ChronoSheaf: Joint Spatio-Temporal Spectral Filtering on Cellular Sheaves

Abeer Mostafa; Zahra Ahmadi

cs.LGOct 15, 2026

ChronoSheaf: Joint Spatio-Temporal Spectral Filtering on Cellular Sheaves

Authors: Abeer Mostafa, Zahra Ahmadi

Organizations: Peter L. Reichertz Institute for Medical Informatics, Hannover Medical School, Germany · Lower Saxony Center for Artificial Intelligence and Causal Methods in Medicine (CAIMed), Hannover, Germany

Abstract

Sheaf neural networks generalize graph neural networks by assigning each edge a learned linear restriction map that specifies how the local representations of adjacent nodes relate to one another, thereby mitigating oversmoothing. Existing sheaf architectures, however, define the sheaf Laplacian as an instantaneous spatial operator and therefore cannot explicitly model how signals evolve over time on a graph. We first introduce a spectral sheaf Laplacian, defined jointly over space and time, such that a single operator regulates signal propagation across both the spatial graph and the temporal domain. Directly extending this operator's receptive field in space or time would require increasingly complex sheaf structures. Instead, we propose a spectral filter that expands the effective receptive field along both dimensions without enlarging the underlying sheaf. Building on this operator, we introduce ChronoSheaf, a spatio-temporal sheaf network with two key components. First, Transitional Restriction Maps (TRM) decompose each node's restriction map into a spatial term and two temporal frames that independently control its relation to neighboring nodes at the preceding and following timestep, enabling asymmetric temporal influence. Second, the Dual Spectral Filter (DSF) applies a single bounded-degree polynomial filter with an adaptively estimated spectral radius to the joint spatio-temporal sheaf operator. We validate ChronoSheaf on a diverse set of spatio-temporal forecasting benchmarks, where it outperforms state-of-the-art baselines. The code is available at: https://anonymous.4open.science/r/ChronoSheaf-1D7D/.

Figures & tables

Figure 1: (A) State-of-the-art graph forecasting models aggregate spatial information from neighboring nodes and enlarge their receptive field by stacking additional layers. (B) Our model performs spectral filtering directly on joint space-time cellular sheaves. Higher-order polynomial filters extend the receptive field to larger neighborhoods and capture deeper temporal context.

Figure 2: (A) Neural sheaf structure Bodnar et al. (2022) . (B) ChronoSheaf’s joint space-time sheaf structure.

Figure 3: Node B’s signal at

t-1

contributes

-A(D,B)(F^{\mathrm{in}}_{D,t})^{\top}F^{\mathrm{out}}_{B,t-1}\,x_{B,t-1}

(Lx)_{D,t}

, transporting B’s signal from its outgoing frame into D’s incoming frame. The analogous term one step later carries D’s present into E’s future.

Figure 4: Test RMSE (left) and Dirichlet energy (right, log scale) versus network depth. Baseline energy collapses with depth, while ChronoSheaf’s remains stable.

Model	LargeST (SD)			LargeST (CA)			PEMS04			PEMS08
Model	MAE	RMSE	MAPE	MAE	RMSE	MAPE	MAE	RMSE	MAPE	MAE	RMSE	MAPE
STGCN Yu et al. (2018)	19.67	34.14	13.86	21.33	36.39	16.53	22.70	35.55	14.59	18.02	27.83	11.40
DCRNN Li et al. (2018b)	21.03	33.37	14.13	21.87	34.41	17.06	24.70	38.12	17.12	17.86	27.83	11.45
GW-Net Wu et al. (2019)	17.74	29.62	11.88	21.72	34.20	17.40	25.45	39.70	17.29	19.13	31.05	12.68
GMAN Zheng et al. (2020)	19.63	30.01	12.20	OOM	OOM	OOM	20.23	32.17	17.06	16.47	25.72	10.48
ASTGCN Guo et al. (2019)	23.70	37.63	15.65	OOM	OOM	OOM	22.80	35.82	16.56	18.63	28.27	13.08

Table 1: Traffic forecasting results on LargeST (SD and CA), PEMS04, and PEMS08, averaged over 12 forecasting steps. Best results are in bold and second-best are underlined. "OOM" denotes out of memory.

Model	Molene (weather)					Air Quality ( $\text{PM}_{2.5}$ )
Model	step-1	step-2	step-3	step-4	step-5	1-12 hr	13-24 hr	25-48 hr
STGCN ( Yu et al., 2018 )	1.066	1.552	1.933	2.225	2.698	22.77	31.59	35.59
DCRNN ( Li et al., 2018b )	0.396	0.635	0.907	1.164	1.419	24.28	32.95	36.11
GW-Net ( Wu et al., 2019 )	0.855	1.124	1.455	1.865	2.104	22.42	30.29	34.73
GMAN ( Zheng et al., 2020 )	0.962	1.424	1.882	2.022	2.417	24.11	32.80	36.25
ASTGCN ( Guo et al., 2019 )	0.981	1.342	1.877	1.906	2.125	25.51	33.98	37.42

Table 2: Weather (Molene) and air quality (PM 2.5 ) forecasting results in MAE across multiple prediction horizons. Best results are in bold and second-best are underlined.

Polynomial Filter Order	$k=1$	$k=2$	$k=3$	$k=4$	$k=5$	$k=6$	$k=7$	$k=8$	$k=9$	$k=10$
RMSE	25.96	23.92	23.26	23.20	23.15	23.15	23.05	23.04	23.01	23.01
Training Time (s/epoch)	8.021	11.63	15.22	19.46	23.86	26.80	30.57	34.80	37.70	40.32

Table 3: Effect of the spectral filter order

k

on test RMSE and training time.

Model	Multimodal PEMS04			Multimodal PEMS08
Model	MAE	RMSE	MAPE	MAE	RMSE	MAPE
STGCN ( Yu et al., 2018 )	21.36	34.42	13.25	17.98	27.52	11.10
DCRNN ( Li et al., 2018b )	21.67	35.91	14.32	18.44	27.93	13.62
GW-Net ( Wu et al., 2019 )	24.17	38.22	16.54	18.88	28.79	14.02
GMAN ( Zheng et al., 2020 )	23.63	37.67	15.50	17.08	28.41	12.66
CITRUS ( Einizade et al., 2024 )	20.34	30.11	14.89	15.56	27.31	12.10

Table 4: Traffic forecasting results on PEMS04 and PEMS08 augmented with NOAA weather covariates from the corresponding regions, averaged over 12 forecasting steps. Best results are in bold and second-best are underlined.

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 5: Ground truth versus 12-step-ahead ChronoSheaf predictions on nine PEMS08 test nodes, with Chebyshev order

K=2

and

K=10

. Per-node MAE is shown above each panel.

	Molene (weather)					Air Quality (PM2.5)
Variant	step-1	step-2	step-3	step-4	step-5	1–12 hr	13–24 hr	25–48 hr
Trivial sheaf	0.36	0.61	0.95	1.32	1.52	23.44	32.91	36.06
$F^{\mathrm{in}}=F^{\mathrm{out}}$	0.32	0.55	0.85	1.07	1.34	21.71	30.83	33.72
Self-loop temporal edges only	0.33	0.56	0.88	1.12	1.40	22.06	31.4	34.76
Fixed $\lambda_{\max}$	0.32	0.55	0.84	1.05	1.29	22.05	30.88	33.41
ChronoSheaf (full)	0.31	0.52	0.80	1.01	1.24	21.31	29.90	32.41

Appendix

Table 5: Ablation study on model variants across prediction horizons in MAE (lower is better).

Polynomial Filter Order	k=1	k=2	k=3	k=4	k=5	k=6	k=7	k=8	k=9	k=10
Molene (N =	1.25	1.82	2.36	3.04	3.71	4.19	4.75	5.43	5.87	6.29
PEMS08 (N =	8.021	11.63	15.22	19.46	23.86	26.80	30.57	34.80	37.70	40.32
PEMS04 (N =	14.73	21.28	28.04	35.61	43.95	49.07	56.31	63.72	69.40	73.89
Air Quality (N =	23.60	34.31	44.65	57.38	70.04	79.02	89.77	102.56	110.71	118.85
LargeST (SD) (N =	34.89	50.47	66.38	84.51	103.96	116.42	133.18	151.14	164.23	175.20
LargeST (CA) (N =	422.17	611.48	802.35	1022.91	1257.40	1408.76	1611.02	1829.35	1986.71	2119.84

Appendix

Table 6: Training time (seconds/epoch) for different polynomial filter orders

k

and different graph sizes

N

Dataset	Domain	Nodes	Time steps	Interval	Period	Target
PEMS04	Traffic	307	16,992	5 min	01/2018–02/2018	Flow
PEMS08	Traffic	170	17,856	5 min	07/2016–08/2016	Flow
LargeST (SD)	Traffic	716	525,888	5 min	01/2017–12/2021	Flow
LargeST (CA)	Traffic	8,600	525,888	5 min	01/2017–12/2021	Flow
Molene	Weather	32	744	1 h	01/2014	Temperature
Air Quality	Air quality	437	8,760	1 h	05/2014–04/2015	PM2.5

Appendix

Table 7: Summary of dataset statistics.

Sheaf Neural Networks

Sheaf neural networks generalize graph learning with local vector spaces and edge maps that model how information is compared and transported.

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Apr 13, 2026J/K move · Enter open · S save

Dynamic Sheaf Diffusion Networks with Adaptive Local Structure for Heterogeneous Spatio-Temporal Graph Learning

Abeer Mostafa, Raneen Younis, Zahra Ahmadi

Peter L. Reichertz Institute for Medical Informatics, Hannover Medical School, Hannover, Germany · Lower Saxony Center for Artificial Intelligence and Causal Methods in Medicine (CAIMed), Hannover, Germany

Over-Smoothing in GNNs Graph Neural Networks Spatiotemporal GNNs Spatiotemporal Forecasting Sheaf Neural Networks

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Spatio-temporal processes often exhibit highly heterogeneous and non-intuitive responses to localized disruptions, limiting the effectiveness of conventional message passing approaches in modeling local heterogeneity. We reformulate spatio-temporal forecasting as the problem of learning information flow over locally structured spaces, rather than propagating globally aligned node representations. To this end, we introduce a spatio-temporal sheaf diffusion graph neural network (ST-Sheaf GNN) that embeds graph topology into sheaf-based vector spaces connected by learned linear restriction maps. Unlike prior approaches relying on static or globally shared transformations, our model learns dynamic restriction maps that evolve over time and adapt to local spatio-temporal patterns, enabling more expressive interactions. The proposed framework both theoretically guarantees and empirically demonstrates evidence that the proposed diffusion mechanism mitigates oversmoothing, preserving discriminative node representations even with increasing diffusion layer depth. Experiments on diverse real-world spatio-temporal forecasting benchmarks across multiple domains demonstrate state-of-the-art performance, highlighting the effectiveness of sheaf topological representations as a principled foundation for spatio-temporal graph learning. The code is available at: https://anonymous.4open.science/r/ST-SheafGNN-6523/.

Abstract

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