Cellular Sheaf Neural Operators for Structure-Preserving Surrogate Modeling of Constrained PDEs
For researchers needing fast, structure-preserving surrogates for constrained multiphysics PDE simulations, this work provides a novel method that enforces geometric and physical constraints via architecture rather than loss penalties.
The paper introduces Cellular Sheaf Neural Operators, a discretization-aware framework for structure-preserving surrogate modeling of constrained PDEs. On turbulent MHD and fusion-equilibrium tasks, it improves structure-sensitive diagnostics such as rollout behavior, divergence control, spectral error, and equilibrium-regression accuracy.
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.