Machine learning assisted coarse-grained molecular dynamics modeling of meso-scale interfacial ﬂuids Pei Ge1Linfeng Zhang2 3and Huan Lei1 4

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Machine learning assisted coarse-grained molecular dynamics modeling of

meso-scale interfacial ﬂuids

Pei Ge,1Linfeng Zhang,2, 3, ∗and Huan Lei1, 4, †

1Department of Computational Mathematics, Science & Engineering,

Michigan State University, East Lansing, MI 48824, USA

2AI for Science Institute, Beijing 100080, China

3DP Technology, Beijing 100080, China

4Department of Statistics & Probability,

Michigan State University, East Lansing, MI 48824, USA

Abstract

A hallmark of meso-scale interfacial ﬂuids is the multi-faceted, scale-dependent interfacial energy, which

often manifests different characteristics across the molecular and continuum scale. The multi-scale nature

imposes a challenge to construct reliable coarse-grained (CG) models, where the CG potential function

needs to faithfully encode the many-body interactions arising from the unresolved atomistic interactions

and account for the heterogeneous density distributions across the interface. We construct the CG models of

both single- and two-component of polymeric ﬂuid systems based on the recently developed deep coarse-

grained potential (DeePCG) [1] scheme, where each polymer molecule is modeled as a CG particle. By only

using the training samples of the instantaneous force under the thermal equilibrium state, the constructed

CG models can accurately reproduce both the probability density function of the void formation in bulk and

the spectrum of the capillary wave across the ﬂuid interface. More importantly, the CG models accurately

predict the volume-to-area scaling transition for the apolar solvation energy, illustrating the effectiveness to

probe the meso-scale collective behaviors encoded with molecular-level ﬁdelity.

∗linfeng.zhang.zlf@gmail.com

†leihuan@msu.edu

arXiv:2210.12482v1 [physics.comp-ph] 22 Oct 2022

I. INTRODUCTION

Molecular dynamics (MD) simulations provide a promising avenue to establish the atomistic-

level understanding of many complex systems relevant to biological and materials science. Despite

the overwhelming success during the past decades, a remaining bottleneck roots in the limitation

of the achievable spatio-temporal scales; the gap between the micro-scale atomistic motions and

many meso-scale emerging phenomena remains large. One important problem is the nano-scale

interfacial ﬂuids, which play a crucial role in the hydration and the assembly of the biomolecules

and functional nano-materials [2, 3]. However, it is well-known that such ﬂuid systems gener-

ally exhibit complex and multifaceted nature on different scales. On the small scale (i.e., the

ﬂuid molecule correlation length), the solvation energy is determined by the molecular reorgani-

zation and scales with the volume of the void space. On the large scale, the solvation energy is

determined by the free energy for maintaining a ﬂuid-void interface and scales with the surface

area. The scale-dependent behavior indicates an cross-over regime of the entropy-enthalpy transi-

tion. While theoretical understandings [4–7] of this ubiquitous phenomenon have been developed,

computational modeling often relies on full micro-scale MD simulations to retain the multifaceted

properties, which, however, remain too expensive to achieve the resolved scale for applications

such as nano-scale assembly.

To accelerate the full MD simulations, many coarse-grained (CG) models have been devel-

oped. By modeling the dynamics in terms of a set of CG variables with reduced dimensionality,

the coarse-grained molecular dynamics (CGMD) simulations, in principle, enable us to probe

the collective behaviors on a broader scale. However, in practice, the construction of truly reli-

able CG models can be highly non-trivial, especially for the meso-scale interfacial ﬂuids. There

are two major challenges. The ﬁrst challenge arises from the many-body nature of CG inter-

actions. Speciﬁcally, the equilibrium density distribution of the CG model needs to match the

marginal density distribution of the CG variables of the full model. Due to the unresolved atom-

istic degrees of freedom, the CG potential generally encodes the many-body interactions even if

the full MD force ﬁeld is governed by two-body interactions [8]. Existing approaches often rely

on various physical intuitions as well as empirical approximations [9–22] that reproduce certain

target thermodynamic quantities and/or structural distributions. For example, the pairwise addi-

tive decomposition based on direct ensemble averaging [11, 12] can recover the thermodynamic

pressure but often fail to recover the pair distribution function. Conversely, the Monte Carlo and

Boltzmann inverse approaches [23–25] can reproduce the pairwise distribution function, which,

however, lead to the biased predictions of the equation of state. Several studies account for the

many-body effects by introducing the conﬁguration-independent volume potential [26–28] and the

local density [21, 22, 29–32] into the pairwise interactions. On the other hand, the accuracy of the

high-order structural correlations as well as the direct applications to interfacial systems remains

under-explored.

Besides the many-body effect, the ﬂuid molecules also exhibit heterogeneous density at the in-

terfacial vicinity. What further complicates the problem is the fact that the interfacial ﬂuid density

distribution is scale-dependent. On the small scale, the molecular reorganization generally leads

to a wet interface with larger density than the bulk value. On the large scale, the ﬂuid-void phase

separation generally leads to a dry interface with lower density. The crossover implies complex

molecular correlations near the interface. To capture this multi-faceted property, the constructed

CG potential needs to properly embody the local particle distribution other than the homogeneous

bulk distribution. Conventional structural-based CG potential functions generally show limitations

to incorporate such information. Similar to the many-body dissipative particle dynamics [14], re-

cent studies employed the local density [33–39] as well as the density gradient [40] as the auxiliary

ﬁeld variables to construct the CG potential functions. While the CG models show signiﬁcant im-

provement to reproduce the interfacial density proﬁle, the scale-dependent interfacial energy and

ﬂuctuations have not been systematically investigated. In Ref. [41], interfacial energy is integrated

into the continuum ﬂuctuation hydrodynamic equation [42] from the top-down perspective. Fluid

particles essentially represent the Lagrangian discretization points based on the smoothed dissi-

pative particle hydrodynamics [43] instead of the CG molecules; the meso-scale ﬂuid structural

properties can not be retained. Currently, the construction of reliable bottom-up CGMD models

that faithfully encode the multifaceted molecular interactions remains largely open.

In this work, we aim to address the above challenges by constructing CG models of meso-scale

interfacial ﬂuids based on the deep molecular dynamics (DeePMD) scheme [44, 45]. DeePMD is

initially developed for learning the many-body interactions from the ab initio molecular dynamics,

and has been applied to construct the deep coarse-grained (DeePCG) model [1] of liquid water in

bulk. Unlike the conventional forms of the inter-molecular potential function, the DeePMD rep-

resents each particle as an agent and the relative positions of its neighboring particles as the local

environment. Rather than approximating the total potential of the full system by an uniﬁed para-

metric function, the DeePMD directly maps the local environment of each agent to the potential

energy of that particle through a neural network that strictly preserves the spatial symmetries and

the particle permutation invariance. Accordingly, the construction does not rely on the empirical

decomposition (e.g., pairwise, three-body) of the high-dimensional particle conﬁguration space.

This unique feature is particularly suited for modeling the many-body potential of CGMD models,

where the ensemble-averaged interaction between two CG particles further depends on the other

neighboring CG particles and can not be represented by a pairwise additive function. Moreover,

the heterogeneous particle density distribution across the ﬂuid interface can be naturally incorpo-

rated into the CG potential function as the local environment of each particle. Accordingly, the

constructed CG models can accurately model the multifaceted, scale-dependent interfacial ﬂuctu-

ations and apolar solvation without additional human intervention.

We demonstrate the effectiveness of the CG models by considering both the single- and two-

component ﬂuids in presence of thermal interfacial ﬂuctuations. As discussed in Ref. [2], the

scale-dependent hydrophobic effects can be general for solvent molecules with attractive interac-

tions; polymeric liquids are therefore used as the benchmark problem. We compare the numerical

results from the full MD simulations and the CG description that represents each molecule as a

single particle located at the center of mass. By merely using training samples under equilibrium

thermal ﬂuctuations, the constructed CG models accurately predict the high-order correlations, the

local compressibility, and the interfacial capillary wave. In contrast, the empirical CG potential

constructed based on the pairwise approximation shows apparent deviations. More importantly,

the CG models accurately predict the probability of void formation in bulk as well as the volume-

to-area scaling transition for the solvation energy, and therefore, pave the way for modeling the

nanoscale assembly in aqueous environment.

Before wrapping up this section, we note that the present work focuses on the collective, quasi-

equilibrium properties determined by the conservative potential function of a set of extensive

CG variables; see Refs. [46, 47] for relevant work. For the conformational free energy of non-

extensive CG variables, several machine-learning based approaches [48–55] have been developed;

see also a recent review [56] and the references therein. Furthermore, to accurately predict the

dynamic properties, memory and coherent noise terms [57, 58] arising from the unresolved vari-

ables need to be properly introduced into the CG model [11, 12, 59, 60], which are left to future

investigations.

II. METHODS AND MODELS

A. Full model of the polymeric ﬂuids

We consider the micro-scale models of the star polymer melt similar to Ref. [12]. The full

system consists of Mmolecules with a total number of Natoms. Each polymer molecule consists

of a “center” atom connected by Naarms with Nbatoms per arm. The positions of the atoms are

denoted by q= [q1,q2,··· ,qN], where qirepresents the position of the i-th atom. The potential

function is governed by the pairwise and bond interactions, i.e.,

V(q) = ∑

i6=j

Vp(qi j) + ∑

Vb(lk),(1)

where Vpis the pairwise interaction between both the intra- and inter-molecular atoms except the

bonded pairs. qi j =kqi−qjkis the distance between the i-th and j-th atoms. Vbis the bond

interaction between the neighboring particles of each polymer arm and lkis the length of the k-th

bond. The bond potential Vbis chosen to be the harmonic potential, i.e.,

Vb(l) = 1

2ks(l−l0)2,(2)

where ksand l0represent the elastic coefﬁcient and the equilibrium length l0, respectively. The

atom mass is chosen to be unity.

We investigate three ﬂuid systems with micro-scale potential governed by Eq. (1). In Sec. III A,

we consider the polymeric ﬂuids in bulk and examine if the CG models can retain the many-body

interactions and the local compressibility. In particular, we choose Na=12, Nb=6, σ=2.415,

ε=1.0, ks=1.714, l0=2.77 similar to Ref. [12]. Vptakes the form of the Lennard–Jones

potential with cut-off rc, i.e.,

Vp(r) = 









VLJ(r)−VLJ(rc),r<rc

0,r≥rc

VLJ(r) = 4εσ

r12 −σ

r6,(3)

where ε=1.0 is the dispersion energy and σ=2.415 is the hardcore distance. Also we choose

rc=21/6σso that Vprecovers the Weeks-Chandler-Andersen potential. The full system consists of

N=2120 polymer molecules in a cubic domain 180×180×180 with periodic boundary condition

imposed along each direction. The Nosé-Hoover thermostat is employed to conduct the canonical

ensemble simulation with kBT=3.96.

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摘要：

Machinelearningassistedcoarse-grainedmoleculardynamicsmodelingofmeso-scaleinterfacialuidsPeiGe,1LinfengZhang,2,3,andHuanLei1,4,1DepartmentofComputationalMathematics,Science&Engineering,MichiganStateUniversity,EastLansing,MI48824,USA2AIforScienceInstitute,Beijing100080,China3DPTechnology,Beijing10...

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Machine learning assisted coarse-grained molecular dynamics modeling of meso-scale interfacial ﬂuids Pei Ge1Linfeng Zhang2 3and Huan Lei1 4

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