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Conformal Online Learning of Deep Koopman Linear Embeddings
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Proposes COLoKe, an online deep Koopman embedding learner that uses a conformal PI-control threshold on multi-step prediction error to trigger parameter updates only when the current model becomes inconsistent with streaming data.
Problem
Learning Koopman-invariant embeddings of nonlinear dynamical systems from streaming data, where existing online methods either use linear/fixed dictionaries or retrain with a fixed number of steps regardless of need, risking overfitting and wasted computation.
Approach
Repurposes conformal prediction as a model-consistency diagnostic: the multi-step prediction conformity score on a sliding buffer is compared to a threshold tracked by conformal PI control, and gradient updates happen only when the score exceeds it; the lifted feature map includes the identity so reconstruction is implicit.
Key results
- On the Lorenz system, COLoKe achieves online error 3.3e-3 versus 1.0e-1 for ODMD and 3.8e-2 for OnlineAE, roughly an order of magnitude or more improvement.
- COLoKe triggers updates in about 46-50% of steps on the synthetic systems, with computation time about 11.5s vs 19.25s for fixed-step OLoKe on the single attractor.
- On the single attractor, COLoKe reaches generalization error 2.4e-7 versus 1.1e-3 for ODMD, and recovers eigenvalues -1.0091, -0.04996, -0.1001 against ground truth -1, -0.05, -0.1.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use conformity-triggered adaptive updates instead of fixed-step updates | COLoKe vs OLoKe · Single Attractor (generalization) | prediction error: 2.1\cdot10^{-6} -> 2.4\cdot10^{-7} | Helped |
| Use conformity-triggered adaptive updates instead of fixed-step updates | COLoKe vs OLoKe · Single Attractor (online) | prediction error: 7.5\cdot10^{-6} -> 7.6\cdot10^{-7} | Helped |
| Use conformity-triggered adaptive updates instead of fixed-step updates | COLoKe vs OLoKe · Duffing oscillator | prediction error: 5.5\cdot10^{-5} -> 3.1\cdot10^{-6}; 2.3\cdot10^{-4} -> 7.3\cdot10^{-5} | Helped |
| Use conformity-triggered adaptive updates instead of fixed-step updates | COLoKe vs OLoKe · Van der Pol oscillator | prediction error: 6.6\cdot10^{-4} -> 3.8\cdot10^{-4}; 9.2\cdot10^{-4} -> 6.0\cdot10^{-4} | Helped |
| Use conformity-triggered adaptive updates instead of fixed-step updates | COLoKe vs OLoKe · Lorenz system | prediction error: 7.6\cdot10^{-3} -> 6.5\cdot10^{-3}; 4.7\cdot10^{-3} -> 3.3\cdot10^{-3} | Helped |
| Use conformity-triggered adaptive updates instead of fixed-step updates | COLoKe vs OnlineAE · ETD | online prediction error: 7.9\cdot10^{-2} -> 7.3\cdot10^{-2} | Helped |
| Use conformity-triggered adaptive updates instead of fixed-step updates | COLoKe vs OEDMD · EEG | online prediction error: 8.0\cdot10^{-3} -> 7.8\cdot10^{-3} | Helped |
| Use conformity-triggered adaptive updates instead of fixed-step updates | COLoKe vs OLoKe · Turbulence | online prediction error: 4.8\cdot10^{-4} -> 4.7\cdot10^{-4} | Helped |
| Use conformity-triggered adaptive updates rather than fixed-step updates | COLoKe vs OLoKe · Single Attractor | computation time (seconds): 19.25 -> 11.50 | Helped |
| Use conformity-triggered adaptive updates rather than fixed-step updates | COLoKe vs OLoKe · Duffing oscillator | computation time (seconds): 19.23 -> 9.72 | Helped |
| Use conformity-triggered adaptive updates rather than fixed-step updates | COLoKe vs OLoKe · Van der Pol oscillator | computation time (seconds): 19.60 -> 10.94 | Helped |
| Use conformity-triggered adaptive updates rather than fixed-step updates | COLoKe vs OLoKe · Lorenz system | computation time (seconds): 266.25 -> 107.96 | Helped |
Surprising findings
- Estimated Koopman eigenvalues on the single attractor closely match ground truth despite purely online updates.
- Reported update triggers occur in roughly 48% of steps across systems, with the trigger rate stable across dynamics despite differing temporal structure.
Limitations
- Key theoretical assumption (A3) on sublinear cumulative conformity thresholds is empirical and not derived.
- Dynamic regret bound is preliminary and relies on strong assumptions (L-smoothness, bounded gradients).
- Baselines are limited to streaming-compatible methods; results on real datasets are from a single trajectory.
Experimental setup
Datasets: Single attractor (simulated) Duffing oscillator (simulated) Van der Pol oscillator (simulated) Lorenz system (simulated) ETTh1 (Electricity Transformer Dataset) PhysioNet EEG Motor Movement/Imagery CASES-99 turbulence
Benchmarks: Single attractor Duffing oscillator Van der Pol oscillator Lorenz system ETTh1 EEG CASES-99
Metrics: generalization error (one-step MSE on held-out trajectories) online prediction error computation time
Baselines: ODMD OEDMD (R-EDMD variant) OnlineAE OLoKe (fixed-step variant)
Builds on: Extended DMD Online DMD Conformal PID control Deep Koopman embeddings Online AE for MPC
Closest papers by meaning
- Deep-Koopman-KANDy: Dictionary Discovery for Deep-Koopman Operators with Kolmogorov-Arnold Networks for Dynamics (NeurIPS 2026 · similarity 0.87)
- Course Correcting Koopman Representations (ICLR 2024 · similarity 0.85)
- ResKoopNet: Learning Koopman Representations for Complex Dynamics with Spectral Residuals (ICML 2025 · similarity 0.84)
- Finding separatrices of dynamical flows with Deep Koopman Eigenfunctions (NeurIPS 2025 · similarity 0.84)
- KoopSTD: Reliable Similarity Analysis between Dynamical Systems via Approximating Koopman Spectrum with Timescale Decoupling (ICML 2025 · similarity 0.84)
- Efficient Parametric SVD of Koopman Operator for Stochastic Dynamical Systems (NeurIPS 2025 · similarity 0.84)
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