AI 中文总结
本研究提出结合Takens延迟嵌入定理、流形学习与通用函数近似器的TAR算法框架，经多类系统验证可重构动力系统，还发布了开源软件包。
AI 中文摘要
嵌入定理可为系统低维观测值与其全维状态及动力学之间的关系提供理论保证，但此类定理未就观测值选择、嵌入构造或学习嵌入与全维状态间映射的方法提供指导。本研究中，我们开发了算法框架TAkens Reconstruction（TAR），该框架结合Takens延迟嵌入定理、流形学习技术及通用函数近似器，用于从低维时间序列分析和重构任意动力系统。我们在多种模拟及实测动力系统的应用中验证了TAR，并利用其探究延迟向量结构如何影响重构精度。在一个生态系统中，我们表明，简单的捕食者-猎物动力学可通过在多种嵌入时间尺度上获取的观测值进行重构；在Villin蛋白的分子动力学模拟中，我们展示了纳入同一观测序列的多个时间延迟如何用于改进具有多个特征时间尺度的系统的重构；在先锋标普500（Vanguard S&P 500）的交易记录中，我们展示了该方法如何揭示数据中的潜在动力学现象，并在无需全维市场观测的情况下对短期收益进行准确预测。我们开发并发布了开源软件包，以支持将TAR应用于任意动力系统。
英文摘要
Embedding theorems can be used to provide theoretical guarantees about the relation between low-dimensional observations of a system and its full-dimensional state and dynamics. Such theorems do not, however, provide guidance on observable choice, embedding construction, or methodologies to learn the mapping between the embedding and full-dimensional state. In this work, we develop an algorithmic framework, TAkens Reconstruction (TAR), to analyze and reconstruct arbitrary dynamical systems from low-dimensional time series using an integration of Takens' Delay Embedding Theorem, manifold learning techniques, and universal function approximators. We validate TAR in applications to a variety of simulated and observed dynamical systems and use it to investigate how delay vector structure impacts reconstruction accuracy. In an ecological system, we show that simple predator-prey dynamics can be reconstructed with observations taken over a wide variety of embedding time scales. In molecular dynamics simulations of the protein Villin, we demonstrate how including multiple time delays of the same observable series can be used to improve reconstruction of systems with multiple characteristic time scales. In the trade record of Vanguard S&P 500, we show how the approach exposes underlying dynamical phenomenologies in the data and accurate return predictions over short time horizons without access to full-dimensional market observations. We develop and release an open-source software package to enable the application of TAR to arbitrary dynamical systems.
