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

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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.