Topological Neural Operators

2026-06-08

AI Summary

Modeling cross-dimensional physical quantities and their conservation laws on unstructured meshes with complex geometries remains highly challenging. This work proposes Topological Neural Operators (TNOs), which generalize neural operators to cell complexes for the first time by integrating discrete exterior calculus to explicitly model couplings among cells of varying dimensions through learnable gradient, curl, and divergence operators. TNOs decouple information propagation pathways from transformation mechanisms and incorporate hierarchically coarsened complexes to capture long-range dependencies and global topological structure. Experiments demonstrate that the method significantly outperforms existing approaches across multiple PDE benchmarks, including fluid dynamics problems on irregular geometries, thereby validating the efficacy of high-rank representations and native topological modeling.

Abstract

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

Problem

• operator learning
• topological domains
• cell complexes
• Discrete Exterior Calculus
• cross-dimensional coupling

Innovation

• Topological Neural Operators
• Discrete Exterior Calculus
• Cell Complexes
• Operator Learning
• Hierarchical TNOs