health Informatics, (7(1))492--3)) 2022.
Kartik Loya and Phanindra Tallapragada. Online learning of koopman operator using streaming data from different dynamical regimes. Onnn,,,, 58(28):90-95, 2024. The 4th Modeling, Estimation, and Control Conference 2024.
Bethany Lusch, J. Nathan Kutz, and Steven L. Brunton. Deep learning for universal linear embeddings of nonlinear dynamics. Nature Communications, 9(1, November 2018.
Jordan Mann and J Nathan Kutz. Dynamic mode decomposition for financial trading strategies. Quantitative Finance, 16(11):1643-1655, 2016.
Alexandre Mauroy, Y Susuki, and Igor Mezic. Koopman operator in systems and control, volume 484. Springer, 2020.
Majid Mazouchi, Subramanya Nageshrao, and Hamidreza Modares. Finite-time koopman identifier: A unified batch-online learning framework for joint learning of koopman structure and parameters. Journal of Machine Learning Research, 24(336):1-35, 2023.
Samuel E. Otto and Clarence W. Rowley. Linearly recurrent autoencoder networks for learning dynamics. SIAM Journal on Applied Dynamical Systems, 18(1):558-59) 2019.
Yaniv Romano, Evan Patterson, and Emmanuel Candes. Conformalized quantile regression. Advances in neural information processing systems, 32, 2019.
Clarence W. Rowley, Igor Mezic, Shervin Bagheri, Philipp Schlatter, and Dan S. Henningson. Spectral analysis of nonlinear flows. Journal of Fluid Mechanics, 641:115-127, 2009.
G. Schalk, D.J. McParland, Hinterberger, N. Birbaumer, and J.R. Wolpaw. Bei2000: general- purpose brain-computer interface (bei) system. IEEE Transactions Biomedical Engineering, 51 (6):1034-104,, 2004.
Subhrajit Sinha, Sai Pushpak Nandanoori, and Enoch Yeung. Online learning of dynamical systems: An operator theoretic approach, 2019.
12

---

Subhrajit Sinha, Sai Pushpak Nandanoori, and David A Barajas-Solano. Online real-time learning of dynamical systems from noisy streaming data. Scientific Reports, 13(1)222564 2023.
Arun Sai Suggala and Praneeth Netrapalli. Online non-convex learning: Following the perturbed leader is optimal. In Aryeh Kontorovich and Gergely Neu, editors, Proceedings of the 31st International Conference on Algorithmic Learning Theory, volume 117 of Proceedings of Machine Learning Research, pages 845-861. PMLR, 08 Feb-11 11 Feb 2020.
Naoya Takeishi, Yoshinobu Kawahara, and Takehisa Yairi. Learning koopman invariant subspaces for dynamic mode decomposition. Advances in neural information processing systems, 30, 2017.
Vladimir Vovk, Alexander Gammerman, and Glenn Shafer. Algorithmic learning in random world, volume 29, Springer, 2005,
Volodya Vovk Alexander Gammerman, and Craig Saunders. Machine-learning applications of algorithmic randomness. In Proceedings of the Sixteenth International Conference Machine Learning, pages 444 453, 1999.
Christoph Wehmeyer and Frank Noé. Time-lagged autoencoders: Deep learning of slow collective variables for molecular kinetics. The Journal of chemical physics, 148(24), 2018.
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, 25:1307-1346, 2015.
Jin Xu, Baranidharan Mohan, and Yao Xie. Conformal prediction for online systems. In Advances in 2021.
Neural Information Processing Systems, volume 34, pages
Yuanchao Xu, Kaidi Shao, Isao Ishikawa, Yuka Hashimoto, Nikos Logothetis, and Zhongwei Shen. A data-driven framework for koopman semigroup estimation in stochastic dynamical systems. arXiv preprint arXiv:2501.13300 2025a
Yuanchao Xu, Kaidi Shao, Nikos Logothetis, and Zhongwei Shen. Reskoopnet: Learning koopman representations for complex dynamics with spectral residuals. arXiv preprint arXiv:2501.0070) 2025b.
Zhipan Xu and Lijun Zhang. Online non-convex learning in dynamic environments, In The Thirty- eighth Annual Conference on Neural Information Processing Systems, 2024.
Enoch Yeung. Soumya Kundu, and Nathan Hodas. Learning deep neural network representations for koopman operators of nonlinear dynamical systems. In 2019 American Control Conference (ACC), pages 4832 4839. IEEE, 2019.
Hao Zhang, Clarence W. Rowley, Eric A. Deem, and Louis N. Cattafesta. Online dynamic mode decomposition for time-varying systems. SIAM Journal on Applied Dynamical Systems, 18(3): 1586 1686,
Haoyi Zhou, Shanghang Zhang, Jieqi Peng. Shuai Zhang, Jianxin Li, Hui Xiong, and Wancai Zhang. Informer: Beyond efficient transformer for long sequence time-series forecasting. In Proceedings of the AAAI conference on artificial intelligence, volume 35, pages 11 11106-11115, 2021.
13

---

NeurIPS Paper Checklist
1. Claims
Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Answer: [Yes]
The main claims presented in the abstract and introduction are accurate and well-aligned with the paper's contributions and scope. They clearly summarize the method- ological innovation (COLoKe), its theoretical guarantees, and empirical performance, all of which are substantiated in the subsequent sections.
Justification:
Guidelines:
The answer NA means that the abstract and introduction do not include the claims made the paper.
The abstract and/or introduction should clearly state the claims made, including the contributions made the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.
The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize other settings.
is fine include aspirational goals as motivation as long is clear that these goals are not attained by the paper.
2. Limitations
Question:
Does the paper discuss the limitations of the work performed by the authors?
Answer: [Yes]
The theoretical foundation of COLoKe is established through Theorem 3.3 whose underlying assumptions are discussed following its statement. The primary limitation lies in the validity of Assumption (A3), which bounds the cumulative conformity thresholds. While this condition is empirically supported, its theoretical justification remains an open question and highlighted in the conclusion as key direction for future work.
Justification:
Guidelines:
The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.
The authors encouraged to create separate "I "Limitations" section in their paper.
The paper should point out any strong assumptions and how robust the results to violations of these assumptions (e.g.. independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.
The authors should reflect the scope of the claims made, e.g., if the approach was only tested on a few datasets or with few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.
The authors should reflect the factors that influence the performance of the approach. For example, facial recognition algorithm may perform poorly when image resolution low or images are taken in low lighting. Or speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.
The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.
If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.
While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, worse outcome might be that reviewers discover limitations that aren't acknowledged the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an impor- tant role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.
14

---

3. Theory assumptions and proofs
Question: For each theoretical result, does the paper provide the full set of assumptions and complete (and correct) proof?
Answer: [Yes]
Justification: All assumptions underlying Theorem 3.3 are explicitly stated within the theorem. A detailed proof is provided the appendix. Hach assumption is discussed in the main text to clarify its practical relevance and interpretability.
Guidelines:
The answer NA means that the paper does include theoretical results. All the theorems, formulas, and proofs in the paper should be numbered and cross- referenced.
All assumptions should be clearly stated or referenced the statement of any theorems.
The proofs can either appear in the main paper or the supplemental material, but they appear in the supplemental material, the authors are encouraged to provide short proof sketch to provide intuition.
Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix supplemental material.
Theorems and Lemmas that the proof relies upon should be properly referenced.
4. Experimental result reproducibility
Does the paper fully disclose all the information needed to reproduce the main ex- perimental results the paper to the extent that affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided not)?
Question:
Answer: [Yes]
Justification: A dedicated section in the appendix provides all necessary information for reproducing the experiments, This includes a precise description of the dynamical systems used, how the datasets are simulated, and full disclosure of model hyperparameters (c.g.. neural network architecture, dictionary choice) and training procedures (e.g., learning rate, update strategy).
Guidelines:
The answer NA means that the paper does not include experiments. If the paper includes experiments, a No answer to this question will not perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.
the contribution is dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable.
Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is a novel architecture, descríbing the architecture fully might suffice, or if the contribution is specific model and empirical evaluation, it may be necessary to either make possible for others to replicate the model with the same dataset, or provide access to the model. In general. releasing code and data is often one good way to accomplish this, but reproducibility can also be provided instructions for how to replicate the results, access to a hosted model (e.g.. in the case of a large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed.
detailed
While NeurlPS does not require releasing code, the conference does require all submis- sions to provide some reasonable avenue for reproducibility, which may depend on nature of the contribution. For example
(a) If the contribution is primarily a new algorithm, the paper should make it clear how reproduce that algorithm.
(b) If the contribution is primarily
model architecture, the paper should describe the architecture clearly and fully.
(c) the contribution a new model (e.g., a large language model), then there should either be way to access this model for reproducing the results or way to reproduce the model (e.g., with an open-source dataset instructions for how to construct the dataset).
15

---

(d)
The authors should provide scripts to reproduce all experimental results for the new reproducible, they We recognize that reproducibility may be tricky in some cases, in which case authors are welcome describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results.
5. Open access to data and code
Question: Does the paper provide open access to the data and code, with sufficient instruc- tions faithfully reproduce the main experimental results, as described in supplemental material?
Answer: [Yes]
Justification: All datasets and code will be made publicly available upon acceptance and are shared via anonymous repository during the review process.
Guidelines:
The answer NA means that paper does not include experiments requiring code.
Please see the NeurlPS code and data submission guidelines (https://nips.. cc/
wublic/guides/CodeSubinn wwtpasummsssssoooooiiooeoo onPolicy) for more details.
While we encourage the release of code and data, we understand that this might not be possible, "No" an acceptable answer. Papers cannot be rejected simply for including code, unless this is central to the contribution (e.g., for new open-source benchmark).
The instructions should contain the exact command and environment needed to run to reproduce the results. See the NeurIPS code and data submission guidelines (https
//nips. craips.cc/puulic//uiiae guwp///ieee//eee/ Codeeubmii ssi ssionPolicy for more details.
The authors should provide instructions on data access and preparation, including how access the raw data, preprocessed data, intermediate data, and generated data,
proposed method and baselines. If only subset of experiments should state which ones are omitted from the script and why.
At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).
Providing as much information as possible supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted.
6. Experimental setting/details
Question: Does the paper specify all the training and test details (e.g., data splits, hyper- parameters, how they were chosen, type of optimizer, etc.) necessary understand the results?
Answer: [Yes]
Justification: All training and testing details are specified in the supplementary material. This includes the data splitting strategy (e.g., 5-fold random train-test splits), the choice and tuning of hyperparameters, and the optimization settings such as learning rates, update strategies, and stopping criteria. These details provided separately for each model to ensure full transparency and facilitate replicability.
Guidelines:
The answer NA means that the paper does not include experiments.
The experimental setting should be presented in the core the paper to a level of detail that is necessary appreciate the results and make sense of them.
The full details can be provided either with the code, in appendix, as supplemental material.
7. Experiment statistical significance
Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
Question:
Answer: [Yes]
16

---

Standard deviations of the means are reported alongside average performance in Table 2 based on five random train-test splits.
Justification:
Guidelines:
The answer NA means that the paper does include experiments.
The authors should answer "Yes" if the results are accompanied by error bars, confi- dence intervals, statistical significance tests, least for the experiments that support the main claims of the paper.
The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions).
The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.)
The assumptions made should be given (e.g.. Normally distributed errors).
It should be clear whether the error bar is the standard deviation the standard error of the mean.
It is OK to report 1-sigma eгror bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have 96% CI, if the hypothesis of Normality of errors is not verified.
For asymmetric distributions, the authors should be careful not to show in tables figures symmetric error bars that would yield results that are out range (e.g. negative error rates).
If error bars are reported in tables or plots, The authors should explain the text how they were calculated and reference the corresponding figures or tables in the text.
8. Experiments compute resources
For each experiment, does the paper provide sufficient information on the com- puter resources (type of compute workers, memory, time of execution) needed reproduce the experiments?
Question:
Answer: [Yes]
Justification: While precise hardware specifications are not critical for reproducing our results, the paper clarifies that the proposed online learning framework computationally lightweight. Most experiments are conducted on well-known dynamical systems with modest data sizes, and training can be performed efficiently on any consumer-gres -aaude laptop. Nonetheless, our hardware specifications (NVIDIA RTX 2000 ADA GPU) are provided in main content of the paper, and timing comparisons are reported when relevant, notably in Section 4.1
Guidelines:
The answer NA means that the paper does not include experiments.
The paper should indicate the type of compute workers CPU GPU, internal cluster, or cloud provider, including relevant memory and storage.
The paper should provide the amount of compute required for each of the individual experimental runs as well estimate the total compute.
The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary failed experiments that didn't make into the paper).
9. Code of ethics
Question: Does the research conducted the paper conform, in every respect, with the NeurlPS Code of Ethics https://neurips: http:///wwrppppocccpppleeeeeeeee httpp////p///ctcccccciieeeeeeele
Answer: [Yes]
Justification: The research adheres to the NeurlPS Code of Ethics. It involves only synthetic and publicly available datasets, poses no foresecable harm to individuals or communities, and promotes transparency, reproducibility, and scientific rigor.
Guidelines:
17

---

The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.
If the authors answer No, they should explain the special circumstances that require deviation from the Code of Ethics.
The authors should make sure to preserve anonymity (e.g.. there a special consid- eration due to laws or regulations in their jurisdiction).
10. Broader impacts
Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
Answer: [NA]
Justification: Given the fundamental and theoretical aspects of the contribution, do not see immediate negative impacts. Positive impacts could be considered from improving the prediction of complex systems.
Guidelines:
The answer NA means that there is no societal impact of the work performed.
If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact.
Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.
The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.
The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.
If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition attacks, mechanisms for monitoring misuse, mechanisms to monitor how system learns from feedback over time, improving the efficiency and accessibility of MI..
11. Safeguards
Does the paper describe safeguards that have been put in place for responsible release of data models that have high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
Question:
Answer: [NA]
Justification: The paper uses datasets corresponding to benchmark dynamical systems commonly used in the Koopman and machine learning literature. Given the fundamental aspect of the contribution, do not see immediate misuse.
Guidelines:
The answer NA means that the paper poses no such risks.
Released models that have high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.
Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.
We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make best faith effort.
18

---

12. Licenses for existing assets
Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and the license and terms of use explicitly mentioned and properly respected?
Question:
Answer: [Yes]
The datasets and setups coming from other papers are cited in the description of the experiment in the main body and in the Appendix [. The models used in the experimental evaluation are cited the introduction as well as in Table
Justification:
Guidelines:
The answer NA means that the paper does use existing assets.
The authors should cite the original paper that produced the code package or dataset.
The authors should state which version of the asset is used and, if possible, include URL.
The name of the license (e.g., CC-BY 4.0) should be included for each asset. For scraped data from a particular source (e.g., website), the copyright and terms of service of that souree should be provided.
assets are released, the license, copyright information, and terms use in package should be provided. For popular datasets, paperswi thcode. com datasets has curated licenses for some datasets. Their licensing guide can help determine the license of dataset.
For existing datasets that re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided.
If this information is not available online, the authors are encouraged to reach out to the asset's s creators.
13. New assets
Question: Are assets introduced in the paper well documented and is the documentation provided alongside the assets?
Answer: [Yes]
Justification: All datasets and code will be made publicly available upon acceptance and are shared via anonymous repository during the review process. Minimum documentation is provided to ensure usability, including dataset descriptions and relevant implementation details.
Guidelines:
The answer NA means that the paper does not release new assets.
Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc.
The paper should discuss whether and how consent was obtained from people whose asset is used.
At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.
Crowdsourcing and research with human subjects
Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well details about compensation (if any)?
Answer: [NA]
Justification: The paper standard datasets/heenhhmark available from the Dynamical Systems and Machine Learning literature.
Guidelines:
The answer NA means that the paper does not involve crowdsoureing nor research with human subjects.
19

---

Including this information in the supplemental material is fine, but if the main contribu- tion of the paper involves human subjects, then as much detail as possible should be included in the main paper.
According to the NeurIPS Code of Ethies, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.
15. Institutional review board (IRB) approvals or equivalent for research with human subjects
Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
Answer: [NA]
Justification: The paper uses standard datasets/benchhmrrk available from the Dynamical Systems and Machine Learning literature.
Guidelines:
The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.
Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.
We recognize that the procedures for this may vary significantly between institutions and locations, and expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution.
For initial submissions, do not include any information that would break anonymity (if' applicable), such the institution conducting the review.
16. Declaration of LLM usage
Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods this research? Note that if the LLM is used only for writing, editing, formatting purposes and does not impact the core methodology, scientific rigorousness, or originality the research, declaration not required.
Answer: [NA]
Justification: The core method development in this research does not involve LLMs.
Guidelines:
The answer NA means that the core method development in this research does not involve LLMs as any important, original, non-standard components.
Please refer to our LLM policy httppp////errrsss.cc///ffeeeeeee for what should or should not be described.
20

---

A Proof of Theorem 3.3
For the reader's convenience, restate Theorem below.
Theorem
3.3 Let (0e. Kt) be the parameters produced by Algorithm and let (0,, K2)
(r. (1 K) denote any dipendepndeett optimal model minimizing the loss at step Further
assume:
(AI) Each Li is L-smooth with 1 (O, & B; (A2) The oracle path has bounded total variation and squared variation:
T
VT = 11000 (2 < co, ST:= Σ (()) < oo;
(A3) The conformity thresholds satisfy Et= at S (ah((T) for some sublinear, nonnegative, nondecreasing function h;
Then the dynamic regret satisfies: Et= (G, K.) - Li(0) K;)) - o (ah((T) + Vr Sr).
Proof. Since le- (Ot, KI)
Kz) S Gr, we have
w(w 1)
Ki) = Σ
(t,, Ki) S [t 4- (0,, Kτ) 2
2 90
(s,))EI,
Let 10(+++)) then we have:
Li(0, - C, K;) c 91 - L, Ki).
Let us define the error:
4 == c. q: L Kī).
We decompose E, as:
Et = c 9:
(1 Ki+1) + Lo Ki-1) C,(0,KK))
(n)
(6)
Step By assumption (A3),
T Σ (c qi - L, (0 Se.
Σ ae S O(ah(T)).
t=1
Step b: We now bound the drift term due to the oracle movement.
Let 2 == (0, K2) and 2²411 = (0%+1 Ki.)). Since L, is L-smooth with |/V S B, we use the standard smoothness bound:
(e(+)+1) (((2,) - (VL((2)), 2441 - - +
2 | 2
- [+,'( || - = + L
|1ii41 - 27112.
Using [/V[(= < B, define A, || =ī'l.
fr(zf41) Lx(2)) & BA, + 1.2
2 A.
Summing over t from 1 to T:
T
Σ - Ca(=)) S > A, +
Σ
A BV + マ
L
- -
2
1=1
Combining both contributions:
T
(e,, K.) L, K;)) ((ah(T)))))))))())))())))))())() + OV7 S7) = ((ah(T) + Vr + ST).
21

---

B Experimental settings
B.1 Datasets and metries
B.1.1 Simulated data
We evaluate performance on four synthetic datasets generated by solving ordinary differential equations (ODEs). For each dynamic, simulate 2000 trajectories and construct 5 random train-test splits { ) For each split, models are trained using the training trajectories while computing the online prediction
EK
Σ
Σ 1
Model4-1(x{_11)
(12)
Train
-
t=t+11
After the training, models are evaluated on the held-out trajectories to compute generalization error
Ek
24
(0)
Modelr(z
(13)
tast
iE
Resuls are reported in Table as averages accross the five splits, along with the standard deviation of the means, i.e.,
k= (Ek- 8)2
(14)
55
k=1
and
5 {
5 km 5-1
(Ex E)2
Ek
(15)
5
55
1=
Datasets are detailed below.
Single Attractor. This system, studied in depth in Section 4.1 and also considered in Section 4.2
is a simple 2D ODE whose dynamic converges to known attracting manifold [Brunton et al.) 2016
2022]. Trajectories are simulated via odeint from SciPy with 100 time steps, equally spaced with step size At 0.01, using initial conditions sampled uniformly from the domain [-2, 2/a The continuous- time eigenvalues reported Section 4.TJare are obtained from the estimated discrete-ting ones via the transformation À = log( (adss) ) At. The ground-truth Koopman eigenfunctions for this
system are given by 1 T2 2- x1. P2
Ii, and y3 = x7.
Duffing Oscillator. This system appears in the study of nonlinear oscillations and serves as a classical benchmark for testing methods in dynamical systems. More specifically, the dynamics follow:
(16) where the parameters 8, 3, and a represent the damping coefficient, the linear stiffness, and the nonlinear stiffness respectively. Here, we choose the parameters = 0.5, 3 = -1, and = 1. following the setting used in [Williams al., 2015 Otto and Rowley, 2019]. This nonlinear system exhibits two stable spirals at = +1, à and saddle at the origin. Trajectories are simulated using odeint, each consisting of 100 time steps with fixed interval At 0.025. Initial conditions are sampled uniformly from the domain 2|
ä - u(B pru?),
-1,
Van der Pol Oscillator. self-sustained oscillations with a stable limit cycle. Its dynamics are governed by the second-order differential equation:
The Van der Pol oscillator is elassical nonlinear system that exhibits
U,
(17)
(1(1 x2 u.
(18)
where 0 controls the nonlinearity and damping strength. We follow the setup in Sinha et al. [019] and set f 0.2. Trajectories are simulated using odeint, with 100 time steps and fixed step size of At = 0.01. Initial conditions are sampled uniformly from the square domain [-4, 4
22

---

Lorenz System. The Lorenz system is classical example of a chaotic dynamical system, originally developed model atmospheric convection, Its dynamics are governed by the following system of nonlinear differential equations:
à = a u) 0 u(p w) U,
(19)
(20)
d = uv Bw.
(21)
where we have chosen o = 10, p p= = 28, and 3 8/3, commonly studied parameter regime known to induce chaotic behavior which was also considered in [Kostic et al.) [022). We simulate the trajectories using RK45, with time step Al = 0.01 over 500 steps. Initial conditions are sampled uniformly from the cube [-10, 10]² centered at the origin.
B.1.2 Real data
We also benchmark on three real datasets. Since the models now evaluated on a single trajectory, we are only interested on online prediction error. Let the online prediction error at time step /. be err: || (x2-e)--(222-)]1 We report in Table 2 the temporal mean of errt
1
CTT
erts
(22)
to
t=to+1
and the standard deviation errt
T
Ōerr
Σ - (rr) t=
(23)
T to
test on ETTh1 which is part of the Electricity Transformer Dataset (ETDataset introduced by Zhou et al. [2021]. ETThl contains data recorded at hourly intervals for roughly two years from an electricity transformer station. The dataset consists of a single trajectory
ETD *
features: HUFL (High Use Frequency Load), HULL (High Use Low Load), MUFL (Medium Use Frequency Load), MULL (Medium Use Low Load), LUFL (Low Use Frequency Load), and LULL (Low Use Low Load). In our setting, retain 200 time steps and aim to learn the transformer's load profile as a dynamical system.
EEG
To assess the COLoKe's performance high-dimensional data, evaluate it on electroen- cephalogram (EEG) recordings with 64 channels from the PhysioNet EEG Motor Movement/Imager We the recording of Subject I in the eyes open condition. The original signal consists
dataset | of one minute of data sampled at 160 Hz. We downsample it to 16 Hz, resulting in trajectory of dimension 64 and length 976.
CASES-99 To further evaluate COLoKe on real-world atmospheric turbulence, use data from the CASES- Co Atmospheric Surface Exchange Study 1999) field experiment. The CASES-99 -9999999----9 examined boundary-layer turbulence using high-frequency measurements from instrumented towers and remote sensing systems. our setting, we use the recordings from the 55m tower. Specifically, time steps. This yields a trajectory of dimension d = 3 and length T 500.
extract the three-dimensional wind components v, and and retain 500
Remark B. For all datasets, data prior to time to are used to initialize the model parameters, including those of the Conformal Pl control for COLoKe (Angelopoulos et al., 2023). the case of synthetic datasets, we set to T/5. resulting in to 20 for the single attractor, Duffing oscillator, and Van der Pol oscillator, and to 100 for the Lorenz system. For all real datasets, we use to = 100.
http://gihhh..cmm///aaaayyyyy/.
https//wwww.oll.uerccccfhlelddd http:///wwweedtoooopee//ssss////
23

---

B.2
Baseline Models
This section details the baseline models used in our experiments, including their initialization, neural network architectures (when applicable), and online training procedures.
Among the various models considered, the most commonly used in the literature ODMD and our variant OHDMD (also known as RR-EDMD), which we briefly recall below.
Online Dynamic Mode Decomposition (Zhng al. 2019 incrementally estimates a linear model from sequential data, enabling efficient updates without storing the entire dataset. Given a trajectory {2, ... Fta available to time to for model initialization. Let X [x), P2.............. Fto- and Yto- = ................................ Utt, The matrix Kt is computed by
ODMD.
Ki = Yi, X
(Xto (Xto X
(24)
Let
Qi = Yig x for and Pro (X X x
(25)
then Kig = Qio Pto We start the online learning when observation 22 for & + arrives:
Q: = Q--1+ +2,2 and P. PT 21-10
(26)
The matrix K, = QtPr is obtained by inverting P.- using Sherman Morrison formula
Pe-1Wt-I0((PPP P (PT) x1-127-1) P 1 PL-T--1
(27)
This online formulation allows for recursive updates of K2 and yields efficient and memory-light algorithm for learning linear dynamics real time. The readers are referred to [Zhang et al.) 2019 for comprehensive explication of ODMD,
OEDMD. We first present Recursive EDMD (Sinha et al.
2019 and point out its limitations for
pure online setting, in order introduce the variant OEDMD. Choose fixed dictionary t :: X' - R.
(x) = (0) (2(x), ,,))))
(28)
Let X (x1). ((x2),.... (x), and Yi, (x2)) F(xs), ((r,,)), ). then the initial-
ization of R-EDMD follows the equations and (25). The recursive updates of K, are given by
equations (26 and (27 To get predictions in the state space X, one should solve least squares problem
min
1 - C$(xx)11
Typically, this involves the storage of all historical data. Therefore, we propose calculate the linear reconstruction matrix C' using the current buffer D;;
min
| - CD(xx))
(30)
A=t-w
Since the modified problem may not have closed form solution due to the buffer size, we use regularized pseudo-inverse to efficiently update C. Rewrite the problem with Ridge regularization
(p = 10-6 for all experiments)
min
||
C*(x))1*+++++000000 C
(31)
k=--w
Let Z [ mt] and Φ { (II=-w),..) (xx+)], then have the closed form solution
C = 2Φ ( pl)
(32)
In the original work (Sinha et al. 2019]. the authors used Radial Basis Functions (RBF) as the fixed dictionary. However, to estimate informative centers for RBF, needs to have access to the
24

---

full trajectory up to time T. When estimating the centers with trajectory up to time ty, the model gives poor results on all datasets for both metrics. Therefore. we choose polynomial dictionary of degree 2, which provides a lifted representation dimension comparable to other baseline models. Increasing the degree beyond 2 offers no significant performance gain. On the contrary, degrades the performance on the real dataset and results in substantially higher computational costs.
We now describe the deep models used in our experiments. All models are trained using the AdamM optimizer with learning rate of 10-3, both during parameter initialization and online training. The initialization phase consists of 4000 epochs for synthetic datasets and 5000 epochs for the real-world dataset. The neural network architectures used in each model are detailed below.
The detailed presentation of COLoKe can be found in Section The neural network is fully connected with architecture {d, 32, 16, 8, m-d} for synthetic datasets and {d, 64, 32, 16, m m-mmmmll. for the real dataset. The dimension of lifted representation
COLoKe.
is chosen to be d - [d/2] for all experiments. Model parameters and initial conformity threshold are initialized with ( ... , as already discussed. The initial threshold 1 is set to be 1 - quantile of the set of scores (sw+1:-.. Stg computed with the initialized model. The model parameters (0,, K2) are updated online according Algorithm 1 For all synthetic datasets, the hyperparameters for Conformal PI procedure are a = 0.5, = 0.1, Cent = 5, and we set Ca = 10 for the real dataset. Note that the coverage guarantees [Angelopoulos et al. 2023 hold for any value of y > O. We set y 0.1 in our experiments as in Angelopoulos et al. 2023 although one may implement adaptive strategies
in [Bhatnagar et al. 2023]. The choice of should reflect the trade-off between computational efficiency and predictive accuracy, depending on the requirements of the application. When a = 1, accuracy is prioritized over speed, whereas for < computational speed more important. In our experiments, we selected a = 0.5 as a balanced compromise between these two objectives.
OLoKe.
The only difference between COLoKe and OLoKe is the online training strategy. For every new buffer D., OLoKe performs fixed number of iterations.
OnlineAE. We implement the model in Liang al. 2022 The original work tackles the prob- lem of Model Predictive Control (MPC). We aim only to perform online learning of dynamics. For synthetic datasets, the encoder architecture is (d. 32, 16, 8, m) and the decoder architecture {m, 8, 16, 32, d}. For the real dataset, the encoder architecture is {d, 64, 32, 16, m] and the decoder architecture is (m, 64, 32, 16, d). The dimension of lifted representation m is chosen be d [d/2] Thus, the model architecture of OnlineAE aligns with COLoKe and OLoKe. The loss function at time is defined as
LD,D,, L1, V1 K,)
Σ k
M (x 2 1 1| o P2(xx) - ==
prediction loss
autoencoding loss
+
K, K, (x - (e (1xx)1
lifted predictioo loss
where o is the encoder and V, is the decoder. The online training strategy consists of fixed iterations with Nuer 100 for synthetic datasets and Niter 500 for the real dataset, which aligns with OLoKe.
C Adaptive update behaviors of COLoKe
We provide supplementary quantitative and qualitative analyses concerning the update frequency of COLoKe. Specifically, we report key summary statistics (see Table 33 that characterize its update behavior and relate it to properties of each dynamical system (sce Appendix B.I.. These statistics include:
1. the number and frequency of update triggers
2. the average interval between two consecutive triggered updates
3. the maximum interval between two consecutive triggered updates.
25

---

Table 3: Update statistics.
Dataset
Total steps
Triggers
Percentage
Avg. interval
Max interval
Single attractor
80
37
0.46
2.16
8
Duffing oscillator
80
38
0,48
2.08
6
VdP oscillator
80
40
0.50
2.00
9
Lorenz system
400
192
0.48
2.07
10
Overall, the proportion steps triggering updates remains consistently close to 48%, but the temporal structure of these updates varies appreciably across systems. This variation illustrates how the conformal triggering mechanism adapts to the underlying stability and complexity of each dynamical regime. In summary:
More iterations are needed initially, but once near the attractor, the model requires fewer corrections.
Single attractor:
Duffing oscillator: Updates are more uniformly distributed, consistent with moderately chaotic behavior and frequent small deviations.
VdP oscillator: Updates tend to cluster, with longer stretches of stability that align with the smoother, low-variability regimes of this system.
Lorenz system: Many updates occur early on due to the highly unstable transient phase, followed by sparser updates as the model adapts.
These behaviors emerge naturally from our conformal- triigerin mechanism, with no dataset-specific tuning heuristics required, highlighting its adaptability across diverse dynamical settings.
D Computation time comparison
To complement the results reported in Table 2 we provide a full comparison of execution time (in seconds) across all datasets. Table shows that COLoKe consistently outperforms neural network Raselines-OnlineAE and OLoKe often by substantial margin. As expected, the purely linear ODMD remains the fastest method overall, but lacks the capacity learn meaningful nonlinear embeddings. OEDMD can approximate nonlinear observables, but it relies on predefined dictionary, which limits its expressivity and scalability. As result, these methods underperform compared to COLoKe (cf. Table (2. Despite its reduced runtime, achieved through the conformal-triggerion mechanism which prevents unnecessary updates, COLoKe maintains best predictive accuracy, highlighting its favorable cost perfoermromm trade-off.
Table 4: Computation time (in seconds) of each method across all datasets.
Dataset
ODMD
OEDMD
OLoKe OnlineAB
COLoKe
Single attractor
1.01
42.25
27.08
19.25
11.50
Duffing oscillator
1.03
43.36
26.77
19.23
9,72
VdP oscillator
1.04
44.06
27.00
19.60
10.94
Lorenz system
5.42
470.53
130.16
266.25
107.96
ETD
0.10
0.52
31.46
22.25
13.31
EEG *
0.03
0.50
4.94
16.55
3.67
Turbulence
*
0.01
0.22
22.83
16.15
11.43
26


Reviews:
[{"review_id": "IFtH4LOhz2", "reviewer": "Reviewer_eTLo", "summary": "This paper proposes an extension of online learning for Koopman embeddings to a conformal online learning framework. The approach is validated on synthetic tasks as well as against several online learning baselines.", "questions": "Can the authors clarify the intuition behind the construction of the prediction score set?\n\nThe paper claims that the prediction set $C_t$ can be used to enhance the interpretability of the model. Have the authors considered any supporting empirical evidence showing how $C_t$ can enhance the interpretability of the model?", "limitations": "The performance gains demonstrated in this paper appear to be largely limited to single-attractor systems, raising questions about the method’s generalizability. Furthermore, the novelty of the proposed prediction score set is difficult to assess in the absence of a detailed comparison to existing conformal or online learning methods. A clearer contextualization of the contribution within the broader literature would significantly strengthen the work.", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Quality. The overall technical quality of the paper is adequate. The experiments are sound, though relatively limited in scope.\n\nClarity. The paper is generally clear and well organized. However, the novelty and significance of the prediction score set could benefit from a more detailed explanation.\n\nSignificance. The method shows meaningful improvements on single-attractor systems, but results on broader or more realistic datasets are limited. The improvements in real-world applications are marginal, and the paper does not fully explore where or why the approach might generalize well.\n\nOriginality. The use of conformal prediction in an online Koopman learning context appears novel. However, it is difficult to assess the distinctiveness of the proposed prediction score set relative to prior work in online learning and conformal prediction. The paper would benefit from a clearer articulation of what makes this construction fundamentally new, and why it is needed beyond existing techniques. This could be supported by an expanded related work discussion that positions the contribution more precisely within the literature on conformal prediction and online learning.", "was_revised": true, "final_justification": "The paper is technically strong and good quality. I find the contributions over existing methods on online time-series predictions compelling and novel. This addressed my concerns with the novelty of the proposed method, which the authors addressed at length—borderline accept, low confidence given my limited exposure to literature on online conformal prediction.", "rebuttal": "We would like to thank the reviewer for their fruitful comments. We have carefully addressed each concern below and believe our clarifications reinforce both the novelty and practical relevance of our contribution. We hope the reviewer finds these clarifications helpful in reassessing the significance of our work. We would be happy to answer any concerns left.\n\n- [L1] *\"The method shows meaningful improvements on single-attractor systems, but results on broader or more realistic datasets are limited. [...]\"*\n\n    We thank the reviewer for this comment and would like to clarify that our evaluation covers both single-attractor and more complex multi-attractor (Duffing oscillator) / chaotic (Lorenz system) systems. Our method achieves significant improvements in online error (which is the primary performance metric in online learning) showing **gains of at least one order of magnitude across all five datasets**. This includes both synthetic and real-world settings present in the manuscript. \n\n    To further support the generality of our approach, we conducted additional experiments on two new challenging real-world datasets: (i) EEG recordings with 64 channels from the PhysioNet EEG Motor Movement/Imagery dataset [Dataset 1], and (ii) turbulence measurements from the CASES-99 atmospheric field experiment [Dataset 2]. The online errors, summarized in the table below (with standard deviations in parentheses), confirm that our method remains effective in broader high-dimensional scenarios. These findings will be included in the revised version of the manuscript.\n\n    | Dataset       | COLoKe         | ODMD           | OnlineAE       | OLoKe          | OEDMD          |\n    |---------------|----------------|----------------|----------------|----------------|----------------|\n    | EEG (x1e-3)  | **7.9 (8.5)** | 8.3 (11.3) | 12.4 (18.1) | 9.3 (10.4) | 8.0 (11.1) |\n    | Turbulence (x1e-4)   | **4.7 (5.9)** | 5.5 (7.5) | 6.4 (8.6) | 5.5 (6.1) | 7.3 (8.9) |  \n\n\n- [L2] *\"The use of conformal prediction in an online Koopman learning context appears novel. [...] positions the contribution more precisely within the literature on conformal prediction and online learning.\"*\n\n    We thank the reviewer for this important point. Our central contribution is to **leverage conformal prediction as a statistically principled event-trigger mechanism** for online learning, a combination that, to our knowledge, has not been previously explored. Our method is the **first that uses conformal scores to actively drive online model updates**. In what follows, to clarify the novelty of our contribution, we first contrast our approach with prior work on online conformal prediction, then discuss related advances in online non-convex learning, and finally position our contribution with respect to recent online time-series forecasting methods.\n\n    - Prior works in **online conformal prediction**, such as Adaptive Conformal Inference (ACI) [Ref. 1], Conformal PID Control [Ref. 2] and their variants (see [Ref. 3] and references therein), focus on dynamically calibrating prediction sets to maintain coverage under distribution shift as new data arrive—either assuming a fixed underlying model, or, when the model is also updated, doing so without using conformal prediction to guide the updates. Unlike these methods, which recalibrate prediction sets in an online manner, our approach uses the conformity score as an indicator of model reliability and triggers model updates only when it exceeds a dynamically calibrated threshold.\n\n    - In parallel, a significant milestone in **online non-convex learning** is the work by Suggala \\& Netrapalli [Ref. 4] who achieves optimal regret even under non-convex, adversarial losses, assuming access to an offline oracle. Later work [Ref. 5] on dynamic regret explores variants to handle changing environment. While they address how to update model parameters every round to minimize regret, they do not address when updates should occur and assume updates happen at every step with a randomized prior.\n\n    - Finally, while recent **online time series forecasting methods** have proposed sophisticated adaptation mechanisms [Ref. 6, Ref. 7], they rely on either ad hoc memory-based heuristics or learned change detection modules, and typically involve black-box neural networks. In contrast, our work combines operator-theoretic modeling with conformal calibration to deliver formal guarantees about prediction quality while limiting unnecessary retraining, thus offering a distinct contribution in both methodology and theory.\n\n    In the paper, we chose to focus our discussions on online Koopman operator learning for clarity. Nevertheless, if the reviewer finds it useful, we would welcome the opportunity to include a paragraph in the introduction positioning our work relative to broader literature.\n    \n- [Q1] *\"[...] clarify the intuition behind the construction of the prediction score set?\"*\n\n    Traditionally, conformal prediction aims to construct a prediction set $C_t$ for the next state $x_t$, given a fixed model. In contrast, our perspective is reversed: given $x_t$, we ask which models produce predictions that fall within $C_t$. In this sense, we repurpose the conformal prediction set as a tool to evaluate the model itself, rather than to quantify uncertainty around its output. Importantly, we do not need to explicitly construct $C_t$ as in standard conformal prediction; it is sufficient to evaluate whether the prediction score falls below a threshold. Accordingly, we define the prediction score set as the set of all acceptable scores attainable by Koopman linear models.\n\n- [Q2] *\"The paper claims that the prediction set can be used to enhance the interpretability of the model. [...]\"*\n\n    We respectfully believe there has been a misunderstanding, as **we do not claim at any point that the prediction set enhances model interpretability**. The only mention of interpretability in the paper refers to the structure of the lifted Koopman embedding, where part of the original state is preserved. Our use of conformal prediction is strictly focused on quantifying uncertainty in the online learning setting, and we make no interpretability-related claims in the manuscript.\n\n\nReferences:\n\n- [Ref. 1] Gibbs, I. and Candes, E. Adaptive conformal inference under distribution shift. NeurIPS 2021.\n- [Ref. 2] Angelopoulos, A.N., Candes, E. and Tibshirani, R.J., Conformal PID Control for Time Series Prediction. NeurIPS 2023.\n- [Ref. 3] Ramalingam, R., The Relationship Between No-Regret Learning and Online Conformal Prediction. ICML 2025.\n- [Ref. 4] Suggala A.S. and Netrapalli P., Online Non-Convex Learning: Following the Perturbed Leader is Optimal, ALT 2020.\n- [Ref. 5] Xu Z. and Lijun Zhang L., Online Non-convex Learning in Dynamic Environments, NeurIPS 2024.\n- [Ref. 6] Pham, Q. et al., Learning Fast and Slow for Online Time Series Forecasting, ICLR 2023.\n- [Ref. 7] Lau, YA. et al, Fast and Slow Streams for Online Time Series Forecasting Without Information Leakage, ICLR 2025.\n- [Dataset 1] https://physionet.org/content/eegmmidb/1.0.0/\n- [Dataset 2] https://www.eol.ucar.edu/field_projects/cases-99"}, {"review_id": "djkBtqnVTl", "reviewer": "Reviewer_4zQ8", "summary": "The authors study the problem of approximating nonlinear dynamical systems using linear Koopman operator theory. Specifically, they focus on the online learning setting, where the goal is to continuously learn a finite-dimensional, linear representation of the dynamics from streaming data.\n\nUnlike traditional approaches that use fixed dictionaries of observables, the authors propose to learn the Koopman-invariant embedding using a neural network, which maps the state into a latent space where the dynamics evolve approximately linearly. This deep Koopman embedding includes both a nonlinear component (learned) and the original state itself, ensuring partial interpretability and reconstruction capability.\n\nTo address the challenges of overfitting and computational inefficiency in online learning, the authors introduce a conformal prediction-based update strategy. Instead of updating the model at every time step, they compute a prediction conformity score based on multi-step prediction errors in the lifted space. Updates are triggered only when this score exceeds a dynamically calibrated threshold.\n\nThis threshold is updated using a Proportional-Integral (PI) controller, inspired by control theory, which tracks the frequency of conformity violations and adjusts sensitivity accordingly. This mechanism ensures that the model adapts only when necessary.", "questions": "Can you make additional analysis on how frequently updates had to be done for table 2.\nUpdates are triggered only when this score exceeds a dynamically calibrated threshold. \nI would suggest to make time series visualisation with trajectories of true dynamics, predictions, and vertical lines that indicate when updates to the model had to be done.\n\nCan you try simple benchmark of reservoir computing for online learning and compare it to your approach?", "limitations": "Author clearly state limitations in their paper.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n- simple framework rooted in strong theory\n- very applied\n\nWeaknesses:\n- no big problems", "was_revised": true, "final_justification": "I would stay with my rating of accept. Authors have done additional work in rebuttal phase, that showed me that accept is a good decision. \nOverall good written paper, good baselines, and solid theoretical framework with applications.", "rebuttal": "We would like to sincerely thank the reviewer for their kind evaluation. Their valuable suggestions will greatly improve the quality of our paper. We will make sure to include them in the revised version of the manuscript.\n\n- [Q1] *\"Can you make additional analysis on how frequently updates had to be done for table 2. Updates are triggered only when this score exceeds a dynamically calibrated threshold. I would suggest to make time series visualisation with trajectories of true dynamics, predictions, and vertical lines that indicate when updates to the model had to be done.\"*\n\n    We thank the reviewer for these great suggestions. Unfortunately, we are unable to share such illustrations during the rebuttal phase, as including images is not permitted. We will include them in the camera-ready version. \n    Nevertheless, we provide below additional quantitative and qualitative analyses on the frequency of model updates. While plots would offer more visual insights, we instead report key statistics summarizing the update behavior. Across all datasets in Table 2, we report: (i) the total number of steps, (ii), the number of update triggers, (ii) the percentage of time steps at which a trigger occurred, as well as the average (iv) and largest (v) time steps between two consecutive triggered updates.\n\n    | Dataset       | Total steps         | Triggers           | Percentage   | Avg interval        | Max interval         | \n    |---------------|----------------|----------------|----------------|----------------|----------------|\n    | Single attractor   |  80  | 37  | 0.46 | 2.16 | 8  | \n    | Duffing oscillator |  80  | 38  | 0.48 | 2.08 | 6  | \n    | VdP oscillator     |  80  | 40  | 0.50 | 2.00 | 9  | \n    | Lorenz system      | 400  | 192 | 0.48 | 2.07 | 10 | \n    | ETD                | 100  | 49  | 0.49 | 2.02 | 6  | \n\n    While the overall frequency of updates is consistently around 48\\%, the timing and distribution of those updates vary significantly across systems. This variation reflects how the conformal triggering mechanism adapts to the complexity and stability of each system’s dynamics. In short:\n\n    - **VdP oscillator and ETD**: Updates are clustered, with long intervals of stability matching the regular, low-variability regimes of these systems. \n    - **Duffing oscillator**: Updates are more uniformly distributed, reflecting moderate chaotic behavior with frequent small deviations.\n    - **Lorenz system**: Frequent early updates reflect the chaotic and unstable initial phase; they become sparser as the model adapts.\n    - **Single Attractor**:  Initial updates require more iterations, but once near the attractor, the model requires fewer corrections.\n       \n    We will include a more detailed discussion in the revised manuscript to highlight how each dataset’s specific dynamics influence the update behavior. Note that these update patterns emerge naturally from our conformal-triggering mechanism, without requiring any dataset-specific tuning or heuristics, highlighting its adaptability across different dynamical regimes.\n\n\n- [Q2] *\"Can you try simple benchmark of reservoir computing for online learning and compare it to your approach?\"*\n\n    We thank the reviewer for this suggestion. We implemented a benchmark using reservoir computing (RC) across all datasets in Table 2. We used the *reservoirpy* package [Ref. 1] and focused on Echo State Networks as the RC architecture. We performed a grid search on two key hyperparameters: (i) the number of internal (reservoir) nodes chosen from the set \\{10, 15, 20, 50, 100, 200, 500, 800, 1000, 2000, 3000, 4000, 5000\\} and (ii) the learning rate of the Recursive Least Squares algorithm [Ref. 2], chosen from the set \\{1e-3, 1e-4, 1e-5, 1e-6, 1e-7\\}. For each dataset, we report the RC configuration that achieved (i) the lowest online error and (ii) the lowest generalization error. The first line gives the generalization error, and the second line the online error.\n\n    | Dataset           | RC (best online error)       | RC (best generalization) | COLoKe                 |\n    |-------------------|------------------------------|--------------------------------|------------------------------|\n    | Single attractor | 8.1 x 1e-2 (9.0 x 1e-3)          | 3.0 x 1e-3 (3.0 x 1e-4)             | **2.4 x 1e-7 (3.6 x 1e-8)**      |\n    |                   | 2.2 x 1e-5 (1.0 x 1e-5)          | 3.7 x 1e-4 (3.8 x 1e-4)             | **7.6 x 1e-7 (9.6 x 1e-8)**      |\n    |                   |                              |                                |                              |\n    | Duffing oscillator | 9.5 x 1e-2 (1.9 x 1e-3)          | 4.0 x 1e-3 (8.2 x 1e-5)             | **3.1 x 1e-6 (2.3 x 1e-7)**      |\n    |                   | 1.0 x 1e-4 (1.2 x 1e-5)          | 6.7 x 1e-4 (4.1 x 1e-4)             | **7.3 x 1e-5 (1.9 x 1e-5)**      |\n    |                   |                              |                                |                              |\n    | VdP oscillator | 3.9 x 1e-1 (5.3 x 1e-2)          | 3.0 x 1e-2 (4.6 x 1e-4)             | **3.8 x 1e-4 (1.2 x 1e-5)**      |\n    |                   | **9.0 x 1e-5 (3.5 x 1e-5)**          | 3.5 x 1e-3 (2.8 x 1e-3)             | 6.0 x 1e-4 (1.4 x 1e-4)      |\n    |                   |                              |                                |                              |\n    | Lorenz system  | 2.8 x 1e1 (2.3 x 1e0)           | 7.3 x 1e0 (9.1 x 1e-1)              | **6.5 x 1e-3 (1.0 x 1e-4)**      |\n    |                   | 2.6 x 1e-2 (2.2 x 1e-2)          | 4.3 x 1e0 (3.4 x 1e0)               | **3.3 x 1e-3 (1.1 x 1e-4)**      |\n    |                   |                              |                                |                              |\n    | ETD            | 2.3 x 1e-1 (2.8 x 1e-1)          | 2.2 x 1e-1 (1.5 x 1e-1)             | **2.1 x 1e-1 (8.6 x 1e-2)**      |\n    |                   | 1.1 x 1e-1 (3.7 x 1e-1)          | 1.2 x 1e-1 (3.6 x 1e-1)             | **7.3 x 1e-2 (6.3 x 1e-2)**      |\n\n    Across all datasets, COLoKe consistently achieves lower generalization and online error than RC, with the sole exception of VdP oscillator, where RC (best online error) slightly outperforms in online error.\n\n    To provide a point of reference on model capacity, we also computed the number of parameters in COLoKe and derived the number of internal nodes that RC would need to match either the total number of parameters (nodes) or just the number of trainable ones (nodes / trainable), with connection density fixed at 0.1:\n\n    | Dataset       | COLoKe (parameters)        | RC (nodes)         | RC (nodes / trainable) |  \n    |---------------|----------------|----------------|----------------|\n    | Single attractor   |  778  | 77  | 388 |\n    | Duffing oscillator |  778  | 77  | 388 |  \n    | VdP oscillator     |  778  | 77  | 388 |  \n    | Lorenz system      | 835  | 75 |  278 |\n    | ETD                | 3188  | 146 | 531 |   \n\n    In summary, our grid search already includes reservoir sizes up to an order of magnitude larger than those required to match COLoKe's parameter count. These preliminary results indicate that COLoKe consistently achieves higher accuracy while maintaining a significantly more compact model architecture.\n\n    As we are not specialists in reservoir computing, we would gladly welcome the opportunity to explore other RC methods, should the reviewer have specific methods or references in mind.\n\nReferences:\n- [Ref. 1] Trouvain, N., Pedrelli, L., Dinh, T.T. and Hinaut, X. ReservoirPy: an Efficient and User-Friendly Library to Design Echo State Networks. ICANN 2020. \n- [Ref. 2] Sussillo, D. and Abbott L.F. Generating Coherent Patterns of Activity from Chaotic Neural Networks. Neuron 2009."}, {"review_id": "LD2x22wOg8", "reviewer": "Reviewer_vz6L", "summary": "This paper introduces COLoKe (Conformal Online Learning of Koopman embeddings), a framework for learning deep Koopman models for nonlinear dynamical systems in an online/streaming data setting. The method combines the learning of deep Koopman embeddings with an adaptive update mechanism inspired by conformal prediction (CP). Instead of using conformal methods for uncertainty quantification, COLoKe leveragesthe conformity score (error from a multi-step prediction) to measure how consistent the incoming data is with the current Koopman model. A gradient-based refinement of the model is performed only if this score exceeds a threshold. The primary advantage of COLoKe is that it reduces computational cost and prevents overfitting by avoiding unnecessary updates. The authors also provide a theoretical analysis establishing a dynamic regret bound under certain assumptions. Evaluations on several benchmark dynamical systems, demonstrate that COLoKe can achieve high prediction accuracy while requiring fewer updates than non-adaptive or offline methods.", "questions": "1. Can you provide some intuition why $\\sum q_t$ should grow sub-linearly? Under what real-world conditions would it fail, e.g., non-stationarity, turbulence, drift?\n\n1. Can you comment regarding the scalability of the COLoKe framework to higher dimensions (e.g. 2D/3D Navier-Stokes systems, or other complex systems exhibiting chaotic and turbulent behavior)?", "limitations": "yes", "rating": 5, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "*Strengths*\n- The paper is technically well-motivated; reducing unnecessary updates of the Koopman bases is a clear goal, and the approach using conformal prediction is quite plausible. The technique to preventing unnecessary updates is repurposing an existing well-established method rather than reinventing the wheel.\n- The empirical evaluation is a key strength. COLoKe shows strong performance across several benchmarks (Table 2), often outperforming baselines. Crucially, it outperforms OLoKe (an ablation with fixed updates), supporting the main claim that adaptive updates are beneficial.\n\n*Weaknesses*\n- The performance of the COLoKe method may be highly dependent on several hyperparameters (e.g. alpha), but the paper lacks discussion or guidance how to go about choosing such parameters.", "was_revised": true, "final_justification": "This work presents the method COLoKe which is a technically sound and well-motivated method for adaptive updates. Strong empirical evaluation validates the method. I believe it is significant contribution to the community.", "rebuttal": "We would like to sincerely thank the reviewer for their insightful and constructive comments. We are grateful for the opportunity to further strengthen our work by adding these points, which we believe will significantly enrich the discussion around our contributions.\n\n\n- [W] *\"The performance of the COLoKe method may be highly dependent on several hyperparameters (e.g. alpha), but the paper lacks discussion or guidance how to go about choosing such parameters.\"*\n\n    We thank the reviewer for this valuable observation. To guide the choice of our two hyperparameters, we propose adding the following paragraphs after Eq. 3 and Eq. 9, respectively\n    - \"Note that classical guarantees in conformal prediction hold for any value of $\\gamma>0$. In our experiments, we will set $\\gamma=0.1$ as in [Angelopoulos et al., 2023a] although one may implement adaptative strategies as in [Ref 1] and references therein.\"\n    - \"The choice of $\\alpha$ should reflect the trade-off between computational efficiency and prediction accuracy, depending on the requirements of the application. When $\\alpha\\approx 1$, accuracy is prioritized over speed, whereas for $\\alpha\\ll 1$, computational speed is more important. In our experiments, we selected $\\alpha=0.5$ as a balanced compromise between these two objectives.\"\n    \n- [Q1] *\"Can you provide some intuition why $\\sum q_t$ should grow sub-linearly? Under what real-world conditions would it fail, e.g., non-stationarity, turbulence, drift?\"*\n\n    We thank the reviewer for their insightful question. This touches on a key research direction that we plan to explore in future work. To provide some intuition, consider a hypothetical scenario in which the cumulative sum grows linearly, this would imply that the $q_t$ remain roughly constant over time. From a conformal prediction perspective, such behavior suggests that the model's predictive uncertainty does not decrease, even as more observations are collected. In other words, the model fails to learn a representation that generalizes or improves with data. As rightfully pointed, such problematic scenario may correspond to cases involving non-autonomous systems or distribution shifts, or cases where the model is insufficiently trainable or expressive to capture complex dynamics, such as those encountered in turbulent regimes.\n\n- [Q2] *\"scalability of the COLoKe framework to higher dimensions (e.g. 2D/3D Navier-Stokes systems, or other complex systems exhibiting chaotic and turbulent behavior\"*\n\n    We thank the reviewer for this important question. To assess the scalability and robustness of COLoKe in higher-dimensional and complex settings, we conducted additional experiments on two real-world datasets: (i) EEG recordings with 64 channels from the PhysioNet EEG Motor Movement/Imagery dataset [Dataset 1], and (ii) real-world turbulence measurements from the CASES-99 atmospheric field experiment [Dataset 2]. Both datasets are governed by complex dynamics and are inherently noisy, making them well-suited to evaluate the practical utility of our method. The table below reports the online errors, with standard deviations shown in parentheses. These results demonstrate that our approach generalizes well beyond low-dimensional synthetic systems and remains effective in realistic high-dimensional scenarios. We will include these findings in the revised version of the manuscript.\n\n    | Dataset       | COLoKe         | ODMD           | OnlineAE       | OLoKe          | OEDMD          |\n    |---------------|----------------|----------------|----------------|----------------|----------------| \n    | EEG (x1e-3)  | **7.9 (8.5)** | 8.3 (11.3) | 12.4 (18.1) | 9.3 (10.4) | 8.0 (11.1) |\n    | Turbulence (x1e-4)   | **4.7 (5.9)** | 5.5 (7.5) | 6.4 (8.6) | 5.5 (6.1) | 7.3 (8.9) |  \n\nReferences:\n\n- [Ref.1] Bhatnagar, A., et al. Improved online conformal prediction via strongly adaptive online learning. In ICML 2023.\n- [Dataset 1] https://physionet.org/content/eegmmidb/1.0.0/ \n- [Dataset 2] https://www.eol.ucar.edu/field_projects/cases-99"}, {"review_id": "sAzIpfjnyc", "reviewer": "Reviewer_vi1N", "summary": "This paper introduces an event triggering mechanism based on conformal prediction to control the frequency of model updates to achieve an adaptive online Koopman embedding learning. The proposed method performs well on 4 synthetic datasets. However, the introduction of event triggering mechanism in online learning methods lacks novelty. The performance of the proposed method on a real dataset is also marginal and lacks validation on more real datasets. And the comparison methods are not enough and outdated.", "questions": "N.A.", "limitations": "See weaknesses above.", "rating": 4, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths:\n\n1.\tIntroduce conformal online learning as an event triggering mechanism to form an adaptive online Koopman embedding learning method to reduce the frequency of model updates.\n2.\tThe proposed method performs well on 4 synthetic datasets.\n\nWeaknesses:\n\n1.\tThe idea of the proposed conformal online learning for controlling the frequency of model updates is quite similar to the event triggering mechanisms and some online learning methods [1] have proposed similar frameworks to control the frequency of model updates, hence it is not novel. \n\n[1] Guan, M., Wen, C., Shan, M., Ng, C. L., & Zou, Y. (2018). Real-time event-triggered object tracking in the presence of model drift and occlusion. IEEE Transactions on Industrial Electronics, 66(3), 2054-2065.\n\n2.\tFrom the tables, the improvement of the proposed method on the real dataset is marginal and even worse than some comparison methods.\n\n3.\tOnly one real dataset is used, which is not enough.\n\n4.\tAuthors claimed that the introduction of conformal prediction to guide adaptive model updates will significantly improve speed and unlock practical Koopman learning in real-time. But there is no speed comparison between the proposed method and other online Koopman embedding learning methods.\n\n5.\tTo show the improvement of the proposed method clearer, introducing some visualization will be better, for example, introducing visualizations of predicted state space trajectories for each system using different methods and MSE curves during training.\n\n6.\tThe comparison methods are not enough and outdated. Although authors listed several comparison methods in Table 1, only ODMD (2019) and OnlineAE (2022) are used in comparison which are quite outdated and the superiority of the proposed method over these methods is not convincing. Some real-time advanced adaptive Koopman embedding learning methods [2,3] are required to compare.\n\n[2] Singh, R., Sah, C. K., & Keshavan, J. (2024). Adaptive Koopman embedding for robust control of complex nonlinear dynamical systems. arXiv preprint arXiv:2405.09101.\n\n[3] Li, B., Ma, Y., Kutz, J. N., & Yang, X. (2023). The adaptive spectral Koopman method for dynamical systems. SIAM Journal on Applied Dynamical Systems, 22(3), 1523-1551.", "was_revised": true, "final_justification": "I'd like to thank the authors for their response. I've raised the overall score.", "rebuttal": "We would like to thank the reviewer for their detailed and thoughtful feedback. Nevertheless, we respectfully believe that the current rating does not adequately reflect the strength of our contribution since our method is 1) theoretically grounded, 2) empirically validated on multiple benchmark datasets commonly studied in the Koopman learning literature and 3) fully reproducible. We have also carried out new experiments to directly address the reviewer’s concerns, including the addition of two real-world datasets (see W3) and a detailed comparison of execution times (see W4). We hope that our clarifications (see W1, W2 and W6) and additional results provide a more complete picture of our contributions and will convince the reviewer to reconsider their evaluation. If there are additional concerns that contributed to the negative score, we would gladly welcome the opportunity to address them as well.\n\n- [W1] *\"The idea of the proposed conformal online learning [...] is not novel.\"*\n\n    We respectfully disagree. While event-triggered mechanisms themselves are not new, the novelty of our contribution lies in introducing conformal prediction as a **principled and statistically grounded approach** to triggering updates in online learning. Unlike works such as [1], which rely on fixed or heuristic thresholds (see their Eq. 20), our method leverages **adaptive thresholdings derived from conformal scores in a bespoke manner**. This provides coverage guarantees and enforces statistical significance into the triggering process. To the best of our knowledge, this is the first work to leverage the power and flexibility of conformal prediction in the context of event-triggered online learning.\n\n- [W2] *\"From the tables, the improvement of the proposed method on the real dataset is marginal and even worse than some comparison methods.\"*\n\n    This may be a misinterpretation of the results.**Our method achieves the best online error**, which is the primary performance metric in online learning. We believe that the results in the table faithfully reflect the model’s ability to make accurate predictions during learning. For completeness, we also report the generalization error (i.e., post-hoc performance on new trajectories), where COLoKe remains competitive and exhibits significantly lower variance than state-of-the-art methods. This robustness further supports the practical reliability of our approach, especially for real-world datasets where classical assumptions in Koopman operator learning (e.g., autonomous system, absence of distribution drift) no longer hold.\n\n- [W3] *\"Only one real dataset is used, which is not enough.\"*\n\n    We thank the reviewer for this helpful suggestion and have extended our numerical study with **two additional experiments on real-world datasets**: (i) EEG recordings with 64 channels from the PhysioNet EEG Motor Movement/Imagery dataset [Dataset 1], and (ii) turbulence measurements from the CASES-99 atmospheric field experiment [Dataset 2]. The table below reports the online errors, with standard deviations shown in parentheses. These results confirm that our method performs reliably in real-world challenging scenarios. We will include these experiments in the revised version of the manuscript.\n\n    | Dataset       | COLoKe         | ODMD           | OnlineAE       | OLoKe          | OEDMD          |\n    |---------------|----------------|----------------|----------------|----------------|----------------|\n    | EEG (x1e-3)  | **7.9 (8.5)** | 8.3 (11.3) | 12.4 (18.1) | 9.3 (10.4) | 8.0 (11.1) |\n    | Turbulence (x1e-4)   | **4.7 (5.9)** | 5.5 (7.5) | 6.4 (8.6) | 5.5 (6.1) | 7.3 (8.9) |  \n\n    We would also like to note that real-world datasets are still rarely used in the Koopman learning literature, where controlled and well-understood systems are typically favored to analyze the spectral properties of the learned Koopman operator. We welcome the opportunity, prompted by your remark, to help bridge this gap and demonstrate how our method can robustly extend to realistic, high-dimensional settings beyond traditional benchmarks.\n\n- [W4] *\"[...] But there is no speed comparison between the proposed method and other online Koopman embedding learning methods.\"*\n\n    We thank the reviewer for this useful observation. While our original analysis provided limited timing insights (Fig. 2b and, to some extent, appendix C2), we now include a full comparison of execution times (in seconds) across all datasets to complement Table. 2: \n\n\n    | Dataset       | ODMD         | OEDMD            | OnlineAE       | OLoKe          | COLoKe         |\n    |---------------|----------------|----------------|----------------|----------------|----------------|\n    | Single attractor  | 1.01 | 42.25 | 27.08 | 19.25 | 11.50 |\n    | Duffing oscillator   | 1.03 | 43.36 | 26.77 | 19.23 | 9.72 |  \n    | VdP oscillator   | 1.04 | 44.06 | 27.00 | 19.60 | 10.94 | \n    | Lorenz system   | 5.42 | 470.53 | 130.16  | 266.25  | 107.96 | \n    | ETD   | 0.10 | 0.52 | 31.46 | 22.25  | 13.31  | \n\n    These results show that **COLoKe consistently outperforms nonlinear baselines** in speed, often by a wide margin, while maintaining superior predictive accuracy (cf. Table 2). Only the purely linear ODMD is faster, yet unable to learn expressive nonlinear embeddings.\n\n    To further illustrate COLoKe’s efficiency, we compared it to OLoKe trained with different fixed iteration budgets on the real high-dimensional EEG dataset [Dataset 1]:\n\n    |        | COLoKe  | OLoKe 5 | OLoKe 10 | OLoKe 50 |OLoKe 100 | OLoKe 500 |\n    |---------------|----------------|----------------|----------------|----------------|----------------|----------------|\n    | Online error (x1e-3)  | **7.9 (8.5)** | 9.4 (10.4) | 8.8 (9.5) | 8.1 (9.5) | 8.2 (9.6)  | 8.3 (8.8) |\n    | Execution time (s)   | **2.98** | 11.12 | 15.29 | 50.44 | 92.81 | 444.84 |\n\n    This confirms that COLoKe not only avoids unnecessary computation through adaptive updates but also yields the best accuracy–speed trade-off without manual tuning. Its data-driven update strategy offers a principled and efficient alternative to fixed-iteration baselines. In the revised manuscript, **we will include Pareto fronts visualizations** (online error vs. execution time) to clearly illustrate the trade-offs between accuracy and speed across all methods and datasets.\n\n- [W5] *\"To show [...] predicted state space trajectories for each system using different methods and MSE curves during training.\"*\n\n    We thank the reviewer for this valuable suggestion. We will include such illustrations in the final version. Unfortunately, we are unable to share them during the rebuttal phase, as including images is not permitted.\n\n- [W6] *\"The comparison methods are not enough and outdated. [...] Some real-time advanced adaptive Koopman embedding learning methods [2,3] are required to compare.\"*\n\n    We thank the reviewer for this comment and would like to clarify the discussion about related works in two points. \n    - First, mentioning [3] which specifically focuses on the numerical integration of ODEs using the Koopman framework is a bit surprising to us, because pertaining to a different type of application. As for [2] and similar adaptive Koopman embedding methods, they operate in a fundamentally different setting: **full trajectories are available ahead of time and these full trajectories are used to fit the model**. Additionally, their adaptive component is designed specifically for model predictive control (MPC). In contrast, our method performs **online Koopman learning from streaming data**, with no access to future states at any point. The comparison is therefore not just unfair, it concerns a completely different problem formulation, both in terms of assumptions and objectives. In the case where the reviewer would find it helpful, we would be happy to add a discussion of MPC in the paper's introduction.\n\n    - Second, to the best of our knowledge, the only works related to our problem are the ones listed in Table 1. We have not included some of them in our experiments, since they cannot fulfill all the conditions of a purely online learning setting (see the \"Baselines\" paragraph in Section 4.2). Please also note that OEDMD appearing in our experiments is just a reimplementation of R-EDMD [Sinha et al., 2023]. While DMD, EDMD and their online variants may appear dated, they remain widely used due to their strong theoretical foundations and performance guarantees, which serve as reliable baselines withstanding the test of time. \n\nReferences:\n\n- [Dataset 1] https://physionet.org/content/eegmmidb/1.0.0/ \n- [Dataset 2] https://www.eol.ucar.edu/field_projects/cases-99"}]

Author rebuttal:

Meta-review: This paper presents an approach to learning deep Koopman representations of non-linear systems in an online setting. This by itself is not a new goal. However, the authors indicate that it may be desirable to limit the number of model updates (or reoptimization steps) due to reasons concerning computational efficiency and overfitting. To accomplish, the work uses conformal prediction to measure how out of character the state evolution seems to be (based on multi-step prediction) for the current estimate of the model, and thus arrives an adaptive schedule of updating the model. 

Although by no means a theory paper, the work also nicely links the conformal coverage guarantee to dynamic regret. In the rebuttal period, the authors present more experimental results that convinced engaged reviewers to increment their score. In my reading, the single remaining negative review rests on two points: the experiments and benchmarking is limited, and the idea of using online/adaptive learning to trigger updates is not new. For the first, the reviewer has not responded to the authors' additional experimental results presented during the rebuttal. We agree with the second point, however, there's value to grounding these model updates in a principled framework like conformal prediction. 

Overall, given the positive evaluation of this work from all engaged reviewers, we are happy to recommend this paper for acceptance.
Decision: Accept (poster)