Search

Graph neural networks have become central to computational materials science, yet their application to periodic crystals remains limited by the absence of robust positional encoding. Without explicit representation of absolute atomic placement, symmetry-equivalent sites with identical local environments collapse into indistinguishable states, undermining predictive accuracy. This limitation becomes critical in large supercells, where extended defects, domain structures, and long-range ordering dominate material behavior. Existing encoding strategies either violate symmetry constraints or fail to scale, leaving a persistent gap between theoretical requirements and practical implementation. This work develops a theory of symmetry-aware positional encoding tailored to large supercell simulations. Six fundamental requirements are identified, spanning invariance, completeness, scalability, differentiability, and interpretability, and are shown to arise directly from the structure of periodic crystals. Analysis of current approaches reveals that none simultaneously satisfies these conditions at scale. A hierarchical framework is therefore introduced, distributing positional information across global, symmetry-aware, and local levels. This structure reconciles representational completeness with computational efficiency while preserving physical consistency. The resulting perspective establishes positional encoding as a central design principle for graph neural networks and provides a foundation for modeling properties that depend on absolute atomic arrangement in complex crystalline systems.