Linear Response and Koopman Modes: Prediction and Criticality - LINK

了解复杂系统对扰动的反应对科学研究和现实应用至关重要。复杂性是各种自然、工程和社会系统（如生态系统、经济、社会网络和气候）的基本特征。复杂系统的特点是在大范围的空间和时间尺度上发生波动。天气在短时间尺度上变化不规则，气候系统是在平稳变化时期和临界点之间交替发展的。我们中的许多人可能在新闻中读到，我们有可能在有生之年经历亚马逊森林的崩溃或大西洋经向环流的倾覆。在系统的自然波动和由于存在强迫而产生的强迫反应之间找到强有力的关系将是关键。找到气候变异和气候变化之间的严格联系意味着能够更好地从历史中预测地球系统的未来状态。数据科学革命正在改变我们对复杂系统建模的方式，人们认识到，基于理论的方法和数据驱动的方法必须集成。这两种方法都在迅速发展，并发现了令人惊讶的共同点。虽然我们可以获得大量的数据，但重要的是要注意，如果没有解释，数据本身就没有意义，正如H. poincar<s:1>所指出的那样。库普曼主义是一个理论框架，它允许通过研究描述可观测演化的线性算子的性质来理解复杂系统如何随时间变化。这种方法非常强大，通过挑选出系统的内在波动模式，可以对系统进行精确的数据驱动分析。我们最近已经能够找到系统自然变异性的库普曼表示和描述其对扰动响应的响应算子之间的理论联系。为复杂系统构造精确的响应算子在理论和计算上都具有挑战性。当系统接近临界行为时，问题变得更加困难，这与这些算子的散度有关。LINK项目旨在发展这一非常有前途的科学思想，利用概念多尺度气候模式上的库曼主义所建议的角度，以简洁而有意义的方式描述大气和海洋的耦合演化，通过构建计算效率高和准确的响应算子来实现。因此，我们将把自由波动和强制波动联系起来。这些模型的特征是亚稳态行为，与临界点的存在有关。然后，我们将研究响应算子如何标记临界的接近程度，从而更好地理解所谓的早期预警指标，通常与系统对扰动的敏感性增加和更长的记忆有关。LINK的结果将与一般复杂系统的研究相关，并将导致使用观测和更高复杂性模型来研究和理解气候危机的新工具。LINK是根据两个工作包构建的，每个工作包都包含旨在实现特定目标的活动，并以本应用程序的专用形式详细说明。LINK将涉及PI， PDRA （John Moroney，目前在都柏林三一学院）和两个外部合作伙伴-魏茨曼研究所（以色列）的Chekroun博士和NORDITA（瑞典）的N. Zagli。外部合作伙伴已承诺为支持这一科学合作提供资源和时间，并将在整个项目期间（每两周举行一次会议）远程指导和监督PDRA，并在将在英国、瑞典和以色列进行的科学访问期间亲自指导和监督PDRA。

Understanding how complex systems respond to perturbations is crucial for scientific research and real-world applications. Complexity is a fundamental characteristic of various natural, engineered, and social systems, such as ecosystems, economics, social networks, and the climate. Complex systems feature fluctuations occurring over a vast range of spatial and temporal scales. The weather changes erratically on short time scales, and the climate system has evolved by alternating between periods of smooth change with the occurrence of tipping points. Many of us might have read in the news that we are at risk of experiencing within our lifetime the collapse of the Amazon Forest or of the Atlantic meridional overturning circulation. It would be key to find robust relations between the natural fluctuations of a system and its forced response resulting from the presence of forcings. Finding a rigorous link between climate variability and climate change would imply being able to better predict the future state of the Earth system from its history. The data science revolution is transforming how we model complex systems, and it is recognized that theory-based and data-driven methods must be integrated. Both approaches are rapidly advancing and discovering surprising commonalities. While we have access to vast amounts of data, it's important to note that data alone lacks significance without interpretation, as suggested by H. Poincaré. Koopmanism is a theoretical framework that allows to understand how complex systems change in time by studying the properties of a linear operator that describes the evolution of observable. This approach is very powerful and allows for accurate data-driven analysis of a system, by singling out its intrinsic modes of fluctuations. We have recently been able to find a theoretical link between the Koopman representation of the natural variability of a system and the response operators describing its response to perturbations. Constructing accurate response operators for complex system has proved to be challenging both theoretically and computationally. The problem becomes even more difficult when the system is close to critical behaviour, which is associated with the divergence of such operators. The LINK project aims at developing this very promising scientific idea by constructing computationally efficient and accurate response operators using the angle suggested by the Koopmanism on conceptual multiscale climate models describing in a succinct yet meaningful way the coupled evolution of the atmosphere and the ocean. Hence, we will link free and forced fluctuations. Such models feature metastable behaviour, associated with the presence of tipping points. We will then study how the response operators flag the proximity of criticality, hence better understanding the so-called early warning indicators, usually associated with the increase in the system's sensitivity to perturbations and longer memory.LINK's results will be of relevance for the study of complex systems in general and will lead to new tools for studying and understanding the climate crisis using observations and higher complexity models. LINK is structured according to two Workpackages, each containing the activities aimed at the achievement of a specific objectives, detailed in the dedicated form of this application. LINK will involve the PI, a PDRA (John Moroney, presently at Trinity College, Dublin), and two external partners - M.D. Chekroun from the Weizmann Institute (Israel) and N. Zagli from NORDITA (Sweden). The external partners have committed resources and time for supporting this scientific collaboration and will contribute to the mentoring and supervision of the PDRA both remotely throughout the project duration (fortnightly meetings) and in person during the scientific visits that will take place in UK, Sweden, and Israel.