GCGNet: Graph-Consistent Generative Network for Time Series Forecasting with Exogenous Variables

Zhengyu Li; Xiangfei Qiu; Yuhan Zhu; Xingjian Wu; Jilin Hu; Chenjuan Guo; Bin Yang

cs.LGMar 9, 2026

GCGNet: Graph-Consistent Generative Network for Time Series Forecasting with Exogenous Variables

Authors: Zhengyu Li, Xiangfei Qiu, Yuhan Zhu, Xingjian Wu, Jilin Hu, Chenjuan Guo, Bin Yang

Organizations: East China Normal University

Abstract

Exogenous variables offer valuable supplementary information for predicting future endogenous variables. Forecasting with exogenous variables needs to consider both past-to-future dependencies (i.e., temporal correlations) and the influence of exogenous variables on endogenous variables (i.e., channel correlations). This is pivotal when future exogenous variables are available, because they may directly affect the future endogenous variables. Many methods have been proposed for time series forecasting with exogenous variables, focusing on modeling temporal and channel correlations. However, most of them use a two-step strategy, modeling temporal and channel correlations separately, which limits their ability to capture joint correlations across time and channels. Furthermore, in real-world scenarios, time series are frequently affected by various forms of noises, underscoring the critical importance of robustness in such correlations modeling. To address these limitations, we propose GCGNet, a Graph-Consistent Generative Network for time series forecasting with exogenous variables. Specifically, GCGNet first employs a Variational Generator to produce coarse predictions. A Graph Structure Aligner then further guides it by evaluating the consistency between the generated and true correlations, where the correlations are represented as graphs, and are robust to noises. Finally, a Graph Refiner is proposed to refine the predictions to prevent degeneration and improve accuracy. Extensive experiments on 12 real-world datasets demonstrate that GCGNet outperforms state-of-the-art baselines.

Figures & tables

Panel of Figure 1 — Figure 1: Illustration of forecasting Electricity Demand (i.e., the endogenous variable) with exogenous variable of Air Temperature. $X^{endo}$ and $Y^{endo}$ are historical and future endogenous variables, and $X^{exo}$ and $Y^{exo}$ are historical and future exogenous variables.

Figure 3: The architecture of GCGNet. (a) The Variational Generator, which produces coarse prediction. (b) The Graph Structure Aligner, which guides the optimization of the Variational Generator. (c) The Graph VAE, which generates the adjacency matrix for the nodes. (d) The Graph Refiner, which refines the final predictions using the learned adjacency matrix.

Models		GCGNet		TimeXer		TFT		TiDE		DUET		CrossLinear		Amplifier		TimeKAN		xPatch		PatchTST
Metrics		mse	mae	mse	mae	mse	mae	mse	mae	mse	mae	mse	mae	mse	mae	mse	mae	mse	mae	mse	mae
NP	24	0.197	0.233	0.236	0.266	0.219	0.249	0.284	0.301	0.246	0.287	0.215	0.261	0.252	0.303	0.273	0.310	0.234	0.278	0.249	0.294
	360	0.496	0.442	0.600	0.475	0.539	0.501	0.601	0.498	0.576	0.528	0.593	0.508	0.587	0.534	0.538	0.529	0.523	0.461	0.530	0.498
	Avg	0.346	0.337	0.418	0.371	0.379	0.375	0.443	0.400	0.411	0.408	0.404	0.384	0.420	0.418	0.405	0.419	0.378	0.370	0.390	0.396
PJM	24	0.058	0.148	0.075	0.166	0.095	0.195	0.106	0.214	0.072	0.166	0.094	0.197	0.096	0.208	0.115	0.244	0.077	0.168	0.116	0.239
	360	0.128	0.223	0.140	0.231	0.133	0.219	0.177	0.279	0.131	0.228	0.142	0.254	0.177	0.285	0.162	0.281	0.131	0.220	0.149	0.279

Table 1: Mean Absolute Error (MAE) and Mean Squared Error (MSE) on 12 real-world datasets with exogenous variables. The best results are Red , and the second-best results are Blue . Avg represents the average results across the two forecasting horizons.

Dataset	NP		PJM		DE		Energy		Average
Metrics	mse	mae	mse	mae	mse	mae	mse	mae	mse	mae
(a) Replace Variational Generator	0.537	0.464	0.140	0.237	0.548	0.471	0.226	0.374	0.363	0.386
(b) Remove $L_{align}$	0.659	0.526	0.137	0.230	0.791	0.584	0.308	0.446	0.474	0.446
(c) Replace Graph VAE	0.691	0.540	0.213	0.290	0.665	0.526	0.228	0.376	0.449	0.433
(d) Remove Graph Refiner	0.853	0.599	0.320	0.387	0.970	0.653	0.278	0.413	0.605	0.513
GCGNet	0.496	0.442	0.128	0.223	0.491	0.445	0.179	0.330	0.323	0.360

Table 2: Ablation studies for GCGNet. The best results are highlighted in bold .

Models	GCGNet		TimeXer		TFT		TiDE		DUET		CrossLinear		Amplifier		TimeKAN		xPatch		PatchTST
Metrics	mse	mae	mse	mae	mse	mae	mse	mae	mse	mae	mse	mae	mse	mae	mse	mae	mse	mae	mse	mae
NP	0.425	0.377	0.440	0.383	0.647	0.488	0.467	0.416	0.444	0.383	0.451	0.394	0.520	0.448	0.484	0.435	0.488	0.402	0.457	0.401
PJM	0.133	0.217	0.141	0.229	0.200	0.270	0.158	0.253	0.140	0.226	0.147	0.241	0.152	0.245	0.175	0.270	0.139	0.223	0.148	0.243
BE	0.458	0.301	0.477	0.301	0.563	0.354	0.547	0.348	0.473	0.305	0.477	0.300	0.502	0.325	0.488	0.315	0.483	0.302	0.485	0.309
FR	0.428	0.245	0.454	0.247	0.535	0.288	0.494	0.290	0.468	0.262	0.476	0.257	0.494	0.286	0.491	0.276	0.477	0.257	0.470	0.264
DE	0.611	0.497	0.659	0.507	0.684	0.515	0.644	0.519	0.660	0.513	0.635	0.508	0.712	0.548	0.669	0.526	0.633	0.500	0.696	0.527

Table 3: Average results under the setting where future exogenous variables are not available. The inputs are (

X^{\text{endo}}

and

X^{\text{exo}}

). The best results are Red , and the second-best results are Blue . Full results are provided in Table 7 of Appendix B.2 .

Mask Type	Zero										Random
Models	GCGNet		TimeXer		CrossLinear		DUET		PatchTST		GCGNet		TimeXer		CrossLinear		DUET		PatchTST
Metrics	mse	mae	mse	mae	mse	mae	mse	mae	mse	mae	mse	mae	mse	mae	mse	mae	mse	mae	mse	mae
Colbun	0.076	0.093	0.084	0.145	0.065	0.102	0.106	0.126	0.070	0.108	0.070	0.096	0.112	0.169	0.066	0.108	0.084	0.118	0.078	0.120
Energy	0.070	0.203	0.122	0.273	0.097	0.241	0.100	0.244	0.108	0.257	0.081	0.217	0.123	0.275	0.098	0.242	0.100	0.245	0.102	0.249
DE	0.292	0.336	0.418	0.403	0.426	0.416	0.428	0.407	0.480	0.445	0.344	0.366	0.433	0.411	0.433	0.418	0.431	0.407	0.490	0.449
PJM	0.064	0.155	0.100	0.198	0.098	0.203	0.079	0.177	0.120	0.246	0.066	0.158	0.102	0.199	0.101	0.201	0.079	0.176	0.112	0.224

Table 4: Average results under the setting where exogenous variables are partial missing. The best results are highlighted in bold . Full results are provided in Table 10 of Appendix B.4 .

Mask Type	Zero				Random
Models	GCGNet		w/o VAE		GCGNet		w/o VAE
Colbun	0.076	0.093	0.094	0.140	0.070	0.096	0.081	0.123
Energy	0.070	0.203	0.142	0.293	0.081	0.217	0.142	0.291
DE	0.292	0.336	0.453	0.430	0.344	0.366	0.505	0.460
PJM	0.064	0.155	0.108	0.208	0.066	0.158	0.094	0.193
NP	0.204	0.235	0.301	0.310	0.208	0.236	0.272	0.283

Table 5: Average results of GCGNet and the version without VAE under the setting where exogenous variables are partial missing. The best results are highlighted in bold .

Figure 4: Prediction results on the NP dataset. (a–b) Exogenous variables: wind power and grid load. (c) Endogenous variables: Electricity Price. (d–f) PatchTST, CrossLinear and GCGNet predictions when future exogenous variables are unavailable. (g–i) PatchTST, CrossLinear and GCGNet predictions when future exogenous variables are available.

Appendix figures & tables8 assets

Supplementary material from the paper’s appendix.

Appendix

Dataset	#Num	Sampling Frequency	Lengths	Split	Ex. Descriptions	En. Descriptions
NP	2	1 Hour	52,416	7:1:2	Grid Load, Wind Power	Nord Pool Electricity Price
PJM	2	1 Hour	52,416	7:1:2	System Load, SyZonal COMED load	Pennsylvania-New Jersey-MarylandElectricity Price
BE	2	1 Hour	52,416	7:1:2	Generation, System Load	Belgium’s Electricity Price
FR	2	1 Hour	52,416	7:1:2	Generation, System Load	France’s Electricity Price
DE	2	1 Hour	52,416	7:1:2	Wind power, Ampirion zonal load	German’s Electricity Price
Energy	5	1 Hour	13,064	7:1:2	Battery, Geothermal, Hydroelectric, Solar, Wind	Thermoelectric

Appendix

Table 6: Statistics of datasets. Ex. and En. are abbreviations for the Exogenous variables and Endogenous variables, respectively.

Figure 5: Visualization of prediction results on NP dataset.

Panel of Figure 6 — Figure 6: Parameter sensitivity studies of main hyper-parameters in GCGNet.

Models		GCGNet		TimeXer		TFT		TiDE		DUET		CrossLinear		Amplifier		TimeKAN		xPatch		PatchTST
Metrics		mse	mae	mse	mae	mse	mae	mse	mae	mse	mae	mse	mae	mse	mae	mse	mae	mse	mae	mse	mae
NP	24	0.234	0.259	0.270	0.292	0.330	0.321	0.297	0.309	0.262	0.271	0.246	0.282	0.302	0.321	0.336	0.344	0.260	0.277	0.283	0.296
	360	0.617	0.495	0.609	0.474	0.964	0.655	0.638	0.523	0.626	0.494	0.655	0.505	0.738	0.574	0.632	0.527	0.717	0.528	0.630	0.505
	Avg	0.425	0.377	0.440	0.383	0.647	0.488	0.467	0.416	0.444	0.383	0.451	0.394	0.520	0.448	0.484	0.435	0.488	0.402	0.457	0.401
PJM	24	0.073	0.168	0.096	0.191	0.129	0.231	0.105	0.213	0.082	0.178	0.096	0.198	0.093	0.195	0.130	0.236	0.085	0.178	0.106	0.212
	360	0.192	0.266	0.186	0.267	0.270	0.310	0.210	0.292	0.198	0.273	0.197	0.283	0.210	0.294	0.219	0.304	0.193	0.267	0.189	0.273

Appendix

Table 7: Results under the setting where future exogenous variables are not available. The inputs are (

X^{\text{endo}}

and

X^{\text{exo}}

). The best results are Red , and the second-best results are Blue . Avg represents the average results across the two forecasting horizons.

Models		GCGNet		TimeXer		TFT		TiDE		DUET		CrossLinear		Amplifier		TimeKAN		xPatch		PatchTST
Metrics		mse	mae	mse	mae	mse	mae	mse	mae	mse	mae	mse	mae	mse	mae	mse	mae	mse	mae	mse	mae
NP	24	0.208	0.237	0.236	0.266	0.219	0.249	0.284	0.301	0.246	0.287	0.210	0.266	0.252	0.303	0.273	0.310	0.234	0.278	0.249	0.294
	360	0.531	0.459	0.600	0.475	0.539	0.501	0.601	0.498	0.576	0.528	0.531	0.508	0.587	0.534	0.538	0.529	0.523	0.461	0.530	0.498
	Avg	0.370	0.348	0.418	0.371	0.379	0.375	0.443	0.400	0.411	0.408	0.371	0.387	0.420	0.418	0.405	0.419	0.378	0.370	0.390	0.396
PJM	24	0.060	0.150	0.075	0.166	0.095	0.195	0.106	0.214	0.072	0.166	0.088	0.191	0.096	0.208	0.115	0.244	0.077	0.168	0.116	0.239
	360	0.129	0.223	0.140	0.231	0.133	0.219	0.177	0.279	0.131	0.228	0.135	0.254	0.177	0.285	0.162	0.281	0.131	0.220	0.149	0.279

Appendix

Table 8: Results under the setting where future exogenous variables are available and each model uses a fixed configuration across forecasting horizons. The inputs are (

X^{\text{endo}}

Y^{\text{endo}}

and

X^{\text{exo}}

). The best results are Red , and the second-best restuls are Blue . Avg represents the average results across the two forecasting horizons.

Models		GCGNet		TimeXer		TFT		TiDE		DUET		CrossLinear		Amplifier		TimeKAN		xPatch		PatchTST
Metrics		mse	mae	mse	mae	mse	mae	mse	mae	mse	mae	mse	mae	mse	mae	mse	mae	mse	mae	mse	mae
NP	24	0.240	0.260	0.270	0.292	0.330	0.321	0.297	0.309	0.262	0.271	0.246	0.282	0.302	0.321	0.336	0.344	0.260	0.277	0.283	0.296
	360	0.620	0.502	0.609	0.474	0.964	0.655	0.638	0.523	0.626	0.494	0.655	0.505	0.738	0.574	0.632	0.527	0.717	0.528	0.630	0.505
	Avg	0.430	0.381	0.440	0.383	0.647	0.488	0.467	0.416	0.444	0.383	0.451	0.394	0.520	0.448	0.484	0.435	0.488	0.402	0.457	0.401
PJM	24	0.072	0.164	0.096	0.191	0.129	0.231	0.105	0.213	0.082	0.178	0.096	0.198	0.093	0.195	0.130	0.236	0.085	0.178	0.106	0.212
	360	0.191	0.269	0.186	0.267	0.270	0.310	0.210	0.292	0.198	0.273	0.197	0.283	0.210	0.294	0.219	0.304	0.193	0.267	0.189	0.273

Appendix

Table 9: Results under the setting where future exogenous variables are not available and each model uses a fixed configuration across forecasting horizons. The inputs are (

X^{\text{endo}}

and

X^{\text{exo}}

). The best results are Red , and the second-best restuls are Blue . Avg represents the average results across the two forecasting horizons.

Mask Type		Zero										Random
Models		GCGNet		TimeXer		CrossLinear		DUET		PatchTST		GCGNet		TimeXer		CrossLinear		DUET		PatchTST
Metrics		mse	mae	mse	mae	mse	mae	mse	mae	mse	mae	mse	mae	mse	mae	mse	mae	mse	mae	mse	mae
Colbun	10	0.066	0.088	0.081	0.145	0.063	0.101	0.128	0.134	0.064	0.109	0.070	0.107	0.115	0.176	0.068	0.111	0.104	0.128	0.081	0.124
	30	0.081	0.093	0.086	0.145	0.063	0.101	0.094	0.122	0.074	0.109	0.068	0.092	0.111	0.168	0.065	0.102	0.070	0.111	0.081	0.120
	50	0.081	0.096	0.084	0.147	0.070	0.105	0.096	0.121	0.071	0.107	0.073	0.090	0.110	0.164	0.065	0.111	0.078	0.115	0.074	0.114
	avg	0.076	0.093	0.084	0.145	0.065	0.102	0.106	0.126	0.070	0.108	0.070	0.096	0.112	0.169	0.066	0.108	0.084	0.118	0.078	0.120

Appendix

Table 10: Results under the setting where exogenous variables are partially missing with missing ratios of 10%, 30%, and 50%. The best results are highlighted in bold .

Panel of Figure 7 — Figure 7: Forecasting performance under varying look-back windows across four datasets (DE, PJM, Energy, and NP).

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XMatch: Enhancing Covariate-Aware Time Series Forecasting through Tree-Structured Exogenous Matching

Ziyang Zhang, Hanyin Cheng, Xiangfei Qiu, Yang Shu+2

East China Normal University

Multivariate Time Series Forecasting Time Series Forecasting Time Series Forecasting with Covariates

Open paper AskPDF arXiv

Future exogenous variables provide valuable information for forecasting endogenous time series. Existing covariate-aware methods primarily learn the direct influence of exogenous variables on endogenous variables. However, these effects can be complex and change with the pattern of the exogenous variables, making them difficult to capture. Beyond this perspective, we observe that a given exogenous pattern often co-occurs with only a small set of endogenous response patterns. These associations motivate a strategy that matches future and historical exogenous patterns and uses the corresponding endogenous patterns to enhance forecasting. However, in real-world forecasting scenarios with multiple exogenous variables, each exogenous variable provides a distinct dimension for matching, creating a dilemma for this strategy between precise matching and sufficient historical support. To bridge this gap, we propose XMatch (EXogenous MATCHing), a covariate-aware forecasting model that realizes the aforementioned strategy through a tree-structured matching process that adaptively adjusts the number of exogenous variables used as matching conditions. Specifically, we first introduce the ProtoTree Creator, which organizes historical correspondences between exogenous and endogenous patterns into a ProtoTree, whose deeper levels incorporate additional exogenous variables for matching. For forecasting, we then design the ProtoTree Matcher, which uses future exogenous variables to query the ProtoTree and adaptively determines how many exogenous variables to use for matching based on exogenous pattern similarity and historical support. Finally, the matched endogenous patterns are used as explicit historical evidence to enhance forecasting. Extensive experiments on 12 real-world datasets demonstrate that XMatch outperforms state-of-the-art baselines.

Abstract

Exogenous Time Series Forecasting

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