Complex dynamical systems-such as climate, ecosystems, and economics-can undergo catastrophic and potentially irreversible regime changes, often triggered by environmental parameter drift and stochastic disturbances. These critical thresholds, known as tipping points, pose a prediction problem of both theoretical and practical significance, yet remain largely unresolved. To address this, we articulate a model-free framework that integrates the measures characterizing the stability and sensitivity of dynamical systems with the reservoir computing (RC), a lightweight machine learning technique, using only observational time series data. The framework consists of two stages. The first stage involves using RC to robustly learn local complex dynamics from observational data segmented into windows. The second stage focuses on accurately detecting early warning signals of tipping points by analyzing the learned autonomous RC dynamics through dynamical measures, including the dominant eigenvalue of the Jacobian matrix, the maximum Floquet multiplier, and the maximum Lyapunov exponent. Furthermore, when these dynamical measures exhibit trend-like patterns, their extrapolation enables ultra-early prediction of tipping points significantly prior to the occurrence of critical transitions. We conduct a rigorous theoretical analysis of the proposed method and perform extensive numerical evaluations on a series of representative synthetic systems and eight real-world datasets, as well as quantitatively predict the tipping time of the Atlantic Meridional Overturning Circulation system. Experimental results demonstrate that our framework exhibits advantages over the baselines in comprehensive evaluations, particularly in terms of dynamical interpretability, prediction stability and robustness, and ultra-early prediction capability.

翻译：复杂动力系统——如气候、生态系统和经济系统——可能经历灾难性且往往不可逆的状态转变，这些转变通常由环境参数漂移和随机扰动触发。这些被称为临界点的关键阈值提出了一个兼具理论与实际意义的预测难题，但至今仍未得到充分解决。为此，我们提出了一种无模型框架，该框架仅利用观测时间序列数据，将表征动力系统稳定性与敏感性的测度与轻量级机器学习技术——储备池计算（RC）相结合。该框架包含两个阶段：第一阶段利用RC从分段窗口化的观测数据中鲁棒地学习局部复杂动力学；第二阶段通过分析学习得到的自主RC动力学，借助包括雅可比矩阵主导特征值、最大弗洛凯乘子和最大李雅普诺夫指数在内的动力学测度，精确检测临界点的早期预警信号。此外，当这些动力学测度呈现趋势性模式时，通过外推可实现临界转变发生前的超早期预测。我们对所提方法进行了严格的理论分析，并在系列代表性合成系统与八个真实世界数据集上进行了广泛的数值评估，同时定量预测了大西洋经向翻转环流系统的临界时间。实验结果表明，我们的框架在综合评价中优于基线方法，尤其在动力学可解释性、预测稳定性与鲁棒性以及超早期预测能力方面表现突出。