FACT: Fine-grained Across-variable Convolution for Multivariate Time Series Forecasting
Authors: Huiqiang Wang
Jieming Shi
Li Qing
Venue: ICLR 2026 Poster (2026)
Keywords: Multivariate time series forecasting
Fine-grained dynamic variable interactions
Multi-dilated depth-wise convolution.
Primary area: learning on time series and dynamical systems
TL;DR: Effective variable interactions modeling from both time and frequency domains; Efficient multi-dilated depth-wise convolution architecture.

Abstract: Modeling the relationships among variables has become increasingly important, particularly in high-dimensional multivariate time series forecasting tasks. However, most existing methods primarily focus on capturing coarse-grained correlations between variables, overlooking a finer and more dynamic aspect: the variable interactions often manifest differently as time progresses.
To address this limitation, we propose FACT, an Fine-grained Across-variable Convolution architecture for multivariate Time series forecasting that explicitly models fine-grained variable interactions from both the time and frequency domains. 
Technically, we introduce a depth-wise convolution block DConvBlock, which leverages a depth-wise convolution architecture with channel-specific kernels to model dynamic variable interactions at each granularity.
To further enhance efficiency, we reconfigure the original one-dimensional variables into a two-dimensional space, reducing the variable distance and the required model layers. Then DConvBlock incorporates multi-dilated 2D convolutions with progressively increasing dilation rates, enabling the model to capture fine-grained and dynamic variable interactions while efficiently attaining a global reception field.
Extensive experiments on twelve benchmark datasets demonstrate that FACT not only achieves state-of-the-art forecasting accuracy but also delivers substantial efficiency gains, significantly reducing both training time and memory consumption compared to attention mechanism.

Published as a conference paper at ICLR 2026
FACT: FINE-GRAINED ACROSS-VARIABLE CONVOLU- TION FOR MULTIVARIATE TIME-SERIES FORECASTING
Huiqiang Wang, Jieming Shi,, Qing Li The Hong Kong Polytechnic University hui huiqiang wongeconnect. . yo. hk, jieming shi @pol yu edu hk esql i comp. pol pulyu. edu.hk hk
ABSTRACT
Modeling the relationships among variables has become increasingly important, particularly in high-dimensional multivariate time series forecasting tasks. How- ever, most existing methods primarily focus capturing coarse-graines correla- tions between variables, overlooking a finer and more dynamic aspect: the variable interactions often manifest differently as time progresses. To address this limita- tion, we propose FACT, an Fine-grained Across-variable Convolution architec- ture for multivariate Time series forecasting that explicitly models fine- fine-grained variable interactions from both the time and frequency domains. Technically. we introduce a depth-wise convolution block DConvBlock, which leverages a depth-wise convolution architecture with channel-specific kernels to model dy- namic variable interactions at each granularity. To further enhance efficiency, we reconfigure the original one-dimensional variables into two-dimensionall space, reducing the variable distance and the required model layers. Then DConvBlock incorporates multi-dilated 2D convolutions with progressively increasing dilation rates, enabling the model to capture fine- fine-graing grained and dynamic variable interac- tions while efficiently attaining global reception field. Extensive experiments twelve benchmark datasets demonstrate that FACT not only achieves state-of-the- art forecasting accuracy but also delivers substantial efficiency gains, significantly reducing both training time and memory consumption compared to attention mechanism. The code is available https: /tapp::///iie t/tubbbb..tttttu com/ wanghq21/ FACT
INTRODUCTION
Multivariate time series forecasting plays a pivotal role in a wide range real-world applications, such as weather prediction (Wu et al. [2023b] Bi et al. (023). traffic management (Yin et al. 20022.. financial market analysis (Liu system optimization (Qian et al.] 2019). In such scenarios, multiple variables interact in intricate and evolving ways, exhibiting correlations, causal relations, covariances, and temporal lags in the time domain, well as resonance, phase synchronization, and frequency-dependent coupling in the frequency domain. Accurately modeling these across-variable dependencies therefore critical to improving forecasting performance.
2024), disease surveillance (Matsubara et al.) 2014)) and energy
wide range of approaches have been developed to capture variable dependencies, including attention-base architectures (c.g.. iTransformer (Liu et al. 2024), TimeXer (Yu et al. 2024)
Crossformer (Zhang & Yan 2023)), clustering- based approaches (e.g.. DUET (Qiu et al. 2025
CCM (Chn et al. (0024)) and CNN- MiNN-baedd models (e.g., Mo (Luo & Wang 2024)). While
effective, most of these models treat multivariate representations as indivisible wholes, thereby over- looking fine-grained and temporally evolving interactions among individual variables.
In practice, such interactions are highly dynamic. For example, the correlation between traffic flows on two adjacent streets may be strong during weekday commuting hours but weak on weekends, when travel patterns become more random. These dynamics often exhibit locality and periodicity,
"Corresoonding Author.

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Published as a conference paper at ICLR 2026
Fllme Demain!
yi IFequsncyy Domain)
Dynamic Comelation
y2
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y2 thequency Daman))
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y3 Irrquencc Domair)
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Frequency (itz)
Time Index
(a) Each frequency.
(b) Each time step.
Figure I: The dynamic interaction at each granularity.
underscoring the importance of modeling variable dependencies at finer granularity and with tem- poral sensitivity. Furthermore, the time and frequency domains offer complementary views: in the time domain, dependencies emerge instantaneous effects at individual time steps, while in the fre- quency domain, they manifest as resonances and phase shifts across specific periodic components.
Figure a) shows four variables along with their spectrograms. Strong resonance patterns are ev- ident between variables y1-y4 and y2-y3 within specific frequency bands, accompanied by clear phase delays in high frequency regions. In contrast, Figure [Tb) displays time domain correlations computed segment of length 100, where variables y yl-y4 and y2-y3 exhibit strong similarity between steps 30 to 70, but weaker correlations elsewhere, reflecting transient dependencies. These observations suggest that combining time and frequency domain interactions provides complemen- tary insights, capturing both transient fluctuations and periodic patterns for more comprehensive modeling of across-variable dependencies. Graph neural networks (GNNs) can represent variable interactions at each time step via adjacency matrices and have shown potential in capturing dynamic vcross-variable relationships. However, they incur substantial computational and memory overhead, particularly for high-dus digg--immnsional datasets, since maintaining large adjacency matrices is required during both training and inference.
To overcome these limitations, we propose FACT, an efficient multi-dilated depth-wise convolu- tion architecture tailored to capture fine-grained across-variable dependencies in both the time and frequency domains. At its core lies the DConvBlock, which draws inspiration from convolution de- signs in computer vision (e.g.. Inception) and adapts them into lightweight depth-wise convolution variants. This design enables the extraction of meaningful across- variable interactions at each gran- ularity without the computational burden of graph-bases structures. Nevertheless, when applied to datasets with large number of variables (c.g., Traffic, PEMS07), conventional ID convolution typi- cally requires stacking numerous layers to achieve a global reception field, leading to high computa- tional costs and training instability. To address this challenge, reconfigure the variable dimension into a two-dimensional space, allowing the use of 2D convolutions to achieve broader coverage with fewer layers. Moreover, by employing multi-dilated convolutions with progressively increasing di- lation rates, we further expand the reception field while simultaneously reducing network depth and parameter count, thereby improving both computational efficiency and training stability.
Extensive experiments on twelve benchmark datasets demonstrate that FACT not only achieves state-of-the--at forecasting accuracy but also offers significant improvements in computational effi ciency over attention-basee models.
Our main contributions are summarized as follows:
We propose FACT, an efficient convolution architecture that jointly leverages time- and frequency-domain variable interactions to achieve complementary modeling of multivariate time series.
We introduce the DConvBlock, which employs depth-wise convolutions with channel- specific kernels to explicitly model variable interactions at each granularity, thereby cap- turing fine-grained dependencies across variables with high efficiency.
We further reconfigure the variable dimension into two-dimensional space and incorpo- rate multi-dilated convolution kernels with progressively increasing dilation rates, effec- tively enlarging the reception field while reducing network depth.
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We conduct extensive experiments twelve high-dimensional multivariate time series benchmarks, demonstrating that FACT consistently outperforms existing methods in fore- casting accuracy while substantially lowering computational cost.
2 PRELIMINARIES
Multivariate time series forecasting Multivariate time series E RCXT is a C-dimensional nu- merical sequence indexed by time, where T is the number of timestamps. and C is the number of variables (C' > 1). The input time series is represented as a matrix, where each row corresponds to variable and each column corresponds to timestamp. The goal of multivariate time series fore- casting is to predict future values of each variables based on past observations and the information from other variables. Specifically, given historical time series X RCXL, where L is the lookback length, the task to predict the future values X(+1++11 E RRCH where H is the prediction length:
X - f(X:-L+1:))
where / is forecasting function. The objective is learn accurate forecasting function ʃ that minimizes the prediction eπror between the predicted time series X and the ground truth Y(+1+H E IR C The prediction error can be quantified using the Mean Squared Error (MSE)
loss function: C
Y1+14+H X4+14+ Our focus is on multivariate time series forecasting, and it is beyond the scope of this work to assume utilize any spatial ordering among variables.
Discrete Fourier transform Given a discrete time series 2 æ[n] E R. its frequency representation be obtained via the Discrete Fourier Transform (DFT), which decomposes the signal into of sinusoids at different frequencies. The DFT is defined as:
F(A]
En
3 kn
h =
0.
N
Each frequency component F[k] a complex number and can be expressed terms of its real and imaginary parts as:
F(k) = R(F(k) + I(FI)) = RR + jIx
(3)
where the real part R2 and imaginary part Ia are computed as:
R
Σ
x|n]
cos
2m
Ix
Σ x|n]
sin
2n
kn
(4)
From these components, we can compute the amplitude [F[k]] and phase øik] of each frequency k as follows:
Is
[F( = v R2 + I2. [[)) arctan
(5)
Rr
Given the magnitude [F[k]] and phase olk), the real and imaginary parts can be reconstructed as:
R₂ = [[]]]| cos( (((+)), In |F(})] (k)())
Thus, the representation of frequency component can be equivalently expressed in either Cartesian form II) or polar form (IF[A)]], ( @k)).. Since cross-variable frequency correlations are often manifested through energy distributions and phase relationships, adopt the amplitude-phase rep- resentation of the frequency spectrum instead of Cartesian decomposition. This choice allows for more direct and interpretable characterization of inter-variable dependencies in the frequency domain. Experimental results in Appendix [ further validate the superiority of this representation over alternative frequency modeling approaches.
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X Lil
Embedding
Reshape Padding
BatchNorm2d
DFT
A
P
Depth-wise Inception
DConvBlock
DConvBlock
DConvBlock
multi-dilated
DConvBlock
DConvBlock
Batchhorm22d
DConvBlock
Ferd-Forwwrd
IDFT
Frequency Domain
Time
Projection
Domain
Reshape Truncate
DConvBlock
The framework of FACT. The time series is projected into both the time and frequency do- mains, where depth-wise DConvBlocks are employed capture fine- one-grained variable interactions. Each DConv Block adopts multi-dilated convolution structure to enlarge the reception field while reducing computational complexity.
Figure 2:
3
METHODOLOGGY
The overall framework of FACT is illustrated in Figure 2 At its core, FACT employs the proposed CNN-based DConvBlock to capture inter-variable dependencies from both the time and frequency domains. The input time series is first projected into high-dimensional embedding space X, RCxD where D denotes the hidden dimension, to enhance representational capacity. We also compared modeling in the raw time-series space and the embedding space in Appendix E.
To achieve comprehensive modeling dependencies, we adopt a dual-domain strategy. the time domain, DConvBlocks are directly applied to the embeddings to capture instantaneous interactions among variables. the frequency domain, the embeddings are transformed via the Fourier trans- form, and DConyBlocks are then employed to model dependencies across amplitude and phase com- ponents. Since Parseval's theorem guarantees energy equivalence between the time and frequency domains, and instantaneous interactions (time domain) and periodic interactions (frequency domain) provide complementary perspectives, we introduce a weighting factor to balance temporal and spectral information, thereby enabling unified characterization of across-variable dependencies. The effects of different a can be found in Appendix B.3
Technically, in each domain, the DConyBlock first reconfigures the one-dimensional variables into a two-dimensional form, reducing the effective distance between variables. It then applies multi- dilated 2D depth-wise convolutions with progressively increasing dilation rates to capture fine- grained dependencies while substantially enlarging the reception field. A subsequent network further enhances the learned variable representations. Structurally, the OConvBlock re- sembles Transformer block, but replaces the attention mechanism with the proposed multi-dilate depth-wise convolutions. This substitution yields a simpler yet more efficient design that retains strong representational power while greatly reducing training cost compared to attention.
3.1
TIME AND FREQUENCY MODELING
Time domain Multivariate time series often exhibit dynamic and time-varying instantaneous in- teractions among all variables. To effectively capture these dependencies, we employ N stacked DConvBlocks. Given the input embedding X., the output of time-domain modeling X E CCCD can be computed as:
X X., X
41 DConeBlock(())) E (0, 1,,,, N 1},
(8)
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Frequency domain Multivariate time series also show dependencies across variables at differ- ent frequencies, such as low- wwwfrequency trends and high-frequency fluctuations. To capture this, we model fine-grained variable interactions in the frequency domain. Specifically, given the input bedding Xe, we apply the Discrete Fourier Transform (DFT) to project into the frequency domain:
F = DFT(X.) - + I(F) E RCX
where R(F) and I(F) denote the real part and imaginary parts, respectively. Then we compute the amplitude and phase of each frequency:
Is
A = [F[]]] V R + [bk] arclan)
arclan
(10)
RR
The amplitude A of each frequency component captures the energy or strength of oscillations at specific frequencies, reflecting how strongly variables co-vary in terms of amplitude. The phase P encodes the relative timing of these oscillations, indicating temporal lead-lag relationships or synchronization patterns between variables.
For simplicity, we separately model the amplitude and phase components using N DConvBlocks, which aligns with the physical interpretation of Fourier components and enhances its expressiveness for complex multivariate temporal dependencies. After N stacked DConvBlocks to reconstruct the amplitude and phase, we get the output of frequency domain and use inverse DFT to transform it into the time domain to get X E CxD The overall process summarized as follows:
Ao A, Po- = P,
(11)
A1+1 DConv At), 0, -1,
(12)
P/11 = DConv Block Pi), = O,....,, 1, 1,
(13)
X iDFT cos(Pi41).. (A4+1 sin(Pi+1))),
(14)
Finally, fuse the outputs from the time and frequency domains using weighting factor to obtain the final representation: X1+1 a xx-X;+1 (1 at) -X The parameter enables flexible control over the relative contributions of each domain. Subsequently, a linear transformation along
the hidden dimension is applied to project X(41 into the predicted sequence X441++H-
3.2 OCONVBLOCE
The DCOnvBlock is designed to capture the dynamic interactions among different variables each granularity, and it mainly consists of two sub-layers: multi-dilated depth-wise convolution struc- ture and a fully connected feed-forward network. Batch normalization and residual connections applied within each sub-layer. Concretely, into 2D space and then select 2D convolution to accelerate the training efficiency. Then the multi- dilated depth-wise convolution structure is employed to capture fine-grained inter-variable interac- tions while significantly expanding the reception field, Subsequently, the feed-for worwwrd network is used enhance the variable representations and the 2D variable reshaped back into 1D space to restore the original data format.
first transform the ID variable representation
Depth-wise structure Convolution structures effectively capture local dependencies among vari- ables, but traditional convolutions compute the filters across all channels, limiting their ability model interactions specific to each granularity. To address this limitation, we adopt depth- wise convolution design, assigning each input channel a separate convolution kernel. By allowing each kernel operate independently on specific granularity, the depth- convolution enables granularity-specife learning of variable interactions.
Meanwhile, is worth noting that several popular convolution architectures, such as standard 2D convolution, Inception (Szegedy et al., 2015). and ResNet (He et al.) 2016)), can serve as the con- volution backbone. In this work, we adopt the Inception structure and modify it into a depth-wise Inception variant due to its stronger representational capacity compared to standard 2D convolu- tion. At the same time, our experiments in Appendix C demonstrate that even standard depth- 2D convolutions achieve competitive performance, though slightly inferior Inception overall. This observation further confirms that the effectiveness of our approach arises not from the specific choice of Inception, but from the underlying architectural design itself.
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2D modeling With the same number of convolution kernel parameters and equal number of model layers. 2D convolution can achieve a larger reception field compared to ID convolution. Assume that the length of a ID tensor is S, which reshape into 2D tensor of size approximately VS x VS. If the kernel size of the ID convolution is 8, then the kernel size of 2D convolution with the same number parameters is approximately vs x vs. For simplicity, we ignore stride and padding effects in the following analysis.
For ID convolution, the number of layers In required to cover the entire length is:
S
In 1 +11
(15)
Similarly, for 2D convolution, the number of layers (2D to cover the entire v5 x VS input is:
12 = VS V8 - +
(16)
Then the percentage reduction in the number of layers when using 2D convolution compared to 1D convolution is given by (if the small effect of [-] on the results ignored and we set R
S,v3 - vS, 11-1 V Vb):
r In 120
VS.
2
2
(17)
LD
v5
From this approximate relation, it is evident that as the length S increases, the relative reduction of the layers required for the 2D convolution also increases. Based this observation, we convert the variables into 2D space to leverage 2D convolution architectures for modeling dynamic inter- variable interactions efficiently. We focus on multivariate time series forecasting, and do not assume or leverage any inherent spatial order among the variables. We reshape the input sequence into a 2D format by row-major order. Specifically, we pad and convert X E RCDD A, P, E RCx 11
X 12
IR HxWxD A1,2D, Pi,2 #xxx 12221 the beginning of DConyBlock, where
H = W = [VC]. Experiments in Appendix C validate the effectiveness of 2D modeling over ID.
Multi-dilated architecture To further enhance the ability to capture variable dependencies with- out increasing parameters count stacking numerous layers, we incorporate dilated convolutions into the architecture. can be significantly expanded, allowing the model to efficiently capture variable interactions while ensuring a more dense and flexible coverage the input space at each scale. This design comple- ments the depth-wise structure to jointly achieve fine-grainee modeling and efficient reception field expansion. Experiments Appendix validate the effectiveness multi-dilated design.
shown Figure by applying multiple dilation rates, the reception field
dilation=1
dilation=2
dilation=3
n=3
Figure 3: Multi-dilated architecture. denotes increasing dilation sizes.
In summary, let MDInception represent the multi-dilated depth-wise Inception module, BN
represent batch normalization and use n distinct increasing dilation rates from 1. Then for X1 A1,p, Ptap, the corresponding outputs, denoted X HxxxD Ates Ptc

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Published as a conference paper at ICLR 2026
R "xx 11 can be computed as:
X X
(DInepption(B((X(.)))) dilationn
(18)
At, = 1 +
IIDInception(NN(A.pp)) Adttiii
(19)
Pt.e P122 +
MDInception(NN(PP.. diaaaiio dilation=
(20)
Feed-Forward Networks After modeling the fine-graines across-variable interactions through the multi-dilated depth-wise convolution structure, we employ a fully connected feed-forward network (FFN) among the hidden dimension further integrate the information. FFN as feature trans- formation module to enhance the variable representations and produce the output X A Pt.- This process can be summarized as follows:
_ X le FFN(BN(X(,.))
(21)
Ar, AI, (FN(BN(AI,)))
(22)
Pir Pt.e + FFN(BN(PL.)).
(23)
Finally, X Ar,F, PIF are reshaped from 2D space back into the original ID space. During this process, we also truncate it along the variable dimension (if necessary) to ensure that its shape matches the input dimension, resulting in the output (-layers DConvBlocks X A1+1, Pi41-
EXPERIMENTS
4.1
SETUP
Datasets. We evaluate the proposed FACT twelve datasets, encompassing diverse variable rela- tionships. For long-term forecasting (prediction lengths: {96, 192, 336, 720)), we use three high- variable real-world datasets: ECL, Traffic, and Solar-Energy (from LSTNet (Lai al.) 2018)). For short-term forecasting (prediction lengths: {12, 24, 48, 96)), we include nine public datasets: PEMS03. PEMS04, PEMS07, PEMS08 (from SCINet (Liu et al. 2022 PEMSD7 (Sen et al. 2019) SEMS-BAY METR-LA (Li et al. 2018). electricity and Traffic2. The lookback length of all datasets are fixed 96 for fair comparison. Dataset details are in Appendix B.I]
Baselines. We include twelve latest models as baselines: one variable dependent models: Times- Net (Wu al. 2023a): two variable attention models: i'Tranformer (Liu et al., 2024). TQNet (Lin et al. 2025) and TimeXer (Wang et al., 2024b); one variable CNN model: ModernTCN (Luo &
Wang 2024); one variable Mamba model: TimePro (Ma et al.] [0025;; one variable clustering
model: DUET (Qiu et al., 2025); two variable MLP models: TSMixer (Ekambaram et al. 2023)
and SOFTS (Han et al. 2024); two variable independent models: PatchTST (Nie et al,, 2023) and
CycleNet (Lin et al. 2024a); and one Graph-based model: TimeFilter (Hu et al. 2025) The hyper-
parameter configuration is provided in Appendix [.2 and the hyperparameter sensitivity analysis is
provided in Appendix B.3
4.2
MAIN RESULTS
Table I and Table summarize the overall performance of FACT and baseline models across all datasets, evaluated using Mean Squared Error (MSE) and Mean Absolute Error (MAE). The re- sults clearly demonstrate that FACT consistently achieves state-of-the-ar performance nearly all datasets. highlighting strong ability to model fine-grained inter-variable interactions effee- tively. Specifically, the PEMS03 dataset (Table I. our FACT achieves remarkable improvements over existing baselines: it reduces the MSE by 23.9% compared to the variable-attentin model iTransformer, by 22.5% compared to the Criable-CNNN model Modem'TN, by 20.4% compared to the graph-basee model TimeFilter and by 27.1 % compared to the variable-independen model CycleNet. And on METR-LA dataset (Table 2 our FACT reduces the MSE by 15.8% compared

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Published as a conference paper at ICLR 2026
Table I: Full results on part of datasets. The input length is fixed as 96 and the results are averaged from all prediction horizons. The results of other models are sourced from i'Transformer (Liu et al., 2024.. The best results are highlighted in bold. 1s denote the count of the best performance.
Models
PEMS03
PEMS04
PEMS07
PEMSOS
PEMSD7
PEMS-BAY
1
MSB
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
TimesNet20238
0,173
0.264
0.148
0.263
0.133
0.241
0.214
0.293
0,569
0.436
0.669
0.375
0
PatcTTST 2023
0.150
0.260
0.171
0.283
0.165
0.267
0.242
0.285
0.412
0.596
0.382
0
CycleNet 20244
0,118
0.225
0.119
0.231
0.113
0.189
0.150
0.228
0,503
0,429
0.604
0.397
0
SOFTS 2024
0.104
0.210
0.102
0.208
0.087
0.184
0.138
0.219
0.503
0.429
0.604
0.397
0
TSMixer
0.158
0.263
0.129
0.247
0.117
0.221
0.201
0.294
0.488
0.394
0.523
0.336
0
MCNe
0,111
0.226
0,103
0.217
0.118
0.208
0.230
0.268
0.374
0,370
0.436
0.330
0
TimePro 2025
0.119
0.225
0.114
0.226
0.096
0.200
0.252
0.259
0.412
0.370
0.577
0,389
0
DUET 2025
0.110
0.219
0.113
0.227
0.091
0.198
0.130
0.229
0.360
0.336
0.447
0.326
0
TQNet 2025
0.097
0.203
0.091
0.197
0.075
0.171
0.142
0.229
0,361
0.343
0.427
0,315
0
iTrnsfoment 2024
0.221
0.119
0.231
0.113
0.189
0.150
0.228
0,382
0.354
0.447
0.319
0
TimeXer 202410
0,111
0.223
0.110
0.227
0.093
0.199
0.233
0.269
0,337
0.319
0.394
0.288
0
TimeFilter 2025
0.108
0.217
0.114
0.226
0.101
0.207
0.139
0.230
0.498
0,404
0.547
0.358
0
FACT
0.086
0.194
0,087
0.194
0.073
0.170
0.107
0.200
0.330
0.316
0.372
0.267
12
Table 2: Full results on other datasets.
ECL
Traffic
Ennery-
electricity
Traffic2
METR-LA
Models
1
MSE
MAB
MSE
MAE
MSB
MAE
MSE
MAE
MSE
MAE
MSE
MAE
TimesNet 2023a
0.192
0.295
0.620
0.336
0
0,319
965.69
69
2.265
0.272
0,258
0.908
0.520
D
PutchTST 2023
0.189
0.276
0.454
0.286
0,236
0.266
410.90
1.648
0.234
0.204
0.762
0,488
0
CycleNet 2024a
0.168
0.259
0.472
0.301
0,210
0.261
178.53
0.988
0.292
0.254
0.748
0.503
0
SOFTS 2024
0.174
0.264
0.409
0.267
0,229
0,256
264.81
1.304
0.224
0.192
0.748
0.503
1
TSMixer 2026
0.186
0.287
0.522
0.357
0,260
0,297
257.00
1.222
0.244
0.232
0.833
0.490
0
McdemnCN 2024
0.197
0.282
0.546
0.348
0.211
0.310
5009.1
1.899
0.252
0.245
0.808
0.500
0
TimePro 2025
0.169
0.262
0.443
0.287
0,232
0,266
200.68
1.093
0.244
0.218
0.787
0.533
DUET 2025
0.170
0.265
0.469
0.289
0,220
0,263
244.13
1.244
0.279
0.244
0.813
0.487
0
TONet 2025
0.164
0.259
0.445
0.276
0.198
0.256
203.41
1.038
0.234
0,200
0.749
0.489
Transforna 2024
0,178
0.270
0.428
0.282
0.233
0.262
384.22
1.656
0.221
0.193
0.837
0.536
0
TimeXer 20246
0.171
0.270
0.466
0.287
0.202
0.269
261.84
1,310
0.217
0.195
0.750
0.496
0
TimeFilten 2025
0.158
0.256
0..07
0.268
0,223
0.250
193.29
1.054
0.227
0.199
0,819
0.493
FACT
0.157
0.254
0.425
0.276
0.196
0.254
159.73
0.922
0.204
0.179
0.704
0.456
9
to iTransformer, by 12.8% compared to ModemnTCN, by 14.0% compared to TimeFilter and by 5.3% compared to CycleNet. These results demonstrate the robustness and effectiveness of our pro- posed architecture in capturing dynamic, fine-grained inter-variable interactions. Experiments under longer lookback lengths in Appendix E.7 demonstrate that FACT can effectively leverage histori- cal information. Furthermore, the performance promotion of our DConvBlock other baselines, presented in Appendix [..2, further indicates that FACT enhances variable representations. Predic- tion showcases are provided in Appendix G.2 illustrating the accuracy of prediction across diverse scenarios.
5 COMPARISON ON THE TARGET FORECASTING HORIZON
Although averaging errors across all forecasting horizons is the standard evaluation protocol in mul- tivariate time series forecasting, as adopted by recent representative works such as PStchTST, iTrans- former, and TimeFilter, evaluation at specific target horizon H (single time step) is also important.
Therefore, we report performance with several different target horizons (target time step ++ +f): 143, (+6, t+12, and t+24. For comprehensive comparison, we include several recent and competitive baselines: the latest attention- models (iTransformer, TimeXer), the latest graph-based model (TimeFilter), and the latest
model (TimePro). The results summarized in Table 3 using MSE as the evaluation metric. Across all baselines and target horizons, FACT consistently demonstrates superior performance. Notably, as the forecasting horizon increases, the performance
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Published as a conference paper at ICLR 2026
gap between FACT and other baselines generally widens. These results further validate the effec-
tiveness of our proposed method under various evaluation settings.
Table 3: Performance comparison on the target forecasting horizon.
H
Model
PEMS03
PEMS04
PEMS08
METR-LA
Traffic2
PEMSD7
PEMS-BAY
FACT
0.049
0.061
0.058
0.273
0.168
0,153
0.162
TimeFilter
0.050
0.064
0.058
0.277
0.187
0.163
0.210
3
TimeXer
0.062
0.070
0.167
0.283
0.181
0.156
0.172
TimePro
0.058
0.069
0.161
0.313
0.183
0.163
0,218
iTransformer
0.063
0.069
0.161
0.326
0.188
0.163
0.175
FACT 0.059
0.068
0.069
0.438
0.184
0.249
0.270
TimeFilter
0.063
0.075
0.070
0.443
0.206
0.284
0.350
TimeXer
0.073
0,078
0.186
0.439
0.196
0.253
0.276
TimePro
0.071
0.079
0.188
0.465
0.200
0.265
0.332
iTransformer
0.075
0.081
0.179
0.468
0,206
0.269
0,290
FACT 0.073
0.077
0.085
0.668
0.203
0.336
0.371
TimeFilter
0.082
0.091
0.092
0.692
0.222
0.441
0.547
TimeXer
0.089
0.090
0,214
0.670
0.211
0.349
0,376
TimePro 0.092
0.097
0,240
0.686
0,218
0.375
0,455
iTransformer
0.098
0.102
0.218
0.677
0,221
0.387
0.413
FACT
0.098
0.097
0.117
0.927
0.222
0.414
0.451
TimeFilter
0.129
0.131
0.146
1.014
0.232
0.661
0.875
24
TimeXer
0.124
0.115
0,268
0.943
0.227
0.427
0.478
TimePro
0.132
0.132
0.352
0.956
0,229
0,482
0.609
iTransformer
0.143
0.143
0.295
0.962
0.233
0.505
0.546
COMPARED WITH ATTENTION MECHANISM
Since our DConvBlock adopts Transformer-He like architecture, its key distinction lies in replacing the attention mechanism with multi- dilltdd depth-wiee wiiee convolutions capture fine- tne-grained variable interactions. To evaluate the efficacy and efficiency of this design, we conduct a detailed comparison between multi-dilated depth- wise convolutions and attention mechanisms in terms forecasting ac- curacy, training time, and memory cost. As shown in Figure g evaluate eight datasets with varying numbers of variables under identical hyperparameter settings to ensure fairness. The results show that method only achieves superior forecasting performance, which is consistent with the dynamic changes of vnter-variable interactions, but also delivers substantial efficiency gains over attention-based approaches. In particular, on large-scale datasets such Traffic (862 variables), PEMS07 (883 variables), and Traffic2 (963 variables), our approach reduces the computational cost by up 50%. These advantages stem from the multi-dilated depth-wise convolution design, whose complexity primarily depends on kernel size and channel dimension, rather than input sequence length as attention. More comparisons with TimeFilter and ModemnTC are provided in Ap- further demonstrating the superior efficiency of our FACT over existing baselines.
pendix
E.5
ettetton
atterle
Mcmory/oa
MOTRLL
MMEF7 M74
mose Ar
MITRIA
Solt Patic
AMMSOT
METRLA
(a) Performance comparison.
(b) Time cost.
(c) Memory cost.
Figure 4: Compared with attention mechanism.

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7 ABLATION ANALYSIS
Modules ablation As illustrated in Figures S we perform an ablation study on two key components of FACT: the multi-dilated depth-wise Inception module and the feed-forward network (FFN), We denote the variant without the Inception module as w/o MDInception and the variant without the FFN as w/o FFN. The multi-dilated depth-wise Inception mod- ule is responsible for capturing variable interactions at each granularity, while the FFN integrates information across granularities to enhance variable representations. The results show that both components are essential, each contributing comple- mentary benefits to modeling different aspects of the data. Notably, removing the Inception module leads to a more substantial performance drop in most cases, high- lighting the critical role of explicitly modeling inter- variable interactions.
MDlsception
#100
6:49
pattes
Time and frequency modeling ablation As shown in Table we further analyze the necessity of modeling variable interactions from both the time and frequency domains. We denote the variant without frequency- domain modeling as w/o Frequency and the variant without time-domain modeling as w/o Time. The results demonstrate that the two domains provide complementary information. Time-domain modeling captures transient and instantaneous variable interactions, while frequency-domall modeling reveals periodic variable dependencies. Combining both perspectives enables a more comprehensive characterization of across-variable interactions. Meanwhile, the effects of the domain weight factor a can be found in Appendix B.3
Datasets
Figure 5: Modules ablation.
Table 4: Ablation results of time domain and frequency domain. We report the average results.
Models
PEMS03
PEMS04
METR-LA
ECL
Traffic
Solar
Traffic2
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
FACT
0.086
0.194
0.087
0.194
0.704
0.456
0.157
0.254
0.425
0.276
0.197
0.259
0,204
0.179
w/o Time
0.089
0.197
0.089
0,198
0.755
0.483
0.162
0.258
0.436
0.286
0,202
0,263
0.215
0.188
w/o Frequency
0.087
0.196
0.090
0.201
0.722
0.459
0.162
0.258
0.426
0.276
0.198
0.256
0.212
0,180
8 CONCLUSION
In this paper, we propose FACT, novel convolution architecture that efficiently captures fine- grained variable interactions in both the time and frequency domains. To this end, we design specialized DConvBlock, which leverages a depth-wise Inception module to dynamically extract fine-grained cross-variable interactions. Additionally, we reconfigure the original ID variables into 2D form to reduce the required model layers and further employs multi-dilated convolution de- sign with progressively increasing dilation rates, substantially enlarging the reception field without increasing the number of parameters. Extensive experiments reveal three key findings: (1) model- ing interactions in both time and frequency domains is crucial for comprehensive characterization of across-variable dependencies; (2) fine- inn-igrined modeling significantly enhances the ability to capture complex variable relationships; and (3) the proposed DConyBloo achieves superior efficiency and effectiveness compared with conventional attention mechanisms.
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ACKNOWLEDGMENTS
This work is supported by grants from the Research Grants Council of Hong Kong Special Ad- ministrative Region, China (No. PolyU 15205224), and Smart Cities Research Institute (SCRI) P0051036-P0050643.
REPRODUCIBILITY STATEMENT
To ensure reproducibility, we provide the implementation details in Appendix Blnd and release the code at the github repository: https:///gith/t http:uub..coo//aagggg////777
REFERENCES
Lei Bai, Lina Yao, Can Li, Xianzhi Wang, and Can Wang. Adaptive graph convolutional recurrent network for traffic forecasting. Advances in neural information processing systems, 33:17804- 17815, 2020.
Kaifeng Bi, Lingxi Xie, Hengheng Zhang, Xin Chen, Xiaotao Gu, and Qi Tian. Accurate medium- range global weather forecasting with 3d neural networks. Nature, 619:1-6 07 2023. doi: 10. 1038//41588-023-6661855
Chao Chen, Karl Petty, Alexander Skabardonis, Pravin Varaiya, and Zhanfeng Jia. Freeway perfor- mance measurement system: mining loop detector data. Transportation research record, 1748 (1):96-102, 2001.
Jialin Chen, Jan Eric Lenssen, Aosong Feng. Weihua Hu, Matthias Fey, Leandros Tassiulas, Jure Leskovec, and Rex Ying. From similarity superiority: Channel clustering for time series fore- casting. Advances in Neural Information Processing Systems, 37:1 17:130335-13066 2024a.
Peng Chen, Yingying ZHANG, Yunyao Cheng, Yang Shu, Yihang Wang, Qingsong Wen, Bin Yang, and Chenjuan Guo. Pathformer: Multi-scale transformers with adaptive pathways for time series forecasting, ICLR, 2024b.
Vijay Ekambaram, Arindam Jati, Nam Nguyen, Phanwadee Sinthong, and Jayant Kalagnanam. Tsmixer: Lightweight mip-mixer model for multivariate time series forecasting. In Proceedings of the 29th ACM SIGKDD conference on knowledge discovery and data mining, pp. 459-469, 2023.
Lu Han, Xu-Yan Yang Chen, Han-Jia Ye, and De-Chuan Zhan. Softs: Efficient multivariate time series forecasting with series-core fusion. Advances Neural Information Processing Systems, 2024.
Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recog- nition. In Proceedings of the IEEE conference computer vision and pattern recognition, pp- 770-778, 2016.
Jie Hu, Li Shen, and Gang Sun. Squeeze-and-exciiation networks. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 7132-7141, 2018.
Yifan Hu, Guibin Zhang, Peiyuan Liu, Disen Lan, Naiqi Li, Dawei Cheng, Tao Dai, Shu-Tao Xia, and Shirui Pan. Timefilter: Patch-specific spatial-temporal graph filtration for time series fore- casting. ICMI, 2025.
Hosagrahar Jagadish, Johannes Gehrke, Alexandros Labrinidis, Yannis Papakonstantinou, Jig- nesh M Patel, Raghu Ramakrishnan, and Cyrus Shahabi. Big data and its technical challenges. Communications of the ACM, 57(7):86 2014.
Diederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization. ICLR, 2015.
Guokun Lai, Wei-Cheng Chang, Yiming Yang, and Hanxiao Liu. Modeling long-and short-term temporal patterns with deep neural networks. In SIGIR, 2018
11

---

Published as a conference paper at ICLR 2026
Yaguang Li, Rose Yu, Cyrus Shahabi, and Yan Liu. Diffusion convolutional recurrent neural net- work: Data-driven traffic forecasting. ICLR, 2018.
Shengsheng Lin, Weiwei Lin, Wentai Wu, Feiyu Zhao, Ruichao Mo, and Haotong Zhang. Segrnm:
Segment recurrent neural network for long-term time series forecasting, 2023. URL https:
//arxiv.ogg/abs/ wttp://gggg.//....aaa// 11200
Shengsheng Lin, Weiwei Lin, Xinyi Hu, Wentai Wu, Ruichao Mo, and Haocheng Zhong. Cyclenet:
enhancing time series forecasting through modeling periodic patterns. NIPS, 2024a.
Shengsheng Lin, Weiwei Lin, Wentai Wu, Haojun Chen, and Junjie Yang. Sparsetsf: Modeling long-term time series forecasting with Ik parameters. ICML, 2024b.
Shengsheng Lin, Haojun Chen, Haijie Wu, Chunyun Qiu, and Weiwei Lin. Temporal query network for efficient multivariate time series forecasting. In Forty-second International Conference on Machine Learning. 2025.
Minhao Liu, Ailing Zeng, Muxi Chen, Zhijian Xu, Qiuxia Lai, Lingna Ma, and Qiang Xu. Scinet: time series modeling and forecasting with sample convolution and interaction. NeurIPS, 2022.
Yong Liu, Tengge Hu, Haoran Zhang, Haixu Wu, Shiyu Wang, Lintao Ma, and Mingsheng Long. itransformer: Inverted transformers effective for time series forecasting. International Con- ference on Learning Representations, 2024.
Dongaa Luo and Xue Wang. ModemTTC: A modern pure convolution structure for gen- eral time series analysis. ICLR, 2024. URL httpps://ppeeere/eeeeeeeetrrrrrr VPJMJerXHU
Xiaowen Ma, Zhenliang Ni, Shuai Xiao, and Xinghao Chen. Timepro: Efficient multivariate long- term time series forecasting with variable- and time-aware hyper-state. ICML,, 2025.
Yasuko Matsubara, Yasushi Sakurai, Willem Van Panhuis, and Christos Faloutsos. Funnel: Auto- matic mining of spatially coevolving epidemics. Proceedings of the ACM SIGKDD International Conference Knowledge Discovery and Data Mining, 08 2014. doi: 10. 00.114522233300222260
Yuqi Nie, Nam H Nguyen, Phanwadee Sinthong, and Jayant Kalagnanam. A time series is worth 64 words: Long-term forecasting with transformers. ICLR, 2023.
Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, et al. Pytorch: An imperative style, high- performance deep learning library. Advances neural information processing systems, 32, 2019.
Xihao Piao, Zheng Chen, Taichi Murayama, Yasuko Matsubara, and Yasushi Sakurai. Fredformer: Frequency debiased transformer for time series forecasting. In Proceedings of the 30th ACM SIGKDD Conference on Knowledge Discovery and Data Mining, pp. 1400-2410 2024.
Zheng Qian, Yan Pei, Hamidreza Zareipour, and Niya Chen. A review and discussion of decomposition based hybrid models for wind energy forecasting applications. Applied Energy, 20w..gy..1111..00000
235:939-953, 02 2019. doi: 10.
Xiangfei Qiu, Xingjian Wu, Yan Lin, Chenjuan Guo, Jilin Hu, and Bin Yang. Duet: Dual clus- tering enhanced multivariate time series forecasting. In Proceedings of the 31st ACM SIGKDD Conference on Knowledge Discovery and Data Mining I. pp. 1185-1196. 2025.
David Salinas, Valentin Flunkert, Jan Gasthaus, and Tim Januschowski. Deepar: Probabilistic fore- casting with autoregressive recurrent networks. International journal of forecasting, 36(3):118) 1191, 2020.
Rajat Sen, Hsiang-Fu Yu, and Inderjit S Dhillon. Think globally, act locally: A deep neural network approach to high-dimensional time series forecasting. Advances in neural information processing systems, 32, 2019.
Christian Szegedy, Wei Liu, Yangging Jia, Pierre Sermanet, Scott Reed, Dragomir Anguelov, Du- mitru Erhan, Vincent Vanhoucke, and Andrew Rabinovich. Going deeper with convolutions. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 1-9, 2015.
12

---

Published as a conference paper at ICLR 2026
Hao Wang, Licheng Pan, Zhichao Chen, Degui Yang, Sen Zhang. Yifei Yang. Xinggao Liu, Haoxuan Li, and Dacheng Tao. Fredf: Learning to forecast in the frequency domain. ICLR, 2025a.
Huiqiang Wang, Jian Peng, Feihu Huang. Jince Wang, Junhui Chen, and Yifei Xiao. Mien: Multi- scale local and global context modeling for long-term series forecasting. International Conference on Learning Representations, 2023.
Shiyu Wang, Haixu Wu, Xiaoming Shi, Tengge Hu, Huakun Luo, Lintao Ma, James Zhang, and Jun Zhou. Timemixer: Decomposable multiscale mixing for time series forecasting. ICIR, 2024a.
Shiyu Wang. Jiawei Li, Xiaoming Shi, Zhou Ye, Baichuan Mo, Wenze Lin, Shengtong Ju, Zhixuan Chu, and Ming Jin. Timemixer++: general time series pattern machine for universal predictive analysis. ICLR, 2025b.
Yuxuan Wang, Haixu Wu, Jiaxiang Dong, Yong Liu, Yunzhong Qiu, Haoran Zhang, Jianmin Wang, and Mingsheng Long. Timexer: Empowering transformers for time series forecasting with ex- ogenous variables. Advances in Neural Information Processing Systems, 2024b.
Zihan Wang, Fanheng Kong, Shi Feng, Ming Wang, Xiaocui Yang, Han Zhao, Daling Wang, and Yifei Zhang. Is mamba effective for time series forecasting? Neurocomputing, 619:129178, 2025c.
Haixu Wu, Jiehui Xu, Jianmin Wang, and Mingsheng Long. Autoformer: Decomposition trans- formers with auto-correlation for long-term series forecasting, In Proceedings of the Advances in Neural Information Processing Systems (NeurIPS), pp. 101-112, 2021.
Haixu Wu, Tengge Hu, Yong Liu, Hang Zhou, Jianmin Wang, and Mingsheng Long. Timesnet:
Temporal 2d-variation modeling for general time series analysis. ICLR, 2023a.
Haixu Wu, Hang Zhou, Mingsheng Long, and Jianmin Wang. Interpretable weather forecasting for worldwide stations with a unified deep model. Nature Machine Intelligence, 5:1-10, 06 2023b. doi: 10.1038//42256-23--000
Zonghan Wu, Shirui Pan, Guodong Long, Jing Jiang, and Chenggi Zhang. Graph wavenet for deep spatial-temporal graph modeling. arXiv preprint arXiv: 1906.001,,,, 2019.
Zonghan Wu, Shirui Pan, Guodong Long, Jing Jiang, Xiaojun Chang, and Chengqi Zhang. Con- necting the dots: Multivariate time series forecasting with graph neural networks. In Proceedings of the 26th ACM SIGKDD international conference on knowledge discovery & data mining, pp- 753-763, 2020.
Zhijian Xu, Ailing Zeng, and Qiang Xu. Fits: Modeling time series with 10k parameters. ICML, 2024.
Wang Xue, Tian Zhou, Qingsong Wen, Jinyang Gao, Bolin Ding, and Rong Jin. Card: Channel aligned robust blend transformer for time series forecasting. ICLR, 2024.
Kun Yi, Qi Zhang, Wei Fan, Shoujin Wang, Pengyang Wang. Hui He. Ning An. Defu Lian, Long- bing Cao, and Zhendong Niu. Frequency-domain mlps are more effective learners in time series forecasting. Advances in Neural Information Processing Systems, 36:76656-76679 2023.
Kun Yi, Jingru Fei, Qi Zhang, Hui He, Shufeng Hao, Defu Lian, and Wei Fan. Filternet: Harnessing frequency filters for time series forecasting. Advances in Neural Information Processing Systems, 11555115-55100 2024.
Xueyan Yin, Genze Wu, Jinze Wei, Yanming Shen, Heng Qi, and Baocai Yin. Deep learning on traffic prediction: Methods, analysis. and future directions. IEEE Transactions on Intelligent Transportation Systems, 23:4927-4943, 06 2022. doi: 10111199TTS.2211100002
Bing Yu, Haoteng Yin, and Zhanxing Zhu. Spatio-temporal graph convolutional networks: A deep learning framework for traffic forecasting. arXiv preprint arXiv: 2017.
Guoqi Yu, Jing Zou, Xiaowei Hu, Angelica I Aviles Riverr, Jing Qin, and Shujun Wang. Revitaliz- ing multivariate time series forecasting: Learnable decomposition with inter-series dependencies and intra-series variations modeling. NIPS, 2024.
13

---

Published as a conference paper at ICLR 2026
Ailing Zeng, Muxi Chen, Lei Zhang, and Qiang Xu. Are transformers effective for time series forecasting? AAAI, 2023.
Yunhao Zhang and Junchi Yan. Crossformer: Transformer utilizing cross-dimension dependency for multivariate time series forecasting. ICLR, 2023.
Haoyi Zhou, Shanghang Zhang, Jieqi Peng, Shuai Zhang, Jianxin Li, Hui Xiong, and Wancai Zhang- Informer: Beyond efficient transformer for long sequence time-series forecasting. In AAAI, 2021.
Tian Zhou, Ziqing Ma, Qingsong Wen, Xue Wang, Liang Sun, and Rong Jin. Fedformer: Frequency enhanced decomposed transformer for long-term series forecastings. In International Conference Machine Learning, 2022.
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A
RELATED WORK
Exploring variable relationships in multivariate time series forecasting has garnered significant atten- tion, with existing methods categorized into Variable Dependent, Variable Independent, Attention- based, cluster-bssed MLP-based, CNN-based, Mamba-based and Graph-based approaches.
Variable Dependent method. Early works embed all variables into hidden vector and focus on capturing the time dependency within the time series, such as cyclical characteristics, trend changes, etc. This method could increase the model's 's representational capability but lack the mod- eling of variable relationship and have poor generalization, such as Informer (Zhou et al., 2 Autoformer (Wu et al., 2021), FEDformer (Zhou et al., 2022). MICN (Wang et al. 2023). Times- Net (Wu al., 2023a). Pathformer (Chen et al., 2024b), TimeMixer (Wang et al., (2024a) and TimeMixxr++ (Wang et al. 2025b). TimeMixer++ serves as the latest model for capturing the temporal dependencies such cycles and trends within time series through sophisticated designs that require great training time and memory costs.
Variable Independent method. Variable Independent method treats the multivariate variables as multiple independent individual variables, including PatchTST (Nie et al.] (023), DLinear (Zeng al. 2023), SegRNN (Lin al. 2023), SparseTSFF (Lin et al. 2024b) and CycleNet (Lin al.) 2024a), which is more generalizable, but the representation capability is reduced and the complex relationship between variables are ignored. The difference between a multivariate time series and a univariate time series is that a multivariate time series contains complex variable relationships, such as correlation, cointegration, causation, etc. Therefore, the modeling of multivariate time series should be different from the modeling of univariate time series and include how to deal with the complex variables relationship.
Attention-based method. Attention mechanism excels in in capturing correlations between differ- ent elements, so many works use to model variable relationship: TQNet (Lin al., 2025), iTrans- former (Liu et al. 2024), Leddam (Yu et al. 2024), CARD (Xue et al.) 2024), Crossformer (Zhang) & Yan 2023), Fredformer (Piao et al., 2024) and TimeXer (Wang et al., 2024b). However, for mul- tivariate time series with a relatively large variable number, attention mechanism will require huge time and memory costs.
Cluster-based method. Clustering methods enhance the modeling of variable relationship by clus- tering variables with strong correlations into the same class (Qiu et al.] [2025] Chen et al., (024a). But the process of finding variable similarity in clustering algorithms requires the same quadratic complexity as the attention mechanism, which can be costly.
MLP-based method. MI.P-based method directly ables, including TSMixer (Ekambaram et al. (023) and SOFTS (Han et al. (024).. This method is lighter, but not sufficient to cope with the complex variable relationship.
MLP model correlations between vari-
CNN-based method. CNN-based method uses CNN to model the variable correlation, including Modern (Luo & Wang 2024), which fuses the modeling of variable correlation into the group- ing operation of convolution but requires high number of parameters and training cost.
Mamba- method. Mamba-based method transfers Mamba the time series forecast- ing to delegates the extraction of inter-variate correlations, including TimePro (Ma et al. 2025)
SMamba (Wang et al. 2025c).
Graph-based method. where each node represents each variable and the edges represent correlations between variables, including MTGNN (Wu et al., 2020) and TimeFilter (Hu et al. 2025). Graph can effectively capture the fine-grained variable relationship at each time step, but how to construct and efficiently store the adjacency matrix is challenge. Moreover, graph structure is not very efficient in handling long sequences and usually requires great memory cost.
Graph-based method uses graph structure to model variable correlations,
Unlike the above approaches that explore variable relationships in terms of the whole variable, this paper focuses on highly efficient fine-grained variable dependencies. We capture the variable in- teractions from both the time and frequency domains and then utilize multi-dilated depth- depth-wise wise volution structure to capture the different grained vnter-variable dependencies at a much lower training time and memory cost.
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B IMPLEMENTATION DETAILS
B.1 DATASET DETAILS
The details of the experimental datasets are summarized as follows: (1) ECL (Wu et al. 2021 :
Records hourly electricity consumption of 321 customers from 2012 to 2014. (2) electricity (Salinas iet al. 2020): The electricity dataset is a hourly time series of electricity consumption of 370 cus- tomers. (3) Traffic (Wu et al., 2021): Hourly road occupancy rates from sensors San Francisco Bay area freeways, collected by the California Department of Transportation. (4) Traffic2 (Salinas ct al. 2020): The traffic dataset contains hourly occupancy rates (between 0 and 1) of 963 car lanes of San Francisco bay area freeways. (5) Solar-Energy (Lai et al., 2018): Solar power production
of 137 PV plants in 2006, sampled every 10 minutes. (6) PEMS (Luu al., 2022): Public traffic network data from California, collected in 5-minute intervals, including subsets PEMS03, PEMS04, PEMS07, and PEMS08. (7) PcMSD7(M) (Chen et al., 2001): Data from the Caltrain PeMS system, containing 228 time-series collected at 5-minute intervals. (8) PEMS-BAY (Li et al.) 2018): Traffic data from 325 sensors in the Bay Area, collected over 6 months (Jan 1, 2017, to May 31, 2017) by CalTrans Performance Measurement System (PeMS). (9) MA (Jagadish al. 2014): Traffic data from 207 sensors on Los Angeles County highways, collected over 4 months (Mar 1, 2012, to Jun 30, 2012).
We adopt the same data processing and saiin-vlldation-teee test split protocol from iTransformer (Liu et al. 2024). For long-term forecasting, the lookback length is fixed at 96 for the ECL, Traffic, and Solar-Energy datasets, with prediction lengths of (96, 192, 336, 7203. For short-term forecasting, the lookback length is also fixed 96 for the PEMS03, PEMS04, PEMS07, PEMS08, PeMSD7(M), PEMS-BAY METR-LAY -LA, electricity and Traffic2 datasets, with prediction lengths of {12, 24, 48, 96). Detailed dataset statistics are provided in Table 5
Table 5: Detailed dataset descriptions. Dim denotes the variable number of each dataset. Dataset Size denotes the total number of time points in (Train, Validation, Test) split respectively. Prediction Length denotes the prediction length in each dataset. Frequency denotes the sampling interval of time points.
Dataset
Dim
Prediction Length
Dataset Size
Frequency
Information
ECL
321
(96. 192, 336, 720}
(18317, 2633, 5261)
Hourly
Electricity
Traftic
862
(96, 192, 336, 720)
(12185, 1757, 3509)
Hourly
Transportation
Solar- Belar-Eneryy
137
(96, 192, 336, 720y
(36601, 5161, 10417)
10min
Energy
electricity
370
(12, 24, 48, 96)
(18104, 2518, 5132)
Hourly
Electricity
Traffic2
963
(12, 24, 48, 96)
(7284, 1045, 2101)
Hourly
Transportation
PEMS03
358
(12, 24, 48, 96}
(15617, 5135, 5135)
5min
Transportation
PEMS04
307
(12, 24, 48, 96}
(10172, 3375, 3375)
5min
Transportation
PEMS07
883
(12, 24, 48, 96)
(16911, 5622, 5622)
5min
Transportation
PEMS08
170
(12, 24, 48, 96]
(10690, 3548, 3548)
Smin
Transportation
PcMSD7(M)
228
(12, 24, 48. 96)
(8763, 1256, 2523)
Smin
Transportation
PEMS-BAN BAY
325
(12, 24, 48, 96)
(36374, 5201, 10412)
5min
Transportation
METR-LA
207
{12, 24, 48, 96)
(23883, 3417, 6843)
5min
Transportation
B.2 EXPERIMENTAL DETAILS
All the experiments are implemented in PyTorch (Paszke et al. 2019) and conducted on NVIDIA A800 GPU. Batch size is uniformly set to 32. The training epoch number is set to 15 and training process is early stopped after three epochs if there is no loss degradation on the valid set. We use L2Loss and ADAM optimizer (Kingma & Baj 2015) with an initial learning rate of 0.001. Considering the computational efficiency and the reception field of Inception module, the number of kernels in Inception is set to 4. We employ 5-layer DConvBlock, with the multi-dilated rates
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set 2,,,,1 by default and can be adjusted according to different datasets. The weight factor a is selected from [0.0, 0.1, 0.3, 0.5, 0.7, 0.9, 1.0] to address the varying variable interactions that exist within different datasets. We repeat the experiments five times with different random seeds and report the mean results in the main text. We use the results of all baselines reported in i'Transfommer (Liu et al. (024) their original papers. For models without corresponding results, run their publicly available code using the default parameter setting. The experiment configuration with best hyperparameter is listed in Table and the hyperparameter sensitivity analysis in the Appendix B.3
Table 6: Experiment configuration of FACT. D denotes the hidden dimension. denotes the dilation size, a denotes the weight factor of time domain and frequency domain. denotes the number of kernels in Inception.
Datascts(Configuratioss
D
π
k
ax
dropout
learning rate
batch size
ECL
512
[1,2,3,2.1]
4
0.1
0.5
0.001
32
Traffic
1024
[1,2,3,2.1]
4 0.5
0.1
0.001
32
Solar-Energy
512
[1,2,1]
4 0.5
0.1
0.001
32
electricity
512
[1,2,3,2,1]
4
0.0
0.5
0.001
32
Traffic2
1024
[1,2.3,2,1]
4
0.3
0.1
0.001
32
PEMS03
512
[1,2.322.1]
4
0.3
0.1
0.001
32
PEMS04
512
[1,2,3,2,1]
4
0.3
0.1
0.001
32
PEMS07
512
[1.2,344.3.2.1]]
4
0.3
0.1
0.001
32
PEMS08
512
[1,2,3,2,1]
4
0.0
0.1
0.001
32
PcMSD7(M)
512
[1,2,3,2.1]
4
1.0
0.7
0.001
32
PEMS-BA
512
[1223,433211]]
4
0.1
0.7
0.001
32
METR-LA
512
[1,2.1]
4
0.9
0.7
0.001
32
B.3 HYPERPARAMETER SENSITIVITY
We evaluate the hyperparameter sensitivity FACT with respect to three critical factors: the weight factor a, the number of kernels in the Inception module, the hidden dimension, and the multiple di- lation rates. These experiments are conducted on four datasets with varying characteristics and scale ensure comprehensive analysis. The hidden dimension directly influences the model's capacity and representational power. The number of kernels in the Inception and the dilation rates determine the basic reception field covered in each model layer, while the weight factor controls the weight of variable interactions in time domain and the frequency domain.
The weight factor a As shown in Figure 6 mance due to the different variable interactions in various datasets. In most cases, modeling both time-domain and frequency-domain variable correlations simultancously yields superior results, as it considers both instantancous interactions at specific data points and periodic patterns. However, in certain specific case (PEMS08 dataset), increasing a (i.e., increasing the weight for time domain modeling) will gradually degrade performance. We attribute this to the fact that the variable interac- tions in this dataset primarily involve periodic pattern co-variation, while instantaneous interactions in the time domain may introduce noise. This further underscores the importance of determining the relative weighting between Lime-domann and rrequency-domane variable modeling.
observe that different yield different perfor-
The number of kernels As shown in Figure 7 increasing the number of kernels in the Inception module enlarges the reception field and tends to enhance performance. However, an overly large number of kernels will introduce optimization challenges and more training cost, which negatively impacts overall results. Considering the balance of performance and computational cost, we set the number of kernels to 4 by default.
The hidden dimension Similarly, as shown in Figure g we observe that increasing the hidden dimension generally improves forecasting performance by enabling richer feature extraction. How-
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0.70
0.23.0
PEMSRE
3.70
0.23775
0:000
0.114
1.2150
MISE
1.0000
0.72
0.002
8.008
0.70
1,2000
Figure 6: Hyperparameter sensitivity of a.
1.40
9.220
100
PEMS34
0.113
PCM53
0.110
0.720
3.303
MSR
6:104
3.700
0.833
0.00
0.002
number of kernels
number of kernels
number of kernels
number of kernels
Figure 7: Hyperparameter sensitivity of the number kernels in Inception.
5.730
PRMSSS
3775
8.225
0.310
0.700
8.225
6.115
MSE
6.110
1034
5,000
hidden dimension
hidden dimension
hidden dimension
hidden dimension
Figure 8: Hyperparameter sensitivity of the hidden dimension.
0722
0.111
$.000
6.007
8327
6.09
3.70
0.24
(2222-4)
(2.2444
(2.3445)
(42.4)
33453
(3222-4)
(3.3343)
(342244)
8101
(122224)
(3.34 (22244)
(4224)
35533
multi-dilation
multi-dilation
multi-dilation
multi-dilation
Figure 9: Hyperparameter sensitivity of the multi-dilated rates.
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Table Sensitivity analysis of model layers.
Model layers
Metric
Traffic2
PEMS04
PEMSO8
METR-LA
MSE
0.210
0.091
0.112
0.704
3
MAE
0.182
0.201
0.207
0.456
MSE
0.207
0.088
0.107
0.705
4
MAE
0.180
0.196
0.202
0.458
MSE MAE
0.204
0.086
0.107
0.711
5
0.179
0.194
0.194
0.459
MSE MAE
0.202
0.086
0.107
0.715
6
0.179
0.194
0.198
0.460
MSE
0.201
0.085
0.109
0.729
7
MAE
0.178
0.192
0.200
0.463
ever, excessively large hidden dimensions may cause training instability and convergence difficul- ties, ultimately yielding diminishing returns or even performance degradation. Considering training stability and computational cost, we the hidden dimension 512 by default.
The multi-dilated rates We adopt a palindromic multi-dilated rate configuration with increasing and then decreasing dilation factors 2, 3, 1] the default setting in this paper. This design sures balanced expansion of the reception field across layers: the dilation factors gradually increase and then decrease, preventing either small- or large-scale dependencies from dominating abruptly and reducing the number of total kernels. Consequently, the model is able to capture both local and global dependencies more smoothly, which promotes stable gradient propagation. For comparison, we also evaluate other dilation configurations, including the increasing format ([1, 2, 4, 5)), consis- tent format (33, 3, 3, 3, 3)), srreessing-incraasing palindromic format (33 2, 1, 2)). ) and decreasing palindromic
format (5, 4, 3, 2, I)), as shown in Figure 9
Among these, the
configuration consistently delivers the best overall performance in most cases.
The number of layer We also conduct sensitivity analysis by varying the number of layers and the corresponding dilation rates n, which are directly related to the number layers. Specifically, we evaluate several configurations: 3 layers (n 21 I)). layers (n 1, 1)), 5 layers (n = 1, 3, 1).. 6 layers (n = 2,,2,3, 3,2. 2, 1)), and layers (n = [1, 2,,,,3,4,3 2, 1)). As the number of layers increases from to 7, the receptive field expands, enabling the model to capture broader dependency ranges. In our model, we employ a 5-layer DConvBloce with default multi- dilation rates of [1,2,3,2,1]. The results summarized in Table 7 Increasing the number of layers enlarges the receptive field, and overall, performance tends to improve or remain stable. Different datasets exhibit varying sensitivities receptive field size due to their distinct characteristics. On most datasets, such as Traffic2, PEMS04, and PEMS08, expanding the receptive field by increasing the number of layers leads stable or slightly improved performance, while on METR-LA, further increasing the receptive field beyond certain point results in performance degradation. Therefore, we set the default number of layers to 5.
These findings underscore the importance of carefully selecting an appropriate balance among these hyperparameters to maintain training efficiency and performance.
C ANALYSIS OF DCONVBLOCK
In this section, analyze the key technical components of the DConvBlock in FACT, focusing 2D modeling, multi-dilated convolution, and the choice of Inception as the convolution backbone.
2D modeling Specifically, the 2D modeling strategy converts the original ID variable into 2D form and utilizes 2D convolution, effectively reducing the number of convolution layers required to achieve global reception field and improving training efficiency. To empirically validate this, we replace the 2D modeling with a ID convolution counterpart and report the results Figure IT (denoted as " 1D").. The consistent superiority of the 2D approach confirms that ID convolutions
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difficult to cover the full reception field with equal number of layers, thus limiting performance. Therefore, we employ 2D modeling to achieve better performance using fewer parameters and fewer model layers.
Multi-dilated structure In addition, we employ a multi-dilated convolution structure to further ex- pand the reception field without increasing the number of parameters. For comparison, we evaluate the model without dilation (dilation rate = 1) and present the results in Figure [10 (denoted as "w/o dilation").. The improved performance with multi-dilated convolutions clearly indicates the critical role of larger reception field in capturing variable interactions.
Inception standard convolution In DConvBlock, we select the Inception the convolution backbone. To demonstrate that our model's effectiveness derives primarily from the architecture rather than specific choice of backbone, we substitute Inception with a standard 2D convolution structure and evaluate the performance, as shown in Figure T0 (denoted as "+Conv"). Although the general convolution backbone achieves competitive results, performs slightly worse than In- ception. This suggests that the multi-dilated depth-wise design is major factor contributing to the superior performance of FACT. More importantly, this result confirms the flexibility and ex- tensibility of our framework, showing that it can be seamlessly integrated with various generalized convolution networks. This adaptability significantly expands the potential of convolution architec- tures in time series forecasting. Mcanwhile, Inception and standard 2D convolution to justify the additional training cost. As shown in Table R the results demonstrate that FACT with the Inception module will introduce only marginal increase in parameters and FLOPs. Since each kernel at each channel has only x H x W parameters, the Inception module increases the number of kernel elements only slightly compared standard 2D convolution. This represents a worthwhile trade-off for improved effectiveness.
also compare the parameter count and FLOPs of
PEMIS4
NETR.LA
3.0075
0.10
MSE
3.0000
6.715
6.710
0.13
6.79
PCT
-10
stanoo
INC
+10
alotton
caol
P6CT
+10
alsson
PACT
+10
dtosos
4aml
Traffit
K0L
Selan
0.01
0.203
0.2075
0.200
2000
3.3005
0.156
3000
8.160
0.197
1.1775
8.158
1.42
1.000
0.184
PATT
-10
dattio
RACT
+10
diattst
PACT
+10
1007
FACT
+10
diaton
Figure 10: Analysis of DConvBlock on different datasets.
Table 8: Parameter and FLOPs comparison of FACT with Inception and standard convolution.
Model
Metric
PEMS03
PEMS08
METR-LA
ECL
Traffic
Solar
parameter (M)
21.94
11.07
3.63
31.05
21.98
3.63
FACT
FLOPS (G)
505.98
138.34
52.08
715.74
1263.74
33.34
parameter (M)
21.59
10.89
3.47
30.43
21.63
3.47
+ Conv
FLOPs (G)
498.52
136.32
50.02
702.46
1225.12
32.02
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Table 9: Analysis of different frequency modeling methods.
Dasasets
PEMSU3
PEM504
PEMSOS
METR-LA
Trifli22
FCT.
Traffic
Solar-Fnergy
Metric
MSE
MAE
MSE
MAB
MSE
MAF
MSF
MAE
MSE
MAF
MSE
MAF
MSF
MAE
MSE
MAR
12
0.058
0.159
0.067
0.169
0.068
0.167
0.425
0.324
0.181
0.165
96
0.131
0.227
0.395
0.261
0.115
0.230
FACT
24
0.072
0.084
0.205
0.178
0,093
0,078
0.202
0.185
0.086
0,117
0.211
0.185
0.601
0.791
0.404
0.503
0.196
0.215
0.173
0.185
339
192
0.149
0.163
0.260
0.264
0.430
0.413
0.77
0.270
0.196
0.197
0.265
0.259
0.123
0.236
0.110
0.222
0,160
0.238
0.998
0.596
0.228
0.195
720
0.188
0.266
0.461
0.296
0.214
0.272
0.086
0,196
0.087
0.194
0.107
0.200
0.704
0.456
0.204
0.179
ang
0.157
0.254
0.425
0.776
0.196
0.254
0.059
0,162
0,068
0.170
0.167
0.181
0.315
0.181
0.164
0.135
0.252
0.393
0.62
0.180
0.238
concat
0.207
0.178
0.095
0.07%
0.202
0.201
00.218
0.192
0.401
0.517
0.212
0.196
0.261
0.252
0.431
0.414
0.201
0.210
0.380
0.258
0.124
0.239
0.112
0.215
0.230
0.605
0.229
0.196
0,284
0,466
0,212
0.269
avg.
0.196
0.018
0.196
0..005
0.459
0.204
Avg
0.161
0.257
0.446
00:000
0.25%
0.081
0.348
0.210
0.376
0.584
0.620
0.352
0.510
0.330
0.226
0.341
0.454
0.270
0.409
share
0.318
0.166
0.138
1.263
0.956
0.809
0.807
0.404
0.298
0.331
0.197
0.270
0.430
0.387
0.180
0.325
00.155
0.286
1.157
03.881
1,002
0.600
00:331
0.525
0.343
0.439
0,550
0,400
0.328
0.447
ayg
0.346
0.167
0.927
0.779
0.461
0,310
0.316
awg
0.372
0.515
0.367
0.299
0.418
0.067
0.070
0.171
0.310
0.185
0.389
0.182
0.241
complex
0.176
0.205
011778
0.094
0,206
0.089
0,118
0.218
0:191
0,500
0.533
0.259
0:199
0.214
0.182
0.133
066
0.265
0.246
0.411
0.431
0267
0.255
0.199
0.204
0.280
0.363
0.276
0.111
0.234
0.160
0.348
1.013
0.600
0.229
0.192
0.207
0.802
0.465
0.704
0.221
0.350
aeg
| 0.086
0.194
010087
0.195
[ 0.109
0.207
0.711
0.460
0.206
0.176
| avg
| 0.165
0.260
0.423
0.273
[ 0.201
0.360
0.058
0.185
0.067
0. 0.169
0.072
0.172
0.424
0.323
0.184
0.134
0.228
0.389
0.257
0.174
0.234
+ real-imug
48
0.071
0.093
0.176
0.205
0.093
0.077
0.201
0.181
0.092
0,121
0.22]
0.193
0.500
0.508
0.507
0.399
0.215
0:198
0.181
0.168
339
0151
0.168
0.263
0.244
0.413
0.430
0.275
0.207
0.207
0.265
0.360
96
0.122
0.236
0.112
0.225
0.165
0.250
1.001
0.595
0.229
0.191
720
0.189
0.286
0.470
0.291
0.204
0.263
avg
0.086
0.193
0.087
0.195
0.112
0,209
0.706
0.456
0.206
0.175
avg
| 0.160
0.255
0.425
0.273
0.198
0.255
D ANALYSIS OF FREQUENCY MODELING
Frequency modeling methods this section, we analyze our frequency-domain modeling ap- proach, which captures variable interactions by separately modeling the amplitude and phase of fre- quency components. We compare it with other four representative alternatives: parameter sharing employs the same model parameters for both the amplitude and phase; concatenation concatenates the amplitude and phase parts before applying a unified model; complex convolution operates di- rectly in the complex domain, where weights, inputs, and outputs are complex valued; real-imag models the real and imaginary parts separately rather than amplitude and phase. Table shows the results of performance comparison on diverse datasets. Although complex convolution and real- imag perform competitively, our separate modeling of amplitude and phase can achieve the better performance across most datasets. To further assess the practical benefits of the amplitude and phase modeling over the Real/Imag alternative, we conducted a convergence analysis under identical pa- rameter settings. While the Amplitude/Phase modeling introduces additional non-linear operations (arctan, cos, sin), it does not increase the number of trainable parameters. We evaluated both meth- ods on four datasets and visualized their test performance throughout training. As shown in Fig- ure two limitations of the Real/Imag alternative are evident: (1) it often fails to converge to the global optimum, negatively impacting final performance; and (2) even when near-optimal solutions reached, Real/Imag exhibits greater instability and larger performance fluctuations compared to Amplitude/Phase. Therefore, we adopt the Amplitude/Phase modeling in practice.
Sea
Anp RacaPtate
RackPiaae
NGC
0.0
spoch
ECL dataset.
METR-LA dataset.
PEMS08 dataset.
PEMS-BAY dataset.
Figure 11: Test performance comparison with real-imag method in the training process.
Compared with other frequency modeling models To further evaluate the effectiveness of our proposed frequency modeling design, we conduct a comprehensive comparison with several repre- sentative frequency- based models, including FreDF (Wang et al. 2025a), FreTS (Yi et al.) 2023), FilterNet (Yi et al.) 2024). FITS (Xu et al., (024)) and Fredformer (Piao et al. 2024). all of which well-known for capturing frequency-domain patterns in time series data. FreDF introduced fre-
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Fropy Prats
Prodformer
PITS
HEMSR3
PEMSIA
Horson
METALL
Butic
Sinu Energy
Datasets
Figure 12: Compared with other frequency- based models.
quency forecast error loss, FreTS employed frequency MLP learn temporal dependencies and channel dependencies, FilterNet designed frequency filters to extract key informative temporal pat- terns by selectively passing or attenuating certain components of time series signals, FITS argued that time series can be manipulated through interpolation in the complex frequency domain with few parameters, and Fredformer proposed use channel-wise attention frequency domain to model the channel dependencies. In contrast, our FACT focuses on capturing fine-gras fine-grained variable interactions by modeling the distinct dependencies variables at each frequency components. As shown in Figure 12 FACT consistently outperforms these models across all datasets. This consis- tent superiority not only demonstrates the effectiveness of our frequency modeling approach but also underscores the advantage of explicitly capturing fine tne-grained cross variable interactions from the rreueencydommii perspective.
E
MODEL ANALYSIS
E
EMBEDDING ANALYSIS
Given multivariate time series X RCXT consisting O variables, first project each length-T' variable into nigh-dimensional embedding space of dimension D) to get its length-D embedding, resulting in X, E CR We then apply the Discrete Fourier Transform (DFT) to X, along the embedding dimension D capture the variable interactions in frequency domain.
important to note that the embedding is computed on a per-variable basis; specifically, each T-length variable is projected into a D-length embedding vector, ensuring that temporal informa- tion implicitly preserved within the embedding space. Compared to applying DFT directly on X along the temporal dimension T, our approach introduces learnable projection step. This pro- vides the flcxibility to adapt to complex temporal patterns within a high-dimensional latent space, yielding more expressive representations for subsequent frequency modeling. Meanwhile, compared FACT against a variant that applies DFT directly along the temporal dimension T of the original series X without embedding. The results, summarized in Table [10] show that FACT sistently outperforms the direct DFT variant without embedding across all datasets, validating the effectiveness of our design choice.
also
Table 10: Comparison of FACT with embedding and the variant applying DFT directly on the original series without embedding.
Model
Metric
PEMS03
PEMS04
PEMS08
METR-LA
Traffic2
ECL
Traffic
Solar
MSE
0.086
0.087
0.107
0,704
0.204
0.157
0.425
0.196
FACT
MAE
0.194
0.194
0.200
0.456
0.179
0.254
0.276
0.254
MSE MAE
0.101
0.102
0.143
0.724
0.216
0.177
0.467
0.203
w/o Embedding
0.213
0.216
0.236
0.475
0.200
0.274
0.312
0.269
E.2 2D VARIABLE RECONSTRUCTION STRATEGIES
For a multivariate time series with C variables, we reshape the input into a 2D grid size x W using row-major order, preserving the original sequence of variables provided in the input.
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Concretely, the first W variables are placed in the first row, the next W in the second row, and so on. C' is not divisible by W, we pad the last row by repeating the first few variables from the beginning of the sequence (former-padding
to complete the H W grid.
Physical topology We do not assume any prior knowledge about the order among the C variables, such physical adjacency, when reshaping them into an H W grid for 2D variable reconstruc- tion. Our method is designed for time series data, regardless whether the variables have inherent physical topology not. This design choice allows our method to be broadly applicable to various multivariate time series forecasting tasks without being constrained by the need for spatial informa- tion.
Meanwhile, we conduct experiments to evaluate the impact of incorporating physical topology by comparing with (1) Clustering- layout, where variables are grouped by correlation strength in ID and then reshaped into 2D grid, positioning highly correlated variables closer together; (2) MDS-based layout, where Multidimensional Scaling (MDS) projects variables into 2D space, preserving correlation-based proximity. While clustering-based layouts partially capture topology, MDS-based layouts offer more rigorous representation. The results Table [1] show that performance of these two variant layouts is comparable to our original FACT model, validating that our model does not rely on physical topology for effective forecasting, and thus is generalizable to various datasets without inherent variable order.
Table 11: Comparison of incorporating physical topology.
Model
Metric
PEMS03
PEMS04
PEMS08
METR-LA
Traflic2
PEMSD7
PEMS-BAY
MSE
0.086
0.087
0.107
0.704
0.204
0.330
0.372
FACT
MAE
0.194
0.194
0.200
0.456
0.179
0.316
0.267
MSE 0.087
0.085
0.113
0.717
0.204
0.328
0.370
+ Cluster
MAE
0.195
0.191
0.204
0.458
0.179
0.315
0.271
MSE
0.086
0.085
0.106
0.699
0.203
0.332
0.371
MDS
MAE
0.194
0.192
0.198
0.457
0.179
0.315
0.271
Variable permutation Moreover, to assess the effect variable permutation, we compare model performance when the variables are randomly shuffled (e.g., swapping variable 5 and variable 50). Additionally, to further demonstrate the robustness of our approach, we evaluate several alternative ID-to-2D variable reshaping methods, including column-mann major. Z-order, snake, spiral, and Hilbert curve layouts. The results (summarized in Table 12) show that random permutation variable order have minimal impact model performance, indicating that method remains stable and effective even when variable indices are swapped. Similarly, performance differences across various 2D reshaping methods negligible, further validating that our approach does not rely on specific variable adjacency or arrangement for accurate forecasting.
Table 12: Comparison of different I ID-to-2D variable reshaping methods.
Model
Metric
PEMS03
PEMS04
PRMS08
METR-LA
Traffic2
PEMSD7
PEMS-BAY
MSE
0.086
0.087
0.107
0.704
0.204
0.330
0.372
FACT (row-major)
MAE
0.194
0.194
0.200
0.456
0.179
0.316
0.267
MSE
0.087
0.088
0.110
0.706
0.206
0.331
0.369
random
MAE
0.196
0.196
0.202
0.458
0.180
0.336
0.270
MSE
0.086
0.086
0.106
0.709
0.204
0.329
0.371
+ column-major
MAE
0.194
0.194
0.201
0.457
0.179
0.315
0.268
0.087
0.086
0.107
0.706
0.206
0.330
0.377
z--order
MAE
0.194
0.202
0.456
0.180
0.316
0.270
MSE
0.087
0.087
0.104
0.707
0.205
0.330
0.369
snake
MAE
0.194
0.195
0.199
0.456
0.179
0.316
0.267
MSE 0.086
0.087
0.108
0.705
0,206
0.332
0.375
spiral
MAE
0.195
0.195
0.201
0.454
0.1
0.317
0.270
MSE 0.086
0.087
0.105
0.715
0.205
0.332
0,380
hilhert
MAF
0.194
0.195
0.197
0.454
0.1
0.318
0.271
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Padding strategies Additionally, evaluated the impact of different padding strategies when C not divisible by W, with results summarized in Table 13 Specifically, we compared Latter- padding, which pads the grid by repeating the last few variables, and Random-padding, which pads by randomly selecting variables from the original C variables. The results demonstrate that these padding methods yield comparable performance, indicating that the choice of padding strategy does not significantly affect model effectiveness in this setting.
Table 13: Comparison of different padding methods.
Model
Metric
PEMS03
PEMS04
PEMS08
METR-LA
Traflic2
BCL
Traflic
Solar
FACT (former-padding
MSE
0.086
0.087
0.107
0.704
0.204
0.157
0,425
0.196
MAE
0.194
0.194
0.200
0.456
0.179
0.254
0.276
0.254
I Latter-paddinn
MSE 0.086
0.086
0.106
0.709
0.205
0.158
0.428
0.197
MAE
0.194
0.194
0.202
0.457
0.179
0.255
0.276
0.256
MSR MAE
0.087
0.087
0.105
0.707
0.203
0.158
0.427
0.198
Candom-paddingg
0.195
0.195
0.199
0.457
0.180
0.256
0.277
0.257
E.3 ENHANCING OTHER FORECASTING BASELINES
To demonstrate the effectiveness of the multi-dilated depth-wise Inception module (MDIInception) in capturing variable interactions, we integrate this module into several representative baselines: the variable-dependent TimesNet, the variable-independent PatchTST, and the attention- based iTrans- former. The results, presented in Figure [3] clearly show that MDInception significantly enhances the performance of all evaluated baselines. Notably, the improvement is most pronounced for PatchTST, which treats multivariate time series as multiple independent univariate series and does not explicitly model variable relationships. Incorporating MDInception into PatchTST leads substantial reduction in Mean Squared Error (MSH) by 15% to 30% across most datasets. These findings validate that the multi-dilated depth-wise Inception module effectively captures dynamic and fine-grained interactions among variables at each granularity. By leveraging fine-graines infor- mation from other variables, MDInception enriches variable representations, thereby substantially boosting the overall predictive capability of diverse multivariate time series models.
Datasets
Datasets
Datasets
Figure 13: Performance improvements of plugging in the multi-dilated depth-wise Inception module into other models.
E.4 COMPARISON WITH GNN BASELINES.
We also conduct experiments to compare with several representative GNN models: MTGNN (Wu al., 2020), DCRNN (Li et al. 2018), AGCRN (Bai al. 2020 Graph WaveNet (Wu al. 2019). STGCN (Yu et al., 2017), and the latest TimeFilter (Hu et al. 2025). As shown in Table 14 we can note that our method achieves superior performance compared to all these GNN baselines, includ- ing the latest strong GNN model TimeFilter. These results indicate that approach is effective even when compared against latest GNN models, and validates the effectiveness of our proposed techniques in the paper.
E.5 COMPARED WITH TIMEFILTER AND MODERNTCN
We also compare the training cost of our FACT with TimeFilter (Hu et al., 2025) and Mod- emTCN (Luo & Wang. 2024). TimeFilter employs graph-based structures capture interactions
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Table 14: Compared with other GNN baselines.
Model
Metric
PEMS03
PEMS04
PEMS07
PEMS08
METR-LA
PEMSD7
PEMS-BAY
MSE
0.086
0.0887
0.073
0.107
0.704
0.330
0.372
FACT
MAE
0.194
0.194
0.170
0.200
0,456
0.316
0.267
MSE
0.273
0.243
0.479
0.542
0.915
0.644
0.704
DCRNN
MAE
0,380
0.360
0.358
0.426
0.602
0.529
0.479
0.186
0.184
0.159
0.261
0,884
0.443
0.499
Gruph WaveNet
MAE
0.289
0.290
0.266
0.304
0.551
0.392
0.345
MSE
0.198
0.195
0.174
0,294
0.895
0.532
0.609
MTGNN
MAE
0.298
0,302
0,281
0.327
0.577
0.439
0,393
MSE
0.157
0.150
0.177
0.266
0.748
0.442
0.510
AGCRN
MAE
0.269
0.267
0.275
0.292
0,501
0.372
0.360
MSE
0.101
0.092
0.094
0.208
0.732
0.343
0.396
STGCN
MAE
0.208
0.203
0.182
0.240
0,473
0.321
0.291
0.108
0.114
0.101
0.139
0,819
0.498
0.547
TimeFilter
MAE
0.217
0.226
0.207
0,230
0.493
0.404
0.358
among variables, thereby enabling fine-grained modeling. ModernTCN also adopts convolution neural network backbone extract inter-variable correlations. In Table [ and Table [ we demon- strated that the superior performance of our FACT compared with TimeFilter and ModernTCN. Here, we further compare the training costs of them to provide comprehensive evaluation.
As illustrated in Figure 14| our proposed FACT consistently demonstrates superior computational efficiency across all evaluated datasets. This advantage holds for datasets with relatively fewer variables, such Solar-Energy and METR-LA, as well as for large-scale datasets with substantially more variables, including PEMS07, Traffic, and Traffic2. The consistent performance and efficiency across different scales of variable dimensionality highlights the robustness and scalability of FACT handling both small- and large-scale multivariate forecasting tasks.
35
TimeF Btor
TimaFilter
ModernTCN
ModernTCN
250
Memory
Time(s)
20
100
PEMS03
PEMS07
PENS-BAY
METR-LA
ECL
Traffic
Solar
Traffi22
PEMS03
PEM507
PEMS-aM BAY
METR-LA
ECL
Traffic
Solar
Traffi2
Datasets
Datasets
(a) Time cost.
(b) Memory cost.
Figure 14: Efficiency comparison with TimeFilter and Modern
E.6 DEPTH-WISH CONVOLUTION WITH EXPLICIT MODELING BETWEEN GRANULARITIES
In depth-wise architecture, there are often alternative approaches to modeling the relationships among different channels, such as Squeeze-and- fore, we also evaluated our method FACT with its variant incorporating the SENet block. The results, summarized Table 15 show that FACT with and without the SENet block achieve com- parable performance across all datasets. We attribute this to point-wise FFN module following the depth-wiss wise Inception block, which already effectively models relationships between channels. Consequently, adding the SENet block does not yield additional performance gains.
(SENet) (Hu et al., 2018) block. There-
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Table 15: Compared with combining SENet block.
Model
Metric
PEMS03
PEMS04
PEMS08
METR-LA
Traffic2
ECL
Traffic
Solar
PEMSD7
PEMS-BAY
0.086
0.204
0.425
0.330
FACT
MAE
0.194
0.194
0.200
0,456
0.179
0.254
0.276
0.254
0.316
0.267
MSE
0.088
0.086
0.106
0.718
0.201
0.158
0.430
0.199
0,330
0,371
SENet
MAE
0.196
0.202
0.473
0.178
0.277
0.261
0.270
E.7 LONGER INPUT LENGTH
Longer input lengths provide models with more historical information, which, when effectively leveraged, can significantly improve forecasting accuracy. To comprehensively evaluate the ability of FACT to utilize extended historical context, conduct experiments varying the input sequence lengths over the set {96, 192, 336, 576, 7203. For fair comparison, we select two benchmark datasets: ECL and Traffic. We also include six recent state-of-the-arr baselines that employ different strategies for modeling variable relationships, including the CNN-based ModernTNN, attention- based I'Transfomme, graph-based TimeFilter, Mamba-based TimePro, variable-dependent TimesNet, PatchTST.
and muriable-independd
Figures 15 depict the forecasting performance of all models under varying lookback lengths on ECL. and Traffic datasets. Our results demonstrate that FACT consistently achieves state-of-the-an aceuracy across all input lengths on both datasets. This stability and robustness indicate that FACT effectively leverages extended historical information to improve prediction quality, outperforming other models regardless of the amount of past data available. These findings validate the strong capability of FACT in capturing complex, long-range dependencies inherent in multivariate time series, making it highly suitable for real-world forecasting tasks where the length of historical data can vary significantly.
NatshTST
maraformer
Tess Timcto
FACT
munafamer
Trmahitt
Muder
Timefibar
ModamICN
Tieeho
0.23
0.65
0.22
0.60
0.21
M5R
0.55
0.20
Input length
Input length
Figure 15: Performance comparison with longer input length on ECL dataset (left) and Traffic dataset (right).
F THE USE OF LARGE LANGUAGE MODELS (LLMS)
Large Language Models (LLMs) are only used to correct grammatical errors and to polish the syntax of the article.
G VISUALIZATION
G.1 ACTIVATION MAPS VISUALIZATION
To further enhance interpretability and verify whether the model truly captures meaningful variable interactions, we provide the visualization analysis.
Specifically, we visualize the overall correlation matrix of all variables the time domain alongside the corresponding activation map of our time-domail convolution, well as the variable amplitude- correlation matrix and variable correlation matrix in the frequency domain together with the activation maps of amplitude and phase interactions from our frequency-domain convolution. We
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include three representative datasets and report the results in Figure [ a), Figure Figure 18 a). These visualizations show that variable pairs with strong correlations exhibit strong responses in the learned activation maps, while weakly correlated pairs show weak activation responses. This demonstrates that our model effectively captures cross-variable interactions, even though the 2D grid physical topology. These results provide direct evidence for the interpretability
is not tied and reliability of our variable interaction modeling, and further show that our model can perform multivariate time series forecasting on real-world datasets without requiring additional information about physical topology among variables.
Further, we also visualize variable correlation and activation maps across multiple granularities, in- cluding different channels in the time domain and each frequencies in the frequency domain. At each granularity, we present the variable correlation matrix the time domain alongside the correspond- ing activation map, and the amplitude-correlan domain together with activation maps capturing amplitude and phase interactions. For clearer il- lustration, we include three different datasets, two different granularities and ten random variables for visualization. The results are reported in Figure T6(b), Figure [7b), Figure 18 b). Our findings show that, for the same dataset, variable correlations which aligns with our model design that explicitly captures dynamic fine- grained variable interac- tions. Moreover, within each granularity, strongly correlated variables consistently exhibit higher activation responses, while weakly correlated variables show weaker responses, both in the time and frequency domains. This consistent pattern demonstrates that our convolution structure effectively captures fine-grate aine-grained cross cross-variaae variable interactions.
and phase-correlation matrices in the frequency
differ substantially across granularities,
Correlations and the corresponding activation maps of all variables.
Vhrivale Commiation
Arthaaion in the domain
Ane Purde Conwartco Artiadio Pute
Phaae Commation
Arthuationn
000
(b) Correlations and activation maps of two granularities (the first line and second line).
Figure 16: The visualizaiton of variable correlation and activation maps on PEMS-BA dataset.
G.2 SHOWCASES
To further evaluate the forecasting performance and illustrate the effectiveness of our model, we present representative prediction showcases from the test dataset. We compare the predictions of FACT against several strong baselines, including CycleNet, TimePro, TimeFilter, i'Transfrmeer, ModernTCN. TimesNet and PatchTST, on several datasets. These baselines adopt a variety of strate- gies capture vnter-variable relationships, which include attention-based, graph-based, convolution- based, Mamba-based, variable dependent-based, and variable independent- methods, thus pro- viding comprehensive and diverse comparison.
Figures 20 Figures 21 Figures 22 Figures 23 and Figures 24 FACT As illustrated in Figures 19
demonstrates superior ability to fit the overall trends of the time series and effectively captures temporal patterns and changes across all datasets. This highlights the model's 's capability to deliver more accurate and stable predictions compared to existing approaches.
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Variadle Combitton
Acthoaton In the coman
Anp huss Conwancon
Activatin hide
PlA Commaatonn
Antarro Pracs
K00
(a) Correlations and the corresponding activation maps of all variables.
Viriaale Combito
Artiatiin In the conann
Are Putk Canwation Actiation Ann
Phage Comiition
Artlamne
(b) Correlations and activation maps of two granularities (the first line and second line).
Figure 17: The visualizaiton of variable correlation and activation maps on PEMS03 dataset.
Vhrivide Combaic
Artiuation
K5
(a) Correlations and the corresponding activation maps of all variables.
Mertodt Conclttonn istiodue Coreati
Actettinn Ampiute
(b) Correlations and activation maps of two granularities (the first line and second line).
Figure 18: The visualizaiton of variable correlation and activation maps on METR-LA dataset.
100
2.50
300
400
6
150
200
300
400
1.50
200
300
400
1:00
200
300
400
(a) FACT
(b) TimePro
(c) iTransformer
(d) TimeFilter
OrundRatth
GroundPuut
Grunnthutt
GroundThutt
-2
(e) CycleNet
ModernTCN
(g) PatchTST
(h) TimesNet
Figure 19: Showcases ECL dataset. The lookback length is 96 and prediction length is 336.
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-1-
-2
GoundPaa7
Gourchuat
Gourdhuth
-2
GroundPuuh
Fresitiion
Fresccin
200
202
50
202
(a) FACT
(b) TimePro
(c) iTransformer
(d) TimeFilter
-1
GoundPu27
GrondeP447
Grouns Prsiicion
Groundhuuth
Prosittio
Prskitiin
50
300
15g
260
300
150
203
50
300
150
203
50
300
150
200
(e) CycleNet
(f) ModernTCN
(g) PatchTST
(h) TimesNet
Figure 20: Showcases on METR-LA dataset. The lookback length 96 and prediction length 96.
100
12
Soundhnd
0.5
0.5
0.5
Fedittin
mediction
Pirdition
.05
1.0
50
150
150
10-
50
150
1.2
160
150
1.0
50
Xdc
150
(a) FACT
(b) TimePro
(c) iTransformer
(d) TimeFilter
1.2
0.5
0.3
0.0
0.2
0.0
(e) CycleNet
ModernTCN
(g) PatchTST
(h) TimesNet
Figure 21: Showcases on Traffi22 dataset. The lookback length is 96 and prediction length is 48.
160
150
200
150
200
50
200
50
100
150
260
(a) FACT
(b) TimePro
(c) iTransformer
TimeFilter
Gruuddrumm
fmuuddum
Smandhum
Preductonn
Preditionn
(e) CycleNet
(f) ModernTCN
(g) PStchTST
(h) TimesNet
Figure 22: Showcases PEMS03 dataset. The lookback length is 96 and prediction length is 96.
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10
15
Croundttath
65
os:
0.5-
0.0
ns-
ea-
0.-
005
-05
-05-
-0.-
-10
-19
-1a
-10
15
1.5
W
W
1m/m///
(a) FACT
(b) TimePro
(c) iTransformer
(d) TimeFilter
Groundhaath
Gruundhuth
Groundhuuh
Groundhuuth Prediction
medcconn
coon
30
120
150
30
120
150
30
100
130
50
300
130
(c) CycleNet
(f) ModernTCN
(g) PatchTST
(h) TimesNet
Figure 23: Showcases on PEMS04 dataset. The lookback length is 96 and prediction length is 48.
Grurdhum
Sranndhum
Granehhm
Pradition
(a) FACT
(b) TimePro
(c) i'Transformer
(d) TimeFilter
Grurdfuu
Groundhvm Peedktton
Sroundhum
Grandhuum teadution
Prdiution
tradation
100
150
200
700
50
100
(e) CycleNet
(f) ModernTCN
(g) PStchTST
(h) TimesNet
Figure 24: Showcases on PEMS07 dataset. The lookback length is 96 and prediction length is 96.
30


Reviews:
[{"review_id": "5m7HnkwP9L", "reviewer": "Reviewer_hCV9", "summary": "This paper proposes FACT, a CNN-based architecture for multivariate time-series forecasting that explicitly models fine-grained inter-variable interactions in both the time and frequency domains. The core module, DConvBlock, uses multi-dilated depth-wise Inception-style 2D convolutions to capture variable interactions at each granularity while keeping computation low. Variables are reshaped from 1D into a 2D grid to enlarge the receptive field with fewer layers; frequency modeling is performed in amplitude–phase form and fused with the time-domain pathway via a learnable weight. Across 12 benchmarks, FACT reports state-of-the-art accuracy and notably reduced training time and memory versus attention, especially on high-dimensional datasets.", "questions": "1. Is the DFT computed along the hidden/embedding dimension D or along time T? If along D, how should we interpret frequency content in that space, and did you compare to a time-axis DFT per variable?\n2. How are variables mapped to the H×W grid (fixed order, padding strategy)? Did you try learned permutations or multiple random layouts with ensembling? Any evidence that adjacency choices matter?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. The proposed method has valid design—depth-wise, multi-dilated 2D convolutions per granularity plus an amplitude–phase frequency path, fused with learnable weight. The 2D reshape argument is analytically motivated (layer-count reduction).\n2. The proposed method shows consistent gains on extensive benchmarks; attention comparison shows lower computations on large datasets; ablations isolate contributions of Inception and FFN and the value of dual-domain modeling.\n3. Contributions are explicitly listed; the pipeline and fusion formulae are precise. Efficiency improvements matter for high-dimensional multivariate forecasting, making the approach practical.", "weaknesses": "1. The DFT is applied to the embedding dimension (not clearly the temporal axis). The physical meaning and potential pitfalls (e.g., phase wrapping, noise sensitivity, leakage) are not deeply analyzed. A comparison to performing frequency modeling strictly along time (per variable) would help.\n2. Reshaping C variables to an H×W grid makes spatial adjacency arbitrary; while dilations broaden coverage, different permutations/layouts might change outcomes. A learned layout or permutation-invariant design could mitigate inductive-bias risks.\n3. The components (depth-wise/dilated convs, Inception-style branching, frequency cues) are each established; the paper’s novelty is primarily in the combination and the dual-domain fine-grained framing. A clearer positioning against ModernTCN/frequency-aware models would help.", "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": "lr01iA4uAQ", "reviewer": "Reviewer_9MyL", "summary": "This paper proposes FACT, a novel convolutional architecture for multivariate time-series forecasting. The model aims to capture fine-grained, time-varying interactions between variables. Its core contributions include: 1) a depth-wise convolution module named DConvBlock to capture dynamic interactions; 2) a strategy of reshaping the 1D variable list into a 2D grid to leverage efficient 2D convolutions; and 3) a dual-domain modeling approach combining time and frequency domains. The authors conduct extensive experiments on 12 benchmark datasets, claiming state-of-the-art (SOTA) performance in both prediction accuracy and computational efficiency.\n\nOn the positive side: Empirically, it achieves SOTA results on 12 benchmarks, which is a strong contribution. Furthermore, its significant improvement in computational efficiency (compared to the $O(N^2)$ complexity of Transformers) holds high practical value and appeal.\nOn the debatable side: The theoretical depth of its core methodology (2D variable reconstruction) needs strengthening. The long-term contribution and guiding effect of this strategy—which trades physical priors for efficiency—on the spatio-temporal data mining field are worthy of further discussion.", "questions": "1.Could the authors further elaborate on the rationale for the \"2D variable reconstruction\" from a theoretical perspective (beyond just \"it works empirically\")? Why would an arbitrary, index-based grid be a better representation for variable relationships than the original 1D list or a true physical graph (as used by GNNs)?\n\n2.Why were GNN models omitted from the baseline comparison? Given that PEMS and METR-LA are standard GNN benchmarks, we strongly suggest the authors add comparisons against mainstream GNN models to truly substantiate the SOTA claim.\n3.Have the authors experimented with other reshaping methods, such as clustering-based reshaping, or projecting the sensors' true physical coordinates (latitude/longitude) onto a 2D space and then applying 2D convolution on this physically meaningful 2D image? This would at least preserve spatial locality.\n\n4.Permutation Sensitivity: The 2D reconstruction strategy seems dependent on the original index order of variables in the dataset. If the order of variables is permuted (e.g., swapping variable 5 and variable 50), would the model's performance change significantly?\n\n5.Generality of GNNs vs. Necessity of Topology: The authors mention GNNs in the related work but seem to avoid them in comparisons. Do the authors perceive GNN models (which require an adjacency matrix) as lacking \"generality,\" whereas FACT (which does not) is more general? If so, does FACT's SOTA performance on these spatio-temporal datasets imply a deeper conclusion: that physical-topological priors are not necessary for forecasting in these specific benchmarks, or perhaps that their importance has been overestimated?\n\n6.Physical Interpretation of 2D Reshaping in Frequency Domain: For frequency domain modeling, the variable dimension C is similarly reshaped into $\\sqrt{C} \\times \\sqrt{C}$. This operation seems even more abstract. In the time domain, \"variable 1\" and \"variable 11\" becoming neighbors is debatable; in the frequency domain, what is the physical meaning of applying a 2D convolution to the \"amplitude/phase of variable 1\" and the \"amplitude/phase of variable 11\"? How does the model learn meaningful patterns in such a highly abstract 2D frequency space?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "1.SOTA Empirical Performance: The most significant strength is that the paper comprehensively outperforms 12 recent models, including iTransformer, across 12 standard benchmarks. This is a very solid engineering achievement.\n2.High Computational Efficiency: Compared to the $O(N^2)$ complexity of attention-based models (where N is the number of variables), the proposed convolutional architecture (especially after 2D reconstruction) offers a substantial advantage for high-dimensional variables. The reported reductions in training time and memory consumption (up to 50%) are highly attractive practical features.\n3.Clever Design of DConvBlock: The use of depth-wise convolution to assign a dedicated kernel for each time/frequency granularity, as a means to capture \"dynamic interactions,\" is a lightweight and effective engineering design.\n4.Thorough Ablation Studies: The authors have conducted detailed ablation studies that demonstrate the necessity of the model's various components (e.g., dual-domain modeling, multi-dilated convolution).", "weaknesses": "1.Questionable Rationale of 2D Variable Reconstruction: A primary concern is the \"2D variable reconstruction\" strategy. This index-based reshaping appears to lack prior support. This operation could potentially place physically unrelated variables adjacent to each other while distancing physically adjacent ones. This approach may not fully leverage the valuable physical-topological priors inherent in spatio-temporal data.\n\n2.Omission of GNN Baselines: This is a noticeable omission. The paper claims SOTA on several datasets with clear spatio-temporal structures (e.g., PEMS, METR-LA, Traffic) but fails to compare against Graph Neural Network (GNN) models specifically designed for such data (e.g., DCRNN, Graph WaveNet, MTGNN). GNNs are a standard baseline in spatio-temporal forecasting, and lacking this comparison somewhat weakens the convincingness of the SOTA claims on these specific datasets.\n\n3.Insufficient Theoretical Discussion: As mentioned, the paper does not provide adequate theoretical justification for the \"2D variable reconstruction.\" It relies heavily on empirical results for validation but lacks a deeper theoretical exploration of why this artificial pseudo-topology is effective, which is a slight pity for an ICLR paper.\n\n4.Interpretability Needs Improvement: As the 2D grid is an artificially constructed \"pseudo-space,\" the weights of the 2D convolution kernels may be difficult to map directly to real-world physical interactions, posing a challenge to the model's interpretability.", "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": "72VysQbBbu", "reviewer": "Reviewer_4AVh", "summary": "This paper proposes the FACT architecture for multivariate time series forecasting (MTSF), designed to capture fine-grained dynamic interactions between variables in both the time and frequency domains. Its core component, the DConvBlock, uses depth-wise convolution to model variable relationships individually at each time step or frequency component (termed 'granularity' in the paper). In the frequency domain, it notably processes the signal by separating it into Amplitude (A) and Phase (P), modeling them independently. Furthermore, it restructures the 1D variable dimension into a 2D space and applies multi-dilated 2D convolution, efficiently achieving a global receptive field across all variables with fewer layers. Experimental results show that FACT achieves SOTA accuracy on 12 benchmarks and significantly reduces training time and memory consumption compared to attention mechanisms.", "questions": "1. You assume no inherent order among variables for 2D reshaping. For datasets like PEMS or METR-LA where actual spatial adjacency between sensors is critical, did you verify if the mapping method (e.g., row-major, z-order curve) from 1D variable IDs to the 2D grid impacts performance?\n2. To support the core claim of 'granularity-specific learning,' could you provide visualizations (e.g., kernel weights, activation maps) to demonstrate that the learned DConvBlock kernels actually capture specific frequency bands or dynamic relationships between variable pairs?\n3. In Appendix D (Table 6), the performance advantage of the amplitude/phase separation method over real/imaginary or complex convolution methods is not significant. Beyond 'physical interpretation,' were there any practical advantages (e.g., faster computation, better training stability) to adopting the amplitude/phase method over the simpler real/imaginary method?\n4. The paper uses a 5-layer DConvBlock. Is there a specific justification or ablation study for setting the number of layers (N) to 5? A sensitivity analysis of performance and receptive field changes based on the number of layers seems necessary.\n5. You used the Inception module as the convolution backbone, but Appendix C shows that standard 2D convolution ('+Conv') also achieves competitive performance. How does the multi-branch structure of Inception compare to standard 2D convolution in terms of parameter count or FLOPs? Is the performance increase sufficient to justify the additional cost?\n6. You used a modified Inception as the backbone for DConvBlock. Did you consider combining the depth-wise convolution with a mechanism that explicitly models relationships between channels (i.e., 'granularity' in this paper), such as a Squeeze-and-Excite (SENet) block?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. Fine-grained Interaction Modeling: Unlike existing models that focus on macro-level relationships of the entire variable set, the idea of capturing dynamic interactions at a fine-grained 'granularity' level (i.e., 'each time step' and 'each frequency component') is highly compelling. The method of assigning independent kernels to each channel (granularity) using depth-wise convolution to achieve this is technically novel and effective.\n2. Robust Dual-Domain Design: The dual-domain architecture, which considers both instantaneous interactions in the time domain and periodic resonances/phase relationships in the frequency domain, is very robust. The approach of modeling the frequency domain by separating it into Amplitude and Phase, which allows for more direct physical interpretation rather than just Real/Imaginary parts, is impressive. The ablation study (Table 3) clearly demonstrates that these two domains are complementary.\n3. Modularity and Generalizability: The core module, MDInception (the convolutional part of DConvBlock), was shown to improve performance not only within FACT but also when integrated into other baseline models (TimesNet, PatchTST, iTransformer) (Appendix E.1). Notably, it significantly boosted the performance of the variable-independent model PatchTST. This suggests the proposed module can function as a general-purpose solution for capturing inter-variable interactions.", "weaknesses": "1. Limitation of the 2D Reshaping Assumption: The paper assumes \"no inherent order among variables\" and restructures the 1D variable list into a 2D grid. However, traffic datasets like PEMS and METR-LA possess an underlying spatial structure (road networks), where the variable (sensor) ID order might reflect adjacency information. This reshaping method, which ignores potential spatial locality, could destroy critical information. There is insufficient discussion on whether this assumption holds true for all 12 datasets.\n2. Lack of Interpretability for 'Granularity-Specific Learning': This is a core claim of the paper. However, there is no qualitative analysis (e.g., visualization) to verify what the depth-wise convolution kernels actually learn. For example, it is not shown whether specific kernels (channels) in the frequency-domain DConvBlock truly specialize in 24-hour periodicity versus 7-day periodicity, or whether time-domain kernels learn to capture interactions between specific variable pairs (like in Fig 1b) or specific time lags.\n3. Questionable Efficacy of A/P Transformation: In Appendix D (Table 6), the paper compares the proposed Amplitude/Phase (A/P) separation modeling with the Real/Imaginary (R/I) alternative. However, the results from both methods are nearly identical (e.g., PEMS03 MSE 0.194 vs 0.193, METR-LA MSE 0.704 vs 0.706). The A/P method requires additional non-linear operations (arctan, cos, sin), whereas R/I is a simple separation. Contrary to the paper's claim of being \"consistently the best,\" the data does not show a clear advantage for the A/P method over the R/I method. This weakens the justification for choosing the more complex A/P representation.", "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": "4TezqJ3sqL", "reviewer": "Reviewer_5JYU", "summary": "This paper proposes FACT, which addresses the limitation where the inter-variable dependency should be modelled dynamically over time. The core design of the paper enables: (1) dependency leveraged across both the time and frequency domains; (2) dependency leveraged at different reception fields; (3) Similar to TimesNet, the 1-dimensional signal is transformed into matrices for better capturing the dependencies.", "questions": "**Q1:** Modelling the dependencies dynamically over time has been proposed, either it is within the same channel [1] or across different channels [2,3], with the dynamics also captured with convolution layers and coarse and fine-grained dependencies captured at different temporal resolutions. Can the authors clarify what the key novelty of the proposed paper is compared to the prior work, and why the advocated way of capturing dynamic dependencies is better?\n\n[1] DeformableTST: Transformer for Time Series Forecasting without Over-reliance on Patching (NeurIPS 2024)\n\n[2] DeformTime: Capturing Variable Dependencies with Deformable Attention for Time Series Forecasting (TMLR 2025)\n\n[3] Adaptive Convolutional Forecasting Network Based on Time Series Feature-Driven\n\n**Q2:**\nThe results are averaged across the full forecasting sequence, i.e., [t+1,t+H]. However, this might over-credit the models that are giving better performance on time steps close to t+1, but bad performance when close to t+H, compared to models that do an average job across all time steps. How is FACT compared to baselines when only evaluated on the target forecasting horizon t+H?\n\n**Q3:** Could the authors clarify the mechanism or provide more details on how the temporal dynamics of dependencies are captured?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "**S1:** The research question raised by the paper is both valid and timely for the community. \n\n**S2:** The paper is easy to follow, with graphs to show the model \\& module designs, and visualised results for straightforward comparison and analysis.", "weaknesses": "**W1:** Inappropriate Averaging Across Horizons. Computing averages across horizons is misleading, as different forecasting horizons are different in forecasting difficulty, where averaging the errors could easily hide poor performance on some horizons. \n\n**W2:** The claim (or the assumption) that there is \"no inherent order among variables in multivariate time series\" (line 290) does not always hold. For example, in weather forecasting, climate indicators at grid points closer to the target by nature have a greater influence than those farther away. Not considering this ordering would limit this work to applications where the covariates are iid, which is OK, as this can be the scope of the paper, but the authors need to make it clearer in the paper.\n\n**W3:** Even if the covariates are iid, there is still concern that the reception field of the paper generally decides which variables are taken into modelling, this means that although the variables does not have an order themselves, the model performance would be highly dependent on the ordering as this influence which variables are modelled and which are not for each target variable.\n\n**W4:** While the paper claims to have the dependency modelled dynamically over time, it is not clear how, just by looking at the modelling design. See Q3.\n\n**W5:** Table 5 shows wildly different configurations per dataset, which raises concern that the good performance presented by the paper might be overfitting to benchmarks rather than a robust general solution.", "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)