On metastability and Markov state models for non-stationary molecular dynamics

Koltai, Péter; Ciccotti, Giovanni; Schütte, Christof

doi:10.1063/1.4966157

Cited by 22 publications

(30 citation statements)

References 19 publications

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“…(15), which is meaningful for both dynamics with and without detailed balance [73]. VAMPnets may, in general, use two distinct network lobes to encode the spectral representation of the left and right singular functions (which is important for non-stationary dynamics [115,116]). EDMD with dictionary learning [117] uses a similar architecture as VAMPnets, but is optimized by minimizing the regression error in latent space.…”

Section: Kinetics: Vampnetsmentioning

confidence: 99%

Machine Learning for Molecular Simulation

Noé

Tkatchenko

Müller

et al. 2020

Annu. Rev. Phys. Chem.

633

476

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Machine learning (ML) is transforming all areas of science. The complex and time-consuming calculations in molecular simulations are particularly suitable for a machine learning revolution and have already been profoundly impacted by the application of existing ML methods. Here we review recent ML methods for molecular simulation, with particular focus on (deep) neural networks for the prediction of quantum-mechanical energies and forces, coarse-grained molecular dynamics, the extraction of free energy surfaces and kinetics and generative network approaches to sample molecular equilibrium structures and compute thermodynamics. To explain these methods and illustrate open methodological problems, we review some important principles of molecular physics and describe how they can be incorporated into machine learning structures. Finally, we identify and describe a list of open challenges for the interface between ML and molecular simulation. 1 arXiv:1911.02792v1 [physics.chem-ph] 7 Nov 2019 complex relationship between the input (the pixels) and the output (the labels) that is unknown in its explicit form but can be inferred by a suitable algorithm. Clearly, such an operating principle can be very useful in the description of atomic and molecular systems as well. We know that atomistic configurations dictate the chemical properties, and the machine can learn to associate the latter to the former without solving first principle equations, if presented with enough examples. Although different machine learning tools are available and have been applied to molecular simulation (e.g., kernel methods [2]), here we mostly focus on the use of neural networks, now often synonymously used with the term "deep learning". We assume the reader has basic knowledge of machine learning and we refer to the literature for an introduction to statistical learning theory [3,4] and deep learning [5,6].One of the first applications of machine learning in Chemistry has been to extract classical potential energy surfaces from quantum mechanical (QM) calculations, in order to efficiently perform molecular dynamics (MD) simulations that can incorporate quantum effects. The seminal work of Behler and Parrinello in this direction [7] has opened the way to a now rapidly advancing area of research [8,9,10,11,12,13,14,15]. In addition to atomistic force fields, it has been recently shown that, in the same spirit, effective molecular models at resolution coarser than atomistic can be designed by ML [16,17,18]. Analysis and simulation of MD trajectories has also been affected by ML, for instance for the definition of optimal reaction coordinates [19,20,21,22,23,24], the estimate of free energy surfaces [25,26,27,22], the construction of Markov State Models [21,23,28], and for enhancing MD sampling by learning bias potentials [29,30,31,32,33] or selecting starting configurations by active learning [34,35,36]. Finally, ML can be used to generate samples from the equilibrium distribution of a molecular system without performing MD altogether, as proposed...

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Section: Kinetics: Vampnetsmentioning

confidence: 99%

Machine Learning for Molecular Simulation

Noé

Tkatchenko

Müller

et al. 2020

Annu. Rev. Phys. Chem.

633

476

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“…For later use, let us remark that for non-autonomous systems (or non-homogeneous, in the stochastic language) the transition kernel depends on the starting time t as well, i.e., p τ (t, x, A), which leads to a time-dependent transfer operator P τ (t) by solving the above integral equation for every t ∈ [0, ∞). The analysis of non-autonomous metastable molecular systems via transfer operators only started recently [22,30], while for atmospheric and fluid dynamic problems they have been developed earlier [10,12,13].…”

Section: A Metastabilitymentioning

confidence: 99%

From metastable to coherent sets— Time-discretization schemes

Fackeldey

Koltai

Névir

et al. 2019

Chaos: An Interdisciplinary Journal of Nonlinear Science

Self Cite

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Given a time-dependent stochastic process with trajectories x(t) in a space Ω, there may be sets such that the corresponding trajectories only very rarely cross the boundaries of these sets. We can analyze such a process in terms of metastability or coherence. Metastable sets M are defined in space M ⊂ Ω, coherent sets M (t) ⊂ Ω are defined in space and time. Hence, if we extend the space Ω by the time-variable t, coherent sets are metastable sets in Ω × [0, ∞). This relation can be exploited, because there already exist spectral algorithms for the identification of metastable sets. In this article we show that these well-established spectral algorithms (like PCCA+) also identify coherent sets of non-autonomous dynamical systems. For the identification of coherent sets, one has to compute a discretization (a matrix T) of the transfer operator of the process using a space-timediscretization scheme. The article gives an overview about different time-discretization schemes and shows their applicability in two different fields of application.

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“…Methods such as umbrella sampling [12,13,14,15], metadynamics [16,17,18,19,20], simulated annealing4 [21,22,23,24,25,26], and replica exchange [27,28,29,30,31,32 have grown increasingly popular by aiming to enhance the sampled configurational space through various techniques. In this work we explore the underpinnings of MSMs through the possibility of different algorithmic techniques for solving what is known as the 'embeddability problem' [33,34] in financial literature, by bypassing the embeddability problem through the use of recursive neural networks, and by utilizing Riemannian manifolds to smooth non-metastable discretizations [35]. Here, we present a comparison between six algorithms, used for predicting bond rating transitions, implemented in Inamura [36] and an extension by Marada [37] of Inamura's work with known free energy estimation methods as well as a rate estimating method developed by Hummer et al [38].…”

Section: Introductionmentioning

confidence: 99%

Addressing the Embeddability Problem in Transition Rate Estimation

Goolsby

Moradi

2019

Preprint

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Markov State Models (MSM) and related techniques have gained significant traction as a tool for analyzing and guiding molecular dynamics simulations due to their ability to extract structural, thermodynamic, and kinetic information on proteins using computationally feasible simulations. Here, we revisit the common practice in extracting the thermodynamic and kinetic information from the empirical transition matrix. We propose to build a rate/generator matrix from the empirical transition matrix to provide an alternative approach for estimating both thermodynamic and kinetic quantities. We also discuss a fundamental issue with this approach, known as the embeddability problem and the ways to address this issue. We use a one-dimensional toy model to show the workings of the proposed methods. We also show that the common method of estimating the rates based on the eigendecomposition and our proposed method based on the rate matrix are complementary in that they can potentially provide an upper and a lower limit for transition rates.

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On metastability and Markov state models for non-stationary molecular dynamics

Cited by 22 publications

References 19 publications

Machine Learning for Molecular Simulation

Machine Learning for Molecular Simulation

From metastable to coherent sets— Time-discretization schemes

Addressing the Embeddability Problem in Transition Rate Estimation

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