Topology-Preserving Neural Operator Learning via Hodge Decomposition

Dongzhe Zheng; Tao Zhong; Christine Allen-Blanchette

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Topology-Preserving Neural Operator Learning via Hodge Decomposition

Dongzhe Zheng, Tao Zhong, Christine Allen-Blanchette

arXiv:2605.13834v2May 13, 2026cs.LGcs.AIcs.CG

TL;DR

Neural operators must jointly handle topological constraints and local geometric dynamics on physical fields defined over complex manifolds. This paper introduces Hodge Spectral Duality, which separates these components through Hodge-based operator decomposition and reports improved accuracy, efficiency, and physical-invariant fidelity on geometric graphs.

Problem
Neural operator learning on manifolds must reconcile global topological constraints with local geometric dynamics in fields defined on curved, bounded, non-trivial geometries.
Method
HSD uses discrete exterior calculus and a dual-branch operator architecture to separate topology-constrained structure from geometry-driven local dynamics.
Results
MSE reached 1.08 × 10−2, a 40% reduction versus FNO-3D, with enstrophy fidelity of 0.7658 and spectral fidelity of 0.8423.
Takeaways & Limitations
Hodge orthogonality provides an additive approximation in which geometry-driven dynamics complement topological structure while completing spectral energy.
Takeaways & Limitations
The method currently requires fixed or mildly perturbed geometry for offline eigendecomposition and is suited to Eulerian simulations on manifolds of dimension three or less.

Abstract

from arXiv · show

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

1. Introduction

The paper frames neural operator learning on manifolds as the joint treatment of topology-preserving structure and geometry- and material-dependent dynamics. It proposes Hodge Spectral Duality (HSD), a dual-branch, operator-splitting framework whose Hodge-orthogonal components enable additive approximation.

Problem Background: Physical fields on bounded Riemannian manifolds are represented as differential forms, while PDE solving becomes reusable operator learning across meshes and geometries.The exterior derivative, codifferential, and Hodge star characterize differential operators used to formulate the PDEs.
Research Problem and Challenges: Existing neural operators exploit regular grids and fast spectral transforms, but manifold extensions face geometry-adaptive overhead, high-frequency limitations, and topological blind spots.The paper identifies embedding the differential complex (d, δ, ∆k) as an efficient architectural inductive bias as an open challenge.
Research Problem and Challenges: Manifold problems require simultaneously preserving global topological structure and resolving local geometric dynamics, creating the method’s central design trade-off.Topology encodes global invariants, whereas metric and material properties govern high-frequency dynamics, anisotropy, and fine-scale structures.
Overview of This Work: HSD transforms PDE solving on oriented simplicial complexes into structured learning on higher-order graphs using a dual-branch architecture coupled by Lie–Trotter type operator splitting.Its framework incorporates discrete exterior calculus as an architectural component for higher-order graph representations.
Overview of This Work: Hodge orthogonality gives manifold operator learning an additive approximation property, allowing geometry-driven dynamics to complement topological structure and complete spectral energy.This is the paper’s stated result concerning the interaction between the decomposed components.

2. Related Work

Related work spans local graph and geometric methods, neural operators on manifolds, and structure-preserving frameworks based on discrete exterior calculus and topological deep learning. These approaches respectively address local PDE coupling, function-space mappings, and algebraic or cohomological structure, while facing limitations in global topology or intrinsic geometric preservation.

Local Methods Based on Graph and Geometric Deep Learning: Graph-based methods use message passing or gauge-equivariant convolution to approximate PDE-induced local coupling on meshes.
Local Methods Based on Graph and Geometric Deep Learning: Local aggregation suffers from oversmoothing and over-squashing when modeling long-range dependencies governed by the Hodge Laplacian kernel.
Neural Operators and Spectral Methods on Manifolds: FNO and DeepONet learn function-space mappings successfully on Euclidean domains, but manifold extensions can struggle to preserve intrinsic metrics and flux conservation discretely.
Higher-Order Graphs, Discrete Exterior Calculus, and Topological Deep Learning: Discrete exterior calculus and finite element exterior calculus provide rigorous frameworks for preserving algebraic and cohomological structure on simplicial complexes.

3. Method: Hodge Spectral Duality Operator

HSD decomposes discrete differential-form fields into topology-dominated harmonic modes and metric-dependent exact/coexact modes, then models them with complementary spectral and ambient branches. Orthogonal projection constrains geometric corrections to the complementary subspace, preserving global topological invariants.

Field Representation: HSD represents physical fields as discrete differential forms: 0-forms on nodes, 1-forms on edges, and 2-forms on faces.These represent scalar potentials, flows, and fluxes, respectively.
Hodge Decomposition: Hodge–de Rham decomposition separates harmonic cohomological degrees of freedom from exact and coexact metric-dependent local variation.Harmonic channels encode conserved circulation and flux, while exact and coexact channels capture local dynamics.
Base Space Branch: The Base branch learns topology-dominated low-frequency responses in a truncated Hodge spectral domain with resolution-independent coefficient dimension.It preserves harmonic and low-frequency non-harmonic modes and provides a topologically consistent anchor for the Fiber branch.
Ambient Fiber Branch: The Fiber branch lifts cochains to an auxiliary Euclidean grid and uses FNO spectral convolution to model metric-related high-frequency correlations.This fixed-grid convolution handles local geometric details while avoiding costly anisotropic manifold convolutions.
Structure Preservation: Orthogonal projection restricts Fiber outputs to the Base complement, eliminating low-frequency artifacts and conservation-violating modes during cross-scale evolution.The constraint ensures geometric corrections affect only high-frequency, metric-dominated degrees of freedom and preserve global topological invariance.

4. Experiments

Experiments evaluate HSD across geometric complexity, topological connectivity, and dynamic evolution using accuracy, conservation, spectral, and topological metrics. HSD improves physical fidelity, high-frequency representation, efficiency, component robustness, and remeshing generalization across these tasks.

Experimental Setup: HSD is evaluated on flow reconstruction, magnetostatic field solving in multiply-connected domains, and transport with periodic topology using multidimensional physical and topological metrics.The evaluation includes accuracy, physical conservation, spectral fidelity, and topological consistency measures.
External Aerodynamics: 1.08 × 10−2 MSE represents a 40% reduction vs FNO-3D in External Aerodynamics, alongside Enstrophy Fidelity (0.7658) and Spectral Fidelity (0.8423).HSD avoids over-smoothing and captures high-frequency vortex features on complex industrial geometries.
Magnetostatics: 1.84 × 10−4 MSE represents a 36% reduction vs DeepONet in Magnetostatics, while Enstrophy Fidelity reaches 0.9444.HSD preserves local flux concentration structures, energy distribution, and topological features despite DeepONet’s lower MSE and optimal spectral fidelity.
Transport Processes: 3.56 × 10−4 MSE represents a 36% reduction vs FNO-3D in transport, with Energy Fidelity (0.6968 vs 0.6365) and β0 Score (0.7829 vs 0.6721).These results indicate preservation of energy dissipation and topological structure during long-time integration without non-physical artifacts.
Spectral Analysis: HSD’s predicted energy spectrum closely matches ground truth at high frequencies, while its Fiber branch compensates the Base branch’s high-frequency deficiencies.DeepONet, Geo-FNO, and GNO exhibit rapid energy decay with increasing eigenfrequency, indicating high-frequency filtering.
Efficiency: HSD’s total training time is 56× faster than MGN in External Aerodynamics (33s vs 1865s) and only 5% of MGN in Magnetostatics (215s vs 3983s).Online complexity is O(Nk) for spectral projection and O(N log N) for FFT, versus O(N|E|) for MGN message passing.
Ablation and Generalization: Removing orthogonal projection introduces non-physical low-frequency noise, removing the commutator MLP most harms multiply-connected domains, and remeshing error varies by at most 30%.When External Aerodynamics mesh density increases from 3000 to 7000 nodes, all baselines show at least a 10× larger error amplification; increasing spectral modes yields diminishing returns.

5. Discussion, Limitations and Scope · A. Notation Reference Table and A Primer on Algebraic Topology

The method is currently scoped to low-dimensional Eulerian simulations because sparse Hodge-Laplacian spectral decomposition benefits from reusable geometry. The appendix standardizes DEC and algebraic-topology concepts through precise linear-algebraic definitions and introduces the base–fiber representation used by HSD.

5. Discussion, Limitations and Scope: Sparse spectral decomposition requires fixed, isomorphic/isometric, or mildly perturbed geometry to amortize eigendecomposition costs offline.Per-step mesh-topology reconstruction remains outside the current scope; future work is proposed around iso-spectral deformation or Functional Maps theory.
5. Discussion, Limitations and Scope: The framework is currently suited to Eulerian-perspective simulations on manifolds of dimension three or less.
A. Notation Reference Table and A Primer on Algebraic Topology: The notation defines continuous manifolds, discrete simplicial complexes, discrete differential forms, Hodge operators, and the discrete Hodge Laplacian.The table identifies Ck(K, R) as the space of k-th order cochains and Lk = dk−1δk + δk+1dk as the k-th order discrete Hodge Laplacian.
A. Notation Reference Table and A Primer on Algebraic Topology: The spectral notation includes Betti numbers, harmonic modes, eigenvectors, eigenvalues, and eigendecomposition of the discrete Hodge Laplacian.The k-th Betti number is defined as the dimension of the harmonic k-form space, while LkΨk = ΨkΛk denotes eigendecomposition.
A. Notation Reference Table and A Primer on Algebraic Topology: The field representation separates each discrete form into a base subspace containing harmonic and low-frequency modes and a fiber component containing high-frequency, metric-dominated content.The base component is associated with span(Φk), and the fiber component is obtained through the complementary projection.
A. Notation Reference Table and A Primer on Algebraic Topology: The appendix defines topology- and geometry-dominated operator components, their base and fiber solution operators, and the network quantities used by the spectral and ambient branches.It also specifies lifting and pullback between discrete forms and an auxiliary Euclidean domain, along with ambient Fourier operators and geometric residual fields.
A. Notation Reference Table and A Primer on Algebraic Topology: The notation primer presents DEC and algebraic topology using standard linear-algebra and graph-signal-processing terminology, prioritizing exact mathematical definitions over heuristic metaphors.

A.1. Data Representation: From Node Signals to Cochains … B.4. Riemannian Geometry and Tangent Bundle

The framework represents physical fields as domain-typed discrete differential forms and separates topology-preserving structure from metric-dependent local dynamics. Its algebraic operators, Hodge decomposition, spectral basis, and tangent-bundle formulation provide the foundations for Hodge Spectral Duality.

A.1. Data Representation: From Node Signals to Cochains: Physical fields are represented as discrete differential forms bound to their integration domains, so velocities use oriented edges rather than vertex features.0-forms represent vertex scalars, 1-forms oriented-edge quantities, and 2-forms oriented-face quantities; edge orientation negates the value.
A.2. The Algebraic Structure: Boundary and Coboundary: Boundary and coboundary operators encode simplex connectivity and discrete gradient or curl actions, while d2 = 0 guarantees curl-of-gradient and divergence-of-curl identities algebraically.The boundary matrix B1 is an oriented edge-to-vertex incidence matrix with entries in {0, 1, −1}.
A.3. The Generalized Laplacian and Hodge Decomposition: The Hodge Laplacian supplies a spectral basis in which discrete fields decompose into orthogonal irrotational, solenoidal, and harmonic components.The decomposition separates gradient-flow, divergence-free, and harmonic subspaces.
A.4. Betti Numbers: Topological Invariants as Null Spaces: Betti numbers equal Hodge-Laplacian null-space dimensions, and explicit projection onto harmonic subspaces preserves global circulation and cavity-related invariants.b0 counts connected components, b1 independent non-contractible loops, and b2 enclosed voids.
B. Mathematical Foundations of Discrete Exterior Calculus and Tangent Bundle: The mathematical foundation models a compact oriented Riemannian manifold with boundary using an oriented simplicial complex whose k-dimensional cochains store integral quantities on k-simplices.Vertices, edges, and faces correspond to 0-, 1-, and 2-dimensional simplices, respectively.
B.2. Discrete Exterior Derivative, Hodge Star, and Codifferential: Discrete exterior derivatives, Hodge stars, and codifferentials define metric-consistent differential operators and a symmetric positive semidefinite generalized Laplacian.d0 acts as a discrete gradient, d1 as a curl-type operator, and δ1 as discrete divergence.
B.3. Hodge–de Rham Decomposition and Hodge Spectrum: Hodge spectral eigenvectors with zero eigenvalues span the harmonic subspace, while smaller nonzero eigenvalues represent large-scale modes and larger ones localized high-frequency modes.A truncated spectral subspace retains all harmonic modes and selected low-frequency modes; truncation error is mainly high-frequency and can be corrected locally.
B.4. Riemannian Geometry and Tangent Bundle: Tangent-bundle coordinates represent metric and material effects, enabling the base branch to encode topology through d and δ while the fiber branch models local metric-dependent dynamics.Physical operators depend on local metrics, curvature, and material tensors; exterior-derivative topology is metric-independent.

C. Spectral Operator Theory and Subspace Derivation … D.3. Total Error Decomposition and Adaptive Correction

The paper derives Hodge-based spectral subspaces and operator splits that preserve topological structure while modeling geometric dynamics. It then establishes embedding, resolution, stability, and total-error conditions, with adaptive correction mechanisms for discretization and aliasing artifacts.

C. Spectral Operator Theory and Subspace Derivation: The appendix formalizes truncated spectral subspaces, orthogonal projections, spectral derivative matrices, and harmonic projections using discrete exterior calculus and Hodge–de Rham structure.
C.1. Truncated Hodge Spectral Basis and Subspace Decomposition: The truncated Hodge spectral space contains all harmonic modes and selected low-frequency modes, while its Hodge-orthogonal complement concentrates remaining high-frequency degrees of freedom.
C.2. Splitting of Topological and Geometric Operators: The control operator is split into topology-dominated terms generated by exterior derivatives, codifferentials, and the Hodge–de Rham Laplacian, plus geometry-dominated diffusion, advection, and source terms.
C.3. Spectral Derivative Matrices and Harmonic Projection: Projected discrete exterior derivatives and codifferentials preserve the differential-complex structure approximately, while harmonic hard constraints exactly preserve corresponding cohomology classes and global conserved quantities.
D. Consistency, Stability, and Resolution Requirements of Ambient Fiber Embedding: The ambient embedding analysis evaluates lifting, spectral convolution, and pullback against the manifold-intrinsic operator, including effects from Whitney derivative discontinuities and anisotropic boundary-layer resolution.
D.1. Geometric Assumptions and Discretization Setup: The embedding requires 0 < ϵ < cτM and haux ≤ min(ϵ/2, c0δres/4), while metric-adaptive kernels use separate normal and tangential bandwidths to mitigate anisotropic-mesh aliasing.
D.2. Stability and Consistency of Whitney Extension: Whitney mollification regularizes interface-induced Dirac-type singularities, controls the gradient norm with C(γ), and balances ϵ bias against h/ϵ roughness, giving the optimal scale ϵ ≍ h1/2.
D.3. Total Error Decomposition and Adaptive Correction: The total error decomposes into geometric error Egeom ≍ ϵ + h/ϵ, voxelization error Evox ≍ haux, and spectral truncation error Espec; projection, learned gating, and finite bandwidth suppress invalid or aliased components.

D.4. Efficiency and Robustness in R3 Embeddings … E.2. Spectral Encoding as Derivative Proxy for Fields

The framework combines efficient, topology-independent R3 embedding with boundary regularization that controls Gibbs artifacts, then justifies a commutator corrector whose derivative information is encoded by spectral interaction features.

D.4. Efficiency and Robustness in R3 Embeddings: Ambient FNO targets R3 physical objects, using a controlled convolution kernel in the lift operator ι to enforce band-limited inputs and avoid severe aliasing.The passage states that efficiency gains outweigh interpolation-related accuracy loss in these scenarios.
D.4. Efficiency and Robustness in R3 Embeddings: Splatting discrete simplices onto the background grid uses local additive operations with complexity O(Nsimplex × K3), independent of manifold curvature or topological complexity.K denotes the kernel width, and the complexity is linear in the number of elements.
D.5. Boundary Regularity and Suppression of Gibbs Artifacts: Direct Fourier transformation of binarized embedded fields causes discontinuities that degrade high-frequency coefficient decay from O(|k|−p) to O(|k|−1), producing spatial ringing.The ringing can confuse high-frequency physical content.
D.5. Boundary Regularity and Suppression of Gibbs Artifacts: Soft masks based on signed distance fields and boundary-consistent extensions restore at least C1 continuity, while the error bound decomposes total error into geometric, voxel, cutoff, and wrap terms.The bound is Etot ≤C [Egeom(ϵn, h) + Evox(haux) + Ecut(r, ξcut) + Ewrap(Lpml)].
E. Theoretical Justification of the Commutator Corrector: Within Hodge Spectral Duality, the interaction features z(ℓ) constitute the complete derivative proxy required by the commutator correction.The section first derives the analytical commutator form before establishing this proxy property.
E.1. Analytical Form of the Commutator: The commutator residual is driven by higher-order derivatives of geometric parameters, ∆κ, and coupling between parameter and field gradients, ∇κ · ∇u.The general correction depends on zeroth-order field and parameter values together with first-order derivative information.
E.2. Spectral Encoding as Derivative Proxy for Fields: For a fixed simplicial complex, z(ℓ) completely encodes the field’s discrete first-order differential structure, enabling an MLP to recover dω and δω from input features.This follows from a linear isomorphism between spectral coefficients with derivative matrices and the field with its first-order derivatives.

E.3. Geometric Interaction in the Fiber Branch · F. Computational Complexity Analysis · F.1. Computational Complexity

The framework combines explicit algebraic derivatives from the Base branch with implicit geometric and material derivatives from the Fiber branch, then projects corrections to preserve topological conservation laws. Its offline-online design concentrates geometry-dependent spectral work offline and achieves approximately linear online complexity with mesh scale.

E.3. Geometric Interaction in the Fiber Branch: The Fiber branch’s convolutional outputs implicitly contain spatial variation rates of input parameters.This derivative information arises from local convolution operations that are equivalent to weighted differences.
E.3. Geometric Interaction in the Fiber Branch: E.3 shows that Base interaction features explicitly provide algebraic derivatives dω and δω, while Fiber features implicitly provide numerical derivatives ∇κ and ∇g.These complementary derivative sources supply local information for modeling geometric interactions.
E.3. Geometric Interaction in the Fiber Branch: A parameterized MLP uses the combined local information to approximate the nonlinear commutator residual Fcomm.The approximation is grounded in the Universal Approximation Theorem.
E.3. Geometric Interaction in the Fiber Branch: Orthogonal projection restricts the learned correction to the complement of the base space, preventing approximation errors from corrupting topological conservation laws.The projection is written as (I −Πk base).
F. Computational Complexity Analysis: The offline-online strategy performs geometry-dependent spectral operations once offline and targets approximately linear online inference complexity with mesh scale.This decouples preprocessing from repeated inference.
F.1. Computational Complexity: Offline DEC assembly uses sparse operators with nnz(Lk) = O(Nk), while extracting mk low-frequency eigenpairs costs O(mk nnz(Lk)) ≈ O(mkNk).Shift-Invert Spectral Transformation with σ ≈0 accelerates Krylov convergence, and base construction from Φk and sparse operators also costs O(mkNk).
F.1. Computational Complexity: Online base-space projections cost O(Nkmk), with mk ≪Nk and typically mk ∼64, while exact-sequence identities can short-circuit certain higher-order derivative compositions to zero.The projections are implemented as parallel GEMM kernels suited to GPU acceleration.
F.1. Computational Complexity: The Fiber branch costs O(Nk) for point-voxel mapping and O(V log V ) for the 3D FNO, yielding a total forward-pass complexity of O(Nkmk) + O(Nk + V log V ) ≈O(Nk).Here V = R3 is fixed independent of mesh scale, and structured FFT operations replace irregular per-vertex tangent-space accesses.

F.2. Scalability and Precomputation Cost · G. Training Details and Hyperparameter Configuration · G.1. Dataset Splitting Strategy

The framework uses offline spectral preprocessing with low online complexity, while reporting training procedures and a strict temporal dataset split across three tasks. Its spectral strategy reaches million-scale meshes within minutes and supports resolution-efficient transfer to high-resolution meshes.

F.2. Scalability and Precomputation Cost: Offline preprocessing is negligible relative to thousands of training iterations, with online complexity O(Nk).The decomposition is motivated by fixed or isometrically deformed physical meshes.
F.2. Scalability and Precomputation Cost: Shift-Invert completes spectral decomposition within minutes for million-scale meshes with N ∼106.Sparse direct factorization exploits bounded fill-in, while spectral-gap magnification accelerates Krylov convergence.
F.2. Scalability and Precomputation Cost: Approximate multilevel or hierarchical-matrix spectral solvers are proposed to reduce preprocessing overhead when mesh topology updates frequently.These extensions are beyond the paper’s scope, which focuses on the foundational architecture.
F.2. Scalability and Precomputation Cost: The base-space branch processes only the first mk spectral coefficients, such as mk = 64 or 128, making its cost nearly independent of Nk.This confines global topological information to a fixed low-dimensional subspace, while the Fiber branch handles high-frequency details through ambient-space FFT.
F.2. Scalability and Precomputation Cost: HSD maintains consistent accuracy under reduced inference mesh density and transfers from coarse training meshes zero-shot to meshes with 7000+ vertices.The transfer incurs only marginal error increase.
G. Training Details and Hyperparameter Configuration: The training-details section documents experimental costs, computational environment, optimization strategies, hyperparameters, and task-specific training procedures.These materials are presented as implementation information for the experiments.
G.1. Dataset Splitting Strategy: Each of the three tasks uses 3,000 simulation samples with a strict temporal partition to prevent data leakage.The test set contains 20% of the total data, and validation uses 15% of the remaining data; split statistics appear in Table 5.

G.2. Auxiliary Evaluations

Auxiliary evaluations test Hodge Spectral Duality on geometric stress tests and topological-invariant prediction. The results show low cross-input variation and improved identification of topological structure over a Euclidean-distance baseline.

Stress tests: Ellipsoid Aero tests boundary-driven external-flow reconstruction, while Torus Helmholtz tests a genus-one Helmholtz response with two independent harmonic 1-form directions and oscillatory forcing.The evaluations vary aspect ratio, curvature, vortex-pair forcing, global moment coupling, and frequency content.
Stress tests: Below 8.5% on Ellipsoid Aero and below 6.0% on Torus Helmholtz, maximum relative variation remains limited across mesh, point-cloud, and graph inputs.The reported variation compares the same evaluation across the three input representations.
Topological identification: 91.0% accuracy, compared with 84.6% for a Euclidean-distance baseline, was achieved when predicting Betti numbers from point-cloud inputs.The classifier uses Hodge-based spectral features and the harmonic-nullspace relation ker ∆k = ker dk ∩ ker δk with bk = dim ker ∆k.

G.3. Detailed Computational Cost Analysis … H.1. Magnetostatics

The paper reports efficient training, stable optimization, and consistent HSD advantages across computational analyses and additional visual evaluations. In Magnetostatics, HSD preserves magnetic-field structure across geometries and boundary conditions while reducing errors in challenging regions.

G.3. Detailed Computational Cost Analysis: Training comparisons use identical hardware, with HSD totals including one-time spectral basis precomputation overhead.The reported comparison measures time in seconds and memory in megabytes under identical hardware conditions.
G.3. Detailed Computational Cost Analysis: HSD remains significantly faster than MGN and GNO while retaining high efficiency on static field tasks.DeepONet trains fastest, whereas MGN is especially slow because of explicit message passing; dynamic tasks add temporal residual correction overhead for HSD.
G.4. Computational Resources: Experiments use PyTorch 2.9.0 with an NVIDIA NGC PyTorch container on nodes equipped with a 32-core AMD CPU and dual RTX PRO 6000 GPUs.Each GPU provides 96 GB of VRAM, totaling 192 GB.
G.5. Optimization and Training Configuration: All tasks use AdamW with cosine annealing, while vector-field losses combine flux and divergence terms with λflux = 1.0 and λdiv = 0.1.The vector-field formulation applies to Magnetostatics and External Aerodynamics; the scalar-field configuration is described separately.
G.6. Model Hyperparameters: HSD and baseline models are controlled to approximately 250k–300k parameters for fair comparison across the three tasks.The External Aerodynamics configuration increases spectral truncation dimension k to 128 to accommodate greater topological and geometric complexity.
G.7. Training Curves: HSD shows stable decreases and the lowest final losses across all three tasks, while competing models oscillate, plateau early, or finish with higher losses.FNO-3D and DeepONet are less stable on complex geometries, whereas GNO and MGN remain comparatively stable but end above HSD.
H. Additional Visualization Results: Additional visualizations use a shared error-map color scale to compare prediction performance across test cases.The scale progresses from black through red and yellow to white as error magnitude increases.
H.1. Magnetostatics: HSD accurately captures magnetic-field spatial distributions across geometries and boundary conditions, maintaining low errors near strong field-strength gradients.FNO-3D shows boundary artifacts from regular-grid interpolation, while GNO and MGN reconstruct high-curvature regions less accurately.

H.2. External Aerodynamics · H.3. Toroidal Transport

The visual evaluations show that HSD preserves accurate flow-field and flux predictions around complex vehicle geometries, while Toroidal Transport performance depends on the spatial frequency and localization of initial conditions.

H.2. External Aerodynamics: HSD accurately predicts near-wall velocity distributions for both streamlined and bluff-body vehicle geometries, including discontinuities around wheel arches and side mirrors.The evaluated geometries exhibit distinctly different flow-field characteristics.
H.2. External Aerodynamics: HSD’s predicted surface flux closely matches Ground Truth, whereas baselines accumulate amplified errors in flow separation and wake confluence regions.Flux is emphasized as a key physical quantity for downstream engineering analysis requiring higher prediction accuracy.
H.2. External Aerodynamics: Figures 14 and 15 compare velocity vector field predictions and errors across models for two External Aerodynamics test geometries.Each figure presents model predictions alongside corresponding errors.
H.2. External Aerodynamics: Figures 16 and 17 compare predicted surface flux and Ground Truth errors for two External Aerodynamics samples.The visualizations report predicted flux in the top row and corresponding errors below.
H.3. Toroidal Transport: Toroidal Transport evaluates advection-diffusion evolution of a scalar field on a torus under three initial conditions with differing spatial frequency and localization.These initial-field properties directly affect the complexity of the final state.
H.3. Toroidal Transport: Under low-frequency initial conditions, most models provide reasonable Toroidal Transport predictions, while increasing initial spatial complexity makes prediction more difficult.The passage describes this progression across the three test samples shown in Figures 18–20.
H.3. Toroidal Transport: Figures 18–20 visualize Toroidal Transport scalar-field predictions and corresponding errors for three test samples with different initial conditions.Each visualization includes the initial condition, Ground Truth, model predictions, and error maps.