Topological Neural Operators

We introduce **Topological Neural Operators** (TNOs), a principled framework for operator learning on cell complexes that lifts neural operators (NOs) from functions on points and/or edges to discrete topological domains. TNOs represent data as features defined on cells of varying dimension and model their interactions through *Discrete Exterior Calculus*, enabling explicit cross-dimensional coupling via gradient-, curl-, and divergence-type operators. The key design principle is to decouple *where* information flows, as governed by fixed topological operators, from *how* it is transformed (which is learned), yielding models that respect the geometric support of physical quantities and expose conservation and compatibility structure. We further propose **Hierarchical TNOs** (HTNOs), which incorporate coarse complexes to propagate long-range and topology-dependent information. Our framework recovers standard point- and graph-based NOs as restricted cases, providing a unified perspective on operator learning across discretizations. Across a range of PDE benchmarks, including problems on irregular geometries, TNOs and HTNOs improve accuracy and physical residuals. Controlled studies further isolate the benefits of incorporating native higher-rank and topological structure.