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One of the main challenges in molecular dynamics is overcoming the “timescale barrier”, a phrase used to describe that in many realistic molecular systems, biologically important rare transitions occur on timescales that are not accessible to direct numerical simulation, not 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 behavior 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 MD. 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.

The recent advances in machine learning in various fields of applications can be largely attributed to the rise of deep learning (DL) methods and architectures. Despite being a key technology behind autonomous cars, image processing, speech recognition, etc., a notorious problem remains the lack of theoretical understanding of DL and related interpretability and (adversarial) robustness issues. Understanding the specifics of DL, as compared to, say, other forms of nonlinear regression methods or statistical learning, is interesting from a mathematical perspective, but at the same time it is of crucial importance in practice: treating neural networks as mere black boxes might be sufficient in certain cases, but many applications require waterproof performance guarantees and a deeper understanding of what could go wrong and why it could go wrong. It is probably fair to say that, despite being mathematically well founded as a method to approximate complicated functions, DL is mostly still more like modern alchemy that is firmly in the hands of engineers and computer scientists. Nevertheless, it is evident that certain specifics of DL that could explain its success in applications demands systematic mathematical approaches. In this work, we review robustness issues of DL and particularly bridge concerns and attempts from approximation theory to statistical learning theory. Further, we review Bayesian Deep Learning as a means for uncertainty quantification and rigorous explainability.

A good deal of molecular dynamics simulations aims at predicting and quantifying rare events, such as the folding of a protein or a phase transition. Simulating rare events is often prohibitive, especially if the equations of motion are high-dimensional, as is the case in molecular dynamics. Various algorithms have been proposed for efficiently computing mean first passage times, transition rates or reaction pathways. This article surveys and discusses recent developments in the field of rare event simulation and outlines a new approach that combines ideas from optimal control and statistical mechanics. The optimal control approach described in detail resembles the use of Jarzynski's equality for free energy calculations, but with an optimized protocol that speeds up the sampling, while (theoretically) giving variance-free estimators of the rare events statistics. We illustrate the new approach with two numerical examples and discuss its relation to existing methods.

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.

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.

Applications of the cross-entropy method to importance sampling and optimal control of diffusions
(2014)

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.

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.

Rare but important transition events between long lived states are a key feature of many molecular systems. In many cases the computation of rare event statistics by direct molecular dynamics (MD) simulations is infeasible even on the most powerful computers because of the immensely long simulation timescales needed. Recently a technique for spatial discretization of the molecular state space designed to help overcome such problems, so-called Markov State Models (MSMs), has attracted a lot of attention. We review the theoretical background and algorithmic realization of MSMs and illustrate their use by some numerical examples. Furthermore we introduce a novel approach to using MSMs for the eﬃcient solution of optimal control problems that appear in applications where one desires to optimize molecular properties by means of external controls.