The limitations of existing neural operator frameworks in handling complex geometries and physical interactions are becoming increasingly apparent. Current methods often have difficulty capturing the inherent topological structure of the data, resulting in suboptimal performance for tasks involving irregular regions or quantities with conservation laws.
Visual TL;DR. The struggle of existing neural operators leads to the capture of topological structures. Capturing the topological structure leads to the introduction of TNO. The introduction of TNO leads to the exploitation of discrete external calculus. The exploitation of discrete external calculus leads to the integration of discrete/continuous physics. Utilizing discrete external calculus can improve the accuracy of partial differential equations. The introduction of TNO leads to the handling of long-distance dependencies.
Capturing topological structure: suboptimal performance in irregular domains or conservation laws
Introducing TNO: Topological Neural Operators for Cellular Complexes
Leverage discrete external calculus: Explicitly model interactions between dimensional cells
Discrete/Continuous Physics Integration: Naturally Bridging Physics Across Dimensions
Improving PDE accuracy: Improving benchmark accuracy in complex domains
Handling long-term dependencies: Hierarchical structures to improve information flow
Visual TL;DR
Bridging discrete and continuous physics using cell complexes
The introduction of topological neural operators (TNOs) has led to significant advances in operator learning. A framework based on this principle extends neural operators beyond simple point or edge functions to process data represented in cellular complexes and naturally capture features across different dimensions. By leveraging discrete external calculus, TNO explicitly models interactions between these dimensional cells through gradient, curl, and divergence type operators. This design elegantly decouples the learned transformation of information from the fixed topology operators that control its flow, ensuring that the model respects the geometric basis of physical quantities and reveals important conserved and compatible structures.
