Collaborative Research: Operator theoretic methods for identification and verification of dynamical systems

自动化在社会许多领域的广泛使用已经产生了大量关于各种动力系统的历史行为的数据，例如无人驾驶航空器，海洋和地面车辆，生物系统和天气系统。该项目旨在开发新的算法来发现解释观察到的行为的管理规则（即，动力系统的轨迹。基础模型的发现虽然对分析和控制有用，但在计算上可能具有挑战性。例如，传统的建模方法依赖于导数，并且可能受到甚至使数值微分偏离轨道的少量测量噪声的阻碍。这样的方法将动态系统的输出的每个测量值视为单独的数据点。然后使用数值微分将数据点与基础模型关联起来。相反，在这个项目中，整个轨迹被视为一个感兴趣的单元。测量数据点的序列被视为该轨迹的采样的、有噪声的表示，并且使用数值积分与底层模型相关。假设将动力系统的轨迹作为数据的基本单位，可以产生更好的数据驱动技术来分析和控制动力系统，本项目旨在开发这种数据驱动的识别和验证技术。为了扩大研究的影响，该团队还将为本科生开发为期一周的研讨会，通过视频游戏教授数据科学和人工智能（AI）概念。为了促进机器学习的早期介绍，该团队还将开发适合在高中夏令营期间提供的人工智能研讨会版本。该项目的具体目标是开发一个新的理论框架来处理大量的时间序列数据，并应用该框架来产生用于非线性动力系统研究的强大而灵活的工具。在所提出的方法中，轨迹信息被嵌入到一个再生核希尔伯特空间（RKHS）通过所谓的占领内核。占领内核绑定到原来的动态通过一个密集定义的运营商，刘维运营商。占领核和刘维算子导致当代的方法，研究有限维非线性优化问题，提升到无限维线性规划的措施空间的非平凡的推广。所提出的方法有利于提升到线性规划的函数空间，而不是测量空间，因此，从函数论和逼近理论的工具成为可用于设计和分析的算法。该项目的具体目标包括：研究RKHS上的占位核和Liouville算子的基本性质，应用于非线性系统识别，研究由Liouville算子对占位核的伴随作用产生的预内积空间，以及该框架在解决运动层析成像问题中的应用。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

Widespread use of automation in many sectors of society has yielded a large amount of data regarding historical behaviors for a variety of dynamical systems, such as unmanned aerial, marine, and ground vehicles, biological systems, and weather systems. This project aims to develop novel algorithms to discover governing rules that explain the observed behaviors (i.e., trajectories) of dynamical systems. Discovery of underlying models, while useful for analysis and control, can be computationally challenging. For example, traditional modeling methods rely on derivatives, and can be hampered by even modest amounts of measurement noise that derails numerical differentiation. Such methods treat each measurement of the output of a dynamical system as a separate data point. The data points are then related back to the underlying model using numerical differentiation. Instead, in this project, the entire trajectory is treated as a unit of interest. The sequence of measured data points is treated as a sampled, noisy representation of that trajectory, and is related back to the underlying model using numerical integration. It is hypothesized that treating trajectories of dynamical systems as the fundamental unit of data can yield better data-driven techniques for analysis and control of dynamical systems, and this project aims to develop such data-driven identification and verification techniques. To broaden the impact of the research, the team will also develop week-long workshops for undergraduate students that teach data science and artificial intelligence (AI) concepts through video games. To facilitate early introduction to machine learning, the team will also develop versions of the AI workshops that are suitable to be offered during high school summer camps. The specific aim of this project is to develop a new theoretical framework to process a large amount of time-series data and to apply the framework to yield robust and flexible tools for the study of nonlinear dynamical systems. In the proposed approach, trajectory information is embedded in a reproducing kernel Hilbert space (RKHS) through what are called occupation kernels. The occupation kernels are tied to the original dynamics through a densely defined operator, the Liouville operator. Occupation kernels and Liouville operators result in a nontrivial generalization of contemporary methods that study finite-dimensional nonlinear optimization problems by lifting them into infinite dimensional linear programs over the spaces of measures. The proposed approach facilitates lifting into linear programs over function spaces instead of measure spaces, and as a result, tools from function theory and approximation theory become available for design and analysis of algorithms. The specific aims of the project include: studying fundamental properties of occupation kernels and Liouville operators over RKHSs, applications to nonlinear system identification, study of the pre-inner product space that results from the action of the adjoint of a Liouville operator on an occupation kernel, and applications of the framework to solve motion tomography problems. The developed tools will be validated by solving identification and verification problems for unmanned ground, air, or underwater vehicles.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.