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
Venue: ICLR 2026 Poster (2026)
Keywords: time series forecasting
Primary area: learning on time series and dynamical systems

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, recorded 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.

Published as a conference paper at ICLR 2026
GCGNET: GRAPH-CONSISTENT GENERATIVE NET- WORK FOR TIME SERIES FORECASTING WITH EXOGE- NOUS VARIABLES
Zhengyu LI, Xiangfei Qiu, Yuhan Zhu, Xingjian Wu, Jilin Hu, Chenjuan Guo, Bin Yang* East China Normal University flizhengyu, xfqiu, yazhu, xjwu) ecnu. edu. en, {jlhu, eiguo, byang) ecnu. edu.cn
ABSTRACT
Exogenous variables offer valuable supplementary information for predicting fu- ture endogenous variables. Forecasting with exogenous variables needs to con- sider both past-to-future dependencies (i.e., temporal correlations) and the influ- ence 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 pro- posed for time series forecasting with exogenous variables, focusing on modeling temporal and channel correlations. However, most of them a two-step strategy, modeling temporal and channel correlations separately, which limits their abil- ity 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 model- Sraph-Consistent
ing. To address these limitations, we propose GCGNet, Generative Network for time series forecasting with exogenous variables. Specif- ically, GCGNet first employs Variational Generator to produce coarse predic- tions. Graph Structure Aligner then further guides it by evaluating the con- sistency between the generated and true correlations, where the correlations are represented as graphs, and are robust to noises. Finally, a Graph Refiner is pro- posed to refine the predictions prevent degeneration and improve accuracy. Extensive experiments on 12 real-world datasets demonstrate that GCGNet out- performs state the- baselines. We make our code and datasets available at https///ghhha..coomcccccoinnincc
INTRODUCTION
Time series forecasting plays crucial role in many domains, including economics (Hu al. 2026
2025a), traffic (Xu et al. 2023 Guo et al. 2014 Ma et al.) 2014). health (Wei et al. 2022), en-
ergy (Alvarez al. 2011 Guo al. 2015), environmental applications TTian al.) 2025 2026).
industrial manufacturing (aang et al. 2023b 2026a Cheng et al. 2026). and AlOps (Campos et al., 2024 Qi et 2023 Pan al. 2023 Accurate predictions not only support better decision-making but also enhance system management and operational efficiency. While many existing forecast- ing methods primarily leverage historical values of the target series (i.e., endogenous variables), real-world applications often benefit from incorporating additional covariate series (i.e., exogenous variables). These covariates provide valuable supplementary information for predicting future en- dogenous values (Quu al.) 2025e). Moreover, in many scenarios, future exogenous variables such as temperature, precipitation, and wind power can be reliably estimated using mature techniques, making them accessible for downstream prediction tasks. These variables are particularly valuable because their horizons align with those of the forecasting targets, offering direct predictive signals. In this work, we foeus forecasting with both historical and future exogenous variables, while also being able to accommodate scenarios where future exogenous variables are unavailable.
"Corresoonding author

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Published as a conference paper at ICLR 2026
Temporal and channel correlations are prevalent in real-world scenarios (Yue et al. 2025 Huang et al.; Ma et al. 2025a 2026), and accurately modeling them is crucial for effective forecasting with exogenous variables Wang al. 2024 Zhou et al.] (2025). For instance, in electricity demand fore- casting, historical electricity demand often exhibits similar temporal dependencies and periodicities to future electricity needs, demonstrating temporal correlations. If we consider temperature as variate in electricity forecasting, as shown in Figure ! we may observe that higher temperatures tend lead to higher electricity demand, which illustrates channel correlations. Many methods have been proposed to capture temporal and channel correlations, but most of them adopt a two-step modeling strategy, in which temporal correlations and channel correlations are modeled separately different steps. As illustrated in Figure 2 these approaches either first model temporal correlations and then channel correlations (e.g.. TimeXer (Wang et al.] 2024). ExoTST (Tayal al.) or first model channel correlations and then temporal correlations (e.g.. TFT
2024). Figure 2p).
[Lim et al., 2021)
CrossLinear (Zhou et al. 2025), Figure [2). However, the two-step modeling strategy may lead to interference between the correlations learned in the two steps (Tu et al. 2025 Chi et al. 2025
which in turn limits the model's capacity to capture temporal and channel correlations, ultimately resulting in suboptimal performance. Graph-based approaches. however,
naturally well-suited
for modeling correlations (Hu et al., 2025b Chen et al. 20233 Shao et al. [025 Liu et al. [2026a]
Yu et al. 2025a). They can capture the relationships between nodes, enabling the joint modeling of temporal and channel correlations instead of modeling them separately.
Moreover, in real-world scenarios, sen- sor failures, transmission errors, and man- ual recording mistakes frequently occur, which introduce diverse types of noises into the data (Wang et al.) 2024] Cheng) et al. 2023). These issues further compli- cate the forecasting task, as the observed data may no longer accurately reflect the true correlations (Qiu et al.] (025a)). As a result, conventional models tend to overfit noisy observations, which prevents them from capturing reliable correlations (Li et al. 2022: Wang et al. 2018). Genera- live models, however, have demonstrated strong performance in many time series analysis tasks (Li 2022) Huang et al.)
History
Now
Future
Temperature
X
Electricity Demand
Xeddo
yendo
Figure 1: Illustration of forecasting Electricity Demand (i.e., the endogenous variable) with exogenous variable of Air Temperature. Xendo and yendo are historical and future endogenous variables, and Xezo and Yezo are historical and future exogenous variables.
pendo
Channel Madeling
Temporal Madeling
Temporal & Channel Modeling
2022 Yu & Oh 2021 DBL 2022). They are designed to learn the underlying data
Temporal Modeling
Channel Modeling
distribution and capture the latent struc- tures of time series, rather than inferring correlations directly from noisy observa- tions, which can be misleading.
X X
X
yezr
dndo
X
x
yeudo
Inspired by the above observations, we propose GCGNet, a Graph- ContisCCnnt Generative Network. The network aims to learn a graph that robustly captures the joint correlations across time and chan- nels. Specifically, we first use a Vari- ational Generator module to produce a coarse prediction. A Graph Structure Aligner module then guides the generator by evaluating the consistency between the generated and true correlations. The correlations are represented as graphs, produced by the Graph VAE module, and are robust to noises. Finally, Graph Refiner module prevents potential degener- ation in the Graph Structure Aligner module and improves the initial predictions by leveraging the learned correlations.
(a) Temporal
Channel
Channel
Temperal
(c) Joint Modeling
Figure 2: Categories of time series forecasting with ex- ogenous variables algorithms: (a) modeling temporal correlations first, then modeling channel correlations; (b) modeling channel correlations first, then model- ing temporal correlations; (c) jointly modeling tempo- ral and channel correlations, where yeedo represents coarse prediction of y yendo detailed in Section 3.2
In summary, our contributions are as follows:
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We propose a general framework called GCGNet, which is designed for time series forecasting with historical and future exogenous variables a robust manner. The framework achieves accurate forecasting through a graph-consistent generative network.
We design the Graph Structure Aligner module, which leverages the consistency of graph struc- tures to guide the generator, ensuring the outputs align with the underlying temporal and channel joint correlations and produce more accurate predictions.
We conduct extensive experiments on 12 real-world datasets with exogenous variables. The results show that GCGNet outperforms state-of-thes art baselines. Additionally, all datasets and code are available at htttss//gggab.oooo/cccooootttttt
RELATED WORK
2.1 TIME SERIES FORECASTING WITH EXOGENOUS VARIABLES
Time series forecasting with exogenous variables has been extensively studied in classical statistics. For example, ARIMAX and SARIMAX (Williams) 2001; Vagropoulos et al. 2016) extend ARIMA by incorporating correlations between exogenous and endogenous variables. Such methods rely on strong parametric assumptions regarding the functional form and distribution of the data, which may limit their flexibility when modeling complex real-world time series. In recent years, many deep learning methods have also been proposed for time series forecasting with exogenous variables. For example, TiDE (Das al. 2023) adopts coarse-graines strategy by directly concatenating exogenous variables with endogenous variables as inputs to the predictor. NBEATSx (Olivares) al. 2023) extends N-BEATS with dedicated branches leverage exogenous signals. TFT (Lim T02T first applies variable selection to obtain step-wise representations and subsequently
et al. models temporal correlations. CrossLinear (Zhou et al., 2025) aggregates channel information ID convolutions before forecasting. TimeXer (Wang et al. 2024) models temporal dependencies first and then incorporates exogenous variables through crossaatention, Similarly, ExoTST (Tayal et al. 2024) also employs cross-attention but introduces additional designs for encoding exogenous variables. Most of these methods adopt a two-step modeling strategy to model temporal and channel correlations, which may result in mutual interference between the two steps. This separation may reduce the model's 's ability to fully capture temporal and channel correlations.
2.2 GENERATIVE MODELS FOR TIME SERIES
Generative models have been widely applied to various time series tasks due to their ability to cap- ture underlying distributions and latent structures. For generation, models such as TimeGAN Yoon et al, 2019) and GT-GAN (DBL. 2022) produce realistic sequences by jointly optimizing supervised and adversarial objectives, while VAE-based methods (Desai al.. 2023 Wu et al.) 2025b) learn latent representations that preserve the statistical properties of the data. For imputation, generative models recover missing values by capturing the latent structures (Yu & Oh 2021 Boquet et al. 2019). In anomaly detection, variational methods capture normal temporal patterns and identify unexpected patterns as anomalies (Huang et al.) 2022 Wang al. 2023a Kieu al. 2022), while GAN-based methods detect anomalies by contrasting generated and observed sequences (Li et al.)
2019. For forecasting, D3VAE (Li et al. (022) combines variational autoencoders with diffusion- based denoising to improve the accuracy of time series prediction. In our work, we leverage gen- crative models for time series forecasting with exogenous variables, enabling robust modeling of temporal and channel correlations.
3 METHODOLOGY
In the context of exogenous-aware time series forecasting, the historical endogenous time series Xendo E XXT, is accompanied by historical exogenous variables e E _g and future exoge- nous variables yow E R2 DxF where N the number of endogenous variables, D is the number of exogenous variables, T is the number of historical time steps, and F is the forecasting horizon. The objective is to forecast the future endogenous series codoo N over the horizon F based on the
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We further introduce the details of Published as a conference paper at ICLR 2026
(d) Graph Refiner
Pendo
(b) Graph Structure Aligner
(c) Graph VAE
Lx
loss
GCN
Flatten Head
Sparsify
Graph Refiner
(S*,
(Se AA)
VAF
Graph
Structure A Aligner
Graph VAF
3
S
SP
Variational Generator
(a) Variational Generator
Patch Embedding
Norn
Instunce
VAE
Concal
Xexo
yeau
X
S
yezo
X
Pexo pendo
s
Graph Leaner
Xendo
Xendo
Vemslo
Xendo
Xando
yendo
history future
exo
X
you
Coptional
predict
adjacency
weight
endo
pand
yanda
patch embedding
matrix
Figure 3: The architecture of GCGNet. (a) The Variational Generator, which produces coarse predic- tion. (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.
given historical endogenous and exogenous series, which can be formally expressed follows.
endo
Fo(X endo Xeao yesa
3.1 STRUCTURE OVERVIEW
In this work, we aim to model the joint temporal and channel correlations in a robust manner by in- tegrating graph-based and generative models. Figure shows the overall architecture of GCGNet, which consists of three main modules: the Variational Generator module, the Graph Structure Aligner module and the Graph Refiner module. First, introduce the Variational Generator mod- ule to generate a coarse prediction, which provides an initial estimate for subsequent modules to guide optimization and correlations modeling. Next, we adopt the Graph Structure Aligner module to guide the optimization of the Variational Generator module. In addition, it generates an adjacency matrix Å, which captures joint correlations over time and channels. Specifically, a Graph VAE is uti- lized within the Graph Structure Aligner module to generate the adjacency matrix Å, which robustly represents these correlations. Finally, we employ the Graph Refiner module to enhance the initial predictions and prevent the degeneration (i.c., the module produces identical outputs regardless of the input) of the Graph Structure Aligner module. By leveraging Å, the Graph Refiner module fa- cilitates information exchange across both temporal steps and channels. After refinement, flatten prediction head is applied to produce the final forecast endo each module in the following sections.
3.2 VARIATIONAL GENERATOR
The Variational Generator module aims to produce coarse generation of the future sequence, facili- tating the subsequent modeling of dependencies. Specifically, we first apply the Instance Normaliza- tion (Ulyanov al. 2016) to unify the distributions of the training and testing data. Subsequently, the historical endogenous and exogenous variables emnko and X are fed into a VAE to generate the coarse predictions eeddn and eo Note that the generation of ÿ is optional: when the future exogenous variables available, the real yexo is directly used instead of the generated one. This design is motivated by the fact that future exogenous variables are not always accessible in real- world scenarios. The generated results then concatenated with eedoo and Xexo construct the
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input for the Graph Structure Aligner module. The process can be formalized as follows.
Greno
VAE
Yox VAB(X
(2)
5=
X
enddo
X
Yendo
E (R(N+D)((T+E),
Z
[re,
Yew
otherwise
if available
where S denotes the generated full sequence. Z takes the observed y if futurre exogenous vari- ables exist, otherwise the predicted Yexo_ S
X
yexo
E R(N+D)x(T+FF) denotes the
Exendo
Yeado
ground-truth full sequence. Both S and S are then fed into the Graph Structure Aligner module.
3.3 GRAPH STRUCTURE ALIGNER
The Graph Structure Aligner module is introduced to constrain the generative process of the Varia- tional Generator module through the alignment of correlations. As shown in Figure Bp. rather than relying solely on point-wise reconstruction losses between predicted and ground-truth series, the aligner enforces consistency at the graph level, which encourages the generator to better focus on the underlying temporal and variable dependencies. Specifically, given the generated full sequence S and the ground-truth full sequence S. first obtain patch- wise representations better model the time series (Cirstea et al., 2022: Nie et al. 2023 Wu et al. 2025a Huang et al. (024).
S" Embeedding(Patchit/y// (S)), S Embeedding(aacchit((()))
(4)
where SP. ST E [NN+))xxxdd Specifically, Patchify module first segments S and S along the temporal dimension into non-overlapping patches with patch size p. resulting in T. [(T + F)/p) patches. Then, the embedding layer maps each patch of length p into a d-dimensional vector repre- sentation.
Graph VAE. As shown in Figure 3c, capture the relational structures among the patches, feed the patch embeddings into the Graph VAE. For illustration, consider an example input X e (NN+ Dxx where in practice the input can be S or ŠP, The adjacency computation is performed
follows.
A' GELU (WV
(5)
Ā (A'= ATT),
(6)
where Á', R4XL W2, E Idd are learnable projection matrices. The symmetrization step, Ā = (A' + ), ensures that the resulting graph is undirected by making the pairwise relationships consistent. While Á encodes pairwise joint correlations among patches, it may contain noises and fail to capture higher-order dependencies. To obtain more robust relational structures, we further employ a VAE (Simonovsky & Komodakis 2018) module, which not only denoises the raw similarity graph but also introduces a probabilistic latent space to model uncertainty in the learned adjacency. By providing a smoothed graph representation, the VAE enables the Graph Structure Aligner module to align structures more reliably, improving stability and generalization.
A AE(A)
Finally, the aligner enforces structural alignment by minimizing the difference between and Á. Here, captures the joint correlations among the ground-truth full sequence S, while Å captures the correlations among the generated full sequence S. Formally, the loss defined as follows.
Latign | - A-1
where I|: denotes the LI distance over elements. Optimizing this loss encourages the generator to produce sequences whose patch-wise relationships are consistent with the ground-trute -tunt structures, thereby considering temporal and channel correlations.
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3.4 GRAPH REFINER
In the Graph Structure Aligner module, both A and are obtained from shared Graph VAE. This setup, however, can lead degeneration, where the Graph VAE produces identical outputs regardless of the input. To mitigate this issue, the Graph Refiner module is introduced. By using Á to refine the predictions, the Graph Refiner module forces the Graph VAE to produce meaningful adjacency matrices, as any degeneration would directly increase the prediction loss. At the same time, the Graph Refiner module integrates dependencies across both temporal steps and channels, facilitating information exchange and resulting in more accurate final predictions.
Specifically, the Graph Refiner module takes two inputs: the generated sequence patch embeddings SP, and the adjacency matrix Á produced by the Graph Structure Aligner module. Here, S serves as the nodes of the graph, and Á serves the edges. Directly utilizing the potentially dense matrix Á may introduce noises from insignificant correlations. To retain only the most significant connections, first apply a Sparsify step. We adopt top-k selection strategy, where for each node, only the edges with the k largest weights are preserved, resulting in sparse adjacency matrix A,.
This Sparsify guides the model focus the significant connections across both temporal steps and channels. With the sparse graph structures A, and the initial node features S we apply multi- layer Graph Convolutional Network (GCN) to propagate information. By stacking multiple GCN layers, each node aggregate information from its neighborhoods, enabling the Graph Refiner module to capture both local and global dependencies and refine feature representations.
H = (S))))
(9)
where H E (NN+D)xLxd shares the same shape as S Finally, the refined representations are flattened and projected through a linear layer to obtain the final output Grendo E NxF.
rendo Linear Linear(Flatten(1 ())
(10)
3.5 Loss FUNCTION
The training objective of GCGNet consists of four components: the aligner loss Lalign introduced in Equation 8 in Section 3.3 which enforces structural alignment between generated and ground- truth full sequences, guiding the optimization of the Variational Generator module. 2) the forecasting loss Ly. which directly supervises the final prediction of future endogenous variables produced by the Graph Refiner module. 3) the variational regularization loss L from the VAE in the Variational Generator module and 4) the graph regularization loss L Kr. from the Graph VAE. The KL terms regularize the latent space of their respective generators.
The forecasting loss is defined as follows.
yendo yrend
(11)
The variational regularization terms defined as follows.
DKL= = DKr.(g(Z'' (X) l) D&1 Dry. D (a(ZA) (|)
(12)
where q(-) and p(-) denote the approximate posterior and prior distributions, zV denotes the latent variable in the Variational Generator VAF, and A denotes the latent variable in the Graph VAE used within the Graph Structure Aligner module.
Finally, the overall loss function is defined as the sum of all components.
Liotal = Ly + Lalign + LK + L
(13)
4 EXPERIMENTS
Datasets We conduct experiments on 12 real-world datasets with exogenous variables to compre- hensively evaluate the performance of GCGNet. In these datasets, the future exogenous variables

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Published as a conference paper at ICLR 2026
Table I: Mcan 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.
Models Messic
OCGNet
TimeNer
TFT
TIDE
DUET
CrossLiteer
Ampliter
TImeKAN
sPasch
PaxhTTT
24
I.197
0.236
0,219
0,292
0.254
0
0.245
0,310
0.254
0.218
0.249
0.294
360
0.600
0,539
0.501
0.601
0.408
0.576
0.393
0.529
0.523
0.461
0.530
0.495
0.579
0.375
0.443
0.400
0.770
0,590
0,345
74
0.148
0.075
0 1166
1,005
0.195
0.106
01214
0.003
0.166
0.044
0.197
0.096
0.288
0115
0.244
0.077
0.168
0.16
0.299
0.133
0.319
0.281
0.131
0.20
0.149
0,279
Aug
0.186
0.108
196
0.114
0.307
0.142
0.246
0.101
0.190
0.118
0.206
00.99
0.346
0.139
0.762
0.104
0.19
0.133
0.289
24
0.323
0.227
0.392
0.253
0.426
0.272
0.426
0.255
0.432
0,272
0.363
0.251
0.471
0.339
0.451
0.319
6.41
0.256
0.452
0.326
160
0.534
0.347
0.512
0.932
[L682
0,310
0.531
0.997
0.4%6
0.554
0.406
0.666
0.462
01645
0.495
0.579
04100
0.002
0.584
Ang
0.423
0.287
0.453
0.200
0.454
0.291
0.486
0.325
0.515
0.354
0.474
0,380
0.590
0,413
0.546
0.407
0.495
0.333
0.577
0,432
24
0.332
0.179
1336
11206
11545
0.253
0.418
0.255
0.364
0.251
01405
0.228
0.459
0.340
01.54
0.245
0.434
0.235
0.51
0.365
360
0.478
0.280
0.469
0.73
0.465
0,261
0.551
0.308
0.607
0,403
0.567
0.371
0.648
0.468
0.641
0.422
0.556
0.350
0.638
0.452
Ang
11,055
0.230
0/422
0.241
1,504
0.257
0.454
0.251
0.445
0.327
0.486
0.300
0.554
0,418
0.547
0.574
0,490
0.298
0.505
0.410
DR
360
24
0,252
11,91
0.445
0.325
01610
0.191
01774
13322
1,599
0,590
0,333
0.509
0.367
0.600
0.511
0.345
0.376
0.589
0,482
0,378
0.364
0.630
0.515
0.390
0,551
0.344
0.407
0,474
0.399
0.547
0.479
0.412
0,399
0,550
11472
11394
0.43
0.599
0.495
0,412
Aug
0.387
0.387
0.475
11.18
0.439
0,445
0.499
0.447
00442
0,430
0.499
0.453
0.473
0.441
0.413
0.445
0,474
0.433
0,501
0.455
Ineegy
26
[L965
0.194
10122
12255
1000
0.785
0.101
01.48
0.117
0.061
0.995
01:44
0.138
0.306
0.135
0.706
12000
11240
0.209
0.10
360
1.177
0.330
0.204
03357
1.167
0,331
0.202
0.355
0,235
0.452
0.213
0.406
0.326
0,472
0.302
0.464
0.204
03367
0.214
0,363
Aug
1.122
0.262
00:66
0.515
0.781
0.153
0.912
0.203
0.167
0.183
0.534
0.231
0.300
0.218
0.581
0.144
11:00
0.226
0.377
Sclepth Stwpfme Saoepiral
24
0.375
0.419
0.558
0.535
0.366
0.421
0.474
0.488
0.551
0.564
0.345
0.433
0.364
0.445
0.418
0.503
0.560
0.580
0.574
0.454
160
R.458
0.498
0.845
11684
1,007
0,578
0.492
0.576
0.645
0.577
01488
0.833
0.510
0.555
1.477
0.566
0.590
01556
(1407
0,590
Aug
0.416
0.449
0.701
0.600
0.482
0.474
0.483
0.907
0.999
0.570
0.417
0.478
0.437
0,400
0.447
0.534
0.630
0.568
0.435
0,502
24
11:31
0.643
0.627
1.570
11611
0.858
0.461
0.092
0.445
0.452
0.570
0.563
0.194
0,462
0.474
0.538
0.300
11.77
0.618
0.5M2
360
0.505
0.514
0.978
0.736
0,541
0,519
0.511
0.540
0.584
0.528
0.755
0.666
0.587
0.563
0.520
0.589
0.523
0.513
0.602
0,503
Aog
1145%
0.479
0,800
0.655
0.476
0.688
0.486
0.56
0.514
0,490
01602
0.615
0,491
0.512
0.497
0.564
00311
11485
0.500
0,547
(.281
0.441
0.611
0.567
0.401
0.402
0.44
0.49
0.527
0.513
0.61
0.576
0,632
0.511
0.581
0.524
0.805
0.423
0.505
300
8.846
0.695
0.841
0.200
0.577
0,23
0.412
0.527
0.551
0,519
0.728
0.600
0.497
0.509
0.645
0.084
004472
11500
0.513
0,545
Ang
R.414
0.468
0.746
0.648
0.470
0.491
0.45
0.908
0.539
0.516
0.671
0.625
0.530
0.598
0.557
0.698
0.498
0.806
0.468
0.521
Stepth
1:426
0.474
0.300
0.677
0.474
0.091
0.519
0.55%
0.629
0:567
03565
0.586
0.471
0.540
0.514
0.64)
0.354
0.577
0.527
0.519
0.529
0.862
0.761
0.657
0.549
0.619
0.614
0.665
0.560
0.656
0.560
0.713
0.720
0.579
0.342
0.64
0.682
Aug
0.472
0.501
0.891
0.719
0.566
0.521
0.599
0.583
0.647
0,566
0.596
0.521
0.647
0.672
0.610
0.351
0.549
0.580
Collen
0.113
0.172
0.092
0.135
0.089
0.131
0.089
0:071
0.061
0.101
0.000
0.104
0.022
0.122
10
8.138
0.224
0.106
11209
0.883
0.460
0.340
0.912
0.907
0.197
01300)
0.325
0.310
0.242
0.317
0.315
0417
0.495
Ang
0.154
0.145
0.235
0.238
0.297
0.164
0.227
0.198
0.366
0.186
0.173
0.246
0.128
0.175
0.153
0.210
0.23
0.309
0.301
0.306
0,201
0.253
0.223
0.271
0.174
0.231
0.161
0.216
U363
0.215
0.327
0.367
0.416
0.409
0.414
0411
0.432
0.365
0.432
0.416
0,500
0.505
0.546
0.374
0.445
Ang
0.212
0,259
0,344
0.302
0,305
0.333
0.320
01351
0.269
0.330
01271
0.257
0.321
0.249
0.311
0,333
0.381
0.209
0.352
are available. We also consider settings where future exogenous variables are unavailable (see Sec- tion 4.2p. Further details of the datasets are provided in Appendix A.
Baselines We conduct a comprehensive comparison against 10 baseline models, which include 1)
methods that inherently support future exogenous variables, such as TimeXer (Wang et al., 2024).
TFT (Lim et al. 202T). and TiDE (Das et al. (023); and 2) methods that do not inherently support
future exogenous variables, including DUET (Qu et al. 2025d), CrossLinear (Zhou et al. 2025
Amplifier (Fei et al.] 2025). TimeKAN (Huang et al. 2025b), xPatch (Stitsyuk & Choi 2025
and PatchTST (Nie et al. 2023). For the latter group, we extend their architectures by integrating future exogenous variables through an MLP fusion module, as detailed in Appendix A.3] To ensure fairness, all models are evaluated using the same inputs, namely historical endogenous variables, historical exogenous variables, and future exogenous variables.
Setup To ensure consistency with previous studies, we adopt Mean Squared Error (MSE) and Mean Absolute Error (MAE) as evaluation metrics. We conduct both short-term and long-term prediction experiments across all datasets. For the Colbun and Raperl datasets, the short-term setting uses a lookback window of 60 with prediction horizon of 10, while the long-term setting uses a lookback window of 180 with a prediction horizon of 30. For the other datasets, the short-term setting uses a lookback window of 168 with prediction horizon of 24, and the long-term setting uses a lookback window of 720 with a prediction horizon of 360.
4.1 MAIN RESULTS
Comprehensive forecasting results are presented in Table [ to demonstrate the performance of GCGNet on the datasets with exogenous variables. Across both long-term and short-term fore-
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Table 2: Ablation studies for GCGNet. The best results are highlighted in bold.
Dataset
NP
PJM
DE
Energy
Average
Metrics
msse
mas
mae
mise
mae
mae
mue
mie
mae
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 Lutiin
0.659
0,526
0.137
0.230
0.791
0.584
0.308
0,446
0.474
0.446
(c) Replace Gruph 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
GCUNet
0.496
0.442
0.223
0.491
0.179
0.330
0.323
0.360
casting settings 12 real-world datasets, GCGNet achieves outstanding performance compared to forecasters of various architectures. Specifically, as shown in Table [ GCGNet achieves 18 first-place rankings in MSE and 20 in MAE. The performance demonstrates its clear superiority over models specifically designed for time series forecasting with exogenous variables that adopt two-step modeling strategy, such as TimeXer and TFT, as well as recent state-of-the-ard forecasting models adapted to this setting, such as DUET and Amplifier. Moreover, it can also be observed that models originally designed to incorporate both historical and future exogenous variables, such as TimeXer and TFT, exhibit relatively strong performance, highlighting the importance of the design of leveraging future exogenous variables in this setting.
4.2 MODEL ANALYSES
Ablation studies To ascertain the impact of different modules within GCGNet, we perform ablation studies focusing on the following components: (a) Replacing the VAE with an MLP in the Varia- tional Generator module. (b) Removing the Latign in the Graph Structure Aligner module, while still using 5 to obtain (S", a)). (c) Replacing the Graph VAE with Graph Learner, directly adopt- ing the adjacency matrix produced by the Graph Learner in the Graph VAE. (d) Removing the entire Graph Refiner module, that the model output is directly generated by the Variational Generator. The prediction remains under the guidance of the Graph Structure Aligner module and the loss is still computed against the ground-truth yendo We have the following observations: 1) Replacing the VAE with an MLP degrades performance, which demonstrates the necessity of variational mod- eling in capturing the uncertainty and diversity of data. 2) When the Lalign term is removed, the performance decreases because the Graph Structure Aligner module no longer provides structural guidance. This indicates that Lutign crucial for encouraging the Variational Generator module to produce coarse forecasts that are consistent with the underlying correlations. 3) Replacing the Graph VAE with a Graph Learner, which directly adopts the adjacency matrix produced by the Graph Learner, performs worse. Unlike the Graph VAE, the Graph Learner generates deterministic graphs and cannot capture the uncertainty and diversity of possible graph structures. Consequently, the learned graphs are less expressive and less robust, limiting the model's 's ability to represent com- plex temporal and channel correlations, 4) Removing the Graph Refiner module results in a signif- icant drop in performance, since the Graph Structure Aligner module is applied but its generated adjacency matrix is not used, the Graph VAE tends to degenerate into producing meaningless out- puts, which turn causes degeneration. The Graph Refiner module prevents this by incorporating Á into the prediction process, thereby foreing the VAE produce informative adjacency matrices and enabling the model to capture temporal and channel correlations.
Forecasting without Future Exogenous Variables Considering that future exogenous variables may not always be available in practice, we conduct experiments using only historical endogenous and exogenous variables to assess the robustness and applicability of GCGNet (see Table 3). When future exogenous variables yexo are unavailable, GCGNet replaces them with predictions y from the Variational Generator module, while keeping the rest of the pipeline unchanged. For all base- line models, we consistently provide only historical variables as input. Methods such as TimeXer and CrossLinear are capable of working without future exogenous variables. DUET, which models channel dependencies, also supports historical endogenous and exogenous inputs. In contrast, the Meannel-independe model PStchTST uses only historical endogenous variables as input. The ex- perimental results show that GCGNet continues to perform excellently. Methods that use historical exogenous variables, such TimeXer and Crossi also perform well. DUET, which flexibly
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Table 3: Average results under the setting where future exogenous variables are not available. The inputs are (Xeedo and X The best results are Red, and the second-best restuls are Blue. Full results are provided in Table Of Appendix B.2
Models
GCGNe:
TaneNer
TIDE
DUET
CrosLinear
Amplitee:
TimeKAN
sPurh
PaxhTST
Motric
1030
maa
NP
0.425
0,377
0.440
0.583
0.677
0.488
0.467
6416
0,444
0.583
0.451
0.394
0.530
0.448
0.454
0.43
0448
0.402
0.47
0,401
PIM
0.135
0.317
0.141
0.229
0,200
0,210
0.155
0.257
0.140
0.226
0.347
0:241
0.152
0.345
0.175
0.270
0.138
0.333
0.448
0,243
BB
0.458
0,301
0,477
0.301
0.503
0.354
0.547
0,348
0.433
0.305
0.477
0.300
0.502
0.325
0,458
0,315
0.483
0,302
0.455
0,309
FR
B:428
0.245
0.454
01244
0.555
0.268
[1004
0,300
[ 0:468
0.262
0.476
0.357
(1:04
0.386
0.491
0.276
0.477
0.257
0.470
0.764
DE
B661
0,497
0.659
0.507
0.584
0.515
0.644
0.319
0.660
0.513
0.635
0.508
0.712
0.545
0.650
0.526
0.633
0.500
0.696
0,527
Energy
0.159
0.209
0.172
0.320
10.37
0.480
11162
0.311
0.100
0.308
0.201
0.336
0.180
0.327
0.147
0.296
0.370
0.314
[ 01203
0,341
Sevpttl
0,200
0,602
0.201
00,009
009964
0.728
0.713
0.633
[
0.724
0.585
0,309
0.694
(1.43
0.085
0.725
0.672
0.737
0.590
0.733
0.061
Stupfne
08815
0.631
0,300
0.65%
1.04
0.767
0.816
0.683
0.330
0.641
0.500
0.662
[1943
0.201
0,890
0.882
0.844
0.714
S4xpthi
0.237
0.643
0.793
0.698
0.808
0.699
0.750
0.637
0.763
0.702
1.100
0.720
0.791
0.630
0.504
0.717
Scwpth2
0.719
10900
0.746
01919
0.751
01200
0.950
0.7774
0.94
0/87
1.005
0,812
Colbun
0.166
0.364
0.132
0.219
10550
0.356
0.188
0.240
0:147
0.19%
(0.19
0.195
0.152
0.310
0.201
0.255
0,238
0.235
0.221
Rapel
0.308
0.515
0/167
0.904
0.200
0.164
0:352
0.194
1 Couss
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 I0 of Appendix B.4
Mask Tppe
Zom
Random
Modals
GCGNes
TimeNer
CrosLines
DUET
Puch757
GCGNNt
TineXer
CrosLincu
DUET
PackTST
Metrics
man
meo
mon
meo
1998
Colbun
0.076
0.093
0.084
0.145
0.065
0.102
0.106
0.126
0,070
0.108
0,070
0.086
0.112
0.169
0,866 0.108
0.084
0.118
0.078
0.120
Enengy
0.000
0.263
0.122
0.273
0.007
0.341
0.000
0.246
0.008
0.081
0.217
0.123
0.275
0.098
0.242
0.102
0.349
DE
0,292
0.336
0.418
0.405
0.426
0.466
0/425
0.907
0,400
0.445
0.344
0.366
0,433
0.411
0,433
0,415
0,431
0.407
0,490
0,429
PFM
6.64
0.158
0.100
0.198
0.008
0.208
0.079
0.137
|
|0.100
0.246
0.066
IL15R
0.102
0.199
0.101
0.201
0.079
0.176
0.112
0.224
NP
A.314
0.216
0.289
0.206
0.426
0:416
| 0.388
0.230
0.266
0.26%
0.318
IL236
0.793
0.297
0.248
01285
0.760
0.270
0.269
0.291
models channel relationships and can leverage historical exogenous information, also performs well. In contrast, PStchTST performs poorly because it cannot exploit historical exogenous variables.
Robustness under Missing Values In complex real-world scenarios, time series data often contain noises due to uncontrollable factors. To further evaluate the generalizability of GCGNet under such conditions, we simulate missing values in exogenous variables by randomly masking the original time series. Specifically, we adopt two masking strategies to assess GCGNet's adaptability: (1) Zeros: masked values are replaced with O; and (2) Random: masked values are replaced with random values sampled from the normal distribution (00 1). We conduct experiments under three masking ratios: 10%, 30%, and 50%. Table Table4reeoo reports the average results under three masking ratios, while the full results are provided in Table of Appendix [B.4] The experimental results show that GCGNet continues to perform excellently, demonstrating that the design of the generative network in GCGNet not only ensures accurate forecasting but also provides robustness in real-world scenarios where the data may contain noises.
Effect of the Generative Network To as- sess the contribution of the generative net- work in GCGNet, we compare the origi- nal model with variant in which all VAE modules are replaced by MLPs (denoted as w/o VAE in the table). Experiments follow the settings in Section 4.2 Ta- ble 5 reports average results across mul- tiple datasets under partially missing ex- ogenous variables. Across all datasets,
Table 5: Average results of GCGNet and the version without VAE under the setting where exogenous vari- ables are partial missing. The best results are high- lighted in bold.
Musk Type
Zero
Random
Models
GOGNN
wlo 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
0.295
0.081
0.217
0.142
0.291
GCGNet performs better than the ver- sion without the VAE. This improvement shows that the generative network helps
DE
0.292
0.336
0.453
0.430
0,344
0.366
0.505
0.460
PIM
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
the model handle missing or noisy inputs, recover useful structure from incomplete data, and provide more stable forecasts. These results also

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Published as a conference paper at ICLR 2026
suggest that the VAE captures hidden correlations that simple architectures cannot, which leads to better overall forecasting accuracy.
The Advantages of Joint Modeling To validate the motivation of GCGNet, visualize its predic- tions on the NP dataset, where the target variable is electricity price (Figure (4) and the exogenous variables are wind power (Figure 4h) and grid load (Figure 4p). PatchTST and CrossLinear serve as comparison models. As shown in Figure [11, as the historical endogenous data exhibit clear periodic patterns, PatchTST, unable consider historical exogenous variables, mainly replicates historical endogenous data patterns. Meanwhile, as shown in Figure 4. CrossLinear, which models channel correlations first and then temporal correlations, causes the two types of correlations to interfere with each other, resulting in neither type of information being well captured. In contrast, GCGNet gen- erates predictions (Figure 4)) that closely match the ground truth. This demonstrates that GCGNet effectively uses both endogenous and historical exogenous variables by jointly modeling temporal and channel correlations. When future exogenous variables are included (Figure ) GCGNet's per- formance improves further. While in Figure Fg an MLP fusion module (Appendix (A.3) is applied to PStchTST, enabling it to incorporate future exogenous variables in a two-step modeling strategy. The first step models temporal correlations, and the second step models channel correlations. We observe that this approach provides only limited improvement, as its predictions closely follow the shape of the future grid load, showing that correlations learned in the second step can override those from the first step. A similar hehavior can also be observed for CrossL.need Figure 4h. Additional cases are provided in Appendix [.I] These results demonstrate the effectiveness of GCGNet's joint modeling strategy in forecasting with exogenous variables.
Exogenous Variables
Future Exogenous Unavailable
Future Exogenous Available
Ground Truth
3530
3530
11:3
Ground Truth Prediction
25.0
27.5
Ground Trush
Prediction
:20
122
(a) Wind Power
(d) PatchTST
(g) PatchTST
Ground
Ground
Ground Truth
(b) Grid Load
(c) CrossLinear
(h) CrossLinear
Ground Truth
Ground Truth
Prodicionn
Prediction
65
[c) Electricity
GCGNet
(1) GCGNet
Figure 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 pre- dictions when future exogenous variables available.
CONCLUSION
In this paper, we introduce GCGNet, graph-consistent generative network designed for time series forecasting with exogenous variables. To summarize, our method first generates coarse predictions, then enforces structural alignment to capture joint temporal and channel correlations robustly, and finally refines the predictions to prevent degeneration and improve accuracy. These designs col- lectively enable GCGNet to capture complex joint temporal and channel correlations while main- taining robustness in real-world scenarios. Extensive experiments diverse real-world datasets demonstrate that GCGNet achieves ofate-of-the-ar forecasting performance.
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ACKNOWLEDGEMENTS
This work was partially supported by the National Natural Science Foundation of China (No. 62372179, No. 62472174) and the Fundamental Research Funds for the Central Universities. Bin Yang is the corresponding author of the work.
ETHICS STATEMENT
This study relies solely on publicly available datasets that contain no personally identifiable formation. The proposed framework is designed for socially beneficial applications in time series forecasting. No human subjects were involved in this research.
REPRODUCIBILITY STATEMENT
The promise that all experimental results can be reproduced. We have released our model code in an anonymous repository: http://ggghl..ccomccccooooolllll
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A EXPERIMENTAL DETAILS
A.1 DATASETS
Table 6: Statistics of datasets. Ex. and En. are abbreviations for the Exogenous variables and Endogenous variables, respectively.
Datuset
#Num
Sampling Frequency
Lengths
Split
Ct. Descripions
Dn. Descriptions
2
Hour
52,416
Load, Power
Nord Pool Electriity Price
PIM
2
Hour
52,416
7:1:2
System Load, SyZonal COMED load
Penncylivania New Jonsey MarylandElettriiity Price
BE
2
Hour
52,416
7:1:2
Genenation System Load
Belgium's Flectricty Prior
2
Hour
52,416
Genentios, System
France's Electiiity Price
2
Hour
52,416
7:1:2
Wind power. Ampirion zonal load
German's Elecricity Price
Energy
5
I Hour
13,064
7:1:2
Banery, Geothermal, Hydroelecricc, Solar, Wind
Thermeleenic
Coibinn
2
Day
Precipitation, Tributary Inflow
Water Level
Rapel
2
3,366
Precipitation, Tributary Intlow
Water Level
Sawpth
6
Houu
14,641
7:1:2
Climate Fearure
Active Power
8dwpfm
Minutes
20,281
Climate Feature
Active Power
We conduct experiments on 12 real-world datasets with exogenous variables, where future exoge- nous variables are either approximately known or can be obtained with high accuracy. These include 5 EPF (Lago al. [02T] Wang et al., (2024) datasets (NP, PJM, BE, FR, DE) and datasets collected (225e) (Energy. Colbún, Rapel, Sdwpfhl, Sdwpfh2, Sdwpfm1, Sdwpfm2). NP records Nord Pool market with hourly electricity price as the endogenous variable and grid load and wind power forecast as exogenous features. PJM records the Pennsylvania- -ewia-Newwa Jersey-Maryland Interconnection market with zonal electricity price in the COMED area as the endogenous variable, and system load and COMED load forecasts as exogenous variables. BE records Belgium's 's electric- ity market with hourly price as the endogenous variable and load forecast in Belgium plus generation forecast from France as exogenous variables. FR records the French electricity market with price as the endogenous variable and generation and load forecasts as exogenous variables. DE records the German electricity market with hourly price as the endogenous variable and zonal load forecast in Amprion, along with wind and solar generation forecasts as exogenous variables. Energy pro- vides hourly power generation data from Chile, with thermoelectric generation as the endogenous variable and battery storage, wind, hydro, and solar as exogenous variables. Colbún and Rapel are daily-resolution hydropower datasets with water level as the endogenous variable and precipitation and tributary inflow as exogenous variables. Sdwpfh1, Sdwp/h2, Sdwpfm1, Sdwpfm2 are wind power datasets from Longyuan wind farm, with active power output (Patv) as the endogenous vari- able and external weather data from ERAS (Hersbach et al. 2020) as exogenous variables, including temperature, surface pressure, relative humidity, wind speed, wind direction, and total precipitation. It is also worth noting that modern deep learning research typically evaluates models diverse collection of datasets to demonstrate generalization across different application domains (Ma et al. 2024a b) 2025e Yang al.. 2023 Wu et al., 2024). This evaluation paradigm is widely adopted not only in natural language processing and computer vision (Ma et al., 2025d c b). but also in time series analysis under various problem settings, including forecasting (Qiu et al. [025c (Shang et al.] 2024 Shang & Chen 2024 Wang et al. 2026b|c 2025c b). anomaly detection (Wu et al.) 2025c), imputation (Wang et [2025a), modeling irregular (Liu et al., 2026b Yu et al. (0256)) ferg-term modeling (Liu et al.) 2026b] Yu et al. (025b), and general modeling frameworks (Wu jet al. 2026, Shang et al., 2006)). Following this evaluation paradigm, the datasets considered this work cover multiple
by DAG (Qiu et al.
domains with heterogeneous data characteristics, thereby enabling comprehensive evaluation of model performance under diverse conditions.
A.2 IMPLEMENTATION DETAILS
The "Drop Last" issue is reported by several researchers (Qiu et al.) 2024, 2025b Li et al. 2025). That is, in some previous works evaluating the model on the test set with drop-last=Trus setting may cause additional errors related to test batch size. In our experiment, to ensure fair comparison, we set drop-last to False for all baselines to avoid this issue. All experiments are conducted using PyTorch (Paszke et al. (211)) in Python 3.8 and execute
NVIDIA Tesla-A800 GPU. We do not use the "Drop Last" operation during testing. To ensure reproducibility and facilitate experimenta- tion, datasets and code are available at: http://siiihlkccoomeecsmmmel///c
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Truth
25000
20000
0930
Ground
Ground
Preciction
23
80
100
60
100
120
100
325
(a) Wind Power (Exogenous)
(b) Grid Load (Exogenous)
(c) PatchTST without Exogenous
27.5
25.0
36-
32.5
20.0
27.5
Pradicaio
Ground Truth
Prediction
Ground Truth
28.
Ground Thuth
Prediction
26
(d) PatchTST with Future Exogenous
(e) GCGNet with Historical Exogenous
(F) GCGNet with All Exogenous
24
Drmund Trush
Prodiction
28
Ground Truth
Prediction
Ground Truth
Prediction
20
172
120
(g) TimeXer with Historical Exogenous
(h) TimeXer with All Exogenous
(1) DUET Exogenous
32-
Ground Truth
Predictin
301
100
125
80
60
302
120
DUET All Exogenous
CrossLinear with Historical Exogenous
CrossLinear with Exogenous
Figure 5: Visualization of prediction results NP dataset.
A.3 MLP FUSION MODULE
We employ an MLP fusion module, which efficient (Huang et al. 2025a). enable traditional forecasting methods to leverage future exogenous variables for fairer comparison. Given histor- ical endogenous variables e E N and historical exogenous variables Xexo PxT the backbone model first generates a latent representation:
Moade (Xemdo Xxex)
(14)
where 2 E R NXF. MMddel denotes the parameters the forecasting model. Subsequently, the latent representation is concatenated with the future exogenous variables Yew DxF and passed through an ML. to produce the final prediction:
endo MLL 0LL (Concatt,,,, you)),
(15)
where fML are the parameters of the fusion module.
B EXPERIMENTAL RESULTS
B. VISUALIZATION RESULTS
We visualize more cases of predictions on the NP dataset in Figure 5 where the target variable is electricity price and the exogenous variables are wind power (Figure 5 and grid load (Figure 5p)). As shown in Figure 5 GCGNet consistently outperforms all baselines, both when future exogenous variables are available and when they are not, indicating that it effectively captures joint temporal and channel correlations. As shown in Figure(5.. PStchTST, which models channels independently, cannot incorporate exogenous variables and therefore mainly reproduces the patterns of historical endogenous variables. As shown in Figure 5d, when MI.P fusion module (Appendix A.3) is
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added to PatchTST, it can include future exogenous variables through two-step modeling strategy- However, this only brings limited improvement, as its predictions largely follow the shape of the future grid load. showing that the modeling in the two steps interferes with each other. As shown in Figune and Figure in the setting with only historical exogenous variables, TimeXer and DUET leverage historical exogenous variables and thus go beyond simply repeating past endogenous patterns, In contrast, Figure 5k shows that CrossLinear first models channel dependencies and then temporal correlations. Since its first step uses a saanseegrained ID convolution for channel modeling, it may distort the periodic information of historical endogenous variables, leading to predictions that deviate from the true patterns. As shown Figure ph, Figure 5J. and Figure [5, when future exogenous variables are introduced, DUET, CrossLinear, and TimeXer all achieve improvements, but these gains remain limited, further confirming the effectiveness of GCGNet.
B.2 FORECASTING WITHOUT FUTURE EXOGENOUS VARIABLES
Considering that future exogenous variables may not always be available in practice, conduct experiments using only historical endogenous and exogenous variables to assess the robustness and applicability of GCGNet. The results in Table[]jiddical that GCGNet maintains strong performance even without future exogenous variables, highlighting its robustness and generalizability.
B.3 CONTROLLED HYPERPARAMETIE EXPERIMENT
We also conduct all experiments under controlled hyperparameter settings, where each model uses fixed configuration across forecasting horizons. The results under the setting where future ogenous variables are available are reported in Table 88 while the results without future exogenous variables are presented in Table 9 It be observed that GCGNet achieves the best performance across both settings.
B.4 FORECASTING WITH MISSING EXOGENOUS VARIABLES
In complex real-world scenarios, time series data are often incomplete due to uncontrollable factors such sensor failures, making robust forecasting models essential. To further evaluate the general- izability of GCGNet under such conditions, we simulate missing values in exogenous variables by randomly masking the original time series. Specifically, we adopt two masking strategies assess GCGNet's adaptability: (1) Zeros: masked values are replaced with O; and (2) Random: masked values are replaced with random values sampled from the normal distribution (/ 1). We conduct experiments under three masking ratios: 10%, 30%, and 50%. We select TimeXer, Crossl Linear, DUET, and PatchTST as baselines for comparison. Specifically, TimeXer and CrossLinear are rep- resentative methods designed to model interactions between endogenous and exogenous variables, DUET is a channel-dependent model, and PatchTST is a heaanel-inndpentee mental results are reported in Table 10] where it can be observed that GCGNet consistently achieves the best performance across different masking strategies and ratios.
model. The experi-
B.5 PARAMETER SENSITIVITY
We also study the parameter sensitivity of the GCGNet, where all datasets use a lookback length of 720 and forecasting horizon of 360 steps. We make the following observations: 1) Figure 6a
Energy
(53
Esergy
MAE
MAE
DE
MP
Onergy
PJM
Patch dimension
VAE latent dimension
GCN layers
Sparsity ratio
(a) Patch dimension
(b) VAE latent dimension
(c) GCN layers
(d) Sparsity ratio
Figure 6: Parameter sensitivity studies of main hyper-parametest in GCGNet.
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Table 7: Results under the setting where future exogenous variables are not available. The inputs are (X and o The best results are Red, and the recond-best results are Blue. Avg represents the average results across the two forecasting horizons.
Model Merrios
GCONet
TimeXer
TFT
TIDE
CrossLitecr
Ampliter
TImcKAN
xPasch
PochTST
ms
11130
1500
2150
ma:
1930
1760
1030
mee
24
0.234
0.259
112770
0.292
0:330
0.531
0.2M)
0.300
11262
0:371
0.282
0 0.002
0.321
0.335
0.364
0.260
0.377
0.285
0.795
300
3.617
0,495
11089
0.474
0.644
0.655
0.038
0.523
0.026
0.044
0.655
0.505
0.738
0.577
0.532
0.527
0.717
0.524
0.030
0,505
A0g
0.425
0.377
1.441
0.50
0.677
0.458
0.467
0,416
0.444
0.383
0.451
0.394
0.520
0.448
0.484
0,435
0.438
0,402
0457
0,403
24
0.073
0.308
0,096
0.191
0.129
0.231
0.105
0.198
0.083
0.195
0.130
0.236
0.085
0.178
0.106
0,212
300
14.193
0.206
11386
0.207
0.270
0.310
0.310
0.292
0.198
0.373
0.197
0.285
0.210
01294
0.219
0.304
0.193
0.287
1.189
0.273
Avg
0.135
0.317
1.141
0.229
0.2100
0.270
0.155
0.255
11140
0.226
01147
0.241
0.152
0.245
0.175
0.270
0.139
0.321
0.148
0.243
24
0.251
0.392
0.253
0.400
0.267
0.492
0.322
0.376
0.253
0.417
0.255
0.424
0.380
0.404
0.258
0.405
0.264
300
10.531
0.351
1.565
0.349
0.727
0.442
0,002
0,375
0.571
0.357
0.564
0.346
0.588
0.367
0.552
0.350
0.562
0.268
0.564
0.354
A0g
0.458
0.301
0.477
0.301
0.563
0.354
0.547
0.348
0.573
0.305
0,477
0.300
0.502
0.325
0.485
0.315
0.433
0.302
0.435
0,309
24
0.195
0,445
0.231
0.415
0.254
0.359
0.204
0.250
0.445
0.250
0.402
0.204
0,997
0,223
360
0.519
0.296
0.542
0.300
0.634
0.345
0.514
0.325
0.577
0.317
0.555
0.310
0.559
0.322
0.534
0.303
0.553
0.306
0.542
0.306
Aug
0.426
0.245
0.454
0.247
0.535
0.258
0.494
0.200
0.468
0.362
0.476
0.257
0.494
0.286
0.491
0.276
0.477
0.257
0.470
0.264
0.901
0.445
0.576
0.422
0.465
0.423
0.493
0.455
0.504
0.462
0:455
0.425
0.908
0.450
360
0.508
0.594
0.818
0.568
0,792
0.578
0.523
0.604
0.895
0.623
0.832
0.992
0.930
0.642
0.834
0.500
0.810
0.575
0.899
0.504
Asg
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
0.096
0.527
0116
0,201
0.350
0.113
0.766
0.246
0,100
0.254
0.109
0.754
0.107
0.389
0.754
360
0.107
0.352
0.206
0.360
0.399
0.407
0.207
0.358
0.211
0.362
0.200
0.425
0.251
01400
0.186
0.388
0.232
0.377
0.008
0.428
Aug
0.150
0.299
0.172
0.325
0.376
0.450
[0.163
0.311
0.160
0.308
0.201
0.536
0.180
0.327
0.147
0.296
0.110
0.314
0.205
0,341
Sdepimm
0.55%
0,533
0.596
0.531
0.553
0.579
0.557
0.508
0.579
0.564
0.500
0.557
0.587
300
0.004
0,712
01845
0.684
1.372
01935
0.904
0,728
11898
0,672
11770
0.834
1.105
03814
0.801
0.764
0910
01772
0.10
0,745
Aug
0,700
00/00
0.201
0.009
0.84
0.728
0.713
0.033
11724
0,581
11000
0.094
0.543
01885
0.725
0,672
0.737
0.110
0/733
0.661
Scoepim22
0.004
0.568
0.677
0,570
07701
0.057
0.605
02552
0.012
0.586
0,009
1.600
0.004
0.21
0.411
0.539
0.608
0,610
360
1,006
0.717
0.918
0.736
1.340
0.877
0.905
0.763
1.028
0.945
0.968
0.788
1,215
0.868
0.067
0.780
1.058
0.736
1.000
0.819
Aug
0.825
0.631
0.805
0.683
1.044
0.767
0.816
0.680
0.800
0.641
0.200
0.687
0.942
0.732
0.836
0.701
00850
0.622
0814
0.714
Slepth
24
3.668
0,583
0.651
0.547
0.754
0.670
0.719
0.641
0.669
0.687
0.652
1,311
0.816
0.755
0.679
0.701
0.579
0.609
0.629
360
0.846
0.674
11811
0.700
0,331
0.727
0.80
0,757
1,890
0,702
03349
0.753
1.089
01823
0.883
0,702
0.82
01402
0.09
0.804
Ang
0.757
0.425
0.746
0.643
0,793
0.698
0,805
0.699
0.760
0.702
01820
0.804
0.720
0.791
0.600
0.004
0.717
Stepth
0.797
0.644
03821
0.677
0:811
0.714
9.781
0,674
11905
0.718
0.834
1.691
0.812
0.740
0.229
0.622
1.769
0,080
360
D.968
0.739
0.962
0.761
1.042
0.779
1,055
0.829
1.036
0.754
1.102
0.830
1.126
0.845
0.990
0.825
1.053
0,742
1.280
0.925
Asg
0,883
0.691
0.891
0.719
0.926
0.746
0.919
0.774
0.980
0.758
0.941
0.782
0.941
0.687
1.025
0.802
Caltes
10
0.064
0,081
0.084
0.146
0.449
0.279
0.005
0.140
0.073
0.068
0.101
0.077
0.120
0.091
0.129
0.071
0.092
0.083
0.125
30
0.165
0,246
1.151
0.293
0.603
0.894
0,282
0,341
0.221
0.290
0.190
0.289
0.227
01301
0,313
0,377
0.405
0.379
0217
0,316
Aug
0.126
0.164
0.132
0.219
0.356
0.386
0,447
0.195
0.152
0.210
0.201
0.253
0.238
0.235
0.150
0,221
Hopel
10
0.191
0.204
0.301
0.30%
0.238
0.214
0.238
0.270
0.221
0.201
0.234
0:240
0.259
0.226
0.238
0.421
0.421
0215
0.322
30
0.525
0.343
0.387
0.416
0.377
0.394
0.409
0.444
0.387
0.396
0.278
0.574
0.433
0.436
0.458
0.437
0.307
0.343
0436
0.436
Asg
0.258
0,273
0.344
0.362
0.308
0.334
0.323
0.304
0.357
0.348
0,342
0.337
0.364
0.382
0.325
0.334
1" Coan
2
2
0
3 2
0
2
0
and Figure [b) illustrate the performance of GCGNet under different settings of the patch dimension and the VAE latent dimension. The results indicate that larger dimensions do not always improve forecasting accuracy because excessively high dimensions can introduce redundancy, increase over- fitting risk, and make optimization more difficult. Instead, dimensions in the range of 64 to 256 generally achieve more favorable performance. 2) Figure 6c| shows that shallower GCN architec- ture with one or two layers performs better. This suggests that the node relationships in the task are relatively direct, and deeper networks may suffer from the over smoothing problem, thereby degrad- ing the forecasting results. 3) Figure (dd shows that the model achieves better performance when the sparsity ratio set to 50%. This indicates that a 50% sparsity strikes a good balance, effectively removing most of the uninformative weights while retaining the critical connections necessary for accurate forecasting.
/hows
B.6 INCREASING LOOK-BACK LENGTH
In general, increasing the look-back window provides the model with more historical information, which can improve forecasting accuracy. However, a longer look-back window also introduces more noises, particularly in forecasting with exogenous variables, which introduces additional noises. To assess the effectiveness of GCGNet, we configure different look-back windows on the DE, PJM, Energy, and NP datasets. We also visualize prediction results for look-back windows of 96, 168, 336, 512, and 720, with forecasting horizon of 24. As shown in Figure 7 GCGNet consistently
20

---

Published as a conference paper at ICLR 2026
Table 8: Results under the setting where future exogenous variables are available and each model uses fixed configuration across forecasting horizons. The inputs are (Xrend, yendo and X The best results are Red, and the second- restuls are Blue. Avg represents the average results across the two forecasting horizons.
Madkh Metrics
OCGNci
TineXer
TFT
TIDE:
DUET
Crcoelinccr
Ampli
TimcKAN
sPaich
PadcTST
mmo
24
0.236
0.266
0,219
0.349
0.264
0.301
0.246
0.210
0.266
0,300
0.234
0.278
0,249
0,294
360
0.459
0,600
0.475
0.539
0.501
0.601
0.498
0.576
0.528
0.508
0.587
0.529
0.523
0.461
0.500
0.495
Ang
1.370
0.348
0.418
0.391
0.599
0.315
0.443
0.400
0.411
0.408
0.911
0.887
0.430
0.418
0.40
0.499
0,378
0.1
0.90
0.195
PMM
360
24
E.060
0.129
0.150
0.223
0.140
0.075
0.166
0.231
0.095
0.133
0.195
0.319
0.106
0.177
0.214
0.279
0.072
0.131
0.228
0.166
0.125
0.058
0.191
0.254
0,096
0.177
0.208
0.285
0.115
0.162
0.244
0.281
0.077
0.131
0.168
0.23
0.16
0.149
0.239
0.279
Aug
0.187
0.108
01 196
0.114
0.307
0.142
0.246
0.101
0.190
0.112
0.205
0.399
0.346
0.139
0.762
0.104
0.998
0.133
0.289
24
0.392
0.255
0426
0,372
0.426
0.255
0.432
0.259
0.319
0.256
0442
0.325
0.340
0.512
0.327
0.883
0.310
0.364
0.517
0.436
0.416
0.457
0.495
0.579
0.410
0.702
0.538
Ang
[1431
0.294
0.482
1.220
0.654
0.001
0.408
0.915
0,515
0.354
0.419
0.397
0.599
0.413
0.548
0.407
0.495
0.318
0.999
0.493
24
0.347
0.185
0,266
0.206
0.543
0.253
0.418
0.255
0.384
0.251
0.350
0.226
0.459
0.348
0.454
0.296
0.424
0.235
0.518
0.365
360
0.452
0.279
0.489
0.233
0.065
0,261
0.551
0.308
0.507
0,413
0.575
0.370
0.648
0.400
0.641
0.452
0.556
0.350
0.655
0.452
Aug
R.418
0.234
0.422
0.341
0.504
0.257
0.484
(1281
0.495
0.322
0.488
0.206
0.554
0.408
0.547
0.534
0.490
0.298
0.588
0,410
24
0,580
0,383
0.367
0.385
0.375
0.399
0.394
0,413
0,412
0.447
0.510
0.474
0,599
0.509
0.630
0.511
0.589
0.462
0..63
0.551
0,474
0.547
0.479
0.500
0.472
0.599
0.495
Ang
[.401
0.309
0.475
0.418
0.489
0.446
0:409
0.447
0.483
0.490
0.485
0.432
0.473
0.441
0.453
0.445
0.474
0.438
0.901
0.455
Inergy
24
0.051
0.221
0.122
0.273
0.093
0.235
0.103
0.248
0.117
0.263
0.241
0.418
0.138
0.306
0.125
0.295
0,092
0.240
0.239
0,300
360
0.182
0.334
0.204
0.357
0.167
0.331
0.202
0.355
0.288
0.452
0.237
0.585
0.328
0.472
0.302
0.464
0.204
0.367
0.114
0,363
Aug
0.131
0.277
0.163
0.315
0.130
0.381
0.153
0.300
0.203
0.367
0.229
0.402
0.233
0.389
0.218
0.581
0.148
0.303
0.226
0,377
Scheprin) Scrppimi
24
0.558
0.533
0,366
0.444
0.488
0.355
0,503
0,374
0.454
360
0.499
0.845
0.684
0.997
0.528
0.492
0.526
0.645
0.577
0.497
0.566
0.592
0.556
0.407
0.550
Any
R.424
0.467
0.701
0.600
0.482
0.474
0.433
0.907
0.599
0.570
0.426
0.902
0.437
0.400
0.447
0.534
0.630
0.568
0.435
0.502
24
B.421
0.441
0.627
0.570
0.411
0.458
0.461
0.892
0.445
0.452
0.477
0.536
0.394
0,462
0.474
0.538
0,308
0.41%
0.418
0.492
360
0.529
0.531
0.978
0.736
0.541
0.519
0.511
0.540
0.584
0.528
0.559
0.611
0.587
0,563
0.520
0.589
0.523
0.513
0.602
0.503
Aug
B.475
0.486
0.303
0.653
0.476
0.488
0.466
0.566
0.514
0.400
0.533
0.575
0.491
0.512
0.497
0.564
0.516
0.495
0,500
0,547
Schepth Schephh
0.511
0.587
0.001
0.400
0.444
0.449
0.527
0.548
0,524
0,505
0.422
0.505
360
0.465
0.514
0.841
0.700
0.537
0.523
0.422
0.527
0.551
0.319
0.366
0.560
0.643
0.694
0.472
0.504
0,513
0.548
Any
0.450
0.502
0.746
0.643
0.470
0.491
0.453
0.508
0.539
0.316
0.357
0.903
0.537
0.577
0.638
0.408
0.305
0.468
0.527
24
B.473
0.505
0,230
0.677
0.674
0.493
0.579
0.555
0.029
0,563
0.868
01540
0,873
0.533
0.550
0.614
0.640
0.354
0.457
0.327
360
0.566
0.565
0.962
0.761
0.657
0.549
0.619
0.614
0.665
0,569
0.608
0.609
0.569
0.628
0.713
0.729
0.579
0.547
0.641
0.632
Ang
0.520
0.536
0.891
0.719
0.566
0.521
0.599
0.585
0.647
0.566
0.538
0.574
0.531
0.581
0.647
0.672
0.610
0.351
0,549
0,580
10
0.105
0.113
0.172
0.135
0.099
0.189
0.134
0.104
0062
0.122
0.243
0.176
0.299
0,383
0.460
0.240
0.322
0.307
0.397
0.182
0.249
0.217
0.315
0.417
0.496
Ang
0.175
0.145
0.235
0.238
0.297
0.164
0.227
0.198
0.366
0.136
0.195
0.173
0.246
0.128
0.175
0.153
0.210
0.239
0.309
Ropel
10
0.211
0.230
0:301
01300
1.201
0.253
0.228
01277
0.174
0,219
0.163
0.309
0.181
0.227
0.174
0.231
0.161
11216
0.133
0.215
30
0.401
0.384
0.367
0.416
0.409
0.414
0.411
0.432
0.365
0,432
0.340
0.417
0.333
0.416
0.325
0,390
0,505
0.546
0.374
0,445
Ang
0.306
0,307
0.344
0.362
0,305
0,333
0.320
0.351
0.269
0.326
0.253
0,313
0.257
0.321
0.349
0.311
0.333
0.381
0.269
0.332
1" Donn
0.45
S00kee
BIFT
0.20
0.432
055
Patce137
Timster
3.33
0.300
Peic1157
Timster
4055
Towder
0375
124
8.50
8:55
104
236
16
166
1600
194
204
700
(a) DE
(b) PJM
(c) Energy
(d) NP
Figure 7: Forecasting performance under varying look-back windows across four datasets (DE, PJM, Energy, and NP).
outperforms the baselines across all four datasets. While other models exhibit large fluctuations in prediction performance as the look-back window increases, GCGNet maintains relatively stable metrics and even shows improvements as the bock-back window increases, indicating its ability to model longer sequences and its robustness to the additional noises introduced by extended look-back windows.
21

---

Published as a conference paper at ICLR 2026
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 (Xondo and X The best results are Red, and the second-best restuls are Blue. Avg represents the average results across the two forecasting horizons.
Mudlh Metrics
OCGNc1
TineXer
TFT
TIDE:
DCET
Cicoelincce
Ampli
TimcKAN
sPaich
PadcTTT
meo
24
0.260
0.270
0.292
0:330
0.321
0.297
0.309
0.262
0.271
0.346
0.292
0.336
0.344
0.260
0.217
0,293
0,296
360
0.502
0.609
0.474
0.964
0.655
0.638
0.523
0.525
0.484
0.655
0.505
0.574
0.527
0.717
0.528
0.630
0.505
Ang
[1:30
0.081
0.442
1.363
0.647
0.488
0:467
0.416
0.444
0.181
0.451
0.944
0.500
0.448
0.444
0.435
0.488
01400
0.457
0.401
PMM
360
24
I.072
0.191
0.164
0.269
0.096
0.196
0.191
0.267
0.129
0.279
0.231
0.310
0.105
0.210
0.292
0.213
0.082
0.195
0.178
0,273
0.197
0.006
0.283
0.196
0,093
0.210
0.195
0.294
0.130
0.219
0.236
0.304
0.085
0.193
0.178
0.267
0.106
0.180
0.212
0,273
Aug
[.131
0.316
0.141
0.239
0,200
0.770
0.158
0.255
0.140
0.326
0.147
0.241
0.153
0.245
0.135
0.220
0.1.0
0.215
0.148
0.363
24
0.259
0.392
0.253
0,400
0.267
0.492
0.522
0.376
0.359
0.280
0.404
0.258
0.405
0,264
0.354
0.503
0.349
0.727
0.442
0.602
0.375
0.571
0.357
0.564
0.346
0.588
0.367
0.350
0.502
0.316
9.564
0.354
Ang
[1459
0.106
00473
0.301
0.563
0.354
0.547
0.346
0.47%
0.105
0.417
0.300
0.507
0.328
0.458
0.35
0.483
0.900
0.485
0.309
24
0.352
0.199
0.266
0.195
D.445
0.231
0.415
0.254
0.359
0.208
0.397
0.204
0.429
0.250
0.448
0.250
0.402
0.208
0.297
0.223
360
0.519
0.293
0.542
1.300
11024
0.345
0.574
0.325
0.577
0,317
0.555
0,310
0.559
0.322
0.534
0.503
0.553
0.300
0,542
0,305
Aug
I1438
0.246
0.454
0.347
0.535
0.788
0.404
0.200
0.468
0.767
0.476
0.257
0.494
0.286
0.491
0.276
0.477
0.257
0.470
0.364
24
0.501
0.445
0.576
0.452
0.465
0.435
0.424
0.438
0.504
0.462
0.455
0.425
0,503
0.450
0.816
0.000
0,818
0.568
0.793
0.578
0.823
0.004
0.855
0.623
0.832
0,592
0.642
0.834
0,590
0.910
0.575
0.89
0.504
Ang
RU614
0.501
0.699
0,800
0.684
0.515
0.644
0.519
0.660
0.513
0.635
0.508
0.712
0.548
0.669
0.536
0.633
0.500
0.006
0.527
Ineryy
24
0.112
0.259
0.128
0.295
0.352
0.463
0.117
0.205
0.105
0.254
0.802
0.346
0.105
0.254
0.109
0.254
0.107
0.252
0.008
0.254
360
0.195
0.351
0.206
0.360
0.399
0.497
0.207
0.358
0.211
0.362
0.259
0.425
0.251
0.400
0.186
0.336
0.232
0.377
0.298
0.425
Aug
0.154
0.305
0.172
0.336
0.576
0.480
0.162
0.311
0.160
0.308
0.201
0.336
0.180
0.322
0L147
0.296
0.170
0.314
0.203
0,341
Screpimi
24
0.509
0.558
0.535
0,996
0.531
0.561
0.540
0.550
0.549
0.598
0.579
0,564
0,509
0.557
0.557
360
0.885
0.682
0.845
0.684
1.372
0:925
0.864
0.725
0.808
0.672
1.070
1.106
0.114
0.853
0.764
0.910
0.672
0,900
0.745
Any
1,211
0.595
0.701
0.600
0.084
0.728
0.713
0.633
0.724
0.583
0.809
0.604
0.843
0.685
0.725
0.672
0.737
0.500
0.733
0.651
Scheprin)
24
0.630
0.547
0.621
0.570
0.749
0.657
0.636
0.562
0.613
0.537
0.025
0.586
0.069
0.600
0.684
0.621
0.641
0.539
0.648
0,510
360
0.953
0.739
0.978
0.736
1,340
0.877
0.995
0.783
1.028
0.745
0.965
0.788
1,215
0.963
0.957
0.780
1,053
0.736
1.020
0.819
Ang
0.807
0.643
0.303
0.653
1.044
0,767
0.366
0.692
0.820
0.641
0.500
0.687
0.942
0,732
0.836
0.701
0.850
0.632
0.34
0.714
Sihephh
24
0.511
0.567
0.754
0.670
0.719
0.641
0.673
0.575
0.755
0.679
0.701
0.579
0.669
0.629
0.830
0.677
0.841
0.700
0.331
0.727
0.897
0.737
0.886
0.712
0.849
0.752
0.823
0.762
0.882
0.600
0.039
0.804
Any
[:741
0.628
0.746
0.648
0.703
0.698
0.808
0.699
0.779
0.644
0.768
0.702
1.100
0.820
0.904
0.720
0.791
0.630
0.004
0.717
Screphh]
24
0.798
0.645
0,230
0.677
0881
0.114
0.783
0.674
0,975
0.004
0.810
0.718
0.834
0.691
0.892
0.740
0,829
0.632
0.709
0,080
360
0.974
0.757
0,962
0.761
1,042
0.779
1,056
0.829
1,039
0.735
1.102
0.630
1.126
0.345
0.950
0.825
1,053
0.742
1,280
0.925
Ang
0.886
0.701
0.391
0.719
0.925
0.746
0.919
0.751
1.007
0.715
0.956
0.774
0.980
0,768
0.941
0.782
0.941
0.687
1.025
0.802
Calua
10
0.094
0.84
0.146
1.449
0.279
0.095
0.140
0,073
0.106
0,968
0.091
0.129
0,071
0
01033
0.125
0.236
0.181
0.298
0.663
0.494
0.252
0.341
0.221
0.290
0.150
0.301
0.311
0.377
0.405
0.379
0217
0.316
Ang
0.119
0.165
0.132
0.219
0.556
0.386
0.188
0.340
0.147
0.198
0.129
0.195
0.210
0.201
0.253
0.238
0.235
0.150
0.221
Ropel
10
0.217
0.221
0:301
11300
11238
0.274
0.238
0.270
0.221
0,225
01201
01.24
0.240
0.259
0.226
0.238
0,421
0.421
0215
0.252
30
0.313
0.345
0.367
0.416
0.577
0.394
0.409
0.444
0.387
0,386
0,278
0.374
0.433
0.436
0.458
0.437
0,307
0.343
0.46
0.436
Ang
0.265
0,284
0.344
0.362
0,305
0.334
0.323
0.357
0.304
0.306
0.240
0.299
0.337
0.348
0.342
0.337
0.364
0.382
0,325
0.334
1" Doum
0
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.
Mask Type
Zem
Random
Mudele
GDGNd
TimeXer
Croudinear
DUET
PascTST
OCGN:1
TineXer
CrooLince
DUET
PaidIST
Metrics
1398
maee
Iise
mac
rise
mac
msse
1110
mee
Iase
mide
1150
rioe
Iise
moe
rise
mae
rise
maa
Coltann
20
10
0.066
0:081
0.153
0.061
0.86
0.145
0.145
0.663
0,063
0.101
0.101
0.17%
0.954
0.122
0.154
0.054
11074
0.109
0.109
0.070
0.058
0.107
0,092
0.111
0.115
0.176
0.165
0.868
0.865
0.102
0.111
0.104
0,000
0.111
0.138
0.061
0.081
0.134
0.120
50
0.081
0.196
0.064
0.147
0.070
0.105
0.006
0.121
0.071
0.107
0.033
0.090
0.110
0.164
0.965
0.131
0.075
0.115
0.074
0.114
0.076
0J133
0.084
0.145
0,065
0.102
0.100
0.125
0,070
0.105
0,070
0,096
0,112
0.169
0.866
0.108
0.064
0.118
0.075
0.120
Etoegy
0,071
0.864
1.192
0.205
0.119
0.121
0.273
0.230
01064
0.015
0.357
0.242
0.102
01992
0,248
0.154
11 11106
11106
0.254
0.369
0
0.199
0.230
0.124
0.119
0.27
0.369
0.003
0.099
0.244
01226
0.091
0.102
0.247
01344
3,099
0.105
0.355
0.244
0,077
0.212
0.125
0.277
0.100
0.244
0.106
0.252
11108
0.255
1085
0.225
0.124
0.500
0.101
0.266
0.107
0.253
0.103
0.248
0.870
0.289
0.123
0.273
0.007
0.341
0.100
0.344
0.108
0.257
0J051
0.217
0.123
0.275
0.008
0.342
0.100
03345
0.102
0.349
0.276
0.325
0.301
0.367
0.410
0.406
0.398
0.391
0.432
0,433
0.286
0.338
0.385
0.343
0.417
0.406
0.413
0.355
0.461
0.435
DE
0.279
0.335
0.407
0.390
04480
0.488
01436
0.411
01466
0.647
1.115
0.364
11475
1140%
0.488
01411
0.483
0.407
0.490
0.453
0,323
0.349
04456
0.424
0.839
0.423
0.440
0.415
11501
0.554
0.309
0,317
03487
0.641
0.440
0.426
0.489
0.418
9,510
0.459
0.292
0.336
0.418
0.403
(1426
0.416
0.426
0.407
01460
0.445
0.344
0.366
0.433
(1411
0.483
0.48
0.451
0.407
0.490
0.449
0,855
0.147
0.090
0.186
0.096
0.200
0.076
0.172
0.131
0.233
0.861
0.151
0.090
0.187
0.005
0.197
0.073
0.171
0.107
0.221
PIM
0.864
11155
0.104
112000
0107
0.300
0.006
0.178
01131
0.348
0.15%
0.105
0.203
0.107
012004
0.000
0.117
0.113
0.222
0,070
0.162
0.108
0,207
0.701
0.207
0.043
0.181
0.128
0.255
0J772
0.164
0,109
0.307
0.104
0.205
0.083
0.179
0.116
0.237
0,864
0.155
0.100
0.158
0.095
0.203
0.079
0.177
0.120
0.245
0
0.158
0.102
0.199
0.301
0.201
0.079
0.176
0.112
0.224
0,194
0.228
0.276
0.290
0.410
0.406
02552
0.264
0.256
0.287
0.193
0.229
0.280
0.293
0.240
0.277
0.254
0.267
0.255
0.257
20
0,200
0.233
0.285
0.296
0.430
0.488
0.259
0.270
0.270
0.295
0.266
0.234
0.296
0.296
0:249
0.257
0.361
0.270
0.27]
0.292
0.218
0.344
0.307
0.303
0.489
0:423
0.363
0.273
0.277
0.295
0.224
0.346
0.302
0.901
0:154
0.202
0:363
0.273
0.278
0.295
0.214
0.235
0.289
0.296
0.426
0.416
0.258
0.270
0.268
0.293
0.248
0.236
0.293
0.297
0.248
0.255
0.260
0.270
0.269
0.291
22

---

Published as a conference paper at ICLR 2026
THE USE OF LARGE LANGUAGE MODELS
We promise not to use Large Language Models in writing.
23


Reviews:
[{"review_id": "xsz25i6GGR", "reviewer": "Reviewer_vFgB", "summary": "The paper **proposes GCGNet**, a *Graph-Consistent Generative Network* that jointly models temporal and channel correlations for robust time series forecasting with exogenous variables. Through a variational generator, graph discriminator, and refiner modules, **GCGNet achieves state-of-the-art performance** across 12 real-world datasets, outperforming prior two-step modeling approaches by effectively capturing joint dependencies and handling noisy data.", "questions": "See weaknesses.", "limitations": "", "rating": 2, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "**Strengths:**\n\n1. The experimental results are significant.\n2. The writing is clear and easy to follow.", "weaknesses": "**Weaknesses:**\n\n1. **Lack of Novelty:** I fail to see significant advantages of this paper over related works. On one hand, graph-based methods for modeling the relationships between variables and the joint modeling of time and variables have been discussed in early literature, such as GWNe[1], DCRNN[2], and STEP[2]. On the other hand, the lack of joint modeling of temporal and channel correlations is not supported by quantitative case analyses. Furthermore, the authors argue that noise affects the model, which motivates the use of generative models, but this lacks persuasiveness. Even generative models are data-driven and cannot fully address the impact of data noise. Overall, the authors should provide more evidence to support their motivation, which is the foundation for the proposed methods and contributions.\n\n2. **Lack of Technical Innovation:** The authors lack a thorough analysis of the challenges encountered while addressing issues in related works. Instead, they directly propose the graph-consistent generative module, including the Variational Generator, Graph Discriminator, Graph Variational Autoencoder, and Graph Refiner modules, without sufficiently justifying each design choice. The overall model design, as well as the design of each part, lacks enough motivation and insights, making the approach feel overly engineered.\n\n3. **Writing Needs Improvement:** As mentioned above, the writing requires significant improvement. The authors should provide more evidence to validate their motivations, the significance of their contributions, and the insights behind the method design.\n\n[1]. Graph WaveNet for Deep Spatial-Temporal Graph Modeling\n[2]. Diffusion Convolutional Recurrent Neural Network: Data-Driven Traffic Forecasting\n[3]. Pre-training Enhanced Spatial-temporal Graph Neural Network for Multivariate Time Series Forecasting", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "EHJeFMAG9O", "reviewer": "Reviewer_wNXH", "summary": "In summary, this paper introduces GCGNet, a graph-consistent generative network for time series forecasting with exogenous variables. GCGNet first generates coarse predictions, then enforces structural alignment to capture joint temporal and channel correlations, and finally refines the predictions to prevent degeneration and improve accuracy. With these designs, GCGNet effectively models complex temporal–channel dependencies while maintaining robustness in real-world scenarios.", "questions": "See weakness", "limitations": "", "rating": 6, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 4, "strengths": "S1. Time series forecasting with exogenous variables is important to various domains.\n\nS2: This paper proposes GCGNet to address the problem of forecasting with exogenous variables.\n\nS3: This paper proposes Graph Discriminator module to ensuring the outputs align with the underlying temporal and channel joint correlation.\n\nS4: The authors released an anonymous GitHub repository, which improves reproducibility.", "weaknesses": "W1: In Figure (c), does it correspond to the framework proposed in this paper? \\(Y^{endo}\\) represents future endogenous variables, so why is \\(Y^{endo}\\) assumed to be accessible in the framework? \n\nW2: In Figure (c), I noticed that the figure labels it as \\(\\tilde{Y}^{endo}\\), but the figure does not provide any explanation of what \\(\\tilde{Y}^{endo}\\) represents.\n\nW3: The description in lines 240–244 is unclear. In Equation (7), the process of obtaining \\(\\hat{A}\\) from \\(A\\) is too vague. Since \\(A\\) represents a graph, the authors should clarify how a graph is processed using the VAE, rather than simply stating that \\(\\hat{A}\\) is obtained from \\(A\\).\n\nW4: In lines 77–84, the authors claim that using a generative network helps improve robustness. However, they only provide the results in Table 3 to validate the overall framework, without isolating the contribution of the generative network itself. It would be more convincing if the authors could provide theoretical justification or additional experiments, such as an ablation study, to demonstrate that the generative network specifically contributes to the performance improvement.\n\nW5: In lines 428–429, the authors mention that in Table 4 the \"random\" setting replaces masked values with random values sampled from the normal distribution N(0,1). This is unreasonable because the range and scale of the original features may differ significantly from N(0,1). Such noise does not reflect realistic scenarios, and therefore the \"random\" experiment is not meaningful.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "iYdrqGYAJd", "reviewer": "Reviewer_soMm", "summary": "The paper proposes GCGNet, a general and robust framework for time series forecasting that incorporates both historical and future exogenous variables. A key component is the Graph Discriminator, which utilizes graph structure consistency to guide the generator, ensuring that the generated forecasts align with temporal and inter-channel dependencies. Extensive experiments are conducted on 12 real-world datasets.", "questions": "1. Since this paper focuses on forecasting with exogenous variables, why weren’t the baselines restricted to algorithms that natively support exogenous variables? \n\n2. Why not use the original model directly, and instead add an MLP fusion approach, since this module might actually degrade the model’s performance?\n\n3. Why are the experimental results for the scenario without future exogenous variables reported only on a subset of datasets?\n\n4. Why is the key module named \"Graph Discriminator\" when it is actually unrelated to an adversarial training paradigm?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "1. The paper is well structured, and it focuses on an important problem that could have real-world applications.\n\n2. Nice figures help to understand the proposed method, and both the figures and tables are clear and well-presented.\n\n3. The proposed method may address some limitations of existing approaches, such as separate modeling and limited robustness.", "weaknesses": "1. The current set of baselines includes many models that do not inherently accommodate exogenous variables; however, this paper focuses on forecasting with exogenous variables.\n\n2. The adaptation of models that inherently lack support for exogenous variables, such as PatchTST, using a simple MLP fusion approach to incorporate future covariates raises concerns about its effectiveness. It remains questionable whether this method preserves the original modeling principles or potentially introduces degradation in performance.\n\n3. The experimental results for the scenario where future exogenous variables are unavailable are only reported on a subset of datasets.  This is questionable, particularly since this scenario represents a more common and practical use case in real-world applications.\n\n4. The naming of the key module as \"Graph Discriminator\" strongly suggests an adversarial training paradigm. However, its loss function L_disc functions more as an L1-based graph structure  reconstruction loss rather than a true discriminator performing binary  classification to distinguish \"real\" from \"generated\" graphs. This nomenclature is misleading.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "c85xhuDQK3", "reviewer": "Reviewer_s4T3", "summary": "This paper introduces GCGNet, a Graph-Consistent Generative Network aimed at time series forecasting with exogenous variables. The framework combines a variational generator for coarse forecasting, a graph-based discriminator enforcing joint temporal and channel consistency, and a graph refiner that leverages learned adjacency structures to refine outputs. Empirical evaluation on 12 real-world datasets, including settings with missing values and ablation studies, demonstrates the proposed method’s improved robustness and competitiveness compared to a suite of baselines.", "questions": "- Please provide a more detailed explanation of the Sparsify operation and conduct a sensitivity analysis on the parameter k  within Sparsify.", "limitations": "", "rating": 8, "confidence": 5, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "- This paper breaks the existing mindset in forecasting with exogenous variables, which typically follows a two-step modeling strategy inherited from traditional forecasting methods. Instead, it jointly models the time and channel, which represent a effective approach for forecasting with exogenous variables.\n\n- The proposed Graph Discriminator in this paper is interesting, as it provides a novel way to align the predicted values with the underlying temporal and channel joint correlations.\n\n- The experimental results demonstrate that the proposed method achieves strong performance.\n\n- The code and datasets has been open, supporting reproducibility.", "weaknesses": "- The description of the Sparsify operation in Section 3.4 only briefly explains its purpose but doesn't  clearly describe how it actually works. More details are needed to help  readers understand what this step does and how it fits into the Graph  Refiner module.\n\n- The Sparsify operation uses a hyperparameter k that controls how much sparsification is applied, and this likely has a big impact on how well the model performs. However, the paper doesn't  explain how the value of k was chosen, or test how different choices of k affect the results.\n\n- Please carefully check the references. References to preprints, such as \"Long-term forecasting with tide: Time-series dense encoder. \" should be updated to their formally published versions.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)