Optimal Reaction Coordinates: Variational Characterization and Sparse Computation

Bittracher, Andreas; Mollenhauer, Mattes; Koltai, Péter; Schütte, Christof

doi:10.48550/arxiv.2107.10158

Cited by 1 publication

(5 citation statements)

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“…The independence of the convergence rate 1/ √ of the dimension of state space is what gives MC methods an edge over conventional methods, but is only effective if the prefactor Var ( ) does not grow too much with the dimension of state space. The following theorem from Bittracher et al (2021) shows that this is not the case. Theorem 5.9 (convergence rate does not on full dimension).…”

Section: Deep Learning Of Reaction Coordinates and Effective Dynamicsmentioning

confidence: 96%

“…). In fact, with small enough, the process is ( , = 1)reducible and -lumpable and -deflatable with respect to ( ) = ; see Bittracher et al (2021). Moreover, the lumped and deflated transition function…”

Section: Slow Variablesmentioning

confidence: 97%

“…Following Bittracher et al (2021), we will now introduce two seemingly different conditions for a system/reaction coordinate pair. Each condition may be taken individually as a definition of a good reaction coordinate.…”

Section: Variational Characterization Of Reaction Coordinatesmentioning

confidence: 99%

“…Under this condition, we know that and LD allow us to approximately reconstruct the effective long-term dynamics of the full system. Bittracher et al (2021) showed that if the process is weakly ( , , )-reducible as defined in (3.43), then it is also -lumpable with respect to the transition manifold reaction coordinate = E(Q(•)), and that -lumpability and -decomposability are equivalent for reversible processes. Conclusively, we can find good reaction coordinates by looking at lumpability or deflatability.…”

Section: Variational Characterization Of Reaction Coordinatesmentioning

confidence: 99%

“…Progress has been achieved, for example, by using kernel-based methods (Williams, Rowley and Kevrekidis 2015b, Schwantes and Pande 2015, Klus, Bittracher, Schuster and Schütte 2018a, utilizing reproducing kernel Hilbert spaces (RKHSs) (Klus, Schuster and Muandet 2019b) and deep learning approaches to molecular kinetics such as VAMPnet (Mardt, Pasquali, Wu andNoé 2020), and their joint software implementation (Hoffmann et al 2021). The enormous progress in data-based approaches has also inspired a variety of new theoretical and algorithmic approaches to the old problem of finding good reaction coordinates for molecular systems, ranging from its theoretical underpinning via transfer operators (Bittracher et al 2018, Bittracher, Mollenhauer, Koltai andSchütte 2021) to the time-lagged autoencoder approach (Wehmeyer and Noé 2018) or the learning the effective dynamics (LED) approach (Vlachas, Zavadlav, Praprotnik and Koumoutsakos 2022) and many others; see Section 5.5 for more details.…”

Section: Introductionmentioning

confidence: 99%

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

Schütte¹,

Klus²,

Hartmann³

2023

Acta Numerica

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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.

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Section: Deep Learning Of Reaction Coordinates and Effective Dynamicsmentioning

confidence: 96%

Section: Slow Variablesmentioning

confidence: 97%

Section: Variational Characterization Of Reaction Coordinatesmentioning

confidence: 99%

Section: Variational Characterization Of Reaction Coordinatesmentioning

confidence: 99%