Conformal Online Learning of Deep Koopman Linear Embeddings

AI summary

Overview

Research area: Machine learning for nonlinear dynamical systems, specifically online (streaming) learning of Koopman operator representations with deep neural embeddings, combined with conformal prediction for update control.

Technical level: Advanced. The paper draws on Koopman operator theory, deep autoencoder-style feature learning, conformal prediction and conformal PID control, and non-convex dynamic regret analysis.

Scope: This paper (arXiv:2511.12760v2 [cs.LG]) proposes COLoKe, a framework for adaptively updating deep Koopman linear embeddings from streaming data, triggers updates only when a dynamically calibrated conformity threshold is exceeded, and validates the approach on synthetic and real sequential datasets.

What This Paper Is About

Most Koopman operator learning methods assume the entire dataset is available in advance, and most online methods either restrict themselves to fixed dictionaries of observables or retrain on a fixed schedule regardless of whether the incoming data actually requires an update. The paper asks how to keep an expressive, deep Koopman embedding accurate over long horizons while updating it only when it genuinely stops being consistent with new observations. Its answer is to repurpose conformal prediction: instead of testing whether a new state conforms to a fixed model, the method tests whether the current Koopman model still conforms to the newly observed data.

Key Contributions

1. A conformal-driven online learning framework. The authors introduce an online learning scheme in which conformal prediction principles decide when to adapt model parameters, rather than only how to adapt them.
2. Instantiation for Koopman operator learning. They build this into a deep Koopman embedding with multi-step prediction consistency, producing expressive online regression of nonlinear dynamical systems through data-adaptive embeddings.
3. A theoretical analysis. They establish a dynamic regret bound for the algorithm under three assumptions (smoothness and bounded gradients, bounded total and squared variation of the oracle path, and sublinear accumulation of the conformity thresholds).
4. A positioning claim and empirical validation. They argue COLoKe is the first principled approach for online learning driven by conformal-based updates, and compare it against online Koopman baselines on synthetic dynamical systems and real datasets.

Main Findings

• Updates become sparse and adaptive. In the illustrative single-attractor experiment, the calibration threshold q_t decays steadily, the score set tightens over time, and updates are triggered only when the score s_t exceeds q_t; the number of updates stays controlled rather than being fixed in advance.
• The threshold accumulation is sublinear. The cumulative conformity thresholds grow sublinearly, reported as order O(√T), which the authors present as empirical support for assumption (A3) in their regret theorem.
• Spectral recovery works. On the analytically tractable system with true Koopman eigenvalues {−1, −0.05, −0.1}, COLoKe recovered {−1.0091, −0.04996, −0.1001}, and the estimated eigenfunctions matched the oracle up to a scalar factor.
• Faster than offline learning with lower test error. Against an offline deep Koopman model with the same neural architecture, measured on an NVIDIA RTX 2000 ADA GPU, COLoKe reached lower test error in significantly less time, with the gap widening as training progressed.
• Best or competitive accuracy against online baselines. Across Single attractor, Duffing oscillator, VdP oscillator, Lorenz system, and the real datasets ETD, EEG, and Turbulence, COLoKe obtained the lowest values among ODMD, OEDMD, OnlineAE, and OLoKe in the reported table cells (the final Turbulence entry for COLoKe is truncated in the provided content).
• Representative numbers:
  • Single attractor generalization error: COLoKe 2.4·10⁻⁷ (±3.6·10⁻⁸); OLoKe 2.1·10⁻⁶ (±6.6·10⁻⁷); ODMD 1.1·10⁻³ (±3.6·10⁻⁵); OnlineAE 1.0·10⁻² (±7.7·10⁻⁴); OEDMD 2.5·10⁻² (±2.8·10⁻⁴).
  • Duffing oscillator generalization error: COLoKe 3.1·10⁻⁶ (±2.3·10⁻⁷); OLoKe 5.5·10⁻⁵ (±1.0·10⁻⁵); ODMD 2.5·10⁻⁴ (±7.8·10⁻⁶); OEDMD 6.8·10⁻³ (±4.5·10⁻³); OnlineAE 8.7·10⁻³ (±2.5·10⁻³).
  • Lorenz system generalization error: COLoKe 6.5·10⁻³ (±1.0·10⁻⁴); OLoKe 7.6·10⁻³ (±1.8·10⁻⁴); ODMD 2.7·10⁻¹ (±1.3·10⁻³); OEDMD 5.5·10⁻¹ (±2.2·10⁻²); OnlineAE 5.9·10⁻¹ (±8.4·10⁻²).
  • EEG: COLoKe 7.8·10⁻³ (±8.5·10⁻³); OEDMD 8.0·10⁻³ (±1.11·10⁻²); ODMD 8.3·10⁻³ (±1.13·10⁻²); OLoKe 8.8·10⁻³ (±9.7·10⁻³); OnlineAE 1.24·10⁻² (±1.81·10⁻²).
  • For synthetic datasets the table reports two rows per method: the first is generalization error and the second is online error.
• Positioning against prior online Koopman methods. In the paper's comparison table, COLoKe is the only listed method marked as needing no history, supporting online updates, using an adaptive embedding, and having built-in reconstruction. ODMD lacks adaptive embeddings; R-EDMD needs history and has neither adaptive embeddings nor built-in reconstruction; DKLT, BatchOnline, and R-SSID are marked "batch only" for online update; OnlineAE has no built-in reconstruction.

Methodology in Plain English

The approach builds a representation of a nonlinear dynamical system in which the dynamics become linear. A neural network learns part of this representation, while the original state variables are kept as part of it, so that reconstruction is automatic and no separate decoder is needed.

At each time step the model receives a rolling window of the most recent observations. It lifts these observations through the learned feature map and asks whether the current lifted state can be predicted by repeatedly applying the learned linear operator to the earlier lifted states in the window. The accumulated discrepancy is a conformity score.

The novelty lies in what happens next. Rather than retraining at every step, the method keeps the previous parameters and compares the score against a threshold that is itself updated online using a conformal PI (proportional-integral) control rule, the same style of rule used in conformal PID control for time series. If the score stays within the threshold, nothing changes. If it exceeds the threshold, gradient steps are taken on the multi-step prediction loss until the score becomes conformant again. The authors formalize this by defining a prediction score set over parameter configurations rather than over candidate states, so the question shifts from "does this observation fit the model?" to "does this model fit the observation?". The full loop is given as Algorithm 1, taking a buffer size, initial parameters, a step size, and an initial threshold as inputs.

Why This Matters

Impact on research. The paper reframes conformal prediction as a model-diagnostic tool rather than only an uncertainty-quantification tool, and it supplies a regret bound that links the cumulative conformity thresholds to dynamic regret. It also claims a first-of-its-kind principled update criterion for online Koopman learning, in contrast to online methods that retrain on fixed schedules or rely on fixed dictionaries. The authors note the framework is not limited to Koopman learning and could apply to general online non-convex learning settings where the open questions are when updates should occur and to what extent.

Real-world applications (as identified in the paper):

• Robotics, where understanding how a system evolves over time is essential.
• Finance, physics, chemistry, and neuroscience, all listed as disciplines that depend on modeling complex system evolution.
• Online monitoring, adaptive control, and real-time forecasting, the three settings the authors explicitly identify as involving sequentially arriving data and potentially non-stationary systems.
• Real sequential measurement streams, exemplified by the ETD, EEG, and Turbulence datasets used in the experiments.

Industry relevance. The method is designed to be memory-efficient, needing only a rolling buffer rather than full history, and to avoid unnecessary computation by skipping updates when the model is still consistent. The authors report that it beats an offline deep Koopman model on test error in less wall-clock time on an NVIDIA RTX 2000 ADA GPU, which is the kind of trade-off that matters for deployed monitoring and control loops.

Future Directions

• Deriving assumption (A3). The sublinear bound on the cumulative conformity thresholds is stated as an assumption and treated as an empirical hypothesis; proving it remains, in the authors' words, an important direction for future work.
• Beyond the current scope. The authors state explicitly that non-autonomous systems under frequent abrupt changes are out of the scope of the present paper, leaving that regime open.
• Handling limited expressivity. The paper suggests that a model failing to learn dynamics because it lacks expressive power can be mitigated by augmenting the dimension of the lifted space.
• Generalization beyond Koopman learning. Since the conformal online learning scheme is presented as applicable to more general online non-convex learning, extending it to other settings, and connecting it to prior dynamic-regret work, is a natural next step.

Target Audience

This paper is best suited to researchers and graduate students working on machine learning for dynamical systems, Koopman operator methods, online and streaming learning, or conformal prediction. It will also interest control theorists and practitioners in real-time forecasting, online monitoring, and adaptive control who need models that update selectively rather than continuously, and readers comfortable with operator-theoretic formulations, neural network training, and non-convex regret analysis.

Authors’ 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.