Topology-Aware Operator Learning

The limitations of existing neural operator frameworks in handling complex geometries and physical interactions are becoming increasingly apparent. Current methods often struggle to capture the inherent topological structure of data, leading to suboptimal performance on tasks involving irregular domains or quantities with conservation laws.

Bridging Discrete and Continuous Physics with Cell Complexes

A significant advancement in operator learning is presented with the introduction of Topological Neural Operators (TNOs). This principled framework extends neural operators beyond simple point or edge functions to handle data represented on cell complexes, which naturally capture features across varying dimensions. By leveraging Discrete Exterior Calculus, TNOs explicitly model interactions between these dimensional cells through gradient-, curl-, and divergence-type operators. This design elegantly decouples the learned transformation of information from the fixed topological operators that govern its flow, ensuring models respect the geometric underpinnings of physical quantities and expose crucial conservation and compatibility structures.

Hierarchical Structures for Long-Range Dependencies

To further enhance the model's capacity for capturing complex dependencies, the researchers introduce Hierarchical TNOs (HTNOs). This extension incorporates learned coarse complexes that facilitate the propagation of long-range and topology-dependent information. The framework is designed to be general, subsuming existing neural operator approaches as a special case and offering a unified perspective on operator learning across diverse discretization schemes. Empirical validation on a range of PDE benchmarks, including challenging irregular-geometry flow problems, demonstrates that TNOs and HTNOs yield improved accuracy. Controlled studies further isolate and confirm the benefits derived from the native incorporation of higher-rank topological structures.