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 - TY - GEN A1 - Hartmann, Carsten A1 - Richter, Lorenz A1 - Schütte, Christof A1 - Zhang, Wei T1 - Variational characterization of free energy: Theory and algorithms N2 - The article surveys and extends variational formulations of the thermodynamic free energy and discusses their information-theoretic content from the perspective of mathematical statistics. We revisit the well-known Jarzynski equality for nonequilibrium free energy sampling within the framework of importance sampling and Girsanov change-of-measure transformations. The implications of the different variational formulations for designing efficient stochastic optimization and nonequilibrium simulation algorithms for computing free energies are discussed and illustrated. T3 - ZIB-Report - 17-52 KW - Importance sampling KW - Donsker-Varadhan principle KW - thermodynamic free energy KW - nonequilibrium molecular dynamics KW - stochastic approximation KW - cross-entropy method Y1 - 2017 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-65045 SN - 1438-0064 ER - TY - JOUR A1 - Donati, Luca A1 - Schütte, Christof A1 - Weber, Marcus T1 - The Kramers turnover in terms of a macro-state projection on phase space JF - Molecular Physics N2 - We have investigated how Langevin dynamics is affected by the friction coefficient using the novel algorithm ISOKANN, which combines the transfer operator approach with modern machine learning techniques. ISOKANN describes the dynamics in terms of an invariant subspace projection of the Koopman operator defined in the entire state space, avoiding approximations due to dimensionality reduction and discretization. Our results are consistent with the Kramers turnover and show that in the low and moderate friction regimes, metastable macro-states and transition rates are defined in phase space, not only in position space. Y1 - 2025 U6 - https://doi.org/10.1080/00268976.2024.2356748 VL - 123 IS - 7-8: Giovanni Ciccotti: A Renaissance Physicist SP - e2356748 PB - Taylor & Francis ER - TY - CHAP A1 - Sunkara, Vikram A1 - Rostami, Atefe A1 - von Tycowicz, Christoph A1 - Schütte, Christof T1 - Stop throwing away your Decoder; extract the learnt local coordinate system using Latent-XAI T2 - The 4th World Conference on Explainable Artificial Intelligence (XAI-2026) Y1 - 2026 ER - TY - JOUR A1 - Wehlitz, Nathalie A1 - Pavliotis, Grigorios A1 - Schütte, Christof A1 - Winkelmann, Stefanie T1 - Data-driven Reduction of Transfer Operators for Particle Clustering Dynamics N2 - We develop an operator-based framework to coarse-grain interacting particle systems that exhibit clustering dynamics. Starting from the particle-based transfer operator, we first construct a sequence of reduced representations: the operator is projected onto concentrations and then further reduced by representing the concentration dynamics on a geometric low-dimensional manifold and an adapted finite-state discretization. The resulting coarse-grained transfer operator is finally estimated from dynamical simulation data by inferring the transition probabilities between the Markov states. Applied to systems with multichromatic and Morse interaction potentials, the reduced model reproduces key features of the clustering process, including transitions between cluster configurations and the emergence of metastable states. Spectral analysis and transition-path analysis of the estimated operator reveal implied time scales and dominant transition pathways, providing an interpretable and efficient description of particle-clustering dynamics. Y1 - 2026 ER - TY - JOUR A1 - Schütte, Christof A1 - Sikorski, Alexander A1 - Kresse, Jakob A1 - Weber, Marcus T1 - On-the-Fly Lifting of Coarse Reaction-Coordinate Paths to Full-Dimensional Transition Path Ensembles N2 - Effective dynamics on a low-dimensional collective-variable (CV) or latent space can be simulated far more cheaply than the underlying high-dimensional stochastic system, but exploiting such coarse predictions requires lifting: turning a coarse CV trajectory into dynamically consistent full-dimensional states and path ensembles, without relying on global sampling of invariant or conditional fiber measures. We present a local, on-the-fly lifting strategy based on guided full-system trajectories. First an effective model in CV space is used to obtain a coarse reference trajectory. Then, an ensemble of full-dimensional trajectories is generated from a guided version of the original dynamics, where the guidance steers the trajectory to track the CV reference path. Because guidance biases the path distribution, we correct it via pathwise Girsanov reweighting, yielding a correct-by-construction importance-sampling approximation of the conditional law of the uncontrolled dynamics. We further connect the approach to stochastic optimal control, clarifying how coarse models can inform variance-reducing guidance for rare-event quantities. Numerical experiments demonstrate that inexpensive coarse transition paths can be converted into realistic full-system transition pathways (including barrier crossings and detours) and can accelerate estimation of transition pathways and statistics while providing minimal bias through weighted ensembles. Y1 - 2026 U6 - https://doi.org/10.48550/arXiv.2603.23266 ER -