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Crystal Graph Convolutional Neural Networks for an Accurate and Interpretable Prediction of Material Properties
Tian Xie, Jeffrey C. Grossman
TL;DR
Existing crystal-property models often require hand-crafted features or coordinate transformations that limit crystal types or chemical interpretability. This paper introduces crystal graph convolutional neural networks that learn properties directly from atomic connectivity, achieving accurate predictions across eight DFT properties while exposing local-environment contributions.
Problem
Existing machine-learning approaches for crystalline materials commonly require manually constructed features or complex coordinate transformations, limiting crystal types or chemical insight.
Method
Crystal graph convolutional neural networks directly learn material properties from atomic connectivity using crystal graphs and convolutional representations of local chemical environments.
Results
The framework accurately predicts eight DFT-calculated crystal properties across varied structures and compositions after training on 10^4 data points.
Takeaways & Limitations
Extracted local-environment contributions support interpretation of global properties and can help identify empirical materials-design rules, as illustrated for perovskites.
Takeaways & Limitations
The supplementary tables treat lanthanide and actinide elements as periods 8 and 9, respectively, and use logarithmic scales for some properties.
Abstract
from arXiv · showhide
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.
Supplemental Materials: Crystal Graph Convolutional Neural Networks for an
This section is identified as supplemental material for the paper on crystal graph convolutional neural networks and accurate, interpretable material-property prediction.
- Supplemental Materials: The section belongs to the paper’s supplemental materials.Its heading explicitly labels the material as supplemental.
- Supplemental Materials: The stated focus is accurate and interpretable prediction of material properties.The heading identifies both accuracy and interpretability as central themes.
Construction of crystal graphs
Crystal graphs connect atoms using geometrically and chemically filtered neighbor relationships, while the model encodes atom and bond attributes as categorical feature vectors. The convolution can reduce sensitivity to the exact initial connectivity.
- Construction of crystal graphs: Neighbors within 6 Å are connected when they share a Voronoi face and satisfy a covalent-radius distance threshold.The threshold includes a 0.25 Å tolerance, retaining strong bonding interactions.
- Construction of crystal graphs: The convolution function can automatically ignore weak bonds, reducing the importance of initial graph connectivity.Connecting the 12 nearest neighbors performs as well in practice as the chemically filtered construction.
- Construction of crystal graphs: Atom and bond properties are represented by one-hot vectors over discrete or binned continuous categories.Continuous ranges are evenly divided into 10 categories.
Illustrative example for differentiating NaCl and KCl
The NaCl–KCl example illustrates how CGCNN converts crystal graphs into pooled feature vectors and learns weights that distinguish the two structures. With more training data, shared weights require optimization rather than exact zero-loss fitting.
- Illustrative example for differentiating NaCl and KCl: CGCNN is illustrated as predicting +1 for NaCl and −1 for KCl using one convolutional and one pooling layer.The task demonstrates binary structural discrimination.
- Illustrative example for differentiating NaCl and KCl: The eight-node crystal representation also demonstrates invariance to unit-cell choice.Although equivalent environments could simplify the graph, the expanded representation is retained for illustration.
- Illustrative example for differentiating NaCl and KCl: Node features are convolved, then normalized pooling averages the resulting vectors over all graph nodes.The pooled representation is used as the crystal-level feature vector.
- Illustrative example for differentiating NaCl and KCl: The one-dimensional crystal feature vectors can directly serve as predictions without another output layer.The example therefore maps the pooled feature directly to the target value.
- Illustrative example for differentiating NaCl and KCl: Weights can produce predictions of 1 for NaCl and −1 for KCl while encoding element identity and occurrence frequency.Center and neighbor weights distinguish chemical roles in the crystal graphs.
- Illustrative example for differentiating NaCl and KCl: More complex training settings require shared weights to be optimized for prediction performance rather than exact zero loss.The passage points to SGD and deeper CGCNN structures as suitable extensions.
Hyperparameter optimization
Hyperparameters are selected separately from trained weights through validation, using random search and performance criteria tailored to the convolution function.
- Hyperparameter optimization: Hyperparameters cover graph construction, network architecture, and training settings, and are chosen through train-validation rather than SGD.Weights are trained by SGD, whereas hyperparameters are selected separately.
- Hyperparameter optimization: The database is split into 60% training, 20% validation, and 20% test data for hyperparameter selection.The chosen settings minimize validation MAE or AOC against DFT-calculated properties.
- Hyperparameter optimization: Random search optimizes all hyperparameters for Eq. 4 but only selected parameters for Eq. 5.For Eq. 5, the optimized parameters are convolutional-layer count, regularization, and Adam step size.
Pooling layer choices
The framework uses different pooling strategies depending on whether prediction performance or interpretability is prioritized. Performance-oriented pooling combines representations from multiple convolutional layers, while interpretability-oriented pooling maps final local features to scalar contributions.
- Pooling layer choices: Normalized summation pools atom representations into a crystal vector that is invariant to crystal size.The pooling function is used throughout the work, while other pooling functions are also possible.
- Pooling layer choices: Performance-oriented pooling transforms and sparsifies feature vectors from all convolutional layers before summing them over layers and atoms.The resulting crystal vector includes chemical environments across different radii.
- Pooling layer choices: Including intermediate feature vectors captures chemical environments at various radii and improves prediction performance.
- Pooling layer choices: Interpretability-oriented pooling maps the final local feature after R convolutions and L1 hidden layers to a scalar before summing site contributions.Training against the global property allows the model to learn contributions from individual local chemical environments.
- Pooling layer choices: With convolution function Eq. 5, the normalized sum of the final feature vector is used because its residual structure passes information to the last layer.This removes the need to add feature vectors from earlier convolutional layers.
SUPPLEMENTARY FIGURES
The supplementary figures illustrate crystal-graph examples, dataset cell-size distribution, hyperparameter effects, predictive performance across properties, training-set scaling, and perovskite formation-energy prediction.
- SUPPLEMENTARY FIGURES: Figure S1 compares the crystal structures and corresponding crystal graphs of NaCl and KCl.
- SUPPLEMENTARY FIGURES: Figure S2 shows the distribution of the number of atoms in primitive cells.
- SUPPLEMENTARY FIGURES: Figure S3 examines validation MAEs versus atom-feature count, convolutional-layer count, and logarithm of the step size for two convolution functions.The atom-feature settings contain 2, 3, or 9 features, with electronegativity added in the 3-feature setting.
- SUPPLEMENTARY FIGURES: Figure S4 visualizes predictive performance for total energy per atom, band gap, Fermi energy, bulk modulus, shear modulus, and Poisson ratio.
- SUPPLEMENTARY FIGURES: Figure S5 plots bulk-modulus MAE against the number of training crystals and marks the DFT-to-experiment MAE baseline of 0.050 Log(GPa).
- SUPPLEMENTARY FIGURES: Figure S6 visualizes perovskite formation-energy prediction using a full pooling layer with Eq. 4 as the convolution function.
SUPPLEMENTARY TABLES
The supplementary tables document the model’s optimized hyperparameters, atom and bond features, prediction benchmarks, and perovskite candidates identified by combinational search.
- SUPPLEMENTARY TABLES: Table I lists the hyperparameters optimized in the study.
- SUPPLEMENTARY TABLES: The study uses L2 for material-property prediction and L1 for learning individual site energies in perovskites.
- SUPPLEMENTARY TABLES: An L2 regularization term λ∥W∥2 is added to the cost function to reduce overfitting.
- SUPPLEMENTARY TABLES: Table II specifies the properties included in the atom feature vector vi, with lanthanides and actinides treated as periods 8 and 9.
- SUPPLEMENTARY TABLES: Table III specifies the properties used in the bond feature vector u(i,j)k.
- SUPPLEMENTARY TABLES: Table IV compares test-set prediction performance for seven properties using different convolution functions.
- SUPPLEMENTARY TABLES: Table V lists perovskites discovered by combinational search with energy above hull below 0.2 eV/atom.