Technical Summary: Topology-Preserving Neural Operator Learning via Hodge Decomposition

1. Problem Statement

The paper addresses the challenge of learning solution operators for physical field equations (governed by partial differential equations) on general Riemannian manifolds with boundaries and non-trivial topology. While neural operators (e.g., FNO, DeepONet) have achieved success in Euclidean domains using regular grids and spectral transforms, they struggle on complex manifolds (e.g., vehicle surfaces, spherical geometries, organ shapes).

The core difficulty lies in the tension between two distinct structural constraints:

1. Topological Constraints: Global invariants (e.g., circulation, net flux) encoded by the cohomology of the manifold. These are governed by the kernel of the Hodge Laplacian and must be preserved exactly.
2. Geometric/Material Constraints: Local high-frequency dynamics, anisotropic diffusion, and boundary layers governed by the Riemannian metric and material tensors.

Existing approaches fail to simultaneously satisfy these constraints: intrinsic geometric methods (e.g., geodesic convolutions) incur prohibitive overhead and struggle with high frequencies; extrinsic spectral methods ignore cohomological structure; and graph-based methods suffer from over-smoothing and lack explicit encoding of differential complexes.

2. Methodology: Hodge Spectral Duality (HSD)

The authors propose the Hodge Spectral Duality (HSD) framework, a hybrid Eulerian-Lagrangian architecture that decomposes the operator learning problem using Hodge theory and Lie-Trotter operator splitting.

2.1. Theoretical Foundation

The method treats physical fields as discrete differential forms (0-forms on nodes, 1-forms on edges, 2-forms on faces) on a simplicial complex. It leverages the Hodge Decomposition, which orthogonally separates any $k$-form $\omega_k$ into:

• Gradient-type (Exact): $im(d_{k-1})$
• Curl-type (Co-exact): $im(\delta_{k+1})$
• Harmonic-type: $ker(\Delta_k)$, representing global topological invariants.

The authors argue that Hodge orthogonality allows for an additive approximation where topological degrees of freedom (unlearnable via standard gradient descent due to their global nature) are isolated from learnable geometric dynamics.

2.2. Architecture

The HSD operator $G_\theta$ is decomposed into two complementary branches:
$$G_\theta = G_{base, \theta} + G_{fiber, \theta}$$

A. Base Space Branch (Global Topology)

• Domain: Operates in a truncated Hodge spectral domain (low-dimensional subspace spanned by the lowest eigenmodes of the discrete Hodge Laplacian).
• Function: Captures topology-dominated, low-frequency responses and harmonic forms.
• Mechanism:
  • Projects discrete fields to spectral coefficients.
  • Uses pre-projected discrete exterior derivative ($d$) and codifferential ($\delta$) matrices to construct spectral features, preserving differential structure.
  • Employs a gated MLP to model nonlinear mode mixing.
  • Hard Constraint: Explicitly preserves harmonic modes (kernel of the Laplacian) by projecting updates back to the original harmonic coefficients at every layer, ensuring exact conservation of global invariants (e.g., Betti numbers).

B. Fiber Branch (Local Geometry)

• Domain: Operates in an auxiliary Euclidean ambient space (a regular voxel grid).
• Function: Captures metric-dominated, high-frequency local dynamics (e.g., boundary layers, anisotropic diffusion).
• Mechanism:
  • Lift: Maps discrete forms to the ambient grid using Whitney forms and kernel density estimation.
  • Convolution: Applies a standard Fourier Neural Operator (FNO) on the regular grid to learn local correlations efficiently.
  • Pullback: Maps the result back to the simplicial complex.
  • Orthogonal Projection: The output is strictly projected onto the orthogonal complement of the Base space ($I - \Pi_{base}$), ensuring that geometric corrections do not violate global topological constraints.

C. Commutator Correction

Since the topological and geometric operators do not commute ($[A_{Topo}, A_{Geom}] \neq 0$), a lightweight correction module $C_\theta$ is introduced. It takes interaction features (combining spectral derivatives and lifted geometric fields) to approximate the splitting residual, constrained to the fiber space to avoid contaminating the base space.

3. Key Contributions

1. Structure-Aware Framework: A neural operator framework on simplicial complexes that incorporates Discrete Exterior Calculus (DEC) as an algebraic inductive bias, ensuring physically consistent learning.
2. Spectral-Geometric Duality: A dual-branch design that explicitly separates topology-constrained global structure from geometry-driven local dynamics, enabling stable and complementary approximation.
3. Exact Topological Preservation: The use of hard constraints on harmonic modes ensures the exact preservation of cohomological invariants (e.g., circulation, net flux) across the network layers.
4. Efficiency: By decoupling offline spectral decomposition from online learning and utilizing ambient FFTs, the method achieves resolution-independent operator approximation with linear scaling relative to mesh size.

4. Experimental Results

The method was evaluated on three tasks involving complex geometries and topologies:

1. External Aerodynamics: Incompressible flow reconstruction on curved automotive meshes (DrivAerNet++).
2. Magnetostatics: Solving for magnetic fields in multiply-connected domains with spherical shell obstacles.
3. Toroidal Transport: Advection-diffusion dynamics on a torus (non-zero genus).

Performance Highlights:

• Accuracy: HSD achieved the lowest Mean Squared Error (MSE) across all tasks, outperforming baselines like FNO-3D, DeepONet, and MeshGraphNets (MGN). For example, in External Aerodynamics, HSD reduced MSE by 40% compared to FNO-3D.
• Physical Consistency: HSD demonstrated superior Enstrophy Fidelity and Energy Fidelity, indicating better preservation of physical conservation laws and energy distributions.
• Topological Fidelity: In the Magnetostatics task, HSD correctly preserved the number of connected components (level set IoU and $\beta_0$ score), whereas other models often smoothed out topological features or failed to capture global flux components.
• Spectral Bias: Analysis showed HSD effectively captures high-frequency spectral energy, overcoming the "spectral bias" (rapid energy decay) observed in DeepONet and Geo-FNO.
• Efficiency: Despite offline preprocessing, HSD training was significantly faster than MGN (e.g., 56x faster in External Aerodynamics) and comparable to Euclidean FNO-3D.

5. Significance and Claims

The paper claims that Hodge orthogonality fundamentally resolves spectral interference by isolating unlearnable topological degrees of freedom from learnable geometric dynamics. This enables an additive approximation confined to structure-preserving subspaces.

The authors assert that their work provides empirical evidence for an intrinsic Hodge-orthogonal additive structure underlying operator learning on manifolds. By embedding the differential complex $(d, \delta, \Delta_k)$ as architectural inductive biases, HSD addresses the core limitation of current methods: the inability to jointly capture high-frequency geometric details and global conservation laws on complex manifolds without sacrificing efficiency or expressiveness.

Limitations Acknowledged:
The method currently relies on fixed geometry or isometric deformations to amortize the cost of eigendecomposition, limiting it to Eulerian simulations on manifolds of dimension three or less. It also assumes smooth structures, as the ambient mollification may not handle strong discontinuities like shock waves. Future work is suggested to incorporate isospectral deformation or Functional Maps for time-varying topologies.