Equivariant Sheaf Neural Networks: Learning Geometric Transport on Graphs

Alessio Borgi; Mario Severino; Fabrizio Silvestri; Pietro Liò

cs.LGAug 28, 2026

Equivariant Sheaf Neural Networks: Learning Geometric Transport on Graphs

Authors: Alessio Borgi, Mario Severino, Fabrizio Silvestri, Pietro Liò

Organizations: Department of Computer Science and Technology, University of Cambridge · Department of Computer, Control and Management Engineering, Sapienza University of Rome · Department of Information Engineering, University of Padua

Abstract

Equivariant graph neural networks provide a principled way to model geometric systems, but efficient first-order architectures remain limited in how vector information can be transformed as it moves across a graph. We introduce \textsc{ESNN}, an Equivariant Sheaf Neural Network that enriches this interaction by learning directed, matrix-valued transport between neighboring vector features while preserving exact Euclidean equivariance. Rather than increasing the order of the representation, ESNN keeps scalar and vector features first-order and places the additional geometric flexibility in the edge transport itself. We characterize this transport theoretically, showing that when relative displacement is the only covariant geometric input, every linear $O(n)$ -equivariant map decomposes into independent radial and tangential components, while learned covariant features enable richer feature-conditioned transformations. We also introduce controlled symmetry relaxation for systems with a preferred ambient direction, which may be prescribed or inferred from data while recovering full $E(n)$ -equivariance when the directional pathway is inactive. Across particle dynamics, mesh-based simulation, point-cloud classification, and molecular property prediction, ESNN improves dynamics prediction, recovers the gravity axis when symmetry is broken, yields substantial gains on selected mesh tasks and long-horizon rollouts, and remains robust to unseen rotations. These results show that learning how geometric information is transported across edges offers a complementary route to expressive equivariant message passing without requiring higher-order representations.

Figures & tables

Panel of Figure 1 — Figure 1: Edge-wise geometric transport in EGNN-style models, generic sheaf networks, and ESNNs. EGNN-style models couple invariant weights with relative displacements, while sheaf networks allow matrix-valued maps between local feature spaces. ESNN combines matrix-valued transport with exact equivariance through $O(n)$ -covariant spatial actions and invariant channel mixing.

Panel of Figure 2 — Figure 2: ESNN transport families. Identity leaves the vector representation unchanged; Diagonal applies an edge-dependent isotropic spatial scaling; Orthogonal learns a feature-conditioned spatial rotation; and Radial–Tangential acts independently on components parallel and orthogonal to the relative displacement.

Panel of Figure 3 — Figure 3: Controlled symmetry relaxation through a preferred ambient direction. When all $\lambda_{\mathbf{g}}^{(\ell)}=0$ , the directional pathway is inactive and ESNN retains full $E(n)$ -equivariance. Activating the directional conditioning $\lambda_{\mathbf{g}}^{(\ell)}\langle\mathbf{r}_{ij},\mathbf{g}\rangle$ reduces the guaranteed symmetry to $E_{\mathbf{g}}(n)=O_{\mathbf{g}}(n)\ltimes\mathbb{R}^{n}$ , where $O_{\mathbf{g}}(n)=\{Q\in O(n):Q\mathbf{g}=\mathbf{g}\}$ . The layerwise relaxation coefficients are initialized at zero; $\mathbf{g}$ may be specified or learned, but is held fixed when the input geometry is transformed.

Table 1: N-body dynamics. Mean squared error (MSE) for charged and gravity-augmented prediction. Charged ESNN and gravity results are mean

\pm

standard deviation over five runs; baseline uncertainties are reported as in the original sources. For gravity,

\max_{\ell}|\lambda_{g}^{(\ell)}|\|\mathbf{g}\|_{2}

measures directional-pathway activation and

A_{g}

alignment with the gravity axis; for Fixed ,

A_{g}=1

by construction. Lower MSE and higher

A_{g}

are better.

Method	$\bm{z/z}$	$\bm{z/\mathrm{SO}(3)}$	$\bm{\mathrm{SO}(3)/\mathrm{SO}(3)}$	$\bm{\Delta_{\mathrm{OOD}}}$ $\downarrow$
PointNet	85.9	19.6	74.7	66.3
RS-CNN	90.3	48.7	82.6	41.6
DGCNN	90.3	33.8	88.6	56.5
RI-Conv	86.5	86.4	86.4	0.1
GC-Conv	89.0	89.1	89.2	0.1
Luo et al. DGCNN	88.4	88.4	88.9	0.0

Table 2: ModelNet40 rotation generalization. Accuracy (%) under

z/z

z/\mathrm{SO}(3)

, and

\mathrm{SO}(3)/\mathrm{SO}(3)

. The

z/\mathrm{SO}(3)

result evaluates the same

z

-trained checkpoint under unseen

\mathrm{SO}(3)

rotations.

\Delta_{\mathrm{OOD}}

is the absolute

z/z

–

z/\mathrm{SO}(3)

accuracy change in percentage points. Literature baselines follow Lippmann et al. (2025) ; EGNN and ESNN use the same evaluation protocol. Higher accuracy and lower

\Delta_{\mathrm{OOD}}

are better.

	CylinderFlow			DeformingPlate			Airfoil
Method	1-Step	50-Step	Full	1-Step	50-Step	Full	1-Step	50-Step	Full
MeshGraphNet	$4.05\pm 0.08$	$12.5\pm 0.6$	$\mathbf{46.14\pm 6.3}$	$0.15\pm 0.02$	$2.2\pm 0.5$	$13.6\pm 1.4$	$1141\pm 68$	$1529\pm 68$	$\mathbf{8033\pm 662}$
EGNN	$9.20\pm 1.80$	$40.5\pm 2.5$	$158.09\pm 2.5$	$0.35\pm 0.01$	$1.6\pm 0.1$	$11.5\pm 0.5$	$5001\pm 134$	$18948\pm 609$	diverged
ESNN-Id	$5.35\pm 0.32$	$20.6\pm 1.0$	$84.09\pm 3.4$	$0.15\pm 0.01$	$1.3\pm 0.1$	$7.7\pm 0.4$	$4744\pm 154$	$7579\pm 287$	$16893\pm 740$
ESNN-Diag	$2.77\pm 0.17$	$8.8\pm 0.4$	$50.84\pm 2.0$	$0.17\pm 0.01$	$1.4\pm 0.1$	$9.2\pm 0.5$	$2413\pm 49$	$4966\pm 126$	$11014\pm 321$
ESNN-Ortho	$2.56\pm 0.15$	$8.5\pm 0.4$	$55.42\pm 2.2$	$0.13\pm 0.01$	$1.1\pm 0.1$	$8.1\pm 0.4$	$1459\pm 27$	$2807\pm 66$	$8821\pm 259$

Table 3: Mesh-based physical dynamics. RMSE (

\times 10^{-3}

) for one-step, 50-step, and full autoregressive rollouts on CylinderFlow , DeformingPlate , and Airfoil . Results are mean

\pm

standard deviation over five matched runs. Best values are bold; Diverged denotes an unstable autoregressive rollout. Lower is better.

Appendix figures & tables6 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 4 — Figure 4: Spatial–channel factorization of ESNN transport. Each term acts on $\mathbf{V}_{j}\in\mathbb{R}^{n\times c_{v}}$ along two independent axes: $\mathbf{S}_{ij}^{(k)}$ acts on the ambient spatial dimension by left multiplication, while $\mathbf{M}_{ij}^{(k)}$ acts on the vector channels by right multiplication. Summing the separable terms gives the canonical transport $\mathcal{T}_{i\leftarrow j}$ . ESNN enforces equivariance by constraining the spatial operators to be $O(n)$ -covariant and the channel maps to be invariant.

Figure 5: Charged-particle dynamics without and with a uniform gravitational field. Translation symmetry is preserved, while the ideal unclipped gravity–Coulomb dynamics retain only orthogonal transformations that preserve

\widehat{\mathbf{g}}_{\mathrm{true}}

Figure 6: ModelNet40 graph construction and rotation protocols. Each centered point cloud is converted into a local $k$ -nearest-neighbor graph and evaluated under the $z/z$ , $z/\mathrm{SO}(3)$ , and $\mathrm{SO}(3)/\mathrm{SO}(3)$ train/test regimes.

Figure 7: Mesh-based dynamics benchmarks: CylinderFlow , DeformingPlate , and Airfoil . Models are trained on one-step targets and evaluated autoregressively.

Method	$\bm{\alpha}$	$\bm{\Delta\epsilon}$	$\bm{\epsilon_{\mathrm{HOMO}}}$	$\bm{\epsilon_{\mathrm{LUMO}}}$	$\bm{\mu}$	$\bm{C_{v}}$	$\bm{G}$	$\bm{H}$	$\bm{\langle R^{2}\rangle}$	$\bm{U}$	$\bm{U_{0}}$	ZPVE
Units	$\mathrm{m}a_{0}^{3}$	meV	meV	meV	mD	$\mathrm{mcal\,mol^{-1}\,K^{-1}}$	meV	meV	$\mathrm{m}a_{0}^{2}$	meV	meV	meV
Invariant models
Cormorant	85	61	34	38	38	26	20	21	961	21	22	2.03
NMP	92	69	43	38	30	40	19	17	180	20	20	1.50
DimeNet++ $\dagger$	44	32.6	24.6	19.5	29.7	23	7.56	6.53	331	6.28	6.32	1.21
ComENet $\dagger$	45	32.4	23.1	19.8	24.5	22	7.98	6.86	259	6.82	6.69	1.20

Appendix

Table 4: QM9 molecular-property prediction. Mean absolute error (MAE) on twelve invariant molecular properties. Baseline values and partition annotations follow Aykent and Xia (2025) ;

\dagger

marks a different data partition. ESNN results are reported for the validation-selected model for each property, with the best ESNN result shown in bold . Units are given in the second header row. Lower is better.

Task	Model	Transport / training regime	$C$	$L$	$V$	Learning rate	Epochs
Charged N-body	ESNN	Identity	64	6	0	$3.38\!\times\!10^{-3}$	10000
	ESNN	Diagonal	32	5	1	$3.25\!\times\!10^{-3}$	3000
	ESNN	Orthogonal	32	7	1	$1.92\!\times\!10^{-3}$	3000
	ESNN	Radial–Tangential	32	5	1	$3.41\!\times\!10^{-3}$	3000
Gravity N-body	ESNN	Diagonal	64	7	1	$1.01\!\times\!10^{-3}$	3000
	ESNN	Orthogonal	32	7	1	$1.24\!\times\!10^{-3}$	3000

Appendix

Table 5: Selected N-body, gravity, and ModelNet40 configurations.

C

L

, and

V

denote scalar width, message-passing layers, and vector groups. For ModelNet40,

z/z

and

z/\mathrm{SO}(3)

use the same

z

-trained checkpoint.

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Symmetry is everywhere in nature and society. Geometric deep learning exploits symmetries in data to improve the performance and efficiency of deep learning systems. In this paper, we extend geometric deep learning to utilize richer symmetry structures. Specifically, we develop order-equivariant neural networks (OENN), which generalize standard graph message passing and sheaf neural networks via the theory of equivariant bundles over face posets (face categories). We (i) characterize all linear order-equivariant maps, (ii) build OENN layers, and (iii) prove universal approximation theorems (UATs) for continuous order-equivariant maps, which are new results even when restricted to sheaf neural networks (for which no UAT was known before). We illustrate the framework on graph and sheaf models. Our results can also be seen as extending the known UAT for graph neural networks to a more general setting that subsumes sheaf neural networks as well. In addition, we show that OENN can be extended further to CENN, Category-Equivariant Neural Network, which gives the general form of equivariant neural networks as well as of equivariant universal approximation theorems, allowing us to leverage categorical symmetry in data (e.g., non-invertible symmetries on multiple objects with compositional relations on those symmetries).

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