Oscillatory Neural Dynamics over Sheaves

Jan-Willem Van Looy; Alessandro Trenta; Alessio Gravina; Alessio Borgi; Ferdinando Zanchetta; Pietro Liò; Davide Bacciu; Rita Fioresi

cs.LGOct 7, 2026

Oscillatory Neural Dynamics over Sheaves

Authors: Jan-Willem Van Looy, Alessandro Trenta, Alessio Gravina, Alessio Borgi, Ferdinando Zanchetta, Pietro Liò, Davide Bacciu, Rita Fioresi

Organizations: FaBiT, University of Bologna · Department of Computer Science, University of Pisa · Department of Computer, Control and Management Engineering, Sapienza University of Rome · Department of Computer Science and Technology, University of Cambridge

Abstract

Effective long-range propagation remains a central challenge in graph neural networks, as increasing a model's propagation depth does not guarantee that distant nodes effectively influence each other. Sheaf neural networks enrich graph propagation through matrix-valued transport between stalks; still, this expressivity alone does not automatically imply effective long-range communication. We introduce ONDA, a long-range graph learning framework based on operator-valued information waves. Stalk-valued representations evolve through second-order dynamics governed by learned sheaf transport operators, combining wave-like propagation with expressive local geometry. We characterize long-range influence through a stalk-wise sensitivity analysis and show that the cross-influence never vanishes. Across long-range propagation, severe graph bottlenecks, graph transfer, and heterophilic benchmarks, ONDA consistently improves over scalar wave propagation, diffusive sheaf baselines, and state-of-the-art models, demonstrating the benefit of coupling wave dynamics with matrix-valued transport.

Figures & tables

Figure 1: Schematic view of ONDA on a single edge. Orange curves show the oscillatory evolution of node states in the stalks

{\mathcal{F}}(u)

and

{\mathcal{F}}(v)

; learned restriction maps express them in the shared edge stalk

{\mathcal{F}}(e)

(dashed orange). The blue curve shows their superposition, highlighting constructive and destructive interference between the mapped signals.

Model	sssp $\downarrow$	ecc $\downarrow$	diam $\downarrow$
A-DGN	$1.176_{\pm 0.140}$	$4.981_{\pm 0.037}$	$1.151_{\pm 0.038}$
DRew	$1.279_{\pm 0.011}$	$4.651_{\pm 0.020}$	$1.243_{\pm 0.047}$
GCN	$2.102_{\pm 0.094}$	$5.233_{\pm 0.034}$	$3.832_{\pm 0.262}$
GCNII	$2.128_{\pm 0.429}$	$5.241_{\pm 0.030}$	$2.005_{\pm 0.093}$
GIN/GINE	$2.234_{\pm 0.271}$	$4.869_{\pm 0.092}$	$1.630_{\pm 0.161}$
GPS	$0.472_{\pm 0.050}$	$4.758_{\pm 0.021}$	$2.160_{\pm 0.098}$

Table 1: ECHO-Synth benchmark results. Performance comparison on MAE (

\downarrow

). Baseline values are reported directly from Miglior et al. (2026) . For each column, the best , second-best , and third-best results are highlighted.

Model	$\bm{N=10}$ $\downarrow$	$\bm{N=20}$ $\downarrow$	$\bm{N=50}$ $\downarrow$
GCN	$884.200_{\pm 6.900}$	$961.100_{\pm 12.300}$	$860.000_{\pm 35.000}$
GAT	$7.900_{\pm 2.100}$	$487.000_{\pm 342.200}$	$852.000_{\pm 29.000}$
GraphSAGE	$455.500_{\pm 199.600}$	$998.200_{\pm 6.500}$	$876.000_{\pm 32.000}$
GIN	$900.500_{\pm 47.900}$	$970.600_{\pm 3.900}$	$985.000_{\pm 123.000}$
GraphTransformer	$670.300_{\pm 306.800}$	$1041.200_{\pm 46.900}$	$831.000_{\pm 13.000}$
ChebNet	$10.600_{\pm 1.200}$	$983.600_{\pm 5.100}$	$922.000_{\pm 47.000}$

Table 2: Barbell-clique Results. Test MSE (

\times 10^{-3}

;

\downarrow

) Results are reported as mean ±std over 5 seeds using the best configuration selected by validation. For each column, the best , second-best , and third-best results are highlighted.

Figure 2: Information transfer performance on (a) Line, (b) Ring, and (c) Crossed-Ring graphs.

Appendix figures & tables7 assets

Supplementary material from the paper’s appendix.

Appendix

Hyperparameter	Candidate values
Hidden dimension $H$	$[36,252]$
Stalk dimension $d$	$\{2,3,4\}$
ONDA Blocks $B$	$\{1,2,3,4\}$
Inner iterations $\mathcal{L}$	$[1,40]$
Step size $h$	$[10^{-4},0.5]$
Learning rate	$[10^{-5},10^{-2}]$

Appendix

Table 3: Hyperparameter search space for the ECHO Benchmark.

Hyperparameter	Candidate values
Hidden dimension $H$	$\{128,192\}$
Stalk dimension $d$	$\{2,6\}$
ONDA Blocks $B$	$\{1,2,4\}$
Inner iterations $\mathcal{L}$	$\{8,10,12,14,16,20\}$
Step size $h$	$\{0.1,0.2,0.25,0.33,0.5,0.6,0.66,0.75,1.0\}$
Learning rate	$\{3\times 10^{-4}\}$

Appendix

Table 4: Hyperparameter search space for the Barbell benchmark.

Hyperparameter	Candidate values
Hidden dimension $H$	$64$ ( Gravina et al., 2025 )
Stalk dimension $d$	$\{1,2,4\}$
ONDA Blocks $B$	$\{1,2\}$
Inner iterations $\mathcal{L}$	distance x $\{1,2\}$ / $B$
Step size $h$	$\{0.05,0.1,0.5,1.0\}$
Learning rate	$10^{-3}$ ( Gravina et al., 2025 )

Appendix

Table 5: Hyperparameter search space for the graph transfer task.

Model	$\bm{N=10}$ $\downarrow$	$\bm{N=20}$ $\downarrow$	$\bm{N=50}$ $\downarrow$
GCN	$884.200_{\pm 6.900}$	$961.100_{\pm 12.300}$	$860.000_{\pm 35.000}$
GAT	$7.900_{\pm 2.100}$	$487.000_{\pm 342.200}$	$852.000_{\pm 29.000}$
GraphSAGE	$455.500_{\pm 199.600}$	$998.200_{\pm 6.500}$	$876.000_{\pm 32.000}$
GIN	$900.500_{\pm 47.900}$	$970.600_{\pm 3.900}$	$985.000_{\pm 123.000}$
GraphTransformer	$670.300_{\pm 306.800}$	$1041.200_{\pm 46.900}$	$831.000_{\pm 13.000}$
ChebNet	$10.600_{\pm 1.200}$	$983.600_{\pm 5.100}$	$922.000_{\pm 47.000}$

Appendix

Table 6: Barbell additional results. Test MSE (

\times 10^{-3}

;

\downarrow

) on the Barbell task for graph sizes

N\in\{10,20,50\}

. All models report the results over the best configuration selected by validation, as mean

{}_{\pm\text{std}}

over 5 seeds. For each column, we report the best , second-best , and third-best results.

Hyperparameter	Candidate values
Hidden dimension $H$	$\{64,128,256,512\}$
Stalk dimension $d$	$\{2,4,8\}$
ONDA Blocks $B$	$\{2,3,4,6,8,10,11\}$
Inner iterations $\mathcal{L}$	$\{1,2,4,6,8,10\}$
Step size $\varepsilon$	$\{0.01,0.025,0.04,0.05,0.075,0.1\}$
Learning rate	$\{5\times 10^{-4},10^{-3},2\times 10^{-3},2\times 10^{-2}\}$

Appendix

Table 7: Hyperparameter search space for the heterophilic benchmarks.

Model	Roman-empire	Amazon-ratings	Minesweeper	Tolokers	Questions
Model	Acc $\uparrow$	Acc $\uparrow$	AUC $\uparrow$	AUC $\uparrow$	AUC $\uparrow$
Luan et al. (2024)
MLP-2	66.04 ±0.71	49.55 ±0.81	50.92 ±1.25	74.58 ±0.75	69.97 ±1.16
SGC-1	44.60 ±0.52	40.69 ±0.42	82.04 ±0.77	73.80 ±1.35	71.06 ±0.92
MLP-1	64.12 ±0.61	38.60 ±0.41	50.59 ±0.83	71.89 ±0.82	70.33 ±0.96
Graph-agnostic

Appendix

Table 8: Mean test set score and std on heterophilic datasets. The higher, the better. First , second , and third best results for each task are color-coded. Baseline results are reported from Finkelshtein et al. (2024) ; Platonov et al. (2023) ; Müller et al. (2024) ; Luan et al. (2024) ; Ribeiro et al. (2026) .

		$\bm{N=10}$			$\bm{N=20}$			$\bm{N=50}$
Model	#Params	Mem.	Train	Infer.	Mem.	Train	Infer.	Mem.	Train	Infer.
		MB	ms/ep.	ms/g	MB	ms/ep.	ms/g	MB	ms/ep.	ms/g
SONAR	100,079	23.1	2400.1	10.245	35.1	2452.0	10.800	120.7	2675.2	11.138
Diag-ONDA	101,405	26.8	2186.3	6.716	50.9	2193.7	6.795	223.0	2270.4	6.722
Ortho-ONDA	103,700	24.5	2229.1	7.849	41.1	2237.4	7.962	159.9	2432.3	7.925
LowRank-ONDA	101,540	27.6	2464.3	8.371	53.9	2459.3	8.290	241.8	2736.4	8.610

Appendix

Table 9: Matched-parameter computational cost on Barbell. All models use approximately

100

k trainable parameters, four blocks, and

12

internal propagation steps per block. We report peak training memory, training time per epoch, and inference latency per graph for

N\in\{10,20,50\}

, corresponding to graphs with

20

40

, and

100

nodes, respectively. Values are means over five seeds; lower is better.

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Deep Graph Neural Networks (GNNs) are essential for capturing complex dependencies in graph-structured data. However, scaling GNNs to depth remains challenging, as stacking layers leads to representation collapse and diminishing sensitivity due to repeated aggregation. While Neural Sheaf Diffusion (NSD) provides strong theoretical guarantees against such collapse, these guarantees do not translate to practice: as depth increases, the disagreement signal of the sheaf Laplacian vanishes, limiting the contribution of deeper layers. We identify mechanisms that hinder NSD effectiveness at depth and propose \emph{Deep Neural Sheaf Diffusion} (DNSD), which replaces the sheaf Laplacian with a sheaf adjacency operator to maintain informative signals across layers. This is complemented by normalization, odd nonlinearities, and gating. To provide a principled explanation of the expected performance improvement, we contrast sheaf diffusion to graph attention mechanisms, highlighting that DNSD replaces scalar attention scores with matrix-valued edge functions and normalizes node representations rather than attention scores. We demonstrate empirically that DNSD effectively utilizes deep aggregation in graph tasks, outperforming GNN and NSD baselines with up to 30pp accuracy on synthetic long-range datasets, and consistently outperforming them on real-world benchmarks. These results position sheaf-based architectures as a promising building block for graph foundation models by supporting effective deep architectures.

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