ATD: Fast Bayesian Anomalies Detection in Dynamical System Time-varying Parameters

该项目旨在通过创建一个复杂的统计框架来检测以常微分方程和偏微分方程为特征的动力系统中的异常，从而推进动力系统分析领域。该系统专注于多地点疾病传播和地理空间物理动力学等应用，将有助于快速准确地识别系统中的关键变化。该项目对科学发现（如物理系统）、公共卫生（如传染病）、自然灾害预测（如天气模型）和隐藏的威胁检测（如地球中的地震波）具有重要意义。它促进科学的进步，提高国民健康福利，并通过检测异常来协助国防，从而改善这些关键领域的决策过程。通过与行业合作伙伴和政府机构的合作，该系统的实用性将直接应用于现实世界。此外，该项目非常重视教育和多样性。它将为各级学生提供机会，特别注重在数学，统计，工程和数据科学领域支持妇女和少数民族，从而通过培养未来几代STEM专业人员进一步促进国家繁荣。该项目的核心目标是构建一个快速贝叶斯方法的基础上高斯过程的有效参数估计和异常检测的时变动力系统。该方法通过将高斯过程约束在常微分方程（ODE）解的流形上，克服了传统数值求解器的局限性，大大节省了计算量，并解决了高斯过程和常微分方程之间的理论不相容性.将使用完整贝叶斯方法、迭代分层聚类方法或两步法进行变点检测或识别时变参数的显著变化。值得注意的是，这个项目的范围扩展到高维常微分方程。该项目的预期贡献包括开发能够检测复杂系统中时变参数变化的统计工具，从而推动当前ODE参数推断研究的界限。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

This project aims to advance the field of dynamical system analysis by creating a sophisticated statistical framework to detect anomalies in dynamical systems characterized by ordinary and partial differential equations. Focusing on applications like multi-locale disease spread and geospatial physical dynamics, this system will serve to identify critical changes in systems quickly and accurately. The project has significant implications for scientific discoveries (e.g. physical systems), public health (e.g. infectious diseases), natural disaster prediction (e.g. weather models), and hidden threat detection (e.g. seismic wave in earth). It promotes the progress of science, advances national health welfare, and assists national defense by detecting anomalies and thus improving decision-making processes in these crucial areas. Through collaborations with industry partners and government agencies, the utility of this system will have direct real-world applications. Moreover, the project places a strong emphasis on education and diversity. It will provide opportunities for students at all levels, particularly focusing on supporting women and minorities in the fields of mathematics, statistics, engineering, and data science, thereby further enhancing the national prosperity by nurturing future generations of STEM professionals. The project's core objective is to construct a fast Bayesian method based on Gaussian processes for efficient parameter estimation and anomaly detection in time-varying dynamical systems. This method, which overcomes the limitations of traditional numerical solver for ordinary differential equation (ODE) inference by constraining Gaussian processes on the manifold of ODE solutions, allows for drastic computational savings and resolves theoretical incompatibilities between Gaussian processes and ODEs. Change point detection, or identifying significant shifts in time-varying parameters, will be executed using a full Bayesian approach, an iterative hierarchical clustering method, or a two-step approach. Notably, the scope of this project extends to high-dimensional ODEs. The anticipated contributions of this project include the development of statistical tools capable of detecting changes in time-varying parameters in complex systems, thus pushing the boundaries of current ODE parameter inference research.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.