The use of machine learning methods for accelerating the design of crystalline materials usually requires manually constructed feature vectors or complex transformation of atom coordinates to input the crystal structure, which either constrains the model to certain crystal types or makes it difficult to provide chemical insights. Here, we develop a crystal graph convolutional neural network (CGCNN) framework to directly learn material properties from the connection of atoms in the crystal, providing a universal and interpretable representation of the crystals structure. Our method achieves the same accuracy as DFT for 8 different properties of crystals with various structure types and compositions after trained with 10 4 data points. Further, our framework is interpretable because one can extract the contributions from local chemical environments to global properties. Using an example of perovskites, we show how this information can be utilized to discover empirical rules for materials design. Machine learning (ML) methods are becoming increasingly popular in accelerating the design of new materials by predicting material properties with accuracy close to ab-initio calculations, but with computational speeds orders of magnitude faster [1] [2] [3] . The arbitrary size of crystal systems poses a challenge as they need to be represented as a fixed length vector in order to be compatible with most ML algorithms. This problem is usually resolved by manually constructing fixed-length feature vectors using simple material properties [1, [3] [4] [5] or designing symmetry-invariant transformations of atom coordinates [6] [7] [8] . However, the former requires case-bycase design for predicting different properties and the latter makes it hard to interpret the models as a result of the complex transformations. In this letter, we present a generalized crystal graph convolutional neural networks (CGCNN) framework for representing periodic crystal systems that provides both material property prediction with DFT accuracy and atomic level chemical insights. Recent advances in "deep learning" have enabled learning from a very raw representation of data, e.g. pixels of an image, making it possible to build general models that outperforms traditionally expert designed representations [9] . By looking into the simplest form of crystal representation, i.e. the connection of atoms in the crystal, we directly build convolutional neural networks (CNN) on top of crystal graphs generated from crystal structures. The CGCNN achieves similar accuracy with respect to DFT calculations as DFT compared with experimental data for eight different properties after being trained with data from the Materials Project[10], indicating the generality of this method. We also demonstrate the interpretability of CGCNN by extracting the energy of each site in the perovskite structure from the total energy, an example of learning the contribution of local chemical environments to the global property. The empirical rules generalized from the results are consistent with the common knowledge for discovering more stable perovskites and can significantly reduce the search space for high throughput screening. The main idea in our approach is to represent the crystal structure by a crystal graph (CG) that encodes both atomic information and bonding interactions between atoms, and then build a convolutional neural network (CNN) on top of the graph to automatically extract representations that are optimum for predicting target properties by training with DFT calculated data. As illustrated in Figure 1 (a) , a crystal graph G is an undirected multigraph which is defined by N nodes and M edges, where N is the number of atoms in the primitive cell and M is the number of connections between atoms (the method for determining atom connectivity is explained in Supplemental Material). The CG is unlike normal graphs since it allows multiple edges between the same pair of end nodes, a characteristic for crystal graphs due to its periodicity, in contrast to molecular graphs. Each node i is represented by a feature vector v i , encoding the property of the atom corresponding to node i. Similarly, each edge (i, j) k is represented by a feature vector u (i,j) k corresponding to the k-th bond connecting atom i and atom j. The CNN built on top of the CG consists of two major components: convolutional layers and pooling layers. Similar architectures have been used for image classification [11] , natural language processing [12] , and molecular fingerprinting[13], but not for crystal property prediction to the best of our knowledge. The convolutional layers repeatedly update the atom feature vector v i by "convolution" with surrounding atoms and bonds, arXiv:1710.10324v1 [cond-mat.mtrl-sci]
Recently, Xie and Grossman introduced crystal graph convolutional networks (CGCNN) for accurate prediction of 8 different DFT calculated properties on crystal structures REF .
Crystal Graph Convolutional Neural Networks for an Accurate and Interpretable Prediction of Material Properties
Phys. Rev. Lett. 120, 145301 (2018)
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