A Space Group Symmetry Informed Network for O(3) Equivariant Crystal Tensor Prediction

TL;DR

This work tackles predicting crystal tensor properties (dielectric, piezoelectric, elastic) while enforcing $O(3)$ equivariance and crystal space‑group invariance. It introduces GMTNet, a symmetry‑aware graph network with four modules: symmetry‑informed crystal graph construction, crystal‑level equivariant feature extraction, equivariant tensor property prediction, and a crystal symmetry enforcement module. A curated JARVIS‑DFT tensor dataset and targeted evaluation metrics demonstrate that GMTNet produces symmetry‑consistent tensor predictions and outperforms baselines in accuracy and efficiency, with strong results across tensor orders up to 4. The approach has practical impact for materials discovery and device design by enabling reliable, symmetry‑constrained predictions of tensor properties; the code is released in the AIRS library for broader use.

Abstract

We consider the prediction of general tensor properties of crystalline materials, including dielectric, piezoelectric, and elastic tensors. A key challenge here is how to make the predictions satisfy the unique tensor equivariance to O(3) group and invariance to crystal space groups. To this end, we propose a General Materials Tensor Network (GMTNet), which is carefully designed to satisfy the required symmetries. To evaluate our method, we curate a dataset and establish evaluation metrics that are tailored to the intricacies of crystal tensor predictions. Experimental results show that our GMTNet not only achieves promising performance on crystal tensors of various orders but also generates predictions fully consistent with the intrinsic crystal symmetries. Our code is publicly available as part of the AIRS library (https://github.com/divelab/AIRS).

Figures (5)

• Figure 1: Overview of GMTNet. GMTNet takes crystal structures represented as $\mathbf{M} = (\mathbf{A}, \mathbf{P}, \mathbf{L})$ as input to predict crystal tensor properties of various orders. It comprises four modules: symmetry-informed crystal graph construction, crystal-level equivariant feature extraction, equivariant tensor property prediction, and symmetry enforcement. GMTNet is carefully designed to generate tensor predictions adhere to the intrinsic symmetries of the input crystal structures. We also include visualizations of crystal structures and tensors with different orders belonging to various crystal systems. These visualizations, generated using matplotlib matplotlib and MTEX mtex, illustrate the correlation between crystal symmetries and tensor property complexities.
• Figure 2: Visualization of piezoelectric tensor predictions on piezoelectric test set. The results underscore GMTNet's effectiveness in generating symmetry-consistent piezoelectric tensor predictions with tensor order 3.
• Figure 3: Comparative visualization of dielectric tensor predictions. This figure presents a model comparison for dielectric tensor prediction, on the dielectric test set comprising various crystal systems: cubic, hexagonal, trigonal, tetragonal, orthorhombic, monoclinic, and triclinic. GMTNet's predictions are highlighted for their alignment with the spatial symmetry characteristics of the ground truth tensors, underscoring its superior performance. In contrast, models such as MEGNET and ETGNN demonstrate a notable discrepancy in capturing these symmetry aspects. The comparison underscores GMTNet's effectiveness in generating symmetry-consistent dielectric tensor predictions.
• Figure 4: Visualization of elastic tensor predictions on elastic test set. The results underscore GMTNet's effectiveness in generating symmetry-consistent elastic tensor predictions with tensor order 4.
• Figure 5: Message passing details for node invariant feature updating and equivariant message passing in GMTNet. (a) Node invariant feature updating that takes node features ($\boldsymbol{f}_j$, $\boldsymbol{f}_i$) and edge feature ($\boldsymbol{f}_{ji}^e$) as input to obtain the updated invariant node feature $\boldsymbol{f}_i^{\text{new}}$. (b) Equivariant message passing that produces high rotation order node features. In this figure, all notations follow Sec. \ref{['sec:global_extract']}.

Theorems & Definitions (1)

• Definition 3.1: Crystal Tensor Property Prediction