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Neural networks exhibit a pronounced spectral bias, preferentially converging to low-frequency components of target functions while struggling with high-frequency details, as governed by the neural tangent kernel and the frequency principle. In the domain of materials science, accurate prediction of atomic forces in periodic crystals is essential because high-frequency force components directly encode short-wavelength phonons that govern thermal transport, lattice thermal conductivity, phonon dispersion relations, and mechanical stability under strain. Periodic crystal graphs, formed by atoms connected through cutoff-based edges under periodic boundary conditions, amplify this spectral bias relative to molecular graphs or Euclidean data: the discrete spectrum of the graph Laplacian introduces pronounced gaps between low- and high-eigenvalue modes, and standard message-passing layers act as repeated low-pass filters that exponentially attenuate high-frequency signals. We develop a theoretical framework that decomposes force fields into graph-Laplacian eigenbasis components, couples the training dynamics to the neural tangent kernel on the crystal graph, and derives bounds on high-frequency force prediction error that grow exponentially with the spectral gap of the crystal graph. This analysis reveals why conventional graph neural networks systematically miss high-frequency forces and provides rigorous mathematical mechanisms—over-smoothing, spectral-gap bottlenecks, pooling-induced filtering, and limited receptive fields—that explain the failure. The resulting bounds and propositions imply concrete architectural modifications, including residual high-frequency pathways and spectral preconditioning, that can mitigate bias while preserving the inductive biases required for periodic materials modeling.

Graph neural networks (GNNs) have transformed the modeling of periodic crystalline materials, enabling rapid prediction of formation energies, band gaps, elastic moduli, and interatomic forces directly from atomic coordinates. Early architectures including CGCNN and MEGNet established that message-passing on crystal graphs could rival density-functional theory (DFT) in speed while maintaining high accuracy for scalar properties. Subsequent advances introduced directional message passing (DimeNet, GemNet), equivariant architectures (E(3)-equivariant networks, MACE), graph attention mechanisms, and multi-scale approaches—each embodying distinct representational choices regarding node-edge features, symmetry handling, and information propagation. This systematic review synthesizes the literature not around incremental performance gains but around the fundamental representational trade-offs and hidden assumptions that govern model behavior in periodic systems. A taxonomy is developed organized by invariance versus equivariance, distance-only versus directional versus angular features, shallow versus deep message passing, and global pooling strategies. Four core trade-offs are analyzed: invariance against equivariance, local against global information capture, model depth against over-smoothing, and computational cost against expressivity. Beyond these, seven rarely articulated hidden assumptions are exposed—cutoff-radius sufficiency, primitive-cell representativeness, distance-only edge sufficiency, global-pooling adequacy, cubic-crystal benchmark bias, DFT-label perfection, and i.i.d. data assumptions—each critically limiting generalization to real-world materials discovery contexts. Empirical comparisons consistently favor newer architectures on standard benchmarks yet rarely isolate the contribution of individual design choices or report failure modes for low-symmetry crystals, extrapolation tasks, or long-range physics. Persistent gaps remain in long-range interaction modeling, uncertainty quantification, extrapolation capability, multi-property trade-offs, and interpretability. By reframing the crystal GNN literature through these lenses, this review offers a structured roadmap for future architecture development that prioritizes robustness and transparency over raw benchmark scores.

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.