Objects whose state changes over time, known as dynamical systems, describe a large number of natural and engineered processes; therefore, developing a deeper understanding of their behavior is of great importance. While sometimes it is possible to derive mathematical models that describe the evolution of a dynamical system, these models are almost always an abstraction of the physical system and, therefore, have a limited ability to predict how the system will change in time. Further, when the system under investigation is large or too complicated with several factors influencing its behavior, it may simply be impossible to describe the system with the corresponding descriptive equations. Consequently, in the absence of adequate analytical models it becomes necessary to instrument the dynamical system with sensors and use the resulting data to understand its characteristics. Specifically, the change in the state of a dynamic system is often governed by an underlying skeleton that gives the overall behavior a shape, and thus the shape of the skeleton directly governs the system behavior. Most of the time, this shape of the underlying skeleton is unknown and can be easily masked by the complicated and rich system signals. The emergent field of topological data analysis (TDA), a branch of mathematics that quantifies the shape of data, is capable of revealing information that is invisible to other existing methods by providing a high level X-ray of the skeleton governing the dynamics. However, the information-rich structures provided by TDA still need to be interpreted in order to classify the dynamics and predict future outcomes. To accomplish this, the principal investigators will leverage ideas from machine learning, a field of study that investigates algorithms that can learn from the data and use the acquired knowledge for classification and prediction. However, the mathematical theory that elucidates how machine learning can operate on the features extracted using TDA currently does not exist. Hence, this work will develop the necessary, novel mathematical and computational tools at the intersection of topological data analysis (TDA), dynamical systems, and machine learning.
The principal investigators seek to understand and formulate the foundations of machine learning when the important features of a dynamical system are summarized by descriptors generated with topological data analysis (TDA). Although these signatures provide an information-rich structure for the evolution of the dynamics, current literature has only been utilizing a fraction of the available information in order to identify, predict, and classify different dynamic behavior. One of the current impediments to further exploring the relationship between TDA and dynamical systems is the lack of machine learning theory that can operate on these structures. Therefore, the success of our effort will lead to (1) the establishment of a novel, general, and robust machine learning framework for studying dynamic signals via topological signatures, (2) better understanding of the relationship between TDA and dynamical systems via the use of these methods on real and synthetic data, and (3) the integration of the new knowledge into the investigators' educational programs, which will provide timely training of well-equipped next generation scientists and engineers.
