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Overcoming the timescale barrier in molecular dynamics: Transfer operators, variational principles and machine learning

Part of: Markov processes Groups and semigroups of linear operators, their generalizations and applications Approximation methods and numerical treatment of dynamical systems Time-dependent statistical mechanics (dynamic and nonequilibrium)

Published online by Cambridge University Press:  11 May 2023

Christof Schütte Open the ORCID record for Christof Schütte [Opens in a new window] ,
Stefan Klus  and
Carsten Hartmann
Christof Schütte
Affiliation:
Zuse Institute Berlin and Freie Universität Berlin, 14195 Berlin, Germany E-mail: schuette@zib.de
Stefan Klus
Affiliation:
Heriot–Watt University, Edinburgh EH14 4AS, UK E-mail: s.klus@hw.ac.uk
Carsten Hartmann
Affiliation:
Brandenburgische Technische Universität Cottbus-Senftenberg, 03046 Cottbus, Germany E-mail: carsten.hartmann@b-tu.de
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Abstract

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One of the main challenges in molecular dynamics is overcoming the ‘timescale barrier’: in many realistic molecular systems, biologically important rare transitions occur on timescales that are not accessible to direct numerical simulation, even on the largest or specifically dedicated supercomputers. This article discusses how to circumvent the timescale barrier by a collection of transfer operator-based techniques that have emerged from dynamical systems theory, numerical mathematics and machine learning over the last two decades. We will focus on how transfer operators can be used to approximate the dynamical behaviour on long timescales, review the introduction of this approach into molecular dynamics, and outline the respective theory, as well as the algorithmic development, from the early numerics-based methods, via variational reformulations, to modern data-based techniques utilizing and improving concepts from machine learning. Furthermore, its relation to rare event simulation techniques will be explained, revealing a broad equivalence of variational principles for long-time quantities in molecular dynamics. The article will mainly take a mathematical perspective and will leave the application to real-world molecular systems to the more than 1000 research articles already written on this subject.

MSC classification

Primary: 37M10: Time series analysis 37M25: Computational methods for ergodic theory (approximation of invariant measures, computation of Lyapunov exponents, entropy) 82C31: Stochastic methods (Fokker-Planck, Langevin, etc.)
Secondary: 47D07: Markov semigroups and applications to diffusion processes 60J35: Transition functions, generators and resolvents 60J60: Diffusion processes
Type
Research Article
Information
Acta Numerica , Volume 32 , May 2023 , pp. 517 - 673
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2023. Published by Cambridge University Press

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