Topological Neural Operators: A New Framework for Learning on Cell Complexes

A team of researchers has proposed Topological Neural Operators (TNOs), a framework designed to generalize neural operator learning beyond standard point- or edge-based representations to richer topological domains called cell complexes. The approach encodes data as features on cells of varying dimension and leverages Discrete Exterior Calculus to enable explicit coupling across dimensions via gradient-, curl-, and divergence-type operators. A core design principle is the decoupling of information flow topology—governed by fixed mathematical operators—from the learned transformation functions, which the authors argue better respects the geometric support of physical quantities and exposes conservation laws. The paper also introduces Hierarchical TNOs (HTNOs), which incorporate learned coarse representations of the complex to capture long-range and topology-dependent interactions. The authors demonstrate that TNOs and HTNOs outperform existing neural operators on a range of PDE benchmarks, including irregular-geometry fluid flow problems, and conduct controlled ablation studies to isolate the contributions of higher-rank and topological structure. The framework is presented as a generalization that subsumes existing neural operators as special cases, offering a unified perspective on operator learning across different discretizations.