Neural network (NN) potentials are a natural choice for coarse-grained (CG) models. Their many-body capacity allows highly accurate approximations of the potential of mean force, promising CG simulations of unprecedented accuracy. CG NN potentials trained bottom-up via force matching (FM), however, suffer from finite data effects: They rely on prior potentials for physically sound predictions outside the training data domain, and the corresponding free energy surface is sensitive to errors in the transition regions. The standard alternative to FM for classical potentials is relative entropy (RE) minimization, which has not yet been applied to NN potentials. In this work, we demonstrate, for benchmark problems of liquid water and alanine dipeptide, that RE training is more data efficient, due to accessing the CG distribution during training, resulting in improved free energy surfaces and reduced sensitivity to prior potentials. In addition, RE learns to correct time integration errors, allowing larger time steps in CG molecular dynamics simulation, while maintaining accuracy. Thus, our findings support the use of training objectives beyond FM, as a promising direction for improving CG NN potential’s accuracy and reliability.

1.
J. D.
McCoy
and
J. G.
Curro
, “
Mapping of explicit atom onto united atom potentials
,”
Macromolecules
31
,
9362
9368
(
1998
).
2.
D.
Reith
,
M.
Pütz
, and
F.
Müller-Plathe
, “
Deriving effective mesoscale potentials from atomistic simulations
,”
J. Comput. Chem.
24
,
1624
1636
(
2003
).
3.
S. J.
Marrink
,
H. J.
Risselada
,
S.
Yefimov
,
D. P.
Tieleman
, and
A. H.
de Vries
, “
The MARTINI force field: Coarse grained model for biomolecular simulations
,”
J. Phys. Chem. B
111
,
7812
7824
(
2007
).
4.
W. G.
Noid
,
J.-W.
Chu
,
G. S.
Ayton
,
V.
Krishna
,
S.
Izvekov
,
G. A.
Voth
,
A.
Das
, and
H. C.
Andersen
, “
The multiscale coarse-graining method. I. A rigorous bridge between atomistic and coarse-grained models
,”
J. Chem. Phys.
128
,
244114
(
2008
).
5.
M. S.
Shell
, “
The relative entropy is fundamental to multiscale and inverse thermodynamic problems
,”
J. Chem. Phys.
129
,
144108
(
2008
).
6.
W. G.
Noid
, “
Perspective: Coarse-grained models for biomolecular systems
,”
J. Chem. Phys.
139
,
090901
(
2013
).
7.
H. I.
Ingólfsson
,
C. A.
Lopez
,
J. J.
Uusitalo
,
D. H.
de Jong
,
S. M.
Gopal
,
X.
Periole
, and
S. J.
Marrink
, “
The power of coarse graining in biomolecular simulations
,”
Wiley Interdiscip. Rev.: Comput. Mol. Sci.
4
,
225
248
(
2014
).
8.
N.
Singh
and
W.
Li
, “
Recent advances in coarse-grained models for biomolecules and their applications
,”
Int. J. Mol. Sci.
20
,
3774
(
2019
).
9.
J.
Behler
and
M.
Parrinello
, “
Generalized neural-network representation of high-dimensional potential-energy surfaces
,”
Phys. Rev. Lett.
98
,
146401
(
2007
).
10.
J.
Behler
, “
Atom-centered symmetry functions for constructing high-dimensional neural network potentials
,”
J. Chem. Phys.
134
,
074106
(
2011
).
11.
K. T.
Schütt
,
P. J.
Kindermans
,
H. E.
Sauceda
,
S.
Chmiela
,
A.
Tkatchenko
, and
K. R.
Müller
, “
SchNet: A continuous-filter convolutional neural network for modeling quantum interactions
,” in
Advances in Neural Information Processing Systems
(
Curran Associates, Inc.
,
2017
), pp. 992–1002.
12.
J.
Gilmer
,
S. S.
Schoenholz
,
P. F.
Riley
,
O.
Vinyals
, and
G. E.
Dahl
, “
Neural message passing for quantum chemistry
,” in
Proceedings of the 34th International Conference on Machine Learning
(
PMLR
,
2017
), pp.
1263
1272
.
13.
J.
Klicpera
,
J.
Groß
, and
S.
Günnemann
, “
Directional message passing for molecular graphs
,” in
8th International Conference on Learning Representations, ICLR
,
2020
.
14.
J.
Klicpera
,
S.
Giri
,
J. T.
Margraf
, and
S.
Günnemann
, “
Fast and uncertainty-aware directional message passing for non-equilibrium molecules
,” in Machine Learning for Molecules Workshop at NeurIPS (
2020
).
15.
Z.
Qiao
,
M.
Welborn
,
A.
Anandkumar
,
F. R.
Manby
, and
T. F.
Miller
, “
OrbNet: Deep learning for quantum chemistry using symmetry-adapted atomic-orbital features
,”
J. Chem. Phys.
153
,
124111
(
2020
).
16.
A. C. P.
Jain
,
D.
Marchand
,
A.
Glensk
,
M.
Ceriotti
, and
W. A.
Curtin
, “
Machine learning for metallurgy III: A neural network potential for Al-Mg-Si
,”
Phys. Rev. Mater.
5
,
053805
(
2021
).
17.
T. W.
Ko
,
J. A.
Finkler
,
S.
Goedecker
, and
J.
Behler
, “
A fourth-generation high-dimensional neural network potential with accurate electrostatics including non-local charge transfer
,”
Nat. Commun.
12
,
398
(
2021
).
18.
S.
Batzner
,
A.
Musaelian
,
L.
Sun
,
M.
Geiger
,
J. P.
Mailoa
,
M.
Kornbluth
,
N.
Molinari
,
T. E.
Smidt
, and
B.
Kozinsky
, “
E(3)-equivariant graph neural networks for data-efficient and accurate interatomic potentials
,”
Nat. Commun.
13
,
2453
(
2022
).
19.
I.
Batatia
,
D. P.
Kovács
,
G. N.
Simm
,
C.
Ortner
, and
G.
Csányi
, “
MACE: Higher order equivariant message passing neural networks for fast and accurate force fields
,” arXiv:2206.07697 (
2022
).
20.
L.
Zhang
,
J.
Han
,
H.
Wang
,
R.
Car
, and
W.
E
, “
DeePCG: Constructing coarse-grained models via deep neural networks
,”
J. Chem. Phys.
149
,
034101
(
2018
).
21.
J.
Wang
,
S.
Olsson
,
C.
Wehmeyer
,
A.
Pérez
,
N. E.
Charron
,
G.
De Fabritiis
,
F.
Noé
, and
C.
Clementi
, “
Machine learning of coarse-grained molecular dynamics force fields
,”
ACS Cent. Sci.
5
,
755
767
(
2019
).
22.
T. D.
Loeffler
,
T. K.
Patra
,
H.
Chan
, and
S. K. R. S.
Sankaranarayanan
, “
Active learning a coarse-grained neural network model for bulk water from sparse training data
,”
Mol. Syst. Des. Eng.
5
,
902
910
(
2020
).
23.
B. E.
Husic
,
N. E.
Charron
,
D.
Lemm
,
J.
Wang
,
A.
Pérez
,
M.
Majewski
,
A.
Krämer
,
Y.
Chen
,
S.
Olsson
,
G.
de Fabritiis
,
F.
Noé
, and
C.
Clementi
, “
Coarse graining molecular dynamics with graph neural networks
,”
J. Chem. Phys.
153
,
194101
(
2020
).
24.
Y.
Chen
,
A.
Krämer
,
N. E.
Charron
,
B. E.
Husic
,
C.
Clementi
, and
F.
Noé
, “
Machine learning implicit solvation for molecular dynamics
,”
J. Chem. Phys.
155
,
084101
(
2021
).
25.
X.
Ding
and
B.
Zhang
, “
Contrastive learning of coarse-grained force fields
,”
J. Chem. Theory Comput.
18
(
10
),
6334
6344
(
2022
).
26.
J.
Köhler
,
Y.
Chen
,
A.
Krämer
,
C.
Clementi
, and
F.
Noé
, “
Force-matching coarse-graining without forces
,” arXiv:2203.11167 (
2022
).
27.
S.
Thaler
and
J.
Zavadlav
, “
Learning neural network potentials from experimental data via differentiable trajectory reweighting
,”
Nat. Commun.
12
,
6884
(
2021
).
28.
I.
Batatia
,
S.
Batzner
,
D. P.
Kovács
,
A.
Musaelian
,
G. N.
Simm
,
R.
Drautz
,
C.
Ortner
,
B.
Kozinsky
, and
G.
Csányi
, “
The design space of E(3)-equivariant atom-centered interatomic potentials
,” arXiv:2205.06643 (
2022
).
29.
J. E.
Herr
,
K.
Yao
,
R.
McIntyre
,
D. W.
Toth
, and
J.
Parkhill
, “
Metadynamics for training neural network model chemistries: A competitive assessment
,”
J. Chem. Phys.
148
,
241710
(
2018
).
30.
M.
Gutmann
and
A.
Hyvärinen
, “
Noise-contrastive estimation: A new estimation principle for unnormalized statistical models
,” in
Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics
, JMLR Workshop and Conference Proceedings (
PMLR
,
2010
), pp.
297
304
.
31.
S. P.
Carmichael
and
M. S.
Shell
, “
A new multiscale algorithm and its application to coarse-grained peptide models for self-assembly
,”
J. Phys. Chem. B
116
,
8383
8393
(
2012
).
32.
S.
Bottaro
,
K.
Lindorff-Larsen
, and
R. B.
Best
, “
Variational optimization of an all-atom implicit solvent force field to match explicit solvent simulation data
,”
J. Chem. Theory Comput.
9
,
5641
5652
(
2013
).
33.
S. Y.
Mashayak
,
M. N.
Jochum
,
K.
Koschke
,
N. R.
Aluru
,
V.
Rühle
, and
C.
Junghans
, “
Relative entropy and optimization-driven coarse-graining methods in VOTCA
,”
PLoS One
10
,
e0131754
(
2015
).
34.
T.
Sanyal
and
M. S.
Shell
, “
Transferable coarse-grained models of liquid–liquid equilibrium using local density potentials optimized with the relative entropy
,”
J. Phys. Chem. B
122
,
5678
5693
(
2018
).
35.
A.
Chaimovich
and
M. S.
Shell
, “
Coarse-graining errors and numerical optimization using a relative entropy framework
,”
J. Chem. Phys.
134
,
094112
(
2011
).
36.
S.
Izvekov
and
G. A.
Voth
, “
Multiscale coarse graining of liquid-state systems
,”
J. Chem. Phys.
123
,
134105
(
2005
).
37.
W. G.
Noid
,
P.
Liu
,
Y.
Wang
,
J.-W.
Chu
,
G. S.
Ayton
,
S.
Izvekov
,
H. C.
Andersen
, and
G. A.
Voth
, “
The multiscale coarse-graining method. II. Numerical implementation for coarse-grained molecular models
,”
J. Chem. Phys.
128
,
244115
(
2008
).
38.
J. W.
Mullinax
and
W. G.
Noid
, “
Generalized Yvon-Born-Green theory for molecular systems
,”
Phys. Rev. Lett.
103
,
198104
(
2009
).
39.
A.
Chaimovich
and
M. S.
Shell
, “
Anomalous waterlike behavior in spherically-symmetric water models optimized with the relative entropy
,”
Phys. Chem. Chem. Phys.
11
,
1901
1915
(
2009
).
40.
A.
Chaimovich
and
M. S.
Shell
, “
Relative entropy as a universal metric for multiscale errors
,”
Phys. Rev. E
81
,
060104
(
2010
).
41.
P.
Español
and
I.
Zúñiga
, “
Obtaining fully dynamic coarse-grained models from MD
,”
Phys. Chem. Chem. Phys.
13
,
10538
10545
(
2011
).
42.
M. S.
Shell
, “
Coarse-graining with the relative entropy
,”
Adv. Chem. Phys.
161
,
395
441
(
2016
).
43.
V.
Harmandaris
,
E.
Kalligiannaki
,
M.
Katsoulakis
, and
P.
Plecháč
, “
Path-space variational inference for non-equilibrium coarse-grained systems
,”
J. Comput. Phys.
314
,
355
383
(
2016
).
44.
E.
Kalligiannaki
,
V.
Harmandaris
,
M. A.
Katsoulakis
, and
P.
Plecháč
, “
The geometry of generalized force matching and related information metrics in coarse-graining of molecular systems
,”
J. Chem. Phys.
143
,
084105
(
2015
).
45.
S.
Kullback
and
R. A.
Leibler
, “
On information and sufficiency
,”
Ann. Math. Stat.
22
,
79
86
(
1951
).
46.
J. F.
Rudzinski
and
W. G.
Noid
, “
Coarse-graining entropy, forces, and structures
,”
J. Chem. Phys.
135
,
214101
(
2011
).
47.
R.
Ramakrishnan
,
P. O.
Dral
,
M.
Rupp
, and
O. A.
Von Lilienfeld
, “
Big data meets quantum chemistry approximations: The Δ-machine learning approach
,”
J. Chem. Theory Comput.
11
,
2087
2096
(
2015
).
48.
L.
Shen
and
W.
Yang
, “
Molecular dynamics simulations with quantum mechanics/molecular mechanics and adaptive neural networks
,”
J. Chem. Theory Comput.
14
,
1442
1455
(
2018
).
49.
L.
Böselt
,
M.
Thürlemann
, and
S.
Riniker
, “
Machine learning in QM/MM molecular dynamics simulations of condensed-phase systems
,”
J. Chem. Theory Comput.
17
,
2641
2658
(
2021
).
50.
H.
Wang
,
C.
Junghans
, and
K.
Kremer
, “
Comparative atomistic and coarse-grained study of water: What do we lose by coarse-graining?
,”
Eur. Phys. J. E
28
,
221
229
(
2009
).
51.
S.
Toxvaerd
, “
Hamiltonians for discrete dynamics
,”
Phys. Rev. E
50
,
2271
(
1994
).
52.
S.
Toxvaerd
, “
Ensemble simulations with discrete classical dynamics
,”
J. Chem. Phys.
139
,
224106
(
2013
).
53.
J. L. F.
Abascal
and
C.
Vega
, “
A general purpose model for the condensed phases of water: TIP4P/2005
,”
J. Chem. Phys.
123
,
234505
(
2005
).
54.
H. J. C.
Berendsen
,
J. P. M.
Postma
,
W. F.
van Gunsteren
, and
J.
Hermans
, “
Interaction models for water in relation to protein hydration
,” in
Intermolecular Forces
(
Springer
,
1981
), pp.
331
342
.
55.
A. K.
Soper
and
C. J.
Benmore
, “
Quantum differences between heavy and light water
,”
Phys. Rev. Lett.
101
,
065502
(
2008
).
56.
A.
Baranyai
and
D. J.
Evans
, “
Three-particle contribution to the configurational entropy of simple fluids
,”
Phys. Rev. A
42
,
849
(
1990
).
57.
B.
Bildstein
and
G.
Kahl
, “
Triplet correlation functions for hard-spheres: Computer simulation results
,”
J. Chem. Phys.
100
,
5882
(
1994
).
58.
D.
Dhabal
,
M.
Singh
,
K. T.
Wikfeldt
, and
C.
Chakravarty
, “
Triplet correlation functions in liquid water
,”
J. Chem. Phys.
141
,
174504
(
2014
).
59.
V.
Rühle
,
C.
Junghans
,
A.
Lukyanov
,
K.
Kremer
, and
D.
Andrienko
, “
Versatile object-oriented toolkit for coarse-graining applications
,”
J. Chem. Theory Comput.
5
,
3211
3223
(
2009
).
60.
C.
Scherer
and
D.
Andrienko
, “
Understanding three-body contributions to coarse-grained force fields
,”
Phys. Chem. Chem. Phys.
20
,
22387
22394
(
2018
).
61.
E.
Kalligiannaki
,
A.
Chazirakis
,
A.
Tsourtis
,
M. A.
Katsoulakis
,
P.
Plecháč
, and
V.
Harmandaris
, “
Parametrizing coarse grained models for molecular systems at equilibrium
,”
Eur. Phys. J.: Spec. Top.
225
,
1347
1372
(
2016
).
62.
B.
Montgomery Pettitt
and
M.
Karplus
, “
The potential of mean force surface for the alanine dipeptide in aqueous solution: A theoretical approach
,”
Chem. Phys. Lett.
121
,
194
201
(
1985
).
63.
D. J.
Tobias
and
C. L.
Brooks
 III
, “
Conformational equilibrium in the alanine dipeptide in the gas phase and aqueous solution: A comparison of theoretical results
,”
J. Phys. Chem.
96
,
3864
3870
(
1992
).
64.
Y.
Duan
,
C.
Wu
,
S.
Chowdhury
,
M. C.
Lee
,
G.
Xiong
,
W.
Zhang
,
R.
Yang
,
P.
Cieplak
,
R.
Luo
,
T.
Lee
,
J.
Caldwell
,
J.
Wang
, and
P.
Kollman
, “
A point-charge force field for molecular mechanics simulations of proteins based on condensed-phase quantum mechanical calculations
,”
J. Comput. Chem.
24
,
1999
2012
(
2003
).
65.
J.
Chen
,
C. L.
Brooks
 III
, and
J.
Khandogin
, “
Recent advances in implicit solvent-based methods for biomolecular simulations
,”
Curr. Opin. Struct. Biol.
18
,
140
148
(
2008
).
66.
R. W.
Zwanzig
, “
High-temperature equation of state by a perturbation method. I. Nonpolar gases
,”
J. Chem. Phys.
22
,
1420
1426
(
1954
).
67.
C.
Chipot
and
A.
Pohorille
,
Free Energy Calculations
(
Springer
,
2007
), Vol. 86.
68.
A. B.
Norgaard
,
J.
Ferkinghoff-Borg
, and
K.
Lindorff-Larsen
, “
Experimental parameterization of an energy function for the simulation of unfolded proteins
,”
Biophys. J.
94
,
182
192
(
2008
).
69.
D.-W.
Li
and
R.
Brüschweiler
, “
Iterative optimization of molecular mechanics force fields from NMR data of full-length proteins
,”
J. Chem. Theory Comput.
7
,
1773
1782
(
2011
).
70.
S. R.
Xie
,
M.
Rupp
, and
R. G.
Hennig
, “
Ultra-fast interpretable machine-learning potentials
,” arXiv:2110.00624 (
2021
).
71.
A.
Barducci
,
M.
Bonomi
, and
M.
Parrinello
, “
Metadynamics
,”
Wiley Interdiscip. Rev.: Comput. Mol. Sci.
1
,
826
843
(
2011
).
72.
L.
Bonati
and
M.
Parrinello
, “
Silicon liquid structure and crystal nucleation from ab initio deep metadynamics
,”
Phys. Rev. Lett.
121
,
265701
(
2018
).
73.
W.
Schneider
and
W.
Thiel
, “
Anharmonic force fields from analytic second derivatives: Method and application to methyl bromide
,”
Chem. Phys. Lett.
157
,
367
373
(
1989
).
74.
M.
Rupp
,
R.
Ramakrishnan
, and
O. A.
Von Lilienfeld
, “
Machine learning for quantum mechanical properties of atoms in molecules
,”
J. Phys. Chem. Lett.
6
,
3309
3313
(
2015
).
75.
S.
Stocker
,
J.
Gasteiger
,
F.
Becker
,
S.
Günnemann
, and
J. T.
Margraf
, “
How robust are modern graph neural network potentials in long and hot molecular dynamics simulations?
,”
Mach. Learn.: Sci. Technol.
3
,
045010
(
2022
).
76.
X.
Fu
,
Z.
Wu
,
W.
Wang
,
T.
Xie
,
S.
Keten
,
R.
Gomez-Bombarelli
, and
T.
Jaakkola
, “
Forces are not enough: Benchmark and critical evaluation for machine learning force fields with molecular simulations
,” arXiv:2210.07237 (
2022
).
77.
J. S.
Smith
,
B.
Nebgen
,
N.
Lubbers
,
O.
Isayev
, and
A. E.
Roitberg
, “
Less is more: Sampling chemical space with active learning
,”
J. Chem. Phys.
148
,
241733
(
2018
).
78.
L.
Zhang
,
D.-Y.
Lin
,
H.
Wang
,
R.
Car
, and
E.
Weinan
, “
Active learning of uniformly accurate interatomic potentials for materials simulation
,”
Phys. Rev. Mater.
3
,
023804
(
2019
).
79.
R.
Jinnouchi
,
K.
Miwa
,
F.
Karsai
,
G.
Kresse
, and
R.
Asahi
, “
On-the-fly active learning of interatomic potentials for large-scale atomistic simulations
,”
J. Phys. Chem. Lett.
11
,
6946
6955
(
2020
).
80.
J. S.
Smith
,
B.
Nebgen
,
N.
Mathew
,
J.
Chen
,
N.
Lubbers
,
L.
Burakovsky
,
S.
Tretiak
,
H. A.
Nam
,
T.
Germann
,
S.
Fensin
 et al, “
Automated discovery of a robust interatomic potential for aluminum
,”
Nat. Commun.
12
,
1257
(
2021
).
81.
J.
Ingraham
,
A.
Riesselman
,
C.
Sander
, and
D.
Marks
, “
Learning protein structure with a differentiable simulator
,” in
7th International Conference on Learning Representations, ICLR
,
2019
.
82.
S. S.
Schoenholz
and
E. D.
Cubuk
, “
JAX, M.D.: A framework for differentiable physics
,” in
Advances in Neural Information Processing Systems
(
Curran Associates, Inc.
,
2020
), Vol. 33.
83.
C. P.
Goodrich
,
E. M.
King
,
S. S.
Schoenholz
,
E. D.
Cubuk
, and
M. P.
Brenner
, “
Designing self-assembling kinetics with differentiable statistical physics models
,”
Proc. Natl. Acad. Sci. U. S. A.
118
,
e2024083118
(
2021
).
84.
S.
Doerr
,
M.
Majewski
,
A.
Pérez
,
A.
Krämer
,
C.
Clementi
,
F.
Noe
,
T.
Giorgino
, and
G.
De Fabritiis
, “
TorchMD: A deep learning framework for molecular simulations
,”
J. Chem. Theory Comput.
17
,
2355
2363
(
2021
).

Supplementary Material

You do not currently have access to this content.