Principled Koopman Representations with Kalman Inference for Efficient Time-Series Prediction

arXiv:2609.17815 · cs.LG, cs.AI · Submitted 2026-09-15 · Read on arXiv

cs.LG, cs.AI

Submitted: 2026-09-15

Updated: 2026-09-15

License: http://creativecommons.org/licenses/by/4.0/

The gist: The Koopman operator has been widely used for time-series prediction in dynamical systems.

Terminology

Abstract

The Koopman operator has been widely used for time-series prediction in dynamical systems. However, prior work that learns latent ``Koopman spaces'' using neural networks often did not construct a valid Koopman space for forecasting, as these representations may be mathematically inconsistent with the operator-theoretic formulation and fail to capture the intrinsic low-rank structure of system dynamics. To address this issue, we introduce K squared SVD, a method that explicitly learns the leading singular functions of the Koopman operator by optimizing a Hilbert-Schmidt objective. This yields a well-defined low-rank approximation of the Koopman operator with an interpretable linear combination, featuring a compact latent space with less than 10% of the dimensions used in previous work. In the learned Koopman space, K squared SVD further captures temporal evolution with a linear Gaussian state-space model and performs inference via Kalman filtering, mitigating noise accumulation during multi-step prediction. Empirical results show that K squared SVD outperforms state-of-the-art methods across multiple datasets, with significantly faster prediction speeds and lower computational cost than previous efficiency-focused models. This highlights the benefits of principled low-rank Koopman representations and opens up broader potential for applications.

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