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

2026-09-15

AI Summary

为解决时间序列预测中Koopman空间数学不一致性和低秩结构捕捉不足的问题，提出K^2SVD方法，通过优化Hilbert-Schmidt目标学习Koopman算子的主要奇异函数，并结合卡尔曼滤波进行推断。

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.

Problem

• Koopman operator
• time-series prediction
• low-rank structure
• dynamical systems

Innovation

• Koopman operator
• Hilbert-Schmidt objective
• low-rank approximation
• Kalman filtering
• time-series prediction