Topology-Preserving Neural Operator Learning via Hodge Decomposition

TL;DR

This paper introduces a topology-preserving neural operator learning method using Hodge decomposition, enabling structure-aware approximations for physical field equations on geometric meshes.

Contribution

It develops a novel operator-level decomposition based on Hodge theory, leading to a hybrid architecture with topology-aware inductive bias called Hodge Spectral Duality.

Findings

Achieves superior accuracy on geometric graph problems.

Enhances fidelity to physical invariants.

Provides an efficient, structure-preserving learning framework.

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