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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