Material Property Prediction using Graphs based on Generically Complete Isometry Invariants

TL;DR

This paper introduces the Distance Distribution Graph (DDG), a graph representation for periodic crystals built from the generically complete Pointwise Distance Distribution (PDD) invariant. By encoding $k$-nearest neighbor distances with weights that reflect atomic types, DDG achieves unit-cell and isometry invariance while substantially reducing graph size compared to conventional Crystal Graphs, leading to improved property predictions on Materials Project and Jarvis-DFT datasets (MAE reductions of up to 12% with 44–88% fewer vertices). The methodology includes adaptations to line graphs (for angular information) and integrates PDD-based weights into CGCNN and ALIGNN architectures, with practical guidance for hyper-parameter selection ($k$ and collapse tolerance). The demonstrated benefits include improved accuracy, reduced computation, and robustness to unit-cell choices, suggesting broad applicability of distribution-graph concepts to crystal learning and potentially other structured point sets. Overall, the work advances crystal property prediction by grounding graph representations in a continuous, generically complete invariants framework and showing meaningful gains in real-world datasets.

Abstract

The structure-property hypothesis says that the properties of all materials are determined by an underlying crystal structure. The main obstacle was the ambiguity of conventional crystal representations based on incomplete or discontinuous descriptors that allow false negatives or false positives. This ambiguity was resolved by the ultra-fast Pointwise Distance Distribution (PDD), which distinguished all periodic structures in the world's largest collection of real materials (Cambridge Structural Database). The state-of-the-art results in property predictions were previously achieved by graph neural networks based on various graph representations of periodic crystals, including the Crystal Graph with vertices at all atoms in a crystal unit cell. This work adapts the Pointwise Distance Distribution for a simpler graph whose vertex set is not larger than the asymmetric unit of a crystal structure. The new Distribution Graph reduces mean-absolute-error by 0.6\%-12\% while having 44\%-88\% of the number of vertices when compared to the crystal graph when applied on the Materials Project and Jarvis-DFT datasets using CGCNN and ALIGNN. Methods for hyper-parameters selection for the graph are backed by the theoretical results of the Pointwise Distance Distribution and are then experimentally justified.

Figures (4)

• Figure 1: Past work used incomplete or discontinuous representations of crystals. This work predicts energies faster by adapting generically complete and continuous invariants widdowson2022resolving.
• Figure 2: (top) The transformation from the crystal structure of LuSi to the Crystal Graph and DDG with $k=2$ nearest neighbors. (bottom) The transformation from the crystal structure of NaCl to the Crystal Graph and DDG using $k=4$ and the conventional unit cell (edges without arrowhead indicate bidirectionality).
• Figure 3: Transformation from unit cell based graph to the DDG including the collapse tolerance which allows for small differences in the edges features while still grouping similar vertices (atoms).
• Figure 4: Absolute Percent difference in prediction of formation energies for the test set using the Crystal Graph after altering samples in the training data to have supercells.

Theorems & Definitions (7)

• Definition 2.1: Periodic Point Set $S=M+\Lambda\subset\mathbb{R}^n$
• Definition 2.2: Isometry Invariants
• Definition 2.3: Pointwise Distance Distribution
• Definition 2.4: Distance Distribution Graph $\mathrm{DDG}$
• Definition 2.5: Crystal Graph
• Definition 2.6: Line Graph
• Definition 2.7: Atomic Mass Weighted Pointwise Distance Distribution