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

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.