Topology-Preserving Neural Operator Learning via Hodge Decomposition

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.

Many scientific and engineering problems require predicting physical fields, such as air flow, magnetic fields, or heat, on complex shapes. A challenge is that these fields are not arbitrary: they must obey conservation rules imposed by the shape itself, including constraints associated with holes, boundaries, and closed surfaces. Our work develops a machine-learning model that represents physical fields in a way that respects these structural constraints. It separates components that are fixed by topology from components that depend on geometry, material properties, and boundary conditions. This leads to predictions that are more accurate and more physically consistent on tasks involving vehicle aerodynamics, magnetostatic fields, and transport on curved surfaces. The broader goal is to make learned simulation tools more reliable for complex real-world geometries.