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

CDS

• 批准号: 1622301
• 负责人: Jose Perea
• 金额: $ 10.5万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-09-01 至 2019-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1622301&HistoricalAwards=false
• 关键词: CDS& Collaborative; Research; Machine; Learning

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.

状态随时间变化的对象（称为动态系统）描述了大量自然和工程过程；因此，深入了解他们的行为非常重要。 虽然有时可以推导出描述动态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，因此预测系统随时间变化的能力有限。 此外，当所研究的系统很大或过于复杂且有多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系统。 因此，在缺乏足够的分析模型的情况下，有必要使用传感器来检测动力系统并使用所得数据来了解其特性。 具体来说，动态系统状态的变化往往由一个底层骨架控制，该骨架赋予整体行为形状，因此骨架的形状直接控制系统行为。 大多数时候，底层骨架的这种形状是未知的，很容易被复杂而丰富的系统信号所掩盖。 拓扑数据分析 (TDA) 是一个新兴的领域，它是量化数据形状的数学分支，能够通过提供控制动力学的骨架的高级 X 射线来揭示其他现有方法看不见的信息。 然而，TDA 提供的信息丰富的结构仍然需要进行解释，以便对动态进行分类并预测未来的结果。 为了实现这一目标，主要研究人员将利用机器学习的思想，机器学习是一个研究可以从数据中学习并使用所获得的知识进行分类和预测的算法的研究领域。 然而，目前还不存在能够阐明机器学习如何对使用 TDA 提取的特征进行操作的数学理论。 因此，这项工作将在拓扑数据分析（TDA）、动力系统和机器学习的交叉点上开发必要的、新颖的数学和计算工具。当动态系统通过拓扑数据分析（TDA）生成的描述符进行总结。 尽管这些特征为动力学的演化提供了信息丰富的结构，但当前的文献仅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。 当前进一步探索 TDA 和动力系统之间关系的障碍之一是缺乏可以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）建立一个新颖的、通用的、鲁棒的机器学习框架，用于通过拓扑特征研究动态信号，（2）通过使用更好地理解 TDA 和动态系统之间的关系这些方法在真实和合成数据上的应用，以及（3）将新知识整合到研究人员的教育计划中，这将为装备精良的下一代科学家和工程师提供及时的培训。