Title: Structure-Preserving Surrogate Modeling of Constrained PDEs via Cellular Sheaf Neural Operators

Abstract

While neural operators serve as efficient surrogate models for solving partial differential equations (PDEs), conventional architectures frequently prioritize field data over geometry and discretization details. Typically, physical states are encoded as grid-channel stacks, despite the fact that distinct quantities often reside on specific geometric entities—such as vertices, edges, faces, cells, boundaries, or interfaces—and must adhere to strict compatibility constraints. To address this, we introduce Cellular Sheaf Neural Operators, a discretization-aware framework designed for structure-preserving neural PDE surrogates.

This approach models PDE states on oriented cell complexes, linking local feature spaces via learned restriction maps. By employing incidence and Hodge-informed message passing, the method aligns with computational geometry principles. Furthermore, learned update heads operate through coboundary or flux maps, enabling certain constraints to emerge directly from the cell-complex topology rather than relying solely on loss-based penalties. In the context of magnetohydrodynamics (MHD), this results in face-based magnetic-flux updates driven by edge electromotive fields, alongside finite-volume-style fluid updates governed by learned face fluxes and cell sources.

Evaluations on tasks involving turbulent MHD and fusion-equilibrium surrogates demonstrate that the method enhances structure-sensitive diagnostics. Improvements are observed in rollout behavior, divergence control, spectral error metrics, and equilibrium-regression accuracy. These findings suggest that cellular-sheaf structures provide a valuable inductive bias for neural PDE surrogates within constrained multiphysics systems.

Source: arXiv Generated at: 2026-06-02 00:00:00 UTC