April  2022, 15(4): 959-982. doi: 10.3934/dcdss.2021096

Effective Mori-Zwanzig equation for the reduced-order modeling of stochastic systems

1. 

Department of Applied Mathematics, University of California, Merced, Merced (CA) 95343, USA

2. 

Department of Computational Mathematics, Michigan State University, East Lansing (MI) 48824, USA

* Corresponding author

Received  February 2021 Revised  June 2021 Published  April 2022 Early access  August 2021

Built upon the hypoelliptic analysis of the effective Mori-Zwanzig (EMZ) equation for observables of stochastic dynamical systems, we show that the obtained semigroup estimates for the EMZ equation can be used to derive prior estimates of the observable statistics for systems in the equilibrium and nonequilibrium state. In addition, we introduce both first-principle and data-driven methods to approximate the EMZ memory kernel and prove the convergence of the data-driven parametrization schemes using the regularity estimate of the memory kernel. The analysis results are validated numerically via the Monte-Carlo simulation of the Langevin dynamics for a Fermi-Pasta-Ulam chain model. With the same example, we also show the effectiveness of the proposed memory kernel approximation methods.

Citation: Yuanran Zhu, Huan Lei. Effective Mori-Zwanzig equation for the reduced-order modeling of stochastic systems. Discrete and Continuous Dynamical Systems - S, 2022, 15 (4) : 959-982. doi: 10.3934/dcdss.2021096
References:
[1]

A. D. Baczewski and S. D. Bond, Numerical integration of the extended variable generalized Langevin equation with a positive Prony representable memory kernel, J. Chem. Phys., 139 (2013), 044107. doi: 10.1063/1.4815917.

[2]

M. BerkowitzJ. D. MorganD. J. Kouri and J. A. McCammon, Memory kernels from molecular dynamics, J. Chem. Phys., 75 (1981), 2462-2463. 

[3]

A. J. ChorinO. H. Hald and R. Kupferman, Optimal prediction and the Mori-Zwanzig representation of irreversible processes, Proc. Natl. Acad. Sci. USA, 97 (2000), 2968-2973.  doi: 10.1073/pnas.97.7.2968.

[4]

W. Chu and X. Li, The Mori–Zwanzig formalism for the derivation of a fluctuating heat conduction model from molecular dynamics, Commun Math Sci., 17 (2019), 539-563.  doi: 10.4310/CMS.2019.v17.n2.a10.

[5]

J. M. Dominy and D. Venturi, Duality and conditional expectations in the Nakajima-Mori-Zwanzig formulation, J. Math. Phys., 58 (2017), 082701. doi: 10.1063/1.4997015.

[6]

J.-P. Eckmann and M. Hairer, Non-equilibrium statistical mechanics of strongly anharmonic chains of oscillators, Commun. Math. Phys., 212 (2000), 105-164.  doi: 10.1007/s002200000216.

[7]

J.-P. Eckmann and M. Hairer, Spectral properties of hypoelliptic operators, Commun. Math. Phys., 235 (2003), 233-253.  doi: 10.1007/s00220-003-0805-9.

[8]

J.-P. EckmannC.-A. Pillet and L. Rey-Bellet, Non-equilibrium statistical mechanics of anharmonic chains coupled to two heat baths at different temperatures, Commun. Math. Phys., 201 (1999), 657-697.  doi: 10.1007/s002200050572.

[9]

P. Español, Hydrodynamics from dissipative particle dynamics, Phys. Rev. E, 52 (1995), 1734.

[10]

P. Español and P. Warren, Statistical mechanics of dissipative particle dynamics, EPL, 30 (1995), 191.

[11]

S. K. J. Falkena, C. Quinn, J. Sieber, J. Frank and H. A. Dijkstra, Derivation of delay equation climate models using the Mori- Zwanzig formalism, Proc. R. Soc. A, 475 (2019), 20190075, 21 pp. doi: 10.1098/rspa.2019.0075.

[12]

D. Funaro, Polynomial Approximation of Differential Equations, volume 8, Springer-Verlag, Berlin, 1992.

[13]

D. GivonR. Kupferman and O. H. Hald, Existence proof for orthogonal dynamics and the Mori-Zwanzig formalism, Isr. J. Math., 145 (2005), 221-241.  doi: 10.1007/BF02786691.

[14]

F. GroganH. LeiX. Li and N. A. Baker, Data-driven molecular modeling with the generalized Langevin equation, J. Comput. Phys., 418 (2020), 109633-109641.  doi: 10.1016/j.jcp.2020.109633.

[15]

B. Helffer and F. Nier, Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians, Springer, 2005. doi: 10.1007/b104762.

[16]

F. Hérau and F. Nier, Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential, Arch. Ration. Mech. Anal, 171 (2004), 151-218.  doi: 10.1007/s00205-003-0276-3.

[17]

T. Hudson and X. H. Li, Coarse-graining of overdamped Langevin dynamics via the Mori–Zwanzig formalism, Multiscale Modeling & Simulation, 18 (2020), 1113-1135.  doi: 10.1137/18M1222533.

[18]

P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, volume 23, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-662-12616-5.

[19]

H. LeiN. A. Baker and X. Li, Data-driven parameterization of the generalized Langevin equation, Proc. Natl. Acad. Sci., 113 (2016), 14183-14188.  doi: 10.1073/pnas.1609587113.

[20]

Z. Li, X. Bian, X. Li and G. E. Karniadakis, Incorporation of memory effects in coarse-grained modeling via the Mori-Zwanzig formalism, J. Chem. Phys., 143 (2015), 243128. doi: 10.1063/1.4935490.

[21]

Z. Li, H. S. Lee, E. Darve and G. E. Karniadakis, Computing the non-Markovian coarse-grained interactions derived from the Mori-Zwanzig formalism in molecular systems: Application to polymer melts, J. Chem. Phys., 146 (2017), 014104. doi: 10.1063/1.4973347.

[22]

K. K. Lin and F. Lu, Data-driven model reduction, Wiener projections, and the Mori-Zwanzig formalism, J. Comput. Phys., 424 (2021), Paper No. 109864, 33 pp. arXiv preprint arXiv: 1908.07725, 2019. doi: 10.1016/j.jcp.2020.109864.

[23]

F. LuK. K. Lin and A. J. Chorin, Data-based stochastic model reduction for the Kuramoto–Sivashinsky equation, Physica D, 340 (2017), 46-57.  doi: 10.1016/j.physd.2016.09.007.

[24]

H. Mori, Transport, collective motion, and Brownian motion, Prog. Theor. Phys., 33 (1965), 423-455.  doi: 10.1143/PTP.33.423.

[25]

T. MoritaH. Mori and K. T. Mashiyama, Contraction of state variables in Non-Equilibrium open systems. II, Prog. Theor. Phys., 64 (1980), 500-521.  doi: 10.1143/PTP.64.500.

[26]

E. J. Parish and K. Duraisamy, Non-Markovian closure models for large eddy simulations using the Mori-Zwanzig formalism, Phys. Rev. Fluids, 2 (2017), 014604. doi: 10.1103/PhysRevFluids.2.014604.

[27]

G. A. Pavliotis, Stochastic Processes and Applications: Diffusion processes, the Fokker-Planck and Langevin Equations, volume 60., Springer, 2014. doi: 10.1007/978-1-4939-1323-7.

[28]

H. Risken, The Fokker-Planck Equation: Methods of Solution and Applications, Second edition. Springer Series in Synergetics, 18. Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-642-61544-3.

[29]

P. Stinis, Stochastic optimal prediction for the Kuramoto–Sivashinsky equation, Multiscale Modeling & Simulation, 2 (2004), 580-612.  doi: 10.1137/030600424.

[30]

R. Tibshirani, Regression shrinkage and selection via the Lasso, J. Royal Stat. Soc. Ser. B, 58 (1996), 267-288.  doi: 10.1111/j.2517-6161.1996.tb02080.x.

[31]

D. Venturi and G. E. Karniadakis, Convolutionless Nakajima-Zwanzig equations for stochastic analysis in nonlinear dynamical systems, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 470 (2014), 1-20.  doi: 10.1098/rspa.2013.0754.

[32]

D. VenturiT. P. SapsisH. Cho and G. E. Karniadakis, A computable evolution equation for the joint response-excitation probability density function of stochastic dynamical systems, Proc. R. Soc. A, 468 (2012), 759-783.  doi: 10.1098/rspa.2011.0186.

[33]

Y. Yoshimoto, I. Kinefuchi, T. Mima, A. Fukushima, T. Tokumasu and S. Takagi, Bottom-up construction of interaction models of non-Markovian dissipative particle dynamics, Phys. Rev. E, 88 (2013), 043305. doi: 10.1103/PhysRevE.88.043305.

[34]

Y. Zhu, J. M. Dominy and D. Venturi, On the estimation of the Mori-Zwanzig memory integral, J. Math. Phys., 59 (2018), 103501. doi: 10.1063/1.5003467.

[35]

Y. Zhu, H. Lei and C. Kim, Generalized second fluctuation-dissipation theorem in the nonequilibrium steady state: Theory and applications, arXiv preprint arXiv: 2104.05222, 2021.

[36]

Y. Zhu and D. Venturi, Faber approximation of the Mori-Zwanzig equation, J. Comp. Phys., 372 (2018), 694-718.  doi: 10.1016/j.jcp.2018.06.047.

[37]

Y. Zhu and D. Venturi, Generalized langevin equations for systems with local interactions, J. Stat. Phys., 178 (2020), 1217-1247.  doi: 10.1007/s10955-020-02499-y.

[38]

Y. Zhu and D. Venturi, Hypoellipticity and the Mori-Zwanzig formulation of stochastic differential equations, arXiv preprint arXiv: 2001.04565, 2020.

[39]

R. Zwanzig, Memory effects in irreversible thermodynamics, Phys. Rev., 124 (1961), 983. doi: 10.1103/PhysRev.124.983.

[40]

R. Zwanzig, Nonlinear generalized Langevin equations, J. Stat. Phys., 9 (1973), 215-220.  doi: 10.1007/BF01008729.

show all references

References:
[1]

A. D. Baczewski and S. D. Bond, Numerical integration of the extended variable generalized Langevin equation with a positive Prony representable memory kernel, J. Chem. Phys., 139 (2013), 044107. doi: 10.1063/1.4815917.

[2]

M. BerkowitzJ. D. MorganD. J. Kouri and J. A. McCammon, Memory kernels from molecular dynamics, J. Chem. Phys., 75 (1981), 2462-2463. 

[3]

A. J. ChorinO. H. Hald and R. Kupferman, Optimal prediction and the Mori-Zwanzig representation of irreversible processes, Proc. Natl. Acad. Sci. USA, 97 (2000), 2968-2973.  doi: 10.1073/pnas.97.7.2968.

[4]

W. Chu and X. Li, The Mori–Zwanzig formalism for the derivation of a fluctuating heat conduction model from molecular dynamics, Commun Math Sci., 17 (2019), 539-563.  doi: 10.4310/CMS.2019.v17.n2.a10.

[5]

J. M. Dominy and D. Venturi, Duality and conditional expectations in the Nakajima-Mori-Zwanzig formulation, J. Math. Phys., 58 (2017), 082701. doi: 10.1063/1.4997015.

[6]

J.-P. Eckmann and M. Hairer, Non-equilibrium statistical mechanics of strongly anharmonic chains of oscillators, Commun. Math. Phys., 212 (2000), 105-164.  doi: 10.1007/s002200000216.

[7]

J.-P. Eckmann and M. Hairer, Spectral properties of hypoelliptic operators, Commun. Math. Phys., 235 (2003), 233-253.  doi: 10.1007/s00220-003-0805-9.

[8]

J.-P. EckmannC.-A. Pillet and L. Rey-Bellet, Non-equilibrium statistical mechanics of anharmonic chains coupled to two heat baths at different temperatures, Commun. Math. Phys., 201 (1999), 657-697.  doi: 10.1007/s002200050572.

[9]

P. Español, Hydrodynamics from dissipative particle dynamics, Phys. Rev. E, 52 (1995), 1734.

[10]

P. Español and P. Warren, Statistical mechanics of dissipative particle dynamics, EPL, 30 (1995), 191.

[11]

S. K. J. Falkena, C. Quinn, J. Sieber, J. Frank and H. A. Dijkstra, Derivation of delay equation climate models using the Mori- Zwanzig formalism, Proc. R. Soc. A, 475 (2019), 20190075, 21 pp. doi: 10.1098/rspa.2019.0075.

[12]

D. Funaro, Polynomial Approximation of Differential Equations, volume 8, Springer-Verlag, Berlin, 1992.

[13]

D. GivonR. Kupferman and O. H. Hald, Existence proof for orthogonal dynamics and the Mori-Zwanzig formalism, Isr. J. Math., 145 (2005), 221-241.  doi: 10.1007/BF02786691.

[14]

F. GroganH. LeiX. Li and N. A. Baker, Data-driven molecular modeling with the generalized Langevin equation, J. Comput. Phys., 418 (2020), 109633-109641.  doi: 10.1016/j.jcp.2020.109633.

[15]

B. Helffer and F. Nier, Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians, Springer, 2005. doi: 10.1007/b104762.

[16]

F. Hérau and F. Nier, Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential, Arch. Ration. Mech. Anal, 171 (2004), 151-218.  doi: 10.1007/s00205-003-0276-3.

[17]

T. Hudson and X. H. Li, Coarse-graining of overdamped Langevin dynamics via the Mori–Zwanzig formalism, Multiscale Modeling & Simulation, 18 (2020), 1113-1135.  doi: 10.1137/18M1222533.

[18]

P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, volume 23, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-662-12616-5.

[19]

H. LeiN. A. Baker and X. Li, Data-driven parameterization of the generalized Langevin equation, Proc. Natl. Acad. Sci., 113 (2016), 14183-14188.  doi: 10.1073/pnas.1609587113.

[20]

Z. Li, X. Bian, X. Li and G. E. Karniadakis, Incorporation of memory effects in coarse-grained modeling via the Mori-Zwanzig formalism, J. Chem. Phys., 143 (2015), 243128. doi: 10.1063/1.4935490.

[21]

Z. Li, H. S. Lee, E. Darve and G. E. Karniadakis, Computing the non-Markovian coarse-grained interactions derived from the Mori-Zwanzig formalism in molecular systems: Application to polymer melts, J. Chem. Phys., 146 (2017), 014104. doi: 10.1063/1.4973347.

[22]

K. K. Lin and F. Lu, Data-driven model reduction, Wiener projections, and the Mori-Zwanzig formalism, J. Comput. Phys., 424 (2021), Paper No. 109864, 33 pp. arXiv preprint arXiv: 1908.07725, 2019. doi: 10.1016/j.jcp.2020.109864.

[23]

F. LuK. K. Lin and A. J. Chorin, Data-based stochastic model reduction for the Kuramoto–Sivashinsky equation, Physica D, 340 (2017), 46-57.  doi: 10.1016/j.physd.2016.09.007.

[24]

H. Mori, Transport, collective motion, and Brownian motion, Prog. Theor. Phys., 33 (1965), 423-455.  doi: 10.1143/PTP.33.423.

[25]

T. MoritaH. Mori and K. T. Mashiyama, Contraction of state variables in Non-Equilibrium open systems. II, Prog. Theor. Phys., 64 (1980), 500-521.  doi: 10.1143/PTP.64.500.

[26]

E. J. Parish and K. Duraisamy, Non-Markovian closure models for large eddy simulations using the Mori-Zwanzig formalism, Phys. Rev. Fluids, 2 (2017), 014604. doi: 10.1103/PhysRevFluids.2.014604.

[27]

G. A. Pavliotis, Stochastic Processes and Applications: Diffusion processes, the Fokker-Planck and Langevin Equations, volume 60., Springer, 2014. doi: 10.1007/978-1-4939-1323-7.

[28]

H. Risken, The Fokker-Planck Equation: Methods of Solution and Applications, Second edition. Springer Series in Synergetics, 18. Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-642-61544-3.

[29]

P. Stinis, Stochastic optimal prediction for the Kuramoto–Sivashinsky equation, Multiscale Modeling & Simulation, 2 (2004), 580-612.  doi: 10.1137/030600424.

[30]

R. Tibshirani, Regression shrinkage and selection via the Lasso, J. Royal Stat. Soc. Ser. B, 58 (1996), 267-288.  doi: 10.1111/j.2517-6161.1996.tb02080.x.

[31]

D. Venturi and G. E. Karniadakis, Convolutionless Nakajima-Zwanzig equations for stochastic analysis in nonlinear dynamical systems, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 470 (2014), 1-20.  doi: 10.1098/rspa.2013.0754.

[32]

D. VenturiT. P. SapsisH. Cho and G. E. Karniadakis, A computable evolution equation for the joint response-excitation probability density function of stochastic dynamical systems, Proc. R. Soc. A, 468 (2012), 759-783.  doi: 10.1098/rspa.2011.0186.

[33]

Y. Yoshimoto, I. Kinefuchi, T. Mima, A. Fukushima, T. Tokumasu and S. Takagi, Bottom-up construction of interaction models of non-Markovian dissipative particle dynamics, Phys. Rev. E, 88 (2013), 043305. doi: 10.1103/PhysRevE.88.043305.

[34]

Y. Zhu, J. M. Dominy and D. Venturi, On the estimation of the Mori-Zwanzig memory integral, J. Math. Phys., 59 (2018), 103501. doi: 10.1063/1.5003467.

[35]

Y. Zhu, H. Lei and C. Kim, Generalized second fluctuation-dissipation theorem in the nonequilibrium steady state: Theory and applications, arXiv preprint arXiv: 2104.05222, 2021.

[36]

Y. Zhu and D. Venturi, Faber approximation of the Mori-Zwanzig equation, J. Comp. Phys., 372 (2018), 694-718.  doi: 10.1016/j.jcp.2018.06.047.

[37]

Y. Zhu and D. Venturi, Generalized langevin equations for systems with local interactions, J. Stat. Phys., 178 (2020), 1217-1247.  doi: 10.1007/s10955-020-02499-y.

[38]

Y. Zhu and D. Venturi, Hypoellipticity and the Mori-Zwanzig formulation of stochastic differential equations, arXiv preprint arXiv: 2001.04565, 2020.

[39]

R. Zwanzig, Memory effects in irreversible thermodynamics, Phys. Rev., 124 (1961), 983. doi: 10.1103/PhysRev.124.983.

[40]

R. Zwanzig, Nonlinear generalized Langevin equations, J. Stat. Phys., 9 (1973), 215-220.  doi: 10.1007/BF01008729.

Figure 1.  Sample path of the tagged oscillator momentum $ p_{50}(t) $. We display the result for the stochastic FPU system (52) with weak ($ \theta = 0.1 $) and strong nonlinearity ($ \theta = 1 $) at high ($ \beta = 1 $) and low ($ \beta = 20 $) temperature
Figure 2.  Temporal auto-correlation function of the tagged oscillator momentum $ p_j(t) $ for weakly nonlinear FPU system at different temperature $ T\propto 1/\beta $. We compare results we obtained by calculating the EMZ memory from first principles using $ 14 $-th order Faber polynomials with results from MC simulation ($ 10^6 $ sample paths). In the subplots, we display $ |C(t)/C(0)| $ and the exponentially decaying upper bound $ ce^{-\alpha t} $ with an estimated decaying rate $ \alpha $
Figure 3.  Approximated EMZ memory kernel corresponding to the tagged particle momentum correlation function $ C(t) $. The subplots display $ |K(t)/K(0)| $ and the exponentially decaying upper bound $ c_{{\mathcal{Q}}}e^{-\alpha_{{\mathcal{Q}}}t} $ with an estimated decaying rate $ \alpha_{{\mathcal{Q}}} $. Other setting is same as Figure 2
Figure 4.  Temporal auto-correlation function of the tagged oscillator momentum $ p_j(t) $ for strongly nonlinear FPU system at different temperature $ T\propto 1/\beta $. The MC simulation results ($ 10^6 $ sample paths) of the correlation function are compared with the one obtained by the data-driven memory kernel using Faber series (20th order) and the standard Laguerre polynomials (20th order). In the subplots, we display $ |C(t)/C(0)| $ and the exponentially decaying upper bound $ ce^{-\alpha t} $ with an estimated decaying rate $ \alpha $
Figure 5.  Comparison of the dynamics of the particle momentum $ p_{50}(t) $ generated by the MC simulation and the ROM (55). The displayed results are for a stochastic FPU system with strong nonlinearity ($ \theta = 1 $) at high temperature $ \beta = 1 $ (first row) and low temperature $ \beta = 20 $ (second row). In the first column, we compare the simulated sample paths. The time autocorrelation functions $ C(t)/C(0) $ (second column) are obtained by averaging a cluster of the sample trajectories. The third column compares the stationary distribution of the stochastic process $ \rho_{p_{50}} $ which are obtained via kernel density estimations
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