Linear Response and Koopman Modes: Prediction and Criticality - LINK

了解复杂系统对扰动的反应对于科学研究和现实世界的应用至关重要。复杂性是各种自然、工程和社会系统的基本特征，如生态系统、经济、社会网络和气候。复杂系统的特征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不规律地变化，气候系统在平稳变化和临界点的出现之间交替演变。我们中的许多人可能已经在新闻中看到，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颠覆。关键是要找到一个系统的自然波动与其因强迫的存在而产生的强迫反应之间的牢固关系。找到气候变化和气候变化之间的严格联系，将意味着能够根据地球系统的历史更好地预测其未来状态。数据科学革命正在改变我们建模复杂系统的方式，人们认识到，基于理论的方法和数据驱动的方法必须整合。这两种方法都在迅速发展，并发现了令人惊讶的共同点。虽然我们可以访问大量数据，但重要的是要注意到，正如H.Poincaré所建议的那样，如果没有解释，数据本身就没有意义。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，通过挑出系统的内在波动模式，允许对系统进行准确的数据驱动分析。我们最近找到了系统自然可变性的库普曼表示和描述其对扰动的响应的响应算符之间的理论联系。构造复杂系统的精确响应算子在理论上和计算上都是具有挑战性的。当系统接近临界行为时，问题变得更加困难，这与这些操作员的分歧有关。LINK项目旨在发展这一非常有希望的科学想法，利用库普曼主义关于概念性多尺度气候模型的角度构建计算高效和准确的响应算子，以简洁而有意义的方式描述大气和海洋的耦合演化。因此，我们将把自由波动和强迫波动联系起来。这类模型的特点是亚稳态行为，与临界点的存在有关。然后，我们将研究响应操作员如何标记临界点的接近程度，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的结果将对一般复杂系统的研究具有相关性，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根据两个工作包构成，每个工作包包含旨在实现特定目标的活动，并以本申请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--来自魏兹曼研究所(以色列)的M.D.Chekroun和来自NORDITA(瑞典)的N.Zagli。外部合作伙伴已承诺投入资源和时间支持这一科学合作，并将在整个项目期间(每两周举行一次会议)和在英国、瑞典和以色列进行科学访问期间亲自对PDRA进行指导和监督。

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.