Conformal Online Learning of Deep Koopman Linear Embeddings

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.

• [W1] "The idea of the proposed conformal online learning [...] is not novel."

  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.

• [W2] "From the tables, the improvement of the proposed method on the real dataset is marginal and even worse than some comparison methods."

  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.

• [W3] "Only one real dataset is used, which is not enough."

  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.

  Dataset	COLoKe	ODMD	OnlineAE	OLoKe	OEDMD
  EEG (x1e-3)	7.9 (8.5)	8.3 (11.3)	12.4 (18.1)	9.3 (10.4)	8.0 (11.1)
  Turbulence (x1e-4)	4.7 (5.9)	5.5 (7.5)	6.4 (8.6)	5.5 (6.1)	7.3 (8.9)

  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.

• [W4] "[...] But there is no speed comparison between the proposed method and other online Koopman embedding learning methods."

  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:

  Dataset	ODMD	OEDMD	OnlineAE	OLoKe	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

  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.

  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]:

  	COLoKe	OLoKe 5	OLoKe 10	OLoKe 50	OLoKe 100	OLoKe 500
  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)
  Execution time (s)	2.98	11.12	15.29	50.44	92.81	444.84

  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.

• [W5] "To show [...] predicted state space trajectories for each system using different methods and MSE curves during training."

  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.

• [W6] "The comparison methods are not enough and outdated. [...] Some real-time advanced adaptive Koopman embedding learning methods [2,3] are required to compare."

  We thank the reviewer for this comment and would like to clarify the discussion about related works in two points.

  • 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.

  • 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.

References:

• [Dataset 1] https://physionet.org/content/eegmmidb/1.0.0/
• [Dataset 2] https://www.eol.ucar.edu/field_projects/cases-99

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.

• [L1] "The method shows meaningful improvements on single-attractor systems, but results on broader or more realistic datasets are limited. [...]"

  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.

  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.

  Dataset	COLoKe	ODMD	OnlineAE	OLoKe	OEDMD
  EEG (x1e-3)	7.9 (8.5)	8.3 (11.3)	12.4 (18.1)	9.3 (10.4)	8.0 (11.1)
  Turbulence (x1e-4)	4.7 (5.9)	5.5 (7.5)	6.4 (8.6)	5.5 (6.1)	7.3 (8.9)

• [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."

  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.

  • 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.

  • 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.

  • 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.

  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.

• [Q1] "[...] clarify the intuition behind the construction of the prediction score set?"

  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.

• [Q2] "The paper claims that the prediction set can be used to enhance the interpretability of the model. [...]"

  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.

References:

• [Ref. 1] Gibbs, I. and Candes, E. Adaptive conformal inference under distribution shift. NeurIPS 2021.
• [Ref. 2] Angelopoulos, A.N., Candes, E. and Tibshirani, R.J., Conformal PID Control for Time Series Prediction. NeurIPS 2023.
• [Ref. 3] Ramalingam, R., The Relationship Between No-Regret Learning and Online Conformal Prediction. ICML 2025.
• [Ref. 4] Suggala A.S. and Netrapalli P., Online Non-Convex Learning: Following the Perturbed Leader is Optimal, ALT 2020.
• [Ref. 5] Xu Z. and Lijun Zhang L., Online Non-convex Learning in Dynamic Environments, NeurIPS 2024.
• [Ref. 6] Pham, Q. et al., Learning Fast and Slow for Online Time Series Forecasting, ICLR 2023.
• [Ref. 7] Lau, YA. et al, Fast and Slow Streams for Online Time Series Forecasting Without Information Leakage, ICLR 2025.
• [Dataset 1] https://physionet.org/content/eegmmidb/1.0.0/
• [Dataset 2] https://www.eol.ucar.edu/field_projects/cases-99

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.

• [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."

  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

  • "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."
  • "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."

• [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?"

  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.

• [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"

  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.

  Dataset	COLoKe	ODMD	OnlineAE	OLoKe	OEDMD
  EEG (x1e-3)	7.9 (8.5)	8.3 (11.3)	12.4 (18.1)	9.3 (10.4)	8.0 (11.1)
  Turbulence (x1e-4)	4.7 (5.9)	5.5 (7.5)	6.4 (8.6)	5.5 (6.1)	7.3 (8.9)

References:

• [Ref.1] Bhatnagar, A., et al. Improved online conformal prediction via strongly adaptive online learning. In ICML 2023.
• [Dataset 1] https://physionet.org/content/eegmmidb/1.0.0/
• [Dataset 2] https://www.eol.ucar.edu/field_projects/cases-99

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.

• [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."

  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. 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.

  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
  ETD	100	49	0.49	2.02	6

  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:

  • VdP oscillator and ETD: Updates are clustered, with long intervals of stability matching the regular, low-variability regimes of these systems.
  • Duffing oscillator: Updates are more uniformly distributed, reflecting moderate chaotic behavior with frequent small deviations.
  • Lorenz system: Frequent early updates reflect the chaotic and unstable initial phase; they become sparser as the model adapts.
  • Single Attractor: Initial updates require more iterations, but once near the attractor, the model requires fewer corrections.

  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.

• [Q2] "Can you try simple benchmark of reservoir computing for online learning and compare it to your approach?"

  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.

  Dataset	RC (best online error)	RC (best generalization)	COLoKe
  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)
  	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)
  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)
  	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)
  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)
  	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)
  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)
  	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)
  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)
  	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)

  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.

  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:

  Dataset	COLoKe (parameters)	RC (nodes)	RC (nodes / trainable)
  Single attractor	778	77	388
  Duffing oscillator	778	77	388
  VdP oscillator	778	77	388
  Lorenz system	835	75	278
  ETD	3188	146	531

  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.

  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.

References:

• [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.
• [Ref. 2] Sussillo, D. and Abbott L.F. Generating Coherent Patterns of Activity from Chaotic Neural Networks. Neuron 2009.