Topology-Preserving Neural Operator Learning via Hodge Decomposition

2026-05-13

AI Summary

This work addresses the spectral interference arising from the coupling between topological degrees of freedom and geometric dynamics when learning solution operators for physical field equations on geometric meshes. Drawing upon Hodge theory and operator splitting, the authors orthogonally decompose the solution operator into topologically dominant and geometrically dominant subspaces. They introduce a Hodge Spectral Duality (HSD) inductive bias to disentangle, at the function space level, the non-learnable topological components from the learnable geometric dynamics. Building on this decomposition, they propose a structure-preserving hybrid Eulerian–Lagrangian neural network architecture. The resulting method significantly improves solution accuracy and computational efficiency on geometric graphs, better preserves physical invariants, and enhances the fidelity of the learned solutions.

Abstract

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. Our code is available at https://github.com/ContinuumCoder/Hodge-Spectral-Duality

Problem

• topology-preserving
• neural operator learning
• Hodge decomposition
• physical field equations
• geometric meshes

Innovation

• Hodge decomposition
• neural operator learning
• topology-preserving
• Hodge Spectral Duality
• discrete differential forms