Data-driven construction of stochastic reduced dynamics encoded with non-Markovian features Zhiyuan She1Pei Ge1and Huan Lei1 2

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Data-driven construction of stochastic reduced dynamics encoded

with non-Markovian features

Zhiyuan She,1Pei Ge,1and Huan Lei1, 2, ∗

1Department of Computational Mathematics,

Science & Engineering, Michigan State University, MI 48824, USA

2Department of Statistics & Probability,

Michigan State University, MI 48824, USA

Abstract

One important problem in constructing the reduced dynamics of molecular systems is the accu-

rate modeling of the non-Markovian behavior arising from the dynamics of unresolved variables.

The main complication emerges from the lack of scale separations, where the reduced dynamics

generally exhibits pronounced memory and non-white noise terms. We propose a data-driven ap-

proach to learn the reduced model of multi-dimensional resolved variables that faithfully retains

the non-Markovian dynamics. Diﬀerent from the common approaches based on the direct con-

struction of the memory function, the present approach seeks a set of non-Markovian features

that encode the history of the resolved variables, and establishes a joint learning of the extended

Markovian dynamics in terms of both the resolved variables and these features. The training is

based on matching the evolution of the correlation functions of the extended variables that can

be directly obtained from the ones of the resolved variables. The constructed model essentially

approximates the multi-dimensional generalized Langevin equation and ensures numerical stability

without empirical treatment. We demonstrate the eﬀectiveness of the method by constructing

the reduced models of molecular systems in terms of both one-dimensional and four-dimensional

resolved variables.

∗leihuan@msu.edu

arXiv:2210.05814v1 [physics.comp-ph] 11 Oct 2022

I. INTRODUCTION

Predictive modeling of multi-scale dynamic systems is a long-standing problem in many

ﬁelds such as biology, materials science, and ﬂuid physics. One essential challenge arises

from the high-dimensionality; numerical simulations of the full models often show limitations

in the achievable spatio-temporal scales. Alternatively, reduced models in terms of a set of

resolved variables are often used to probe the evolution on the scale of interest. However, the

construction of reliable reduced models remains a highly non-trivial problem. In particular,

for systems without a clear scale separation, the reduced dynamics often exhibits non-

Markovian memory eﬀects, where the analytic form is generally unknown. To close the

reduced dynamics, existing methods are primarily based on the following two approaches.

The ﬁrst approach seeks various numerical approximations of the memory term by projecting

the full dynamics onto the resolved variables based on frameworks such as the Mori-Zwanzig

formalism [1, 2] or canonical models such as the generalized Langevin equation (GLE) [3].

Examples include the t-model approximation [4], the Galerkin discretization [5], regularized

integral equation discretization [6], the hierarchical construction [7–11], and so on. Recent

studies [12–15] based on the recurrent neural networks [16] provide a promising approach to

learn the memory term of deterministic dynamics. Yet, for ergodic dynamics, how to impose

the coherent noise term compensating for the unresolved variables remains open. The second

approach parameterizes the memory term by certain ansatz, e.g., the ﬁctitious particle [17],

continued fraction [18, 19], rational function [20], such that the memory and the noise terms

can be embedded in an extended Markovian dynamics [17, 19, 21–28]. In addition, non-

Markovian models are represented by discrete dynamics with exogenous inputs in form of

NARMAX (nonlinear autoregression moving average with exogenous input) [29, 30] and

SINN (statistics information neural network) [31] and parameterized for each speciﬁc time

step. Despite the overall success, most studies focus on the cases with a scalar memory

function. Notably, the reduced model of a two-dimensional GLE is constructed in Ref. [25].

To the best of our knowledge, the systematic construction of stochastic reduced dynamics

of multi-dimensional resolved variables remains under-explored.

Ideally, to obtain a reliable reduced model, the construction needs to accurately retain

the non-Markovian features, enable certain modeling ﬂexibility (e.g., the dimensionality of

the resolved variables) and adaptivity (e.g., the order of approximation), and guarantee

the numerical stability and robustness. In a recent study, we developed a Petrov-Galerkin

approach [32] to construct the non-Markovian reduced dynamics by projecting the full dy-

namics into a subspace spanned by a set of projection bases in form of the fractional deriva-

tives of the resolved variables. The obtained reduced model is parameterized as extended

stochastic diﬀerential equations by introducing a set of test bases. Diﬀerent from most

existing approaches, the construction does not rely on the direct ﬁtting of the memory func-

tion. Non-local statistical properties can be naturally matched by choosing the appropriate

bases, and the model accuracy can be systematically improved by introducing more basis

functions to expand the projection subspace. Despite these appealing properties, the con-

struction relies on the heuristic choices of the projection and test bases. Given the target

number of basis, how to choose the optimal basis functions for the best representation of the

non-Markovian dynamics remains an open problem. Furthermore, the numerical stability

needs to be treated empirically. These issues limit the applications in complex systems with

multi-dimensional resolved variables.

In this work, we aim to address the above issues by developing a new data-driven approach

to construct the stochastic reduced dynamics of multi-dimensional resolved variables. The

method is based on the joint learning of a set of non-Markovian features and the extended

dynamic equation in terms of both the resolved variables and these features. Unlike the

empirically chosen projection bases adopted in the previous work [32], the non-Markovian

features take an interpretable form that encodes the history of the resolved variables, and

are learned along with the extended Markovian dynamic such that they are optimal for

the reduced model representation. In this sense, they represent the optimal subspace that

embodies the non-Markovian nature of the resolved variables. The learning process enables

the adaptive choices of the number of features and is easy to implement by matching the

evolution of the correlation functions of the extended variables. In particular, the explicit

form of the encoder function enables us to obtain the correlation functions of these features

directly from the ones of the resolved variables rather than the time-series samples. The

constructed model automatically ensures numerical stability, strictly satisﬁes the second

ﬂuctuation-dissipation theorem [33], and retains the consistent invariant distribution [34, 35].

We demonstrate the method by modeling the dynamics of a tagged particle immersed

in solvents and a polymer molecule. With the same number of features (or equivalently,

the projection bases), the present method yields more accurate reduced models than the

previous methods [23, 32] due to the concurrent learning of the non-Markovian features.

More importantly, reduced models with respect to multi-dimensional resolved variables can

be conveniently constructed without the cumbersome eﬀorts of matrix-valued kernel ﬁtting

and stabilization treatment. This is well-suited for model reduction in practical applications,

where the constructed reduced models often need to retain the non-local correlations among

the resolved variables. It provides a convenient approach to construct meso-scale mod-

els encoded with molecular-level ﬁdelity and paves the way towards constructing reliable

continuum-level transport model equations [36, 37].

Finally, it is worthwhile to mention that the present study focuses on the model reduction

of ergodic dynamic systems where the full or part of the resolved variables are speciﬁed as

known quantities that either retain a clear physical interpretation (e.g., the tagged particle

position), or are experimentally accessible (e.g., the polymer end-to-end distance, the radius

of gyration). Another relevant direction focuses on learning the slow or Markovian dynamics

from the complex dynamic systems where the resolved variables are unknown a priori; we

refer to Refs. [38–43] on learning resolved variables that retain the Markovianity, Refs. [44–

49] on learning the slow dynamics on a non-linear manifold, and Refs. [50–53] on model

reduction of the transfer operator.

II. METHODS

A. Problem Setup

Let us consider the full model as a Hamiltonian system represented by a 6N-dimensional

phase vector Z= [Q;P], where Qand Pare the position and momentum vectors, respec-

tively. The equation of motion follows

Z=S∇H(Z),(1)

where S=

0I

−I0

is the symplectic matrix, and H(Z) is the Hamiltonian function and

initial condition is given by Z(0) = Z0. For high-dimensional systems with N1, the

numerical simulation of Eq. (1) can be computational expensive. It is often desirable to

construct a reduced model with respect to a set of low-dimensional resolved variables z(t) :=

φ(Z(t)) where φ:R6N→Rmrepresents the mapping from the full to the coarse-grained

state space with mN. With the explicit form of H(Z) and φ(Z), the evolution of z(t) can

be mapped from the initial values via the Koopman operator [54], i.e., z(t)=etLz(0), where

the Liouville operator Lφ(Z) = −((∇H(Z0))TS∇Z0)φ(Z) depends on the full-dimensional

phase vector Z. Using the Duhamel–Dyson formula, the evolution of z(t) can be further

formulated in terms of zbased on the Mori-Zwanzig (MZ) projection formalism [1, 2].

However, the numerical evaluation of the derived model relies on solving the full-dimensional

orthogonal dynamics [4], which can be still computational expensive.

In practice, the resolved variables are often deﬁned by the position vector Q. The MZ-

formed reduced dynamics is often simpliﬁed into the GLEs, i.e.,

q=M−1p

p=−∇U(q)−Zt

θ(t−τ)˙

q(τ) dτ+R(t),

(2)

where q∈Rmis the so-called collective variables, Mis the mass matrix, U(q) is the free

energy function, θ(t) : R+→Rm×mis a matrix-valued function representing the memory

kernel, and R(t) is a stationary colored noise related to θ(t) through the second ﬂuctuation-

dissipation condition [55], i.e., R(t)R(0)T=kBTθ(t). Numerical simulation of Eq. (2)

requires the explicit knowledge of both the free energy U(q) and the memory function θ(t).

Several methods based on importance sampling [56–58] and temperature elevation [59–61]

have been developed to construct the multi-dimensional free energy function. In real applica-

tions, the main challenge often lies in the treatment of the memory kernel θ(t). In particular,

for multi-dimensional collective variables q, the eﬃcient construction of numerically stable

matrix-valued memory function remains under-explored.

In this study, we develop an alternative approach to learn the reduced model. Rather than

directly constructing the memory function θ(t) in Eq. (2), we seek a set of non-Markovian

features from the full model, denoted by {ζi}n

i=1, and establish a joint learning of the reduced

Markovian dynamics in terms of both the resolved variables and these features, i.e.,

d˜

z=g(˜

z) dt+ΣdWt,(3)

where ˜

z:= [q;p;ζ1;··· ;ζn] represents the extended variables and Wtrepresents the stan-

dard Wiener process. In principle, any such extended system would generally lead to a

non-Markovian dynamics for the resolved variables z= [q;p]. However, the essential chal-

lenge is to determine {ζi}n

i=1 so that the non-local statistical properties of zcan be preserved

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Data-drivenconstructionofstochasticreduceddynamicsencodedwithnon-MarkovianfeaturesZhiyuanShe,1PeiGe,1andHuanLei1,2,1DepartmentofComputationalMathematics,Science&Engineering,MichiganStateUniversity,MI48824,USA2DepartmentofStatistics&Probability,MichiganStateUniversity,MI48824,USAAbstractOneimportant...

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Data-driven construction of stochastic reduced dynamics encoded with non-Markovian features Zhiyuan She1Pei Ge1and Huan Lei1 2

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