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

arXiv: https://arxiv.org/abs/2606.00937 · HTML: https://mlsanity.com/paper/2606.00937/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 31 of its 64 references that are in the map.
- First posted on arXiv 31 May 2026.

## Builds on (most cited first, 30 of 31)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [PDEBENCH: An Extensive Benchmark for Scientific Machine Learning](https://mlsanity.com/paper/2210.07182/) (2022)
- [Graph networks as learnable physics engines for inference and control](https://mlsanity.com/paper/1806.01242/) (2018)
- [Neural Sheaf Diffusion: A Topological Perspective on Heterophily and Oversmoothing in GNNs](https://mlsanity.com/paper/2202.04579/) (2022)
- [The Well: a Large-Scale Collection of Diverse Physics Simulations for Machine Learning](https://mlsanity.com/paper/2412.00568/) (2024)
- [Simplicial Neural Networks](https://mlsanity.com/paper/2010.03633/) (2020)
- [Sheaf Neural Networks](https://mlsanity.com/paper/2012.06333/) (2020)
- [Signal Processing on Higher-Order Networks: Livin' on the Edge ... and Beyond](https://mlsanity.com/paper/2101.05510/) (2021)
- [Principled Simplicial Neural Networks for Trajectory Prediction](https://mlsanity.com/paper/2102.10058/) (2021)
- [Cell Complex Neural Networks](https://mlsanity.com/paper/2010.00743/) (2020)
- [Position: Topological Deep Learning is the New Frontier for Relational Learning](https://mlsanity.com/paper/2402.08871/) (2024)
- [Sheaf Neural Networks with Connection Laplacians](https://mlsanity.com/paper/2206.08702/) (2022)
- [Towards Stability of Autoregressive Neural Operators](https://mlsanity.com/paper/2306.10619/) (2023)
- [HodgeNet: Graph Neural Networks for Edge Data](https://mlsanity.com/paper/1912.02354/) (2019)
- [Simplicial Representation Learning with Neural k-Forms](https://mlsanity.com/paper/2312.08515/) (2023)
- [The Sample Complexity of Learning Lipschitz Operators with respect to Gaussian Measures](https://mlsanity.com/paper/2410.23440/) (2024)
- [Structure-Preserving Digital Twins via Conditional Neural Whitney Forms](https://mlsanity.com/paper/2508.06981/) (2025)
- [Structure-Preserving Learning Improves Geometry Generalization in Neural PDEs](https://mlsanity.com/paper/2602.02788/) (2026)
- [Diagnosing Failure Modes of Neural Operators Across Diverse PDE Families](https://mlsanity.com/paper/2601.11428/) (2026)
- [One Operator to Rule Them All? On Boundary-Indexed Operator Families in Neural PDE Solvers](https://mlsanity.com/paper/2603.01406/) (2026)
- [E(n) Equivariant Message Passing Cellular Networks](https://mlsanity.com/paper/2406.03145/) (2024)

## Abstract (arXiv)

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.