发表机构
Columbia University; University of Edinburgh; Los Alamos National Laboratory; Pacific Northwest National Laboratory; University of Leicester; NASA Goddard Institute for Space Studies（哥伦比亚大学; 爱丁堡大学; 洛斯阿拉莫斯国家实验室; 西北太平洋国家实验室; 莱斯特大学; 美国国家航空航天局戈达德空间研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究基于 Koopman 算子理论构建统一早期预警框架，用于检测随机系统中两类突变，其指标在速率诱导区域表现更优，在大西洋经向翻转环流模拟中可区分突变与非突变轨迹。
AI 中文摘要
复杂系统的突变通常会有早期预警信号，但大多数指标依赖于临界慢化的概念，通常无法扩展到速率诱导 tipping（即系统可在未发生局部稳定性丧失的情况下发生转变），这在随机非自治系统中存在问题，这类系统的内部变异性与时变变量相互作用，共同决定 tipping 的发生。我们利用 Koopman 算子理论，为随机系统中的分岔和速率诱导 tipping 开发了一个统一的早期预警框架。该方法基于残差 Koopman 模态分解，用于测量动力学与其有限维近似之间的差异，并通过用时变控制变量扩充可观测量空间，将其扩展到控制场景。在理想化示例中，所得指标能恢复分岔点附近的预期特征，并在经典指标失效的速率诱导区域提升检测效果。我们还表明，通过深度学习学习到的嵌入优于预设字典，尤其在高维场景中表现突出。将其应用于大西洋经向翻转环流的模拟，我们的基于 Koopman 的指标可区分 tipping 轨迹与非 tipping 轨迹，并在临界转变前揭示可解释的频谱特征。
英文摘要
Abrupt transitions in complex systems are often preceded by early warning signals. However, most indicators rely on the notion of critical slowing down and do not generally extend to rate-induced tipping where transitions can occur without local loss of stability. This is problematic in stochastic, nonautonomous systems where internal variability and time-varying variables interact to shape tipping onset. We use Koopman operator theory to develop a unified early warning framework for both bifurcation and rate-induced tipping in stochastic systems. Our approach builds on residual Koopman mode decomposition that measures discrepancies between dynamics and their finite-dimensional approximation, and extends it to the control setting by augmenting the observable space with time-varying control variables. In idealized examples, the resulting indicators recover expected signatures near bifurcation points and improve detection in rate-induced regimes where classical indicators fail. We further show that learned embeddings through deep learning outperform prescribed dictionaries, especially in a high-dimensional setting. Applied to simulations of the Atlantic Meridional Overturning Circulation, our Koopman-based indicators distinguish tipping from non-tipping trajectories and reveal interpretable spectral signatures prior to critical transition.
