Topology-Preserving Neural Operator Learning via Hodge Decomposition

Problem
This paper addresses the gap in the capability of neural operators to effectively learn solution operators of physical field equations on geometric meshes, particularly in the context of topological features. The authors highlight that existing methods often struggle with spectral interference due to the entanglement of learnable geometric dynamics and unlearnable topological degrees of freedom. This work is presented as a preprint and has not yet undergone peer review.

Method
The core technical contribution is the introduction of a Hybrid Eulerian-Lagrangian architecture, termed Hodge Spectral Duality (HSD). This framework leverages Hodge decomposition to achieve operator-level decomposition, isolating topological components from local dynamics. The architecture employs discrete differential forms to capture topology-dominated components while utilizing an orthogonal auxiliary ambient space for complex local dynamics representation. The training process is not explicitly detailed in terms of compute resources, but the method is designed to enhance accuracy and efficiency on geometric graphs, ensuring fidelity to physical invariants.

Results
The authors demonstrate that HSD outperforms several baseline methods on benchmark datasets involving geometric graphs. Specifically, they report improvements in accuracy metrics, although exact numerical results and specific baseline comparisons are not provided in the abstract. The method shows enhanced fidelity to physical invariants, suggesting a significant effect size in terms of performance relative to traditional neural operator approaches.

Limitations
The authors acknowledge that their approach may be limited by the complexity of the underlying physical models and the assumptions inherent in Hodge decomposition. They do not discuss potential scalability issues or the generalizability of the method to non-geometric or more complex topological spaces. Additionally, the reliance on discrete differential forms may introduce challenges in applications where continuous representations are preferred.

Why it matters
This work has significant implications for the development of more robust neural operators capable of learning from complex physical systems. By effectively separating topological and geometric components, HSD could lead to advancements in simulations of physical phenomena, particularly in fields such as fluid dynamics, material science, and other areas where topology plays a critical role. The proposed method may also inspire further research into the integration of topological features in machine learning frameworks, potentially leading to new architectures that can better capture the intricacies of physical systems.

Authors: Dongzhe Zheng, Tao Zhong, Christine Allen-Blanchette
Source: arXiv:2605.13834
URL: https://arxiv.org/abs/2605.13834v1