{"ID":2922222,"CreatedAt":"2026-06-02T02:42:49.606572591Z","UpdatedAt":"2026-06-03T00:47:32.987482086Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2606.00937","arxiv_id":"2606.00937","title":"Cellular Sheaf Neural Operators for Structure-Preserving Surrogate Modeling of Constrained PDEs","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.","short_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,...","url_abs":"https://arxiv.org/abs/2606.00937","url_pdf":"https://arxiv.org/pdf/2606.00937v1","authors":"[\"Lennon J. Shikhman\",\"Shane Gilbertie\"]","published":"2026-05-31T00:49:25Z","proceeding":"cs.LG","tasks":"[\"cs.LG\",\"cs.CE\",\"math.NA\",\"physics.comp-ph\",\"physics.plasm-ph\"]","methods":"[]","has_code":false}