CDS&E: Collaborative Research: Machine Learning on Dynamical Systems via Topological Features

状态随时间变化的对象称为动力学系统，描述了大量的自然和工程过程；因此，深入了解它们的行为是非常重要的。虽然有时可以推导出描述动力系统演化的数学模型，但这些模型几乎总是物理系统的抽象，因此预测系统将如何随时间变化的能力有限。此外，当被调查的系统太大或太复杂，且有几个因素影响其行为时，可能根本不可能用相应的描述方程来描述该系统。因此，在缺乏适当的分析模型的情况下，有必要用传感器测量动力系统，并使用所得到的数据来了解其特征。具体地说，动态系统状态的改变通常由底层骨架控制，该底层骨架赋予整体行为一个形状，因此骨架的形状直接控制系统行为。大多数情况下，底层骨架的这种形状是未知的，可以很容易地被复杂而丰富的系统信号掩盖。拓扑数据分析(TDA)是量化数据形状的数学分支，它通过提供支配动力学的骨架的高水平X射线，能够揭示其他现有方法看不到的信息。然而，TDA提供的信息丰富的结构仍然需要解释，以便对动态进行分类并预测未来结果。为了做到这一点，主要调查人员将利用机器学习的想法，机器学习是一个研究算法的领域，可以从数据中学习，并使用获得的知识进行分类和预测。然而，解释机器学习如何对使用TDA提取的特征进行操作的数学理论目前还不存在。因此，这项工作将在拓扑数据分析(TDA)、动态系统和机器学习的交叉点上开发必要的、新的数学和计算工具。主要研究人员试图理解和制定机器学习的基础，即通过由拓扑数据分析(TDA)生成的描述符来总结动态系统的重要特征。虽然这些特征为动力学的演化提供了一个信息丰富的结构，但目前的文献只利用了可用信息的一小部分来识别、预测和分类不同的动态行为。目前，进一步探索TDA和动力系统之间的关系的障碍之一是缺乏能够对这些结构进行操作的机器学习理论。因此，我们的努力的成功将导致(1)建立一个新颖、通用和健壮的机器学习框架，用于通过拓扑签名来研究动态信号；(2)通过使用这些针对真实和合成数据的方法，更好地理解TDA和动态系统之间的关系；以及(3)将新知识整合到研究人员的教育计划中，这将及时培训装备精良的下一代科学家和工程师。

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.