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Learning Discrete Riemannian Metrics for Physical Fields with Cochain-Frame Equivarianc

Dongzhe Zheng, Christine Allen-Blanchette

arXiv:2608.14556v1cs.LG

TL;DR

Mesh-based physical-field models must preserve exact topological identities while learning geometry-dependent behavior, but existing surrogates often mix these roles. RHMP fixes coboundaries, learns SPD cochain metrics, and enforces cochain-frame equivariance; across seven benchmarks, it achieves the best overall performance, especially where topology, learned geometry, and field structure interact.

  • Problem

    Mesh-field learning must preserve exact topological identities while adapting to geometry, material response, anisotropy, and constitutive variation.

  • Method

    RHMP fixes cellular coboundaries, learns SPD cochain metrics for propagation, and enforces invariance to orthogonal hidden cochain-frame changes.

  • Results

    Across seven benchmarks, RHMP achieves the best overall performance, with the largest gains when topology, learned geometry, and field structure interact.

  • Takeaways & Limitations

    Fixed coboundaries and learned metrics jointly support PSD Hodge operators, cochain-frame-equivariant layers, and exact Abelian curvature invariance.

  • Takeaways & Limitations

    Future work includes full non-Abelian gauge covariance, better-conditioned SPD parameterizations for heterogeneous media, and more scalable metric activations.

Abstract

from arXiv · show

Physical fields on meshes require a separation between topology and geometry: conservation laws are topological and should be exact, while geometry, material response, and anisotropic coupling must be learned from data. Existing neural surrogates often mix these roles inside unconstrained message passing. We introduce Riemannian Hodge Message Passing (RHMP), which turns this separation into an architectural principle. RHMP fixes the cellular coboundaries ($d_k$) determined by oriented incidence and learns symmetric positive-definite cochain metrics ($H_k$) for geometry-dependent propagation. Treating $H_k$ as the learned metric motivates cochain-frame equivariance: physical propagation should be invariant to orthogonal changes of the hidden cochain feature basis. RHMP implements this principle with metric-weighted Hodge blocks ($d_k^\top H_{k+1}d_k$), yielding exact cochain-complex identities ($d_{k+1}d_k=0$), nonnegative Hodge energies, positive-semidefinite operators, and exact Abelian curvature invariance. Across seven physical benchmarks spanning fluids, electromagnetism, gauge fields, and variable-mesh CFD, RHMP achieves the best overall performance, with the largest gains when topology, learned geometry, and field structure interact.

1 Introduction

RHMP separates exact mesh topology from learned geometry by fixing coboundary maps and learning positive-definite cochain metrics. It further makes hidden cochain features equivariant to orthogonal frame changes, embedding topology, geometry, and symmetry into message passing.

  • Motivation: Physical fields occupy different cochain degrees, so models must preserve exact topological identities while adapting to geometric and material responses.Scalars, fluxes, circulations, vorticity, and field strengths naturally live on vertices, edges, faces, or higher-dimensional cells.
  • Topology and geometry: Hodge theory fixes coboundaries d_k from oriented incidence, ensuring d_{k+1}d_k = 0, while positive-definite metrics H_k encode geometry.The coboundaries contain no material parameters; geometry enters through the cochain metrics.
  • RHMP principle: RHMP fixes d_k and learns H_k, concentrating geometric freedom in metric-weighted Hodge propagation while keeping the cochain complex exact.The coboundary determines differentiated quantities and closed conservation laws, whereas the metric determines measurement and coupling.
  • Cochain-frame symmetry: RHMP enforces cochain-frame equivariance by making propagation invariant to orthogonal changes of the hidden channel basis.For x_k ∈ R^{n_k×C}, an orthogonal frame change R ∈ O(C) acts as x_k 7→ x_kR^⊤, and equivariant layers commute with this action.
  • Constructive architecture: The architecture assigns features to cochain degrees, freezes coboundaries, predicts H_k ≻ 0 from invariant statistics, and uses norm-gated nonlinearities.These design choices build topology and symmetries into message passing while learning geometry from data.

2 Related Work

RHMP builds on topology–geometry separation from discrete exterior calculus and related compatible discretizations, fixing coboundaries while learning metric-dependent cochain inner products. It differs from higher-order cellular networks, learned Hodge operators, gauge-equivariant models, and neural operators by combining input-dependent SPD cochain metrics with fixed topology, O(C)-equivariance, and conservation/gauge-field guarantees.

  • Discrete exterior calculus and compatible discretization: Discrete exterior calculus and finite element exterior calculus separate cochain topology, encoded by coboundaries satisfying d_k+1d_k = 0, from metric-dependent Hodge stars or mass matrices.This separation also appears in compatible and mimetic discretizations and computational electromagnetism.
  • Cellular, simplicial, and sheaf neural networks: Cellular and simplicial neural networks extend graph filtering and message passing to higher-order cochains using incidence maps, Hodge Laplacians, orientation-aware aggregation, and attention.The passage names SNN, MPSN, SCCNN, CW Net, SAT, and Clifford-SMPN among these methods.
  • Learned Hodge operators and mesh spectral geometry: RHMP differs from HodgeNet, HodgeFormer, and HSD by learning input-dependent SPD cochain metrics inside physical message passing while keeping coboundaries fixed and providing O(C)-equivariance and conservation/gauge-field guarantees.These prior methods learn Hodge or Hodge-like matrices for mesh spectral or transformer operators.
  • Gauge-equivariant and surface networks: Gauge-equivariant surface models use local tangent-frame transport, connection-Laplacian filters, or surface diffusion, whereas lattice-gauge models impose gauge structure on link variables, Wilson-loop features, or flow-based samplers.RHMP targets a separate symmetry: orthogonal changes of the hidden cochain feature basis.
  • Equivariant graph networks and physical surrogates: Equivariant graph networks and physical surrogates use coordinate-aware geometric features or learn dynamics on particles and meshes, while neural operators learn function-space input–output maps through discretization-consistent kernels.The cited neural-operator families include FNO, DeepONet, graph-kernel and multipole operators, Galerkin Transformer, F-FNO, and WNO.

3 Discrete Geometry on Cell Complexes

This section separates fixed cellular topology from learned symmetric positive-definite cochain metrics: d_k provides discrete differentiation, while H_k measures and propagates geometry-dependent fields. Metric-weighted Hodge operators preserve exact cochain identities, positivity, cochain-frame equivariance, spatial equivariance, and Abelian curvature invariance.

  • Discrete cochain geometry: Coboundaries d_k are fixed incidence operators on oriented cells, mapping k-cochains to (k + 1)-cochains and encoding discrete derivatives such as vertex differences and oriented face sums.A C-channel k-cochain assigns one C-dimensional feature vector to each k-cell.
  • Discrete cochain geometry: Learning H_k learns the discrete geometry used to measure and propagate cochain fields, with diagonal entries representing local weights and off-diagonal entries representing anisotropic or nonlocal couplings.The metrics are positive definite, so they define energy measurements on cochain spaces.
  • Metric-weighted Hodge operators: Metric-weighted Hodge blocks combine topological differentiation from d_k with learned geometry from H_k, and positive-definite metrics make the resulting Hodge operators positive semidefinite.For any input, the induced quadratic form is nonnegative, certifying the learned geometry as an energy metric.
  • Metric-induced properties: Theorem 1 guarantees exact d_{k+1}d_k = 0 independently of training, positive-semidefinite Hodge operators, O(C) cochain-frame equivariance, and E(n)-equivariant vector readouts.These properties follow from the fixed cochain complex, commuting cell- and channel-index actions, and Euclidean-invariant geometric quantities.
  • Gauge invariance: Abelian curvature observables are exactly invariant under gauge shifts A → A + d_0λ because curvature F = d_1A and d_1d_0 = 0.For compact U(1) phases, the equality is understood modulo 2π.

4 Riemannian Hodge Message Passing

RHMP factorizes topology and geometry by fixing oriented coboundaries while learning SPD cochain metrics for metric-weighted Hodge message passing. Its invariant metric construction and cochain-degree-aware architecture preserve structural constraints while enabling geometry-dependent propagation.

  • Architecture: RHMP lifts fields to 0-, 1-, and 2-cochains, applies metric-weighted Hodge message passing, and reads out task-specific scalar or vector quantities.The learned geometric object is H_k, while d_k remains fixed by the oriented cell complex.
  • Cochain representation: Physical quantities are assigned by cochain degree: scalars to 0-cochains, flux-like fields to 1-cochains, and intensity-like fields to 2-cochains.The coboundaries are fixed once the oriented cell complex is fixed.
  • Topology–geometry factorization: The topology–geometry factorization fixes d_k to determine communication and preserve the cochain complex, while learned SPD H_k determine communication geometry.Learning therefore leaves identities such as d_2 = 0 intact.
  • Cochain-frame equivariance: Metric prediction uses channel-basis invariants, so deterministic predictors built from norms and inner products are cochain-frame invariant.Neighbor aggregation is permutation-invariant, and these statistics are unchanged under x 7→xR⊤.
  • Structural verification: RHMP satisfies all ten tested structural constraints, spanning spatial E(n), topology, cochain-frame O(C), and Abelian U(1) curvature invariance families.The topology tests include d_2 = 0, while the cochain-frame tests include channel rotations, permutations, and sign flips.
  • Spectral flexibility: Metric parameters provide first-order control over simple non-harmonic eigenvalues, giving diagonal metrics greater local spectral flexibility than the unweighted Hodge Laplacian near identity.This result is established from the spectral viewpoint in the appendix propositions and corollary.

5 Experiments

RHMP is evaluated across seven physical systems spanning fixed and variable meshes, with the strongest overall results on all six fixed-mesh tasks and especially large gains on gauge-theoretic benchmarks. Ablations show that learned metrics and cross-dimensional transport drive performance, while preserving the model’s structural properties.

  • Fixed-mesh results: RHMP achieves the best performance on all six fixed-mesh tasks, with the largest margins over the strongest baseline on gauge-theoretic settings.The fixed-mesh comparison covers six tasks and evaluates RHMP against multiple graph, topological, geometric, and operator-learning baselines.
  • Structured grids: 0.984 SSIM / 0.009 NRMSE is achieved by RHMP on Navier–Stokes vorticity, versus FNO’s 0.976 SSIM / 0.011 NRMSE.This regular-grid result suggests that Hodge factorization remains useful even when spectral-grid methods are naturally applicable.
  • Cochain support and topology: Node-based graph baselines perform poorly when targets naturally live on edges or faces, while cell-complex baselines improve by representing fields on the correct geometric support.The affected examples include vorticity, surface flow, and gauge curvature; node encodings must otherwise infer cochain type, incidence structure, and identities such as d2 = 0 from data.
  • Learned geometry beyond fixed incidence: 0.993 SSIM is achieved by RHMP on the U(1) Wilson-loop task, compared with CW Net’s 0.977, demonstrating gains from learning geometry beyond fixed incidence.RHMP additionally learns the metric Hk controlling geometry-dependent propagation, whereas both methods benefit from fixed cochain incidence.
  • Variable-mesh generalization: RHMP achieves the best performance across all three AirfRANS metrics and transfers across meshes with differing size and connectivity.Its metric Hk depends only on cell-level O(C)-invariants and not nk, enabling application to per-sample variable meshes.
  • Ablations: Setting Hk = I or disabling cross-dimensional transport sharply reduces R2, including a Wilson-loop drop from 0.96 to 0.50 with Hk = I.Ellipsoid surface flow is unaffected within ∆R2 ≤0.05, while learning scalar dk or replacing norm-gated nonlinearities with element-wise ReLU changes R2 by at most 0.04.

6 Conclusion and Limitations · A Broader Impacts

RHMP separates topology from learned geometry through fixed coboundaries and SPD cochain metrics, producing structurally constrained, equivariant operators and consistent gains across seven benchmarks. The paper identifies extensions for non-Abelian gauge covariance, heterogeneous media, and scalable metric parameterizations, while recommending validation and uncertainty quantification for safety-critical deployment.

  • 6 Conclusion and Limitations: RHMP fixes coboundaries and learns SPD cochain metrics, preserving the cochain complex while controlling geometry-dependent propagation.This metric-centered design yields PSD Hodge operators, cochain-frame equivariant layers, and exact Abelian curvature invariance.
  • 6 Conclusion and Limitations: Across seven benchmarks, RHMP delivers consistent gains, especially when conservation, learned geometry, topology, and gauge structure interact.The conclusion attributes these gains to the architecture’s combined structural constraints.
  • 6 Conclusion and Limitations: Future work includes group-valued transport to move beyond fitting SU(2) field strength toward full non-Abelian gauge covariance.This is identified as a specialization of the metric layer.
  • 6 Conclusion and Limitations: Further limitations motivate better-conditioned SPD parameterizations for highly heterogeneous media and compressed or basis-tied per-cell metric activations for larger simulations.Both are listed as natural next steps for improving the learned metric layer.
  • A Broader Impacts: The most immediate applications are reducing CFD simulation costs in engineering design and accelerating routine computations in materials science and computational physics.Examples include airfoil optimization, thermal management, and property inference for inhomogeneous media.
  • A Broader Impacts: The work uses no personal data, user behavior, decision recommendations, surveillance, or automated decision-making capabilities.These properties are stated in the broader-impacts discussion.
  • A Broader Impacts: For safety-critical engineering simulations, the authors recommend cross-validation against high-fidelity solvers before deployment and quantified prediction uncertainty estimates.The examples include aerospace structures and nuclear reactors.

B Primer: From Continuous Metrics to Discrete k-Form Metrics … D.2 Metric Parameterization and PSD Guarantee

RHMP separates fixed topological coboundaries from learned cochain metrics that encode geometry and material response. Its metric-weighted Hodge operators preserve exact cochain identities while guaranteeing positive-semidefinite propagation.

  • B.2 Continuous k-form Background: Continuous metrics determine inner products and Hodge stars, while the exterior derivative remains topological and independent of coordinates and material parameters.Changing the metric changes diffusion, wave propagation, constitutive response, and curvature-dependent terms.
  • B.1 Worked Example: One Triangle: On an oriented cell complex, cochains discretize k-forms and Stokes’ theorem yields coboundaries satisfying d_k+1d_k = 0 exactly.For the triangle example, d0 is a discrete gradient, d1 is a discrete curl, and d1d0 = 0 by cancellation.
  • B.2 Continuous k-form Background: RHMP keeps coboundaries fixed while learning symmetric positive-definite cochain metrics, preserving topology while adapting geometry.The resulting Hodge messages inherit cochain-frame equivariance and spatial invariance or equivariance from metric invariance to feature-basis and coordinate transformations.
  • C Cochain Complex Data Flow: Each Hodge return loop lifts features with d_k, reweights them using H_k+1, and projects them back with d_k^T.The forward coboundaries are frozen oriented-incidence maps, and their composition remains exactly zero throughout training.
  • D.1 Hyperparameters: The default metric uses C = 128 channels, L = 4 Hodge message-passing layers, hidden dimension 16, rank r = 8, and approximately 200K parameters.The parameter count varies with n0, n1, and n2.
  • D.2 Metric Parameterization and PSD Guarantee: The metric parameterization H_k = B_kB_k^T + diag(softplus(h_k) + 10^-6) guarantees SPD by construction without eigenvalue clipping or post-hoc projection.The associated Hodge operator is positive semidefinite through its energy identity, while r = 0 and full Cholesky forms are special cases.

D.3 Empirical Comparison of Metric Parameterizations … E.1 Design Principles

The study selects a diagonal-plus-r=8 metric as the Pareto-optimal parameterization and evaluates RHMP across structurally diverse physical systems using specialized batching, lifting, and compute setups. The benchmark suite spans established PDE and CFD tasks alongside advection–diffusion, curved-manifold, conservation-constrained, and lattice gauge systems.

  • D.3 Empirical Comparison of Metric Parameterizations: The scalar metric reaches R2=0.495, only 1.5 points behind the default, while larger r=16 capacity loses ground.The r=8 basis captures additional useful structure while remaining well-conditioned; the gap is task-dependent.
  • D.3 Empirical Comparison of Metric Parameterizations: The diagonal-plus-r=8-basis metric is the Pareto-optimal choice under matched optimization, combining the highest test R2 among mesh-dependent variants with the smallest parameter budget.The default configuration is used in all main-table experiments.
  • D.4 Batched Forward Pass: For same-mesh samples, RHMP merges the batch into the feature dimension through sparse matrix multiplication, whereas per-sample meshes require separate forward passes.The airfoil pressure task uses per-sample meshes.
  • D.5 Lifting Encoder Details: The lifting encoder supplies 0-cochains with field, degree, and average-neighbor-distance features, while 2-cochains use signed boundary aggregation and cell-area features.Boundary signs σe ∈ {+1, −1} come from the sparsity structure of d1.
  • D.6 Compute Environment: Experiments use Python 3.12, PyTorch 2.10, CUDA 12.8, and native torch.sparse on two NVIDIA RTX PRO 6000 Blackwell GPUs.Each individual run uses a single GPU and no third-party geometric deep learning libraries.
  • D.7 Training Cost: RHMP training across all seven tasks takes approximately 2.5 hours, while the full experiment with 12 baselines takes about 24 GPU hours.Per-task RHMP training times range from 5 minutes for Torus Adv-Diff to 35 minutes for Airfoil.
  • E.1 Design Principles: The seven benchmark tasks are organized by independently testable structural capabilities, with diagnostic signals and external anchors mapped for each task.Public benchmarks provide external anchors, while additional tasks cover structural dimensions beyond currently public data.
  • E.1 Design Principles: The benchmark spectrum combines established compressible Navier–Stokes and external-aerodynamics surrogates with harder advection–diffusion, curved-manifold, conservation-constrained, and U(1) / SU(2) lattice gauge systems.The underlying physics follows established discretization schemes, with per-task numerical setups detailed separately.

E.2 Dataset Composition and Generation Protocol … F Full Proof of Theorem 1

The paper specifies reproducible generation, numerical, baseline, and evaluation protocols across seven benchmarks, including robustness tests for resolution and parameter shifts. It also proves that RHMP preserves positive semidefiniteness, cochain-frame equivariance, Euclidean symmetries, and exact cellular identities.

  • E.2 Dataset Composition and Generation Protocol: All tasks use 70/15/15 train/validation/test splits, and results are averaged across 3 seeds.
  • E.2 Dataset Composition and Generation Protocol: The benchmark suite comprises seven tasks with task-specific inputs, outputs, reference methods, mesh sizes, sample counts, sampling distributions, and simulation details.The protocol includes torus advection–diffusion with 15-step IMEX integration and SU(2) Yang–Mills connections sampled independently on the SU(2) Lie algebra.
  • E.3 Implementation Details: Reference fields are computed to numerical tolerance ≤10^-6 using validated FEM and sparse-direct-solve routines, with code, scripts, and datasets released for reproduction.The implementation uses SciPy SuperLU and cotangent-weight FEM, while per-task numerical schemes are listed in §E.
  • E.3 Implementation Details: Baselines retain the input/output forms of their original papers and are evaluated in regimes matched to fixed meshes, regular grids, or per-sample variable meshes.The fixed-mesh regime evaluates 11 baselines admitting irregular meshes, while FNO is additionally compared for NS Vorticity.
  • E.4 Robustness Checks: RHMP’s R2 remained above 0.94 on U(1) Wilson loop when node counts changed from 1024 to 512, 2048, and 4096.This resolution test probes sensitivity across four resolutions.
  • E.4 Robustness Checks: On Maxwell–Poisson, RHMP’s NRMSE degradation was ∆=0.013 when test source-density magnitudes expanded from Unif[0.5, 1.5] to Unif[0.2, 2.5].
  • F Full Proof of Theorem 1: The proof establishes that metric-weighted Hodge operators are positive semidefinite when the surrounding matrices are positive semidefinite, including RHMP’s learned positive-definite metrics.The argument applies channel by channel and then sums over channels.

G Symmetry Verification Protocol … H.3 Tangent Bundle Convolution

The protocol evaluates symmetry on topology, geometry, and cochain-channel transformations across regular and irregular meshes, while verifying exact coboundary and Abelian curvature identities. The curvature comparisons position RHMP’s learned cochain metrics as richer or complementary geometric representations than existing discrete curvature and tangent-bundle approaches.

  • G Symmetry Verification Protocol: Symmetry tests act on the cell complex for E(n) and edge orientation, and on C-dimensional cochain channels for fiber O(C) transformations.Spatial transformations rebuild K after acting on vertex coordinates, while edge flips synchronously update d0[e, :] and d1[:, e].
  • G Symmetry Verification Protocol: The protocol uses a 4×4 regular Delaunay mesh with (n0, n1, n2) = (16, 33, 18) and a 6-point irregular Delaunay mesh with (n0, n1, n2) = (6, 11, 6).Each (model, subgroup) cell is averaged over 5 independent random transformations, with err < 10−3 as the numerical pass threshold.
  • H Discrete Curvature Comparison: Fixed coboundaries satisfy d2 =0 exactly, while 1-cochain message passing preserves F = dA under A 7→A + dλ.The U(1) curvature check is distinct from the O(C) cochain-frame action, and methods without cochain-frame design have no corresponding inductive bias.
  • H.1 Ollivier-Ricci Curvature: Ollivier-Ricci curvature is a scalar edge quantity defined through optimal transport, whereas RHMP’s Hk is a matrix over all k-cells.Thus Hk provides a richer geometric parameterization than Ollivier’s edgewise scalar curvature.
  • H.2 Forman-Ricci Curvature: Forman curvature uses fixed combinatorial structure, while RHMP learns Hk adaptively as geometric weights through training.The diagonal entries of H0 and H1 play roles analogous to graph vertex and edge weights in the Forman construction.
  • H.2 Forman-Ricci Curvature: Table 8 evaluates ten symmetry subgroups on the primary 4×4 Delaunay mesh, and Table 9 repeats the protocol on the auxiliary 6-point irregular Delaunay mesh.Both tables mark d2 =0 as an exact fixed-coboundary topology check and U(1) as Abelian curvature invariance; FNO’s FFT parameterization is tied to regular grids.
  • H.3 Tangent Bundle Convolution: Tangent-bundle convolution represents geometry through convolution on the tangent bundle, while RHMP represents it through inner-product weights on cochains.The two approaches are characterized as complementary duals.

I Expressivity Analysis · J Computational Complexity and Scalability

RHMP’s learnable metrics provide local first-order spectral control while preserving harmonic modes, making the diagonal metric family locally more expressive than the unweighted Hodge operator. Its rank-8 implementation retains sparse message-passing complexity and achieves speed comparable to cell-complex baselines, though it trails simpler graph and operator-learning methods.

  • I Expressivity Analysis: The eigenvalue map is differentiable near the identity metric for simple eigenvalues.This follows because the weighted Hodge operator depends affinely on SPD metric variables and is self-adjoint.
  • I Expressivity Analysis: Non-harmonic eigenvalues admit nonzero first-order changes through symmetric metric perturbations.Perturbations can be chosen from either adjacent metric, depending on whether d_k v_j or d_{k−1}^T v_j is nonzero.
  • I Expressivity Analysis: Harmonic modes are first-order stationary under all symmetric metric perturbations.When v_j lies in both relevant kernels, the eigenvalue derivative vanishes in every symmetric perturbation direction.
  • I Expressivity Analysis: The diagonal metric family is locally more expressive than the unweighted Hodge operator near the identity metric.A diagonal perturbation changes any simple non-harmonic eigenvalue at first order.
  • I Expressivity Analysis: The expressivity result is local: metrics add geometric degrees of freedom while the cohomological harmonic component remains protected by d2 = 0.The passage characterizes first-order spectral control around the identity metric rather than global universal approximation.
  • J Computational Complexity and Scalability: RHMP uses a diagonal-plus-low-rank-basis metric parameterization with rank r = 8 in the main experiments.The pure diagonal metric costs O(n_k) to apply, while the rank-r variant adds O(n_k r C).
  • J Computational Complexity and Scalability: Sparse applications of d_k and d_k^T dominate at O(nnz(d_k) · C), matching standard cell-complex message passing.Metric application consists of diagonal scaling plus a low-rank update.
  • J Computational Complexity and Scalability: RHMP is comparable in training speed to SCCNN and CW Net, faster than GAT, EGNN, and SchNet, and slower than GCN, FNO, and DeepONet.These comparisons are based on single-epoch training time across three representative tasks under the same GPU and data pipeline.

J.1 Large-Mesh Scalability

RHMP scales near-linearly to 100K-cell meshes because sparse Hodge operators avoid quadratic propagation costs, while maintaining a moderate memory footprint and mesh-invariant trainable weights. At 100× benchmark size, it preserves accuracy and achieves a substantial speedup over CW Net.

  • Runtime scaling: A 100× mesh-size increase raises forward time approximately 10×, from 16.7 ms to 169.2 ms, reflecting near-linear sparse-operator scaling.The dominant cost is sparse matrix–dense matrix multiplication at O(nnz(d_k) · C), which scales with the number of nonzeros rather than quadratically with mesh size.
  • Memory scaling: At 100K cells, peak GPU memory is 3.9 GB, while linear extrapolation estimates ∼39 GB for ∼1M cells; SCCNN requires 83.8 GB at 100K cells.RHMP therefore fits comfortably on a single 80 GB accelerator at 100K cells, whereas SCCNN exceeds available memory at that scale.
  • Parameter scaling: Trainable weights remain fixed at ∼0.05M across mesh sizes because metric parameters and transforms are shared across cells.Per-pass activations grow linearly with n_k, but mesh-invariant weights enable direct transfer across meshes with differing size and connectivity.
  • Large-mesh accuracy: At 100K cells, RHMP reaches test R2 = 0.9823 versus 0.9231 for CW Net at matched parameter counts of 10.80M versus 11.30M.This comparison uses the same single GPU and 70/15/15 split on the U(1) Wilson loop benchmark.
  • Large-mesh efficiency: RHMP requires ∼10 ms per step versus ∼700 ms for CW Net, and saturates near R2 ≈0.97 by epoch 20 while CW Net is still rising at epoch 30.Both models improve on the larger mesh, with RHMP increasing from R2 ≈0.955 to 0.982 and CW Net from ≈0.875 to 0.923.

K Ablation Studies … M Additional Qualitative Comparisons

Ablations show that RHMP’s learned metrics and cross-dimensional transport drive accuracy, while fixed coboundaries and norm-gated updates preserve exact structural properties. Statistical checks and additional qualitative samples further assess uncertainty, seed variation, and task-wide field fidelity.

  • K Ablation Studies: Learned cochain metrics and cross-dimensional transport dominate accuracy, with identity metrics causing ΔR2 losses of −0.37, −0.46, and −0.49 on CNS vorticity, Wilson loop, and Yang–Mills.Turning off cross-dimensional transport gives ΔR2 losses of −0.31, −0.08, and −0.39 on those tasks, while ellipsoid flow changes only by approximately −0.04.
  • K Ablation Studies: Fixed coboundaries and norm-gated nonlinearities have small R2 effects but preserve structure that learnable coboundaries and ReLU violate by orders of magnitude.Learnable coboundaries produce ∥d1d0∥F drift of approximately 10^1, while ReLU raises relative cochain-frame equivariance error from approximately 10^-6 to approximately 10^0.
  • K Ablation Studies: The strongest ablations visibly lose field structure, whereas ReLU and learnable coboundaries remain visually close to the full model despite diagnostic symmetry deviations.The structural losses are reported for Hk = I and cross-dimensional-transport-off variants; ReLU and learnable dk deviations appear in Table 14 diagnostics.
  • L.1 Test-set bootstrap statistics: Test-set uncertainty is estimated with 1000 bootstrap resamples for every task–method metric combination, while main-text estimates use seed means over {42, 1, 2}.Inference uses each representative seed-42 best-epoch checkpoint and recomputes every metric on resampled test indices.
  • L.2 Cross-seed training variance: Cross-seed standard deviations quantify training-time variance from initialization and stochastic optimization, complementing bootstrap finite-size variance.The RHMP ranking in Table 2 is preserved across seeds on every task.
  • M Additional Qualitative Comparisons: Additional qualitative samples are independently drawn from the same per-sample winner pool and span each task’s value distribution, indicating trends are not tied to one example.Four additional test samples are shown for each task.
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