Machine-Learning Algorithms for Transition Path Theory to Study Rare Transition Events

The study of rare transition events on energy landscape is one of the important research themes in computational and applied mathematics, with far-reaching implications for statistical physics, materials science, and theoretic chemistry. Transition Path Theory (TPT) creates an important theoretical framework based on potential theory to precisely characterize rare events. Although TPT is extremely successful, it faces challenges when applied to complex, high-dimensional systems. Traditional algorithms struggle with the curse of dimensionality and complex landscape scenario, which limits the practical applications. This research aims to develop new algorithms for studying rare transition events on energy landscape, based firmly on theoretical foundations of TPT and leveraging advances in machine learning to extend their reach. The goal is to advance state-ofthe-art computation of key TPT constructs — committor functions — to characterize rare events in challenging new problem domains. The following methodology is the focus of the project. First, using an optimizationbased method, the committor function associated with over-damped Langevin dynamics is calculated using the variational formulation. Two ideas are explored: (i) the importance sampling based adaptive training of committor function by using deep learning and normalizing flow techniques, and (ii) the localized representation of reactive current function by incorporating the multitple pathways. Second, a selfconsistent iterative approach assisted with reactive distribution is suggested to train the committor using flow-based generative models. Validation is based on committer analysis and reaction rate. Comparisons are also carried out using benchmark examples. The application of biology and organic reactions to the real world is also pursued. Overall, this work aims to overcome current limitations and accelerate characterization of rare events in complex, high-dimensional systems by integrating machine learning into the theoretical foundation of TPT. The algorithms developed promise to extend the reach of TPT to previously intractable problems, advancing the frontier of rare event simulation.