Koopman models with Kalman filtering speed up time-series prediction
Principled Koopman Representations with Kalman Inference for Efficient Time-Series Prediction
Machine LearningArtificial Intelligence
Summary
Time-series prediction tries to guess what will happen next based on past data. The authors found that past methods using Koopman operators with neural networks often made mistakes by not following the right math rules, causing less accurate predictions. They created a new method called K2SVD that finds a simpler, more accurate way to represent the system's behavior and uses Kalman filtering to reduce errors when predicting many steps ahead. Their approach is faster and uses fewer resources than previous models, working well on several datasets.
What this means in practice
- •For data scientists: Speed up and improve accuracy of forecasting models for complex system data by using a compact Koopman representation with Kalman inference.
- •For robotics engineers: Use efficient and noise-robust Koopman-based models to better predict future states of robotic systems in real time.
Authors
Ruiquan Li, Yuheng Bu
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$^2$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$^2$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$^2$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.