Holistic Analysis and Control of High-Dimensional Dynamical Systems via Operator-Theoretic and Data-Driven Approaches

在大数据和物联网时代，经济实惠的无处不在的计算能力和测量数据的可用性迅速增加，为自然界和工程中的各种动态系统的新型控制工程和自动化解决方案提供了前所未有的机会。该项目将开展基础研究，为如何最好地利用数据采集和计算技术的最新进展提供新知识，以显着提高性能，并拓宽控制工程设计方法和原则的范围。该项目开创的新公式和框架将允许对日益复杂和高维的动力系统进行更全面的理解和处理，这些系统超出了最先进方法的范围。这也将为更抽象的定义和广泛涵盖的高级任务提供坚实而严谨的基础，例如“模拟或控制大脑中的动态”，以系统的方式处理。该项目的研究从各种不同的学科，如系统与控制、数据科学与工程、细胞生物学、脑科学和医疗保健，汲取并承诺做出新的贡献，并将反过来加强这些学科的研究和教育基础设施。通过华盛顿大学学校伙伴关系研究所，将共同努力吸引代表性不足的群体参与这一多学科研究项目，并吸引普通公众和大学前K-12学生参与科学研究。该项目将启动一项基础的、理论驱动的研究，旨在通过将先进的算子理论和基于密度的方法与微分几何和代数几何技术相结合，实现高维非线性系统的整体分析和控制方法。具体来说，研究小组将研究并利用Koopman算子建立一个新的数据集成框架，用于将定义在有限维流形上的非线性控制系统转换为定义在高维（可能是无限维）向量空间上的线性系统。使用基于密度的系统描述和基于矩的表示对双重问题的研究将使数据驱动的框架能够促进更全面的控制设计方法。该项目的理论和计算进步的可行性将在科学和工程的各个前沿领域得到强调，例如神经科学中的脑动力学研究和细胞生物学中的癌症治疗。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The rapid increase in availability of affordable ubiquitous computing power and measurement data in the age of Big Data and Internet of Things is presenting unprecedented opportunities for novel control engineering and automation solutions for a wide range of dynamic systems in nature and engineering. This project will conduct fundamental research that will contribute new knowledge on how to best utilize these recent advances in data acquisition and computing technologies to significantly enhance the performance, as well as broaden the scope of, control engineering design methodologies and principles. New formulations and frameworks pioneered in this project will allow for a more holistic understanding and treatment of increasingly complicated and high-dimensional dynamical systems that escape the scope of state-of-the-art approaches. This will also provide a solid and rigorous basis for more abstractly defined and widely encompassing high-level tasks, such as "modelling or controlling the dynamics in the brain", to be tackled in a systematic manner. The investigation in this project draws from and promises new contributions to a variety of different disciplines, such as, systems and control, data science and engineering, cell biology, brain science, and healthcare, and will in turn enhance the infrastructure for research and education across these disciplines. Concerted effort will be made to attract underrepresented groups in this multi-disciplinary research program and to engage the general public and pre-college K-12 students in scientific research through the Institute of School Partnerships at Washington University.This project will initiate a fundamental, theory-driven investigation aimed at enabling holistic analysis and control methodologies for high-dimensional nonlinear systems via merging advanced operator-theoretic and density-based approaches with differential geometric and algebraic geometric techniques. Specifically, the research team will investigate and leverage Koopman operators to establish a novel data-integrated framework for transforming nonlinear control systems defined on a on a finite-dimensional manifold to linear systems defined on a higher dimensional, possibly infinite-dimensional, vector space. The study of the dual problem using density-based system descriptions and moment-based representations will enable a data-driven framework that facilitates a more holistic control design methodology. The feasibility of the theoretical and computational advances in this project will be highlighted in diverse cutting-edge areas in science and engineering, such as in the study of brain dynamics in neuroscience and cancer treatment in cell biology.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.