Gauge-Equivariant Graph Neural Networks for Lattice Gauge Theories

Local gauge structures play a central role in a wide range of condensed matter systems and synthetic quantum platforms, where they emerge as effective descriptions of strongly correlated phases and engineered dynamics. We introduce a gauge-invariant graph neural network (GNN) architecture for Abelian lattice gauge models, in which symmetry is enforced explicitly through local gauge-invariant inputs, such as Wilson loops, and preserved throughout message passing, eliminating redundant gauge degrees of freedom while retaining expressive power. We benchmark the approach on both $\mathbb{Z}_2$ and $\mathrm{U}(1)$ lattice gauge models, achieving accurate predictions of global observables and spatially resolved quantities despite the nonlocal correlations induced by gauge-matter coupling. We further demonstrate that the learned model serves as an efficient surrogate for semiclassical dynamics in $\mathrm{U}(1)$ quantum link models, enabling stable and scalable time evolution without repeated fermionic diagonalization, while faithfully reproducing both local dynamics and statistical correlations. These results establish gauge-invariant message passing as a compact and physically grounded framework for learning and simulating Abelian lattice gauge systems.

Equivariant network architectures are a well-established tool for predicting invariant or equivariant quantities. However, almost all learning problems considered in this context feature a global symmetry, i.e. each point of the underlying space is transformed with the same group element, as opposed to a local ``gauge'' symmetry, where each point is transformed with a different group element, exponentially enlarging the size of the symmetry group. Gauge equivariant networks have so far mainly been applied to problems in quantum chromodynamics. Here, we introduce a novel application domain for gauge-equivariant networks in the theory of topological condensed matter physics. We use gauge equivariant networks to predict topological invariants (Chern numbers) of multiband topological insulators. The gauge symmetry of the network guarantees that the predicted quantity is a topological invariant. We introduce a novel gauge equivariant normalization layer to stabilize the training and prove a universal approximation theorem for our setup. We train on samples with trivial Chern number only but show that our models generalize to samples with non-trivial Chern number. We provide various ablations of our setup. Our code is available at https://github.com/sitronsea/GENet/tree/main.

Symmetry is everywhere in nature and society. Geometric deep learning exploits symmetries in data to improve the performance and efficiency of deep learning systems. In this paper, we extend geometric deep learning to utilize richer symmetry structures. Specifically, we develop order-equivariant neural networks (OENN), which generalize standard graph message passing and sheaf neural networks via the theory of equivariant bundles over face posets (face categories). We (i) characterize all linear order-equivariant maps, (ii) build OENN layers, and (iii) prove universal approximation theorems (UATs) for continuous order-equivariant maps, which are new results even when restricted to sheaf neural networks (for which no UAT was known before). We illustrate the framework on graph and sheaf models. Our results can also be seen as extending the known UAT for graph neural networks to a more general setting that subsumes sheaf neural networks as well. In addition, we show that OENN can be extended further to CENN, Category-Equivariant Neural Network, which gives the general form of equivariant neural networks as well as of equivariant universal approximation theorems, allowing us to leverage categorical symmetry in data (e.g., non-invertible symmetries on multiple objects with compositional relations on those symmetries).