Accurate neural-network-based tting of full-dimensional two-body potential energy surfaces

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Accurate neural-network-based ﬁtting of

full-dimensional two-body potential energy

surfaces

Artem A. Finenko∗,†,‡,¶

†Department of Chemistry, Lomonosov Moscow State University, GSP-1, 1-3 Leninskiye

Gory, Moscow 119991, Russia

‡Institute of Quantum Physics, Irkutsk National Research Technical University, 83

Lermontov str., Irkutsk 664074, Russia

¶Institute of Applied Physics of the Russian Academy of Sciences, 46 Ulyanov str., Nyzhny

Novgorod 603950, Russia

E-mail: artﬁn@mail.ru

Abstract

We describe the development of machine-learned potentials of atmospheric gases

with ﬂexible monomers for molecular simulations. A recently suggested permutation-

ally invariant polynomial neural network (PIP-NN) approach is utilized to represent

the full-dimensional two-body component of the dimer energy. To ensure the asymp-

totic zero-interaction limit, a tailored subset of the full invariant polynomial basis set

is utilized and their variables are modiﬁed to achieve a better ﬁt of the correct asymp-

totic behavior at a long range. The new technique is used to build full-dimensional

potentials for the two-body N2−Ar and N2−CH4interactions by ﬁtting databases of

ab initio energies calculated at the coupled-cluster level of theory. The second virial

coeﬃcients with full account of molecular ﬂexibility eﬀects are then calculated within

arXiv:2210.09970v1 [physics.chem-ph] 18 Oct 2022

the classical framework using the PIP-NN potential surfaces. To showcase the advan-

tages of the PIP-NN method, we compare its accuracy and computational eﬃciency to

several kernel-based and neural-network-based approaches using the MD17 database of

energies and forces for ethanol. For large training set sizes, the PIP-NN models attain

the best accuracy among examined models, and the computation time is shown to be

comparable to that of the PIP regression model and several orders of magnitude faster

than the quickest alternatives.

Introduction

A potential energy surface (PES) is the key concept in computational chemistry that enables

understanding the processes on an atomistic scale. The continuous changes within atomistic

systems are often thought of as being driven by a multidimensional energy landscape, which

is determined by the spatial positions of the atoms. Oﬀering unprecedented insights into

complex physical and chemical phenomena, molecular dynamics simulation is indispensable

in the computational chemistry toolbox. One of the widely recognized issues in MD sim-

ulations is the lack of accuracy of underlying interatomic potentials, which hinders truly

predictive modeling of molecular systems’ properties. Although the potential energy values

could, in principle, be obtained on-the-ﬂy using pertinent electronic structure calculations,

more eﬃcient approaches are often required to characterize relevant observables, for example,

(ro-)vibrational spectra, rate constants, etc. Thanks to modern computational resources, a

large number of highly precise electronic structure calculations can be carried out at various

points across the conﬁguration space.1Despite signiﬁcant advances, representing a global

PES from these ab initio points poses a diﬃcult challenge. Thus, one of the key focuses

of computational chemistry has been on the development of eﬃcient methods to generate

high-quality representations of global PESs for molecules. Various non-ML approaches, such

as modiﬁed Sheppard interpolation,2interpolating moving least-squares (IMLS),3reproduc-

ing kernel Hilbert space interpolations4and PIP regression,5,6 to name a few, have been

successful at constructing global PESs. Machine learning, particularly neural networks, has

recently shown great promise as tools for building both ﬂexible and computationally eﬃcient

models of PESs.7–9 Combining techniques coming from ML and non-ML perspectives has

been demonstrated to be eﬀective; PIP-IMLS10 and PIP-NN11,12 are excellent examples of

this synergy.

In the current study, we undertake the development of ML PES for molecular simulations

of atmospheric gases with ﬂexible monomers. Collision processes in atmospheric gases mod-

ify the radiative properties of moieties by shifting and broadening spectral lines as well as

leading to the emergence of collision-induced bands. The advancement of the observational

and retrieval capabilities of terrestrial and planetary remote-sensing missions poses new chal-

lenges for theoretical research to improve the representation of these phenomena. Starting

from potential energy and dipole moment surfaces, contemporary state-of-the-art theories

permit ﬁrst-principle simulations of spectral features that emerge in real gases (see review

by Hartmann et al. 13 and references therein). One of the hurdles aﬀecting remote-sensing

retrievals is the lack of or insuﬃcient accuracy of models describing continuum absorption.

In Refs.,14,15 we developed a ﬁrst-principle trajectory-based approach to simulate collision-

induced absorption. In light of the approach’s eﬀectiveness in simulating far-infrared absorp-

tion in the cases of N2−N2,14 CO2−Ar15 and N2−CH4,16 we seek to simulate the forbidden

rotovibrational fundamental and overtone bands of molecule pairs consisting of dipoleless

moieties. Following many-body expansion,17 we speciﬁcally focus on the full-dimensional

representation of the two-body term, V2b, which corresponds to the interaction energy be-

tween two molecules. The two-body component is deﬁned as the diﬀerence between the total

energy of the dimer Vdimer and the one-body energies of isolated monomers Vmon1and Vmon2

V2b =Vdimer −Vmon1−Vmon2.(1)

Note that in this context dimer refers to a molecular pair that is not necessarily in a bound

state. Keep in mind that the two-body component of the PES can be paired with various

monomer potentials depending on the level of ﬁdelity required for a particular application,

making the global PES model conﬁgurable and adaptable.

The machine-learning potential is typically built from two components: the molecular

descriptor and the ML model. The descriptor provides a mapping that associates an atomic

conﬁguration, which is identiﬁed by the positions and chemical identities of atoms, with a

point in a feature space.18 Although the Cartesian coordinates of the atoms encode all the

information about the structure of a molecular system, it is evident that they cannot be used

directly as input to the ML model. In principle, the potential energy surface is invariant

to overall translation and rotation as well as to permutations of identical atoms. Generally

speaking, ML models that account for symmetry either using descriptors or through clever

architecture tend to be more data-eﬃcient and accurate. In this paper, we use PIPs as a type

of descriptor that will allow us to account for translational, rotational, and permutational

symmetry.

In recent years, numerous ML methods have been applied to ﬁt electronic energies (see

review19 and references therein). These ML methods produce mappings that can be charac-

terized as parametric or nonparametric.20 The assumption made by parametric algorithms

is that the mapping function has a predeﬁned functional form with a ﬁxed number of pa-

rameters. The typical example is a neural network. Nonparametric algorithms, on the other

hand, do not make this assumption. The more training data is given, the more complex

the algorithm becomes and the more parameters it contains. Such is the case with kernel

methods like Gaussian process regression and kernel ridge regression. Assuming the NN’s

structure is ﬁxed, the prediction cost for NN-based methods becomes independent of the

size of the training set, in contrast to kernel-based approaches where the prediction cost

scales linearly with the size of the training set. Given our goal of performing large-scale

collision simulations with ML potentials, which necessitates robust force models, we chose

an NN-based method.

To assess the computational eﬃciency of the PIP-NN method, we conducted a showcase

study on the ethanol molecule using the MD17 database of energies and forces. A range

of ML approaches have been compared by Pinheiro et al.21 based on the MD17-ethanol

database. Using the suggested protocol based on the analysis of learning curves, we compare

our implementation of the PIP-NN method to other approaches, in particular, symmetrized

gradient-domain machine learning (sGDML),22,23 deep neural network with PhysNet archi-

tecture,9Gaussian approximation potential,24 and PIP regression6on the basis of precision

and prediction times.

The rest of the paper is organized as follows. The structure of the PIP-NN model and a

few modiﬁcations that we explore in this paper are described in the next section, along with

the generation of interaction energy data sets and an explanation of the training procedure.

The Results section discusses how PIP-NN models performed on these data sets as well as

on the MD17-ethanol data set. The Summary and Conclusions section covers concluding

remarks.

Methods

In this work, we adapt the PIP-NN approach put forward by Guo and co-workers11,25 to

represent multidimensional PESs, characterized by two-body, non-covalent interactions. We

begin by outlining the PIP-NN model’s structure and how the PIP basis could be altered

to be able to reproduce essential properties pertinent to the two-body component of the

PES. Next, we explain how the variables of invariant polynomials could be changed to more

accurately map the long-range region. Finally, we discuss the training procedure and the

development of interaction energy data sets.

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Accurateneural-network-basedttingoffull-dimensionaltwo-bodypotentialenergysurfacesArtemA.Finenko,y,z,{yDepartmentofChemistry,LomonosovMoscowStateUniversity,GSP-1,1-3LeninskiyeGory,Moscow119991,RussiazInstituteofQuantumPhysics,IrkutskNationalResearchTechnicalUniversity,83Lermontovstr.,Irkutsk664074...

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Accurate neural-network-based tting of full-dimensional two-body potential energy surfaces

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