Hostname: page-component-669899f699-vbsjw Total loading time: 0 Render date: 2025-04-29T01:36:06.635Z Has data issue: false hasContentIssue false
  • English
  • Français

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

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

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: [email protected]
Stefan Klus
Affiliation:
Heriot–Watt University, Edinburgh EH14 4AS, UK E-mail: [email protected]
Carsten Hartmann
Affiliation:
Brandenburgische Technische Universität Cottbus-Senftenberg, 03046 Cottbus, Germany E-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

Agapiou, S., Papaspiliopoulos, O., Sanz-Alonso, D. and Stuart, A. M. (2015), Importance sampling: Computational complexity and intrinsic dimension, Statist. Sci. 32, 405–431.Google Scholar
Alford-Lago, D. J., Curtis, C. W., Ihler, A. T. and Issan, O. (2022), Deep learning enhanced dynamic mode decomposition, Chaos 32, 033116.CrossRefGoogle ScholarPubMed
Allen, R. J., Frenkel, D. and Ten Wolde, P. R. (2006), Forward flux sampling-type schemes for simulating rare events: Efficiency analysis, J. Chem. Phys. 124, 194111.Google ScholarPubMed
Allen, R. J., Valeriani, C. and Ten Wolde, P. R. (2009), Forward flux sampling for rare event simulations, J. Phys. Condensed Matter 21, 463102.CrossRefGoogle ScholarPubMed
Allshouse, M. R. and Peacock, T. (2015), Lagrangian based methods for coherent structure detection, Chaos 25, 097617.CrossRefGoogle ScholarPubMed
Andrew, G., Arora, R., Bilmes, J. and Livescu, K. (2013), Deep canonical correlation analysis, in Proceedings of the 30th International Conference on Machine Learning, Vol. 28 of Proceedings of Machine Learning Research, PMLR, pp. 1247–1255.Google Scholar
Arampatzis, G., Katsoulakis, M. A. and Rey-Bellet, L. (2016), Efficient estimators for likelihood ratio sensitivity indices of complex stochastic dynamics, J. Chem. Phys. 144, 104107.CrossRefGoogle ScholarPubMed
Aristoff, D., Lelièvre, T., Mayne, C. G. and Teo, I. (2015), Adaptive multilevel splitting in molecular dynamics simulations, ESAIM Proc. Surveys 48, 215–225.CrossRefGoogle ScholarPubMed
Asmussen, S., Dupuis, P., Rubinstein, R. Y. and Wang, H. (2013), Rare event simulation, in Encyclopedia of Operations Research and Management Science (Gass, S. I. and Fu, M. C., eds), Springer, pp. 1264–1279.CrossRefGoogle Scholar
Avila, A. M. and Mezić, I. (2020), Data-driven analysis and forecasting of highway traffic dynamics, Nature Commun. 11, 2090.CrossRefGoogle ScholarPubMed
Awad, H. P., Glynn, P. W. and Rubinstein, R. Y. (2013), Zero-variance importance sampling estimators for Markov process expectations, Math. Oper. Res. 38, 358–388.CrossRefGoogle Scholar
Ayaz, C., Scalfi, L., Dalton, B. A. and Netz, R. R. (2022), Generalized Langevin equation with a nonlinear potential of mean force and nonlinear memory friction from a hybrid projection scheme, Phys. Rev. E 105, 054138.CrossRefGoogle ScholarPubMed
Ayaz, C., Tepper, L., Brünig, F. N., Kappler, J., Daldrop, J. O. and Netz, R. R. (2021), Non-Markovian modeling of protein folding, Proc . Nat. Acad. Sci. USA 118, e2023856118.CrossRefGoogle Scholar
Bachouch, A., Huré, C., Langrené, N. and Pham, H. (2022), Deep neural networks algorithms for stochastic control problems on finite horizon: Numerical applications, Methodol . Comput. Appl. Probab. 24, 143–178.CrossRefGoogle Scholar
Badowski, T. (2016), Adaptive importance sampling via minimization of estimators of cross-entropy, mean square, and inefficiency constant. Doctoral thesis, Freie Universität Berlin.Google Scholar
Banisch, R. and Hartmann, C. (2016), A sparse Markov chain approximation of LQ-type stochastic control problems, Math. Control Related Fields 6, 363–389.CrossRefGoogle Scholar
Banisch, R. and Koltai, P. (2017), Understanding the geometry of transport: Diffusion maps for Lagrangian trajectory data unravel coherent sets, Chaos 27, 035804.CrossRefGoogle ScholarPubMed
Banisch, R., Conrad, N. D. and Schütte, C. (2015), Reactive flows and unproductive cycles for random walks on complex networks, Eur. Phys. J. Spec. Top. 224, 2369–2387.CrossRefGoogle Scholar
Bartels, C. and Karplus, M. (1997), Multidimensional adaptive umbrella sampling: Applications to main chain and side chain peptide conformations, J. Comput. Chem. 18, 1450–1462.3.0.CO;2-I>CrossRefGoogle Scholar
Beauchamp, K. A., Bowman, G. R., Lane, T. J., Maibaum, L., Haque, I. S. and Pande, V. S. (2011), MSMBuilder2: Modeling conformational dynamics at the picosecond to millisecond scale, J. Chem. Theory Comput. 7, 3412–3419.CrossRefGoogle ScholarPubMed
Beccara, S., Skrbic, T., Covino, R. and Faccioli, P. (2012), Dominant folding pathways of a WW domain, Proc . Nat. Acad. Sci. USA 109, 2330–2335.CrossRefGoogle Scholar
Beck, C., Jentzen, A. et al. (2019), Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations, J. Nonlinear Sci. 29, 1563–1619.CrossRefGoogle Scholar
Belkacemi, Z., Gkeka, P., Lelièvre, T. and Stoltz, G. (2022), Chasing collective variables using autoencoders and biased trajectories, J. Chem. Theory Comput. 18, 59–78.CrossRefGoogle ScholarPubMed
Bello-Rivas, J. and Elber, R. (2015), Exact milestoning, J. Chem. Phys. 142, 094102.CrossRefGoogle ScholarPubMed
Beltran, J. and Landim, C. (2010), Tunneling and metastability of continuous time Markov chains, J. Statist. Phys. 140, 1065–1114.CrossRefGoogle Scholar
Beltran, J. and Landim, C. (2013), A martingale approach to metastability, Probab . Theory Related Fields 161, 267–307.CrossRefGoogle Scholar
Bender, C. and Moseler, T. (2010), Importance sampling for backward SDEs, Stoch . Anal. Appl. 28, 226–253.Google Scholar
Bender, C. and Steiner, J. (2012), Least-squares Monte Carlo for backward SDEs, in Numerical Methods in Finance (Carmona, R. A. et al., eds), Springer, pp. 257–289.Google Scholar
Bengtsson, T., Bickel, P. and Li, B. (2008), Curse-of-dimensionality revisited: Collapse of the particle filter in very large scale systems, in Probability and Statistics: Essays in Honor of David A. Freedman, Institute of Mathematical Statistics, pp. 316–334.Google Scholar
Berezhkovskii, A. M. and Szabo, A. (2019), Committors, first-passage times, fluxes, Markov states, milestones, and all that, J. Chem. Phys. 150, 054106.CrossRefGoogle Scholar
Berglund, N. (2013), Kramers’ law: Validity, derivations and generalisations, Markov Process . Related Fields 19, 459–490.Google Scholar
Bhakat, S. (2022), Collective variable discovery in the age of machine learning: Reality, hype and everything in between, RSC Adv. 12, 25010–25024.CrossRefGoogle ScholarPubMed
Bianchi, A. and Gaudillière, A. (2016), Metastable states, quasi-stationary distributions and soft measures, Stoch . Process. Appl. 126, 1622–1680.CrossRefGoogle Scholar
Birrell, J. and Rey-Bellet, L. (2020), Uncertainty quantification for Markov processes via variational principles and functional inequalities, SIAM/ASA J. Uncertain. Quantif. 8, 539–572.CrossRefGoogle Scholar
Birrell, J., Katsoulakis, M. A. and Rey-Bellet, L. (2021), Quantification of model uncertainty on path-space via goal-oriented relative entropy, ESAIM Math. Model. Numer. Anal. 55, 131–169.CrossRefGoogle Scholar
Bittracher, A. and Schütte, C. (2020), A weak characterization of slow variables in stochastic dynamical systems, in Advances in Dynamics, Optimization and Computation (SON 2020) (Junge, O. et al., eds), Springer, pp. 132–150.CrossRefGoogle Scholar
Bittracher, A., Koltai, P., Klus, S., Banisch, R., Dellnitz, M. and Schütte, C. (2018), Transition manifolds of complex metastable systems, J. Nonlinear Sci. 28, 471–512.CrossRefGoogle ScholarPubMed
Bittracher, A., Mollenhauer, M., Koltai, P. and Schütte, C. (2021), Optimal reaction coordinates: Variational characterization and sparse computation. Available at arXiv:2107.10158 (to appear in SIAM J. Multiscale Model. Simul.).Google Scholar
Bolhuis, P. G. and Swenson, D. W. H. (2021), Transition path sampling as Markov chain Monte Carlo of trajectories: Recent algorithms, software, applications, and future outlook, Adv . Theory Simul. 4, 2000237.CrossRefGoogle Scholar
Bolhuis, P. G., Chandler, D., Dellago, C. and Geissler, P. (2002), Transition path sampling: Throwing ropes over mountain passes, in the dark, Annu . Rev. Phys. Chem. 59, 291–318.CrossRefGoogle Scholar
Bonati, L., Piccini, G. and Parrinello, M. (2021), Deep learning the slow modes for rare events sampling, Proc . Nat. Acad. Sci. USA 118, e2113533118.CrossRefGoogle Scholar
Bond, S. D., Benedict, B. B. L. and Leimkuhler, J. (1999), The Nosé–Poincaré method for constant temperature molecular dynamics, J. Comput. Phys. 151, 114–134.CrossRefGoogle Scholar
Borrell, E. R., Quer, J., Richter, L. and Schütte, C. (2022), Improving control based importance sampling strategies for metastable diffusions via adapted metadynamics. Available at arXiv:2206.06628.Google Scholar
Bou-Rabee, N. and Vanden-Eijnden, E. (2010), Pathwise accuracy and ergodicity of Metropolized integrators for SDEs, Commun . Pure Appl. Math. 63, 655–696.Google Scholar
Boué, M. and Dupuis, P. (1998), A variational representation for certain functionals of Brownian motion, Ann . Probab. 26, 1641–1659.CrossRefGoogle Scholar
Bovier, A. and Den Hollander, F. (2016), Metastability: A Potential-Theoretic Approach, Vol. 351 of Grundlehren der mathematischen Wissenschaften, Springer.Google Scholar
Bovier, A., Eckhoff, M., Gayrard, V. and Klein, M. (2002a), Metastability and low lying spectra in reversible Markov chains, Comm. Math. Phys. 228, 219–255.CrossRefGoogle Scholar
Bovier, A., Eckhoff, M., Gayrard, V. and Klein, M. (2002b), Metastability in reversible diffusion processes I: Sharp asymptotics for capacities and exit times, J. Eur. Math. Soc. 6, 399–424.Google Scholar
Bovier, A., Gayrard, V. and Klein, M. (2002c), Metastability in reversible diffusion processes II: Precise asymptotics for small eigenvalues, J. Eur. Math. Soc. 7, 69–99.Google Scholar
Bowman, G. R., Pande, V. S. and Noé, F., eds (2014), An Introduction to Markov State Models and Their Application to Long Timescale Molecular Simulation, Vol. 797 of Advances in Experimental Medicine and Biology, Springer.CrossRefGoogle Scholar
Bowman, G. R., Volez, V. and Pande, V. S. (2011), Taming the complexity of protein folding, Curr . Opinion Struct. Biol. 21, 4–11.CrossRefGoogle Scholar
Bris, C. L., Lelièvre, T., Luskin, M. and Perez, D. (2012), A mathematical formalization of the parallel replica dynamics, Monte Carlo Methods Appl. 18, 119–146.Google Scholar
Brunton, S. L., Proctor, J. L. and Kutz, J. N. (2016), Discovering governing equations from data by sparse identification of nonlinear dynamical systems, Proc . Nat. Acad. Sci. USA 113, 3932–3937.CrossRefGoogle Scholar
Bussi, G., Laio, A. and Tiwary, P. (2020), Metadynamics: A unified framework for accelerating rare events and sampling thermodynamics and kinetics, in Handbook of Materials Modeling – Methods: Theory and Modeling (Andreoni, W. and Yip, S., eds), Springer, pp. 565–595.CrossRefGoogle Scholar
Carroll, J. D. and Chang, J. J. (1970), Analysis of individual differences in multidimensional scaling via an N-way generalization of ‘Eckart–Young’ decomposition, Psychometrika 35, 283–319.CrossRefGoogle Scholar
Cérou, F. and Guyader, A. (2007), Adaptive multilevel splitting for rare event analysis, Stoch . Anal. Appl. 25, 417–443.Google Scholar
Cérou, F., Del Moral, P., Furon, T. and Guyader, A. (2012), Sequential Monte Carlo for rare event estimation, Statist . Comput. 22, 795–808.Google Scholar
Chandler, D. (1998), Finding Transition Pathways: Throwing Ropes Over Rough Mountain Passes, in the Dark, World Scientific.Google Scholar
Chodera, J. D., Swope, W. C., Pitera, J. W. and Dill, K. A. (2006), Long-time protein folding dynamics from short-time molecular dynamics simulations, Multiscale Model . Simul. 5, 1214–1226.Google Scholar
Chorin, A. J., Hald, O. H. and Kupferman, R. (2000), Optimal prediction and the Mori–Zwanzig representation of irreversible processes, Proc . Nat. Acad. Sci. USA 97, 2968–2973.CrossRefGoogle Scholar
Comer, J., Gumbart, J. C., Hénin, J., Lelièvre, T., Pohorille, A. and Chipot, C. (2015), The adaptive biasing force method: Everything you always wanted to know but were afraid to ask, J. Phys. Chem. B 119, 1129–1151.CrossRefGoogle ScholarPubMed
Conrad, N. D., Sarich, M. and Schütte, C. (2012), Estimating the eigenvalue error of Markov state models, Multiscale Model . Simul. 10, 61–81.Google Scholar
Conrad, N. D., Weber, M. and Schütte, C. (2015), Finding dominant structures of nonreversible Markov processes, SIAM J. Mult. Model. Simul. 14, 1319–1340.CrossRefGoogle Scholar
Crooks, G. (1999), Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences, Phys. Rev. E 60, 2721–2726.CrossRefGoogle ScholarPubMed
Pra, P. Dai, Meneghini, L. and Runggaldier, W. (1996), Connections between stochastic control and dynamic games, Math. Control Signals Systems 9, 303–326.CrossRefGoogle Scholar
Davies, E. B. (1982a), Metastable states of symmetric Markov semigroups I, Proc. London Math. Soc. s3-45, 133–150.CrossRefGoogle Scholar
Davies, E. B. (1982b), Metastable states of symmetric Markov semigroups II, J. London Math. Soc. s2-26, 541–556.CrossRefGoogle Scholar
Davis, C. and Kahan, W. M. (1970), The rotation of eigenvectors by a perturbation III, SIAM J. Numer. Anal. 7, 1–46.CrossRefGoogle Scholar
Del Moral, P. (2004), Feynman–Kac Formulae: Genealogical and Interacting Particle Systems with Applications, Probability and its Applications, Springer.CrossRefGoogle Scholar
Delbaen, F., Hu, Y. and Richou, A. (2011), On the uniqueness of solutions to quadratic BSDEs with convex generators and unbounded terminal conditions, Ann . Inst. H. Poincaré Probab. Statist. 47, 559–574.Google Scholar
Dellnitz, M. and Junge, O. (1998), An adaptive subdivision technique for the approximation of attractors and invariant measures, Comput . Vis. Sci. 1, 63–68.CrossRefGoogle Scholar
Dellnitz, M. and Junge, O. (1999), On the approximation of complicated dynamical behavior, SIAM J. Numer. Anal. 36, 491–515.CrossRefGoogle Scholar
Deuflhard, P. and Weber, M. (2005), Robust Perron cluster analysis in conformation dynamics, Linear Algebra Appl. 398, 161–184.CrossRefGoogle Scholar
Deuflhard, P., Dellnitz, M., Junge, O. and Schütte, C. (1999), Computation of essential molecular dynamics by subdivision techniques, in Computational Molecular Dynamics: Challenges, Methods, Ideas, Vol. 4 of Lecture Notes in Computational Science and Engineering, Springer, pp. 98–115.CrossRefGoogle Scholar
Di Gesu, G., Lelièvre, T., Peutrec, D. and Nectoux, B. (2016), Jump Markov models and transition state theory: The quasi-stationary distribution approach, Faraday Discuss. 195, 469–495.CrossRefGoogle ScholarPubMed
Donati, L., Heida, M., Keller, B. G. and Weber, M. (2018), Estimation of the infinitesimal generator by square-root approximation, J. Phys. Condensed Matter 30, 425201.CrossRefGoogle ScholarPubMed
Donati, L., Weber, M. and Keller, B. G. (2021), Markov models from the square root approximation of the Fokker–Planck equation: Calculating the grid-dependent flux, J. Phys. Condensed Matter 33, 115902.CrossRefGoogle ScholarPubMed
Donsker, M. D. and Varadhan, S. R. S. (1975), On a variational formula for the principal eigenvalue for operators with maximum principle, Proc . Nat. Acad. Sci. USA 72, 780–783.CrossRefGoogle Scholar
Doob, J. L. (1984), Classical Potential Theory and its Probabilistic Counterpart, Vol. 262 of Grundlehren der Mathematischen Wissenschaften, Springer.CrossRefGoogle Scholar
Down, D., Meyn, S. P. and Tweedie, R. L. (1995), Exponential and uniform ergodicity of Markov processes, Ann . Probab. 23, 1671–1691.CrossRefGoogle Scholar
Dupuis, P. and Wang, H. (2004), Importance sampling, large deviations, and differential games, Stochastics 76, 481–508.Google Scholar
Dupuis, P. and Wang, H. (2007), Subsolutions of an Isaacs equation and efficient schemes for importance sampling, Math. Oper. Res. 32, 723–757.CrossRefGoogle Scholar
Dupuis, P., Katsoulakis, M. A., Pantazis, Y. and Plechác̆, P. (2016), Path-space information bounds for uncertainty quantification and sensitivity analysis of stochastic dynamics, SIAM/ASA J. Uncertain. Quantif. 4, 80–111.CrossRefGoogle Scholar
Dupuis, P., Katsoulakis, M. A., Pantazis, Y. and Rey-Bellet, L. (2020), Sensitivity analysis for rare events based on Rényi divergence, Ann . Appl. Probab. 30, 1507–1533.Google Scholar
Dupuis, P., Liu, Y., Plattner, N. and Doll, J. D. (2012), On the infinite swapping limit for parallel tempering, Multiscale Model . Simul. 10, 986–1022.Google Scholar
Dupuis, P., Spiliopoulos, K. and Zhou, X. (2015), Escaping from an attractor: Importance sampling and rest points I, Ann . Appl. Probab. 25, 2909–2958.Google Scholar
W, E and Vanden-Eijnden, E. (2004), Metastability, conformation dynamics, and transition pathways in complex systems, in Multiscale Modelling and Simulation, Vol. 39 of Lecture Notes in Computational Science and Engineering, Springer, pp. 35–68.Google Scholar
W, E and Vanden-Eijnden, E. (2006), Towards a theory of transition paths, J. Statist. Phys. 123, 503–523.Google Scholar
W, E and Vanden-Eijnden, E. (2010), Transition-path theory and path-finding algorithms for the study of rare events, Annu . Rev. Phys. Chem. 61, 391–420.Google Scholar
W, E, Han, J. and Jentzen, A. (2017), Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations, Commun . Math. Statist. 5, 349–380.Google Scholar
W, E, Han, J. and Jentzen, A. (2021), Algorithms for solving high dimensional PDEs: From nonlinear Monte Carlo to machine learning, Nonlinearity 35, 278–310.Google Scholar
Earl, D. J. and Deem, M. W. (2005), Parallel tempering: Theory, applications, and new perspectives, Phys. Chem. Chem. Phys. 7, 3910–3916.CrossRefGoogle ScholarPubMed
Ellis, R. S. (1985), Entropy, Large Deviations and Statistical Mechanics, Springer.CrossRefGoogle Scholar
Eyring, H. (1935), The activated complex in chemical reactions, J. Chem. Phys. 3, 107–115.CrossRefGoogle Scholar
Faccioli, P., Lonardi, A. and Orland, H. (2010), Dominant reaction pathways in protein folding: A direct validation against molecular dynamics simulations, J. Chem. Phys. 133, 045104.CrossRefGoogle ScholarPubMed
Faradjian, A. K. and Elber, R. (2004), Computing time scales from reaction coordinates by milestoning, J. Chem. Phys. 120, 10880–10889.CrossRefGoogle ScholarPubMed
Federer, H. (1969), Geometric Measure Theory, Springer.Google Scholar
Fleming, W. H. (2006), Risk sensitive stochastic control and differential games, Commun . Inf. Syst. 6, 161–177.Google Scholar
Fleming, W. H. and Soner, H. M. (2006), Controlled Markov Processes and Viscosity Solutions, Springer.Google Scholar
Frank, A.-S., Sikorski, A. and Röblitz, S. (2022), Spectral clustering of Markov chain transition matrices with complex eigenvalues. Available at arXiv:2206.14537.Google Scholar
Freidlin, M. and Wentzell, A. D. (1998), Random Perturbations of Dynamical Systems, Springer.CrossRefGoogle Scholar
Froyland, G. (2013), An analytic framework for identifying finite-time coherent sets in time-dependent dynamical systems, Phys. D 250, 1–19.Google Scholar
Froyland, G. and Junge, O. (2018), Robust FEM-based extraction of finite-time coherent sets using scattered, sparse, and incomplete trajectories, SIAM J. Appl. Dyn. Syst. 17, 1891–1924.CrossRefGoogle Scholar
Froyland, G., Gottwald, G. and Hammerlindl, A. (2013), A computational method to extract macroscopic variables and their dynamics in multiscale systems, SIAM J. Appl. Dyn. Syst. 13, 1816–1846.CrossRefGoogle Scholar
Ge, Y. and Voelz, V. A. (2021), Markov state models to elucidate ligand binding mechanism, Methods Mol . Biol. 2266, 239–259.Google Scholar
Gelß, P. (2017), The tensor-train format and its applications: Modeling and analysis of chemical reaction networks, catalytic processes, fluid flows, and Brownian dynamics. Doctoral thesis, Freie Universität Berlin.Google Scholar
Gobet, E. and Turkedjiev, P. (2016), Linear regression MDP scheme for discrete backward stochastic differential equations under general conditions, Math. Comp. 85, 1359–1391.CrossRefGoogle Scholar
Gobet, E., Lemor, J.-P. and Warin, X. (2005), A regression-based Monte Carlo method to solve backward stochastic differential equations, Ann . Appl. Probab. 15, 2172–2202.Google Scholar
Grabert, H. (1982), Projection Operator Techniques in Nonequilibrium Statistical Mechanics, Springer Tracts in Modern Physics, Springer.CrossRefGoogle Scholar
Graham, C. and Talay, D. (2013), Stochastic Simulation and Monte Carlo Methods, Springer.CrossRefGoogle Scholar
Grubmüller, H. (1995), Predicting slow structural transitions in macromolecular systems: Conformational flooding, Phys. Rev. E 52, 2893.CrossRefGoogle ScholarPubMed
Gyöngy, I. and Martínez, T. (2001), On stochastic differential equations with locally unbounded drift, Czechoslovak Math. J. 51, 763–783.CrossRefGoogle Scholar
Hackbusch, W. (2014), Numerical tensor calculus, Acta Numer. 23, 651–742.CrossRefGoogle Scholar
Hamelberg, D., Mongan, J. and McCammon, J. A. (2004), Accelerated molecular dynamics: A promising and efficient simulation method for biomolecules, J. Chem. Phys. 120, 11919.CrossRefGoogle ScholarPubMed
Han, J., Lu, J. and Zhou, M. (2020), Solving high-dimensional eigenvalue problems using deep neural networks: A diffusion Monte Carlo like approach, J. Comput. Phys. 423, 109792.CrossRefGoogle Scholar
Hartmann, C. and Richter, L. (2021), Nonasymptotic bounds for suboptimal importance sampling. Available at arXiv:2102.09606.Google Scholar
Hartmann, C. and Schütte, C. (2012), Efficient rare event simulation by optimal nonequilibrium forcing, J. Statist. Mech. Theory Exp. 2012, P11004.CrossRefGoogle Scholar
Hartmann, C., Kebiri, O., Neureither, L. and Richter, L. (2019), Variational approach to rare event simulation using least-squares regression, Chaos 29, 063107.CrossRefGoogle ScholarPubMed
Hartmann, C., Neureither, L. and Sharma, U. (2020), Coarse graining of nonreversible stochastic differential equations: Quantitative results and connections to averaging, SIAM J. Math. Anal. 52, 2689–2733.CrossRefGoogle Scholar
Hartmann, C., Richter, L., Schütte, C. and Zhang, W. (2017), Variational characterization of free energy: Theory and algorithms, Entropy 19, 626.CrossRefGoogle Scholar
Hartmann, C., Schütte, C. and Zhang, W. (2016), Model reduction algorithms for optimal control and importance sampling of diffusions, Nonlinearity 29, 2298.CrossRefGoogle Scholar
Heida, M. (2018), Convergences of the squareroot approximation scheme to the Fokker–Planck operator, Math. Models Methods Appl. Sci. 28, 2599–2635.CrossRefGoogle Scholar
Hérau, F., Hitrik, M. and Sjöstrand, J. (2008), Tunnel effect for Kramers–Fokker–Planck type operators: Return to equilibrium and applications, Int. Math. Res. Not. 2008, rnn057.Google Scholar
Hérau, F., Hitrik, M. and Sjöstrand, J. (2010), Tunnel effect and symmetries for Kramers–Fokker–Planck type operators. Available at arXiv:1007.0838v1 [math.SP].CrossRefGoogle Scholar
Hijón, C., Español, P., Vanden-Eijnden, E. and Delgado-Buscalioni, R. (2010), Mori–Zwanzig formalism as a practical computational tool, Faraday Discuss. 144, 301–322.CrossRefGoogle ScholarPubMed
Hoffmann, M., Scherer, M., Hempel, T., Mardt, A., de Silva, B., Husic, B. E., Klus, S., Wu, H., Kutz, N., Brunton, S. L. and Noé, F. (2021), Deeptime: A Python library for machine learning dynamical models from time series data, Mach . Learn. Sci. Technol. 3, 015009.CrossRefGoogle Scholar
Hotelling, H. (1936), Relations between two sets of variates, Biometrika 28, 321–377.CrossRefGoogle Scholar
Hua, J., Noorian, F., Moss, D., Leong, P. H. W. and Gunaratne, G. H. (2017), High-dimensional time series prediction using kernel-based Koopman mode regression, Nonlinear Dynam. 90, 1785–1806.CrossRefGoogle Scholar
Huang, X., Yao, Y., Bowman, G., Sun, J., Guibas, L. J., Carlsson, G. and Pande, V. (2010), Constructing multi-resolution Markov state models (MSMs) to elucidate RNA hairpin folding mechanisms, in Pacific Symposium on Biocomputing 2010 (PSB 2010) (Altman, R. B. et al., eds), World Scientific, pp. 228–239.Google Scholar
Huisinga, W. (2001), Metastability of Markovian systems: A transfer operator approach in application to molecular dynamics. Doctoral thesis, Freie Universität Berlin.Google Scholar
Huisinga, W. and Schmidt, B. (2002), Metastability and dominant eigenvalues of transfer operators, in Advances in Algorithms for Macromolecular Simulation (Chipot, C. et al., eds), Vol. 49 of Lecture Notes in Computational Science and Engineering, Springer.Google Scholar
Huisinga, W., Meyn, S. and Schütte, C. (2004), Phase transitions & metastability in Markovian and molecular systems, Ann . Appl. Probab. 14, 419–458.Google Scholar
Huré, C., Pham, H., Bachouch, A. and Langrené, N. (2021), Deep neural networks algorithms for stochastic control problems on finite horizon: Convergence analysis, SIAM J. Numer. Anal. 59, 525–557.CrossRefGoogle Scholar
Husic, B. E. and Pande, V. S. (2018), Markov state models: From an art to a science, J. Amer. Chem. Soc. 140, 2386.CrossRefGoogle ScholarPubMed
Hussain, S. and Haji-Akbari, A. (2020), Studying rare events using forward-flux sampling: Recent breakthroughs and future outlook, J. Chem. Phys. 152, 060901.CrossRefGoogle ScholarPubMed
Invernizzi, M. and Parrinello, M. (2020), Rethinking metadynamics: From bias potentials to probability distributions, J. Phys. Chem. Lett. 11, 2731–2736.CrossRefGoogle ScholarPubMed
Jarzynski, C. (1997), Nonequilibrium equality for free energy differences, Phys. Rev. Lett. 78, 2690–2693.CrossRefGoogle Scholar
Juneja, S. and Shahabuddin, P. (2006), Rare-event simulation techniques: An introduction and recent advances, in Simulation (Henderson, S. G. and Nelson, B. L., eds), Vol. 13 of Handbooks in Operations Research and Management Science, Elsevier, pp. 291–350.Google Scholar
Kappler, J., Daldrop, J. O., Brünig, F. N., Boehle, M. D. and Netz, R. (2018), Memory-induced acceleration and slowdown of barrier crossing, J. Chem. Phys. 148, 014903.CrossRefGoogle ScholarPubMed
Karatzas, I. and Shreve, S. E. (1991), Brownian Motion and Stochastic Calculus, Graduate Texts in Mathematics, Springer.Google Scholar
Kato, T. (1995), Perturbation Theory for Linear Operators, Springer.CrossRefGoogle Scholar
Kebiri, O., Neureither, L. and Hartmann, C. (2019), Adaptive importance sampling with forward–backward stochastic differential equations, in Stochastic Dynamics Out of Equilibrium (IHPStochDyn 2017) (Giacomin, G. et al., eds), Vol. 282 of Proceedings in Mathematics & Statistics, Springer, pp. 265–281.CrossRefGoogle Scholar
Keller, B., Prinz, J.-H. and Noé, F. (2011), Markov models and dynamical fingerprints: Unraveling the complexity of molecular kinetics, Chem. Phys. 396, 92–107.CrossRefGoogle Scholar
Kevrekidis, I. G. and Samaey, G. (2009), Equation-free multiscale computation: Algorithms and applications, Annu . Rev. Phys. Chem. 60, 321–344.CrossRefGoogle Scholar
Klus, S. and Conrad, N. D. (2022), Koopman-based spectral clustering of directed and time-evolving graphs, J. Nonlinear Sci. 33, 8.CrossRefGoogle Scholar
Klus, S., Bittracher, A., Schuster, I. and Schütte, C. (2018a), A kernel-based approach to molecular conformation analysis, J. Chem. Phys. 149, 244109.CrossRefGoogle ScholarPubMed
Klus, S., Husic, B. E., Mollenhauer, M. and Noé, F. (2019a), Kernel methods for detecting coherent structures in dynamical data, Chaos 29, 123112.CrossRefGoogle ScholarPubMed
Klus, S., Koltai, P. and Schütte, C. (2016), On the numerical approximation of the Perron–Frobenius and Koopman operator, J. Comput. Dyn. 3, 51–79.Google Scholar
Klus, S., Nüske, F. and Hamzi, B. (2020a), Kernel-based approximation of the Koopman generator and Schrödinger operator, Entropy 22, 722.CrossRefGoogle ScholarPubMed
Klus, S., Nüske, F. and Peitz, S. (2022), Koopman analysis of quantum systems, J. Phys. A Math. Theor. 55, 314002.CrossRefGoogle Scholar
Klus, S., Nüske, F., Koltai, P., Wu, H., Kevrekidis, I., Schütte, C. and Noé, F. (2018b), Data-driven model reduction and transfer operator approximation, J. Nonlinear Sci. 28, 985–1010.CrossRefGoogle Scholar
Klus, S., Nüske, F., Peitz, S., Niemann, J.-H., Clementi, C. and Schütte, C. (2020b), Data-driven approximation of the Koopman generator: Model reduction, system identification, and control, Phys. D 406, 132416.CrossRefGoogle Scholar
Klus, S., Schuster, I. and Muandet, K. (2019b), Eigendecompositions of transfer operators in reproducing kernel Hilbert spaces, J. Nonlinear Sci. 30, 283–315.CrossRefGoogle Scholar
Kobylanski, M. (2000), Backward stochastic differential equations and partial differential equations with quadratic growth, Ann . Probab. 28, 558–602.CrossRefGoogle Scholar
Koltai, P., Wu, H., Noé, F. and Schütte, C. (2018), Optimal data-driven estimation of generalized Markov state models for non-equilibrium dynamics, Computation 6, 22.CrossRefGoogle Scholar
Kontoyiannis, I. and Meyn, S. P. (2003), Spectral theory and limit theorems for geometrically ergodic Markov processes, Ann . Appl. Probab. 13, 304–362.Google Scholar
Kontoyiannis, I. and Meyn, S. P. (2012), Geometric ergodicity and the spectral gap of non-reversible Markov chains, Probab . Theory Related Fields 154, 327–339.CrossRefGoogle Scholar
Korda, M. and Mezić, I. (2018a), Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control, Automatica 93, 149–160.CrossRefGoogle Scholar
Korda, M. and Mezic, I. (2018b), On convergence of extended dynamic mode decomposition to the Koopman operator, J. Nonlinear Sci. 28, 687–710.CrossRefGoogle Scholar
Kressner, D. (2006), Block algorithms for reordering standard and generalized Schur forms, ACM Trans. Math. Softw. 32, 521–532.CrossRefGoogle Scholar
Krivov, S. V. (2018), Protein folding free energy landscape along the committor: The optimal folding coordinate, J. Chem. Theory Comput. 14, 3418–3427.CrossRefGoogle ScholarPubMed
Krylov, N. V. (2019), A few comments on a result of A. Novikov and Girsanov’s theorem, Stochastics 91, 1186–1189.CrossRefGoogle Scholar
Lange, O. F. and Grubmüller, H. (2006), Collective Langevin dynamics of conformational motions in proteins, J. Chem. Phys. 124, 214903.CrossRefGoogle ScholarPubMed
Lasota, A. and Mackey, M. C. (1994), Chaos, Fractals and Noise, Vol. 97 of Applied Mathematical Sciences, second edition, Springer.CrossRefGoogle Scholar
Latorre, J., Metzner, P., Hartmann, C. and Schütte, C. (2011), A structure-preserving numerical discretization of reversible diffusions, Commun . Math. Sci. 9, 1051–1072.Google Scholar
L’Ecuyer, P., Mandjes, M. and Tuffin, B. (2009), Importance Sampling in Rare Event Simulation, Wiley, pp. 17–38.Google Scholar
Legoll, F. and Lelièvre, T. (2010), Effective dynamics using conditional expectations, Nonlinearity 23, 2131.CrossRefGoogle Scholar
Li, B., Bengtsson, T. and Bickel, P. (2005), Curse-of-dimensionality revisited: Collapse of importance sampling in very high-dimensional systems. Technical report 696, Department of Statistics, UC Berkeley.Google Scholar
Lie, H. C. (2016), On a strongly convex approximation of a stochastic optimal control problem for importance sampling of metastable diffusions. Doctoral thesis, Freie Universität Berlin.Google Scholar
Lie, H. C. (2021), Fréchet derivatives of expected functionals of solutions to stochastic differential equations. Available at https://arxiv.org/abs/2106.09149arXiv:2106.09149.Google Scholar
Lie, H. C. and Quer, J. (2017), Some connections between importance sampling and enhanced sampling methods in molecular dynamics, J. Chem. Phys. 147, 194107.CrossRefGoogle ScholarPubMed
Lu, J. and Vanden-Eijnden, E. (2014), Exact dynamical coarse-graining without time-scale separation, J. Chem. Phys. 141, 044109.CrossRefGoogle ScholarPubMed
Lücke, M. and Nüske, F. (2022), tgEDMD: Approximation of the Kolmogorov operator in tensor train format, J. Nonlinear Sci. 32, 44.CrossRefGoogle Scholar
Mardt, A., Pasquali, L., Wu, H. and Noé, F. (2018), VAMPnets for deep learning of molecular kinetics, Nature Commun. 9, 5.CrossRefGoogle ScholarPubMed
Marinari, E. and Parisi, G. (1992), Simulated tempering: A new Monte Carlo scheme, Europhys . Lett. 19, 451.Google Scholar
Marrouch, N., Slawinska, J., Giannakis, D. and Read, H. L. (2020), Data-driven Koopman operator approach for computational neuroscience, Ann . Math. Artif. Intell. 88, 1155–1173.CrossRefGoogle Scholar
Martinsson, A., Lu, J., Leimkuhler, B. and Vanden-Eijnden, E. (2019), The simulated tempering method in the infinite switch limit with adaptive weight learning, J. Statist. Mech. Theory Exp. 2019, 013207.CrossRefGoogle Scholar
Mattingly, J. C., Stuart, A. M. and Higham, D. J. (2002), Ergodicity for SDEs and approximations: Locally Lipschitz vector fields and degenerate noise, Stoch . Process. Appl. 101, 185–232.CrossRefGoogle Scholar
Mauroy, A. and Goncalves, J. (2016), Linear identification of nonlinear systems: A lifting technique based on the Koopman operator, in 2016 IEEE 55th Conference on Decision and Control (CDC), IEEE, pp. 6500–6505.Google Scholar
Melzer, T., Reiter, M. and Bischof, H. (2001), Nonlinear feature extraction using generalized canonical correlation analysis, in Artificial Neural Networks (ICANN 2001) (Dorffner, G. et al., eds), Springer, pp. 353–360.CrossRefGoogle Scholar
Metzner, P. (2007), Transition path theory for Markov processes: Application to molecular dynamics. Doctoral thesis, Freie Universität Berlin.Google Scholar
Metzner, P., Noé, F. and Schütte, C. (2009a), Estimating the sampling error: Distribution of transition matrices and functions of transition matrices for given trajectory data, Phys. Rev. E 80, 021106.CrossRefGoogle ScholarPubMed
Metzner, P., Schütte, C. and Vanden-Eijnden, E. (2006), Illustration of transition path theory on a collection of simple examples, J. Chem. Phys. 125, 084110.CrossRefGoogle ScholarPubMed
Metzner, P., Schütte, C. and Vanden-Eijnden, E. (2009b), Transition path theory for Markov jump processes, Multiscale Model . Simul. 7, 1192–1219.Google Scholar
Metzner, P., Weber, M. and Schütte, C. (2010), Observation uncertainty in reversible Markov chains, Phys. Rev. E 82, 031114.CrossRefGoogle ScholarPubMed
Meyn, S. and Tweedie, R. (1993), Markov Chains and Stochastic Stability, Springer.CrossRefGoogle Scholar
Molgedey, L. and Schuster, H. G. (1994), Separation of a mixture of independent signals using time delayed correlations, Phys. Rev. Lett. 72, 3634–3637.CrossRefGoogle ScholarPubMed
Mollenhauer, M. (2022), On the statistical approximation of conditional expectation operators. Doctoral thesis, Freie Universität Berlin.Google Scholar
Mollenhauer, M. and Koltai, P. (2020), Nonparametric approximation of conditional expectation operators. Available at https://arxiv.org/abs/2012.12917arXiv:2012.12917.Google Scholar
Mollenhauer, M., Mücke, N. and Sullivan, T. J. (2022), Learning linear operators: Infinite-dimensional regression as a well-behaved non-compact inverse problem. Available at arXiv:2211.08875.Google Scholar
Mori, H. (1965), Transport, collective motion, and Brownian motion, Prog . Theor. Phys. 33, 423–455.CrossRefGoogle Scholar
Moroni, D., van Erp, T. and Bolhuis, P. (2004), Investigating rare events by transition interface sampling, Phys. A 340, 395–401.CrossRefGoogle Scholar
Noé, F. (2008), Probability distributions of molecular observables computed from Markov models, J. Chem. Phys. 128, 244103.CrossRefGoogle ScholarPubMed
Noé, F. and Nüske, F. (2013), A variational approach to modeling slow processes in stochastic dynamical systems, Multiscale Model . Simul. 11, 635–655.Google Scholar
Noé, F., Schütte, C., Vanden-Eijnden, E., Reich, L. and Weikl, T. (2009), Constructing the full ensemble of folding pathways from short off-equilibrium trajectories, Proc . Nat. Acad. Sci. USA 106, 19011–19016.CrossRefGoogle Scholar
Nüske, F., Gelß, P., Klus, S. and Clementi, C. (2021), Tensor-based computation of metastable and coherent sets, Phys. D 427, 133018.CrossRefGoogle Scholar
Nüske, F., Schneider, R., Vitalini, F. and Noé, F. (2016), Variational tensor approach for approximating the rare-event kinetics of macromolecular systems, J. Chem. Phys. 144, 054105.CrossRefGoogle ScholarPubMed
Nüske, F., Wu, H., Prinz, J.-H., Wehmeyer, C., Clementi, C. and Noé, F. (2017), Markov state models from short non-equilibrium simulations: Analysis and correction of estimation bias, J. Chem. Phys. 146, 094104.CrossRefGoogle Scholar
Nüsken, N. and Richter, L. (2021), Solving high-dimensional Hamilton–Jacobi–Bellman PDEs using neural networks: Perspectives from the theory of controlled diffusions and measures on path space, Partial Differ . Equ. Appl. 2, 48.Google Scholar
Øksendal, B. (2003), Stochastic Differential Equations: An Introduction with Applications, Springer.CrossRefGoogle Scholar
Olivieri, E. and Vares, M. E. (2005), Large Deviations and Metastability, Encyclopedia of Mathematics and its Applications, Cambridge University Press.CrossRefGoogle Scholar
Oseledets, I. (2011), Tensor-train decomposition, SIAM J. Sci. Comput. 33, 2295–2317.CrossRefGoogle Scholar
Oseledets, I. and Tyrtyshnikov, E. (2010), TT-cross approximation for multidimensional arrays, Linear Algebra Appl. 432, 70–88.CrossRefGoogle Scholar
Pande, V., Beauchamp, K. and Bowman, G. (2010), Everything you wanted to know about Markov state models but were afraid to ask, Methods 52, 99–105.CrossRefGoogle ScholarPubMed
Pardoux, E. and Peng, S. (1990), Adapted solution of a backward stochastic differential equation, Systems Control Lett. 14, 55–61.CrossRefGoogle Scholar
Pardoux, E. and Tang, S. (1999), Forward–backward stochastic differential equations and quasilinear parabolic PDEs, Probab . Theory Related Fields 114, 123–150.CrossRefGoogle Scholar
Paul, F., Wehmeyer, C., Abualrous, E. T., Wu, H., Crabtree, M. D., Schöneberg, J., Clarke, J., Freund, C., Weikl, T. R. and Noé, F. (2017), Protein–peptide association kinetics beyond the seconds timescale from atomistic simulations, Nature Commun. 8, 1095.CrossRefGoogle ScholarPubMed
Peitz, S. and Klus, S. (2019), Koopman operator-based model reduction for switched-system control of PDEs, Automatica 106, 184–191.CrossRefGoogle Scholar
Perez-Hernandez, G., Paul, F., Giorgino, T., De Fabritiis, G. and Noé, F. (2013), Identification of slow molecular order parameters for Markov model construction, J. Chem. Phys. 139, 015102.CrossRefGoogle ScholarPubMed
Pfau, D., Spencer, J. S., Matthews, A. G. D. G. and Foulkes, W. M. C. (2020), Ab initio solution of the many-electron Schrödinger equation with deep neural networks, Phys. Rev. Res. 2, 033429.CrossRefGoogle Scholar
Pham, H. (2009), Continuous-Time Stochastic Control and Optimization with Financial Applications, Vol. 61 of Stochastic Modelling and Applied Probability, Springer.CrossRefGoogle Scholar
Pham, H., Warin, X. and Germain, M. (2021), Neural networks-based backward scheme for fully nonlinear PDEs, SN Partial Differ . Equ. Appl. 2, 16.Google Scholar
Pinamonti, G., Zhao, J., Condon, D. E., Paul, F., Noé, F., Turner, D. H. and Bussi, G. (2017), Predicting the kinetics of RNA oligonucleotides using Markov state models, J. Chem. Theory Comput. 13, 926–934.CrossRefGoogle ScholarPubMed
Pinski, F. and Stuart, A. (2010), Transition paths in molecules: Gradient descent in path space, J. Chem. Phys. 132, 184104.CrossRefGoogle Scholar
Pinsky, R. G. (1985), On the convergence of diffusion processes conditioned to remain in a bounded region for large time to limiting positive recurrent diffusion processes, Ann . Probab. 13, 363–378.CrossRefGoogle Scholar
Prinz, J. H., Keller, B. and Noé, F. (2011), Probing molecular kinetics with Markov models: Metastable states, transition pathways and spectroscopic observables, Phys. Chem. Chem. Phys. 13, 16912–16927.CrossRefGoogle ScholarPubMed
Quer, J., Donati, L., Keller, B. G. and Weber, M. (2018), An automatic adaptive importance sampling algorithm for molecular dynamics in reaction coordinates, SIAM J. Sci. Comput. 40, A653–A670.CrossRefGoogle Scholar
Rabben, R. J., Ray, S. and Weber, M. (2020), ISOKANN: Invariant subspaces of Koopman operators learned by a neural network, J. Chem. Phys. 153, 114109.CrossRefGoogle ScholarPubMed
Ren, W. and Vanden-Eijnden, E. (2002), String method for the study of rare events, Phys. Rev. B 66, 052301.Google Scholar
Ribeiro, J., Bravo, P., Wang, Y. and Tiwary, P. (2018), Reweighted autoencoded variational Bayes for enhanced sampling (RAVE), J. Chem. Phys. 149, 072301.CrossRefGoogle ScholarPubMed
Richter, L. (2022), Solving high-dimensional PDEs, approximation of path space measures and importance sampling of diffusions. Doctoral thesis, BTU Cottbus–Senftenberg.Google Scholar
Risken, H. (1996), The Fokker–Planck Equation, second edition, Springer.CrossRefGoogle Scholar
Röblitz, S. and Weber, M. (2013), Fuzzy spectral clustering by PCCA+: Application to Markov state models and data classification, Adv . Data Anal. Classif. 7, 147–179.CrossRefGoogle Scholar
Röder, K. and Wales, D. J. (2022), The energy landscape perspective: Encoding structure and function for biomolecules, Front . Molecular Biosci. 9, 820792.CrossRefGoogle Scholar
Rogers, L. C. G. and Williams, D. (2000), Diffusions , Markov Processes and Martingales, Vol. 2: Itô Calculus, Cambridge University Press.Google Scholar
Roux, B. (2021), String method with swarms-of-trajectories, mean drifts, lag time, and committor, J. Phys. Chem. A 125, 7558–7571.CrossRefGoogle ScholarPubMed
Roux, B. (2022), Transition rate theory, spectral analysis, and reactive paths, J. Chem. Phys. 156, 134111.CrossRefGoogle ScholarPubMed
Sanz-Alonso, D. (2018), Importance sampling and necessary sample size: An information theory approach, SIAM/ASA J. Uncertain. Quantif. 6, 867–879.CrossRefGoogle Scholar
Sarich, M. (2011), Projected transfer operators. Doctoral thesis, Freie Universität Berlin.Google Scholar
Sarich, M., Noé, F. and Schütte, C. (2010), On the approximation quality of Markov state models, Multiscale Model . Simul. 8, 1154–1177.Google Scholar
Scherer, M. K., Trendelkamp-Schroer, B., Paul, F., Pérez-Hernández, G., Hoffmann, M., Plattner, N., Wehmeyer, C., Prinz, J.-H. and Noé, F. (2015), PyEMMA 2: A software package for estimation, validation, and analysis of Markov models, J. Chem. Theory Comput. 11, 5525.CrossRefGoogle ScholarPubMed
Schmid, P. J. (2010), Dynamic mode decomposition of numerical and experimental data, J. Fluid Mech. 656, 5–28.CrossRefGoogle Scholar
Schölkopf, B. and Smola, A. J. (2001), Learning with Kernels: Support Vector Machines, Regularization, Optimization and Beyond, MIT Press.Google Scholar
Schütte, C. (1998), Conformational dynamics: Modelling, theory, algorithm, and application to biomolecules. Habilitation thesis, Freie Universität Berlin.Google Scholar
Schütte, C. and Huisinga, W. (2000), On conformational dynamics induced by Langevin processes, in EQUADIFF 99: International Conference on Differential Equations (Fiedler, K., Gröger and Sprekels, J., eds), World Scientific, pp. 1247–1262.Google Scholar
Schütte, C. and Sarich, M. (2014), Metastability and Markov State Models in Molecular Dynamics: Modeling, Analysis, Algorithmic Approaches, Vol. 32 of Courant Lecture Notes, American Mathematical Society.Google Scholar
Schütte, C., Fischer, A., Huisinga, W. and Deuflhard, P. (1999), A direct approach to conformational dynamics based on hybrid Monte Carlo, J. Comput. Phys. 151, 146–168.CrossRefGoogle Scholar
Schütte, C., Huisinga, W. and Meyn, S. (2003), Metastability of diffusion processes, in IUTAM Symposium on Nonlinear Stochastic Dynamics (Namachchivaya, N. S. and Lin, Y. K., eds), Springer, pp. 71–81.CrossRefGoogle Scholar
Schütte, C., Noé, F., Lu, J., Sarich, M. and Vanden-Eijnden, E. (2011), Markov state models based on milestoning, J. Chem. Phys. 134, 204105.CrossRefGoogle ScholarPubMed
Schwantes, C. and Pande, V. (2013), Improvements in Markov state model construction reveal many non-native interactions in the folding of NTL9, J. Chem. Theory Comput. 9, 2000–2009.CrossRefGoogle ScholarPubMed
Schwantes, C. R. and Pande, V. S. (2015), Modeling molecular kinetics with tICA and the kernel trick, J. Chem. Theory Comput. 11, 600–608.CrossRefGoogle ScholarPubMed
Senne, M., Trendelkamp-Schroer, B., Mey, A., Schütte, C. and Noé, F. (2012), EMMA: A software package for Markov model building and analysis, J. Chem. Theory Comput. 8, 2223–2238.CrossRefGoogle Scholar
Shawe-Taylor, J. and Cristianini, N. (2004), Kernel Methods for Pattern Analysis, Cambridge University Press.CrossRefGoogle Scholar
Sidky, H., Chen, W. and Ferguson, A. L. (2020), Machine learning for collective variable discovery and enhanced sampling in biomolecular simulation, Molecular Phys. 118, e1737742.CrossRefGoogle Scholar
Sikorski, A. (2015), PCCA+ and its application to spatial time series clustering. Bachelor thesis, Freie Universität Berlin.Google Scholar
Sikorski, A. (2023), Reduced dynamics of high dimensional stochastic systems. Doctoral thesis, Freie Universität Berlin.Google Scholar
Singhal, N. and Pande, V. S. (2005), Error analysis in Markovian state models for protein folding, J. Chem. Phys. 123, 204909.CrossRefGoogle Scholar
Sirignano, J. and Spiliopoulos, K. (2018), DGM: A deep learning algorithm for solving partial differential equations, J. Comput. Phys. 375, 1339–1364.CrossRefGoogle Scholar
Steinwart, I. and Christmann, A. (2008), Support Vector Machines, first edition, Springer.Google Scholar
Suarez, E., Wiewiora, R. P., Wehmeyer, C., Noé, F., Chodera, J. D. and Zuckerman, D. M. (2021), What Markov state models can and cannot do: Correlation versus path-based observables in protein-folding models, J. Chem. Theory Comput. 17, 3119–3133.CrossRefGoogle ScholarPubMed
Swendsen, R. H. and Wang, J.-S. (1986), Replica Monte Carlo simulation of spin-glasses, Phys. Rev. Lett. 57, 2607–2609.CrossRefGoogle ScholarPubMed
Swenson, D. W. H. and Bolhuis, P. G. (2014), A replica exchange transition interface sampling method with multiple interface sets for investigating networks of rare events, J. Chem. Phys. 141, 044101.CrossRefGoogle ScholarPubMed
Tian, W. and Wu, H. (2021), Kernel embedding based variational approach for low-dimensional approximation of dynamical systems, Comput . Methods Appl. Math. 21, 635–659.Google Scholar
Tsourtis, A., Pantazis, Y., Katsoulakis, M. A. and Harmandaris, V. (2015), Parametric sensitivity analysis for stochastic molecular systems using information theoretic metrics, J. Chem. Phys. 143, 014116.CrossRefGoogle ScholarPubMed
Tucker, L. R. (1964), The extension of factor analysis to three-dimensional matrices, in Contributions to Mathematical Psychology (Gulliksen, H. and Frederiksen, N., eds), Holt, Rinehart & Winston, pp. 110–127.Google Scholar
Turkedjiev, P. (2013), Numerical methods for backward stochastic differential equations of quadratic and locally Lipschitz type. Doctoral thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II.Google Scholar
Valsson, O. and Parrinello, M. (2014), Variational approach to enhanced sampling and free energy calculations, Phys. Rev. Lett. 113, 090601.CrossRefGoogle ScholarPubMed
Van Erp, T. S., Moroni, D. and Bolhuis, P. G. (2003), A novel path sampling method for the calculation of rate constants, J. Chem. Phys. 118, 7762–7774.CrossRefGoogle Scholar
Vanden-Eijnden, E. and Weare, J. (2012), Rare event simulation of small noise diffusions, Commun . Pure Appl. Math. 65, 1770–1803.CrossRefGoogle Scholar
Villén-Altamirano, M. and Villén-Altamirano, J. (1994), Restart: A straightforward method for fast simulation of rare events, in Proceedings of the 26th Conference on Winter Simulation (WSC ’94), Society for Computer Simulation International, pp. 282–289.Google Scholar
Vlachas, P. R., Zavadlav, J., Praprotnik, M. and Koumoutsakos, P. (2022), Accelerated simulations of molecular systems through learning of effective dynamics, J. Chem. Theory Comput. 18, 538–549.CrossRefGoogle ScholarPubMed
Voter, A. F. (1998), Parallel replica method for dynamics of infrequent events, Phys. Rev. B 57, R13985–R13988.CrossRefGoogle Scholar
Wales, D. J. (2003), Energy Landscapes, Cambridge University Press.Google Scholar
Wales, D. J. (2005), Energy landscapes and properties of biomolecules, Phys. Biol. 2, S86–S93.CrossRefGoogle ScholarPubMed
Wang, Y., Ribeiro, J. M. L. and Tiwary, P. (2019), Past–future information bottleneck for sampling molecular reaction coordinate simultaneously with thermodynamics and kinetics, Nature Commun. 10, 3573.CrossRefGoogle ScholarPubMed
Weber, M. and Ernst, N. (2017), A fuzzy-set theoretical framework for computing exit rates of rare events in potential-driven diffusion processes. Available at arXiv:1708.00679.Google Scholar
Weber, M. and Fackeldey, K. (2014), Computing the minimal rebinding effect included in a given kinetics, Multiscale Model . Simul. 12, 318–334.Google Scholar
Wehmeyer, C. and Noé, F. (2018), Time-lagged autoencoders: Deep learning of slow collective variables for molecular kinetics, J. Chem. Phys. 148, 241703.CrossRefGoogle ScholarPubMed
West, A., Elber, R. and Shalloway, D. (2007), Extending molecular dynamics time scales with milestoning: Example of complex kinetics in a solvated peptide, J. Chem. Phys. 126, 145104.CrossRefGoogle Scholar
Wigner, E. (1938), The transition state method, Trans . Faraday Soc. 34, 29–41.CrossRefGoogle Scholar
Williams, M. O., Kevrekidis, I. G. and Rowley, C. W. (2015a), A data-driven approximation of the Koopman operator: Extending dynamic mode decomposition, J. Nonlinear Sci. 25, 1307–1346.CrossRefGoogle Scholar
Williams, M. O., Rowley, C. W. and Kevrekidis, I. G. (2015b), A kernel-based method for data-driven Koopman spectral analysis, J. Comput. Dyn. 2, 247–265.CrossRefGoogle Scholar
Wu, H. and Noé, F. (2020), Variational approach for learning Markov processes from time series data, J. Nonlinear Sci. 30, 23–66.CrossRefGoogle Scholar
Wu, H., Nüske, F., Paul, F., Klus, S., Koltai, P. and Noé, F. (2017), Variational Koopman models: Slow collective variables and molecular kinetics from short off-equilibrium simulations, J. Chem. Phys. 146, 154104.CrossRefGoogle ScholarPubMed
Yang, Z., Zang, Y., Wang, H., Kang, Y., Zhang, J., Li, X., Zhang, L. and Zhang, S. (2022), Recognition between CD147 and cyclophilin A deciphered by accelerated molecular dynamics simulations, Phys. Chem. Chem. Phys. 24, 18905–18914.CrossRefGoogle ScholarPubMed
Yeung, E., Kundu, S. and Hodas, N. (2019), Learning deep neural network representations for Koopman operators of nonlinear dynamical systems, in 2019 American Control Conference (ACC), IEEE, pp. 4832–4839.Google Scholar
Yu, Y., Wang, T. and Samworth, R. J. (2015), A useful variant of the Davis–Kahan theorem for statisticians, Biometrika 102, 315–323.CrossRefGoogle Scholar
Zhang, W. and Schütte, C. (2017), Reliable approximation of long relaxation timescales in molecular dynamics, Entropy 19, 367.CrossRefGoogle Scholar
Zhang, W., Hartmann, C. and Schütte, C. (2016), Effective dynamics along given reaction coordinates, and reaction rate theory, Faraday Discuss. 195, 365–394.CrossRefGoogle ScholarPubMed
Zhang, W., Li, T. and Schütte, C. (2022), Solving eigenvalue PDEs of metastable diffusion processes using artificial neural networks, J. Comput. Phys. 465, 111377.CrossRefGoogle Scholar
Zhang, W., Wang, H., Hartmann, C., Weber, M. and Schütte, C. (2014), Applications of the cross-entropy method to importance sampling and optimal control of diffusions, SIAM J. Sci. Comput. 36, A2654–A2672.CrossRefGoogle Scholar
Zhou, D.-X. (2008), Derivative reproducing properties for kernel methods in learning theory, J. Comput. Appl. Math. 220, 456–463.CrossRefGoogle Scholar
Zhuang, Y., Bureau, H. R., Quirk, S. and Hernandez, R. (2021), Adaptive steered molecular dynamics of biomolecules, Molecular Simul. 47, 408–419.CrossRefGoogle Scholar
Zwanzig, R. (1973), Nonlinear generalized Langevin equations, J. Statist. Phys. 9, 215–220.CrossRefGoogle Scholar