Fundamental Limits of Crystalline Equivariant Graph Neural Networks: A Circuit Complexity Perspective
Yang Cao, Zhao Song, Jiahao Zhang, Jiale Zhao
TL;DR
This work investigates the fundamental expressive limits of equivariant GNNs for crystalline structures through circuit complexity. By formalizing crystalline EGNN computations and embedding them into a concrete circuit class, the authors show that under polynomial precision, embedding width $d=O(n)$, a constant number of layers, and $O(n)$‑width, $O(1)$‑depth MLPs, the EGNN family can be realized by a uniform $\mathsf{TC}^0$ circuit of polynomial size, with an explicit per‑layer depth bound. This provides a rigorous ceiling on the decision and prediction power of such models under realistic hardware constraints and clarifies how to surpass this regime (e.g., deeper networks or richer geometric primitives). The results complement WL‑style analyses by offering a complexity‑theoretic foundation for symmetry‑aware learning in crystals and guide architectural choices for scalable, physically informed graph models. Practically, the work informs which architectural changes are necessary to transcend the $\mathsf{TC}^0$ barrier and what remains provably in reach for standard EGNN backbones in crystalline contexts.
Abstract
Graph neural networks (GNNs) have become a core paradigm for learning on relational data. In materials science, equivariant GNNs (EGNNs) have emerged as a compelling backbone for crystalline-structure prediction, owing to their ability to respect Euclidean symmetries and periodic boundary conditions. Despite strong empirical performance, their expressive power in periodic, symmetry-constrained settings remains poorly understood. This work characterizes the intrinsic computational and expressive limits of EGNNs for crystalline-structure prediction through a circuit-complexity lens. We analyze the computations carried out by EGNN layers acting on node features, atomic coordinates, and lattice matrices, and prove that, under polynomial precision, embedding width $d=O(n)$ for $n$ nodes, $O(1)$ layers, and $O(1)$-depth, $O(n)$-width MLP instantiations of the message/update/readout maps, these models admit a simulation by a uniform $\mathsf{TC}^0$ threshold-circuit family of polynomial size (with an explicit constant-depth bound). Situating EGNNs within $\mathsf{TC}^0$ provides a concrete ceiling on the decision and prediction problems solvable by such architectures under realistic resource constraints and clarifies which architectural modifications (e.g., increased depth, richer geometric primitives, or wider layers) are required to transcend this regime. The analysis complements Weisfeiler-Lehman style results that do not directly transfer to periodic crystals, and offers a complexity-theoretic foundation for symmetry-aware graph learning on crystalline systems.