Topological Neural Operators
2026-06-08 • Machine Learning
Machine LearningArtificial Intelligence
AI summaryⓘ
The authors propose Topological Neural Operators (TNOs), a new method that learns from data arranged in complex shapes rather than just points or edges. They use mathematical tools from Discrete Exterior Calculus to connect information across different dimensions, like points, lines, and surfaces, in a way that respects the underlying geometry and physics. They also introduce Hierarchical TNOs (HTNOs) to handle long-range connections in complex shapes. Their approach generalizes previous neural operators and shows better accuracy on various physics problems involving partial differential equations (PDEs).
Topological Neural OperatorsNeural OperatorsDiscrete Exterior CalculusCell ComplexesGradientCurlDivergencePartial Differential EquationsHierarchical ModelsOperator Learning
Authors
Lennart Bastian, Samuel Leventhal, Mustafa Hajij, Tolga Birdal
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