AI 中文总结
本研究针对从单个含噪标量时间序列重构未知动力学系统的问题，提出基于差分嵌入坐标与弱形式回归的方法，在无噪场景下可实现长期预测，且能恢复可解释的闭包系数。
AI 中文摘要
我们研究从单个含噪标量时间序列重构未知动力学系统的问题，目标是恢复潜在动力学以用于预测。我们提出一种利用差分嵌入坐标直接从数据中识别嵌入动力学有理闭包的方法，该闭包通过弱形式回归流程识别，可避免对含噪数据进行不稳定的逐点求导。将该方法应用于无噪的Lorenz和Rössler系统时，其恢复的闭包可在大量实现集合上支持长期预测，对应分别为18.1和7.1个李雅普诺夫时间。在15%至30%的加性高斯噪声下，性能随系统变化：对于Lorenz系统，即使在最优情况下预测时长仍较短；而Rössler系统的绝对表现通常更好，但归一化至李雅普诺夫时间后则不然。我们的方法可恢复可直接解释的闭包系数，并将其与Lorenz和Rössler系统的已知解析闭包进行了比较。
英文摘要
We study the reconstruction of an unknown dynamical system from a single noisy scalar time series. The goal is to recover the underlying dynamics for forecasting. We introduce a method that uses differential embedding coordinates to identify a rational closure of the embedding dynamics directly from data. The closure is identified through a weak-form regression pipeline, which avoids unstable pointwise differentiation of noisy data. When applied to noise-free Lorenz and Rössler systems, the method recovers closures that support long forecasts across a broad ensemble of realizations ($18.1$ and $7.1$ Lyapunov times respectively). Under $15$--$30\%$ additive Gaussian noise, performance becomes system-dependent. For the Lorenz system, forecast horizons remain short even in the best cases, whereas the Rössler system generally performs better in absolute terms, though not once normalized by the Lyapunov time. Our proposed method recovers directly interpretable closure coefficients which we compared against the known analytic closures of the Lorenz and Rössler systems.
