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.

中文翻译：

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克服分子动力学中的时间尺度障碍：转移算子、变分原理和机器学习

分子动力学的主要挑战之一是克服“时间尺度障碍”：在许多现实的分子系统中，生物学上重要的罕见转变发生在直接数值模拟无法获得的时间尺度上，即使在最大或专门的超级计算机上也是如此。本文讨论了如何通过过去二十年从动力系统理论、数值数学和机器学习中出现的一系列基于传输算子的技术来规避时间尺度障碍。我们将重点关注如何使用转移算子来近似长时间尺度上的动力学行为，回顾这种方法在分子动力学中的引入，并概述各自的理论，以及早期基于数值的方法的算法发展，通过变分重构，到现代基于数据的技术，利用和改进机器学习的概念。此外，还将解释它与罕见事件模拟技术的关系，揭示分子动力学中长期量的变分原理的广泛等价性。这篇文章将主要从数学角度出发，将实际分子系统的应用留给已经写成的 1000 多篇关于该主题的研究文章。