发表机构
UC Santa Barbara（加州大学圣塔芭芭拉分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对现有Koopman表示学习缺乏数学一致性和低秩结构的问题，提出K$^2$SVD方法，通过优化Hilbert-Schmidt目标学习主要奇异函数，结合Kalman滤波推断，实现高效且准确的时间序列预测。
AI 中文摘要
Koopman算子已广泛用于动力系统中的时间序列预测。然而，先前使用神经网络学习潜在“Koopman空间”的工作往往未能构建有效的Koopman空间用于预测，因为这些表示可能在数学上与算子理论公式不一致，并且无法捕捉系统动力学的内在低秩结构。为解决这一问题，我们提出了K$^2$SVD方法，通过优化Hilbert-Schmidt目标显式学习Koopman算子的主要奇异函数。这产生了具有可解释线性组合的明确定义的Koopman算子低秩近似，其紧凑潜在空间维度不到先前工作的$10\%$。在学习到的Koopman空间中，K$^2$SVD进一步通过线性高斯状态空间模型捕捉时间演化，并通过Kalman滤波进行推断，减轻了多步预测中的噪声累积。实验结果表明，K$^2$SVD在多个数据集上优于最先进的方法，且预测速度显著更快，计算成本低于先前注重效率的模型。这凸显了原则性低秩Koopman表示的益处，并为其应用开辟了更广泛的潜力。
英文摘要
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.
