We introduce opological Neural Operators (TNOs), a principled framework for operator learning on cell complexes that lifts neural operators (NOs) from functions on points to 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—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 learned coarse complexes to propagate long-range and topology-dependent information. Our framework subsumes existing NOs as a special case, providing a unified perspective on operator learning across discretizations. Across a range of PDE benchmarks, TNOs and HTNOs achieve improved accuracy while exhibiting stronger physical consistency in structured settings.
AI-generated Podcast (Paper Introduction)
Topological Neural Operators operate on cell complexes (i) whose physical signals are cochains at multiple ranks (ii). A TNO layer (iii) couples ranks through fixed DEC operators (d^k, \delta^k) and learns the rank-wise channel mixing in four blocks, signifying gradient, curl, harmonic, and self maps, producing a predicted PDE field on the complex (iv).
Topological Neural Operators
Lennart Bastian, Samuel Leventhal, Mustafa Hajij, Tolga Birdal
arXiv 2026
@article{bastian2026TNO,
title = {Topological Neural Operators},
author = {Bastian, Lennart and Leventhal, Samuel and Hajij, Mustafa and Birdal, Tolga},
journal = {arXiv preprint arXiv:},
year = {2026}
}Please contact l.bastian@imperial.ac.uk or tbirdal@imperial.ac.uk for any inquiries related to this work.