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Conformal Online Learning of Deep Koopman Linear Embeddings

Ben Gao, Jordan Patracone, Stéphane Chrétien, Olivier Alata

TL;DR

COLoKe addresses online learning of Koopman-invariant representations for nonlinear dynamical systems from streaming data by integrating deep Koopman embeddings with a conformal prediction-based update trigger. The method maintains a bounded memory buffer and optimizes a multi-step lifted-space loss while updating only when a conformity score signals temporal inconsistency, yielding a dynamic regret bound. Empirically, COLoKe recovers the Koopman spectrum and outperforms online baselines across synthetic and real datasets, achieving strong predictive performance with fewer updates and lower computation. This yields a practical, real-time approach for adaptive, linear embeddings of nonlinear dynamics in evolving environments, with potential extensions to non-autonomous settings.

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

TL;DR

COLoKe addresses online learning of Koopman-invariant representations for nonlinear dynamical systems from streaming data by integrating deep Koopman embeddings with a conformal prediction-based update trigger. The method maintains a bounded memory buffer and optimizes a multi-step lifted-space loss while updating only when a conformity score signals temporal inconsistency, yielding a dynamic regret bound. Empirically, COLoKe recovers the Koopman spectrum and outperforms online baselines across synthetic and real datasets, achieving strong predictive performance with fewer updates and lower computation. This yields a practical, real-time approach for adaptive, linear embeddings of nonlinear dynamics in evolving environments, with potential extensions to non-autonomous settings.

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.

Paper Structure

This paper contains 31 sections, 1 theorem, 39 equations, 4 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Mathematical background
  3. Data-driven learning of the Koopman operator
  4. Conformal prediction for online data
  5. Online conformal learning of deep Koopman embedding
  6. From online learning…
  7. … To conformal online learning
  8. Numerical experiments
  9. Illustration and validation
  10. Comparison with online baselines
  11. Conclusion
  12. Proof of Theorem \ref{['thm:colok_regret']}
  13. Experimental settings
  14. Datasets and metrics
  15. Simulated data
  16. ...and 16 more sections

Key Result

Theorem 3.3

Let $(\theta_t, K_t)$ be the parameters produced by Algorithm alg:colok and let $(\theta_t^*, K_t^*) \in \mathrm{argmin}_{(\theta, K)} \mathcal{L}_t(\theta, K)$ denote any time-dependent optimal model minimizing the loss at step $t$. Further assume: Then the dynamic regret satisfies: $\sum_{t=1}^T \left[\mathcal{L}_t(\theta_t, K_t) - \mathcal{L}_t(\theta_t^*, K_t^*)\right] \le \mathcal{O}\left( \

Figures (4)

  • Figure 1: Schematic representation of COLoKe. The model receives a rolling window of observations, lifts them via a partially-learned feature map, computes a conformity score based on multi-step prediction error, and updates its parameters only if the score exceeds a conformal threshold.
  • Figure 2: Illustration and empirical support for COLoKe's adaptive learning strategy.
  • Figure 3: Convergence of the Koopman eigenvalue and eigenfunction estimates in the online setting.
  • Figure 4: Pareto front comparing COLoKe to fixed-step OLoKe variants.

Theorems & Definitions (7)

  • Definition 3.1: Online Koopman training loss
  • Remark 3.1: Prediction conformity score
  • Definition 3.2: Prediction score set
  • Theorem 3.3: Dynamic regret of COLoKe
  • proof
  • proof
  • Remark B.1