Conformal Online Learning of Deep Koopman Linear Embeddings

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.

💡 Research Summary

The paper introduces COLoKe (Conformal Online Learning of Koopman embeddings), a novel framework for continuously updating deep Koopman representations of nonlinear dynamical systems from streaming data. Traditional Koopman approaches such as DMD, EDMD, and their deep extensions typically operate offline or require fixed dictionaries, limiting expressiveness and adaptability. Existing online methods either rely on linear observables or retrain the model at every step, leading to unnecessary computation and potential over‑fitting.

COLoKe addresses these shortcomings by integrating three key components: (1) a hybrid feature map Φθ(x) =