Cellular Sheaf Neural Operators for Structure-Preserving Surrogate Modeling of Constrained PDEs

• URL: http://arxiv.org/abs/2606.00937v1
• Date: Sun, 31 May 2026 00:49:25 GMT
• Title: Cellular Sheaf Neural Operators for Structure-Preserving Surrogate Modeling of Constrained PDEs
• Abstract summary: We propose a discretization-aware framework for structure-preserving neural PDE surrogates.<n>We show that cellular-sheaf structure is a useful inductive bias for neural PDE surrogates in constrained multiphysics systems.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Neural operators provide fast surrogate models for PDE simulations, but standard architectures often treat geometry and discretization as secondary to field data. Physical states are usually represented as grid-channel stacks, even when different quantities naturally belong on vertices, edges, faces, cells, boundaries, or interfaces and must satisfy compatibility constraints. We propose Cellular Sheaf Neural Operators, a discretization-aware framework for structure-preserving neural PDE surrogates. The method represents PDE states on oriented cell complexes, couples local feature spaces through learned restriction maps, and uses incidence/Hodge-informed message passing to follow computational geometry. Learned update heads pass through coboundary or flux maps, allowing selected constraints to arise from cell-complex structure rather than only from loss penalties. For magnetohydrodynamics, this yields face-based magnetic-flux updates driven by edge electromotive fields and finite-volume-style fluid updates driven by learned face fluxes and cell sources. On turbulent MHD and fusion-equilibrium surrogate tasks, the method improves structure-sensitive diagnostics, including rollout behavior, divergence control, spectral error, and equilibrium-regression accuracy. These results indicate that cellular-sheaf structure is a useful inductive bias for neural PDE surrogates in constrained multiphysics systems.

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