PDDFormer: Pairwise Distance Distribution Graph Transformer for Crystal Material Property Prediction

TL;DR

This work tackles crystal data fuzziness by introducing atom-weighted and unit-cell pairwise distance distributions (WPDD/UPDD) to form a continuous, geometrically complete crystal graph. By modeling PDD as global information and integrating it into matrix-based message passing, the authors present PDDFormer, a transformer-based framework that achieves state-of-the-art performance on Materials Project and JARVIS benchmarks while maintaining computational efficiency. Theoretical guarantees are provided for continuity under perturbations via Earth Mover's Distance and for geometric completeness (WPDD) and continuity (UPDD), with practical proofs and ablations supporting the design. The approach bridges traditional crystal descriptors and dynamic atomic behavior, enabling more reliable and scalable crystal property predictions with robust invariance to perturbations.

Abstract

Crystal structures can be simplified as a periodic point set that repeats across three-dimensional space along an underlying lattice. Traditionally, crystal representation methods characterize the structure using descriptors such as lattice parameters, symmetry, and space groups. However, in reality, atoms in materials always vibrate above absolute zero, causing their positions to fluctuate continuously. This dynamic behavior disrupts the fundamental periodicity of the lattice, making crystal graphs based on static lattice parameters and conventional descriptors discontinuous under slight perturbations. Chemists proposed the pairwise distance distribution (PDD) method to address this problem. However, the completeness of PDD requires defining a large number of neighboring atoms, leading to high computational costs. Additionally, PDD does not account for atomic information, making it challenging to apply it directly to crystal material property prediction tasks. To tackle these challenges, we introduce the atom-Weighted Pairwise Distance Distribution (WPDD) and Unit cell Pairwise Distance Distribution (UPDD) and apply them to the construction of multi-edge crystal graphs. We demonstrate the continuity and general completeness of crystal graphs under slight atomic position perturbations. Moreover, by modeling PDD as global information and integrating it into matrix-based message passing, we significantly reduce computational costs. Comprehensive evaluation results show that WPDDFormer achieves state-of-the-art predictive accuracy across tasks on benchmark datasets such as the Materials Project and JARVIS-DFT.

Figures (5)

• Figure 1: Illustrations of different unit cells and lattice representations of the same crystal structure. Figure (a) shows several possible choices among the infinitely many unit cells for the same crystal structure in the undisturbed case. Figure (b) illustrates that for almost any perturbation, the symmetry group and any reduced unit cell (with minimal volume) will undergo discontinuous changes.
• Figure 2: Schematic diagram of the selected neighbors in PDD. Figure (a) represents the 3D unit cell structure. The edges in Figure (b) show the neighbor selection for atom $i$ in WPDD. By comparing Figures (c) and (d), we can see that we construct the unit cell centered around each atom and select neighbors, rather than being limited to the unit cell where the atoms are located.
• Figure 3: Architecture Overview. PDDFormer accepts an input crystal structure $S$. During the prediction process, it first undergoes a graph construction step to generate a continuous and general complete crystal graph structure, followed by an embedding block, then multiple blocks of node-wise Transformer and PDD Message Passing, and finally, an output block.
• Figure 4: The coefficient of determination for WPDDFormer's predictions is presented. The scatter plots reflect the differences between our predicted values and the actual values, while the bar charts show the frequency of the values.
• Figure 5: The different neighbor selection under slight perturbations.

Theorems & Definitions (10)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Definition 5
• Definition 6
• Proposition 1
• proof
• Proposition 2
• proof