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Data-driven parameterization of the generalizedLangevin equationHuan Leia,1, Nathan A. Bakera,b, and Xiantao Lic

aAdvanced Computing, Mathematics & Data, Pacific Northwest National Laboratory, Richland, WA 99352; bDivision of Applied Mathematics, BrownUniversity, Providence, RI 02912; and cDepartment of Mathematics, The Pennsylvania State University, University Park, PA 16802

Edited by Alexandre J. Chorin, University of California, Berkeley, CA, and approved October 27, 2016 (received for review June 15, 2016)

We present a data-driven approach to determine the memorykernel and random noise in generalized Langevin equations. Tofacilitate practical implementations, we parameterize the kernelfunction in the Laplace domain by a rational function, withcoefficients directly linked to the equilibrium statistics of thecoarse-grain variables. We show that such an approximation canbe constructed to arbitrarily high order and the resulting gen-eralized Langevin dynamics can be embedded in an extendedstochastic model without explicit memory. We demonstrate howto introduce the stochastic noise so that the second fluctuation-dissipation theorem is exactly satisfied. Results from severalnumerical tests are presented to demonstrate the effectivenessof the proposed method.

generalized Langevin dynamics | data-driven parameterization |coarse-grained molecular models | reaction rate | model reduction

Generalized Langevin equations (GLEs) have recentlyreemerged in the area of molecular modeling as a promis-

ing description of reduced-dimension coarse-grained variables.In principle, GLEs can be derived using the Mori–Zwanzigprojection formalism (1, 2). Examples of such derivations canbe found for a variety of applications (3–10), for example,climate modeling (11, 12). The GLE approach eliminates alarge number of irrelevant degrees of freedom, reducing sys-tem dimensions to make direct computation feasible. Becausethis elimination often projects out high-frequency modes, theGLE can also extend the time scale of simulations. The GLEdoes this by describing the dynamics for explicit quantities ofinterest and implicitly describing the remaining degrees of free-dom through a memory term and a random noise term. The ran-dom noise term is often strongly correlated in time.

However, practical implementations of GLEs require specifi-cation of the memory function, which can be difficult to obtain,even when the full dynamics of the system is known. For example,the memory functions obtained in past studies (8, 9, 13, 14) haveinvolved functions of high-dimensional matrices. Darve et al.(14) proposed a more efficient algorithm to compute the mem-ory kernel by solving an equation for the orthogonal dynam-ics derived from the Mori–Zwanzig formalism. However, theorthogonal dynamics equation can be expensive to solve whenthe original system is large. Furthermore, even when the mem-ory kernel function is available, direct evaluation of the mem-ory term can be costly because it requires the history of thecoarse-grained (CG) variables at every time step and the asso-ciated numerical integration. Sampling of the random noise isalso a challenging component of GLEs: To generate the correctequilibrium statistics for the CG model, the random noise hasto obey the second fluctuation-dissipation theorem (FDT) (15).The theory of stationary processes (16) states that the randomprocess is uniquely determined by the correlation function, whichis proportional to the memory kernel; however, sampling the ran-dom noise is nontrivial in practice. Methods based on matrix fac-torization are computationally challenging because they requiredecomposition of a correlation matrix with dimension propor-tional to the total simulation period. Alternatively, more efficient

methods based on fast Fourier transforms may create artificialperiodicity (17).

In addition to the direct derivation of memory kernels (8, 9, 13,14), there have been numerous attempts to compute the memorykernel from full molecular dynamics (MD) simulations (18–20),especially for systems with zero net mean force. Such analyseslead to integral equations of the first kind, which are numericallyunstable without additional regularization. Another approachfor estimating the kernel uses Kalman filtering and assumes func-tions of exponential form (21, 22) such that the GLE can beembedded in a Markovian dynamics framework. In recent work,Chorin and Lu (23) considered a time-discrete representation,representing the memory effects using the NARMAX (nonlinearautoregression moving average with exogenous input) method.Voth and coworkers (24) proposed an alternative approach torecover the CG dynamics by introducing fictitious particles thatinteract with the variables, effectively introducing an approxima-tion of the kernel function.

In this work, we present a hierarchical approach to obtainGLE kernel functions from simulation data. Such a data-driven approach is particularly useful for complex models (e.g.,biomolecules or climate) in which full dynamics models are typi-cally unavailable or inaccessible with finite computing resources.The key idea is parameterization of the kernel function Laplacetransform by a rational approximation. The parameters in ouransatz can be computed directly from statistical properties ofthe CG variables. Additionally, this ansatz makes it possible toeliminate memory from the GLE by introducing auxiliary vari-ables. In particular, we will show how to introduce inexpensivewhite-noise terms into the extended dynamics to approximate the

Significance

The generalized Langevin equation (GLE) provides a precisedescription of coarse-grained variable dynamics in reduceddimension models. However, computation of the memorykernel poses a major challenge to the practical use of theGLE. This paper presents a data-driven approach to computethe memory kernel, relying on a hierarchy of parameterizedrational approximations in terms of the Laplace transform,which can be expanded to arbitrarily high order as neces-sary. This parameterization makes it convenient to representthe GLE via an extended stochastic model where the memoryterm is eliminated by properly introducing auxiliary variables.The present method is well-suited for constructing reducedmodels for nonequilibrium properties of complex systemssuch as biomolecules, chemical reaction networks, and climatesimulations.

Author contributions: H.L., N.A.B., and X.L. designed research, performed research, ana-lyzed data, and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.1To whom correspondence should be addressed. Email: huan.lei@pnnl.gov.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1609587113/-/DCSupplemental.

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random noise while satisfying the second FDT is exactly. Asa result, no memory term needs to be evaluated, and no col-ored noise needs to be sampled. The first two approximations inthe rational approximation hierarchy correspond to the Marko-vian approximation (6, 25, 26) and approximation of noise by aOrnstein–Uhlenbeck process, which is the ansatz used in someprevious work. As we will show, these two approximations areoften insufficient to predict dynamics properties; however, ourhierarchy can be used to construct arbitrarily high-order mod-els to characterize long-time behaviors and complex transitiondynamics. The technique of using auxiliary variables has beendemonstrated by others (27–29) to treat the memory term inthe GLE. However, the current work makes a direct connectionbetween the parameters in the extended system and the statisticsof the CG variables. As a result, our method does not requirethe knowledge of the full model: Only the time series of the CGvariables are needed.

Data-Driven Parameterization of GLEThe GLE can be expressed in the following form:{

q = M−1p,

p = F(q)−∫ t

0θ(t − τ)υ(τ)dτ + R(t),

[1]

where q and p are the CG coordinates and momenta, andF(q) = −∇U (q) is the conservative force term determined bythe potential of mean force U (q). θ(t) denotes a memory kernelfunction, which is the main focus of this paper. The noise R(t)is a stationary Gaussian process with zero mean, satisfying thesecond FDT (15):

〈R(t)R(t ′)T 〉=β−1θ(t − t ′), [2]

where β= 1/kBT . Such equations have been derived in previouswork (8, 13) using the Mori–Zwanzig formalism (1, 30). We viewthe construction of reduced models as a two-step procedure. Thefirst step starts with a parametric form of the mean force termF (q). Many numerical methods have been proposed to estimateF (q) with respect to the CG variables q, such as iterative Boltz-mann inversion (31), force-matching (32, 33), inverse MonteCarlo (34), relative entropy (35), umbrella sampling (36), andmetadynamics (37) approaches. These methods produce goodchoices for CG variables but do not generally include accuratedescriptions of the system dynamics. In the present work, ourmain focus is on the second step: data-driven estimation of thefriction kernel and noise terms. We assume that the conservativeforce term F(q) for the CG variables is known a priori (i.e., ithas been estimated from the coarse-graining methods describedabove).

Calculation of the Memory Kernel. We assume that we have a timeseries dataset of v and F(q) (v =M−1p) drawn from an equilib-rium simulation such that the time series corresponds to a sta-tionary random process. We right-multiply the second equationin Eq. 1 by v(0)T to obtain

g(t) =

∫ t

0

θ(t − τ)h(τ)dτ. [3]

Here we have defined the correlation matrices,

g(t) = 〈Mv(t)−F (q(t)), v(0)T 〉, h(t) = 〈v(t), v(0)T 〉, [4]

and we have made the assumption that 〈R(t)v(0)T 〉= 0, whichwas verified in previous work (13).

Given the correlation functions, Eq. 3 can be regarded as anintegral equation from which the memory function can be com-puted. However, this is an integral equation of the first kind, andit is not well-posed, leading to unreliable solutions. Instead of adetermining the kernel function directly in the time-domain, we

can instead parameterize its Laplace transform. We define theLaplace transform as

G(λ) =

∫ +∞

0

g(t)e−t/λdt . [5]

Similarly, we denote the Laplace transforms of h(t) and θ(t)by H(λ) and Θ(λ), respectively. Taking the Laplace transform ofEq. 3, we arrive at

G(λ) = −Θ(λ)H(λ). [6]

We use the values of Θ at specific time points to constructΘ(λ). By taking λ→ +∞, we obtain

Θ(+∞) = −G(+∞)H (+∞)−1. [7]

It is clear that

G(+∞) =

∫ +∞

0

g(t)dt , H (+∞) =

∫ +∞

0

h(t)dt [8]

which makes it possible to incorporate a Green–Kubo type offormula in the approximation and model accuracy over long timescales.

For short or intermediate time scales, we use the point λ = 0.Using Eq. 6, one can find the limiting values of the kernel andits derivatives as λ → 0. This calculation amounts to computingG′(0), and similarly H′(0), which is straightforward. For instance,by integrating by parts repeatedly in Eq. 5, we find that

G ′(0) = g(0),G ′′(0)

2= g ′(0), · · · , G

(j)(0)

j != g(j−1)(0). [9]

For example, we have H′(0) = h(0) = kBT I. In addition, wecan find Θ(0) = 0,

Θ′(0) =−βg ′(0),

Θ′′(0) =−2β[g ′′(0) + βg(0)h ′′(0)

],

Θ′′′(0) =−6β[g ′′′(0) + βg ′(0)h ′′(0)

]. [10]

In the derivations above, we have incorporated the values atλ = 0 and λ = +∞. However, the ansatz of the rational approx-imation is quite flexible, and other interpolation points can beused as well. For stationary process of Hamiltonian system,〈v(0)q(0)T 〉 = 0; therefore, g′′(0) = 〈m v(0)−F(q(0)), v(0)T 〉 ≡0 and Θ′′(0) ≡ 0.

Rational Approximations. Given limiting values available extractedfrom the data, we seek a rational function approximation forΘ(λ), in the form of

Θ(λ) ≈[I − λB0 − λ2B1− · · · −λnBn−1

]−1

×[A0 + λA1+ · · · +λn−1An−1

]λ. [11]

The coefficients {Ai ,Bi} can be determined by matching thelimits of Θ(λ). The matching conditions lead to a linear system ofequations, which can be solved analytically for small n or numer-ically for large n .

The zeroth-order approximation treats Θ(λ)≡ θ0 as a con-stant matrix set to Θ(+∞). Accordingly, one gets a Markovianapproximation by a Langevin dynamics with damping coefficientgiven by γ = θ0. We can determine θ0 by Eq. 7. In fact, H(+∞)is proportional to the diffusion tensor, that is, the matching con-dition recovers the Einstein relation D = kBT

γ(15, 38).

For the first-order approximation (n = 1), we have

Θ(λ) =[I − λB0

]−1A0λ. [12]

By matching Eqs. 7 and 10, we find that

A0 = θ(0), B0 =− θ(0)Θ(+∞)−1.

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()

10-2 10-1 100 10110-2

10-1

100

ExactZero orderFirst orderSecond orderThird order

0.6 0.7 0.8 0.9 1 1.10.45

0.5

0.55

0.6

Fig. 1. Laplace transform of the memory kernel Θ(λ) modeling a taggedparticle attached to a one-dimensional harmonic chain. (Inset) Plot showingthe close-up view of Θ(λ) within the rectangular region.

In this case, the memory function in the time domain is given by

θ(t)≈ etB0A0. [13]

Depending on the eigenvalues of B0, the memory function canexhibit both exponential decay and oscillations.

Extended Dynamics. A computational difficulty in GLE simu-lation is that the integral has to be evaluated at every step.However, this difficulty can be removed by introducing auxil-iary equations based on the rational approximation of the mem-ory function Eq. 11. More specifically, we can define d(t) =∫ t

0θ(t − τ)v(τ)dτ . Then, Eq. 12 implies that the approximate

GLE can be written asq =M−1p,p =F (q) + d ,

d =B0d −A0v + W (t),[14]

where we added a white-noise term W(t) satisfying 〈W(t)

W(t ′)T 〉=−β−1(B0A0 + A0BT

0 )δ(t − t ′). We pick the ini-tial state of the auxiliary variable d to satisfy 〈d(0)d(0)T 〉=β−1A0.

Fig. 2. Laplace transform Θ(λ) for a tagged particle in a particle bath obtained from full MD data and calculated with different orders of rational approxi-mation. Case (I) (Left) and case (II) (Right).

We can show that this new memoryless dynamics correspondsto an approximation of the GLE Eq. 1. The memory function isapproximated by the rational function in the frequency space,which is precisely Eq. 13. More importantly, with this properchoice of the initial condition for d, the random noise R(t) isgiven by R(t) =

∫ t

0eB0(t−s)W(s)ds + d(0)eB0t , which is a station-

ary Gaussian process that satisfies the second FDT Eq. 2 exactlywith an invariant distribution (see Supporting Information fordetails) given by

ρ(q , p, d)∼ e−β( 12M−1p2+U (q)+ 1

2dTA−1

0 d) [15]

The procedure above can be extended to arbitrarily high order,and the extended system can be written as follows:

q =M−1p,p =F (q) + ZTd ,

d =Bd −QZv + W (t),[16]

where B is a matrix and W is added white noise. For example, forthe fourth-order method, B is given by

B =

0 0 0 B3

I 0 0 B2

0 I 0 B1

0 0 I B0

. [17]

The matrices Q and Z can be determined by matchingthe Laplace transform of the memory kernel given by θ(t) =ZTeBtQZ with the rational approximation Eq. 11. Similar to Eq.14, we can also show that, by choosing the white noise W andthe initial conditions of d properly, that is, 〈d(0)d(0)T 〉=β−1Q,〈W(t)W(s)T 〉=−β−1(BQ + QBT )δ(t − s), the colored noisegenerated by the extended dynamics also satisfies the secondFDT Eq. 2 exactly. Eq. 16 has an invariant distribution (see Sup-porting Information for details) given by

ρ(q , p, d)∼ e−β( 12M−1p2+U (q)+ 1

2dTQ−1d). [18]

Numerical ResultsA One-Dimensional Chain Model. We demonstrate our methodthrough coarse-graining the dynamics of a tagged particle withina one-dimensional harmonic chain. In particular, we consider thefirst particle (on the free end) as the target particle and treatthe remaining particles as the heat bath. As shown in refs. 39and 40, the dynamics of the target particle can be modeled bya GLE with kernel given by θ(t) =

√mKt

J1

(2Kt√m

), where m is

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Fig. 3. Simulations of a tagged particle in a particle bath. (Upper) Velocity correlation function in short time scale (t between 0 and 2) obtained from MDdata and different orders of the rational approximation for cases (I) (Left) and (II) (Right). (Middle) Velocity correlation function (for t between 0 and 20) inlog–log scale for cases (I) (Left) and (II) (Right). (Lower) Mean square displacement obtained from MD data and different order of the rational approximationfor cases (I) (Left) and (II) (Right).

the mass of each particle, K is the force constant of harmonicinteraction, and J1 is a Bessel function of the first kind. We cal-culated the trajectory of the tagged particle in a harmonic chainconsisting of N = 1,000 particles with K , m and β set to unity(see Supporting Information for details). Fig. 1 shows the numer-ical results for Θ(λ) obtained using the rational approximationmethod described above. As the approximation increases to thirdorder, Θ(λ) agrees well with the exact formulation, which is given

by√

1+4λ2−1

2λ.

A Tagged Particle in Solvent. We also studied a tagged particleimmersed in a fluid system governed by pairwise-conservativeforces similar to those in dissipative particle dynamics (DPD)simulations (41, 42), that is, defined by

Fij =

{a(1.0− rij/rc)eij , rij < rc ,0, rij > rc ,

[19]

where rij = ri − rj , rij = |rij | and eij = rij/rij and a is the forcemagnitude and rc is the cutoff radius beyond which all interac-tions vanish. One tagged particle and 2,999 solvent particles wereplaced in a cubic box 10× 10× 10r3

c , with a periodic bound-ary condition imposed in each direction. The mass m of boththe tagged particle and solvent particle were chosen to be unity.A Nose–Hoover thermostat was used to equilibrate the system(additional details are provided as Supporting Information).

Following the Mori–Zwanzig method, the dynamics of the tar-get particle can be modeled by a GLE with zero mean force.We considered two cases: (I) β= 0.5, a = 25.0 and (II) β= 1.0,a = 50.0. Based on the velocity correlation function 〈v(0)v(t)T 〉obtained from MD simulation data, we can compute the dif-ferent orders of kernel approximation Θ(λ) by Eq. 11. Asshown in Fig. 2, the exact kernel function Θ(λ) agrees wellwith the numerical result directly obtained from 〈v(0)v(t)T 〉as the approximation order n increases to second orderor above.

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t

<v(t)v(0)>/<v(0)2 >,(t)/(0)

0 0.4 0.8 1.2 1.6 2-0.2

0

0.2

0.4

0.6

0.8

1

(I)(II)

Fig. 4. Simulations of a tagged particle in a particle bath. Velocity correla-tion (dashed line) and the fourth-order approximation of θ(t) (solid line) forcases (I) and (II).

Unlike case (I), Θ(λ) in case (II) shows a pronouncedpeak near λ= 0.2, indicating significant oscillations in thetime domain of θ(t). Fig. 3 shows the velocity correlationfunction 〈v(0)v(t)〉= Tr

[〈v(0)v(t)T 〉

]/3 and the mean-squa-

red displacement 〈(q(t)− q(0))2〉= Tr[〈(q(t) − q(0))(q(t)−

q(0))T 〉]/3 obtained from solving Eq. 1 with kernel θ(t)

constructed by different orders of rational approximations.The zeroth-order approximation corresponds to the Markovianapproximation of the kernel term∫ t

0

θ(s)v(t − s)ds ≈[∫ ∞

0

θ(s)ds

]v(t). [20]

In case (I), Eq. 1 can reproduce the velocity correlation atshort timescales (e.g., t between 0 and 2) fairly well for rationalapproximations of second order (and above). In contrast, for case(II), 〈v(0)v(t)〉 obtained from the second-order approximationexhibits artificial oscillation and deviations from the dynamicsresults; the third- and fourth-order approximations yield much

Fig. 5. (Left) Transition flux correlations for a particle in a double-well potential obtained from full MD and GLE simulations with different orders ofapproximation. (Right) Plateau reaction rate obtained by using GLEs with different orders of approximation. Order “−1” refers to the reaction rate predictedby transition state theory.

better agreement. Moreover, the low-order approximations areinsufficient to capture the long correlation tail for both cases.As shown in Fig. 2, a third-order approximation is required tocapture the velocity correlation function up for t up to 6 and afourth-order approximation is required for t up to 20.

The different performance for cases (I) and (II) can be under-stood by examining the time scale separation of θ(t) and v(t)

in the memory term∫ t

0θ(s)v(t − s)ds . As shown in Fig. 4, the

velocity correlation 〈v(0)v(t)〉 of case (I) decays much moreslowly than for case (II). The plateau region of Θ(λ) of case(I) shown in Fig. 2 illustrates the similarities between θ(t) andθ0δ(t). These similarities explain why the Markovian approxima-tion by Eq. 20 yields fairly good agreement for case (I). In con-trast, there is no apparent time scale separation between v(t) andθ(t) for case (II), explaining the need for higher-order approxi-mations to characterize the coupling between θ(t) and v(t).

Non-Markovian Effects on Transition Dynamics. To further validateour method, we simulated the transition rate of a tagged particlein a double-well potential using both by GLE and full MD. Thedouble-well potential had the form

U (x ) = U0

[1−

(x

x0

)2]2

[21]

where U0 = 25 and x0 = 2.5 refer to the depth and width of thepotential field. The tagged particle interacted with solvent parti-cles through interactions defined in Eq. 19 with β= 1 and a = 50,interactions that drove the transitions between the two energyminima at x1 =− x0 and x2 = x0. The instantaneous transitionrate κ12(t) was computed by

κ12(t) = 〈δ[S(0)−S0]Sχ(t)〉/QR, [22]

where S is a collective variable defining the dividing iso-surfaceS −S0 = 0 between the states and χ(t) is the characteristic func-tion for the reaction trajectory. For the double-well system, S = xand χ(t) = Hs(S(t))Hs(−S(−t)), where Hs is the Heavisidefunction. QR =

∫Γ1e−βH is the reaction partition function over

the phase space of state 1. The reaction flux correlation functionCRF (t) is given by CRF (t) = κ12(t)/κTST , where κTST is thereaction rate predicted from transition state theory. Fig. 5 showsCRF (t) obtained from the full MD the GLE systems with kernelsmodeled by different orders of rational approximation (simula-tion details are presented in Supporting Information). The zero-order approximation (corresponding to Langevin dynamics witha constant friction coefficient) yielded a CRF (t) that deviatedfrom the full MD result, indicating a pronounced non-Markovian

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effect. In contrast, CRF (t) obtained from the GLE using thethird- and fourth-order rational approximations agree well withthe full MD results. Moreover, as also shown in Fig. 5, bothtransition state theory and the zero-order (Langevin dynamics)model overestimate the reaction rate; κ12 approaches the fullMD results only when using the third- and fourth-order approx-imations where the memory kernel can be more accuratelymodeled.

ConclusionWe have presented a data-driven approach to obtain the mem-ory kernel for a GLE through rational-function approximationin the Laplace transform domain. The data-driven nature of thismethod arises through connection of the rational function coef-ficients with equilibrium statistics of the CG variables—statisticsthat can be calculated through simulation time series data. Thezero- and first-order approximations recover the Langevin andOrnstein–Uhlenbeck stochastic processes, respectively. Higher-order approximations have also been systematically derivedfor systems with significant memory effects. Unlike the time-domain kernel function representation, numerical simulations ofthe GLE using the rational approximation can be convenientlyimplemented by introducing auxiliary variables that follow lin-ear stochastic dynamics with no memory, eliminating the needfor expensive calculations of history-dependent memory terms.We have also shown that the second FDT Eq. 2 can be satisfiedautomatically using our approach. This method has been tested

with simple systems but is applicable to much more complicatedbiological and material systems with pronounced memory effects[e.g., subdiffusion in single-molecule measurements (43, 44) ortransition dynamics of chemical and biological reaction systems(45, 46)]. Many of these systems will be high-dimensional, evenafter coarse graining. Although our GLE-based method has nodimensionality restrictions, it can still suffer from the curse ofdimensionality. In particular, high-dimensional systems will becomputationally expensive and require time series for all CGvariables. The rational function approximations (Eqs. 14 and 16)of the GLE enable us to analyze the transition dynamics viathe extended dynamics in an augmented phase space (q, p, d),where direct analysis via the GLE could be difficult/inaccessible.In general, it is possible to introduce more interpolation pointsfor the rational approximation rather than simply using the lim-iting values at 0 and ∞. The introduction of additional inter-polation points is likely to be important for more complexdynamics when intermediate timescales are of importance. Sometheory and algorithms available for matrix-valued rational inter-polations (47) could guide future extension of our method toincorporate additional sampling points.

ACKNOWLEDGMENTS. We thank George Karniadakis, Eric Darve, WilliamNoid, Panos Stinis, Gregory Schenter, Lei Wu, Dave Sept, J. AndrewMcCammon, and Zhen Li for informative discussions and advice and the twoanonymous reviewers for very helpful suggestions. This work was supportedby the US Department of Energy, Office of Science, Office of Advanced Sci-entific Computing Research as part of the Collaboratory on Mathematics forMesoscopic Modeling of Materials (CM4).

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