Zhuge Chengzuo, Li Jiawei, Chen Wei
School of Mathematical Sciences, Beihang University, Beijing 100191, People's Republic of China.
Key Laboratory of Mathematics, Informatics and Behavioral Semantics (LMIB), Beihang University, Beijing 100191, People's Republic of China.
R Soc Open Sci. 2025 Jul 16;12(7):242240. doi: 10.1098/rsos.242240. eCollection 2025 Jul.
Tipping points occur in many real-world systems, at which the system shifts suddenly from one state to another. The ability to predict the occurrence of tipping points from time series data remains an outstanding challenge and a major interest in a broad range of research fields. Particularly, the widely used methods based on bifurcation theory are neither reliable in prediction accuracy nor applicable for irregularly sampled time series which are commonly observed from real-world systems. Here, we address this challenge by developing a deep learning algorithm for predicting the occurrence of tipping points in untrained systems, by exploiting information about normal forms. Our algorithm not only outperforms traditional methods for regularly sampled model time series but also achieves accurate predictions for irregularly sampled model time series and empirical time series. Our ability to predict tipping points for complex systems paves the way for mitigation risks, prevention of catastrophic failures and restoration of degraded systems, with broad applications in social science, engineering and biology.
临界点出现在许多现实世界的系统中，在这些点上系统会突然从一种状态转变为另一种状态。从时间序列数据预测临界点的出现仍然是一个突出的挑战，也是广泛研究领域中的一个主要兴趣点。特别是，基于分岔理论的广泛使用的方法在预测准确性方面既不可靠，也不适用于从现实世界系统中常见的不规则采样时间序列。在这里，我们通过开发一种深度学习算法来应对这一挑战，该算法通过利用关于范式的信息来预测未训练系统中临界点的出现。我们的算法不仅在规则采样的模型时间序列上优于传统方法，而且在不规则采样的模型时间序列和经验时间序列上也能实现准确预测。我们预测复杂系统临界点的能力为降低风险、预防灾难性故障和恢复退化系统铺平了道路，在社会科学、工程和生物学中有广泛应用。
