[Paper] Topological Neural Operators
Source: arXiv - 2606.09806v1
Overview
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 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 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, including irregular-geometry flow problems, TNOs and HTNOs improve accuracy; controlled studies further isolate the benefits of native higher-rank and topological structure. Project page: https://circle-group.github.io/research/TNO
Key Contributions
This paper presents research in the following areas:
- cs.LG
- cs.AI
Methodology
Please refer to the full paper for detailed methodology.
Practical Implications
This research contributes to the advancement of cs.LG.
Authors
- Lennart Bastian
- Samuel Leventhal
- Mustafa Hajij
- Tolga Birdal
Paper Information
- arXiv ID: 2606.09806v1
- Categories: cs.LG, cs.AI
- Published: June 8, 2026
- PDF: Download PDF