TY - JOUR A1 - Hartmann, Carsten A1 - Schütte, Christof A1 - Weber, Marcus A1 - Zhang, Wei T1 - Importance sampling in path space for diffusion processes with slow-fast variables JF - Probability Theory and Related Fields N2 - Importance sampling is a widely used technique to reduce the variance of a Monte Carlo estimator by an appropriate change of measure. In this work, we study importance sampling in the framework of diffusion process and consider the change of measure which is realized by adding a control force to the original dynamics. For certain exponential type expectation, the corresponding control force of the optimal change of measure leads to a zero-variance estimator and is related to the solution of a Hamilton–Jacobi–Bellmann equation. We focus on certain diffusions with both slow and fast variables, and the main result is that we obtain an upper bound of the relative error for the importance sampling estimators with control obtained from the limiting dynamics. We demonstrate our approximation strategy with an illustrative numerical example. Y1 - 2017 UR - http://rdcu.be/oA51 U6 - https://doi.org/10.1007/s00440-017-0755-3 N1 - If you are not subscribing to the journal please use the http link below as part of the Springer Nature SharedIt initiative SP - 1 EP - 52 ER - TY - GEN A1 - Zhang, Wei A1 - Hartmann, Carsten A1 - Schütte, Christof T1 - Effective Dynamics Along Given Reaction Coordinates, and Reaction Rate Theory N2 - In molecular dynamics and related fields one considers dynamical descriptions of complex systems in full (atomic) detail. In order to reduce the overwhelming complexity of realistic systems (high dimension, large timescale spread, limited computational resources) the projection of the full dynamics onto some reaction coordinates is examined in order to extract statistical information like free energies or reaction rates. In this context, the effective dynamics that is induced by the full dynamics on the reaction coordinate space has attracted considerable attention in the literature. In this article, we contribute to this discussion: We first show that if we start with an ergodic diffusion processes whose invariant measure is unique then these properties are inherited by the effective dynamics. Then, we give equations for the effective dynamics, discuss whether the dominant timescales and reaction rates inferred from the effective dynamics are accurate approximations of such quantities for the full dynamics, and compare our findings to results from approaches like Zwanzig-Mori, averaging, or homogenization. Finally, by discussing the algorithmic realization of the effective dynamics, we demonstrate that recent algorithmic techniques like the ”equation-free” approach and the ”heterogeneous multiscale method” can be seen as special cases of our approach. T3 - ZIB-Report - 16-35 KW - Ergodic diffusion KW - reaction coordinate KW - effective dynamics KW - model reduction KW - equation-free KW - heterogeneous multiscale method Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-59706 SN - 1438-0064 ER - TY - JOUR A1 - Zhang, Wei A1 - Schütte, Christof T1 - Reliable approximation of long relaxation timescales in molecular dynamics JF - Entropy Y1 - 2017 U6 - https://doi.org/10.3390/e19070367 VL - 19 IS - 7 ER - TY - JOUR A1 - Zhang, Wei A1 - Hartmann, Carsten A1 - Schütte, Christof T1 - Effective dynamics along given reaction coordinates, and reaction rate theory JF - Faraday Discussions Y1 - 2016 U6 - https://doi.org/10.1039/C6FD00147E IS - 195 SP - 365 EP - 394 ER - TY - JOUR A1 - Hartmann, Carsten A1 - Richter, Lorenz A1 - Schütte, Christof A1 - Zhang, Wei T1 - Variational characterization of free energy: theory and algorithms JF - Entropy Y1 - 2017 U6 - https://doi.org/10.3390/e19110626 VL - 19 IS - 11 ER - TY - JOUR A1 - Zhang, Wei A1 - Wang, Han A1 - Hartmann, Carsten A1 - Weber, Marcus A1 - Schütte, Christof T1 - Applications of the cross-entropy method to importance sampling and optimal control of diffusions JF - Siam Journal on Scientific Computing Y1 - 2014 U6 - https://doi.org/10.1137/14096493X VL - 36 IS - 6 SP - A2654 EP - A2672 ER - TY - JOUR A1 - Hartmann, Carsten A1 - Schütte, Christof A1 - Zhang, Wei T1 - Model reduction algorithms for optimal control and importance sampling of diffusions JF - Nonlinearity N2 - We propose numerical algorithms for solving optimal control and importance sampling problems based on simplified models. The algorithms combine model reduction techniques for multiscale diffusions and stochastic optimization tools, with the aim of reducing the original, possibly high-dimensional problem to a lower dimensional representation of the dynamics, in which only a few relevant degrees of freedom are controlled or biased. Specifically, we study situations in which either a reaction coordinate onto which the dynamics can be projected is known, or situations in which the dynamics shows strongly localized behavior in the small noise regime. No explicit assumptions about small parameters or scale separation have to be made. We illustrate the approach with simple, but paradigmatic numerical examples. Y1 - 2016 U6 - https://doi.org/10.1088/0951-7715/29/8/2298 VL - 29 IS - 8 SP - 2298 EP - 2326 ER - TY - GEN A1 - Zhang, Wei A1 - Wang, Han A1 - Hartmann, Carsten A1 - Weber, Marcus A1 - Schütte, Christof T1 - Applications of the cross-entropy method to importance sampling and optimal control of diffusions N2 - We study the cross-entropy method for diffusions. One of the results is a versatile cross-entropy algorithm that can be used to design efficient importance sampling strategies for rare events or to solve optimal control problems. The approach is based on the minimization of a suitable cross-entropy functional, with a parametric family of exponentially tilted probability distributions. We illustrate the new algorithm with several numerical examples and discuss algorithmic issues and possible extensions of the method. T3 - ZIB-Report - 14-10 KW - important sampling KW - optimal control KW - cross-entropy method KW - rare events KW - change of measure Y1 - 2014 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-49720 SN - 1438-0064 ER - TY - JOUR A1 - Petkova, Mariela D. A1 - Januszewski, Michał A1 - Blakely, Tim A1 - Herrera, Kristian J. A1 - Schuhknecht, Gregor F.P. A1 - Tiller, Robert A1 - Choi, Jinhan A1 - Schalek, Richard L. A1 - Boulanger-Weill, Jonathan A1 - Peleg, Adi A1 - Wu, Yuelong A1 - Wang, Shuohong A1 - Troidl, Jakob A1 - Vohra, Sumit Kumar A1 - Wei, Donglai A1 - Lin, Zudi A1 - Bahl, Armin A1 - Tapia, Juan Carlos A1 - Iyer, Nirmala A1 - Miller, Zachary T. A1 - Hebert, Kathryn B. A1 - Pavarino, Elisa C. A1 - Taylor, Milo A1 - Deng, Zixuan A1 - Stingl, Moritz A1 - Hockling, Dana A1 - Hebling, Alina A1 - Wang, Ruohong C. A1 - Zhang, Lauren L. A1 - Dvorak, Sam A1 - Faik, Zainab A1 - King, Jr., Kareem I. A1 - Goel, Pallavi A1 - Wagner-Carena, Julian A1 - Aley, David A1 - Chalyshkan, Selimzhan A1 - Contreas, Dominick A1 - Li, Xiong A1 - Muthukumar, Akila V. A1 - Vernaglia, Marina S. A1 - Carrasco, Teodoro Tapia A1 - Melnychuck, Sofia A1 - Yan, TingTing A1 - Dalal, Ananya A1 - DiMartino, James A1 - Brown, Sam A1 - Safo-Mensa, Nana A1 - Greenberg, Ethan A1 - Cook, Michael A1 - Finley, Samantha A1 - Flynn, Miriam A. A1 - Hopkins, Gary Patrick A1 - Kovalyak, Julie A1 - Leonard, Meghan A1 - Lohff, Alanna A1 - Ordish, Christopher A1 - Scott, Ashley L. A1 - Takemura, Satoko A1 - Smith, Claire A1 - Walsh, John J. A1 - Berger, Daniel R. A1 - Pfister, Hanspeter A1 - Berg, Stuart A1 - Knecht, Christopher A1 - Meissner, Geoffrey W. A1 - Korff, Wyatt A1 - Ahrens, Misha B A1 - Jain, Viren A1 - Lichtman, Jeff W. A1 - Engert, Florian T1 - A connectomic resource for neural cataloguing and circuit dissection of the larval zebrafish brain JF - bioRxiv N2 - We present a correlated light and electron microscopy (CLEM) dataset from a 7-day-old larval zebrafish, integrating confocal imaging of genetically labeled excitatory (vglut2a) and inhibitory (gad1b) neurons with nanometer-resolution serial section EM. The dataset spans the brain and anterior spinal cord, capturing >180,000 segmented soma, >40,000 molecularly annotated neurons, and 30 million synapses, most of which were classified as excitatory, inhibitory, or modulatory. To characterize the directional flow of activity across the brain, we leverage the synaptic and cell body annotations to compute region-wise input and output drive indices at single cell resolution. We illustrate the dataset’s utility by dissecting and validating circuits in three distinct systems: water flow direction encoding in the lateral line, recurrent excitation and contralateral inhibition in a hindbrain motion integrator, and functionally relevant targeted long-range projections from a tegmental excitatory nucleus, demonstrating that this resource enables rigorous hypothesis testing as well as exploratory-driven circuit analysis. The dataset is integrated into an open-access platform optimized to facilitate community reconstruction and discovery efforts throughout the larval zebrafish brain. Y1 - 2025 U6 - https://doi.org/10.1101/2025.06.10.658982 ER - TY - GEN A1 - Zhang, Wei A1 - Schütte, Christof T1 - Reliable approximation of long relaxation timescales in molecular dynamics N2 - Many interesting rare events in molecular systems like ligand association, protein folding or con- formational changes happen on timescales that often are not accessible by direct numerical simulation. Therefore rare event approximation approaches like interface sampling, Markov state model building or advanced reaction coordinate based free energy estimation have attracted huge attention recently. In this article we analyze the reliability of such approaches: How precise is an estimate of long relaxation timescales of molecular systems resulting from various forms of rare event approximation methods? Our results give a theoretical answer to this question by relating it with the transfer operator approach to molecular dynamics. By doing so they also allow for understanding deep connections between the different approaches. T3 - ZIB-Report - 17-19 Y1 - 2017 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-63718 SN - 1438-0064 ER - TY - JOUR A1 - Lelièvre, Tony A1 - Zhang, Wei T1 - Pathwise estimates for effective dynamics: the case of nonlinear vectorial reaction coordinates JF - Multiscale Modeling and Simulation N2 - Effective dynamics using conditional expectation was proposed in [F. Legoll and T. Lelièvre, Nonlinearity, 2010] to approximate the essential dynamics of high-dimensional diffusion processes along a given reaction coordinate. The approximation error of the effective dynamics when it is used to approximate the behavior of the original dynamics has been considered in recent years. As a continuation of the previous work [F. Legoll, T. Lelièvre, and S. Olla, Stoch. Process. Appl, 2017], in this paper we obtain pathwise estimates for effective dynamics when the reaction coordinate function is either nonlinear or vector-valued. Y1 - 2018 U6 - https://doi.org/10.1137/18M1186034 IS - 17 SP - 1019 EP - 1051 ER - TY - JOUR A1 - Hartmann, Carsten A1 - Schütte, Christof A1 - Zhang, Wei T1 - Jarzynski's equality, fluctuation theorems, and variance reduction: Mathematical analysis and numerical algorithms JF - Journal of Statistical Physics N2 - In this paper, we study Jarzynski's equality and fluctuation theorems for diffusion processes. While some of the results considered in the current work are known in the (mainly physics) literature, we review and generalize these nonequilibrium theorems using mathematical arguments, therefore enabling further investigations in the mathematical community. On the numerical side, variance reduction approaches such as importance sampling method are studied in order to compute free energy differences based on Jarzynski's equality. Y1 - 2018 U6 - https://doi.org/10.1007/s10955-019-02286-4 VL - 175 IS - 6 SP - 1214 EP - 1261 ER - TY - JOUR A1 - Zhang, Wei T1 - Some new results on relative entropy production, time reversal, and optimal control of time-inhomogeneous diffusion processes JF - Journal of Mathematical Physics N2 - This paper studies time-inhomogeneous nonequilibrium diffusion processes, including both Brownian dynamics and Langevin dynamics. We derive upper bounds of the relative entropy production of the time-inhomogeneous process with respect to the transient invariant probability measures. We also study the time reversal of the reverse process in Crooks' fluctuation theorem. We show that the time reversal of the reverse process coincides with the optimally controlled forward process that leads to zero variance importance sampling estimator based on Jarzynski's equality. Y1 - 2021 U6 - https://doi.org/10.1063/5.0038740 VL - 62 IS - 4 ER - TY - JOUR A1 - Lelievre, Tony A1 - Stoltz, Gabriel A1 - Zhang, Wei T1 - Multiple projection MCMC algorithms on submanifolds JF - IMA Journal of Numerical Analysis N2 - We propose new Markov Chain Monte Carlo algorithms to sample probability distributions on submanifolds, which generalize previous methods by allowing the use of set-valued maps in the proposal step of the MCMC algorithms. The motivation for this generalization is that the numerical solvers used to project proposed moves to the submanifold of interest may find several solutions. We show that the new algorithms indeed sample the target probability measure correctly, thanks to some carefully enforced reversibility property. We demonstrate the interest of the new MCMC algorithms on illustrative numerical examples. Y1 - 2022 U6 - https://doi.org/10.1093/imanum/drac006 ER - TY - JOUR A1 - Zhang, Wei A1 - Li, Tiejun A1 - Schütte, Christof T1 - Solving eigenvalue PDEs of metastable diffusion processes using artificial neural networks JF - Journal of Computational Physics N2 - In this paper, we consider the eigenvalue PDE problem of the infinitesimal generators of metastable diffusion processes. We propose a numerical algorithm based on training artificial neural networks for solving the leading eigenvalues and eigenfunctions of such high-dimensional eigenvalue problem. The algorithm is useful in understanding the dynamical behaviors of metastable processes on large timescales. We demonstrate the capability of our algorithm on a high-dimensional model problem, and on the simple molecular system alanine dipeptide. Y1 - 2021 U6 - https://doi.org/10.1016/j.jcp.2022.111377 VL - 465 ER - TY - JOUR A1 - Zhang, Wei A1 - Klus, Stefan A1 - Conrad, Tim A1 - Schütte, Christof T1 - Learning chemical reaction networks from trajectory data JF - SIAM Journal on Applied Dynamical Systems (SIADS) N2 - We develop a data-driven method to learn chemical reaction networks from trajectory data. Modeling the reaction system as a continuous-time Markov chain and assuming the system is fully observed,our method learns the propensity functions of the system with predetermined basis functions by maximizing the likelihood function of the trajectory data under l^1 sparse regularization. We demonstrate our method with numerical examples using synthetic data and carry out an asymptotic analysis of the proposed learning procedure in the infinite-data limit. Y1 - 2019 U6 - https://doi.org/10.1137/19M1265880 VL - 18 IS - 4 SP - 2000 EP - 2046 ER - TY - JOUR A1 - Zhao, Yue A1 - Zhang, Wei A1 - Li, Tiejun T1 - EPR-Net: Constructing non-equilibrium potential landscape via a variational force projection formulation JF - National Science Review N2 - We present EPR-Net, a novel and effective deep learning approach that tackles a crucial challenge in biophysics: constructing potential landscapes for high-dimensional non-equilibrium steady-state (NESS) systems. EPR-Net leverages a nice mathematical fact that the desired negative potential gradient is simply the orthogonal projection of the driving force of the underlying dynamics in a weighted inner-product space. Remarkably, our loss function has an intimate connection with the steady entropy production rate (EPR), enabling simultaneous landscape construction and EPR estimation. We introduce an enhanced learning strategy for systems with small noise, and extend our framework to include dimensionality reduction and state-dependent diffusion coefficient case in a unified fashion. Comparative evaluations on benchmark problems demonstrate the superior accuracy, effectiveness, and robustness of EPR-Net compared to existing methods. We apply our approach to challenging biophysical problems, such as an 8D limit cycle and a 52D multi-stability problem, which provide accurate solutions and interesting insights on constructed landscapes. With its versatility and power, EPR-Net offers a promising solution for diverse landscape construction problems in biophysics. Y1 - 2024 U6 - https://doi.org/10.1093/nsr/nwae052 VL - 11 IS - 7 ER - TY - JOUR A1 - Akhyar, Fatima-Zahrae A1 - Zhang, Wei A1 - Stoltz, Gabriel A1 - Schütte, Christof T1 - Generative modeling of conditional probability distributions on the level-sets of collective variables N2 - Given a probability distribution $\mu$ in $\mathbb{R}^d$ represented by data, we study in this paper the generative modeling of its conditional probability distributions on the level-sets of a collective variable $\xi: \mathbb{R}^d \rightarrow \mathbb{R}^k$, where $1 \le k