@inproceedings{bibcite_111,
  author = {Dongzhe Zheng and Tao Zhong and Christine Allen-Blanchette},
  title = {Topology-Preserving Neural Operator Learning via Hodge Decomposition},
  abstract = {<p>In this paper, we study solution operators of physical field equations on geometric meshes from a function-space perspective. We reveal that Hodge orthogonality fundamentally resolves spectral interference by isolating unlearnable topological degrees of freedom from learnable geometric dynamics, enabling an additive approximation confined to structure-preserving subspaces. Building on Hodge theory and operator splitting, we derive a principled operator-level decomposition. The result is a Hybrid Eulerian-Lagrangian architecture with an algebraic-level inductive bias we call Hodge Spectral Duality (HSD). In our framework, we use discrete differential forms to capture topology-dominated components and an orthogonal auxiliary ambient space to represent complex local dynamics. Our method achieves superior accuracy and efficiency on geometric graphs with enhanced fidelity to physical invariants.</p>},
  year = {2026},
  journal = {International Conference on Machine Learning (ICML)},
}