Hodge Decomposition Enhances Neural Operator Learning on Geometric Meshes

A recent paper published on arXiv (2605.13834) presents a novel neural operator learning approach that preserves topology through Hodge decomposition. The researchers demonstrate that Hodge orthogonality effectively addresses spectral interference by distinguishing between unlearnable topological degrees of freedom and learnable geometric dynamics. They establish a systematic operator-level decomposition grounded in Hodge theory and operator splitting, leading to the creation of a Hybrid Eulerian-Lagrangian architecture featuring an algebraic inductive bias termed Hodge Spectral Duality (HSD). This framework employs discrete differential forms for topology-centric components and utilizes an orthogonal auxiliary ambient space to manage intricate local dynamics. The method shows improved accuracy and efficiency on geometric graphs while maintaining a strong fidelity to physical invariants. The source code is accessible via the provided GitHub repository.