TY - JOUR A1 - von Kleist, Max A1 - Schütte, Christof A1 - Zhang, Wei T1 - Statistical analysis of the first passage path ensemble of jump processes JF - Journal of Statistical Physics N2 - The transition mechanism of jump processes between two different subsets in state space reveals important dynamical information of the processes and therefore has attracted considerable attention in the past years. In this paper, we study the first passage path ensemble of both discrete-time and continuous-time jump processes on a finite state space. The main approach is to divide each first passage path into nonreactive and reactive segments and to study them separately. The analysis can be applied to jump processes which are non-ergodic, as well as continuous-time jump processes where the waiting time distributions are non-exponential. In the particular case that the jump processes are both Markovian and ergodic, our analysis elucidates the relations between the study of the first passage paths and the study of the transition paths in transition path theory. We provide algorithms to numerically compute statistics of the first passage path ensemble. The computational complexity of these algorithms scales with the complexity of solving a linear system, for which efficient methods are available. Several examples demonstrate the wide applicability of the derived results across research areas. Y1 - 2018 U6 - https://doi.org/10.1007/s10955-017-1949-x VL - 170 SP - 809 EP - 843 ER - TY - JOUR A1 - Zhang, Wei A1 - Schütte, Christof T1 - Understanding recent deep-learning techniques for identifying collective variables of molecular dynamics BT - Special Issue: 93rd Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM) JF - Proceedings in Applied Mathematics and Mechanics N2 - High-dimensional metastable molecular dynamics (MD) can often be characterised by a few features of the system, that is, collective variables (CVs). Thanks to the rapid advance in the area of machine learning and deep learning, various deep learning-based CV identification techniques have been developed in recent years, allowing accurate modelling and efficient simulation of complex molecular systems. In this paper, we look at two different categories of deep learning-based approaches for finding CVs, either by computing leading eigenfunctions of transfer operator associated to the underlying dynamics, or by learning an autoencoder via minimisation of reconstruction error. We present a concise overview of the mathematics behind these two approaches and conduct a comparative numerical study of these two approaches on illustrative examples. Y1 - 2023 U6 - https://doi.org/10.1002/pamm.202300189 VL - 23 IS - 4 ER -