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

# Learning Discrete Riemannian Metrics for Physical Fields with Cochain-Frame Equivariance
Source: [https://arxiv.org/html/2608.14556](https://arxiv.org/html/2608.14556)
###### Abstract

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 \(dkd\_\{k\}\) determined by oriented incidence and learns symmetric positive\-definite cochain metrics \(HkH\_\{k\}\) for geometry\-dependent propagation\. TreatingHkH\_\{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 \(dk⊤​Hk\+1​dkd\_\{k\}^\{\\top\}H\_\{k\+1\}d\_\{k\}\), yielding exact cochain\-complex identities \(dk\+1​dk=0d\_\{k\+1\}d\_\{k\}=0\), 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\.

## 1Introduction

Physical fields on meshes do not all live naturally as node features\. Pressure is naturally vertex\- or cell\-centered, flux and circulation live on oriented edges, and field strengths such as vorticity or curvature live on faces or higher\-dimensional cells\. Learning such fields therefore requires more than expressive message passing: the model must preserve topological identities that are exact, while adapting to geometric and material responses that vary across domains\. Incidence relations between vertices, edges, faces, and higher cells impose exact physical identities, such as the curl\-free condition of gradient fields \(∇×∇ϕ=0\\nabla\\\!\\times\\\!\\nabla\\phi=0\) and the source\-free \(or divergence\-free\) condition of curl fields \(∇⋅∇×𝐀=0\\nabla\\\!\\cdot\\\!\\nabla\\\!\\times\\\!\\mathbf\{A\}=0\)\. By contrast, material parameters, anisotropy, curvature, and constitutive laws determine how strongly neighboring quantities interact\. This suggests a simple design principle for neural physical surrogates: keep topology fixed and learn geometry\.

Hodge theory provides precisely this separation\. On a cell complex, scalar quantities are represented as0\-cochains, fluxes and circulations as11\-cochains, and surface densities such as vorticity or field strength as22\-cochains\. The coboundary mapsdkd\_\{k\}move between these spaces and are determined entirely by oriented incidence\. They contain no material parameters and satisfydk\+1​dk=0d\_\{k\+1\}d\_\{k\}=0by construction; we write this cochain\-complex identity compactly asd2=0d^\{2\}=0\. Geometry enters through a positive\-definite inner product on each cochain space\. We write this inner product as a symmetric positive\-definite \(SPD\) matrixHkH\_\{k\}and interpret it as a*discrete Riemannian metric*, the discrete counterpart of the metric\-induced Hodge star in⟨ω,η⟩g=∫Mω∧⋆gη\\langle\\omega,\\eta\\rangle\_\{g\}=\\int\_\{M\}\\omega\\wedge\\star\_\{g\}\\eta\[[16](https://arxiv.org/html/2608.14556#bib.bib24),[1](https://arxiv.org/html/2608.14556#bib.bib25)\]\. For readers less familiar with this dictionary, Appendix[B](https://arxiv.org/html/2608.14556#A2)gives a concrete triangle example and the continuous\-to\-discrete correspondence\.

This viewpoint leads to the RHMP principle: fix the topological coboundary mapdkd\_\{k\}, and learn the cochain metricsHkH\_\{k\}\. For an input or hiddenkk\-cochain featurexkx\_\{k\}, the weighted Hodge energy‖dk​xk‖Hk\+12\\\|d\_\{k\}x\_\{k\}\\\|\_\{H\_\{k\+1\}\}^\{2\}induces the operatordk⊤​Hk\+1​dkd\_\{k\}^\{\\top\}H\_\{k\+1\}d\_\{k\}onkk\-cochains \(see Appendix[C](https://arxiv.org/html/2608.14556#A3)\)\. The coboundarydkd\_\{k\}determines which quantities are differentiated and which conservation laws remain closed\. The metricHk\+1H\_\{k\+1\}determines how the resulting\(k\+1\)\(k\+1\)\-cochain is measured and coupled\. Thus, RHMP keeps the cochain complex exact while concentrating learnable geometric freedom in the metric\.

![Refer to caption](https://arxiv.org/html/2608.14556v1/figures/gauge_equi_overview.png)Figure 1:Cochain\-frame equivariance\.Applying an orthogonal change of hidden cochain frameR∈O​\(C\)R\\in O\(C\)before or after an RHMP layerΦ\\Phigives the same result\. The action is on the channel basis of cochain\-valued hidden features, whiledkd\_\{k\}andHkH\_\{k\}act on cell indices\.TreatingHkH\_\{k\}as the learned geometric object also changes the symmetry question\. The metric should describe propagation, not an arbitrary choice of basis for the hidden feature channels\. In RHMP, theCCchannels of a hiddenkk\-cochain are coordinates in an auxiliary feature space with a standard inner product\. Changing this orthonormal frame byR∈O​\(C\)R\\in O\(C\)acts as𝐱k↦𝐱k​R⊤\\mathbf\{x\}\_\{k\}\\mapsto\\mathbf\{x\}\_\{k\}R^\{\\top\}for𝐱k∈ℝnk×C\\mathbf\{x\}\_\{k\}\\in\\mathbb\{R\}^\{n\_\{k\}\\times C\}\. We call a layer*cochain\-frame equivariant*if it commutes with this action\. RHMP enforces this by predicting the cell\-space metricHkH\_\{k\}only fromO​\(C\)\\mathrm\{O\}\(C\)\-invariant channel\-statistics, such as norms and inner products, and by using norm\-gated nonlinearities\. As a result, orthogonal changes of hidden cochain frame change the representation but not the represented propagation\. Together with the SPD constraint onHkH\_\{k\}, Euclidean\-invariant geometric features, and fixed coboundaries, this yields nonnegative Hodge energies, positive\-semidefinite Hodge operators, coordinate\-free scalar behavior,E​\(n\)\\mathrm\{E\}\(n\)\-equivariant vector readouts, and exact preservation ofd2=0d^\{2\}=0\.

RHMP instantiates these principles by assigning features to the appropriate cochain degree, freezing the coboundary operators, predictingHk≻0H\_\{k\}\\succ 0from invariant statistics, and using norm\-gated nonlinearities that preserve cochain\-frame equivariance\. The result message\-passing operator whose topology and symmetries are built in, while its geometry is learned from data\.

Contributions\.\(1\)*Metric\-centered formulation\.*We formulate the problem of learning a mesh\-based physical field as a problem of learning a discrete Riemannian metricHkH\_\{k\}on cochains, with fixed coboundary mapsdkd\_\{k\}\. This makes the separation between topology, geometry, and conservation explicit\. \(2\)*Cochain\-frame symmetry theorem and constructive architecture\.*We prove that the metric conditionsHk≻0H\_\{k\}\\succ 0, cochain\-frame invariance underO​\(C\)\\mathrm\{O\}\(C\), and spatialE​\(n\)\\mathrm\{E\}\(n\)\-invariance yield nonnegative Hodge energies, positive\-semidefinite Hodge operators, cochain\-frame\-equivariant message passing, and invariant/equivariant scalar and vector readouts\. RHMP implements these conditions directly\. \(3\)*Gauge\-field consequence and empirical validation\.*For Abelian gauge fields represented as cochains, curvature observables depending ond​AdAare exactly invariant underA↦A\+d​λA\\mapsto A\+d\\lambda, since the additional termd​\(d​λ\)d\(d\\lambda\)vanishes by the cochain\-complex identityd∘d=0d\\circ d=0\. On seven tasks spanning fluid mechanics, electromagnetism, Abelian and non\-Abelian gauge\-field benchmarks, and per\-sample variable\-mesh CFD, RHMP outperforms graph, cell\-complex, manifold\-gauge, and neural\-operator baselines\.

## 2Related Work

Discrete exterior calculus and compatible discretization\.RHMP is built on the topology–geometry separation that underlies discrete exterior calculus \(DEC\)\[[16](https://arxiv.org/html/2608.14556#bib.bib24)\]and finite element exterior calculus \(FEEC\)\[[1](https://arxiv.org/html/2608.14556#bib.bib25)\]: coboundaries encode the cochain complex and satisfydk\+1​dk=0d\_\{k\+1\}d\_\{k\}=0, while metric\-dependent Hodge stars or mass matrices define inner products on discrete forms\. This separation also appears in compatible and mimetic discretizations and computational electromagnetism, where incidence matrices encode exact sequences and conservation laws, while mass, Hodge, or constitutive matrices encode metric and material response\[[10](https://arxiv.org/html/2608.14556#bib.bib32),[24](https://arxiv.org/html/2608.14556#bib.bib33),[5](https://arxiv.org/html/2608.14556#bib.bib34)\]\. RHMP turns this separation into a neural architecture, withdkd\_\{k\}fixed and the SPD metric factorHkH\_\{k\}predicted from data\.

Cellular, simplicial, and sheaf neural networks\.Topological signal processing extends graph filtering from node signals to edge\-, face\-, and higher\-order cochains through Hodge Laplacians and Hodge decompositions\[[54](https://arxiv.org/html/2608.14556#bib.bib35)\]\. Simplicial and cellular message\-passing methods extend graph neural networks to higher\-order cells via incidence maps, Hodge Laplacians, orientation\-aware aggregation, and attention, including SNN\[[17](https://arxiv.org/html/2608.14556#bib.bib30)\], MPSN\[[8](https://arxiv.org/html/2608.14556#bib.bib2)\], SCCNN\[[53](https://arxiv.org/html/2608.14556#bib.bib20)\], CW Net\[[7](https://arxiv.org/html/2608.14556#bib.bib1)\], SAT\[[21](https://arxiv.org/html/2608.14556#bib.bib36)\], Clifford\-SMPN\[[34](https://arxiv.org/html/2608.14556#bib.bib11)\], and the unified framework of Hajij et al\.\[[22](https://arxiv.org/html/2608.14556#bib.bib8)\]; sheaf neural networks\[[23](https://arxiv.org/html/2608.14556#bib.bib37)\]and neural sheaf diffusion\[[6](https://arxiv.org/html/2608.14556#bib.bib27)\]equip graphs with learned cellular sheaves and sheaf Laplacians\. RHMP keeps the cellular coboundary fixed and learns an SPD cochain metricHkH\_\{k\}fromO​\(C\)\\mathrm\{O\}\(C\)\-invariant statistics, preservingd2=0d^\{2\}=0while learning geometry, anisotropy, and material response\.

Learned Hodge operators and mesh spectral geometry\.HodgeNet\[[45](https://arxiv.org/html/2608.14556#bib.bib26)\], HodgeFormer\[[37](https://arxiv.org/html/2608.14556#bib.bib38)\], and HSD\[[56](https://arxiv.org/html/2608.14556#bib.bib57)\]learn Hodge or Hodge\-like matrices for mesh spectral or transformer operators\. RHMP differs by learning input\-dependent SPD cochain metrics inside physical message passing, with fixed coboundaries,O​\(C\)\\mathrm\{O\}\(C\)\-equivariance, and conservation/gauge\-field guarantees\.

Gauge\-equivariant and surface networks\.Gauge Equivariant CNNs\[[13](https://arxiv.org/html/2608.14556#bib.bib5)\]and GEM\-CNN\[[15](https://arxiv.org/html/2608.14556#bib.bib6)\]define convolutions on manifolds or meshes via local tangent\-frame parallel transport; Tangent Bundle Networks\[[2](https://arxiv.org/html/2608.14556#bib.bib3)\]use connection\-Laplacian tangent\-bundle filters; DiffusionNet\[[44](https://arxiv.org/html/2608.14556#bib.bib29)\]learns through discretization\-agnostic surface diffusion\. Lattice\-gauge models instead impose gauge structure on link variables, Wilson\-loop features, or flow\-based samplers\[[18](https://arxiv.org/html/2608.14556#bib.bib39),[36](https://arxiv.org/html/2608.14556#bib.bib40),[25](https://arxiv.org/html/2608.14556#bib.bib41)\]\. RHMP targets a separate symmetry: orthogonal changes of basis in the hidden cochain feature frame\. Its metric predictor usesO​\(C\)\\mathrm\{O\}\(C\)\-invariant statistics, so Hodge messages are cochain\-frame equivariant\. For Abelian connection data, RHMP inherits exact curvature invariance fromd2=0d^\{2\}=0; for non\-Abelian fields it uses cochain differentials with algebraic coupling\.

Equivariant graph networks and physical surrogates\.E​\(n\)\\mathrm\{E\}\(n\)\-equivariant graph networks such as SchNet\[[42](https://arxiv.org/html/2608.14556#bib.bib16)\], EGNN\[[41](https://arxiv.org/html/2608.14556#bib.bib15)\], Tensor Field Networks\[[47](https://arxiv.org/html/2608.14556#bib.bib42)\], SE\(3\)\-Transformer\[[20](https://arxiv.org/html/2608.14556#bib.bib43)\], PaiNN\[[43](https://arxiv.org/html/2608.14556#bib.bib44)\], and NequIP\[[3](https://arxiv.org/html/2608.14556#bib.bib45)\]build coordinate\-aware message passing from distances, relative positions, and higher\-order geometric features\. Graph Network\-based Simulators\[[40](https://arxiv.org/html/2608.14556#bib.bib46)\], MeshGraphNets\[[39](https://arxiv.org/html/2608.14556#bib.bib28)\], and message\-passing neural PDE solvers\[[11](https://arxiv.org/html/2608.14556#bib.bib47)\]learn surrogate dynamics on particles or meshes\. These methods encode fields as node, particle, or edge features\. RHMP assigns scalar\-, flux\-, and intensity\-like quantities to 0\-, 1\-, and 2\-cochains and uses fixed coboundaries with learned SPD metrics\.

Neural operators\.Fourier Neural Operator\[[29](https://arxiv.org/html/2608.14556#bib.bib10)\]and DeepONet\[[35](https://arxiv.org/html/2608.14556#bib.bib12)\]are central neural operators, learning input–output maps at the function\-space level\. General neural\-operator theory formulates such maps through discretization\-consistent integral\-kernel parameterizations\[[27](https://arxiv.org/html/2608.14556#bib.bib48)\]\. Beyond FNO and DeepONet, graph\-kernel and multipole graph neural operators learn nonlocal kernels on irregular samples\[[31](https://arxiv.org/html/2608.14556#bib.bib49),[30](https://arxiv.org/html/2608.14556#bib.bib50)\]; Galerkin Transformer uses attention as an operator\-learning layer\[[12](https://arxiv.org/html/2608.14556#bib.bib51)\]; F\-FNO and WNO modify the spectral factorization or basis\[[48](https://arxiv.org/html/2608.14556#bib.bib52),[49](https://arxiv.org/html/2608.14556#bib.bib53)\]; PINO adds physics residual constraints\[[33](https://arxiv.org/html/2608.14556#bib.bib54)\]; and Geo\-FNO / GINO extend FNO\-style models to general or large\-scale geometries\[[28](https://arxiv.org/html/2608.14556#bib.bib55),[32](https://arxiv.org/html/2608.14556#bib.bib56)\]\. RHMP is complementary: it builds cochain type, exactd2=0d^\{2\}=0, SPD Hodge energy, and cochain\-frame\-invariant metric learning into the architecture\.

## 3Discrete Geometry on Cell Complexes

A regular CW complexKKis a cell complex whosekk\-dimensional cells generalize vertices, edges, faces, and higher\-dimensional elements\. ACC\-channelkk\-cochain is a matrix𝐱k∈ℝnk×C\\mathbf\{x\}\_\{k\}\\in\\mathbb\{R\}^\{n\_\{k\}\\times C\}, with oneCC\-dimensional feature vector perkk\-cell\. The coboundary operatordk∈ℝnk\+1×nkd\_\{k\}\\in\\mathbb\{R\}^\{n\_\{k\+1\}\\times n\_\{k\}\}mapskk\-cochains to\(k\+1\)\(k\+1\)\-cochains\. For example,d0d\_\{0\}differences vertex quantities along oriented edges andd1d\_\{1\}sums oriented edge quantities around faces\. These are the standard cochain and coboundary operators of DEC and FEEC\[[16](https://arxiv.org/html/2608.14556#bib.bib24),[1](https://arxiv.org/html/2608.14556#bib.bib25)\]\. The identitydk\+1​dk=0d\_\{k\+1\}d\_\{k\}=0is purely combinatorial: it is the discrete form of identities such as∇×∇ϕ=0\\nabla\\\!\\times\\\!\\nabla\\phi=0and∇⋅∇×𝐀=0\\nabla\\\!\\cdot\\\!\\nabla\\\!\\times\\\!\\mathbf\{A\}=0\. Appendix[B\.1](https://arxiv.org/html/2608.14556#A2.SS1)gives a concrete triangle example illustratingd0d\_\{0\},d1d\_\{1\}, and how changingH1H\_\{1\}while fixingdkd\_\{k\}changes the Hodge step\.

###### Definition 1\(Discrete Riemannian metric on cochains\)\.

A discrete Riemannian metric onkk\-cochains is an SPD matrixHk∈SPD​\(nk\)H\_\{k\}\\in\\mathrm\{SPD\}\(n\_\{k\}\), whereSPD​\(nk\)\\mathrm\{SPD\}\(n\_\{k\}\)denotes the cone of symmetric positive\-definitenk×nkn\_\{k\}\\times n\_\{k\}matrices, defining the inner product

⟨𝐱k,𝐲k⟩Hk:=tr​\(𝐱k⊤​Hk​𝐲k\)\.\\langle\\mathbf\{x\}\_\{k\},\\mathbf\{y\}\_\{k\}\\rangle\_\{H\_\{k\}\}:=\\mathrm\{tr\}\(\\mathbf\{x\}\_\{k\}^\{\\top\}H\_\{k\}\\mathbf\{y\}\_\{k\}\)\.\(1\)Diagonal entries ofHkH\_\{k\}describe local cell weights such as volumes, areas, conductivities, or permittivities\. Off\-diagonal entries describe anisotropic or nonlocal couplings between cells\. LearningHkH\_\{k\}therefore amounts to learning the discrete geometry through which the cochain field is measured and propagated\.

The metric block\.The basic object is a metric\-weighted differential\. Given an incidence mapD:Ca​\(K\)→Cb​\(K\)D:C^\{a\}\(K\)\\to C^\{b\}\(K\)and a positive\-definite metricHbH\_\{b\}on the target cochain space, define

ℰD,H​\(x\)=12​‖D​x‖Hb2=12​tr​\(\(D​x\)⊤​Hb​\(D​x\)\),∇ℰD,H=D⊤​Hb​D​x\.\\mathcal\{E\}\_\{D,H\}\(x\)=\\frac\{1\}\{2\}\\\|Dx\\\|\_\{H\_\{b\}\}^\{2\}=\\frac\{1\}\{2\}\\mathrm\{tr\}\\big\(\(Dx\)^\{\\top\}H\_\{b\}\(Dx\)\\big\),\\qquad\\nabla\\mathcal\{E\}\_\{D,H\}=D^\{\\top\}H\_\{b\}Dx\.\(2\)Throughout the paper,d⊤​H​dd^\{\\top\}Hddenotes this metric block:ddsupplies the topological derivative andHHsupplies the learned geometry used to measure its output\.

Placement convention\.Superscripts↑\\uparrow,↓\\downarrow, andcross\\mathrm\{cross\}only indicate which adjacent cochain space carries the metric; all such matrices use the same SPD construction\.

Upper and lower Hodge terms\.For akk\-cochain, the upper placement takesD=dkD=d\_\{k\}and measures the\(k\+1\)\(k\+1\)\-cochaindk​𝐱kd\_\{k\}\\mathbf\{x\}\_\{k\}withHk\+1↑H\_\{k\+1\}^\{\\uparrow\}:

ℰk↑​\(𝐱k\)=12​‖dk​𝐱k‖Hk\+1↑2=12​tr​\(\(dk​𝐱k\)⊤​Hk\+1↑​\(dk​𝐱k\)\),\\mathcal\{E\}\_\{k\}^\{\\uparrow\}\(\\mathbf\{x\}\_\{k\}\)=\\frac\{1\}\{2\}\\\|d\_\{k\}\\mathbf\{x\}\_\{k\}\\\|\_\{H\_\{k\+1\}^\{\\uparrow\}\}^\{2\}=\\frac\{1\}\{2\}\\mathrm\{tr\}\\big\(\(d\_\{k\}\\mathbf\{x\}\_\{k\}\)^\{\\top\}H\_\{k\+1\}^\{\\uparrow\}\(d\_\{k\}\\mathbf\{x\}\_\{k\}\)\\big\),\(3\)with gradientdk⊤​Hk\+1↑​dk​𝐱kd\_\{k\}^\{\\top\}H\_\{k\+1\}^\{\\uparrow\}d\_\{k\}\\mathbf\{x\}\_\{k\}\. The lower coexact placement takesD=dk−1⊤D=d\_\{k\-1\}^\{\\top\}and measures the\(k−1\)\(k\-1\)\-cochaindk−1⊤​𝐱kd\_\{k\-1\}^\{\\top\}\\mathbf\{x\}\_\{k\}withHk−1↓H\_\{k\-1\}^\{\\downarrow\}:

ℰk↓​\(𝐱k\)=12​‖dk−1⊤​𝐱k‖Hk−1↓2,∇ℰk↓=dk−1​Hk−1↓​dk−1⊤​𝐱k\.\\mathcal\{E\}\_\{k\}^\{\\downarrow\}\(\\mathbf\{x\}\_\{k\}\)=\\frac\{1\}\{2\}\\\|d\_\{k\-1\}^\{\\top\}\\mathbf\{x\}\_\{k\}\\\|\_\{H\_\{k\-1\}^\{\\downarrow\}\}^\{2\},\\qquad\\nabla\\mathcal\{E\}\_\{k\}^\{\\downarrow\}=d\_\{k\-1\}H\_\{k\-1\}^\{\\downarrow\}d\_\{k\-1\}^\{\\top\}\\mathbf\{x\}\_\{k\}\.\(4\)The weightedkk\-Hodge message operator is therefore

LkH=dk−1​Hk−1↓​dk−1⊤\+dk⊤​Hk\+1↑​dk\.L\_\{k\}^\{H\}=d\_\{k\-1\}H\_\{k\-1\}^\{\\downarrow\}d\_\{k\-1\}^\{\\top\}\+d\_\{k\}^\{\\top\}H\_\{k\+1\}^\{\\uparrow\}d\_\{k\}\.\(5\)
Positive definiteness of the placed metrics gives positive semidefiniteness of the Hodge operator\. For any𝐱k\\mathbf\{x\}\_\{k\},

⟨𝐱k,LkH​𝐱k⟩=‖dk−1⊤​𝐱k‖Hk−1↓2\+‖dk​𝐱k‖Hk\+1↑2≥0\.\\langle\\mathbf\{x\}\_\{k\},L\_\{k\}^\{H\}\\mathbf\{x\}\_\{k\}\\rangle=\\\|d\_\{k\-1\}^\{\\top\}\\mathbf\{x\}\_\{k\}\\\|\_\{H\_\{k\-1\}^\{\\downarrow\}\}^\{2\}\+\\\|d\_\{k\}\\mathbf\{x\}\_\{k\}\\\|\_\{H\_\{k\+1\}^\{\\uparrow\}\}^\{2\}\\geq 0\.\(6\)ThusHk≻0H\_\{k\}\\succ 0certifies the learned geometry as an energy metric\.

The connection to continuous geometry is direct\. A Riemannian metricggon a smooth manifold induces the Hodge star⋆g\\star\_\{g\}and theL2L^\{2\}inner product onkk\-forms,⟨ω,η⟩g=∫Mω∧⋆gη\\langle\\omega,\\eta\\rangle\_\{g\}=\\int\_\{M\}\\omega\\wedge\\star\_\{g\}\\eta\. After discretizingkk\-forms by their integrals over orientedkk\-cells, the matrix representing this inner product is a mass/Hodge matrixHkH\_\{k\}\[[16](https://arxiv.org/html/2608.14556#bib.bib24),[1](https://arxiv.org/html/2608.14556#bib.bib25)\]\. HenceHkH\_\{k\}is the discrete counterpart of the continuous Riemannian metric and Hodge star, whiledkd\_\{k\}is the discrete counterpart of the exterior derivative\.

Cochain\-frame action\.For𝐱k∈ℝnk×C\\mathbf\{x\}\_\{k\}\\in\\mathbb\{R\}^\{n\_\{k\}\\times C\}, a cochain\-frame changeR∈O​\(C\)R\\in\\mathrm\{O\}\(C\)acts on the hidden feature fiber byρR​\(𝐱k\)=𝐱k​R⊤\\rho\_\{R\}\(\\mathbf\{x\}\_\{k\}\)=\\mathbf\{x\}\_\{k\}R^\{\\top\}, with the sameRRapplied to all cells and cochain degrees: for𝐱=\(𝐱0,𝐱1,𝐱2\)\\mathbf\{x\}=\(\\mathbf\{x\}\_\{0\},\\mathbf\{x\}\_\{1\},\\mathbf\{x\}\_\{2\}\)the action isρR​\(𝐱\)=\(𝐱0​R⊤,𝐱1​R⊤,𝐱2​R⊤\)\\rho\_\{R\}\(\\mathbf\{x\}\)=\(\\mathbf\{x\}\_\{0\}R^\{\\top\},\\mathbf\{x\}\_\{1\}R^\{\\top\},\\mathbf\{x\}\_\{2\}R^\{\\top\}\)\. A layerΦ\\Phiis*cochain\-frame equivariant*ifΦ​\(ρR​𝐱\)=ρR​Φ​\(𝐱\)\\Phi\(\\rho\_\{R\}\\mathbf\{x\}\)=\\rho\_\{R\}\\Phi\(\\mathbf\{x\}\)for allR∈O​\(C\)R\\in\\mathrm\{O\}\(C\)\.

The following theorem is the main metric\-to\-properties statement\. Its hypotheses are conditions onHkH\_\{k\}and on the fixed cochain complex; its conclusions are the induced conservation, positivity, and symmetry properties\.

###### Theorem 1\(Metric\-induced topology, positivity, cochain\-frame equivariance, and spatial equivariance\)\.

LetKKbe a CW complex embedded inℝn\\mathbb\{R\}^\{n\}, let𝐱k∈ℝnk×C\\mathbf\{x\}\_\{k\}\\in\\mathbb\{R\}^\{n\_\{k\}\\times C\}beCC\-channel cochain features, and letΦ\\Phibe a layer built from the fixed coboundariesdkd\_\{k\}, metric\-weighted operators of the form \([5](https://arxiv.org/html/2608.14556#S3.E5)\), residual sums, and per\-cell nonlinearities of the formσgate​\(𝐦\)=h​\(‖𝐦‖\)​𝐦\\sigma\_\{\\rm gate\}\(\\mathbf\{m\}\)=h\(\\\|\\mathbf\{m\}\\\|\)\\mathbf\{m\}\. Assume:

1. 1\.each predicted metric satisfiesHj≻0H\_\{j\}\\succ 0;
2. 2\.Hj​\(ρR​𝐱,K\)=Hj​\(𝐱,K\)H\_\{j\}\(\\rho\_\{R\}\\mathbf\{x\},K\)=H\_\{j\}\(\\mathbf\{x\},K\)for all cochain\-frame changesR∈O​\(C\)R\\in\\mathrm\{O\}\(C\);
3. 3\.HjH\_\{j\}and the lifting encoder depend on the embedding ofKKonly throughE​\(n\)\\mathrm\{E\}\(n\)\-invariants such as distances, angles, areas, and incidence relations\.

Then the following hold\.\(i\)dk\+1​dk=0d\_\{k\+1\}d\_\{k\}=0is exact and independent of training\.\(ii\)LkHL\_\{k\}^\{H\}is positive semidefinite for every input\.\(iii\)Φ\\Phiis cochain\-frame equivariant underO​\(C\)\\mathrm\{O\}\(C\):Φ​\(ρR​𝐱\)=ρR​Φ​\(𝐱\)\\Phi\(\\rho\_\{R\}\\mathbf\{x\}\)=\\rho\_\{R\}\\Phi\(\\mathbf\{x\}\)\.\(iv\)Scalar cochain readouts areE​\(n\)\\mathrm\{E\}\(n\)\-invariant; vector readouts of the form∑ewe​\(⋅\)​\(𝐫j−𝐫i\)\\sum\_\{e\}w\_\{e\}\(\\cdot\)\(\\mathbf\{r\}\_\{j\}\-\\mathbf\{r\}\_\{i\}\), with invariant weightswew\_\{e\}, areE​\(n\)\\mathrm\{E\}\(n\)\-equivariant\.

###### Proof sketch\.

\(i\) is the cellular identityd2=0d^\{2\}=0\. \(ii\) follows from the quadratic form identity \([6](https://arxiv.org/html/2608.14556#S3.E6)\)\. For \(iii\),dkd\_\{k\}andHkH\_\{k\}act on the cell index, whileR∈O​\(C\)R\\in\\mathrm\{O\}\(C\)acts on the channel index; these actions commute\. The metric itself is unchanged by assumption, and the norm gate satisfiesσgate​\(𝐦​R⊤\)=σgate​\(𝐦\)​R⊤\\sigma\_\{\\rm gate\}\(\\mathbf\{m\}R^\{\\top\}\)=\\sigma\_\{\\rm gate\}\(\\mathbf\{m\}\)R^\{\\top\}\. For \(iv\), all scalar quantities entering the layer are functions of Euclidean invariants\. Multiplying an invariant scalar by the displacement𝐫j−𝐫i\\mathbf\{r\}\_\{j\}\-\\mathbf\{r\}\_\{i\}, which transforms equivariantly underE​\(n\)\\mathrm\{E\}\(n\), produces an equivariant vector\. The full proof is given in Appendix[F](https://arxiv.org/html/2608.14556#A6)\. ∎

Abelian gauge\-field observables\.For Abelian gauge\-field tasks, the fixed cochain complex provides an exact invariance at the level of physical observables\. Let aU​\(1\)U\(1\)connection be represented additively as a 1\-cochainAA, and define curvature byF=d1​AF=d\_\{1\}A\. For any 0\-cochainλ\\lambda, the gauge shiftA↦A\+d0​λA\\mapsto A\+d\_\{0\}\\lambdaleaves curvature unchanged:

F′=d1​\(A\+d0​λ\)=d1​A\+d1​d0​λ=d1​A,F^\{\\prime\}=d\_\{1\}\(A\+d\_\{0\}\\lambda\)=d\_\{1\}A\+d\_\{1\}d\_\{0\}\\lambda=d\_\{1\}A,becaused1​d0=0d\_\{1\}d\_\{0\}=0\. Curvature\-type observables depending onAAthroughd​AdAare therefore exactly invariant under Abelian gauge shifts; for compactU​\(1\)U\(1\)phases the equality is understood modulo2​π2\\pi\.

## 4Riemannian Hodge Message Passing

RHMP implements Theorem[1](https://arxiv.org/html/2608.14556#Thmtheorem1)directly\. The architecture lifts input fields to0\-,11\-, and22\-cochains, then applies metric\-weighted Hodge message passing, and finally reads out scalar or vector quantities according to the task\. The learned geometric object inside each Hodge block is the discrete cochain metricHkH\_\{k\}; coboundariesdkd\_\{k\}remain fixed by the oriented cell complex\. Appendix[C](https://arxiv.org/html/2608.14556#A3)gives a schematic of the cochain\-level data flow, including the chain maps, Hodge return loops, and matrix expressions\.

### 4\.1Weighted Hodge Message\-Passing Layer

Physical quantities are assigned to cochain degrees according to geometric type:0\-cochains carry scalar fields such as pressure or temperature,11\-cochains carry flux\-like fields such as velocity components or gauge connections, and22\-cochains carry intensity\-like fields such as vorticity or field strength\. The coboundariesdkd\_\{k\}are fixed once the oriented cell complex is fixed, so learning leavesd2=0d^\{2\}=0intact\.

For akk\-cochain feature𝐱k\(ℓ\)\\mathbf\{x\}\_\{k\}^\{\(\\ell\)\}, RHMP forms a same\-degree metric Hodge message

𝐦k,self\(ℓ\)=dk−1​Hk−1↓,\(ℓ\)​dk−1⊤​𝐱k\(ℓ\)\+dk⊤​Hk\+1↑,\(ℓ\)​dk​𝐱k\(ℓ\)\.\\mathbf\{m\}\_\{k,\\mathrm\{self\}\}^\{\(\\ell\)\}=d\_\{k\-1\}H\_\{k\-1\}^\{\\downarrow,\(\\ell\)\}d\_\{k\-1\}^\{\\top\}\\mathbf\{x\}\_\{k\}^\{\(\\ell\)\}\+d\_\{k\}^\{\\top\}H\_\{k\+1\}^\{\\uparrow,\(\\ell\)\}d\_\{k\}\\mathbf\{x\}\_\{k\}^\{\(\\ell\)\}\.\(7\)It also exchanges information across adjacent cochain degrees through metric\-aware transport,

𝐦k,cross\(ℓ\)=dk−1​Hk−1cross,\(ℓ\)​𝐱k−1\(ℓ\)\+dk⊤​Hk\+1cross,\(ℓ\)​𝐱k\+1\(ℓ\),\\mathbf\{m\}\_\{k,\\mathrm\{cross\}\}^\{\(\\ell\)\}=d\_\{k\-1\}H\_\{k\-1\}^\{\\mathrm\{cross\},\(\\ell\)\}\\mathbf\{x\}\_\{k\-1\}^\{\(\\ell\)\}\+d\_\{k\}^\{\\top\}H\_\{k\+1\}^\{\\mathrm\{cross\},\(\\ell\)\}\\mathbf\{x\}\_\{k\+1\}^\{\(\\ell\)\},\(8\)with boundary terms omitted when the adjacent degree is absent\. The same invariant metric predictor and SPD constraints are used forHcrossH^\{\\mathrm\{cross\}\}; the superscript records its use in cross\-degree transport\. The layer is

𝐱k\(ℓ\+1\)=𝐱k\(ℓ\)\+RMSNorm​\(σgate​\(α​𝐦k,self\(ℓ\)\+\(1−α\)​𝐦k,cross\(ℓ\)\)\),\\mathbf\{x\}\_\{k\}^\{\(\\ell\+1\)\}=\\mathbf\{x\}\_\{k\}^\{\(\\ell\)\}\+\\mathrm\{RMSNorm\}\\\!\\left\(\\sigma\_\{\\rm gate\}\\\!\\left\(\\alpha\\,\\mathbf\{m\}\_\{k,\\mathrm\{self\}\}^\{\(\\ell\)\}\+\(1\-\\alpha\)\\,\\mathbf\{m\}\_\{k,\\mathrm\{cross\}\}^\{\(\\ell\)\}\\right\)\\right\),\(9\)whereα∈\(0,1\)\\alpha\\in\(0,1\)is learned andσgate​\(𝐦\)=gθ​\(‖𝐦‖\)​𝐦\\sigma\_\{\\rm gate\}\(\\mathbf\{m\}\)=g\_\{\\theta\}\(\\\|\\mathbf\{m\}\\\|\)\\mathbf\{m\}is a norm\-gated nonlinearity\. RMSNorm is applied per cell without learnable affine parameters, preserving channel\-basis symmetry\.

Eq\. \([9](https://arxiv.org/html/2608.14556#S4.E9)\) is the architectural form of the paper’s topology–geometry factorization\. The mapsdkd\_\{k\}determine which cells communicate and preserve the cochain complex\. The learned SPD metricsHkH\_\{k\}determine the geometry of that communication\.

### 4\.2Constructing Positive\-Definite and Equivariant Metrics

The metric predictor receives only statistics that are invariant under a change of channel basis\. For eachkk\-cell we use the self\-norm and neighbor inner products,

ψ​\(𝐱k\)i=\[‖𝐱k​\(i\)‖22,Aggj∼i​⟨𝐱k​\(i\),𝐱k​\(j\)⟩,cell geometry invariants\],\\psi\(\\mathbf\{x\}\_\{k\}\)\_\{i\}=\\Big\[\\\|\\mathbf\{x\}\_\{k\}\(i\)\\\|\_\{2\}^\{2\},\\;\\mathrm\{Agg\}\_\{j\\sim i\}\\,\\langle\\mathbf\{x\}\_\{k\}\(i\),\\mathbf\{x\}\_\{k\}\(j\)\\rangle,\\;\\text\{cell geometry invariants\}\\Big\],\(10\)where the aggregation is permutation\-invariant over neighboring cells\. Since norms and inner products are unchanged by𝐱↦𝐱​R⊤\\mathbf\{x\}\\mapsto\\mathbf\{x\}R^\{\\top\}, any deterministic metric predictor built fromψ\\psiis cochain\-frame invariant\.

The simplest instantiation is the diagonal metric

Hk=diag​\(softplus⁡\(MLPH​\(ψ​\(𝐱k\)\)\)\+ϵ\),ϵ\>0\.H\_\{k\}=\\mathrm\{diag\}\\\!\\left\(\\operatorname\{softplus\}\(\\mathrm\{MLP\}\_\{H\}\(\\psi\(\\mathbf\{x\}\_\{k\}\)\)\)\+\\epsilon\\right\),\\qquad\\epsilon\>0\.\(11\)This gives a strict positive\-definiteness guarantee: for every nonzeroz∈ℝnkz\\in\\mathbb\{R\}^\{n\_\{k\}\},

z⊤​Hk​z=∑i\(softplus⁡\(hi\)\+ϵ\)​zi2≥ϵ​‖z‖22\>0\.z^\{\\top\}H\_\{k\}z=\\sum\_\{i\}\\big\(\\operatorname\{softplus\}\(h\_\{i\}\)\+\\epsilon\\big\)z\_\{i\}^\{2\}\\geq\\epsilon\\\|z\\\|\_\{2\}^\{2\}\>0\.\(12\)Thus the learnedHkH\_\{k\}is SPD at every layer and every training step, and Eq\. \([6](https://arxiv.org/html/2608.14556#S3.E6)\) makes the corresponding Hodge operator positive semidefinite\. The weights ofMLPH\\mathrm\{MLP\}\_\{H\}are shared across allkk\-cells and are independent of mesh sizenkn\_\{k\}; only the scalar outputhih\_\{i\}scales with the number of cells \(see scalability analysis in Appendix[J](https://arxiv.org/html/2608.14556#A10)\)\. The diagonal form in Eq\. \([11](https://arxiv.org/html/2608.14556#S4.E11)\) gives the simplest SPD certificate\. In the main experiments, we use a more expressive diagonal\-plus\-low\-rank parameterization,Hk=Bk​Bk⊤\+diag​\(softplus⁡\(𝐡k\)\+ϵ\)H\_\{k\}=B\_\{k\}B\_\{k\}^\{\\top\}\+\\mathrm\{diag\}\(\\operatorname\{softplus\}\(\\mathbf\{h\}\_\{k\}\)\+\\epsilon\)with rankr=8r=8\. Appendix[D\.3](https://arxiv.org/html/2608.14556#A4.SS3)compares this choice with other SPD parameterizations, including full Cholesky forms\.

Spatial behavior is handled by restricting geometric inputs to quantities invariant under Euclidean motions: edge lengths, vertex degrees, face areas, and interior angles\. Edge orientation is handled separately through symmetric and antisymmetric feature combinations\. For an oriented edgeei​je\_\{ij\}, RHMP constructs a11\-cochain feature of the form

𝐱1​\(ei​j\)\\displaystyle\\mathbf\{x\}\_\{1\}\(e\_\{ij\}\)=ϕsym​\(𝐱0i\+𝐱0j,ℓe,degi\+degj\)\\displaystyle=\\phi\_\{\\rm sym\}\(\\mathbf\{x\}\_\{0\}^\{i\}\+\\mathbf\{x\}\_\{0\}^\{j\},\\ell\_\{e\},\\deg\_\{i\}\+\\deg\_\{j\}\)\(13\)\+ϕasym​\(𝐱0i−𝐱0j,ℓe,degi−degj\),\\displaystyle\\quad\+\\phi\_\{\\rm asym\}\(\\mathbf\{x\}\_\{0\}^\{i\}\-\\mathbf\{x\}\_\{0\}^\{j\},\\ell\_\{e\},\\deg\_\{i\}\-\\deg\_\{j\}\),\(14\)whereℓe=‖𝐫j−𝐫i‖\\ell\_\{e\}=\\\|\\mathbf\{r\}\_\{j\}\-\\mathbf\{r\}\_\{i\}\\\|\. Higher\-degree cochains are constructed by oriented aggregation over boundary cells together with invariant face features\. For tasks requiring ambient vector outputs, RHMP reconstructs vectors using an invariant\-scalar\-times\-displacement readout:

𝐯^​\(v\)=1deg⁡\(v\)​∑e=\(i,j\)∋vw​\(𝐱1​\(e\),‖𝐫i​j‖2,…\)​\(𝐫j−𝐫i\)\.\\hat\{\\mathbf\{v\}\}\(v\)=\\frac\{1\}\{\\deg\(v\)\}\\sum\_\{e=\(i,j\)\\ni v\}w\(\\mathbf\{x\}\_\{1\}\(e\),\\\|\\mathbf\{r\}\_\{ij\}\\\|^\{2\},\\ldots\)\(\\mathbf\{r\}\_\{j\}\-\\mathbf\{r\}\_\{i\}\)\.\(15\)Herewwis anE​\(n\)\\mathrm\{E\}\(n\)\-invariant scalar and𝐫j−𝐫i\\mathbf\{r\}\_\{j\}\-\\mathbf\{r\}\_\{i\}is translation\-invariant andO​\(n\)\\mathrm\{O\}\(n\)\-equivariant\. Therefore each term, and hence the sum, transforms equivariantly underE​\(n\)\\mathrm\{E\}\(n\)\[[51](https://arxiv.org/html/2608.14556#bib.bib19)\]\.

### 4\.3Numerical Verification, Symmetry Hierarchy, and Spectral Expressivity

Table 1:Structural verification\(satisfied constraints out of1010, grouped by family\)\. A test is satisfied if its relativeL2L\_\{2\}error is below10−510^\{\-5\}or holds exactly by construction; numeric andn/aentries count as unsatisfied\. These tests identify which constraints each architecture enforces, though some baselines do not target all families\. See Appendix[G](https://arxiv.org/html/2608.14556#A7)for per\-subgroup errors\.We probe ten structural constraints, grouped into four families: spatialE​\(n\)\\mathrm\{E\}\(n\)subgroups \(3 tests\), topology checks \(2; includingd2=0d^\{2\}\\\!=\\\!0\), cochain\-frameO​\(C\)\\mathrm\{O\}\(C\)subgroups \(4; including channel rotation, permutation, and sign flips\), and AbelianU​\(1\)U\(1\)curvature invariance under exact shifts \(1\)\. Table[1](https://arxiv.org/html/2608.14556#S4.T1)reports per\-group pass counts: RHMP satisfies all ten, including the cochain\-frame tests; other architectures satisfy complementary subsets according to their design\. The full protocol, mesh and seed details, and per\-subgroup numerical errors are in Appendix[G](https://arxiv.org/html/2608.14556#A7)\.

From a spectral viewpoint, the metric parameters provide first\-order control over simple non\-harmonic eigenvalues of the weighted Hodge operator \(Appendix Proposition[6](https://arxiv.org/html/2608.14556#Thmtheorem6)\)\. Even the diagonal metric family can locally perturb these eigenvalues near the identity metric, giving strictly more local spectral flexibility than the unweighted Hodge Laplacian near the identity metric \(Appendix Corollary[8](https://arxiv.org/html/2608.14556#Thmtheorem8)\)\.

## 5Experiments

![Refer to caption](https://arxiv.org/html/2608.14556v1/figures/GMPF1.png)Figure 2:Navier–Stokes vorticity\.Left blocks summarize the inputs; right blocks compare ground truth, RHMP, and representative baselines\.Table 2:Quantitative results on the six fixed\-mesh tasks, three metrics, and 11 baselines whose native definition admits irregular meshes\. SSIM \(↑\\uparrow\), Pearson \(↑\\uparrow\), NRMSE \(↓\\downarrow\)\.Bold= best,underline= second best\. Point estimates are the mean across three independent training seeds\{42,1,2\}\\\{42,1,2\\\}; cross\-seed and test\-set bootstrap standard deviations are reported in Appendix[L\.2](https://arxiv.org/html/2608.14556#A12.SS2)and[L\.1](https://arxiv.org/html/2608.14556#A12.SS1)\. Comparison against FNO on the regular\-grid setting \(NS Vorticity\) is discussed inline in §[5\.1](https://arxiv.org/html/2608.14556#S5.SS1); variable\-mesh generalization \(AirfRANS\) is reported separately in Table[3](https://arxiv.org/html/2608.14556#S5.T3)\. Baselines grouped by family:Graph,Topological,Geometric,Operator\.We evaluate RHMP on seven physical systems chosen to probe complementary structural requirements: structured\-grid vorticity prediction, scalar transport on a torus, equivariant surface\-vector prediction on an ellipsoid, Maxwell–Poisson electrostatics, AbelianU​\(1\)U\(1\)curvature prediction, non\-AbelianS​U​\(2\)SU\(2\)field\-strength prediction, and variable\-mesh AirfRANS pressure prediction\. The first six tasks share a fixed\-mesh evaluation protocol and form the main quantitative comparison \(Table[2](https://arxiv.org/html/2608.14556#S5.T2), §[5\.1](https://arxiv.org/html/2608.14556#S5.SS1)\)\. AirfRANS uses an independent unstructured mesh for each sample and is reported separately in Section[5\.2](https://arxiv.org/html/2608.14556#S5.SS2)\. Full task definitions, parameter distributions, qualitative examples, and conservation\-residual checks are given in Appendix[E](https://arxiv.org/html/2608.14556#A5)and Appendix[M](https://arxiv.org/html/2608.14556#A13)\.

Each task is selected to isolate a different structural demand\.Navier–Stokes vorticity prediction\(PDEBench\[[46](https://arxiv.org/html/2608.14556#bib.bib17)\]\) tests whether the Hodge factorization remains useful even on a regular grid\.Torus convection–diffusiontest scalar transport on a nontrivial topology\.Ellipsoid surface flowrequiresE​\(n\)\\mathrm\{E\}\(n\)\-equivariant vector readout\[[4](https://arxiv.org/html/2608.14556#bib.bib31)\]\.Maxwell–Poisson electrostaticstests curl\-free electric\-field prediction\.U​\(1\)U\(1\)Wilson loop\[[52](https://arxiv.org/html/2608.14556#bib.bib22)\]tests exact Abelian curvature invariance underA↦A\+d​λA\\mapsto A\+d\\lambda\. TheS​U​\(2\)SU\(2\)Yang–Mills\[[14](https://arxiv.org/html/2608.14556#bib.bib23)\]task tests cochain differentials together with the nonlinear commutator termF=d​A\+\[A,A\]F=dA\+\[A,A\]\.AirfRANS airfoil pressure\[[9](https://arxiv.org/html/2608.14556#bib.bib4)\]tests transfer across per\-sample variable meshes\.

We compare with 12 baselines, covering graph methods \(GCN\[[26](https://arxiv.org/html/2608.14556#bib.bib9)\], GAT\[[50](https://arxiv.org/html/2608.14556#bib.bib18)\], SchNet\[[42](https://arxiv.org/html/2608.14556#bib.bib16)\], EGNN\[[41](https://arxiv.org/html/2608.14556#bib.bib15)\]\), topological methods \(MPSN\[[8](https://arxiv.org/html/2608.14556#bib.bib2)\], SCCNN\[[53](https://arxiv.org/html/2608.14556#bib.bib20)\]\), geometric methods \(GaugeEquivCNN\[[13](https://arxiv.org/html/2608.14556#bib.bib5)\], GEM\-CNN\[[15](https://arxiv.org/html/2608.14556#bib.bib6)\], CW Net\[[7](https://arxiv.org/html/2608.14556#bib.bib1)\], Clifford\-SMPN\[[34](https://arxiv.org/html/2608.14556#bib.bib11)\]\), and operator learning methods \(FNO\[[29](https://arxiv.org/html/2608.14556#bib.bib10)\], DeepONet\[[35](https://arxiv.org/html/2608.14556#bib.bib12)\]\)\. All models are trained for 100 epochs under the same data split \(Adam, cosine annealing10−3→10−510^\{\-3\}\\to 10^\{\-5\}, three independent training seeds\{42,1,2\}\\\{42,1,2\\\}; point estimates are the seed mean, with test\-set bootstrap and cross\-seed standard deviations in Appendix[L\.1](https://arxiv.org/html/2608.14556#A12.SS1)and[L\.2](https://arxiv.org/html/2608.14556#A12.SS2)\)\. Baselines are matched to RHMP in parameter count up to a\+20%\+20\\%headroom\.

### 5\.1Main Fixed\-Mesh Results

Table[2](https://arxiv.org/html/2608.14556#S5.T2)reports results on the six fixed\-mesh tasks\. RHMP achieves the best performance on all six tasks, with the largest margins over the strongest baseline on the gauge\-theoretic settings\. For objectivity, each baseline is evaluated in the form proposed in its original paper\. We discuss the results below, grouped by structural regime\.

Structured grids\.Navier–Stokes vorticity is the only regular\-grid task and therefore the only fixed\-mesh benchmark where the spectral\-grid baseline FNO is naturally applicable\. FNO achieves SSIM 0\.976 / NRMSE 0\.011 and RHMP achieves SSIM 0\.984 / NRMSE 0\.009, suggesting that the Hodge factorization remains useful even on structured grids\.

![Refer to caption](https://arxiv.org/html/2608.14556v1/figures/GMPF2.png)Figure 3:Torus advection–diffusion\.Left blocks summarize the inputs; right blocks compare ground truth, RHMP, and representative baselines\.Table 3:Variable\-mesh generalization on AirfRANS\. SSIM \(↑\\uparrow\), Pearson \(↑\\uparrow\), NRMSE \(↓\\downarrow\)\.Bold= best,underline= second best\.
![Refer to caption](https://arxiv.org/html/2608.14556v1/figures/GMPF3.png)
Figure 4:Ellipsoid surface flow\.Left blocks summarize the inputs; right blocks compare ground truth, RHMP, and representative baselines\. Additional qualitative comparisons in Appendix[M](https://arxiv.org/html/2608.14556#A13)\.
Cochain support and topology\.Node\-based graph baselines perform poorly on tasks whose target quantities live naturally on edges or faces, such as vorticity, surface flow, and gauge curvature\. Encoding these quantities as ordinary node features leaves cochain type, incidence structure, and identities such asd2=0d^\{2\}=0to be inferred from data\. Cell\-complex baselines improve substantially on these tasks, confirming the value of representing fields on the correct geometric support\.

Learned geometry beyond fixed incidence\.Among methods with access to higher\-order topology, RHMP gains most when the strength of geometric coupling must be learned from data\. On theU​\(1\)U\(1\)Wilson\-loop task, CW Net reaches SSIM 0\.977, while RHMP reaches SSIM 0\.993\. Both methods benefit from fixed cochain incidence, but RHMP additionally learns the metricHkH\_\{k\}controlling geometry\-dependent propagation\. The gap widens on theS​U​\(2\)SU\(2\)Yang–Mills task, where the model must capture both the differential termd​AdAand the nonlinear commutator\[A,A\]\[A,A\]\. On Maxwell–Poisson, methods respecting cochain structure outperform graph and operator\-learning baselines, consistent with the curl\-free constraint∇×𝐄=0\\nabla\\\!\\times\\\!\\mathbf\{E\}=0encoded byd2=0d^\{2\}=0\.

Frame symmetries\.Gauge\-equivariant surface methods such as GaugeEquivCNN and GEM\-CNN are strong on the ellipsoid surface\-flow task, where local tangent\-frame equivariance is well aligned with the output geometry\. The gauge\-field benchmarks instead favor cell\-complex methods\. RHMP combines fixed cochain topology with cochain\-frame\-invariant metric prediction, separating manifold\-frame symmetry from the hidden cochain\-frame symmetry introduced in this work\.

### 5\.2Mesh Generalization and Scalability

AirfRANS\[[9](https://arxiv.org/html/2608.14556#bib.bib4)\]tests a regime in which each sample carries its own unstructured mesh\. Methods that are tied to a fixed spectrum, or fixed mesh are not directly applicable, so we compare against those whose native formulation supports per\-sample variable meshes\. RHMP achieves the best performance across all three metrics\. This transfer follows from the way the metric is parameterized\. Because the RHMP metricHkH\_\{k\}depends only on cell\-levelO​\(C\)\\mathrm\{O\}\(C\)\-invariants, and notnkn\_\{k\}, the same model can be applied across meshes of differing size and connectivity \(see Figure[16](https://arxiv.org/html/2608.14556#A13.F16)\)\. Larger\-scale profiling and a 100K\-cell comparison with CW Net are in Appendix[J](https://arxiv.org/html/2608.14556#A10)\.

![Refer to caption](https://arxiv.org/html/2608.14556v1/figures/GMPF5.png)Figure 5:Maxwell–Poisson electrostatics\.Left blocks summarize the inputs; right blocks compare ground truth, RHMP, and representative baselines\.![Refer to caption](https://arxiv.org/html/2608.14556v1/figures/GMPF4.png)Figure 6:U\(1\) Wilson\-loop curvature\.Left blocks summarize the inputs; right blocks compare ground truth, RHMP, and representative baselines\.
### 5\.3Ablation Studies

We ablate four RHMP design choices on four tasks: CNS vorticity, ellipsoid surface flow,U​\(1\)U\(1\)Wilson loop, andS​U​\(2\)SU\(2\)Yang–Mills\. Each variant changes one component at a time and is retrained for5050epochs with the same split and parameter budget\. Full protocols and tables are in Appendix[K](https://arxiv.org/html/2608.14556#A11)\.

The ablations separate performance\-driving components from structure\-preserving components\. SettingHk=IH\_\{k\}=Ior disabling cross\-dimensional transport \(α=1\\alpha=1\) sharply reducesR2R^\{2\}on gauge\-curvature and structured\-grid tasks\. For example, Wilson loop drops from0\.960\.96to0\.500\.50withHk=IH\_\{k\}=I\. Ellipsoid surface flow is unaffected \(Δ​R2≤0\.05\\Delta R^\{2\}\\leq 0\.05\), consistent with its one\-step coexact target\. By contrast, learning scalardkd\_\{k\}or replacing the norm\-gated nonlinearity with element\-wise ReLU changesR2R^\{2\}by at most0\.040\.04, but breaks structural diagnostics:‖d1​d0‖F\\\|d\_\{1\}d\_\{0\}\\\|\_\{F\}grows from0to∼101\\sim 10^\{1\}, and cochain\-frame equivariance error from∼10−6\\sim 10^\{\-6\}to∼100\\sim 10^\{0\}\. Thus, fixeddkd\_\{k\}and theO​\(C\)\\mathrm\{O\}\(C\)\-equivariant nonlinearity are essential for topology and cochain\-frame symmetry preservation\.

## 6Conclusion and Limitations

Conclusion\.RHMP is a metric\-centered architecture for mesh fields: fixed coboundaries preserve the cochain complex, while learned SPD cochain metrics control geometry\-dependent propagation\. This yields positive semidefinite \(PSD\) Hodge operators, cochain\-frame equivariant layers, and exact Abelian curvature invariance\. Across seven benchmarks, these constraints bring consistent gains, especially when conservation, learned geometry, topology, and gauge structure interact\.

Limitations\.RHMP shows that learning SPD cochain metrics with fixed coboundaries can capture geometry, anisotropy, variable\-mesh transfer, and gauge\-field structure\. Natural next steps are to specialize the metric layer further: group\-valued transport to move beyond fitting SU\(2\) field strength toward full non\-Abelian gauge covariance, better\-conditioned SPD parameterizations for highly heterogeneous media \(Appendix\.[D\.3](https://arxiv.org/html/2608.14556#A4.SS3)\), and compressed or basis\-tied per\-cell metric activations for larger simulations \(Appendix\.[J](https://arxiv.org/html/2608.14556#A10)\)\.

## References

- \[1\]D\. N\. Arnold, R\. S\. Falk, and R\. Winther\(2006\)Finite element exterior calculus, homological techniques, and applications\.Acta Numerica15,pp\. 1–155\.Cited by:[§B\.2](https://arxiv.org/html/2608.14556#A2.SS2.SSS0.Px1.p1.10),[§1](https://arxiv.org/html/2608.14556#S1.p2.8),[§2](https://arxiv.org/html/2608.14556#S2.p1.3),[§3](https://arxiv.org/html/2608.14556#S3.p1.19),[§3](https://arxiv.org/html/2608.14556#S3.p6.10)\.
- \[2\]C\. Battiloro, Z\. Wang, H\. Riess, P\. Di Lorenzo, and A\. Ribeiro\(2023\)Tangent bundle convolutional learning: from manifolds to cellular sheaves and back\.External Links:2303\.11323Cited by:[§H\.3](https://arxiv.org/html/2608.14556#A8.SS3.p1.1),[§2](https://arxiv.org/html/2608.14556#S2.p4.2)\.
- \[3\]S\. Batzner, A\. Musaelian, L\. Sun, M\. Geiger, J\. P\. Mailoa, M\. Kornbluth, N\. Molinari, T\. E\. Smidt, and B\. Kozinsky\(2022\)E\(3\)\-equivariant graph neural networks for data\-efficient and accurate interatomic potentials\.Nature Communications13\(1\),pp\. 2453\.External Links:[Document](https://dx.doi.org/10.1038/s41467-022-29939-5)Cited by:[§2](https://arxiv.org/html/2608.14556#S2.p5.1)\.
- \[4\]H\. Bhatia, G\. Norgard, V\. Pascucci, and P\. Bremer\(2013\)The Helmholtz–Hodge decomposition—a survey\.IEEE Transactions on Visualization and Computer Graphics19\(8\),pp\. 1386–1404\.Cited by:[Table 5](https://arxiv.org/html/2608.14556#A5.T5.2.2.2.2.2.2.2.2.3),[§5](https://arxiv.org/html/2608.14556#S5.p2.5)\.
- \[5\]P\. B\. Bochev and J\. M\. Hyman\(2006\)Principles of mimetic discretizations of differential operators\.InCompatible Spatial Discretizations,pp\. 89–119\.External Links:[Document](https://dx.doi.org/10.1007/0-387-38034-5%5F5)Cited by:[§2](https://arxiv.org/html/2608.14556#S2.p1.3)\.
- \[6\]C\. Bodnar, F\. Di Giovanni, B\. P\. Chamberlain, P\. Liò, and M\. M\. Bronstein\(2022\)Neural sheaf diffusion: a topological perspective on heterophily and oversmoothing in GNNs\.InAdvances in Neural Information Processing Systems,Cited by:[§2](https://arxiv.org/html/2608.14556#S2.p2.3)\.
- \[7\]C\. Bodnar, F\. Frasca, N\. Otter, Y\. G\. Wang, P\. Liò, G\. F\. Montúfar, and M\. Bronstein\(2021\)Weisfeiler and Lehman go cellular: CW networks\.InAdvances in Neural Information Processing Systems,Cited by:[§2](https://arxiv.org/html/2608.14556#S2.p2.3),[§5](https://arxiv.org/html/2608.14556#S5.p3.3)\.
- \[8\]C\. Bodnar, F\. Frasca, Y\. G\. Wang, N\. Otter, G\. F\. Montúfar, P\. Liò, and M\. Bronstein\(2021\)Weisfeiler and Lehman go topological: message passing simplicial networks\.InInternational Conference on Machine Learning,Cited by:[§2](https://arxiv.org/html/2608.14556#S2.p2.3),[§5](https://arxiv.org/html/2608.14556#S5.p3.3)\.
- \[9\]F\. Bonnet, J\. Mazari, P\. Cinnella, and P\. Gallinari\(2022\)AirfRANS: high fidelity computational fluid dynamics dataset for approximating Reynolds\-averaged Navier–Stokes solutions\.InAdvances in Neural Information Processing Systems Datasets and Benchmarks Track,Cited by:[§E\.1](https://arxiv.org/html/2608.14556#A5.SS1.p2.2),[Table 5](https://arxiv.org/html/2608.14556#A5.T5.8.8.8.8.8.8.8.11.2.2),[Table 6](https://arxiv.org/html/2608.14556#A5.T6.9.9.9.9.9.9.9.9.5),[§5\.2](https://arxiv.org/html/2608.14556#S5.SS2.p1.3),[§5](https://arxiv.org/html/2608.14556#S5.p2.5)\.
- \[10\]A\. Bossavit\(1998\)Computational electromagnetism: variational formulations, complementarity, edge elements\.Academic Press,San Diego\.External Links:ISBN 9780121187101Cited by:[§2](https://arxiv.org/html/2608.14556#S2.p1.3)\.
- \[11\]J\. Brandstetter, D\. E\. Worrall, and M\. Welling\(2022\)Message passing neural pde solvers\.InInternational Conference on Learning Representations,Cited by:[§2](https://arxiv.org/html/2608.14556#S2.p5.1)\.
- \[12\]S\. Cao\(2021\)Choose a transformer: fourier or galerkin\.InAdvances in Neural Information Processing Systems,Vol\.34,pp\. 24924–24940\.Cited by:[§2](https://arxiv.org/html/2608.14556#S2.p6.1)\.
- \[13\]T\. S\. Cohen, M\. Weiler, B\. Kicanaoglu, and M\. Welling\(2019\)Gauge equivariant convolutional networks and the icosahedral CNN\.InInternational Conference on Machine Learning,Cited by:[§2](https://arxiv.org/html/2608.14556#S2.p4.2),[§5](https://arxiv.org/html/2608.14556#S5.p3.3)\.
- \[14\]M\. Creutz\(1980\)Monte Carlo study of quantized SU\(2\) gauge theory\.Physical Review D21\(8\),pp\. 2308–2315\.Cited by:[§E\.2](https://arxiv.org/html/2608.14556#A5.SS2.p1.14),[Table 5](https://arxiv.org/html/2608.14556#A5.T5.7.7.7.7.7.7.7.7.1),[Table 6](https://arxiv.org/html/2608.14556#A5.T6.7.7.7.7.7.7.7.7.4),[§5](https://arxiv.org/html/2608.14556#S5.p2.5)\.
- \[15\]P\. de Haan, M\. Weiler, T\. Cohen, and M\. Welling\(2021\)Gauge equivariant mesh CNNs: anisotropic convolutions on geometric graphs\.InInternational Conference on Learning Representations,Cited by:[§2](https://arxiv.org/html/2608.14556#S2.p4.2),[§5](https://arxiv.org/html/2608.14556#S5.p3.3)\.
- \[16\]M\. Desbrun, A\. N\. Hirani, M\. Leok, and J\. E\. Marsden\(2005\)Discrete exterior calculus\.External Links:math/0508341Cited by:[§B\.2](https://arxiv.org/html/2608.14556#A2.SS2.SSS0.Px1.p1.10),[§E\.1](https://arxiv.org/html/2608.14556#A5.SS1.p2.2),[§E\.2](https://arxiv.org/html/2608.14556#A5.SS2.p1.14),[Table 6](https://arxiv.org/html/2608.14556#A5.T6.4.4.4.4.4.4.4.4.4),[§1](https://arxiv.org/html/2608.14556#S1.p2.8),[§2](https://arxiv.org/html/2608.14556#S2.p1.3),[§3](https://arxiv.org/html/2608.14556#S3.p1.19),[§3](https://arxiv.org/html/2608.14556#S3.p6.10)\.
- \[17\]S\. Ebli, M\. Defferrard, and G\. Spreemann\(2020\)Simplicial neural networks\.InNeurIPS Workshop on Topological Data Analysis and Beyond,Cited by:[§2](https://arxiv.org/html/2608.14556#S2.p2.3)\.
- \[18\]M\. Favoni, A\. Ipp, D\. I\. Müller, and D\. Schuh\(2022\)Lattice gauge equivariant convolutional neural networks\.Physical Review Letters128\(3\),pp\. 032003\.External Links:[Document](https://dx.doi.org/10.1103/PhysRevLett.128.032003)Cited by:[§2](https://arxiv.org/html/2608.14556#S2.p4.2)\.
- \[19\]R\. Forman\(2003\)Bochner’s method for cell complexes and combinatorial Ricci curvature\.Discrete & Computational Geometry29\(3\),pp\. 323–374\.Cited by:[§H\.2](https://arxiv.org/html/2608.14556#A8.SS2.p1.7)\.
- \[20\]F\. B\. Fuchs, D\. E\. Worrall, V\. Fischer, and M\. Welling\(2020\)SE\(3\)\-transformers: 3d roto\-translation equivariant attention networks\.InAdvances in Neural Information Processing Systems,Vol\.33,pp\. 1970–1981\.Cited by:[§2](https://arxiv.org/html/2608.14556#S2.p5.1)\.
- \[21\]C\. W\. J\. Goh, C\. Bodnar, and P\. Liò\(2022\)Simplicial attention networks\.Cited by:[§2](https://arxiv.org/html/2608.14556#S2.p2.3)\.
- \[22\]M\. Hajij, G\. Zamzmi, T\. Papamarkou, N\. Miolane, A\. Guzmán\-Sáenz, K\. N\. Ramamurthy, T\. Birdal, T\. K\. Dey, S\. Mukherjee, S\. N\. Samaga, N\. Livesay, R\. Walters, P\. Rosén, and M\. T\. Schaub\(2022\)Topological deep learning: going beyond graph data\.External Links:2206\.00606Cited by:[§2](https://arxiv.org/html/2608.14556#S2.p2.3)\.
- \[23\]J\. Hansen and T\. Gebhart\(2020\)Sheaf neural networks\.InNeurIPS 2020 Workshop on Topological Data Analysis and Beyond,Cited by:[§2](https://arxiv.org/html/2608.14556#S2.p2.3)\.
- \[24\]R\. Hiptmair\(2002\)Finite elements in computational electromagnetism\.Acta Numerica11,pp\. 237–339\.External Links:[Document](https://dx.doi.org/10.1017/S0962492902000041)Cited by:[§2](https://arxiv.org/html/2608.14556#S2.p1.3)\.
- \[25\]G\. Kanwar, M\. S\. Albergo, D\. Boyda, K\. Cranmer, D\. C\. Hackett, S\. Racanière, D\. J\. Rezende, and P\. E\. Shanahan\(2020\)Equivariant flow\-based sampling for lattice gauge theory\.Physical Review Letters125\(12\),pp\. 121601\.External Links:[Document](https://dx.doi.org/10.1103/PhysRevLett.125.121601)Cited by:[§2](https://arxiv.org/html/2608.14556#S2.p4.2)\.
- \[26\]T\. N\. Kipf and M\. Welling\(2017\)Semi\-supervised classification with graph convolutional networks\.InInternational Conference on Learning Representations,Cited by:[§5](https://arxiv.org/html/2608.14556#S5.p3.3)\.
- \[27\]N\. B\. Kovachki, Z\. Li, B\. Liu, K\. Azizzadenesheli, K\. Bhattacharya, A\. M\. Stuart, and A\. Anandkumar\(2023\)Neural operator: learning maps between function spaces with applications to PDEs\.Journal of Machine Learning Research24\(89\),pp\. 1–97\.Cited by:[§2](https://arxiv.org/html/2608.14556#S2.p6.1)\.
- \[28\]Z\. Li, D\. Z\. Huang, B\. Liu, and A\. Anandkumar\(2023\)Fourier neural operator with learned deformations for PDEs on general geometries\.Journal of Machine Learning Research24\(388\),pp\. 1–26\.Cited by:[§2](https://arxiv.org/html/2608.14556#S2.p6.1)\.
- \[29\]Z\. Li, N\. Kovachki, K\. Azizzadenesheli, B\. Liu, K\. Bhattacharya, A\. Stuart, and A\. Anandkumar\(2021\)Fourier neural operator for parametric partial differential equations\.InInternational Conference on Learning Representations,Cited by:[§2](https://arxiv.org/html/2608.14556#S2.p6.1),[§5](https://arxiv.org/html/2608.14556#S5.p3.3)\.
- \[30\]Z\. Li, N\. Kovachki, K\. Azizzadenesheli, B\. Liu, A\. Stuart, K\. Bhattacharya, and A\. Anandkumar\(2020\)Multipole graph neural operator for parametric partial differential equations\.Vol\.33,pp\. 6755–6766\.Cited by:[§2](https://arxiv.org/html/2608.14556#S2.p6.1)\.
- \[31\]Z\. Li, N\. B\. Kovachki, K\. Azizzadenesheli, B\. Liu, K\. Bhattacharya, A\. M\. Stuart, and A\. Anandkumar\(2020\)Neural operator: graph kernel network for partial differential equations\.arXiv preprint arXiv:2003\.03485\.Cited by:[§2](https://arxiv.org/html/2608.14556#S2.p6.1)\.
- \[32\]Z\. Li, N\. Kovachki, C\. Choy, B\. Li, J\. Kossaifi, S\. Otta, M\. A\. Nabian, M\. Stadler, C\. Hundt, K\. Azizzadenesheli,et al\.\(2023\)Geometry\-informed neural operator for large\-scale 3d pdes\.Vol\.36,pp\. 35836–35854\.Cited by:[§2](https://arxiv.org/html/2608.14556#S2.p6.1)\.
- \[33\]Z\. Li, H\. Zheng, N\. B\. Kovachki, D\. Jin, H\. Chen, B\. Liu, K\. Azizzadenesheli, and A\. Anandkumar\(2024\)Physics\-informed neural operator for learning partial differential equations\.ACM/IMS Journal of Data Science1\(3\),pp\. 1–27\.External Links:[Document](https://dx.doi.org/10.1145/3648506)Cited by:[§2](https://arxiv.org/html/2608.14556#S2.p6.1)\.
- \[34\]C\. Liu, D\. Ruhe, F\. Eijkelboom, and P\. Forré\(2024\)Clifford group equivariant simplicial message passing networks\.InInternational Conference on Learning Representations,Cited by:[§2](https://arxiv.org/html/2608.14556#S2.p2.3),[§5](https://arxiv.org/html/2608.14556#S5.p3.3)\.
- \[35\]L\. Lu, P\. Jin, G\. Pang, Z\. Zhang, and G\. E\. Karniadakis\(2021\)Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators\.Nature Machine Intelligence3\(3\),pp\. 218–229\.Cited by:[§2](https://arxiv.org/html/2608.14556#S2.p6.1),[§5](https://arxiv.org/html/2608.14556#S5.p3.3)\.
- \[36\]D\. Luo, G\. Carleo, B\. K\. Clark, and J\. Stokes\(2021\)Gauge equivariant neural networks for quantum lattice gauge theories\.Physical Review Letters127\(27\),pp\. 276402\.External Links:[Document](https://dx.doi.org/10.1103/PhysRevLett.127.276402)Cited by:[§2](https://arxiv.org/html/2608.14556#S2.p4.2)\.
- \[37\]A\. Nousias and S\. Nousias\(2026\)HodgeFormer: transformers for learnable operators on triangular meshes through data\-driven hodge matrices\.Cited by:[§2](https://arxiv.org/html/2608.14556#S2.p3.1)\.
- \[38\]Y\. Ollivier\(2009\)Ricci curvature of Markov chains on metric spaces\.Journal of Functional Analysis256\(3\),pp\. 810–864\.Cited by:[§H\.1](https://arxiv.org/html/2608.14556#A8.SS1.p1.3)\.
- \[39\]T\. Pfaff, M\. Fortunato, A\. Sanchez\-Gonzalez, and P\. W\. Battaglia\(2021\)Learning mesh\-based simulation with graph networks\.InInternational Conference on Learning Representations,Cited by:[§2](https://arxiv.org/html/2608.14556#S2.p5.1)\.
- \[40\]A\. Sanchez\-Gonzalez, J\. Godwin, T\. Pfaff, R\. Ying, J\. Leskovec, and P\. Battaglia\(2020\)Learning to simulate complex physics with graph networks\.InProceedings of the 37th International Conference on Machine Learning,Proceedings of Machine Learning Research, Vol\.119,pp\. 8459–8468\.Cited by:[§2](https://arxiv.org/html/2608.14556#S2.p5.1)\.
- \[41\]V\. G\. Satorras, E\. Hoogeboom, and M\. Welling\(2021\)E\(n\) equivariant graph neural networks\.InInternational Conference on Machine Learning,Cited by:[§2](https://arxiv.org/html/2608.14556#S2.p5.1),[§5](https://arxiv.org/html/2608.14556#S5.p3.3)\.
- \[42\]K\. T\. Schütt, P\. Kindermans, H\. E\. Sauceda, S\. Chmiela, A\. Tkatchenko, and K\. Müller\(2017\)SchNet: a continuous\-filter convolutional neural network for modeling quantum interactions\.InAdvances in Neural Information Processing Systems,Cited by:[§2](https://arxiv.org/html/2608.14556#S2.p5.1),[§5](https://arxiv.org/html/2608.14556#S5.p3.3)\.
- \[43\]K\. T\. Schütt, O\. T\. Unke, and M\. Gastegger\(2021\)Equivariant message passing for the prediction of tensorial properties and molecular spectra\.InProceedings of the 38th International Conference on Machine Learning,Proceedings of Machine Learning Research, Vol\.139,pp\. 9377–9388\.Cited by:[§2](https://arxiv.org/html/2608.14556#S2.p5.1)\.
- \[44\]N\. Sharp, S\. Attaiki, K\. Crane, and M\. Ovsjanikov\(2022\)DiffusionNet: discretization agnostic learning on surfaces\.ACM Transactions on Graphics41\(3\),pp\. 27:1–27:16\.Cited by:[§2](https://arxiv.org/html/2608.14556#S2.p4.2)\.
- \[45\]D\. Smirnov and J\. Solomon\(2021\)HodgeNet: learning spectral geometry on triangle meshes\.ACM Transactions on Graphics40\(4\),pp\. 166:1–166:11\.Cited by:[§2](https://arxiv.org/html/2608.14556#S2.p3.1)\.
- \[46\]M\. Takamoto, T\. Praditia, R\. Leiteritz, D\. MacKinlay, F\. Alesiani, D\. Pflüger, and M\. Niepert\(2022\)PDEBench: an extensive benchmark for scientific machine learning\.InAdvances in Neural Information Processing Systems Datasets and Benchmarks Track,Cited by:[§E\.1](https://arxiv.org/html/2608.14556#A5.SS1.p2.2),[Table 5](https://arxiv.org/html/2608.14556#A5.T5.8.8.8.8.8.8.8.10.1.2),[Table 6](https://arxiv.org/html/2608.14556#A5.T6.2.2.2.2.2.2.2.2.5),[§5](https://arxiv.org/html/2608.14556#S5.p2.5)\.
- \[47\]N\. Thomas, T\. Smidt, S\. Kearnes, L\. Yang, L\. Li, K\. Kohlhoff, and P\. Riley\(2018\)Tensor field networks: rotation\- and translation\-equivariant neural networks for 3d point clouds\.arXiv preprint arXiv:1802\.08219\.Cited by:[§2](https://arxiv.org/html/2608.14556#S2.p5.1)\.
- \[48\]A\. Tran, A\. Mathews, L\. Xie, and C\. S\. Ong\(2023\)Factorized fourier neural operators\.InThe Eleventh International Conference on Learning Representations,Cited by:[§2](https://arxiv.org/html/2608.14556#S2.p6.1)\.
- \[49\]T\. Tripura and S\. Chakraborty\(2023\)Wavelet neural operator for solving parametric partial differential equations in computational mechanics problems\.Computer Methods in Applied Mechanics and Engineering404,pp\. 115783\.External Links:[Document](https://dx.doi.org/10.1016/j.cma.2022.115783)Cited by:[§2](https://arxiv.org/html/2608.14556#S2.p6.1)\.
- \[50\]P\. Veličković, G\. Cucurull, A\. Casanova, A\. Romero, P\. Liò, and Y\. Bengio\(2018\)Graph attention networks\.InInternational Conference on Learning Representations,Cited by:[§5](https://arxiv.org/html/2608.14556#S5.p3.3)\.
- \[51\]S\. Villar, D\. W\. Hogg, K\. Storey\-Fisher, W\. Yao, and B\. Blum\-Smith\(2021\)Scalars are universal: equivariant machine learning, structured like classical physics\.InAdvances in Neural Information Processing Systems,Cited by:[§4\.2](https://arxiv.org/html/2608.14556#S4.SS2.p3.8),[Remark 9](https://arxiv.org/html/2608.14556#Thmtheorem9.p1.3.3)\.
- \[52\]K\. G\. Wilson\(1974\)Confinement of quarks\.Physical Review D10\(8\),pp\. 2445–2459\.Cited by:[§E\.2](https://arxiv.org/html/2608.14556#A5.SS2.p1.14),[Table 5](https://arxiv.org/html/2608.14556#A5.T5.4.4.4.4.4.4.4.4.1),[Table 6](https://arxiv.org/html/2608.14556#A5.T6.6.6.6.6.6.6.6.6.4),[§5](https://arxiv.org/html/2608.14556#S5.p2.5)\.
- \[53\]M\. Yang, E\. Isufi, and G\. Leus\(2022\)Simplicial convolutional neural networks\.InIEEE International Conference on Acoustics, Speech and Signal Processing \(ICASSP\),Cited by:[§2](https://arxiv.org/html/2608.14556#S2.p2.3),[§5](https://arxiv.org/html/2608.14556#S5.p3.3)\.
- \[54\]M\. Yang, E\. Isufi, M\. T\. Schaub, and G\. Leus\(2022\)Simplicial convolutional filters\.IEEE Transactions on Signal Processing70,pp\. 4633–4648\.External Links:[Document](https://dx.doi.org/10.1109/TSP.2022.3207045)Cited by:[§2](https://arxiv.org/html/2608.14556#S2.p2.3)\.
- \[55\]D\. Yarotsky\(2022\)Universal approximations of invariant maps by neural networks\.Constructive Approximation55,pp\. 407–474\.Cited by:[Remark 9](https://arxiv.org/html/2608.14556#Thmtheorem9.p1.3.3)\.
- \[56\]D\. Zheng, T\. Zhong, and C\. Allen\-Blanchette\(2026\)Topology\-preserving neural operator learning via hodge decomposition\.InInternational conference on machine learning,Cited by:[§2](https://arxiv.org/html/2608.14556#S2.p3.1)\.

## Appendix ABroader Impacts

This work is directly aimed at surrogate modeling for physical simulation\. The most immediate application is reducing the computational cost of CFD simulation in engineering design scenarios \(e\.g\., airfoil optimization, thermal management\), as well as accelerating routine computations in materials science \(property inference for inhomogeneous media\) and computational physics \(surrogate solvers for lattice gauge theory\)\. The method itself addresses the approximation of physical operators\. It involves no personal data, user behavior, or decision recommendations, and introduces no surveillance or automated decision\-making capabilities\. If surrogate models are used in safety\-critical engineering simulations \(aerospace structures, nuclear reactors, etc\.\), we recommend cross\-validation against high\-fidelity solvers prior to deployment, together with quantified uncertainty estimates for the predictions\.

## Appendix BPrimer: From Continuous Metrics to Discretekk\-Form Metrics

This appendix gives the geometric dictionary behind the main text and clarifies whyHkH\_\{k\}is the natural object to learn\.

### B\.1Worked Example: One Triangle

We illustrate the constructions of Section[3](https://arxiv.org/html/2608.14556#S3)on the simplest nontrivial cell complex: a single triangleKKwith three verticesv0,v1,v2v\_\{0\},v\_\{1\},v\_\{2\}, three oriented edgese1=\(v0→v1\)e\_\{1\}=\(v\_\{0\}\\\!\\to\\\!v\_\{1\}\),e2=\(v1→v2\)e\_\{2\}=\(v\_\{1\}\\\!\\to\\\!v\_\{2\}\),e3=\(v0→v2\)e\_\{3\}=\(v\_\{0\}\\\!\\to\\\!v\_\{2\}\), and one facef1f\_\{1\}oriented counterclockwise, so that\(n0,n1,n2\)=\(3,3,1\)\(n\_\{0\},n\_\{1\},n\_\{2\}\)=\(3,3,1\)\.

#### Cochains and coboundaries\.

A 0\-cochainx0∈ℝ3x\_\{0\}\\in\\mathbb\{R\}^\{3\}assigns one scalar to each vertex; a 1\-cochainx1∈ℝ3x\_\{1\}\\in\\mathbb\{R\}^\{3\}assigns one scalar to each edge; a 2\-cochainx2∈ℝx\_\{2\}\\in\\mathbb\{R\}assigns a scalar to the face\. The coboundaryd0∈ℝ3×3d\_\{0\}\\in\\mathbb\{R\}^\{3\\times 3\}maps vertex values to oriented edge differences: the row corresponding to edgee=\(vi→vj\)e=\(v\_\{i\}\\\!\\to\\\!v\_\{j\}\)has−1\-1in columniiand\+1\+1in columnjj, sod0d\_\{0\}acts as a discrete gradient\. The coboundaryd1∈ℝ1×3d\_\{1\}\\in\\mathbb\{R\}^\{1\\times 3\}maps edge values to the face by summing oriented edge values around∂f1\\partial f\_\{1\}, sod1d\_\{1\}acts as a discrete curl\. The matrices and their correspondence to the oriented mesh are shown in Figure[7](https://arxiv.org/html/2608.14556#A2.F7)\.

The identityd1​d0=0d\_\{1\}d\_\{0\}=0holds combinatorially: each vertex on∂f1\\partial f\_\{1\}appears in two boundary edges with opposite signs, so its contributions cancel\. This is the discrete counterpart of∇×∇φ=0\\nabla\\\!\\times\\\!\\nabla\\varphi=0and holds for every inputx0x\_\{0\}\.

![Refer to caption](https://arxiv.org/html/2608.14556v1/figures/triangle_coboundary.png)Figure 7:The triangle complexKKand its coboundary matrices\. Left: three vertices \(blue\), three oriented edges \(orange, arrows indicate orientation\), and one face \(green, counterclockwise\)\. Right: the coboundary matricesd0d\_\{0\}\(discrete gradient,3×33\\times 3\) andd1d\_\{1\}\(discrete curl,1×31\\times 3\), with row/column labels matching the mesh elements\. The identityd1​d0=0d\_\{1\}d\_\{0\}=0holds by construction: every vertex on the face boundary appears in exactly two boundary edges with opposite signs\.
#### A numerical Hodge step\.

We evaluate one upper Hodge stepd0⊤​H1​d0​x0d\_\{0\}^\{\\top\}H\_\{1\}d\_\{0\}x\_\{0\}onx0=\(2\.0,5\.0,3\.0\)x\_\{0\}=\(2\.0,5\.0,3\.0\)\(Figure[8](https://arxiv.org/html/2608.14556#A2.F8)\)\.

\(i\) Coboundaryd0d\_\{0\}\.The coboundary computes oriented differences along each edge; for example,e1=\(v0→v1\)e\_\{1\}=\(v\_\{0\}\\\!\\to\\\!v\_\{1\}\)yieldsx0​\(v1\)−x0​\(v0\)=5\.0−2\.0=3\.0x\_\{0\}\(v\_\{1\}\)\-x\_\{0\}\(v\_\{0\}\)=5\.0\-2\.0=3\.0\. This step is determined by the mesh and contains no learnable parameters\.

\(ii\) Metric weightingH1H\_\{1\}\.The diagonal metricH1=diag​\(h1,h2,h3\)H\_\{1\}=\\mathrm\{diag\}\(h\_\{1\},h\_\{2\},h\_\{3\}\)scales each edge’s gradient by a per\-edge weight\. Withh1=1\.5h\_\{1\}=1\.5,h2=0\.3h\_\{2\}=0\.3,h3=2\.1h\_\{3\}=2\.1, the weighted gradient one1e\_\{1\}is1\.5×3\.0=4\.51\.5\\times 3\.0=4\.5\. Eachhih\_\{i\}has the interpretation of a local material parameter \(e\.g\., conductivity, permeability\) along edgeeie\_\{i\}, andH1H\_\{1\}is the only learnable component of the operator\.

\(iii\) Adjointd0⊤d\_\{0\}^\{\\top\}\.The adjoint coboundary aggregates the weighted gradients back to vertices, with signs determined byd0⊤d\_\{0\}^\{\\top\}: each edge contributes−1\-1to its source vertex and\+1\+1to its target\. The result isd0⊤​H1​d0​x0=\(−6\.6,5\.1,1\.5\)d\_\{0\}^\{\\top\}H\_\{1\}d\_\{0\}x\_\{0\}=\(\-6\.6,\\ 5\.1,\\ 1\.5\)\.

The composited0⊤​H1​d0d\_\{0\}^\{\\top\}H\_\{1\}d\_\{0\}is a weighted graph Laplacian whose connectivity pattern is determined byd0d\_\{0\}and whose edge weights are determined byH1H\_\{1\}\. ReplacingH1H\_\{1\}by the permuted variantH1′=diag​\(0\.3,2\.1,1\.5\)H\_\{1\}^\{\\prime\}=\\mathrm\{diag\}\(0\.3,2\.1,1\.5\)on the same complex yields\(−2\.4,5\.1,−2\.7\)\(\-2\.4,\\ 5\.1,\\ \-2\.7\)\(inset of Figure[8](https://arxiv.org/html/2608.14556#A2.F8)\);d0d\_\{0\}and the identityd1​d0=0d\_\{1\}d\_\{0\}=0are unchanged, so the change in output is attributable entirely toH1H\_\{1\}\.

![Refer to caption](https://arxiv.org/html/2608.14556v1/figures/triangle_hodge_step.png)Figure 8:One upper Hodge stepd0⊤​H1​d0​x0d\_\{0\}^\{\\top\}H\_\{1\}d\_\{0\}x\_\{0\}on the triangle of Figure[7](https://arxiv.org/html/2608.14556#A2.F7), withx0=\(2\.0,5\.0,3\.0\)x\_\{0\}=\(2\.0,5\.0,3\.0\)\. Panel 1: the coboundaryd0d\_\{0\}computes oriented differences along edges \(discrete gradient, fixed\)\. Panel 2: the diagonal metricH1=diag​\(1\.5,0\.3,2\.1\)H\_\{1\}=\\mathrm\{diag\}\(1\.5,0\.3,2\.1\)weights each edge’s gradient by a learned material parameter \(red\); edge thickness reflects the metric value\. Panel 3: the adjointd0⊤d\_\{0\}^\{\\top\}aggregates weighted gradients to vertices; arrows from each edge midpoint show the sign \(−\-at source,\+\+at target\)\. Inset: the samed0d\_\{0\}with a permuted metricH1′=diag​\(0\.3,2\.1,1\.5\)H\_\{1\}^\{\\prime\}=\\mathrm\{diag\}\(0\.3,2\.1,1\.5\)produces a different output, illustrating the topology–geometry separation\.

### B\.2Continuouskk\-form Background

#### Continuouskk\-forms\.

Akk\-formω\\omegais an object that can be integrated over orientedkk\-dimensional pieces of a domain\. Functions are0\-forms, line circulations and fluxes are represented by11\-forms, and surface intensities such as vorticity or field strength are represented by22\-forms\. The exterior derivativeddmapskk\-forms to\(k\+1\)\(k\+1\)\-forms and satisfiesd2=0d^\{2\}=0\. This identity is topological, independent of the coordinate system and material parameters; the standard discrete counterparts are developed in DEC\[[16](https://arxiv.org/html/2608.14556#bib.bib24)\]and FEEC\[[1](https://arxiv.org/html/2608.14556#bib.bib25)\]\.

#### The Riemannian metric and the Hodge star\.

A Riemannian metricggdefines lengths, angles, volumes, and the Hodge star⋆g\\star\_\{g\}\. The metric\-induced inner product onkk\-forms is

⟨ω,η⟩g=∫Mω∧⋆gη\.\\langle\\omega,\\eta\\rangle\_\{g\}=\\int\_\{M\}\\omega\\wedge\\star\_\{g\}\\eta\.\(16\)Changingggchanges this inner product and therefore changes diffusion, wave propagation, constitutive response, and curvature\-dependent terms\. The exterior derivativeddstays the same; the geometry is in⋆g\\star\_\{g\}\.

#### Discretization by cochains\.

Given an oriented cell complex, a smoothkk\-form is discretized by integrating it over eachkk\-cell:

xi=∫σikω,i=1,…,nk\.x\_\{i\}=\\int\_\{\\sigma\_\{i\}^\{k\}\}\\omega,\\qquad i=1,\\ldots,n\_\{k\}\.\(17\)The vectorx∈ℝnkx\\in\\mathbb\{R\}^\{n\_\{k\}\}is akk\-cochain\. Stokes’ theorem gives the discrete coboundary matricesdkd\_\{k\}and preservesdk\+1​dk=0d\_\{k\+1\}d\_\{k\}=0exactly\.

#### The discrete metric matrix\.

Choose basis functions\{φik\}i=1nk\\\{\\varphi\_\{i\}^\{k\}\\\}\_\{i=1\}^\{n\_\{k\}\}for discretekk\-forms\. The matrix representation of the continuous inner product is

\(Hk\)i​j=∫Mφik∧⋆gφjk\.\(H\_\{k\}\)\_\{ij\}=\\int\_\{M\}\\varphi\_\{i\}^\{k\}\\wedge\\star\_\{g\}\\varphi\_\{j\}^\{k\}\.\(18\)For any coefficient vectorzz,z⊤​Hk​zz^\{\\top\}H\_\{k\}zis theLg2L^\{2\}\_\{g\}norm of the corresponding discretekk\-form\. HenceHkH\_\{k\}is symmetric positive definite whenever the basis has no null mode\. This is the finite\-dimensional version of a Riemannian metric\.

#### FromHkH\_\{k\}to the Hodge operator\.

For akk\-cochainxkx\_\{k\}, the upper Hodge energy is

‖dk​xk‖Hk\+12=xk⊤​dk⊤​Hk\+1​dk​xk\.\\\|d\_\{k\}x\_\{k\}\\\|\_\{H\_\{k\+1\}\}^\{2\}=x\_\{k\}^\{\\top\}d\_\{k\}^\{\\top\}H\_\{k\+1\}d\_\{k\}x\_\{k\}\.\(19\)Thus the operatordk⊤​Hk\+1​dkd\_\{k\}^\{\\top\}H\_\{k\+1\}d\_\{k\}is positive semidefinite wheneverHk\+1≻0H\_\{k\+1\}\\succ 0\. The lower coexact term has the analogous formdk−1​Hk−1​dk−1⊤d\_\{k\-1\}H\_\{k\-1\}d\_\{k\-1\}^\{\\top\}\. RHMP learns these metric matrices while keeping the coboundaries fixed, preserving the topology while adapting the geometry\.

#### Why this implies the symmetry constraints\.

A metric is a coordinate\-free object\. In the network this is implemented by making the predictedHkH\_\{k\}invariant to feature\-basis changes \(O​\(C\)\\mathrm\{O\}\(C\)invariance\) and to translations, rotations, and reflections of the spatial coordinate system \(E​\(n\)\\mathrm\{E\}\(n\)invariance\)\. The corresponding Hodge messages then inherit cochain\-frame equivariance and spatial invariance/equivariance\. This is the logic used in Theorem[1](https://arxiv.org/html/2608.14556#Thmtheorem1)\.

## Appendix CCochain Complex Data Flow

![Refer to caption](https://arxiv.org/html/2608.14556v1/figures/cochain_dataflow.png)Figure 9:Schematic of the cochain\-complex data flow within one RHMP layer\. The horizontal arrows are the fixed coboundariesd0d\_\{0\}andd1d\_\{1\}\(oriented incidence; satisfyd1∘d0=0d\_\{1\}\\circ d\_\{0\}=0\)\. Each upper loopdk⊤​Hk\+1​dkd\_\{k\}^\{\\top\}H\_\{k\+1\}d\_\{k\}lifts features to the adjacent cochain space viadkd\_\{k\}, re\-weights them with the learned SPD metricHk\+1H\_\{k\+1\}, and projects back viadk⊤d\_\{k\}^\{\\top\}\.Figure[9](https://arxiv.org/html/2608.14556#A3.F9)provides a schematic view of the cochain complex data flow within one RHMP layer\. The three cochain spacesC0​\(K\)C^\{0\}\(K\),C1​\(K\)C^\{1\}\(K\),C2​\(K\)C^\{2\}\(K\)carry features on cells of increasing dimension\. The forward coboundariesd0d\_\{0\}\(discrete gradient\) andd1d\_\{1\}\(discrete curl\) are determined by oriented incidence and remain frozen throughout training; their composition satisfiesd1∘d0=0d\_\{1\}\\circ d\_\{0\}=0exactly\.

Geometry enters through two upper Hodge return loops\. The upper loop onC0C^\{0\}is the composition

C0​\(K\)→d0C1​\(K\)→H1C1​\(K\)→d0⊤C0​\(K\),x0↦d0⊤​H1​d0​x0,C^\{0\}\(K\)\\xrightarrow\{\\;d\_\{0\}\\;\}C^\{1\}\(K\)\\xrightarrow\{\\;H\_\{1\}\\;\}C^\{1\}\(K\)\\xrightarrow\{\\;d\_\{0\}^\{\\top\}\\;\}C^\{0\}\(K\),\\qquad x\_\{0\}\\mapsto d\_\{0\}^\{\\top\}H\_\{1\}\\,d\_\{0\}\\,x\_\{0\},\(20\)and the upper loop onC1C^\{1\}is

C1​\(K\)→d1C2​\(K\)→H2C2​\(K\)→d1⊤C1​\(K\),x1↦d1⊤​H2​d1​x1\.C^\{1\}\(K\)\\xrightarrow\{\\;d\_\{1\}\\;\}C^\{2\}\(K\)\\xrightarrow\{\\;H\_\{2\}\\;\}C^\{2\}\(K\)\\xrightarrow\{\\;d\_\{1\}^\{\\top\}\\;\}C^\{1\}\(K\),\\qquad x\_\{1\}\\mapsto d\_\{1\}^\{\\top\}H\_\{2\}\\,d\_\{1\}\\,x\_\{1\}\.\(21\)In each loop,dkd\_\{k\}lifts features to the adjacent cochain space, the learned metricHk\+1∈SPD​\(nk\+1\)H\_\{k\+1\}\\in\\mathrm\{SPD\}\(n\_\{k\+1\}\)re\-weights them, anddk⊤d\_\{k\}^\{\\top\}projects back\. The compositedk⊤​Hk\+1​dkd\_\{k\}^\{\\top\}H\_\{k\+1\}d\_\{k\}is positive semidefinite wheneverHk\+1≻0H\_\{k\+1\}\\succ 0\(Eq\. \([6](https://arxiv.org/html/2608.14556#S3.E6)\)\)\. A concrete numerical walkthrough on a single triangle is given in §[B\.1](https://arxiv.org/html/2608.14556#A2.SS1)\.

## Appendix DArchitecture and Training Details

### D\.1Hyperparameters

All tasks use unified hyperparameters:C=128C=128\(number of cochain channels\),L=4L=4\(number of Hodge MP layers\), MP layer MLP hidden dimension 16, diagonal\-plus\-low\-rank\-basis metricHk=Bk​Bk⊤\+diag​\(softplus⁡\(𝐡k\)\+ϵ\)H\_\{k\}=B\_\{k\}B\_\{k\}^\{\\top\}\+\\mathrm\{diag\}\(\\operatorname\{softplus\}\(\\mathbf\{h\}\_\{k\}\)\+\\epsilon\)with basis rankr=8r=8\(selected in the metric\-parameterization sweep of §[D\.3](https://arxiv.org/html/2608.14556#A4.SS3)\)\. The total parameter count is approximately 200K \(varying withn0,n1,n2n\_\{0\},n\_\{1\},n\_\{2\}\)\.

### D\.2Metric Parameterization and PSD Guarantee

The main experiments use the diagonal\-plus\-low\-rank\-basis form

Hk=Bk​Bk⊤\+diag​\(softplus⁡\(𝐡k\)\+ϵ\),ϵ=10−6,Bk∈ℝnk×r,r=8\.H\_\{k\}=B\_\{k\}B\_\{k\}^\{\\top\}\+\\mathrm\{diag\}\(\\operatorname\{softplus\}\(\\mathbf\{h\}\_\{k\}\)\+\\epsilon\),\\qquad\\epsilon=10^\{\-6\},\\qquad B\_\{k\}\\in\\mathbb\{R\}^\{n\_\{k\}\\times r\},\\qquad r=8\.The metric predictor outputs the diagonal scalars𝐡k\\mathbf\{h\}\_\{k\}perkk\-cell and the basisBkB\_\{k\}from the cell\-level invariants\. SPD is guaranteed by construction: for any nonzeroz∈ℝnkz\\in\\mathbb\{R\}^\{n\_\{k\}\},

z⊤​Hk​z=‖Bk⊤​z‖22\+∑i\(softplus⁡\(hi\)\+ϵ\)​zi2≥ϵ​‖z‖22\>0\.z^\{\\top\}H\_\{k\}z=\\\|B\_\{k\}^\{\\top\}z\\\|\_\{2\}^\{2\}\+\\sum\_\{i\}\\big\(\\operatorname\{softplus\}\(h\_\{i\}\)\+\\epsilon\\big\)z\_\{i\}^\{2\}\\geq\\epsilon\\\|z\\\|\_\{2\}^\{2\}\>0\.The PSD property of the Hodge operator then follows from the energy identity

z⊤​dk⊤​Hk\+1​dk​z=\(dk​z\)⊤​Hk\+1​\(dk​z\)≥0\.z^\{\\top\}d\_\{k\}^\{\\top\}H\_\{k\+1\}d\_\{k\}z=\(d\_\{k\}z\)^\{\\top\}H\_\{k\+1\}\(d\_\{k\}z\)\\geq 0\.No eigenvalue clipping or post\-hoc projection is used\. The pure diagonal form \(r=0r\{=\}0\) and the full Cholesky formHk=Lk​Lk⊤\+ϵ​IH\_\{k\}=L\_\{k\}L\_\{k\}^\{\\top\}\+\\epsilon Iare special cases of the same algebraic pattern; their empirical comparison against ther=8r\{=\}8default is given in §[D\.3](https://arxiv.org/html/2608.14556#A4.SS3)\.

### D\.3Empirical Comparison of Metric Parameterizations

To justify the choice of metric parameterization, we sweep five SPD variants ofHkH\_\{k\}under matched training conditions onS​U​\(2\)SU\(2\)Yang–Mills \(256\-node Delaunay lattice; 1000 train / 200 val / 500 test, 40 epochs\); the rest of the architecture and all hyperparameters are held fixed\.

Table 4:Metric parameterization sweep onS​U​\(2\)SU\(2\)Yang–Mills \(256\-node mesh; 1000 train / 200 val / 500 test, 40 epochs\)\. The diagonal\-plus\-low\-rank\-basis form withr=8r\{=\}8is the configuration used in all main\-table experiments\.Diagonal\-plus\-low\-rank\-basis is Pareto\-optimal\.The defaultr=8r\{=\}8variant attains the highest testR2R^\{2\}at the smallest parameter budget among the configurations with mesh\-dependent capacity\. Both larger\-capacity variants in the same family \(r=16r\{=\}16, full Cholesky\) and the smaller\-capacity variants \(scalar,r=0r\{=\}0\) trail it\.

Full Cholesky is harder to optimize despite higher nominal capacity\.Hk=Lk​Lk⊤\+ϵ​IH\_\{k\}=L\_\{k\}L\_\{k\}^\{\\top\}\+\\epsilon Ihas∼\\sim555×555\\timesthe parameter count of the default; its training loss reaches lower values, but validationR2R^\{2\}saturates5\.15\.1points below the default \(testR2R^\{2\}gap5\.55\.5\)\. Off\-diagonal coupling expands the optimization landscape without expanding the useful capacity for this task\.

Capacity is not the only axis\.The scalar variant \(one learnedhhper layer\) reachesR2=0\.495R^\{2\}\{=\}0\.495, only1\.51\.5points behind the default\. This is consistent with Proposition[6](https://arxiv.org/html/2608.14556#Thmtheorem6): a per\-cell\-varying diagonal is strictly more expressive than a uniform scaling, but the size of the gap is task\-dependent\. Ther=8r\{=\}8basis appears to capture the additional useful structure while remaining well\-conditioned; pushing tor=16r\{=\}16overshoots and loses ground\.

The diagonal\-plus\-r=8r\{=\}8\-basis choice in the main experiments is the Pareto\-optimal point of an architecture\-level sweep under matched optimization\.

### D\.4Batched Forward Pass

For samples sharing the same mesh, we merge the batch into the feature dimension by exploiting the linearity of sparse matrix multiply:spmm​\(d,\[𝐱1,…,𝐱B\]\)=\[d​𝐱1,…,d​𝐱B\]\\mathrm\{spmm\}\(d,\[\\mathbf\{x\}\_\{1\},\\dots,\\mathbf\{x\}\_\{B\}\]\)=\[d\\mathbf\{x\}\_\{1\},\\dots,d\\mathbf\{x\}\_\{B\}\], avoiding per\-sample loops\. The airfoil pressure task \(per\-sample meshes\) requires a per\-sample forward pass\.

### D\.5Lifting Encoder Details

The 0\-cochain MLP input is\[𝐟0​‖deg⁡\(v\)‖​d¯​\(v\)\]\[\\mathbf\{f\}\_\{0\}\\\|\\deg\(v\)\\\|\\bar\{d\}\(v\)\], whered¯​\(v\)\\bar\{d\}\(v\)is the average distance to the neighbors\. The 2\-cochain MLP input is\[∑e∈∂fσe​𝐱1​\(e\)​‖Af‖​α0,α1,α2\]\[\\sum\_\{e\\in\\partial f\}\\sigma\_\{e\}\\mathbf\{x\}\_\{1\}\(e\)\\\|A\_\{f\}\\\|\\alpha\_\{0\},\\alpha\_\{1\},\\alpha\_\{2\}\], whereσe∈\{\+1,−1\}\\sigma\_\{e\}\\in\\\{\+1,\-1\\\}is given by the sparsity structure ofd1d\_\{1\}\.

### D\.6Compute Environment

All experiments were conducted on a workstation with two NVIDIA RTX PRO 6000 Blackwell GPUs \(96 GB each\); each individual run uses a single GPU\. Software environment: Python 3\.12, PyTorch 2\.10, CUDA 12\.8\. No third\-party geometric deep learning library dependencies \(torch\-scatter, PyG, etc\.\) are used; sparse matrix operations use PyTorch’s nativetorch\.sparse\.

### D\.7Training Cost

Each model\-task pair is trained for 100 epochs, and the total training time of RHMP per task is as follows: NS Vorticity 20 min, Torus Adv\-Diff 5 min, Ellipsoid Flow 24 min, Maxwell 14 min, Wilson Loop 21 min, Yang\-Mills 27 min, Airfoil 35 min\. The total training time of RHMP across all 7 tasks is approximately 2\.5 hours\. The full experiment including 12 baselines takes about 24 GPU hours\. Code and pre\-trained checkpoints will be released with the paper\.

## Appendix EBenchmark Tasks

### E\.1Design Principles

The 7 tasks are organized around the structural capability being tested\. Public benchmarks serve as external anchors, and the remaining tasks fill in structural dimensions beyond currently public data\. Table[5](https://arxiv.org/html/2608.14556#A5.T5)lists, for each task, the corresponding physical domain, the structural capability being tested, the diagnostic signal associated with that capability, and the external anchor\.

Table 5:Design mapping of the benchmark tasks\. Each task corresponds to a structural capability that can be characterized independently; the diagnostic\-signal column lists the structural cue measured by that task\.The seven physical systems span a spectrum of structural difficulty\. Compressible Navier–Stokes\[[46](https://arxiv.org/html/2608.14556#bib.bib17)\]and external aerodynamics\[[9](https://arxiv.org/html/2608.14556#bib.bib4)\]place the architecture against established surrogates on common PDEs and engineering CFD; the remaining tasks reach into more demanding regimes \(advection–diffusion on closed surfaces, surface flow on curved manifolds, electrostatics with hard conservation constraints, andU​\(1\)U\(1\)/S​U​\(2\)SU\(2\)lattice gauge theory\), where standard benchmarks do not yet probe the relevant differential and gauge structure\. For these systems we follow the established discretization schemes for the underlying physics\[[16](https://arxiv.org/html/2608.14556#bib.bib24)\], with the per\-task numerical setup detailed below\.

### E\.2Dataset Composition and Generation Protocol

Table[6](https://arxiv.org/html/2608.14556#A5.T6)gives the input/output, reference method, mesh size, and number of samples for each task\. The following lists per\-task sampling distributions, hyperparameters, and implementation details that complement §[5](https://arxiv.org/html/2608.14556#S5)\.Torus advection–diffusion: IMEX time integration over 15 steps atΔ​t=0\.02\\Delta t=0\.02,ν=0\.01\\nu=0\.01; implicit diffusion is solved by sparse LU factorization of the lumped\-mass\-plus\-stiffness system, and advection uses an explicit upwind flux\.Ellipsoid surface flow: the stream functionψ\\psiis sampled from spherical harmonics, and𝐯tan=𝐧×∇ψ\\mathbf\{v\}\_\{\\text\{tan\}\}=\\mathbf\{n\}\\times\\nabla\\psiis evaluated by cotangent\-weighted discrete gradients\[[16](https://arxiv.org/html/2608.14556#bib.bib24)\]\.Maxwell–Poisson: the electric field𝐄=−∇ϕ\\mathbf\{E\}=\-\\nabla\\phiis recovered by the discrete gradient applied to the SuperLU solution on the same mesh\.U​\(1\)U\(1\)Wilson loop: edge phasesθ\\thetaare sampled standard\-Gaussian and reduced modulo2​π2\\pi;F=d​θF=d\\thetais the oriented plaquette product\[[52](https://arxiv.org/html/2608.14556#bib.bib22)\]\.S​U​\(2\)SU\(2\)Yang–Mills: the connectionAAis sampled independently on theS​U​\(2\)SU\(2\)Lie algebra;F=d​A\+\[A,A\]F=dA\+\[A,A\]is evaluated on each plaquette following\[[14](https://arxiv.org/html/2608.14556#bib.bib23)\]\. The splits are all70/15/1570/15/15\(train/validation/test\), and each task is repeated with 3 seeds and reported as an average\.

Table 6:Full description of the 7 benchmark tasks\.
### E\.3Implementation Details

Numerical setup\.All reference fields are computed at numerical tolerance≤10−6\\leq 10^\{\-6\}using validated FEM and sparse\-direct\-solve routines \(SciPy SuperLU, cotangent\-weight FEM\); per\-task numerical schemes are listed in §[E](https://arxiv.org/html/2608.14556#A5)\. Code, simulation scripts, and datasets are released together for reproduction\.

Baseline protocol\.Each baseline is used under the input/output form of its original paper, without any core architectural modification to adapt to the tasks in this paper\. We separate three evaluation regimes by the native domain of each method: \(i\) the six fixed\-mesh tasks \(Table[2](https://arxiv.org/html/2608.14556#S5.T2)\), evaluated on the 11 baselines whose native definition admits irregular meshes; \(ii\) the regular\-grid task NS Vorticity, where FNO is additionally compared inline in §[5\.1](https://arxiv.org/html/2608.14556#S5.SS1); \(iii\) per\-sample variable meshes \(AirfRANS, §[5\.2](https://arxiv.org/html/2608.14556#S5.SS2), Table[3](https://arxiv.org/html/2608.14556#S5.T3)\), where only methods that admit a different mesh per sample participate\. All runnable models are trained for100100epochs under the same data split, and each baseline is matched to RHMP in parameter count up to a\+20%\+20\\%headroom\.

### E\.4Robustness Checks

We probe two robustness axes that bear directly on the architecture’s claims, on the tasks where each axis is most diagnostic\. \(i\)Resolution sensitivity: onU​\(1\)U\(1\)Wilson loop we changed the node count from10241024to512,2048,4096512,2048,4096, and theR2R^\{2\}of RHMP remained above0\.940\.94at all four resolutions\. \(ii\)Parameter OOD: on Maxwell–Poisson we extended the source\-density magnitude of the test set from the training distributionUnif​\[0\.5,1\.5\]\\mathrm\{Unif\}\[0\.5,1\.5\]toUnif​\[0\.2,2\.5\]\\mathrm\{Unif\}\[0\.2,2\.5\], and the NRMSE degradation of RHMP \(Δ=0\.013\\Delta=0\.013\) was smaller than that of the second\-best cell\-complex baseline \(CW NetΔ=0\.024\\Delta=0\.024\)\.

## Appendix FFull Proof of Theorem[1](https://arxiv.org/html/2608.14556#Thmtheorem1)

Conventions\.𝐱k∈ℝnk×C\\mathbf\{x\}\_\{k\}\\in\\mathbb\{R\}^\{n\_\{k\}\\times C\}is in row\-vector format \(each row is aCC\-dimensional feature of akk\-cell\)\.R∈O​\(C\)R\\in\\mathrm\{O\}\(C\)acts on the channel \(column\) index via right\-multiplication𝐱k↦𝐱k​R⊤\\mathbf\{x\}\_\{k\}\\mapsto\\mathbf\{x\}\_\{k\}R^\{\\top\}, with the sameRRapplied to all cells and cochain degrees\. We*use normalization layers without learnable affine \(scale/shift\) parameters*: theO​\(C\)\\mathrm\{O\}\(C\)action treats all channel bases symmetrically, and a per\-channel learnableγ\\gammaorβ\\betawould select a preferred direction in channel space\. This is the condition used in our implementation \(Section[4\.1](https://arxiv.org/html/2608.14556#S4.SS1)\)\.

###### Lemma 2\(Positive semidefiniteness\)\.

IfA⪰0A\\succeq 0andB⪰0B\\succeq 0, thenLk=dk−1​A​dk−1⊤\+dk⊤​B​dkL\_\{k\}=d\_\{k\-1\}Ad\_\{k\-1\}^\{\\top\}\+d\_\{k\}^\{\\top\}Bd\_\{k\}is positive semidefinite\. IfA≻0A\\succ 0andB≻0B\\succ 0, this holds in particular for the learned metric matrices used by RHMP\.

###### Proof\.

For any𝐱∈ℝnk×C\\mathbf\{x\}\\in\\mathbb\{R\}^\{n\_\{k\}\\times C\},

tr​\(𝐱⊤​Lk​𝐱\)=‖dk−1⊤​𝐱‖A2\+‖dk​𝐱‖B2≥0\.\\mathrm\{tr\}\(\\mathbf\{x\}^\{\\top\}L\_\{k\}\\mathbf\{x\}\)=\\\|d\_\{k\-1\}^\{\\top\}\\mathbf\{x\}\\\|\_\{A\}^\{2\}\+\\\|d\_\{k\}\\mathbf\{x\}\\\|\_\{B\}^\{2\}\\geq 0\.The same argument applies channel by channel and then sums over channels\. ∎

###### Lemma 3\(Cochain\-frame equivariance\)\.

LetΦ\(ℓ\)\\Phi^\{\(\\ell\)\}denote one layer of the architecture in Theorem[1](https://arxiv.org/html/2608.14556#Thmtheorem1), composed of metric\-weighted Hodge operators, cochain\-frame\-invariant metricsHkH\_\{k\}, norm\-gated activationσgate\\sigma\_\{\\mathrm\{gate\}\}, per\-cell RMSNorm \(without learnable affine parameters\), and a residual connection\. Then for allR∈O​\(C\)R\\in\\mathrm\{O\}\(C\),

Φ\(ℓ\)​\(𝐱​R⊤\)=Φ\(ℓ\)​\(𝐱\)​R⊤,Hk​\(𝐱​R⊤\)=Hk​\(𝐱\)\.\\Phi^\{\(\\ell\)\}\(\\mathbf\{x\}R^\{\\top\}\)=\\Phi^\{\(\\ell\)\}\(\\mathbf\{x\}\)\\,R^\{\\top\},\\qquad H\_\{k\}\(\\mathbf\{x\}R^\{\\top\}\)=H\_\{k\}\(\\mathbf\{x\}\)\.

###### Proof\.

The proof rests on the following standard algebraic fact, which we state explicitly for clarity\.

###### Lemma 4\(Kronecker product commutation\)\.

Let𝐱∈ℝm×C\\mathbf\{x\}\\in\\mathbb\{R\}^\{m\\times C\},A∈ℝm′×mA\\in\\mathbb\{R\}^\{m^\{\\prime\}\\times m\}acting on the row \(cell\) index,B∈ℝC×CB\\in\\mathbb\{R\}^\{C\\times C\}acting on the column \(channel\) index\. Identifying𝐱\\mathbf\{x\}with a vector inℝm⊗ℝC\\mathbb\{R\}^\{m\}\\otimes\\mathbb\{R\}^\{C\}, the mixed\-product property of Kronecker products\(A⊗B\)​\(C⊗D\)=\(A​C\)⊗\(B​D\)\(A\\otimes B\)\(C\\otimes D\)=\(AC\)\\otimes\(BD\)gives

A​\(𝐱​B⊤\)=\(A⊗IC\)​𝐱​\(Im⊗B⊤\)=\(A​𝐱\)​B⊤\.A\\,\(\\mathbf\{x\}\\,B^\{\\top\}\)=\(A\\otimes I\_\{C\}\)\\,\\mathbf\{x\}\\,\(I\_\{m\}\\otimes B^\{\\top\}\)=\(A\\,\\mathbf\{x\}\)\\,B^\{\\top\}\.\(22\)Operators on orthogonal tensor indices commute\.

We now verify each component of the layer\.

Linear Hodge operators\.EachMj∈ℝnk×nk′M\_\{j\}\\in\\mathbb\{R\}^\{n\_\{k\}\\times n\_\{k^\{\\prime\}\}\}\(e\.g\.dk−1​Hk−1↓​dk−1⊤d\_\{k\-1\}H\_\{k\-1\}^\{\\downarrow\}d\_\{k\-1\}^\{\\top\}ordk⊤​Hk\+1↑​dkd\_\{k\}^\{\\top\}H\_\{k\+1\}^\{\\uparrow\}d\_\{k\}\) acts on the cell index, whileR⊤∈ℝC×CR^\{\\top\}\\in\\mathbb\{R\}^\{C\\times C\}acts on the channel index\. By Lemma[4](https://arxiv.org/html/2608.14556#Thmtheorem4):

Mj​\(𝐱​R⊤\)=\(Mj​𝐱\)​R⊤\.M\_\{j\}\\,\(\\mathbf\{x\}\\,R^\{\\top\}\)=\(M\_\{j\}\\,\\mathbf\{x\}\)\\,R^\{\\top\}\.\(23\)
Frame\-invariance ofHkH\_\{k\}\.HkH\_\{k\}is driven by the row\-wise statisticsψ\(𝐱k\)=\{∥𝐱i∥22,𝐱i𝐱j⊤:j∼i\}\\psi\(\\mathbf\{x\}\_\{k\}\)=\\\{\\\|\\mathbf\{x\}\_\{i\}\\\|\_\{2\}^\{2\},\\;\\mathbf\{x\}\_\{i\}\\mathbf\{x\}\_\{j\}^\{\\top\}:j\\sim i\\\}together with geometric invariants\. For the Frobenius norm, the cyclic property of trace and orthogonalityR⊤​R=IR^\{\\top\}R=Igive

‖𝐱k​R⊤‖F2=tr​\(R​𝐱k⊤​𝐱k​R⊤\)​=cyc​tr​\(𝐱k⊤​𝐱k​R⊤​R\)=tr​\(𝐱k⊤​𝐱k\)=‖𝐱k‖F2\.\\\|\\mathbf\{x\}\_\{k\}R^\{\\top\}\\\|\_\{F\}^\{2\}=\\mathrm\{tr\}\\\!\\bigl\(R\\,\\mathbf\{x\}\_\{k\}^\{\\top\}\\mathbf\{x\}\_\{k\}\\,R^\{\\top\}\\bigr\)\\overset\{\\mathrm\{cyc\}\}\{=\}\\mathrm\{tr\}\\\!\\bigl\(\\mathbf\{x\}\_\{k\}^\{\\top\}\\mathbf\{x\}\_\{k\}\\,R^\{\\top\}R\\bigr\)=\\mathrm\{tr\}\(\\mathbf\{x\}\_\{k\}^\{\\top\}\\mathbf\{x\}\_\{k\}\)=\\\|\\mathbf\{x\}\_\{k\}\\\|\_\{F\}^\{2\}\.For the per\-cell inner product, with𝐱i∈ℝ1×C\\mathbf\{x\}\_\{i\}\\in\\mathbb\{R\}^\{1\\times C\}denoting theii\-th row:\(𝐱i​R⊤\)​\(𝐱j​R⊤\)⊤=𝐱i​R⊤​R​𝐱j⊤=𝐱i​𝐱j⊤\(\\mathbf\{x\}\_\{i\}R^\{\\top\}\)\(\\mathbf\{x\}\_\{j\}R^\{\\top\}\)^\{\\top\}=\\mathbf\{x\}\_\{i\}R^\{\\top\}R\\,\\mathbf\{x\}\_\{j\}^\{\\top\}=\\mathbf\{x\}\_\{i\}\\mathbf\{x\}\_\{j\}^\{\\top\}\. Henceψ​\(𝐱​R⊤\)=ψ​\(𝐱\)\\psi\(\\mathbf\{x\}R^\{\\top\}\)=\\psi\(\\mathbf\{x\}\), and sinceHk=f​\(ψ​\(𝐱\)\)H\_\{k\}=f\(\\psi\(\\mathbf\{x\}\)\)for a deterministicff, we obtainHk​\(𝐱​R⊤\)=Hk​\(𝐱\)H\_\{k\}\(\\mathbf\{x\}R^\{\\top\}\)=H\_\{k\}\(\\mathbf\{x\}\)\.

Norm\-gated activation\.For the row\-wise norm‖𝐦i​R⊤‖2=‖𝐦i‖2\\\|\\mathbf\{m\}\_\{i\}R^\{\\top\}\\\|\_\{2\}=\\\|\\mathbf\{m\}\_\{i\}\\\|\_\{2\}\(sinceRRis orthogonal\), we haveσgate​\(𝐦​R⊤\)=g​\(‖𝐦​R⊤‖\)​𝐦​R⊤=g​\(‖𝐦‖\)​𝐦​R⊤=σgate​\(𝐦\)​R⊤\\sigma\_\{\\mathrm\{gate\}\}\(\\mathbf\{m\}\\,R^\{\\top\}\)=g\(\\\|\\mathbf\{m\}\\,R^\{\\top\}\\\|\)\\,\\mathbf\{m\}\\,R^\{\\top\}=g\(\\\|\\mathbf\{m\}\\\|\)\\,\\mathbf\{m\}\\,R^\{\\top\}=\\sigma\_\{\\mathrm\{gate\}\}\(\\mathbf\{m\}\)\\,R^\{\\top\}\.

Per\-cell RMSNorm \(without learnable affine parameters\)\.RMSNorm​\(𝐱i\)=𝐱i⋅C/‖𝐱i‖2\\mathrm\{RMSNorm\}\(\\mathbf\{x\}\_\{i\}\)=\\mathbf\{x\}\_\{i\}\\cdot\\sqrt\{C\}\\,/\\,\\\|\\mathbf\{x\}\_\{i\}\\\|\_\{2\}, whereRMS​\(𝐱i\)=‖𝐱i‖2/C\\mathrm\{RMS\}\(\\mathbf\{x\}\_\{i\}\)=\\\|\\mathbf\{x\}\_\{i\}\\\|\_\{2\}/\\sqrt\{C\}\. Then:

RMSNorm​\(𝐱i​R⊤\)=𝐱i​R⊤⋅C‖𝐱i​R⊤‖2=𝐱i​R⊤⋅C‖𝐱i‖2=RMSNorm​\(𝐱i\)​R⊤\.\\mathrm\{RMSNorm\}\(\\mathbf\{x\}\_\{i\}R^\{\\top\}\)=\\frac\{\\mathbf\{x\}\_\{i\}R^\{\\top\}\\cdot\\sqrt\{C\}\}\{\\\|\\mathbf\{x\}\_\{i\}R^\{\\top\}\\\|\_\{2\}\}=\\frac\{\\mathbf\{x\}\_\{i\}R^\{\\top\}\\cdot\\sqrt\{C\}\}\{\\\|\\mathbf\{x\}\_\{i\}\\\|\_\{2\}\}=\\mathrm\{RMSNorm\}\(\\mathbf\{x\}\_\{i\}\)\\,R^\{\\top\}\.*Remark\.*Standard LayerNorm with mean\-subtraction uses the all\-ones vector𝟏∈ℝC\\mathbf\{1\}\\in\\mathbb\{R\}^\{C\}as a preferred channel direction; for a generalR∈O​\(C\)R\\in\\mathrm\{O\}\(C\),∑c\(𝐱i​R⊤\)c≠∑cxi​c\\sum\_\{c\}\(\\mathbf\{x\}\_\{i\}R^\{\\top\}\)\_\{c\}\\neq\\sum\_\{c\}x\_\{ic\}\. Learnable scaleγ∈ℝC\\gamma\\in\\mathbb\{R\}^\{C\}or shiftβ∈ℝC\\beta\\in\\mathbb\{R\}^\{C\}similarly selects a channel basis\. Our implementation therefore uses RMSNorm without learnable affine parameters; the measured equivariance error is2×10−72\\times 10^\{\-7\}\(Table[7](https://arxiv.org/html/2608.14556#A7.T7)\)\.

Residual connection\.\(𝐱\+σ​\(M​𝐱\)\)​R⊤=𝐱​R⊤\+σ​\(M​𝐱\)​R⊤\(\\mathbf\{x\}\+\\sigma\(M\\mathbf\{x\}\)\)\\,R^\{\\top\}=\\mathbf\{x\}\\,R^\{\\top\}\+\\sigma\(M\\mathbf\{x\}\)\\,R^\{\\top\}, since right\-multiplication byR⊤R^\{\\top\}distributes over addition\. Composing all components, each layer isO​\(C\)\\mathrm\{O\}\(C\)\-equivariant\. ∎

###### Lemma 5\(E​\(n\)\\mathrm\{E\}\(n\)\-invariance and equivariance\)\.

IfHkH\_\{k\}depends only onE​\(n\)\\mathrm\{E\}\(n\)\-invariant quantities ofKK’s embedding \(Theorem[1](https://arxiv.org/html/2608.14556#Thmtheorem1)\(iii\)\), then all cochain features𝐱k\(ℓ\)\\mathbf\{x\}\_\{k\}^\{\(\\ell\)\}areE​\(n\)\\mathrm\{E\}\(n\)\-invariant, scalar readouts areE​\(n\)\\mathrm\{E\}\(n\)\-invariant, and vector reconstructions areE​\(n\)\\mathrm\{E\}\(n\)\-equivariant\.

###### Proof\.

LetT=\(Rs,t\)∈E​\(n\)T=\(R\_\{s\},t\)\\in\\mathrm\{E\}\(n\)act on the embedding by𝐫v↦Rs​𝐫v\+t\\mathbf\{r\}\_\{v\}\\mapsto R\_\{s\}\\mathbf\{r\}\_\{v\}\+t\.

Lifting encoder \(base case\)\.All encoder inputs areE​\(n\)\\mathrm\{E\}\(n\)\-invariant: edge lengthsℓe=‖𝐫j−𝐫i‖\\ell\_\{e\}=\\\|\\mathbf\{r\}\_\{j\}\-\\mathbf\{r\}\_\{i\}\\\|are preserved becauseRsR\_\{s\}preserves norms andttcancels in differences; vertex degreesdeg⁡\(v\)\\deg\(v\)are combinatorial; face areasAf=12​‖\(𝐫1−𝐫0\)×\(𝐫2−𝐫0\)‖A\_\{f\}=\\tfrac\{1\}\{2\}\\\|\(\\mathbf\{r\}\_\{1\}\-\\mathbf\{r\}\_\{0\}\)\\times\(\\mathbf\{r\}\_\{2\}\-\\mathbf\{r\}\_\{0\}\)\\\|are preserved because forRs∈O​\(n\)R\_\{s\}\\in\\mathrm\{O\}\(n\),Rs​𝐚×Rs​𝐛=\(detRs\)​Rs​\(𝐚×𝐛\)R\_\{s\}\\mathbf\{a\}\\times R\_\{s\}\\mathbf\{b\}=\(\\det R\_\{s\}\)\\,R\_\{s\}\(\\mathbf\{a\}\\times\\mathbf\{b\}\), so‖Rs​𝐚×Rs​𝐛‖=\|detRs\|⋅‖𝐚×𝐛‖=‖𝐚×𝐛‖\\\|R\_\{s\}\\mathbf\{a\}\\times R\_\{s\}\\mathbf\{b\}\\\|=\|\\det R\_\{s\}\|\\cdot\\\|\\mathbf\{a\}\\times\\mathbf\{b\}\\\|=\\\|\\mathbf\{a\}\\times\\mathbf\{b\}\\\|\(the norm absorbs the sign under reflections\); interior angles are determined by normalized inner products, preserved byRsR\_\{s\}\. Therefore𝐱k\(0\)\\mathbf\{x\}\_\{k\}^\{\(0\)\}isE​\(n\)\\mathrm\{E\}\(n\)\-invariant\.

Inductive step \(layerℓ→ℓ\+1\\ell\\to\\ell\+1\)\.Assume𝐱k\(ℓ\)\\mathbf\{x\}\_\{k\}^\{\(\\ell\)\}isE​\(n\)\\mathrm\{E\}\(n\)\-invariant\. Then:

1. \(a\)ψ​\(𝐱k\(ℓ\)\)\\psi\(\\mathbf\{x\}\_\{k\}^\{\(\\ell\)\}\)is invariant \(norms and inner products of invariant vectors are invariant\);
2. \(b\)Hk\(ℓ\)=f​\(ψ​\(𝐱k\(ℓ\)\)\)H\_\{k\}^\{\(\\ell\)\}=f\(\\psi\(\\mathbf\{x\}\_\{k\}^\{\(\\ell\)\}\)\)is invariant \(ffis a deterministic function of invariant inputs\);
3. \(c\)Mj​𝐱k\(ℓ\)M\_\{j\}\\mathbf\{x\}\_\{k\}^\{\(\\ell\)\}is invariant \(dkd\_\{k\}is a topological operator independent of the embedding, andHk\(ℓ\)H\_\{k\}^\{\(\\ell\)\}is invariant by \(b\)\);
4. \(d\)σgate​\(Mj​𝐱k\(ℓ\)\)\\sigma\_\{\\mathrm\{gate\}\}\(M\_\{j\}\\mathbf\{x\}\_\{k\}^\{\(\\ell\)\}\)is invariant \(g​\(‖𝐦‖\)​𝐦g\(\\\|\\mathbf\{m\}\\\|\)\\,\\mathbf\{m\}with‖𝐦‖\\\|\\mathbf\{m\}\\\|and𝐦\\mathbf\{m\}both invariant\);
5. \(e\)RMSNorm​\(⋅\)\\mathrm\{RMSNorm\}\(\\cdot\)preserves invariance \(‖𝐱i‖\\\|\\mathbf\{x\}\_\{i\}\\\|is invariant, so the normalized output inherits invariance\);
6. \(f\)the residual sum𝐱k\(ℓ\)\+\(gated, normalized update\)\\mathbf\{x\}\_\{k\}^\{\(\\ell\)\}\+\(\\text\{gated, normalized update\}\)is invariant \(a sum of invariant quantities is invariant\)\.

By induction,𝐱k\(ℓ\)\\mathbf\{x\}\_\{k\}^\{\(\\ell\)\}isE​\(n\)\\mathrm\{E\}\(n\)\-invariant for allℓ\\ell\.

Scalar readout\.y^​\(v\)=h​\(𝐱0\(L\)​\(v\)\)\\hat\{y\}\(v\)=h\(\\mathbf\{x\}\_\{0\}^\{\(L\)\}\(v\)\)is a function of invariant inputs, hence invariant\.

Vector reconstruction\.𝐯^​\(v\)=1deg⁡\(v\)​∑e∋vw​\(𝐱1​\(e\),‖𝐫i​j‖2,‖𝐱0i‖2,‖𝐱0j‖2\)⋅\(𝐫j−𝐫i\)\\hat\{\\mathbf\{v\}\}\(v\)=\\tfrac\{1\}\{\\deg\(v\)\}\\sum\_\{e\\ni v\}w\(\\mathbf\{x\}\_\{1\}\(e\),\\,\\\|\\mathbf\{r\}\_\{ij\}\\\|^\{2\},\\,\\\|\\mathbf\{x\}\_\{0\}^\{i\}\\\|^\{2\},\\,\\\|\\mathbf\{x\}\_\{0\}^\{j\}\\\|^\{2\}\)\\cdot\(\\mathbf\{r\}\_\{j\}\-\\mathbf\{r\}\_\{i\}\)\. All arguments ofwwareE​\(n\)\\mathrm\{E\}\(n\)\-invariant, sowwis invariant\. The displacement𝐫j−𝐫i↦Rs​\(𝐫j−𝐫i\)\\mathbf\{r\}\_\{j\}\-\\mathbf\{r\}\_\{i\}\\mapsto R\_\{s\}\(\\mathbf\{r\}\_\{j\}\-\\mathbf\{r\}\_\{i\}\)isE​\(n\)\\mathrm\{E\}\(n\)\-equivariant \(translations cancel\)\. The product of an invariant scalar and an equivariant vector is equivariant:𝐯^↦Rs​𝐯^\\hat\{\\mathbf\{v\}\}\\mapsto R\_\{s\}\\hat\{\\mathbf\{v\}\}\. ∎

Proof thatd2=0d^\{2\}=0\.The identityd1​d0=0d\_\{1\}d\_\{0\}=0is a purely algebraic consequence of the CW complex structure\. Each column ofd0d\_\{0\}is indexed by a vertexvvand records±1\\pm 1on the edges incident tovv\. Each row ofd1d\_\{1\}is indexed by a faceffand records±1\\pm 1on the boundary edges offf\(with signs from the chosen orientation\)\. The\(f,v\)\(f,v\)\-entry ofd1​d0d\_\{1\}d\_\{0\}equals∑e\(d1\)f,e​\(d0\)e,v\\sum\_\{e\}\(d\_\{1\}\)\_\{f,e\}\\,\(d\_\{0\}\)\_\{e,v\}\. Ifv∉∂fv\\notin\\partial f, every term vanishes\. Ifv∈∂fv\\in\\partial f, exactly two boundary edges offfare incident tovv\. The boundary orientation makes the two signed incidence products opposite in sign, so their contributions cancel\. This is the cellular statement that the boundary of an oriented boundary is empty\. The cancellation holds for every\(f,v\)\(f,v\)pair, henced1​d0=0d\_\{1\}d\_\{0\}=0exactly\. The operatorsd0,d1d\_\{0\},d\_\{1\}are frozen throughout training, so the identity is preserved at all times\.

## Appendix GSymmetry Verification Protocol

Each symmetry subgroup test selects its scope according to the natural object of action\. The spatial groupE​\(n\)\\mathrm\{E\}\(n\)and theℤ2\\mathbb\{Z\}\_\{2\}edge\-orientation act on the cell complexKKitself: translation/rotation/reflection act on the vertex coordinates ofKKafter whichKKis rebuilt, and theℤ2\\mathbb\{Z\}\_\{2\}edge orientation is realized by flipping the endpoints of a single edge and synchronously updating the signs ofd0​\[e,:\]d\_\{0\}\[e,:\]andd1​\[:,e\]d\_\{1\}\[:,e\]\. FiberO​\(C\)\\mathrm\{O\}\(C\)\-type symmetries act on theCC\-dimensional cochain channels, whose natural test object is the single\-layer weighted Hodge operator described in Theorem[1](https://arxiv.org/html/2608.14556#Thmtheorem1); for fair comparison with baselines, for each baseline we extract its corresponding message passing module, apply the channel\-space transformation to a random hidden stateh0∈ℝn0×Ch\_\{0\}\\in\\mathbb\{R\}^\{n\_\{0\}\\times C\}, and compare outputs\. LetΦ\\Phidenote the operator under test andT^\\hat\{T\}the action ofTTon the output side \(identity for invariance tests,T−1T^\{\-1\}for equivariance tests\); then the reported quantity is

err​\(T\)=‖T^​Φ​\(T⋅input\)−Φ​\(input\)‖F‖Φ​\(input\)‖F,\\mathrm\{err\}\(T\)=\\frac\{\\\|\\hat\{T\}\\,\\Phi\(T\\\!\\cdot\\\!\\mathrm\{input\}\)\-\\Phi\(\\mathrm\{input\}\)\\\|\_\{F\}\}\{\\\|\\Phi\(\\mathrm\{input\}\)\\\|\_\{F\}\},and each \(model, subgroup\) cell is averaged over55independent random transformations\.

We use two test meshes to balance comparability and generality: the primary test mesh is a4×44\\\!\\times\\\!4regular Delaunay \(\(n0,n1,n2\)=\(16,33,18\)\(n\_\{0\},n\_\{1\},n\_\{2\}\)=\(16,33,18\)\), whose regular structure allows FNO to participate on its native grid as well; the auxiliary test mesh is a Delaunay of66uniformly random points \(\(n0,n1,n2\)=\(6,11,6\)\(n\_\{0\},n\_\{1\},n\_\{2\}\)=\(6,11,6\)\), used to verify the robustness of the same protocol on irregular connectivity, with full results listed in Table[9](https://arxiv.org/html/2608.14556#A7.T9)\. Both test meshes useC=32C=32channels,L=2L=2layers, evaluation mode, and initialization seed4242\. The specific distributions of the transformations are as follows: translationsv∼𝒩​\(0,0\.01​I2\)v\\sim\\mathcal\{N\}\(0,0\.01\\,I\_\{2\}\), rotationsR​\(θ\)R\(\\theta\)withθ∼Unif​\[0,2​π\)\\theta\\sim\\mathrm\{Unif\}\[0,2\\pi\), reflections via random\-direction Householder; forO​\(C\)\\mathrm\{O\}\(C\),QQis taken from the QR decomposition of a32×3232\\\!\\times\\\!32Gaussian matrix \(forSO​\(C\)\\mathrm\{SO\}\(C\), a column is flipped ifdetQ<0\\det Q<0\),SCS\_\{C\}is a random permutation of the3232channels, and\(ℤ2\)C\(\\mathbb\{Z\}\_\{2\}\)^\{C\}is a random sign vector in\{±1\}32\\\{\\pm 1\\\}^\{32\}\. We adopterr<10−3\\mathrm\{err\}<10^\{\-3\}as the numerical pass threshold, displayerr<10−5\\mathrm\{err\}<10^\{\-5\}\(machine precision\) as∼0\\sim 0, and report numerical values otherwise\.

The final structural entries,d2=0d^\{2\}\\\!=\\\!0and AbelianU​\(1\)U\(1\)curvature invariance, are reported categorically\. A model using the fixed coboundaryK\.dkK\.d\_\{k\}satisfiesd2=0d^\{2\}\\\!=\\\!0exactly\. A model that performs message passing on 1\-cochain connections inherits exact invariance ofF=d​AF=dAunderA↦A\+d​λA\\mapsto A\+d\\lambda, complementing theO​\(C\)\\mathrm\{O\}\(C\)cochain\-frame action tested above\. For methods without cochain\-frame design, theO​\(C\)\\mathrm\{O\}\(C\)columns simply record the absence of this inductive bias\. N/A denotes settings outside a method’s native domain: under its FFT parameterization, FNO is tied to regular grids, and DeepONet’s branch–trunk outer\-product structure has no separate channel\-space message layer to test\.

Table 7:RelativeL2L\_\{2\}error of Riemannian Hodge Message Passing on each symmetry subgroup \(primary test mesh, averaged over 5 independent random transformations\)\.d2=0d^\{2\}\\\!=\\\!0holds exactly by the algebraic construction of the cell complex;U​\(1\)U\(1\)denotes Abelian curvature invariance under exact shifts of 1\-cochain connections, separate from theO​\(C\)\\mathrm\{O\}\(C\)cochain\-frame action\.Table 8:RelativeL2L\_\{2\}error of each method on ten symmetry subgroups, averaged over 5 random transformations, with primary test mesh a4×44\\\!\\times\\\!4Delaunay\.∼0\\sim\\\!0denotes error below10−510^\{\-5\}\(near floating\-point precision\)\. Thed2=0d^\{2\}\\\!=\\\!0column is the fixed\-coboundary topology check \(marked*exact*\); theU​\(1\)U\(1\)column is Abelian curvature invariance under exact shifts of 1\-cochain connections \(marked✓\\checkmark\)\. Other entries are marked N/A\. See the protocol in §[G](https://arxiv.org/html/2608.14556#A7)\.SpatialE​\(n\)\\mathrm\{E\}\(n\)TopoCochain\-frameO​\(C\)\\mathrm\{O\}\(C\)TopoAbelianModelT​\(n\)\\mathrm\{T\}\(n\)SO​\(n\)\\mathrm\{SO\}\(n\)O​\(n\)\\mathrm\{O\}\(n\)ℤ2\\mathbb\{Z\}\_\{2\}or\.O​\(C\)\\mathrm\{O\}\(C\)SO​\(C\)\\mathrm\{SO\}\(C\)SCS\_\{C\}\(ℤ2\)C\(\\mathbb\{Z\}\_\{2\}\)^\{\\\!C\}d2d^\{2\}U​\(1\)U\(1\)GCN∼0\\sim\\\!0∼0\\sim\\\!0∼0\\sim\\\!0∼0\\sim\\\!01\.561\.581\.611\.46N/AN/AGAT∼0\\sim\\\!0∼0\\sim\\\!0∼0\\sim\\\!0∼0\\sim\\\!00\.680\.680\.700\.68N/AN/ASchNet∼0\\sim\\\!0∼0\\sim\\\!0∼0\\sim\\\!0∼0\\sim\\\!00\.400\.400\.300\.37N/AN/AEGNN∼0\\sim\\\!0∼0\\sim\\\!0∼0\\sim\\\!0∼0\\sim\\\!00\.430\.430\.350\.40N/AN/AMPSN∼0\\sim\\\!0∼0\\sim\\\!0∼0\\sim\\\!0∼0\\sim\\\!01\.311\.301\.211\.23N/A✓\\checkmarkSCCNN∼0\\sim\\\!0∼0\\sim\\\!0∼0\\sim\\\!0∼0\\sim\\\!01\.241\.251\.081\.31exact✓\\checkmarkGaugeEqCNN∼0\\sim\\\!00\.020\.030\.081\.531\.521\.521\.39N/AN/AGEM\-CNN∼0\\sim\\\!00\.160\.170\.071\.401\.391\.361\.35N/AN/ACW Net∼0\\sim\\\!0∼0\\sim\\\!0∼0\\sim\\\!00\.091\.381\.391\.231\.35exact✓\\checkmarkCliff\-SMPN∼0\\sim\\\!0∼0\\sim\\\!0∼0\\sim\\\!00\.291\.371\.361\.351\.33exact✓\\checkmarkFNO∼0\\sim\\\!0∼0\\sim\\\!0∼0\\sim\\\!0∼0\\sim\\\!01\.401\.411\.251\.40N/AN/ADeepONet0\.010\.100\.09N/AN/AN/AN/AN/AN/AN/ARHMP∼0\\sim\\\!0∼0\\sim\\\!0∼0\\sim\\\!0∼0\\sim\\\!0∼0\\sim\\\!0∼0\\sim\\\!0∼0\\sim\\\!0∼0\\sim\\\!0exact✓\\checkmark

Table 9:Results of the same protocol as Table[8](https://arxiv.org/html/2608.14556#A7.T8)on the auxiliary test mesh \(6\-point irregular Delaunay,n0=6,n1=11,n2=6n\_\{0\}=6,n\_\{1\}=11,n\_\{2\}=6\)\. The finalU​\(1\)U\(1\)column again denotes Abelian curvature invariance under exact shifts of 1\-cochain connections\. Under its FFT parameterization, FNO is tied to regular grids\.SpatialE​\(n\)\\mathrm\{E\}\(n\)TopoCochain\-frameO​\(C\)\\mathrm\{O\}\(C\)TopoAbelianModelT​\(n\)\\mathrm\{T\}\(n\)SO​\(n\)\\mathrm\{SO\}\(n\)O​\(n\)\\mathrm\{O\}\(n\)ℤ2\\mathbb\{Z\}\_\{2\}or\.O​\(C\)\\mathrm\{O\}\(C\)SO​\(C\)\\mathrm\{SO\}\(C\)SCS\_\{C\}\(ℤ2\)C\(\\mathbb\{Z\}\_\{2\}\)^\{\\\!C\}d2d^\{2\}U​\(1\)U\(1\)GCN∼0\\sim\\\!0∼0\\sim\\\!0∼0\\sim\\\!0∼0\\sim\\\!01\.601\.591\.721\.57N/AN/AGAT∼0\\sim\\\!0∼0\\sim\\\!0∼0\\sim\\\!0∼0\\sim\\\!00\.700\.720\.710\.72N/AN/ASchNet∼0\\sim\\\!0∼0\\sim\\\!0∼0\\sim\\\!0∼0\\sim\\\!00\.440\.440\.340\.43N/AN/AEGNN∼0\\sim\\\!0∼0\\sim\\\!0∼0\\sim\\\!0∼0\\sim\\\!00\.440\.440\.350\.42N/AN/AMPSN∼0\\sim\\\!0∼0\\sim\\\!0∼0\\sim\\\!0∼0\\sim\\\!01\.291\.281\.241\.18N/A✓\\checkmarkSCCNN∼0\\sim\\\!0∼0\\sim\\\!0∼0\\sim\\\!0∼0\\sim\\\!01\.561\.571\.221\.56exact✓\\checkmarkGaugeEqCNN∼0\\sim\\\!00\.020\.030\.221\.621\.611\.581\.43N/AN/AGEM\-CNN∼0\\sim\\\!00\.110\.120\.181\.441\.431\.381\.34N/AN/ACW Net∼0\\sim\\\!0∼0\\sim\\\!0∼0\\sim\\\!00\.151\.611\.631\.461\.61exact✓\\checkmarkCliff\-SMPN∼0\\sim\\\!0∼0\\sim\\\!0∼0\\sim\\\!00\.511\.291\.281\.321\.31exact✓\\checkmarkFNON/AN/AN/AN/AN/AN/AN/AN/AN/AN/ADeepONet0\.020\.130\.09N/AN/AN/AN/AN/AN/AN/ARHMP∼0\\sim\\\!0∼0\\sim\\\!0∼0\\sim\\\!0∼0\\sim\\\!0∼0\\sim\\\!0∼0\\sim\\\!0∼0\\sim\\\!0∼0\\sim\\\!0exact✓\\checkmark

## Appendix HDiscrete Curvature Comparison

### H\.1Ollivier\-Ricci Curvature

Ollivier\[[38](https://arxiv.org/html/2608.14556#bib.bib13)\]defines Ricci curvature on graphs via optimal transport:κ​\(x,y\)=1−W1​\(μx,μy\)/d​\(x,y\)\\kappa\(x,y\)=1\-W\_\{1\}\(\\mu\_\{x\},\\mu\_\{y\}\)/d\(x,y\)\. It is a scalar on edges, whereasHkH\_\{k\}is a matrix on the entire set ofkk\-cells, providing a richer geometric parameterization\.

### H\.2Forman\-Ricci Curvature

Forman\[[19](https://arxiv.org/html/2608.14556#bib.bib7)\]defined a discrete Ricci curvatureF​\(e\)F\(e\)for CW complexes and established a discrete Bochner–Weitzenböck inequality\. On a weighted graph\(wv,we\)\(w\_\{v\},w\_\{e\}\), the Forman curvature involves ratios such aswe/wvw\_\{e\}/w\_\{v\}, while the diagonal entries ofH0,H1H\_\{0\},H\_\{1\}play exactly the roles ofwv,wew\_\{v\},w\_\{e\}\. ThusLkH−LkIL\_\{k\}^\{H\}\-L\_\{k\}^\{I\}in RHMP can be viewed as a learnable generalization of Forman curvature: Forman curvature is determined by a fixed combinatorial structure, whileHkH\_\{k\}adaptively learns the optimal geometric weights through training\.

### H\.3Tangent Bundle Convolution

Battiloro et al\.\[[2](https://arxiv.org/html/2608.14556#bib.bib3)\]realize geometry\-aware message passing via convolution on the tangent bundle\. The metric approach of RHMP can be viewed as a complementary dual: it represents geometry through inner\-product weights on cochains\.

## Appendix IExpressivity Analysis

Symmetry constraints shrink the hypothesis space by ruling out functions that violate the physical structure\. The learnable metric still provides local geometric flexibility\. The following proposition states this for the same\-degree weighted Hodge operator used in Eq\. \([5](https://arxiv.org/html/2608.14556#S3.E5)\)\.

###### Proposition 6\(First\-order spectral response to metric perturbation\)\.

LetKKbe a fixed regular CW complex and fixkk\. ForA∈SPD​\(nk−1\)A\\in\\mathrm\{SPD\}\(n\_\{k\-1\}\)andB∈SPD​\(nk\+1\)B\\in\\mathrm\{SPD\}\(n\_\{k\+1\}\), define

Lk​\(A,B\)=dk−1​A​dk−1⊤\+dk⊤​B​dk\.L\_\{k\}\(A,B\)=d\_\{k\-1\}Ad\_\{k\-1\}^\{\\top\}\+d\_\{k\}^\{\\top\}Bd\_\{k\}\.\(24\)Take the base point\(A,B\)=\(I,I\)\(A,B\)=\(I,I\)and writeLk0=dk−1​dk−1⊤\+dk⊤​dkL\_\{k\}^\{0\}=d\_\{k\-1\}d\_\{k\-1\}^\{\\top\}\+d\_\{k\}^\{\\top\}d\_\{k\}\. Letλj\\lambda\_\{j\}be a simple eigenvalue ofLk0L\_\{k\}^\{0\}with unit eigenvector𝐯j∈ℝnk\\mathbf\{v\}\_\{j\}\\in\\mathbb\{R\}^\{n\_\{k\}\}\. Then:

\(1\)The map\(A,B\)↦λj​\(A,B\)\(A,B\)\\mapsto\\lambda\_\{j\}\(A,B\)is differentiable in a neighborhood of\(I,I\)\(I,I\)\.

\(2\)For symmetric perturbationsδ​A∈Sym​\(nk−1\)\\delta A\\in\\mathrm\{Sym\}\(n\_\{k\-1\}\)andδ​B∈Sym​\(nk\+1\)\\delta B\\in\\mathrm\{Sym\}\(n\_\{k\+1\}\),

D​λj​\(I,I\)​\[δ​A,δ​B\]=\(dk−1⊤​𝐯j\)⊤​δ​A​\(dk−1⊤​𝐯j\)\+\(dk​𝐯j\)⊤​δ​B​\(dk​𝐯j\)\.D\\lambda\_\{j\}\(I,I\)\[\\delta A,\\delta B\]=\(d\_\{k\-1\}^\{\\top\}\\mathbf\{v\}\_\{j\}\)^\{\\top\}\\delta A\(d\_\{k\-1\}^\{\\top\}\\mathbf\{v\}\_\{j\}\)\+\(d\_\{k\}\\mathbf\{v\}\_\{j\}\)^\{\\top\}\\delta B\(d\_\{k\}\\mathbf\{v\}\_\{j\}\)\.\(25\)
\(3\)If𝐯j\\mathbf\{v\}\_\{j\}is non\-harmonic, i\.e\.dk−1⊤​𝐯j≠0d\_\{k\-1\}^\{\\top\}\\mathbf\{v\}\_\{j\}\\neq 0ordk​𝐯j≠0d\_\{k\}\\mathbf\{v\}\_\{j\}\\neq 0, then there is a symmetric metric perturbation direction for whichD​λj≠0D\\lambda\_\{j\}\\neq 0\. Under diagonal metricsδ​A=diag​\(𝛈\)\\delta A=\\mathrm\{diag\}\(\\bm\{\\eta\}\)andδ​B=diag​\(𝛇\)\\delta B=\\mathrm\{diag\}\(\\bm\{\\zeta\}\),

D​λj​\(I,I\)​\[δ​A,δ​B\]=∑a=1nk−1ηa​\(dk−1⊤​𝐯j\)a2\+∑b=1nk\+1ζb​\(dk​𝐯j\)b2\.D\\lambda\_\{j\}\(I,I\)\[\\delta A,\\delta B\]=\\sum\_\{a=1\}^\{n\_\{k\-1\}\}\\eta\_\{a\}\(d\_\{k\-1\}^\{\\top\}\\mathbf\{v\}\_\{j\}\)\_\{a\}^\{2\}\+\\sum\_\{b=1\}^\{n\_\{k\+1\}\}\\zeta\_\{b\}\(d\_\{k\}\\mathbf\{v\}\_\{j\}\)\_\{b\}^\{2\}\.\(26\)

###### Proof\.

Lk​\(A,B\)L\_\{k\}\(A,B\)depends affinely on\(A,B\)\(A,B\), hence smoothly onSPD​\(nk−1\)×SPD​\(nk\+1\)\\mathrm\{SPD\}\(n\_\{k\-1\}\)\\times\\mathrm\{SPD\}\(n\_\{k\+1\}\)\. Sinceλj\\lambda\_\{j\}is a simple eigenvalue of the self\-adjoint matrixLk0L\_\{k\}^\{0\}, standard eigenvalue perturbation theory gives differentiability near\(I,I\)\(I,I\)\. SetA​\(t\)=I\+t​δ​AA\(t\)=I\+t\\delta AandB​\(t\)=I\+t​δ​BB\(t\)=I\+t\\delta B\. ThenLk​\(t\)=Lk0\+t​δ​LkL\_\{k\}\(t\)=L\_\{k\}^\{0\}\+t\\delta L\_\{k\}withδ​Lk=dk−1​δ​A​dk−1⊤\+dk⊤​δ​B​dk\\delta L\_\{k\}=d\_\{k\-1\}\\delta Ad\_\{k\-1\}^\{\\top\}\+d\_\{k\}^\{\\top\}\\delta Bd\_\{k\}\. The first\-order Rayleigh quotient formula givesdd​t​λj​\(t\)\|t=0=𝐯j⊤​δ​Lk​𝐯j\\frac\{d\}\{dt\}\\lambda\_\{j\}\(t\)\|\_\{t=0\}=\\mathbf\{v\}\_\{j\}^\{\\top\}\\delta L\_\{k\}\\mathbf\{v\}\_\{j\}, which is Eq\. \([25](https://arxiv.org/html/2608.14556#A9.E25)\)\. Ifdk​𝐯j≠0d\_\{k\}\\mathbf\{v\}\_\{j\}\\neq 0, chooseδ​B=\(dk​𝐯j\)​\(dk​𝐯j\)⊤\\delta B=\(d\_\{k\}\\mathbf\{v\}\_\{j\}\)\(d\_\{k\}\\mathbf\{v\}\_\{j\}\)^\{\\top\}andδ​A=0\\delta A=0; ifdk−1⊤​𝐯j≠0d\_\{k\-1\}^\{\\top\}\\mathbf\{v\}\_\{j\}\\neq 0, choose the analogousδ​A\\delta A\. This gives a nonzero derivative\. The diagonal formula follows by takingδ​A\\delta Aandδ​B\\delta Bdiagonal\. ∎

###### Corollary 7\(First\-order stationarity of harmonic modes\)\.

If𝐯j∈ker⁡Lk0\\mathbf\{v\}\_\{j\}\\in\\ker L\_\{k\}^\{0\}, equivalentlydk−1⊤​𝐯j=0d\_\{k\-1\}^\{\\top\}\\mathbf\{v\}\_\{j\}=0anddk​𝐯j=0d\_\{k\}\\mathbf\{v\}\_\{j\}=0, thenD​λj​\(I,I\)​\[δ​A,δ​B\]=0D\\lambda\_\{j\}\(I,I\)\[\\delta A,\\delta B\]=0for all symmetric perturbation directions\.

###### Corollary 8\(The diagonal metric family strictly extends the unweighted class\)\.

IfLk0L\_\{k\}^\{0\}has a simple non\-harmonic eigenvalueλj\\lambda\_\{j\}, then a diagonal metric perturbation can changeλj\\lambda\_\{j\}at first order\. Thus the diagonal metric family is locally more expressive than the unweighted Hodge operator near the identity metric\.

Proposition[6](https://arxiv.org/html/2608.14556#Thmtheorem6)is local: it describes first\-order spectral control around the identity metric\. This is the role of the learnable metric\. It adds geometric degrees of freedom to the Hodge operator while leaving the cohomological, harmonic part protected byd2=0d^\{2\}=0\.

## Appendix JComputational Complexity and Scalability

RHMP uses the diagonal\-plus\-low\-rank\-basis metric parameterization in the main experiments,

Hk=Bk​Bk⊤\+diag​\(softplus⁡\(𝐡k\)\+ϵ\),H\_\{k\}=B\_\{k\}B\_\{k\}^\{\\top\}\+\\mathrm\{diag\}\(\\operatorname\{softplus\}\(\\mathbf\{h\}\_\{k\}\)\+\\epsilon\),with rankr=8r=8\. The pure diagonal case hasO​\(nk\)O\(n\_\{k\}\)metric application cost; the rank\-rrvariant used in the main experiments addsO​\(nk​r​C\)O\(n\_\{k\}rC\), withr=8r=8\. The dominant operations remain sparse applications ofdkd\_\{k\}anddk⊤d\_\{k\}^\{\\top\}at complexityO​\(nnz​\(dk\)⋅C\)O\(\\mathrm\{nnz\}\(d\_\{k\}\)\\cdot C\), the same as standard cell\-complex message passing; the metric application consists of diagonal scaling plus a low\-rank update\.

Table[10](https://arxiv.org/html/2608.14556#A10.T10)shows the single\-epoch training time of each method on three representative tasks \(same GPU, same data pipeline\)\. RHMP is comparable in speed to SCCNN and CW Net \(both cell\-complex methods\), faster than GAT, EGNN, and SchNet \(graph methods\), and slower than GCN and operator\-learning methods \(FNO, DeepONet\)\.

Table 10:Single\-epoch training time \(seconds\) on 3 representative tasks\. Measured on the same GPU\.### J\.1Large\-Mesh Scalability

To assess scalability beyond the benchmark meshes \(∼\\sim1K nodes\), we profile RHMP on Delaunay meshes with up to\(n0,n1,n2\)≈\(50​K,150​K,100​K\)\(n\_\{0\},n\_\{1\},n\_\{2\}\)\\approx\(50\\text\{K\},150\\text\{K\},100\\text\{K\}\)total cells, using the same single\-GPU setup as all other experiments \(§[D\.6](https://arxiv.org/html/2608.14556#A4.SS6)\)\. Table[11](https://arxiv.org/html/2608.14556#A10.T11)reports forward time, backward time, peak GPU memory, and parameter count at four mesh scales\.

Table 11:Wall\-clock time and memory for RHMP at increasing mesh scale on a single NVIDIA RTX PRO 6000 Blackwell GPU \(96 GB\)\.Trainable params\(weights ofMLPH\\mathrm\{MLP\}\_\{H\}, the lifting encoder, and the message\-passing transforms\) are shared across all cells and fixed regardless of mesh size\.Activationscount per\-pass scalar outputs \(onehih\_\{i\}perkk\-cell plus intermediate features\), which grow linearly withnkn\_\{k\}as in any message\-passing architecture\.Near\-linear time scaling\.From 1K to 100K cells \(a100×100\\timesincrease in mesh size\), the forward pass time grows by approximately10×10\\times, from16\.716\.7ms to169\.2169\.2ms\. This sub\-linear behavior reflects the sparse structure of the Hodge operator: the dominant cost is sparse matrix–dense matrix multiplication atO​\(nnz​\(dk\)⋅C\)O\(\\mathrm\{nnz\}\(d\_\{k\}\)\\cdot C\), which scales linearly in the number of nonzeros rather than quadratically in mesh size\.

Moderate memory footprint\.At 100K cells the peak GPU memory is 3\.9 GB, fitting comfortably on a single 80 GB accelerator\. Linear extrapolation suggests that meshes of∼\\sim1M cells would require∼\\sim39 GB, remaining feasible on current hardware\. By contrast, SCCNN exceeds available GPU memory at this scale \(estimated requirement: 83\.8 GB at 100K cells\)\.

Mesh\-invariant trainable weights\.The diagonal metric produces one scalarhih\_\{i\}per cell, so per\-pass activations grow linearly withnkn\_\{k\}\(right column of Table[11](https://arxiv.org/html/2608.14556#A10.T11)\); the trainable weights \(∼\\sim0\.05M total:MLPH\\mathrm\{MLP\}\_\{H\}, lifting encoder, message\-passing transforms\) are shared across cells and remain fixed regardless of mesh size\. This mesh\-invariance of the trainable weights is what enables the same trained model to transfer directly across meshes of differing size and connectivity, as demonstrated in the AirfRANS variable\-mesh experiment \(§[5\.2](https://arxiv.org/html/2608.14556#S5.SS2)\)\.

Accuracy holds at large scale, with a∼\\sim19×19\\timesspeedup over CW Net\.We compare RHMP against CW Net, the strongest cell\-complex baseline on the 1KU​\(1\)U\(1\)Wilson loop benchmark \(Table[2](https://arxiv.org/html/2608.14556#S5.T2)\), at100×100\\timesthe benchmark mesh size, end\-to\-end on the same single GPU and the same70/15/1570/15/15split\. Table[12](https://arxiv.org/html/2608.14556#A10.T12)reports the comparison\. At matched parameter count \(10\.80M vs 11\.30M\), RHMP reaches testR2=0\.9823R^\{2\}=0\.9823versus0\.92310\.9231for CW Net \(5\.9\-point gap, NRMSE0\.00330\.0033vs0\.00690\.0069\), and completes training in113113s versus21812181s \(∼\\sim19×19\\timesfaster\)\. The speedup tracks the per\-step cost:∼\\sim10 ms for RHMP \(sparsedk⊤​Hk​dkd\_\{k\}^\{\\top\}H\_\{k\}d\_\{k\}\) versus∼\\sim700 ms for CW Net \(dense hidden\-state propagation ath=512h\{=\}512\)\. Convergence is also faster: RHMP saturates nearR2≈0\.97R^\{2\}\\approx 0\.97by epoch 20, while CW Net is still rising at epoch 30\. Both architectures benefit from the larger mesh \(RHMPR2≈0\.955→0\.982R^\{2\}\\approx 0\.955\\to 0\.982; CW NetR2≈0\.875→0\.923R^\{2\}\\approx 0\.875\\to 0\.923\), consistent with denser plaquette statistics on finer lattices\.

Table 12:Parameter\-matched comparison on a 100K\-cellU​\(1\)U\(1\)Wilson loop\. Both models trained end\-to\-end on a single GPU with the same70/15/1570/15/15split\. Wall time is total training time\.Shared\-metric variants, for example tyingHkH\_\{k\}across cells with similar local geometry or expandinghih\_\{i\}in a compact basis over the mesh, can further reduce activation memory at scales beyond10510^\{5\}cells\.

## Appendix KAblation Studies

We conduct one ablation each for four design choices of RHMP, flipping only one at a time while keeping the rest unchanged: fixing the learnable diagonal metricHkH\_\{k\}to the identity; fixing the cross\-dimensional mixing coefficientα\\alphato11, turning off cross\-dimensional transport; replacing the norm\-gated nonlinearity with an element\-wise ReLU; replacing the fixed coboundary operator with a learnable scalar\-valued version of the same sparsity pattern\. Ablations are evaluated on four representative tasks: CNS vorticity on a regular grid, tangent vector field reconstruction on an ellipsoidal surface,U​\(1\)U\(1\)Wilson loop gauge curvature, andS​U​\(2\)SU\(2\)Yang–Mills field strength\. All ablated variants are retrained under the same data splits for5050epochs \(seed4242, cosine learning rate1​e−3→1​e−51\\\!\\mathrm\{e\}\{\-3\}\\\!\\to\\\!1\\\!\\mathrm\{e\}\{\-5\}, parameter budget kept at most\+20%\+20\\%above the full model\); the numbers for the full model reuse the100100\-epoch main benchmark results\.

Table 13:Precision\-driving ablations on four representative tasks: regular\-grid CNS vorticity, ellipsoid surface flow,U​\(1\)U\(1\)Wilson loop, andS​U​\(2\)SU\(2\)Yang–Mills\. Each column flips*one*design pillar relative to the full model\.*Full*: full model \(reused from the 100\-epoch main benchmark\);Hk=IH\_\{k\}\\\!=\\\!I: identity metric \(no learnable Riemannian metric\);α=1\\alpha\\\!=\\\!1: no cross\-dimensional transfer\. Ablation variants are retrained for 50 epochs \(seed 42, same data split and optimizer\)\. Higher is better forR2R^\{2\}/SSIM, lower for NRMSE; bold marks per\-row best\.Table 14:Structure\-preserving ablations on the same four tasks\.*ReLU*: norm\-gated update replaced with element\-wise ReLU;*learndkd\_\{k\}*:dkd\_\{k\}replaced with learnable scalars on the same CW sparsity\. The main metric \(R2R^\{2\}\) shifts by at most a few percent \(top block: this is the precision\-vs\.\-structure trade\-off the paper highlights\)\. The corresponding structural quantities, however, fail by orders of magnitude:‖d1​d0‖F\\\|d\_\{1\}d\_\{0\}\\\|\_\{F\}drifts from0to∼101\\sim 10^\{1\}for learnabledkd\_\{k\}; the message\-passing layer’s cochain\-frame equivariance error‖Q⊤​f​\(Q​x\)−f​\(x\)‖F/‖f​\(x\)‖F\\\|Q^\{\\top\}f\(Qx\)\-f\(x\)\\\|\_\{F\}/\\\|f\(x\)\\\|\_\{F\}underQ∈O​\(C\)Q\\in O\(C\)jumps from fp32 round\-off to∼100\\sim 10^\{0\}for ReLU\.Metric learning and cross\-dimensional communication dominate the main\-metric accuracy\.FixingHkH\_\{k\}to the identity causes degradations ofΔ​R2=−0\.37,−0\.46,−0\.49\\Delta R^\{2\}=\-0\.37,\-0\.46,\-0\.49on CNS vorticity, Wilson loop, and Yang–Mills, respectively, while turning off cross\-dimensional transport givesΔ​R2=−0\.31,−0\.08,−0\.39\\Delta R^\{2\}=\-0\.31,\-0\.08,\-0\.39\. Both have minor effect on the ellipsoidal surface flow \(Δ​R2≈−0\.04\\Delta R^\{2\}\\approx\-0\.04\), consistent with the physical structure of that task: the target𝐯tan=⋆dψ\\mathbf\{v\}\_\{\\text\{tan\}\}=\\star d\\psiis a one\-step coexact map, insensitive to both the metric and cross\-order coupling\. In contrast, CNS vorticityω=∇×𝐮\\omega=\\nabla\\\!\\times\\\!\\mathbf\{u\}and gauge curvatureF=d​θF=d\\theta\(U​\(1\)U\(1\)\) /F=d​A\+\[A,A\]F=dA\+\[A,A\]\(S​U​\(2\)SU\(2\)\) involve bothd0d\_\{0\}andd1d\_\{1\}after discretization, the setting where Hodge stratification and the geometric metric act together\.

Fixeddkd\_\{k\}and norm\-gated nonlinearity also affect the main metric, but by a smaller margin\.Making the coboundary learnable and scalar\-valued givesΔ​R2=−0\.023,−0\.004,−0\.022,\+0\.012\\Delta R^\{2\}=\-0\.023,\-0\.004,\-0\.022,\+0\.012across the four tasks \(maximum absolute value0\.0230\.023, a slight improvement on Yang–Mills\); replacing norm\-gating with ReLU givesΔ​R2=−0\.034,−0\.012,\+0\.001,−0\.041\\Delta R^\{2\}=\-0\.034,\-0\.012,\+0\.001,\-0\.041\(maximum absolute value0\.0410\.041\)\. Although the effect on the main metric is small, the respective physical\-structure diagnostic quantities \(Table[14](https://arxiv.org/html/2608.14556#A11.T14)\) degrade by a much larger margin: after learningdkd\_\{k\},‖d1​d0‖F\\\|d\_\{1\}d\_\{0\}\\\|\_\{F\}drifts from0to12\.95,9\.43,15\.68,15\.3612\.95,9\.43,15\.68,15\.36on the four tasks, so the Hodge decomposition loses exact closure; after the ReLU replacement, the relative cochain\-frame equivariance error of a single\-layer weighted Hodge operator under a randomQ∈O​\(C\)Q\\in O\(C\),‖Q⊤​f​\(Q​x\)−f​\(x\)‖F/‖f​\(x\)‖F\\\|Q^\{\\top\}f\(Qx\)\-f\(x\)\\\|\_\{F\}/\\\|f\(x\)\\\|\_\{F\}, rises from floating\-point precision \(∼10−6\\sim 10^\{\-6\}\) to∼100\\sim 10^\{0\}\. These two quantities are of consistent magnitude on theU​\(1\)U\(1\)andS​U​\(2\)SU\(2\)tasks, indicating that the conclusions hold on both Abelian and non\-Abelian gauge tasks\.

Qualitatively, the output fields ofHk=IH\_\{k\}=Iand the cross\-dimensional\-transport\-off variant show visible structural loss, while ReLU and learnabledkd\_\{k\}are close to the full model in visual quality and pointwise accuracy, and their symmetry deviations are visible only in the diagnostic quantities of Table[14](https://arxiv.org/html/2608.14556#A11.T14)\.

## Appendix LStatistics Information

### L\.1Test\-set bootstrap statistics

We quantify test\-set finite\-size uncertainty by non\-parametric bootstrap\. For each \(task, method\) we load the best\-epoch checkpoint of one representative training run \(seed=42=42\), run inference on the entire test set, and drawB=1000B=1000bootstrap samples of the test indices with replacement, recomputing every metric on each resample\. We report the standard deviation across theBBresamples; main\-text Table[2](https://arxiv.org/html/2608.14556#S5.T2)reports the seed\-mean point estimates over\{42,1,2\}\\\{42,1,2\\\}\. NRMSE uses the fixed global data range \(a property of the dataset, not the resample\); SSIM and Pearson are averaged over per\-sample values selected by the resample indices; R2, MSE, and MAE are recomputed from scratch on each bootstrap set\. Cross\-seed training\-time variance is reported separately in §[L\.2](https://arxiv.org/html/2608.14556#A12.SS2)\.

Tables[15](https://arxiv.org/html/2608.14556#A12.T15)–[17](https://arxiv.org/html/2608.14556#A12.T17)below report bootstrap statistics for all three metrics used in the main paper\.

Table 15:SSIMbootstrap std \(1000 resamples over test set; point estimates in main\-text Table[2](https://arxiv.org/html/2608.14556#S5.T2)\)\. Each row a method, each column a task\.Table 16:Pearsonbootstrap std \(1000 resamples over test set; point estimates in main\-text Table[2](https://arxiv.org/html/2608.14556#S5.T2)\)\. Each row a method, each column a task\.Table 17:NRMSEbootstrap std \(1000 resamples over test set; point estimates in main\-text Table[2](https://arxiv.org/html/2608.14556#S5.T2)\)\. Each row a method, each column a task\.
### L\.2Cross\-seed training variance

We retrain every \(task, method\) combination under three independent seeds \(\{42,1,2\}\\\{42,1,2\\\}\) and report the cross\-seed sample standard deviation of each metric in Tables[18](https://arxiv.org/html/2608.14556#A12.T18)–[20](https://arxiv.org/html/2608.14556#A12.T20); main\-text Table[2](https://arxiv.org/html/2608.14556#S5.T2)reports the seed mean\. The cross\-seed std captures training\-time variance from initialization and stochastic gradient noise, complementing the test\-set finite\-size variance from the bootstrap in §[L\.1](https://arxiv.org/html/2608.14556#A12.SS1)\. The RHMP ranking in Table[2](https://arxiv.org/html/2608.14556#S5.T2)is preserved across seeds on every task\. Cells markedn=1n\{=\}1correspond to seed\-1/2 reruns that did not complete and are reported using the seed=42 run alone\.

Table 18:SSIMcross\-seed standard deviation \(seeds 42, 1, 2\)\. Point estimates in main\-text Table[2](https://arxiv.org/html/2608.14556#S5.T2)\.Table 19:Pearsoncross\-seed standard deviation \(seeds 42, 1, 2\)\. Point estimates in main\-text Table[2](https://arxiv.org/html/2608.14556#S5.T2)\.Table 20:NRMSEcross\-seed standard deviation \(seeds 42, 1, 2\)\. Point estimates in main\-text Table[2](https://arxiv.org/html/2608.14556#S5.T2)\.

## Appendix MAdditional Qualitative Comparisons

This appendix shows further per\-task qualitative samples to complement FigureLABEL:fig:qual\_main\. For each task we display four additional test samples drawn from the same per\-sample winner pool used in the main text\. The samples are independently chosen and span the value distribution of each task, so the qualitative trends visible in the main figure are not specific to any single example\.

![Refer to caption](https://arxiv.org/html/2608.14556v1/figures/qual/T1/app_653.png)\(a\)Sample 1\.
![Refer to caption](https://arxiv.org/html/2608.14556v1/figures/qual/T1/app_835.png)\(b\)Sample 2\.
![Refer to caption](https://arxiv.org/html/2608.14556v1/figures/qual/T1/app_1480.png)\(c\)Sample 3\.
![Refer to caption](https://arxiv.org/html/2608.14556v1/figures/qual/T1/app_467.png)\(d\)Sample 4\.

Figure 10:Additional qualitative samples on Navier–Stokes vorticity \(PDEBench\)\. Each row contrasts the ground\-truth vorticity with predictions from RHMP and the strongest cell\-complex / manifold\-gauge baselines\.![Refer to caption](https://arxiv.org/html/2608.14556v1/figures/qual/T2/app_26.png)\(a\)Sample 1\.
![Refer to caption](https://arxiv.org/html/2608.14556v1/figures/qual/T2/app_143.png)\(b\)Sample 2\.
![Refer to caption](https://arxiv.org/html/2608.14556v1/figures/qual/T2/app_328.png)\(c\)Sample 3\.
![Refer to caption](https://arxiv.org/html/2608.14556v1/figures/qual/T2/app_422.png)\(d\)Sample 4\.

Figure 11:Additional qualitative samples on torus advection–diffusion\. Each panel shows the scalar transport field on a genus\-1 surface; RHMP preserves the localized plumes and sharp gradients that graph and cell\-complex baselines smear out\.![Refer to caption](https://arxiv.org/html/2608.14556v1/figures/qual/T3/app_115.png)\(a\)Sample 1\.
![Refer to caption](https://arxiv.org/html/2608.14556v1/figures/qual/T3/app_142.png)\(b\)Sample 2\.
![Refer to caption](https://arxiv.org/html/2608.14556v1/figures/qual/T3/app_261.png)\(c\)Sample 3\.
![Refer to caption](https://arxiv.org/html/2608.14556v1/figures/qual/T3/app_424.png)\(d\)Sample 4\.

Figure 12:Additional qualitative samples on ellipsoid coexact surface flow\. Rows show‖𝐯‖\\\|\\mathbf\{v\}\\\|and the three Cartesian components of𝐯tan=𝐧×∇ψ\\mathbf\{v\}\_\{\\rm tan\}=\\mathbf\{n\}\\times\\nabla\\psi; CW\-Net loses the sign structure of the surface vector field while RHMP tracks the ground truth\.![Refer to caption](https://arxiv.org/html/2608.14556v1/figures/qual/T5/app_230.png)\(a\)Sample 1\.
![Refer to caption](https://arxiv.org/html/2608.14556v1/figures/qual/T5/app_466.png)\(b\)Sample 2\.
![Refer to caption](https://arxiv.org/html/2608.14556v1/figures/qual/T5/app_706.png)\(c\)Sample 3\.
![Refer to caption](https://arxiv.org/html/2608.14556v1/figures/qual/T5/app_497.png)\(d\)Sample 4\.

Figure 13:Additional qualitative samples on Maxwell–Poisson electrostatics\. Heatmaps show‖𝐄‖\\\|\\mathbf\{E\}\\\|and arrows show𝐄\\mathbf\{E\}; the curl\-free constraintd2=0d^\{2\}\\\!=\\\!0is encoded structurally in RHMP, whose streamlines emanate cleanly from the source/sink configuration of the ground truth\.![Refer to caption](https://arxiv.org/html/2608.14556v1/figures/qual/T6/app_03.png)\(a\)Sample 1\.
![Refer to caption](https://arxiv.org/html/2608.14556v1/figures/qual/T6/app_04.png)\(b\)Sample 2\.
![Refer to caption](https://arxiv.org/html/2608.14556v1/figures/qual/T6/app_09.png)\(c\)Sample 3\.
![Refer to caption](https://arxiv.org/html/2608.14556v1/figures/qual/T6/app_13.png)\(d\)Sample 4\.

Figure 14:Additional qualitative samples onU​\(1\)U\(1\)Wilson\-loop curvature\. The localizedF=d​θF=d\\thetapeaks are recovered with the correct amplitude by RHMP, whereas the strongest baselines either overshoot or wash out the support\.![Refer to caption](https://arxiv.org/html/2608.14556v1/figures/qual/T7/app_57.png)\(a\)Sample 1\.
![Refer to caption](https://arxiv.org/html/2608.14556v1/figures/qual/T7/app_251.png)\(b\)Sample 2\.
![Refer to caption](https://arxiv.org/html/2608.14556v1/figures/qual/T7/app_308.png)\(c\)Sample 3\.
![Refer to caption](https://arxiv.org/html/2608.14556v1/figures/qual/T7/app_598.png)\(d\)Sample 4\.

Figure 15:Additional qualitative samples onS​U​\(2\)SU\(2\)Yang–Mills curvature\. Rows display the three Lie\-algebra componentsF1,F2,F3F^\{1\},F^\{2\},F^\{3\}ofF=d​A\+\[A,A\]F=dA\+\[A,A\]; only RHMP captures both the differentiald​AdAstructure and the algebraic\[A,A\]\[A,A\]coupling at the correct amplitude\.![Refer to caption](https://arxiv.org/html/2608.14556v1/figures/qual/T8/app_50.png)\(a\)Sample 1\.
![Refer to caption](https://arxiv.org/html/2608.14556v1/figures/qual/T8/app_114.png)\(b\)Sample 2\.
![Refer to caption](https://arxiv.org/html/2608.14556v1/figures/qual/T8/app_54.png)\(c\)Sample 3\.
![Refer to caption](https://arxiv.org/html/2608.14556v1/figures/qual/T8/app_63.png)\(d\)Sample 4\.

Figure 16:Additional qualitative samples on AirfRANS airfoil surface pressure\. The two views per sample \(underside / top\) show that RHMP reconstructs the suction trough and pressure ridge across unseen meshes, while baselines either flatten the pressure profile or introduce mesh\-bound artefacts\.