Learning Ordering in Crystalline Materials with Symmetry-Aware Graph Neural Networks

TL;DR

Various neural network architectures are benchmarked for capturing the ordering-dependent energetics of multicomponent materials in a custom-made dataset generated with high-throughput atomistic simulations.

Abstract

Graph convolutional neural networks (GCNNs) have become a machine learning workhorse for screening the chemical space of crystalline materials in fields such as catalysis and energy storage, by predicting properties from structures. Multicomponent materials, however, present a unique challenge since they can exhibit chemical (dis)order, where a given lattice structure can encompass a variety of elemental arrangements ranging from highly ordered structures to fully disordered solid solutions. Critically, properties like stability, strength, and catalytic performance depend not only on structures but also on orderings. To enable rigorous materials design, it is thus critical to ensure GCNNs are capable of distinguishing among atomic orderings. However, the ordering-aware capability of GCNNs has been poorly understood. Here, we benchmark various neural network architectures for capturing the ordering-dependent energetics of multicomponent materials in a custom-made dataset generated with high-throughput atomistic simulations. Conventional symmetry-invariant GCNNs were found unable to discern the structural difference between the diverse symmetrically inequivalent atomic orderings of the same material, while symmetry-equivariant model architectures could inherently preserve and differentiate the distinct crystallographic symmetries of various orderings.

Figures (12)

• Figure 1: Existing knowledge gap and high-throughput DFT dataset. While GCNNs have been widely shown to learn the dependence of materials properties on chemical compositions, it is unknown to what degree they can capture the dependence on atomic orderings. Color scheme: A, grey; B, cyan or green; B', brown or purple; O, white. b,c, Occurrences of A-site (b) and B-site elements (c) in our high-throughput DFT dataset of multicomponent perovskite oxides (Table \ref{['table:dataset_comparison']}), which encompasses a variety of elements, including (but not limited to) commonly observed A-site (e.g., alkaline earth and rare earth metals) and B-site cations (e.g., transition metals), suggesting that this dataset covers compositions across a broad space of multicomponent perovskite oxides. d--f, Histograms for the occurrences of A-site (d) and B-site stoichiometries (e) and the numbers of other symmetrically inequivalent cation orderings (f) examined computationally for all oxide structures in our dataset. As the occurrences of AxA'8--x and ByB'8--y are equal to those of A8--xA'x and B8--yB'y, respectively, we only showed the stoichiometries of $8 \geq x, y \geq 4$ (d,e) for clarity.
• Figure 2: Representation learning with symmetry-invariant and symmetry-equivariant GCNNs across compositional space.a, Message passing and graph convolution in invariant and equivariant GCNNs, where the information of coordination environments is encoded as scalars and vectors (or higher-order tensors), respectively. Regarding input perovskite oxide data, unrelaxed, idealized structures result in minimum computational cost yet low data fidelity, while DFT-relaxed, distorted perovskite structures possess higher data fidelity but also greater computational expense. Color scheme: A, grey; B, cyan; B', brown; O, white. b--e, GCNN-predicted vs. DFT-computed $E_\mathrm{hull}$ for a test set of $1,261$ perovskite oxides, where CGCNN and e3nn were trained on either unrelaxed (b,c) or DFT-relaxed structures (d,e). f, Test set MAE as a function of training set size. The baseline implies the MAE from simply predicting $E_\mathrm{hull}$ as a linear interpolation of the DFT-computed $E_\mathrm{hull}$ of ternary perovskites. All data points and error bars denote the mean and standard deviation of the three best-performing models, respectively.
• Figure 3: Representation learning with symmetry-invariant and symmetry-equivariant GCNNs across ordering space.a, All possible symmetrically unique cation arrangements in an idealized cubic supercell of AB0.5B'0.5O3 perovskites with 40 atoms, when distributing four B and four B' cations among the eight B-site locations in the supercell. Color scheme: A, grey; B, cyan; B', brown; O, white. The relative DFT-computed energies of these symmetrically unique cation arrangements can be well correlated with the experimental orderings of AB0.5B'0.5O3 across a broad oxide space Peng:2024. b--f, GCNN-predicted vs. DFT-computed $E_\mathrm{hull}$ of all symmetrically unique cation arrangements relative to those of GCNN-predicted or DFT-computed ground-state arrangements, respectively, for a holdout set of $100$ AB0.5B'0.5O3 perovskite compositions, where each composition has six symmetrically unique arrangements (a), and CGCNN and e3nn were trained on either unrelaxed (b,c) or DFT-relaxed structures (d,e), with holdout set $R^2$ score and MAE shown for all models and inputs (f). All data points denote the mean of the three best-performing models.
• Figure 4: Differentiating symmetrically inequivalent orderings by capturing crystallographic symmetries with GCNNs.a, Equivariant GCNNs can contrast the distinct symmetries of crystal structures with the same composition but different orderings more easily than their invariant counterparts. Color scheme: A, grey; B, cyan; B', brown; O, white. b--f, 2D PCA for the GCNN embeddings after graph convolution layers for all the six possible symmetrically distinct cation arrangements (Fig. \ref{['fig:Main_ordering_dependence']}a) of the holdout set of $100$ AB0.5B'0.5O3 oxides. We trained CGCNN and e3nn on either unrelaxed (b,c) or DFT-relaxed structures (d,e) and further selected the best-performing model of each case for this latent embedding analysis, where the arrows show the embedding spread (multiplied by a factor of 2, for clarity) of an ordering relative to the mean of all orderings averaged over these $100$ perovskites. The embedding spread of different orderings for the same composition was quantified as the average Euclidean vector from each embedding to the centroid and further normalized by the spreads of all latent embeddings (f).
• Figure 5: Generalizability of ordering-sensitive GCNNs across model architectures and perovskite compositions.a, GCNN-predicted vs. DFT-computed $E_\mathrm{hull}$ for a holdout set of $863$ AxA'1--xB0.5B'0.5O3 perovskite structures with various symmetrically inequivalent cation arrangements, where CGCNN and e3nn were trained on either unrelaxed or DFT-relaxed structures. Violin plots show the kernel density estimation of the energy distribution of these symmetrically distinct cation arrangements. b, PaiNN-predicted vs. DFT-computed $E_\mathrm{hull}$ for a test set of $1,261$ perovskites oxides. c, PaiNN-predicted vs. DFT-computed $E_\mathrm{hull}$ of all symmetrically distinct cation arrangements relative to those of PaiNN-predicted or DFT-computed ground-state arrangements, respectively, for a holdout set of $100$ AB0.5B'0.5O3 perovskite oxide compositions, where each composition has six symmetrically inequivalent cation arrangements. PaiNN was trained on either unrelaxed or DFT-relaxed structures. All data points denote the mean of the three best-performing models.