Crystal Graph Convolutional Neural Networks for an Accurate and Interpretable Prediction of Material Properties

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 networks framework to directly learn material properties from the connection of atoms in the crystal, providing a universal and interpretable representation of crystalline materials. Our method provides a highly accurate prediction of density functional theory calculated properties for eight different properties of crystals with various structure types and compositions after being 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.

Figure 1

Illustration of the crystal graph convolutional neural networks. (a) Construction of the crystal graph. Crystals are converted to graphs with nodes representing atoms in the unit cell and edges representing atom connections. Nodes and edges are characterized by vectors corresponding to the atoms and bonds in the crystal, respectively. (b) Structure of the convolutional neural network on top of the crystal graph. $R$ convolutional layers and $L_1$ hidden layers are built on top of each node, resulting in a new graph with each node representing the local environment of each atom. After pooling, a vector representing the entire crystal is connected to $L_2$ hidden layers, followed by the output layer to provide the prediction.

Figure 2

Performance of CGCNN on the Materials Project database [11]. (a) Histogram representing the distribution of the number of elements in each crystal. (b) Mean absolute error as a function of training crystals for predicting formation energy per atom using different convolution functions. The shaded area denotes the MAEs of DFT calculations compared with experiments [28]. (c) 2D histogram representing the predicted formation per atom against DFT calculated value. (d) Receiver operating characteristic curve visualizing the result of metal-semiconductor classification. It plots the proportion of correctly identified metals (true positive rate) against the proportion of wrongly identified semiconductors (false positive rate) under different thresholds.

Figure 3

Extraction of site energy of perovskites from total formation energy. (a) Structure of perovskites. (b) 2D histogram representing the predicted total energy above hull against DFT calculated value. (c),(d) Periodic table with the color of each element representing the mean of the site energy when the element occupies $A$ site (c) or $B$ site (d).