Data-driven parameterization of the generalized Langevin equation

Lei, Huan; Baker, Nathan A.; Li, Xiantao

doi:10.1073/pnas.1609587113

Title: Data-driven parameterization of the generalized Langevin equation

Abstract

Significance The generalized Langevin equation (GLE) provides a precise description of coarse-grained variable dynamics in reduced dimension models. However, computation of the memory kernel poses a major challenge to the practical use of the GLE. This paper presents a data-driven approach to compute the memory kernel, relying on a hierarchy of parameterized rational approximations in terms of the Laplace transform, which can be expanded to arbitrarily high order as necessary. This parameterization makes it convenient to represent the GLE via an extended stochastic model where the memory term is eliminated by properly introducing auxiliary variables. The present method is well-suited for constructing reduced models for nonequilibrium properties of complex systems such as biomolecules, chemical reaction networks, and climate simulations.

Authors:

Lei, Huan ^[1]; Baker, Nathan A. ^[2]; Li, Xiantao ^[3]

Advanced Computing, Mathematics &, Data, Pacific Northwest National Laboratory, Richland, WA 99352,
Advanced Computing, Mathematics &, Data, Pacific Northwest National Laboratory, Richland, WA 99352,, Division of Applied Mathematics, Brown University, Providence, RI 02912,
Department of Mathematics, The Pennsylvania State University, University Park, PA 16802

Publication Date:: Tue Nov 29 00:00:00 EST 2016

Research Org.:: Pacific Northwest National Laboratory (PNNL), Richland, WA (United States)

Sponsoring Org.:: USDOE

OSTI Identifier:: 1333885

Alternate Identifier(s):: OSTI ID: 1339841

Report Number(s):: PNNL-SA-118385
Journal ID: ISSN 0027-8424

Grant/Contract Number:: CM4; AC05-76RL01830

Resource Type:: Published Article

Journal Name:: Proceedings of the National Academy of Sciences of the United States of America

Additional Journal Information:: Journal Name: Proceedings of the National Academy of Sciences of the United States of America Journal Volume: 113 Journal Issue: 50; Journal ID: ISSN 0027-8424

Publisher:: Proceedings of the National Academy of Sciences

Country of Publication:: United States

Language:: English

Subject:: 97 MATHEMATICS AND COMPUTING; fluctuation-dissipation theorem; generalized langevin equation; generalized Langevin dynamics; data-driven parameterization; coarse-grained molecular models; reaction rate; model reduction

Citation Formats


                    Lei, Huan, Baker, Nathan A., and Li, Xiantao. Data-driven parameterization of the generalized Langevin equation.  United States: N. p., 2016. 
Web.  doi:10.1073/pnas.1609587113.

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                    Lei, Huan, Baker, Nathan A., & Li, Xiantao. Data-driven parameterization of the generalized Langevin equation.  United States.  https://doi.org/10.1073/pnas.1609587113

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                    Lei, Huan, Baker, Nathan A., and Li, Xiantao. Tue .  
"Data-driven parameterization of the generalized Langevin equation".  United States.  https://doi.org/10.1073/pnas.1609587113.

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@article{osti_1333885,

  title        = {Data-driven parameterization of the generalized Langevin equation},

  author       = {Lei, Huan and Baker, Nathan A. and Li, Xiantao},

  abstractNote = {Significance The generalized Langevin equation (GLE) provides a precise description of coarse-grained variable dynamics in reduced dimension models. However, computation of the memory kernel poses a major challenge to the practical use of the GLE. This paper presents a data-driven approach to compute the memory kernel, relying on a hierarchy of parameterized rational approximations in terms of the Laplace transform, which can be expanded to arbitrarily high order as necessary. This parameterization makes it convenient to represent the GLE via an extended stochastic model where the memory term is eliminated by properly introducing auxiliary variables. The present method is well-suited for constructing reduced models for nonequilibrium properties of complex systems such as biomolecules, chemical reaction networks, and climate simulations.},

  doi          = {10.1073/pnas.1609587113},

  journal      = {Proceedings of the National Academy of Sciences of the United States of America},

  number       = 50,

  volume       = 113,

  place        = {United States},

  year         = {Tue Nov 29 00:00:00 EST 2016},

  month        = {Tue Nov 29 00:00:00 EST 2016}

}

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Works referenced in this record:

Physics constrained nonlinear regression models for time series
journal, November 2012

Majda, Andrew J.; Harlim, John
Nonlinearity, Vol. 26, Issue 1
DOI: 10.1088/0951-7715/26/1/201

Variational boundary conditions for molecular dynamics simulations of crystalline solids at finite temperature: Treatment of the thermal bath
journal, September 2007

Li, Xiantao; E., Weinan
Physical Review B, Vol. 76, Issue 10
DOI: 10.1103/PhysRevB.76.104107

Simulating Microscopic Hydrodynamic Phenomena with Dissipative Particle Dynamics
journal, June 1992

Hoogerbrugge, P. J.; Koelman, J. M. V. A.
Europhysics Letters (EPL), Vol. 19, Issue 3
DOI: 10.1209/0295-5075/19/3/001

Mori–Zwanzig formalism as a practical computational tool
journal, January 2010

Hijón, Carmen; Español, Pep; Vanden-Eijnden, Eric
Faraday Discuss., Vol. 144
DOI: 10.1039/B902479B

Extracting macroscopic dynamics: model problems and algorithms
journal, August 2004

Givon, Dror; Kupferman, Raz; Stuart, Andrew
Nonlinearity, Vol. 17, Issue 6
DOI: 10.1088/0951-7715/17/6/R01

Langevin Equation with Colored Noise for Constant-Temperature Molecular Dynamics Simulations
journal, January 2009

Ceriotti, Michele; Bussi, Giovanni; Parrinello, Michele
Physical Review Letters, Vol. 102, Issue 2
DOI: 10.1103/PhysRevLett.102.020601

Statistical Mechanics of Dissipative Particle Dynamics
journal, May 1995

Español, P.; Warren, P.
Europhysics Letters (EPL), Vol. 30, Issue 4
DOI: 10.1209/0295-5075/30/4/001

Dynamic force matching: A method for constructing dynamical coarse-grained models with realistic time dependence
journal, April 2015

Davtyan, Aram; Dama, James F.; Voth, Gregory A.
The Journal of Chemical Physics, Vol. 142, Issue 15
DOI: 10.1063/1.4917454

A coarse-grained molecular dynamics model for crystalline solids
journal, August 2010

Li, Xiantao
International Journal for Numerical Methods in Engineering, Vol. 83, Issue 8-9
DOI: 10.1002/nme.2892

Generalized Langevin equation approach for atom/solid‐surface scattering: Collinear atom/harmonic chain model
journal, November 1974

Adelman, S. A.; Doll, J. D.
The Journal of Chemical Physics, Vol. 61, Issue 10
DOI: 10.1063/1.1681723

Bottom-up coarse-graining of a simple graphene model: The blob picture
journal, February 2011

Kauzlarić, David; Meier, Julia T.; Español, Pep
The Journal of Chemical Physics, Vol. 134, Issue 6
DOI: 10.1063/1.3554395

Calculation of effective interaction potentials from radial distribution functions: A reverse Monte Carlo approach
journal, October 1995

Lyubartsev, Alexander P.; Laaksonen, Aatto
Physical Review E, Vol. 52, Issue 4
DOI: 10.1103/PhysRevE.52.3730

Generalized Langevin dynamics simulations with arbitrary time‐dependent memory kernels
journal, March 1983

Berkowitz, M.; Morgan, J. D.; McCammon, J. Andrew
The Journal of Chemical Physics, Vol. 78, Issue 6
DOI: 10.1063/1.445244

THE weighted histogram analysis method for free-energy calculations on biomolecules. I. The method
journal, October 1992

Kumar, Shankar; Rosenberg, John M.; Bouzida, Djamal
Journal of Computational Chemistry, Vol. 13, Issue 8
DOI: 10.1002/jcc.540130812

Numerical integration of the extended variable generalized Langevin equation with a positive Prony representable memory kernel
journal, July 2013

Baczewski, Andrew D.; Bond, Stephen D.
The Journal of Chemical Physics, Vol. 139, Issue 4
DOI: 10.1063/1.4815917

Multi-level Dynamical Systems: Connecting the Ruelle Response Theory and the Mori-Zwanzig Approach
journal, March 2013

Wouters, Jeroen; Lucarini, Valerio
Journal of Statistical Physics, Vol. 151, Issue 5
DOI: 10.1007/s10955-013-0726-8

Discrete approach to stochastic parametrization and dimension reduction in nonlinear dynamics
journal, July 2015

Chorin, Alexandre J.; Lu, Fei
Proceedings of the National Academy of Sciences, Vol. 112, Issue 32
DOI: 10.1073/pnas.1512080112

A Multiscale Coarse-Graining Method for Biomolecular Systems
journal, February 2005

Izvekov, Sergei; Voth, Gregory A.
The Journal of Physical Chemistry B, Vol. 109, Issue 7
DOI: 10.1021/jp044629q

Problem reduction, renormalization, and memory
journal, January 2006

Chorin, Alexandre; Stinis, Panagiotis
Communications in Applied Mathematics and Computational Science, Vol. 1, Issue 1
DOI: 10.2140/camcos.2006.1.1

Generalized Langevin Equation with Fractional Gaussian Noise: Subdiffusion within a Single Protein Molecule
journal, October 2004

Kou, S. C.; Xie, X. Sunney
Physical Review Letters, Vol. 93, Issue 18
DOI: 10.1103/PhysRevLett.93.180603

Dynamics of an Inhomogeneously Coarse Grained Multiscale System
journal, June 2002

Curtarolo, Stefano; Ceder, Gerbrand
Physical Review Letters, Vol. 88, Issue 25
DOI: 10.1103/PhysRevLett.88.255504

A generalized Langevin dynamics approach to model solvent dynamics effects on proteins via a solvent-accessible surface. The carboxypeptidase A inhibitor protein as a model
journal, December 2000

Oliva, Baldomero; Daura, Xavier; Querol, Enrique
Theoretical Chemistry Accounts: Theory, Computation, and Modeling (Theoretica Chimica Acta), Vol. 105, Issue 2
DOI: 10.1007/s002140000183

Parametric reduced models for the nonlinear Schrödinger equation
journal, May 2015

Harlim, John; Li, Xiantao
Physical Review E, Vol. 91, Issue 5
DOI: 10.1103/PhysRevE.91.053306

Computation of the memory functions in the generalized Langevin models for collective dynamics of macromolecules
journal, August 2014

Chen, Minxin; Li, Xiantao; Liu, Chun
The Journal of Chemical Physics, Vol. 141, Issue 6
DOI: 10.1063/1.4892412

Nonlinear generalized Langevin equations
journal, November 1973

Zwanzig, Robert
Journal of Statistical Physics, Vol. 9, Issue 3
DOI: 10.1007/BF01008729

Dynamic friction on rigid and flexible bonds
journal, October 1990

Berne, B. J.; Tuckerman, M. E.; Straub, John E.
The Journal of Chemical Physics, Vol. 93, Issue 7
DOI: 10.1063/1.458647

Incorporation of memory effects in coarse-grained modeling via the Mori-Zwanzig formalism
journal, December 2015

Li, Zhen; Bian, Xin; Li, Xiantao
The Journal of Chemical Physics, Vol. 143, Issue 24
DOI: 10.1063/1.4935490

Transport, Collective Motion, and Brownian Motion
journal, March 1965

Mori, Hazime
Progress of Theoretical Physics, Vol. 33, Issue 3
DOI: 10.1143/PTP.33.423

Direct construction of mesoscopic models from microscopic simulations
journal, February 2010

Lei, Huan; Caswell, Bruce; Karniadakis, George Em
Physical Review E, Vol. 81, Issue 2
DOI: 10.1103/PhysRevE.81.026704

Time-Domain Methods for Diffusive Transport in Soft Matter
journal, January 2009

Fricks, John; Yao, Lingxing; Elston, Timothy C.
SIAM Journal on Applied Mathematics, Vol. 69, Issue 5
DOI: 10.1137/070695186

Modeling real dynamics in the coarse-grained representation of condensed phase systems
journal, October 2006

Izvekov, Sergei; Voth, Gregory A.
The Journal of Chemical Physics, Vol. 125, Issue 15
DOI: 10.1063/1.2360580

The relative entropy is fundamental to multiscale and inverse thermodynamic problems
journal, October 2008

Shell, M. Scott
The Journal of Chemical Physics, Vol. 129, Issue 14
DOI: 10.1063/1.2992060

Observation of a Power-Law Memory Kernel for Fluctuations within a Single Protein Molecule
journal, May 2005

Min, Wei; Luo, Guobin; Cherayil, Binny J.
Physical Review Letters, Vol. 94, Issue 19
DOI: 10.1103/PhysRevLett.94.198302

Dynamic solvent effects on activated chemical reactions. I. Classical effects of reaction path curvature
journal, December 1992

Schenter, Gregory K.; McRae, Robin P.; Garrett, Bruce C.
The Journal of Chemical Physics, Vol. 97, Issue 12
DOI: 10.1063/1.463970

A note on matrix-valued rational interpolants
journal, October 1999

Zhu, Gong-Qin; Tan, Jie-Qing
Journal of Computational and Applied Mathematics, Vol. 110, Issue 1
DOI: 10.1016/S0377-0427(99)00203-4

The Elementary Gaussian Processes
journal, September 1944

Doob, J. L.
The Annals of Mathematical Statistics, Vol. 15, Issue 3
DOI: 10.1214/aoms/1177731234

The derivation and approximation of coarse-grained dynamics from Langevin dynamics
journal, November 2016

Ma, Lina; Li, Xiantao; Liu, Chun
The Journal of Chemical Physics, Vol. 145, Issue 20
DOI: 10.1063/1.4967936

Colored-Noise Thermostats à la Carte
journal, March 2010

Ceriotti, Michele; Bussi, Giovanni; Parrinello, Michele
Journal of Chemical Theory and Computation, Vol. 6, Issue 4
DOI: 10.1021/ct900563s

Memory kernels from molecular dynamics
journal, September 1981

Berkowitz, Max; Morgan, John D.; Kouri, Donald J.
The Journal of Chemical Physics, Vol. 75, Issue 5
DOI: 10.1063/1.442269

Escaping free-energy minima
journal, September 2002

Laio, A.; Parrinello, M.
Proceedings of the National Academy of Sciences, Vol. 99, Issue 20
DOI: 10.1073/pnas.202427399

Deriving effective mesoscale potentials from atomistic simulations: Mesoscale Potentials from Atomistic Simulations
journal, August 2003

Reith, Dirk; Pütz, Mathias; Müller-Plathe, Florian
Journal of Computational Chemistry, Vol. 24, Issue 13
DOI: 10.1002/jcc.10307

Collective Langevin dynamics of conformational motions in proteins
journal, June 2006

Lange, Oliver F.; Grubmüller, Helmut
The Journal of Chemical Physics, Vol. 124, Issue 21
DOI: 10.1063/1.2199530

Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen
journal, January 1905

Einstein, A.
Annalen der Physik, Vol. 322, Issue 8
DOI: 10.1002/andp.19053220806

Computing generalized Langevin equations and generalized Fokker-Planck equations
journal, June 2009

Darve, E.; Solomon, J.; Kia, A.
Proceedings of the National Academy of Sciences, Vol. 106, Issue 27
DOI: 10.1073/pnas.0902633106

The fluctuation-dissipation theorem
journal, January 1966

Kubo, R.
Reports on Progress in Physics, Vol. 29, Issue 1
DOI: 10.1088/0034-4885/29/1/306

The multiscale coarse-graining method. I. A rigorous bridge between atomistic and coarse-grained models
journal, June 2008

Noid, W. G.; Chu, Jhih-Wei; Ayton, Gary S.
The Journal of Chemical Physics, Vol. 128, Issue 24
DOI: 10.1063/1.2938860

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Title: Data-driven parameterization of the generalized Langevin equation

Abstract

Citation Formats

Physics constrained nonlinear regression models for time series journal, November 2012

Variational boundary conditions for molecular dynamics simulations of crystalline solids at finite temperature: Treatment of the thermal bath journal, September 2007

Simulating Microscopic Hydrodynamic Phenomena with Dissipative Particle Dynamics journal, June 1992

Mori–Zwanzig formalism as a practical computational tool journal, January 2010

Extracting macroscopic dynamics: model problems and algorithms journal, August 2004

Langevin Equation with Colored Noise for Constant-Temperature Molecular Dynamics Simulations journal, January 2009

Statistical Mechanics of Dissipative Particle Dynamics journal, May 1995

Dynamic force matching: A method for constructing dynamical coarse-grained models with realistic time dependence journal, April 2015

A coarse-grained molecular dynamics model for crystalline solids journal, August 2010

Generalized Langevin equation approach for atom/solid‐surface scattering: Collinear atom/harmonic chain model journal, November 1974

Bottom-up coarse-graining of a simple graphene model: The blob picture journal, February 2011

Calculation of effective interaction potentials from radial distribution functions: A reverse Monte Carlo approach journal, October 1995

Generalized Langevin dynamics simulations with arbitrary time‐dependent memory kernels journal, March 1983

THE weighted histogram analysis method for free-energy calculations on biomolecules. I. The method journal, October 1992

Numerical integration of the extended variable generalized Langevin equation with a positive Prony representable memory kernel journal, July 2013

Multi-level Dynamical Systems: Connecting the Ruelle Response Theory and the Mori-Zwanzig Approach journal, March 2013

Discrete approach to stochastic parametrization and dimension reduction in nonlinear dynamics journal, July 2015

A Multiscale Coarse-Graining Method for Biomolecular Systems journal, February 2005

Problem reduction, renormalization, and memory journal, January 2006

Generalized Langevin Equation with Fractional Gaussian Noise: Subdiffusion within a Single Protein Molecule journal, October 2004

Dynamics of an Inhomogeneously Coarse Grained Multiscale System journal, June 2002

A generalized Langevin dynamics approach to model solvent dynamics effects on proteins via a solvent-accessible surface. The carboxypeptidase A inhibitor protein as a model journal, December 2000

Parametric reduced models for the nonlinear Schrödinger equation journal, May 2015

Computation of the memory functions in the generalized Langevin models for collective dynamics of macromolecules journal, August 2014

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Dynamic friction on rigid and flexible bonds journal, October 1990

Incorporation of memory effects in coarse-grained modeling via the Mori-Zwanzig formalism journal, December 2015

Transport, Collective Motion, and Brownian Motion journal, March 1965

Direct construction of mesoscopic models from microscopic simulations journal, February 2010

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Modeling real dynamics in the coarse-grained representation of condensed phase systems journal, October 2006

The relative entropy is fundamental to multiscale and inverse thermodynamic problems journal, October 2008

Observation of a Power-Law Memory Kernel for Fluctuations within a Single Protein Molecule journal, May 2005

Dynamic solvent effects on activated chemical reactions. I. Classical effects of reaction path curvature journal, December 1992

A note on matrix-valued rational interpolants journal, October 1999

The Elementary Gaussian Processes journal, September 1944

The derivation and approximation of coarse-grained dynamics from Langevin dynamics journal, November 2016

Colored-Noise Thermostats à la Carte journal, March 2010

Memory kernels from molecular dynamics journal, September 1981

Escaping free-energy minima journal, September 2002

Deriving effective mesoscale potentials from atomistic simulations: Mesoscale Potentials from Atomistic Simulations journal, August 2003

Collective Langevin dynamics of conformational motions in proteins journal, June 2006

Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen journal, January 1905

Computing generalized Langevin equations and generalized Fokker-Planck equations journal, June 2009

The fluctuation-dissipation theorem journal, January 1966

The multiscale coarse-graining method. I. A rigorous bridge between atomistic and coarse-grained models journal, June 2008

Physics constrained nonlinear regression models for time series
journal, November 2012

Variational boundary conditions for molecular dynamics simulations of crystalline solids at finite temperature: Treatment of the thermal bath
journal, September 2007

Simulating Microscopic Hydrodynamic Phenomena with Dissipative Particle Dynamics
journal, June 1992

Mori–Zwanzig formalism as a practical computational tool
journal, January 2010

Extracting macroscopic dynamics: model problems and algorithms
journal, August 2004

Langevin Equation with Colored Noise for Constant-Temperature Molecular Dynamics Simulations
journal, January 2009

Statistical Mechanics of Dissipative Particle Dynamics
journal, May 1995

Dynamic force matching: A method for constructing dynamical coarse-grained models with realistic time dependence
journal, April 2015

A coarse-grained molecular dynamics model for crystalline solids
journal, August 2010

Generalized Langevin equation approach for atom/solid‐surface scattering: Collinear atom/harmonic chain model
journal, November 1974

Bottom-up coarse-graining of a simple graphene model: The blob picture
journal, February 2011

Calculation of effective interaction potentials from radial distribution functions: A reverse Monte Carlo approach
journal, October 1995

Generalized Langevin dynamics simulations with arbitrary time‐dependent memory kernels
journal, March 1983

THE weighted histogram analysis method for free-energy calculations on biomolecules. I. The method
journal, October 1992

Numerical integration of the extended variable generalized Langevin equation with a positive Prony representable memory kernel
journal, July 2013

Multi-level Dynamical Systems: Connecting the Ruelle Response Theory and the Mori-Zwanzig Approach
journal, March 2013

Discrete approach to stochastic parametrization and dimension reduction in nonlinear dynamics
journal, July 2015

A Multiscale Coarse-Graining Method for Biomolecular Systems
journal, February 2005

Problem reduction, renormalization, and memory
journal, January 2006

Generalized Langevin Equation with Fractional Gaussian Noise: Subdiffusion within a Single Protein Molecule
journal, October 2004

Dynamics of an Inhomogeneously Coarse Grained Multiscale System
journal, June 2002

A generalized Langevin dynamics approach to model solvent dynamics effects on proteins via a solvent-accessible surface. The carboxypeptidase A inhibitor protein as a model
journal, December 2000

Parametric reduced models for the nonlinear Schrödinger equation
journal, May 2015

Computation of the memory functions in the generalized Langevin models for collective dynamics of macromolecules
journal, August 2014

Nonlinear generalized Langevin equations
journal, November 1973

Dynamic friction on rigid and flexible bonds
journal, October 1990

Incorporation of memory effects in coarse-grained modeling via the Mori-Zwanzig formalism
journal, December 2015

Transport, Collective Motion, and Brownian Motion
journal, March 1965

Direct construction of mesoscopic models from microscopic simulations
journal, February 2010

Time-Domain Methods for Diffusive Transport in Soft Matter
journal, January 2009

Modeling real dynamics in the coarse-grained representation of condensed phase systems
journal, October 2006

The relative entropy is fundamental to multiscale and inverse thermodynamic problems
journal, October 2008

Observation of a Power-Law Memory Kernel for Fluctuations within a Single Protein Molecule
journal, May 2005

Dynamic solvent effects on activated chemical reactions. I. Classical effects of reaction path curvature
journal, December 1992

A note on matrix-valued rational interpolants
journal, October 1999

The Elementary Gaussian Processes
journal, September 1944

The derivation and approximation of coarse-grained dynamics from Langevin dynamics
journal, November 2016

Colored-Noise Thermostats à la Carte
journal, March 2010

Memory kernels from molecular dynamics
journal, September 1981

Escaping free-energy minima
journal, September 2002

Deriving effective mesoscale potentials from atomistic simulations: Mesoscale Potentials from Atomistic Simulations
journal, August 2003

Collective Langevin dynamics of conformational motions in proteins
journal, June 2006

Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen
journal, January 1905

Computing generalized Langevin equations and generalized Fokker-Planck equations
journal, June 2009

The fluctuation-dissipation theorem
journal, January 1966

The multiscale coarse-graining method. I. A rigorous bridge between atomistic and coarse-grained models
journal, June 2008