将延迟微分方程的无限维系统转化为可学习的有限维表示，实现精准预测。
On Data-Driven Koopman Representations of Nonlinear Delay Differential Equations
- 通过历史离散化与重构算子，构建有限维柯普曼近似框架
- 理论证明预测误差由离散、插值和回归三部分构成，可量化
- 适用于非线性延迟系统的建模与未来控制研究
本文建立了无限维延迟动力系统与有限维柯普曼学习之间的严格桥梁，提供明确且可解释的误差保证。尽管柯普曼分析在常微分方程（ODE）和部分偏微分方程（PDE）中已较成熟，但其向延迟微分方程（DDE）的扩展受限于DDE的无限维相空间。我们提出基于历史离散化与合适重构算子的有限维柯普曼近似框架，通过核基扩展动态模态分解（kEDMD）实现柯普曼算子的可计算表示。推导出学习预测器的确定性误差界，将总误差分解为历史离散化、核插值和数据驱动回归三部分贡献。此外，开发了基于核的重构方法，从升维后的柯普曼坐标恢复离散状态，并提供理论保证。数值结果表明该方法能可靠预测非线性延迟系统，具有潜在控制应用价值。
原文摘要 · Abstract (English)
This work establishes a rigorous bridge between infinite-dimensional delay dynamics and finite-dimensional Koopman learning, with explicit and interpretable error guarantees. While Koopman analysis is well-developed for ordinary differential equations (ODEs) and partially for partial differential equations (PDEs), its extension to delay differential equations (DDEs) remains limited due to the infinite-dimensional phase space of DDEs. We propose a finite-dimensional Koopman approximation framework based on history discretization and a suitable reconstruction operator, enabling a tractable representation of the Koopman operator via kernel-based extended dynamic mode decomposition (kEDMD). Deterministic error bounds are derived for the learned predictor, decomposing the total error into contributions from history discretization, kernel interpolation, and data-driven regression. Additionally, we develop a kernel-based reconstruction method to recover discretized states from lifted Koopman coordinates, with provable guarantees. Numerical results demonstrate reliable prediction of nonlinear delay systems, with potential relevance to future control applications.
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