The combination of high throughput computation and machine learning has led to a new paradigm in materials design by allowing for the direct screening of vast portions of structural, chemical, and property spaces. The use of these powerful techniques leads to the generation of enormous amounts of data, which in turn calls for new techniques to efficiently explore and visualize the materials space to help identify underlying patterns. In this work, we develop a unified framework to hierarchically visualize the compositional and structural similarities between materials in an arbitrary material space with representations learned from different layers of graph convolutional neural networks. We demonstrate the potential for such a visualization approach by showing that patterns emerge automatically that reflect similarities at different scales in three representative classes of materials: perovskites, elemental boron, and general inorganic crystals, covering material spaces of different compositions, structures, and both. For perovskites, elemental similarities are learned that reflects multiple aspects of atom properties. For elemental boron, structural motifs emerge automatically showing characteristic boron local environments. For inorganic crystals, the similarity and stability of local coordination environments are shown combining different center and neighbor atoms. The method could help transition to a data-centered exploration of materials space in automated materials design.

1.
G.
Niu
,
X.
Guo
, and
L.
Wang
, “
Review of recent progress in chemical stability of perovskite solar cells
,”
J. Mater. Chem. A
3
,
8970
8980
(
2015
).
2.
H. J.
Snaith
, “
Perovskites: The emergence of a new era for low-cost, high-efficiency solar cells
,”
J. Phys. Chem. Lett.
4
,
3623
3630
(
2013
).
3.
M.
Xu
,
T.
Liang
,
M.
Shi
, and
H.
Chen
, “
Graphene-like two-dimensional materials
,”
Chem. Rev.
113
,
3766
3798
(
2013
).
4.
S. Z.
Butler
,
S. M.
Hollen
,
L.
Cao
,
Y.
Cui
,
J. A.
Gupta
,
H. R.
Gutiérrez
,
T. F.
Heinz
,
S. S.
Hong
,
J.
Huang
,
A. F.
Ismach
 et al, “
Progress, challenges, and opportunities in two-dimensional materials beyond graphene
,”
ACS Nano
7
,
2898
2926
(
2013
).
5.
O.
Madelung
,
Physics of III-V Compounds
(
J. Wiley
,
1964
).
6.
J.
Greeley
,
T. F.
Jaramillo
,
J.
Bonde
,
I.
Chorkendorff
, and
J. K.
Nørskov
, “
Computational high-throughput screening of electrocatalytic materials for hydrogen evolution
,”
Nat. Mater.
5
,
909
(
2006
).
7.
S. M.
Senkan
, “
High-throughput screening of solid-state catalyst libraries
,”
Nature
394
,
350
(
1998
).
8.
R.
Potyrailo
,
K.
Rajan
,
K.
Stoewe
,
I.
Takeuchi
,
B.
Chisholm
, and
H.
Lam
, “
Combinatorial and high-throughput screening of materials libraries: Review of state of the art
,”
ACS Comb. Sci.
13
,
579
633
(
2011
).
9.
S.
Curtarolo
,
G. L.
Hart
,
M. B.
Nardelli
,
N.
Mingo
,
S.
Sanvito
, and
O.
Levy
, “
The high-throughput highway to computational materials design
,”
Nat. Mater.
12
,
191
(
2013
).
10.
G.
Hautier
,
C. C.
Fischer
,
A.
Jain
,
T.
Mueller
, and
G.
Ceder
, “
Finding natures missing ternary oxide compounds using machine learning and density functional theory
,”
Chem. Mater.
22
,
3762
3767
(
2010
).
11.
R.
Gómez-Bombarelli
,
J.
Aguilera-Iparraguirre
,
T. D.
Hirzel
,
D.
Duvenaud
,
D.
Maclaurin
,
M. A.
Blood-Forsythe
,
H. S.
Chae
,
M.
Einzinger
,
D.-G.
Ha
,
T.
Wu
 et al, “
Design of efficient molecular organic light-emitting diodes by a high-throughput virtual screening and experimental approach
,”
Nat. Mater.
15
,
1120
(
2016
).
12.
F. A.
Faber
,
A.
Lindmaa
,
O. A.
Von Lilienfeld
, and
R.
Armiento
, “
Machine learning energies of 2 million elpasolite (ABC2D6) crystals
,”
Phys. Rev. Lett.
117
,
135502
(
2016
).
13.
M.
Rupp
,
A.
Tkatchenko
,
K.-R.
Müller
, and
O. A.
Von Lilienfeld
, “
Fast and accurate modeling of molecular atomization energies with machine learning
,”
Phys. Rev. Lett.
108
,
058301
(
2012
).
14.
G.
Pilania
,
C.
Wang
,
X.
Jiang
,
S.
Rajasekaran
, and
R.
Ramprasad
, “
Accelerating materials property predictions using machine learning
,”
Sci. Rep.
3
,
2810
(
2013
).
15.
B.
Meredig
,
A.
Agrawal
,
S.
Kirklin
,
J. E.
Saal
,
J.
Doak
,
A.
Thompson
,
K.
Zhang
,
A.
Choudhary
, and
C.
Wolverton
, “
Combinatorial screening for new materials in unconstrained composition space with machine learning
,”
Phys. Rev. B
89
,
094104
(
2014
).
16.
L.
Ward
,
A.
Agrawal
,
A.
Choudhary
, and
C.
Wolverton
, “
A general-purpose machine learning framework for predicting properties of inorganic materials
,”
npj Comput. Mater.
2
,
16028
(
2016
).
17.
L. M.
Ghiringhelli
,
J.
Vybiral
,
S. V.
Levchenko
,
C.
Draxl
, and
M.
Scheffler
, “
Big data of materials science: Critical role of the descriptor
,”
Phys. Rev. Lett.
114
,
105503
(
2015
).
18.
J.
Behler
, “
Atom-centered symmetry functions for constructing high-dimensional neural network potentials
,”
J. Chem. Phys.
134
,
074106
(
2011
).
19.
F.
Pietrucci
and
W.
Andreoni
, “
Graph theory meets ab initio molecular dynamics: Atomic structures and transformations at the nanoscale
,”
Phys. Rev. Lett.
107
,
085504
(
2011
).
20.
A.
Sadeghi
,
S. A.
Ghasemi
,
B.
Schaefer
,
S.
Mohr
,
M. A.
Lill
, and
S.
Goedecker
, “
Metrics for measuring distances in configuration spaces
,”
J. Chem. Phys.
139
,
184118
(
2013
).
21.
A. P.
Bartók
,
R.
Kondor
, and
G.
Csányi
, “
On representing chemical environments
,”
Phys. Rev. B
87
,
184115
(
2013
).
22.
K.
Schütt
,
H.
Glawe
,
F.
Brockherde
,
A.
Sanna
,
K.
Müller
, and
E.
Gross
, “
How to represent crystal structures for machine learning: Towards fast prediction of electronic properties
,”
Phys. Rev. B
89
,
205118
(
2014
).
23.
F. A.
Faber
,
A. S.
Christensen
,
B.
Huang
, and
O. A.
von Lilienfeld
, “
Alchemical and structural distribution based representation for universal quantum machine learning
,”
J. Chem. Phys.
148
,
241717
(
2018
).
24.
A.
Glielmo
,
C.
Zeni
, and
A.
De Vita
, “
Efficient nonparametric n-body force fields from machine learning
,”
Phys. Rev. B
97
,
184307
(
2018
).
25.
L.
Ward
,
R.
Liu
,
A.
Krishna
,
V. I.
Hegde
,
A.
Agrawal
,
A.
Choudhary
, and
C.
Wolverton
, “
Including crystal structure attributes in machine learning models of formation energies via voronoi tessellations
,”
Phys. Rev. B
96
,
024104
(
2017
).
26.
O.
Isayev
,
D.
Fourches
,
E. N.
Muratov
,
C.
Oses
,
K.
Rasch
,
A.
Tropsha
, and
S.
Curtarolo
, “
Materials cartography: Representing and mining materials space using structural and electronic fingerprints
,”
Chem. Mater.
27
,
735
743
(
2015
).
27.
S.
De
,
A. P.
Bartók
,
G.
Csányi
, and
M.
Ceriotti
, “
Comparing molecules and solids across structural and alchemical space
,”
Phys. Chem. Chem. Phys.
18
,
13754
13769
(
2016
).
28.
F.
Musil
,
S.
De
,
J.
Yang
,
J. E.
Campbell
,
G. M.
Day
, and
M.
Ceriotti
, “
Machine learning for the structure–energy–property landscapes of molecular crystals
,”
Chem. Sci.
9
,
1289
1300
(
2018
).
29.
P.
Das
,
M.
Moll
,
H.
Stamati
,
L. E.
Kavraki
, and
C.
Clementi
, “
Low-dimensional, free-energy landscapes of protein-folding reactions by nonlinear dimensionality reduction
,”
Proc. Natl. Acad. Sci. U. S. A.
103
,
9885
9890
(
2006
).
30.
M.
Ceriotti
,
G. A.
Tribello
, and
M.
Parrinello
, “
Simplifying the representation of complex free-energy landscapes using sketch-map
,”
Proc. Natl. Acad. Sci. U. S. A.
108
,
13023
13028
(
2011
).
31.
V.
Spiwok
and
B.
Králová
, “
Metadynamics in the conformational space nonlinearly dimensionally reduced by isomap
,”
J. Chem. Phys.
135
,
224504
(
2011
).
32.
M. A.
Rohrdanz
,
W.
Zheng
, and
C.
Clementi
, “
Discovering mountain passes via torchlight: Methods for the definition of reaction coordinates and pathways in complex macromolecular reactions
,”
Annu. Rev. Phys. Chem.
64
,
295
316
(
2013
).
33.
F.
Pietrucci
and
R.
Martoňák
, “
Systematic comparison of crystalline and amorphous phases: Charting the landscape of water structures and transformations
,”
J. Chem. Phys.
142
,
104704
(
2015
).
34.
E. A.
Engel
,
A.
Anelli
,
M.
Ceriotti
,
C. J.
Pickard
, and
R. J.
Needs
, “
Mapping uncharted territory in ice from zeolite networks to ice structures
,”
Nat. Commun.
9
,
2173
(
2018
).
35.
A.
Seko
,
A.
Togo
,
H.
Hayashi
,
K.
Tsuda
,
L.
Chaput
, and
I.
Tanaka
, “
Prediction of low-thermal-conductivity compounds with first-principles anharmonic lattice-dynamics calculations and bayesian optimization
,”
Phys. Rev. Lett.
115
,
205901
(
2015
).
36.
O.
Isayev
,
C.
Oses
,
C.
Toher
,
E.
Gossett
,
S.
Curtarolo
, and
A.
Tropsha
, “
Universal fragment descriptors for predicting properties of inorganic crystals
,”
Nat. Commun.
8
,
15679
(
2017
).
37.
J.
Behler
and
M.
Parrinello
, “
Generalized neural-network representation of high-dimensional potential-energy surfaces
,”
Phys. Rev. Lett.
98
,
146401
(
2007
).
38.
V.
Botu
,
R.
Batra
,
J.
Chapman
, and
R.
Ramprasad
, “
Machine learning force fields: Construction, validation, and outlook
,”
J. Phys. Chem. C
121
,
511
522
(
2016
).
39.
D. K.
Duvenaud
,
D.
Maclaurin
,
J.
Iparraguirre
,
R.
Bombarell
,
T.
Hirzel
,
A.
Aspuru-Guzik
, and
R. P.
Adams
, “
Convolutional networks on graphs for learning molecular fingerprints
,” in
Advances in Neural Information Processing Systems
(
Neural Information Processing Systems Foundation
,
2015
), pp.
2224
2232
.
40.
S.
Kearnes
,
K.
McCloskey
,
M.
Berndl
,
V.
Pande
, and
P.
Riley
, “
Molecular graph convolutions: Moving beyond fingerprints
,”
J. Comput.-Aided Mol. Des.
30
,
595
608
(
2016
).
41.
J.
Gilmer
,
S. S.
Schoenholz
,
P. F.
Riley
,
O.
Vinyals
, and
G. E.
Dahl
, “
Neural message passing for quantum chemistry
,”
Proc. Mach. Learn. Res.
70
,
1263
1272
(
2017
), available at http://proceedings.mlr.press/v70/gilmer17a.html.
42.
K. T.
Schütt
,
F.
Arbabzadah
,
S.
Chmiela
,
K. R.
Müller
, and
A.
Tkatchenko
, “
Quantum-chemical insights from deep tensor neural networks
,”
Nat. Commun.
8
,
13890
(
2017
).
43.
T.
Xie
and
J. C.
Grossman
, “
Crystal graph convolutional neural networks for an accurate and interpretable prediction of material properties
,”
Phys. Rev. Lett.
120
,
145301
(
2018
).
44.
K. T.
Schütt
,
H. E.
Sauceda
,
P.-J.
Kindermans
,
A.
Tkatchenko
, and
K.-R.
Müller
, “
SchNet—A deep learning architecture for molecules and materials
,”
J. Chem. Phys.
148
,
241722
(
2018
).
45.
I.
Goodfellow
,
Y.
Bengio
, and
A.
Courville
,
Deep Learning
(
MIT Press
,
2016
), http://www.deeplearningbook.org.
46.
Z.
Wu
,
B.
Ramsundar
,
E. N.
Feinberg
,
J.
Gomes
,
C.
Geniesse
,
A. S.
Pappu
,
K.
Leswing
, and
V.
Pande
, “
MoleculeNet: A benchmark for molecular machine learning
,”
Chem. Sci.
9
,
513
530
(
2018
).
47.
T. B.
Blank
,
S. D.
Brown
,
A. W.
Calhoun
, and
D. J.
Doren
, “
Neural network models of potential energy surfaces
,”
J. Chem. Phys.
103
,
4129
4137
(
1995
).
48.
V. L.
Deringer
,
C. J.
Pickard
, and
G.
Csányi
, “
Data-driven learning of total and local energies in elemental boron
,”
Phys. Rev. Lett.
120
,
156001
(
2018
).
49.
J.
Schmidt
,
J.
Shi
,
P.
Borlido
,
L.
Chen
,
S.
Botti
, and
M. A.
Marques
, “
Predicting the thermodynamic stability of solids combining density functional theory and machine learning
,”
Chem. Mater.
29
,
5090
5103
(
2017
).
50.
Q.
Zhou
,
P.
Tang
,
S.
Liu
,
J.
Pan
,
Q.
Yan
, and
S.-C.
Zhang
, “
Learning atoms for materials discovery
,”
Proc. Natl. Acad. Sci. U. S. A.
115
,
E6411
(
2018
).
51.
M. J.
Willatt
,
F.
Musil
, and
M.
Ceriotti
, “
A data-driven construction of the periodic table of the elements
,” preprint arXiv:1807.00236 (
2018
).
52.
A.
Jain
,
S. P.
Ong
,
G.
Hautier
,
W.
Chen
,
W. D.
Richards
,
S.
Dacek
,
S.
Cholia
,
D.
Gunter
,
D.
Skinner
,
G.
Ceder
, and
K. a.
Persson
, “
The materials project: A materials genome approach to accelerating materials innovation
,”
APL Mater.
1
,
011002
(
2013
).
53.
I. E.
Castelli
,
D. D.
Landis
,
K. S.
Thygesen
,
S.
Dahl
,
I.
Chorkendorff
,
T. F.
Jaramillo
, and
K. W.
Jacobsen
, “
New cubic perovskites for one-and two-photon water splitting using the computational materials repository
,”
Energy Environ. Sci.
5
,
9034
9043
(
2012
).
54.
I. E.
Castelli
,
T.
Olsen
,
S.
Datta
,
D. D.
Landis
,
S.
Dahl
,
K. S.
Thygesen
, and
K. W.
Jacobsen
, “
Computational screening of perovskite metal oxides for optimal solar light capture
,”
Energy Environ. Sci.
5
,
5814
5819
(
2012
).
55.
S. P.
Ong
,
W. D.
Richards
,
A.
Jain
,
G.
Hautier
,
M.
Kocher
,
S.
Cholia
,
D.
Gunter
,
V. L.
Chevrier
,
K. A.
Persson
, and
G.
Ceder
, “
Python materials genomics (pymatgen): A robust, open-source python library for materials analysis
,”
Comput. Mater. Sci.
68
,
314
319
(
2013
).
56.
T.
Ogitsu
,
E.
Schwegler
, and
G.
Galli
, “
β-Rhombohedral boron: At the crossroads of the chemistry of boron and the physics of frustration
,”
Chem. Rev.
113
,
3425
3449
(
2013
).
57.
L.
van der Maaten
and
G.
Hinton
, “
Visualizing data using t-SNE
,”
J. Mach. Learn. Res.
9
,
2579
2605
(
2008
).
58.
H.-J.
Zhai
,
Y.-F.
Zhao
,
W.-L.
Li
,
Q.
Chen
,
H.
Bai
,
H.-S.
Hu
,
Z. A.
Piazza
,
W.-J.
Tian
,
H.-G.
Lu
,
Y.-B.
Wu
 et al, “
Observation of an all-boron fullerene
,”
Nat. Chem.
6
,
727
(
2014
).
59.
N. E. R.
Zimmermann
,
M. K.
Horton
,
A.
Jain
, and
M.
Haranczyk
, “
Assessing local structure motifs using order parameters for motif recognition, interstitial identification, and diffusion path characterization
,”
Front. Mater.
4
,
34
(
2017
).
60.
M.
Hellenbrandt
, “
The inorganic crystal structure database (ICSD): Present and future
,”
Crystallogr. Rev.
10
,
17
22
(
2004
).
61.
J. E.
Saal
,
S.
Kirklin
,
M.
Aykol
,
B.
Meredig
, and
C.
Wolverton
, “
Materials design and discovery with high-throughput density functional theory: The open quantum materials database (OQMD)
,”
Jom
65
,
1501
1509
(
2013
).
62.
F.
Faber
,
A.
Lindmaa
,
O. A.
von Lilienfeld
, and
R.
Armiento
, “
Crystal structure representations for machine learning models of formation energies
,”
Int. J. Quantum Chem.
115
,
1094
1101
(
2015
).
63.
J.
Bonnet
and
J.
Daou
, “
Study of the hydrogen solid solution in thulium
,”
J. Phys. Chem. Solids
40
,
421
425
(
1979
).

Supplementary Material

You do not currently have access to this content.