Graph Neural Network for Hamiltonian-Based Material PropertyPrediction

Hexin Bai1, Peng Chu2, Jeng-Yuan Tsai3, Nathan Wilson4, Xiaofeng Qian5, Qimin Yan6, andHaibin Ling7

1Temple University, hexin.bai@temple.edu2Temple University, peng.chu@temple.edu

3Temple University, jeng-yuan.tsai@temple.edu4Texas A&M University, wilsonnater@tamu.edu

5Texas A&M University, feng@tamu.edu, corresponding author6Temple University, qiminyan@temple.edu, corresponding author

7Stony Brook University, hling@cs.stonybrook.edu, corresponding author

May 28, 2020

1 abstract

Development of next-generation electronic devices forapplications call for the discovery of quantum materi-als hosting novel electronic, magnetic, and topologicalproperties. Traditional electronic structure methods re-quire expensive computation time and memory consump-tion, thus a fast and accurate prediction model is desiredwith increasing importance. Representing the interac-tions among atomic orbitals in any material, a materialHamiltonian provides all the essential elements that con-trol the structure-property correlations in inorganic com-pounds. Effective learning of material Hamiltonian by de-veloping machine learning methodologies therefore offersa transformative approach to accelerate the discovery anddesign of quantum materials. With this motivation, wepresent and compare several different graph convolutionnetworks that are able to predict the band gap for inor-ganic materials. The models are developed to incorporatetwo different features: the information of each orbital it-self and the interaction between each other. The infor-mation of each orbital includes the name, relative coordi-nates with respect to the center of super cell and the atom

number, while the interaction between orbitals are repre-sented by the Hamiltonian matrix. The results show thatour model can get a promising prediction accuracy withcross-validation.

2 Introduction

A key bottleneck limiting the applications of current ar-tificial intelligent (AI) techniques, such as convolutionalneural networks (CNN) [21], for quantum material dis-covery is the great diversity in condensed morphologyof materials. To accelerate the discovery of functionalinorganic materials using a data-driven approach, effec-tive learning methods to establish the connections amongmaterials are essential. In recent years, successful CNN-based methods have been developed for signals, e.g. im-ages [20], where the local neighborhood for convolutionoperation is unique and fixed. CNNs have shown promis-ing achievements on real-space wavefunction-based anal-ysis in small molecules and orthogonal systems. How-ever, for nonorthogonal grids and large molecules, CNN-based methods are found to be insufficient for the proper

1

arX

iv:2

005.

1335

2v1

[ph

ysic

s.co

mp-

ph]

27

May

202

0

modeling since the locality in infinite or nonorthogonalsystems differs substantially from that in finite orthogo-nal systems. Moreover, the diversity in grid and scale alsoprohibits generalization of models trained on one systemto another.

Containing comprehensive and complete informationof material systems, a Hamiltonian matrix governs thefundamental physics and electronic structure related com-plex material behaviors such as electronic, magnetic, andtopological properties. It is a subject of continuous inter-est to compute Hamiltonian matrices directly from first-principles calculations in a tight-binding setup . Accu-rate tight-binding Hamiltonian can be extracted from first-principles calculations using the quasi-atomic minimal-basis-set orbitals [26] readily available in the Wannier90code [1, 24]. These first-principles tight-binding Hamil-tonian matrices provide a direct and transparent pictureof chemical bonding, revealing physical quantities suchas local densities of states and bond orders. Further-more, the development of the Hamiltonian matrix enablesus to perform tight-binding electronic structure calcula-tions for large systems which requires much less com-putational time and memory than full electronic struc-ture calculations. As a fundamental set of material in-formation, Hamiltonian matrix can serve as a novel in-put to completely represent inorganic crystalline systemsand be incorporated into a network-based machine learn-ing framework to provide the fundamental understandingof orbital-based quantum mechanical interactions and ac-celerate quantum material discovery.

The basis of atoms and mutual interactions in quantummaterials naturally encourages using the more generalgraph representation for materials. In particular, local-ity in quantum material systems can be defined by graphnodes independently from their actual grid and scale inreal-space with the interaction between elements repre-sented by graph edges. The recent emerging graph convo-lutional networks [3, 5, 17, 19] enable a flexible modelingof various grid and atomic structures. Furthermore, sym-metry information of the condensed system can be easilyintegrated into the spectral formulation of graph convolu-tion, which is essential for determining the physical prop-erty of a solid-state quantum material.

In this paper, we explore two different Graph Convolu-tional Network (GCN) architectures to predict the mate-rial properties of the quantum materials using their Hamil-

tonian representation. The Message Passing Graph Net-work conducts the information aggregation of neighbor-ing nodes in the graph in the real space. On the other hand,by transferring the graph Laplacian into Fourier domain,Chebyshev Convolution leverages the Chebyshev polyno-mial to accelerate the convolutional operation in the spec-tral domain. The two methods are tested on our collecteddataset to predict the band gap. Compared with traditionalhand-crafted feature-based methods, the GCN-based onesgenerate clearly better performance on this binary classi-fication task.

3 Related WorkMachine learning techniques provide a novel opportu-nity to significantly reduce computational costs and speedup the pace of materials discovery by utilizing data-driven paradigms. [13,25,27] Instead of numerically solv-ing complex systems with quantum interactions, physicalquantities are statistically estimated based on a referenceset of known solutions. Machine learning, especially su-pervised learning, has been effectively applied to mate-rials property predictions, including phase stability (forboth molecules and crystals), [23] crystal structure, [7,16]electronic structure, [22, 25] molecule atomization ener-gies, [15] effective potential for molecule dynamics simu-lations, [2,4] and energy functional for density functionaltheory based simulations. [29] Neural network has beenapplied to inorganic crystal systems with limited applica-tions. Band gaps of given classes of inorganic compoundshave been predicted using deep neural networks. [9, 28]Recently, machine learning has been applied to 3D vol-ume data such as electron charge densities to predict elec-tronic properties. [6]

Recently, there has been great advance in the apply-ing Graph neural network (GNN) for pattern recognitionand data mining. Zhou et al. gave a comprehensive lit-erature review in [30] which covered recent methods andapplications of GNN in modeling physics system, learn-ing molecular fingerprints, and predicting protein inter-face. In [3], Battaglia et al. applied relational inductivebiases in deep learning and presented a highly general-ized framework for GNN. With the increasing number ofGNN architectures appeared recently, [10] proposed a re-producible benchmark framework for GNN to gauge their

2

Figure 1: The Hamiltonian matrix of Li-Al-Si

effectiveness and compare with peer methods.

4 Method

4.1 Problem FormulationIn materials sciences, the material band gap is an impor-tant property governing whether the material is metal ornon-metal. In this study, we aim to use GCN to predictthe band gap given the Hamiltonian of the material. Bandgap is described by a non-negative real number, Eg ∈ RandEg ≥ 0. To simplify the problem, threshold is appliedto sort Eg into two categories c ∈ {0, 1} to represent themetal and non-metal classes as the learning target, respec-tively. Finally, the learning problem is defined as:

c = arg maxθ

f(c; θ, x

), (1)

where c is the prediction, x the input representation ofHamiltonian, and f(·) the function with trainable param-eters θ to be determined in this study.

4.2 Hamiltonian MatrixWe will focus on the 2D Hamiltonian matrix that gov-erns the fundamental physics of the quantum materials.Specially, Hamiltonian of a physical system contains all

operators corresponding to the kinetic and potential ener-gies. Let the Hamiltonian operator H for an N -particlecondensed matter system be

H =

N∑n=1

Tn + V, (2)

where Tn indicates the kinetic energy operators for eachparticle, and V = V (r1, r2, . . . , rN , t) is the potential en-ergy between particles. To facilitate the calculation, theHamiltonian operator can be represented as a 2D numeri-cal matrix. In detail, the representation of an operator canbe obtained through the integral with the basis of a Hilbertspace. In the condense matter system, the wavefunctions{ϕi} of orbitals from all atoms in the material system canform a Hilbert space. Therefore, the element in the matrixrepresentation of Hamiltonian Hij can be calculated from

Hij =

∫ϕiHϕjd

3r, (3)

resulting in an M ×M Hamiltonian matrix H = {Hij}with M as the total number of orbitals. In this paper, theHamiltonian is computed from a super cell of 7×7×7 =343 unit cells. Each cell contains around 3 atoms withcertain number of orbitals. In Fig. 1, we visualize theHamiltonian matrix for a sample in the Li-Al-Si materialsystem with M = 432. Note that in order to keep con-sistency with our experiment, we consider only the center3 × 3 × 3 = 27 for computation cost. In the example ofFig. 1, with 9 orbitals, we got a square matrix of dimen-sion 27×9 = 243 filled with the real value which is quiteclosed to the norm given the imaginary part is very small.

4.3 Graph Representation of HamiltonianHamiltonian contains more concentrated information andhas much lower input dimension than wavefunctions orcharge density when used for physical property predic-tion. Thus, directly using Hamiltonian for prediction mayreduce the model complexity and further relieve the de-mand of big data for model training. However, due todifferent atom composition in each condensed matter sys-tem, the size of Hamiltonian varies greatly. To handle thediversity of input dimension, we propose using a graph tostore the Hamiltonian matrix for this learning task.

3

A weighted undirected graph can be constructed fromthe Hamiltonian matrix to encode all interactions andsymmetric information of the quantum system. As in-dicated in Eq. 3, the matrix element Hij represents theinteraction intensity between the i-th orbital and the j-thorbital in the condensed matter system. Then, if the or-bitals are represented by M vertices in a graph, such asV = {vi}i=1:M , naturally, the interaction between or-bitals can be described by the edges in the graph. Not-ing that, according to Eq. 3, interactions exist for eachpair of orbitals in the system, which results in a com-plete graph. In this work, we only consider the inter-action stronger than a threshold τh to reduce the com-putational burden. Finally, a graph G = (c,V, E), withE = {(erksk , rk, sk)}k=1:N for N edges, is built for eachHamiltonian matrix. Specifically, let rk = i and sk = jindicate the two nodes associated with the k-th edge; anedge (erksk , rk, sk) captures the interaction between i-thand j-th orbitals; its weight eij = real(Hij) stands forthe real part of complex number Hij in Hamiltonian ma-trix. In practice the scale of the imaginary part is negligi-ble comparing with the real part. Hence the real part of thecomplex number is a close approximation to the modulusof the complex number.

Figure 2: Visualization of node features for Li-Al-Si.Each row of the matrix represents a node feature in the3 × 3 × 3 unit cells. Each node feature contains atomicnumber, coordinates and orbit type with detail in Tab. 1

For each node inside the graph, we have the followingnode feature vector representation. In Tab. 1, a summaryof feature components for the node and edges is listed.Specifically, for each node, features encode the informa-tion for each orbital, which fuse both atom and orbitalinformation. The collection of node features for all nodesin 3×3×3 unit cells of a Li-Al-Si sample is visualized inFig. 2. Edges, as we discussed above, represent the inter-action between orbits, thus directly use the real part Hij

as feature.With the graph represented Hamiltonian, we investigate

two types of GCNs to explore and learn the informationin the Hamiltonian for the band gap prediction.

4.4 Message Passing Graph Network

A single graph convolutional block is composed of updatefunctions φ and aggregation functions ρ, such as

xik = ψ(eik,vk), k ∈ N (i)

xi = ρlocal

( ⋃k∈N (i)

xik

)vi = φv(xi,vi)

v = ρglobal

(⋃i

vi

)c = φc(v)

(4)

where N (i) stands for the neighborhood of node i. Theorder-invariant aggregation functions ρ must provide thesame output no matter what permutation of input is. ρlocalaggregates information from all the edges connecting tothe node iwhile ρglobal summarizes the information glob-ally. The update functions φv and φc update the node andglobal attribute respectively. The xik, updated by functionψ, is a learned vector representation for each edge whilethe original edge attribute is kept unchanged to providefurther critical information. Note that this learned edgefeature vector is also updated due to refreshing the noderepresentation.

This general framework can be implemented with dif-ferent flexible variants. In [14], the Message Passing Neu-ral Network is proposed to allow long range interactionsbetween nodes in the graph for molecular properties pre-diction. A modified version under this general frame-

4

Table 1: Feature component used in the GCN methods.Feature Type Description

NodeAtomic Number 1 Integer Number of protons

Atom Coordinates 3 Real 3D location in the unit cellOrbit Type 1× 16 One hot s, p, d, . . .

Edge Interaction 1 Real Real part of Hij

work is implemented using Eq. 4 as explained in detailin Alg. 1.

Function MPNN {eki,vi, c}:for m = 0, 1 do

foreach node i, doforeach k ∈ N (i) do

xki = eki · vmk ;xki = LReLU(Wmxki + bm) ;

endxi = 1

|N (i)|∑k∈N (i) xki ;

vmi = LReLU(Wmvmi + bm) ;vmi = vmi + xi;vmi = Dropout(vmi )

endendv = 1

M

∑i v

1i ;

v = W2v + b2 ;c = softmax(v) ;return c

endAlgorithm 1: Message Passing Neural Network.

In Alg. 1, LReLU stands for leaky ReLU activation andDropout is the random dropout algorithm. The weightWm and bias bm, where m = 0, 1, 2, are learnable pa-rameters. Each node vector marked with m = 0, 1 areupdated twice. In each updating, Wm and bm are de-signed such that the length of each node vector gets longerand hence incorporates more information before the lastglobal averaging. Besides, an affine function based onlearned representation frees us from a direct update onthe original edge attribute eki. Finally, random dropoutis used to battle over-fitting when training the network onthe small dataset.

4.5 Chebyshev Convolution

With vertex-wised signal x, the convolution operationon graph G such as x ∗ G can also be defined in theFourier domain. To analysis graph G in Fourier domain,one essential operation is to obtain the graph LaplacianL = D − E where E = {ek} = {eij} and D isthe diagonal degree matrix with Dii =

∑j eij or as

a normalized form L = In − D(−1/2)ED(−1/2). TheLaplacian can be diagonalized by its graph Fourier modesU = [u0, u1, . . . , uM−1] such that L = UΛUT whereΛ = diag

([λ0, λ1, . . . , λM−1]

)are the frequencies of the

graph. Finally, a signal x is filtered by gθ as

y = gθ(L)x = gθ

(UΛUT

)x = Ugθ(Λ)UTx. (5)

Convolution with constant neighbors could use non-parametric filter such as gθ(Λ) = diag(θ) where θ ∈ RMis trainable variables. However, it is not localized inspace. Therefore, polynomial parametrization is usedto construct the localized filters gθ(Λ) =

∑K−1k=0 θkΛk

which limit the shortest path distance to K (e.g. withinthe K-th order neighbors of a vertex) [8]. In order to fur-ther accelerate the computation of the multiplication withthe Fourier basis U , Chebyshev polynomial Tk(x) is usedto build the filter

gθ(Λ)

=

K−1∑k=0

θkTk(Λ). (6)

As a result, the filtering operation can be written as

y = gθ(L)x =

K−1∑k=0

θkTk(L)x (7)

where θk is the trainable parameters for a single outputchannel.

5

Pooling operation is another essential operation forbuilding the convolution neural network (CNN), whichgradually increases the receipt field size in each layer toenhance the hierarchical representation of features. Thepooling operation on graphs is to cluster similar verticesdepending on the meaningful neighborhoods. After clus-tering, all vertices within the same cluster will be repre-sented as one vertex. We also require the clustering tech-niques that can reduce the size of the graph by a factorof two at each level, which offers a precise control of thecoarsening and polling size. Since graph clustering is NP-hard, approximation based algorithm are usually adaptedin this calculation [12].

Beside the convolution and pooling layers, other op-erations such as activation, fully-connected, dropout andloss function are directly portable from the ordinary CNNdefined on the rigid neighborhood.

5 Experiments

5.1 Dataset

We collected 530 half-Heusler compounds from the Ma-terials Project database [18] using the data mining ap-proach. Each of the generated raw Hamiltonian sam-ple contains three atoms, each with maximum 16 orbitalsconsisting of one s, three p, five d, and seven f or-bitals. The Hamiltonian matrices are all calculated within7× 7× 7 = 343 unit cells. The real values of target bandgap fall in the range of

[0, 5.6

]. We choose a threshold 0.2

to produce binary labels (Eg > 0.2 for = 1 and Eg < 0.1as c = 0), which results 116 positive and 414 negativesamples. For balance, 117 negative samples are sampledfrom the original 414 ones to match the number of posi-tive samples. Finally, a balanced subset with 233 samplesare used in the experiments.

5.2 Features

We generate two groups of features for GCN based meth-ods and shallow methods respectively. The feature usedin GCN based approaches is already presented before.

For shallow methods, a set of fixed length features iscreated to include both atomic and interaction informa-tion. To limit the total dimension of feature, only the

Hamiltonian from the center unit cell (relative coordinate[0, 0, 0]) is selected for use. Zeros padding is used to ac-commodate size variation of the Hamiltonian for differentsamples. In detail, all Hamiltonian are embedded in thesquare matrix of size 48×48 where each side correspondsto the three atoms each with 16 orbitals. The interactionfeatures are the vectorization of the square matrix, whichresults features with a dimension of 2,304. The atomicfeatures are the concatenation of the atomic number andatom coordinate feature in Tab. 1. Finally, the combina-tion of the two features results in a set of features with adimension of 3 + 3× 3 + 2304 = 2316.

5.3 Experiment Settings

Five popular shallow classification methods are evaluated,including Decision Tree, NaiveBayes, MultiLayer Per-ceptron, SVM and RandomForest. Three variations ofthe features are used in these experiments: (1) Interactiononly, (2) Atom only, and (3) Atom+Interaction. Since theclass distribution is balanced, the baseline performance ofrandom output is 50%. Reported results are binary classi-fication accuracy by 5-fold cross-validation.

A two-layer MPNN and a three layer Chebyshev con-volutional network are built upon the Hamiltonian data.For Chebyshev convolutional network, the convolutionalfilter size in those three layers is chosen as {1, 2, 2}respectively to gradually increase the receptive field.Leaky-ReLU is adapted for nonlinear activation. At last, aglobal average pooling layer followed by a softmax layeroutputs the probabilities of input graph belong to the twocategories. For both methods, Adam optimizer with afixed learning rate of 0.001 and weight decay of 5e-4 isused. The training is performed up to 2,000 epochs. Thetwo GCNs are implemented in PyTorch Geometric [11]and trained on an Nvidia Titan X GPU.

5.4 Results

We report the classification accuracy of the shallow meth-ods using the five-fold cross-validation in Tab. 2. Amongthe different variations of features, the Atom+Interactionusually achieves the best performance, while Atom fea-ture alone performs worst. The Interaction features aloneachieve the similar performance with the combination of

6

Table 2: Results on the success rates (%) of band gap classification.Method Interaction Atom Atom+Interaction

shal

low

Decision Tree 52.75 63.59 57.83NaiveBayes 57.03 57.45 61.78

MultiLayer Perceptron 64.91 52.84 66.16SVM 67.89 55.40 66.23

RandomForest 66.16 64.80 67.84

GC

N

MPNN w/o neighbor cell - - 69.12Cheb. Conv. w/o neighbor cell - - 70.39

Cheb. Conv. w/ 1st neighbor cells - - 70.43

Figure 3: Learning curves of the two GCN based methodson one fold.

Interaction and Atom feature, which demonstrates the im-portance of using the Hamiltonian to represent the wholecharacteristic of a material. The experiments on shal-low methods confirm that the interaction embedded in theHamiltonian contains the key information for the bandgap prediction. Therefore, using graph to model this in-teraction structure of the Hamiltonian and applying GCNfor learning process provide an advanced solution to thistask.

The learning curves of MPNN and Chebyshev Networkare shown in the Fig. 3. The same cross-validation setgenerated with the same random seed are used here forfair comparison. Without the random dropout module, theChebyshev Network clearly has over fitting on the valida-tion accuracy curve with nearly 68% accuracy while it is

98% accuracy on the training set. By applying the randomdropout module, there is only 10% difference of accuracyin training and validation for MPNN. We also report nu-merical performance of all five folds in Tab. 2.

We also explore the case that includes the neighbor-ing unit cell for the classification task. When only centerunit cell was used, one can observe from the results thatthe GCN based method is obviously better than traditionalshallow methods benefited by the more accurate and com-pact graph representation of the Hamiltonian data. Byfurther including the neighboring unit cells, feature di-mension will dramatically increases to 20,000 if follow-ing the same strategy mentioned above to unify the size ofthe input Hamiltonian for shallow methods. This high di-mension input is unacceptable for shallow methods beforeeffective dimension reduction technique is applied. Butthe GCN based methods can handle this case naturally.As shown in Tab. 2, Chebyshev convolution based net-work achieves similar performance with the eight timeslarger input. With the limited training samples, the highdimensional input may not directly benefit the final per-formance. When training with sufficient data and includ-ing other essential layers, such as local pooling and batchnormalization layers, we can expect the GCN to achievemuch better performance in learning the periodic structureinformation in the Hamiltonian for the physical propertiesprediction.

6 Conclusion

In this paper, we investigate two different GCN algo-rithms on the Hamiltonian data for band gap classifica-

7

tion. The five-fold cross-validation results show that usinggraph based representation and learning techniques forthe Hamiltonian data achieve advanced performance overother shallow methods combining hand-crafted dense fea-tures. Moreover, compared with traditional methods thatrequire fixed input dimension, graph formulates the in-teraction structure in the Hamiltonian more naturally andcompactly, which is scalable to include the neighboringcells for exploring the periodic pattern in the condensedmatter system.

Acknowledgement: N.W. and X.Q. gratefully ac-knowledge the support by the National Science Founda-tion (NSF) under award number OAC-1835690. Portionsof this research were conducted with the advanced com-puting resources provided by Texas A&M High Perfor-mance Research Computing. Q.Y. acknowledges the sup-port by the U.S. Department of Energy, Office of Science,under award number DE-SC0020310.

References[1] An updated version of wannier90: A tool for obtaining

maximally-localised wannier functions. Computer PhysicsCommunications, 185(8):2309 – 2310, 2014.

[2] A. P. Bartok, M. C. Payne, R. Kondor, and G. Csanyi.Gaussian approximation potentials: The accuracy of quan-tum mechanics, without the electrons. Physical ReviewLetters, 104(13):136403, 2010.

[3] Peter W Battaglia, Jessica B Hamrick, Victor Bapst, Al-varo Sanchez-Gonzalez, Vinicius Zambaldi, Mateusz Ma-linowski, Andrea Tacchetti, David Raposo, Adam San-toro, Ryan Faulkner, et al. Relational inductive bi-ases, deep learning, and graph networks. arXiv preprintarXiv:1806.01261, 2018.

[4] J. Behler. Neural network potential-energy surfaces inchemistry: a tool for large-scale simulations. PhysicalChemistry Chemical Physics, 13(40):17930–17955, 2011.

[5] Michael M Bronstein, Joan Bruna, Yann LeCun, ArthurSzlam, and Pierre Vandergheynst. Geometric deep learn-ing: going beyond euclidean data. IEEE Signal ProcessingMagazine, 34(4):18–42, 2017.

[6] Anand Chandrasekaran, Deepak Kamal, Rohit Batra,Chiho Kim, Lihua Chen, and Rampi Ramprasad. Solv-ing the electronic structure problem with machine learn-ing. npj Computational Materials, 5(1):22, 2019.

[7] S. Curtarolo, D. Morgan, K. Persson, J. Rodgers, andG. Ceder. Predicting crystal structures with data min-ing of quantum calculations. Physical Review Letters,91(13):135503, 2003.

[8] Michael Defferrard, Xavier Bresson, and Pierre Van-dergheynst. Convolutional neural networks on graphs withfast localized spectral filtering. In Advances in neural in-formation processing systems, pages 3844–3852, 2016.

[9] Yuan Dong, Chuhan Wu, Chi Zhang, Yingda Liu, JianlinCheng, and Jian Lin. Bandgap prediction by deep learn-ing in configurationally hybridized graphene and boron ni-tride. npj Computational Materials, 5(1):26, 2019.

[10] Vijay Prakash Dwivedi, Chaitanya K Joshi, Thomas Lau-rent, Yoshua Bengio, and Xavier Bresson. Benchmarkinggraph neural networks. arXiv preprint arXiv:2003.00982,2020.

[11] Matthias Fey and Jan E. Lenssen. Fast graph represen-tation learning with PyTorch Geometric. In ICLR Work-shop on Representation Learning on Graphs and Mani-folds, 2019.

[12] Matthias Fey and Jan Eric Lenssen. Fast graph repre-sentation learning with pytorch geometric. arXiv preprintarXiv:1903.02428, 2019.

[13] L. M. Ghiringhelli, J. Vybiral, S. V. Levchenko, C. Draxl,and M. Scheffler. Big data of materials science: Crit-ical role of the descriptor. Physical Review Letters,114(10):105503, 2015.

[14] Justin Gilmer, Samuel S Schoenholz, Patrick F Riley, OriolVinyals, and George E Dahl. Neural message passing forquantum chemistry. In Proceedings of the 34th Interna-tional Conference on Machine Learning-Volume 70, pages1263–1272. JMLR. org, 2017.

[15] Katja Hansen, Grgoire Montavon, Franziska Biegler, Sia-mac Fazli, Matthias Rupp, Matthias Scheffler, O. Anatolevon Lilienfeld, Alexandre Tkatchenko, and Klaus-RobertMller. Assessment and validation of machine learningmethods for predicting molecular atomization energies.Journal of Chemical Theory and Computation, 9(8):3404–3419, 2013.

[16] G. Hautier, C. C. Fischer, A. Jain, T. Mueller, andG. Ceder. Finding nature’s missing ternary oxide com-pounds using machine learning and density functional the-ory. Chemistry of Materials, 22(12):3762–3767, 2010.

[17] Mikael Henaff, Joan Bruna, and Yann LeCun. Deep convo-lutional networks on graph-structured data. arXiv preprintarXiv:1506.05163, 2015.

8

[18] Anubhav Jain, Shyue Ping Ong, Geoffroy Hautier,Wei Chen, William Davidson Richards, Stephen Dacek,Shreyas Cholia, Dan Gunter, David Skinner, GerbrandCeder, et al. Commentary: The materials project: A ma-terials genome approach to accelerating materials innova-tion. APL Materials, 1(1):011002, 2013.

[19] Thomas N Kipf and Max Welling. Semi-supervised clas-sification with graph convolutional networks. 2017.

[20] Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton.Imagenet classification with deep convolutional neural net-works. In Advances in neural information processing sys-tems, pages 1097–1105, 2012.

[21] Yann LeCun, Leon Bottou, Yoshua Bengio, and PatrickHaffner. Gradient-based learning applied to documentrecognition. Proceedings of the IEEE, 86(11):2278–2324,1998.

[22] Joohwi Lee, Atsuto Seko, Kazuki Shitara, KeitaNakayama, and Isao Tanaka. Prediction model of bandgap for inorganic compounds by combination of densityfunctional theory calculations and machine learning tech-niques. Physical Review B, 93(11):115104, 2016.

[23] C. J. Long, J. Hattrick-Simpers, M. Murakami, R. C. Sri-vastava, I. Takeuchi, V. L. Karen, and X. Li. Rapid struc-tural mapping of ternary metallic alloy systems using thecombinatorial approach and cluster analysis. Review ofScientific Instruments, 78(7):072217, 2007.

[24] Nicola Marzari and David Vanderbilt. Maximally local-ized generalized wannier functions for composite energybands. Physical Review B, 56:12847–12865, 1997.

[25] G. Pilania, A. Mannodi-Kanakkithodi, B. P. Uberuaga,R. Ramprasad, J. E. Gubernatis, and T. Lookman. Ma-chine learning bandgaps of double perovskites. ScientificReports, 6:19375, 2016.

[26] Xiaofeng Qian, Ju Li, Liang Qi, Cai-Zhuang Wang, Tzu-Liang Chan, Yong-Xin Yao, Kai-Ming Ho, and SidneyYip. Quasiatomic orbitals for ab initio tight-binding anal-ysis. Physical Review B, 78:245112, 2008.

[27] K. Rajan. Materials informatics. Materials Today,8(10):38–45, 2005.

[28] Zhe Shi, Evgenii Tsymbalov, Ming Dao, Subra Suresh,Alexander Shapeev, and Ju Li. Deep elastic strain engi-neering of bandgap through machine learning. Proceed-ings of the National Academy of Sciences, 116(10):4117,2019.

[29] J. C. Snyder, M. Rupp, K. Hansen, K. R. Muller, andK. Burke. Finding density functionals with machine learn-ing. Physical Review Letters, 108(25):253002, 2012.

[30] Jie Zhou, Ganqu Cui, Zhengyan Zhang, Cheng Yang,Zhiyuan Liu, and Maosong Sun. Graph neural networks:A review of methods and applications. arXiv preprintarXiv:1812.08434, 2018.

9