EAGER: Data-driven Koopman Operator Techniques for Chaotic and Non-Autonomous Dynamical Systems

该项目旨在开发数据驱动的Koopman方法，用于分析和识别相干模式的频谱特性，并预测其在具有振荡模式和连续频谱以及服从长期趋势的动力系统中的动态演化。20世纪30年代初，库普曼算子被用来证明关于理想化动力系统行为的一个重要定理。但在动力系统理论之外，它们在很大程度上仍然鲜为人知，直到21世纪初，人们认识到它们对复杂的数据密集型问题的有用性。从那时起，库普曼算子与机器学习算法相结合的思想被应用于流体动力学、电力网络稳定性分析、金融交易算法的开发和大脑活动研究的各种问题。与这些应用程序一样，气候研究通常是由数据驱动的，涉及对复杂系统的分析，因此库普曼操作员可能是气候研究工具包中的一个有价值的补充。该项目在复杂系统模式提取和预测能力方面的进展将对非线性动力系统领域产生重大影响，并将适用于从气候科学、流体动力学到神经科学和经济学的各种处理时间演化现象的学科。该项目将通过培训博士后和在首席研究人员所在研究所开发关于数据驱动的动力系统建模的新课程来促进STEM劳动力的发展。该项目将推进机器学习技术和谱Galerkin方法，用于在不了解基本控制方程和边界条件的情况下，从时间序列和时空数据中识别受随机强迫和外部强迫(如地球气候系统)影响的复杂动力系统的Koopman特征频率和特征函数。该团队将把数据驱动的方法应用于厄尔尼诺南方涛动的气候模拟，以研究气候系统的自然变异性和强迫响应之间的相互作用。这将展示该方法揭示和预测数据中主要模式随时间演变的许多方面的能力，这是传统的基于协方差的方法无法实现的。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This project aims to develop data-driven Koopman methodologies for analyzing and identifying spectral properties of coherent patterns and predicting their dynamical evolutions in dynamical systems with both oscillatory modes and continuous spectrum as well subject to secular trends. Koopman operators were derived in the early 1930s and used to prove an important theorem on the behavior of idealized dynamical systems. But they remained largely unknown beyond dynamical systems theory until the early 2000s, when their usefulness for complicated, data intensive problems was recognized. Since then, the idea of Koopman operators in combination with machine learning algorithms has been applied to various problems in fluid dynamics, stability analysis of power networks, the development of financial trading algorithms, and brain activity research. Like these applications, climate research is often data-driven and involves analysis of a complex system, thus Koopman operators could be a valuable addition to the climate research toolkit. The advances in pattern extraction and predictive capabilities for complex systems stemming from this project will have great impacts in the field of nonlinear dynamical systems and will be applicable to various disciplines that deal with time-evolving phenomena, from climate science, fluid dynamics, to neuroscience and economics. The project will contribute towards STEM workforce development through the training of a postdoc and new course development for on data-driven dynamical systems modeling at Principal Investigator's home institution.This project will advance machine learning techniques and spectral Galerkin methods for identifying Koopman eigen-frequencies and eigenfunctions of complex dynamical systems that are subject to both stochastic forcing and external forcing (such as the Earth's climate system) from time series and spatiotemporal data without knowledge of the underlying governing equations and boundary conditions. The team will apply the data-driven methods to climate simulations of El Nino Southern Oscillation to study the interactions of natural variability and forced response of the climate system. This will demonstrate the ability of the methods to reveal and predict many aspects of the temporal evolutions of dominant modes in data, which cannot be achieved from conventional covariance-based methods.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.