Simplicial Sheaf Diffusion: Learned Anisotropic Transport on 2-Complexes

Simplicial neural networks propagate through Hodge Laplacians whose coupling between coincident simplices is a fixed incidence sign, so the transport they learn is isotropic. Sheaf neural networks learn a map per edge but operate only on graphs. We introduce **Simplicial Sheaf Diffusion (SSD)**, which diffuses node, edge and face cochains jointly through the Hodge Laplacians of a cellular sheaf whose restriction maps are learned from the features and recomputed at every layer. Placing one map on each vertex, one on each triangle and, optionally, a unit-determinant gauge on each edge makes successive coboundaries compose to zero for every value the learner produces, so the operator is the Hodge Laplacian of a sheaf at every layer and its equilibria are the sheaf cohomology. We prove permutation and orientation equivariance for this learned operator and an exact per-coordinate formula for its harmonic dimension, which the normalization we use preserves. In one shared harness with ten neural baselines across five benchmarks, SSD has the highest accuracy on SHREC mesh classification and the lowest error among the neural models on a real ocean-current complex, where removing its triangles raises its error to that of the zero predictor; on the remaining three benchmarks it has the highest mean or is level with the best model. A seed-paired ablation on the two trajectory sets attributes $13.5$ and $8.4$ points of accuracy to the learned maps. Under a random node relabeling the trained model's outputs change only at floating-point noise on every benchmark; on PROTEINS, where we run the same test on the baselines, four simplicial models' outputs do not.