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

自动化在社会的许多部门的广泛使用已经产生了关于各种动力系统的历史行为的大量数据，例如无人驾驶航空、海洋和地面车辆、生物系统和天气系统。这个项目旨在开发新的算法来发现解释动态系统观察到的行为(即轨迹)的支配规则。发现基础模型虽然对分析和控制有用，但在计算上可能是一项挑战。例如，传统的建模方法依赖于导数，即使是少量的测量噪声也会阻碍数值微分。这种方法将动力系统输出的每一次测量都视为一个单独的数据点。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。测量的数据点序列被视为该轨迹的采样、噪声表示，并使用数值积分与基础模型相关联。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态系统，本项目的目的是开发这种数据驱动的辨识和验证技术。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引入，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的版本。这个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稳健和灵活的工具来研究非线性动力系统。在该方法中，轨迹信息通过所谓的占领核嵌入到再生核-希尔伯特空间(RKHS)中。占领核通过一个密集定义的算符--Liouville算符与原始动力学联系在一起。占领核和Liouville算子导致了当代方法的非平凡推广，这些方法通过将有限维非线性优化问题提升到测度空间上的无限维线性规划来研究有限维非线性优化问题。该方法便于在函数空间上而不是在度量空间上提升为线性规划，因此，函数论和逼近论中的工具可用于算法的设计和分析。该项目的具体目标包括：研究RKHS上的占领核和Liouville算子的基本性质，在非线性系统辨识中的应用，研究由Liouville算子的伴随在占领核上的作用所产生的前内积空间，以及该框架在解决运动断层成像问题中的应用。开发的工具将通过解决无人地面、空中或水下航行器的识别和验证问题进行验证。该奖项反映了NSF的法定使命，并已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。

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.