DecompNet: Enhancing Time Series Forecasting Models with Implicit Decomposition
Authors: Donghao Luo
Xue Wang
Venue: NeurIPS 2025 poster ()
Keywords: Time series forecasting
Time series enhancement framework
Time series decomposition
Deep learning
Primary area: other
TL;DR: We pioneer the idea of implicit decomposition and propose a powerful decomposition-based enhancement framework based on it, which can consistently enhance the forecasting performance of various state-of-the-art time series models.

Abstract: In this paper, we pioneer the idea of implicit decomposition. And based on this idea, we propose a powerful decomposition-based enhancement framework, namely DecompNet. Our method converts the time series decomposition into an implicit process, where it can give a time series model the decomposition-related knowledge during inference, even though this model does not actually decompose the input time series. Thus, our DecompNet can enable a model to inherit the performance promotion brought by time series decomposition but will not introduce any additional inference costs, successfully enhancing the model performance while enjoying better efficiency. Experimentally, our DecompNet exhibits promising enhancement capability and compelling framework generality. Especially, it can also enhance the performance of the latest and state-of-the-art models, greatly pushing the performance limit of time series forecasting. Through comprehensive comparisons, DecompNet also shows excellent performance and efficiency superiority, making the decomposition-based enhancement framework surpass the well-recognized normalization-based frameworks for the first time. Code is
available at this repository: https://github.com/luodhhh/DecompNet.

DecompNet: Enhancing Time Series Forecasting Models with Implicit Decomposition
Donghao Luo, Xue Wang
Department of Precision Instrument, Tsinghua University, Beijing 100084, China 1aa21mmail... tsinghua. cn, mangxue@mail. tsinghua. edu.
Abstract
In this paper, we pioneer the idea of implicit decomposition. And based this idea, we propose a powerful decomposition- hased enhancement framework, namely De- compNet. Our method converts the time series decomposition into an implicit process, where it can give time series model the decomposition-rrlltee knowl- edge during inference, even though this model does not actually decompose the input time series. Thus, our DecompNet can enable model to inherit the perfor- mance promotion brought by time series decomposition but will not introduce any additional inference costs, successfully enhancing the model performance while enjoying better efficiency. Experimentally. our DecompNet exhibits promising enhancement capability and compelling framework generality. Especially, it can also enhance the performance of the latest and state-of-the-ee models, greatly pushing the performance limit of time series forecasting. Through comprehen- sive comparisons, DecompNet also shows excellent performance and efficiency superiority, making the decomposition-based enhancement framework surpass the well-recognized normalization-basel frameworks for the first time. Code is available this repository: https: http://ggtttt/t/ com wwt.c/commcocceeeeee http///oohhhh//ooooeeeeeet
Introduction
Time series forecasting is widely used in various real-world scenarios (e.g., transportation manage- ment [4], industrial monitoring [35 25 26), energy planning [36 29 and weather forecasting [40), helping the society make better decisions. Because of the immense practical value, various time series models are developed in recent years, bringing great prosperity to time series forecasting
33
28
24 27
Apart from the proposal of time series models, designing model-agnostic enhancement frameworks is another hot research topic. Enhancement frameworks can assist the existing time series models to handle the non-stationary issue in time series data, effectively improving their performance. And recently, the research of enhancement frameworks is still dominated by normalization-bart methods 20 43)
On the other hand, time series decomposition is also powerful tool to handle the non-stationary issue [0] 34), because it can unravel the entangled temporal patterns in raw time series and highlight the stationary components, making it easier to capture the temporal dependency. However, there is still no popularized enhancement framework based on time series decomposition. This can be attributed to the following reasons: (i) Less framework generality: Previous decomposition-hanl methods are only designed as specific time series models, not meeting the needs of model-agnostic frameworks. (ii) Severe efficiency issues. As shown in Figure (a. previous decomposition-bann methods need to explicitly decompose the time series into seasonal and trend components and maintain two expert models (namely seasonal model and trend model) for inference. This process will multiple the computation and inevitably introduce additional inference costs. Considering the
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Figure I: (a) Comparison between implicit decomposition and explicit decomposition. (b) Method overview of our DecompNet.
budget compute and latency, these previous explicit decomposition methods are too costly to be an ideal enhancement framework.
Targeted to these issues, we propose the idea of implicit decomposition, in which we aim to give time series model the decomposition-rele relattd knowledge during inference, even though it does not actually decompose the input time series. As shown in Figure I (a.2), our idea brings following advantages, meeting the needs of a powerful enhancement framework: (i) Efficiency. We only deploy one enhanced model for inference and no more need to explicitly decompose the input time series, successfully solving the efficiency issue in previous explicit decomposition methods. (ii) Performance. Although directly handling the undecomposed input time series, the enhanced model can still utilize the already-injected decomposition related knowledge to enhance the performance. (iii) Framework generality. We can inject the decomppoition--relaee knowledge into any time series models, regardless of the model-specific macro designs, nicely meeting the needs of aaggosig frameworks.
Based on this idea, we propose a powerful decomposition- based enhancement framework named DecompNet. Technically, as shown T when given a time series model to be enhanced, we first pretrain two copies of it on the decomposed seasonal and trend data, helping them grasp the decomposition--rlltte knowledge and making them be the seasonal and trend expert models. Then we fuse these two expert models back as one single model, merging all of their decomposition- related knowledge into the fused model. related knowledge into single model (i.e., the fused model). And this model can be seen as an enhanced version of the original given model. During inference, since the decompositionn-eel related knowledge has been injected into the fused model, it can utilize the deccopposition-relate knowledge to enhance the performance without actually decomposing the input time series, enjoying great performance improvement while not introducing any additional inference costs. Experimentally. when compared with other enhancement frameworks and explicit decomposition methods, our method shows comprehensive superiority in performance, efficiency, enhancement capability and framework generality. Especially, our method can also enhance the performance of the latest and state-of-the-an models, greatly pushing the performance limit of time series forecasting. Our contributions are:
this way, we make it to inject the decomposition-
We pioneer the idea of implicit decomposition and provide guidance to upgrade the time series decomposition into a model-agnostic enhancement framework. Based on this idea, we propose powerful decomposition-baset enhancement framework called DecompNet, which features two key designs: Seasonal- -rend Re-parameterization and decoupled training strategy.
DecompNet shows promising enhancement capability, framework generality and perfor- mance superiority, making the decomposition-bun based enhancement frameworks surpass the well-recognized normalization-ban frameworks for the first time.
Our method can enhance the model performance without introducing any additional infer- ence costs, enjoying totally the same inference costs the original model. This inference- cost-free property demonstrates the efficiency superiority of our method.
As brand new idea, our implicit decomposition can bring some fresh perspectives and provide better solution to the classic research topic of time series decomposition.

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2
Related Work
2.1 Time Series Decomposition
Time series decomposition is an important technique in time series community. In the latest decomposition methods, the mainstream procedure is to decompose the time series into seasonal and trend components and process them with two expert models respectively. In terms of specific implementations, most methods adopt the classic moving average to decompose the time series 45| 46]. As improvement, [37] adopts
multi-kernel decomposition and [44 proposes learnable decomposition. Besides, 388 proposes to decompose the time series in frequency domain. However, these explicit decomposition methods are mainly designed specific time series models and cannot serve as model-agnostic enhancement frameworks. To fill this gap, we pioneer the idea of implicit decomposition, providing guidance upgrade the time series decomposition into model-agnostic enhancement framework. Please refer to Appendix F.
for more discussion.
2.2
Time Series Enhancement Framework
Time series enhancement framework can assist time series models to solve the non-stationary issue in time series data and improve their performance. The mainstream enhancement frameworks can be categorized as normalization-bart based and prior- based ones. In detail, normalization- based frameworks can alleviate the non-stationary issue from the perspective of statistical measures [[] g 20 43
And prior-based frameworks can also improve the model performance, provided that the periodicity
prior of dataset is known in advance [15 [6]. In this paper, we establish a new category of decomposition-bie -hased enhancement framework and propose DecompNet as its representative, which can greatly improve the model performance and bring completely no additional inference costs.
It's worth noting that, previous studies generally consider normalization-bass enhancement frame- works as better solutions for the non-stationary issue [9] 20 43 And we are the first to propose a decomposition- based enhancement framework that achieves better performance than normalization- based ones, which is great breakthrough in this direction. More discussion is in Appendix F.4
2.3 Structural Re-parameterization
Structural re- re-praameterization a methodology to convert model structures via fusing model param- eters. For example. 66 g
17 23 adopt structural c--aarameerrzaaton to simplify the structure of multi-branch convolution networks, which achieved by fusing the multi-branch part as one single branch, successfully solving their efficiency issues while maintaining their performance. M2PT [47] further extends its application to fusing two individual neural networks, which achieved by fusing each of the layers corresponding positions of two models. As variant of structural re- parametcrization specially designed for implicit decomposition, we propose Seasonal- -ranonal-Trenn Re- parammtteriiatioonn in this paper, which brings efficiency superiority to our method.
3
DecompNet
3.1
Preliminaries
Time Series Forecasting Given length-/ multivariate time series with M variates as input, time series forecasting aims to predict the length-T' future series (Y E RTM based on this length- input series (X E ( To bring better forecasting performance, we propose model-agnostic enhancement framework, namely DecompNet, to assist existing time series forecasters, which effectively improve their performance and greatly push the performance limit of time series forecasting.
Time Series Decomposition Time series decomposition is mainly carried out based on moving average. In detail, we first smooth out the raw time series and obtain the trend component. Then we obtain the seasonal component by subtracting the estimated trend component from the raw time series. For length- input time series X E R xxM the decomposition process can be formulated as:
X, = AvgPPol(Pa
X, x X,
(2)
where X,, X, E I denote the seasonal and the trend components, respectively. We adopt the Avglool(-) for moving average and set the moving average window size as 25 by default. And we adopt the Padding(-) operation keep the series length unchanged, which employed using terminal values. For simplicity, we use X,, X, = Series to summarize above equations.

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3.2 Framework Overview
Figure (b) shows the overall workflow of DecompNet. During training, the main purpose of our method is to give model the recomposition-relatee knowledge and enhance its performance. And we propose first-decouple-then--
multi-stage training process to achieve this goal.
When given a time series model to be enhanced, we first pretrain two copies of it on the decomposed seasonal and trend data, helping them grasp the decomposition-rclated knowledge and making them to be the seasonal and trend expert models. During this pretraining stage for two expert models, we propose decoupled training strategy help them better grasp the decomposition-relacee knowledge, which will be introduced in Section 3.3
Then we fuse these two pretrained expert models back decomposition--rlatee knowledge into the fused model. In this way, we make it to inject the decomposition--reattee knowledge into single model (i.e., the fused model). And this fused model can be seen as an enhanced version of the original given model. And the key design of this fusion peeearameterizatinn, variant of structural oe-parameterization specially
single model, merging all of their
stage is Seasonal-Trem designed for our implicit decomposition, which will be introduced in Section 3.4
During inference, we only deploy this single fused model for inference and no more need to explicitly decompose the input time series. Since the decomposition-relle related knowledge has been injected into the fused model, it utilize the decomposition related knowledge to enhance the performance without actually decomposing the input time series. Thus, our method can enhance the model performance without introducing any additional inference costs, enjoying totally the same inference costs as the original model.
3.3 Decoupled Training Strategy
We propose a decoupled training strategy to bet- ter pretrain the seasonal expert model and trend expert model. As shown in Figure 2 (a), our strategy decouples the whole-series prediction task into seasonal prediction task and a trend prediction task respectively, which can provide more direct and clearer supervision signals for two expert models.
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Given X E R as the input time series, first decompose it into the seasonal and trend inputs X., X, E / Then they passed into the two expert models to predict the sea- sonal and trend outputs respectively:
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X,, X, ScriesDecomp(X)
(3) Figure Illustration of our decoupled training strategy. We also provide a figure of previous joint (4) training strategy for comparison.
Y, Model,(X,)
Y, Model,
(5)
where Model, and Model, are the seasonal model and trend model respectively. And Y., Y, T denote the predicted seasonal output and trend output.
Then in terms of loss calculation, we also decompose the ground truth Y to obtain the seasonal ground truth and trend ground truth, denoted as Y.. Y, E TMM,, Then we can calculate losses for the seasonal part and trend part separately, which are L. and L. this way, our strategy can provide separate supervision signals for two expert models, making two models trained in a decoupled manner:
Y,, 5, Y, = Series DerriesDecoop(Y
(6)
C, Y Y Y.
L. = IY, Y, Y.
(8)
Advantages of Decoupled Training Strategy The goal of decomposition--asee methods is to train two expert models specialized in the trend prediction task and seasonal prediction task, respec- tively. By decoupling the training objective from the previous whole-series prediction task into

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seasonal prediction task and a trend prediction task, our strategy can provide more direct and clearer supervision signals for two expert models, thus bringing better training results and fully unleashing methods. And our strategy can also contribute to
the performance potential of decomposition-basee faster convergence, since it unravels Y into more predictable components and makes the training process easier. Please refer to Appendix D for details.
3.4
Seasonal- Trend Re-parameterization
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Figure Illustration of Seasonal-Trend Re-parameterizatioon. (a) The actual structure of two pretrained expert models, which infers in an explicit decomposition manner and suffers from heavier inference costs. (b) The conceptual structure for extra training of (c) The final deployed structure. It infers in an implicit decomposition manner, thus solving the efficiency issue.
After above decoupled pretraining, if we directly deploy the two pretrained expert models for inference in an explicit decomposition manner (Figure 3 (a)), it will bring heavier inference costs. To solve this efficiency issue, propose Seasonal- Trend Re- pe-aammtteriztiionn to fuse these two expert models and convert the decomposition process into an implicit manner.
Fusion of linear layers As shown in Figure we start from the fusion of two one-layer expert models to introduce our Seasonal-Trend Re-parameterizatioo. Let X,, X, be the seasonal and trend inputs decomposed from input series X. And let W, and W, be the weight matrices for the linear layers in seasonal model and trend model, respectively. If we directly deploy these two expert models for inference in explicit decomposition manner (Figure (a)), we have:
Y= X. X.W.
Here, we use the matrix multiplication between an input data and weight matrix to indicate the forward process in a linear layer. And we omit the bias term for brevity.
We observe that X X, + X, (Equation 2}.
So the data part is naturally mergeable. And the convert above explicit decomposition into an implicit
weight matrix part can also be merged, manner (Figure(((c) Inspired by (47), we propose Seasonal-
Re-parameterization to achieve this goal, where intent find learnable fusion factor À fuse the two pretrained weight matrices W,, W, as single matrix W, letting W
W. + XW, and making following target hold:
Y YxX.W. + X,
=(X, (W.. + A
(10)
-X(W. N.W.+AW. +AW,)
XW
From above process, we make it to convert the explicit decomposition into an implicit process, in which we can directly process the undecomposed raw time series X and only need to deploy one linear layer with W as the weight matrix for inference (Figure 3 (c)). This linear layer is the fusion of the two linear layers from seasonal and trend models, so it can inherit their pretrained decomposition--rlltee knowledge to enhance its performance.
5

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To make our target described in Equation I0 hold, we should introduce an extra training process to learn the optimal fusion faccorr A. suggested by [47.. In detail, since we have Y = X(W. e AW;)= = XW. XXW, according to Equation I0 we can convert the actual structure of two expert models (Figure (a)) into a conceptual structure for this extra training process (Figure (b)). where both expert models turn to directly receive the raw time series X as inputs, and the output is their weighted sum scaled by A. In this stage, we let W, and W. continue to be trained along with A. Since W. and W, are already well pretrained by decomposition-related knowledge, it helps to make this extra training process converge quickly.
After above extra training. we merge the parameters by computing W = W, + AW, and only save this fused model parameterized by W. Thus we only need to deploy one model for inference and no longer explicitly decompose the input series, making our method not introduce any additional inference costs, successfully solving the efficiency issue.
Note that the fusion logic of convolution layer is similar to linear layer in convolution layer can be indicated by the convolution operation between an input data and a weight matrix, which similar to the matrix multiplication operation the forward process in a linear layer. So we can realize the fusion of convolution layers in a similar way we fuse linear layers.
42 since the forward process
Fusion of complicated models Based on some foundational researches on structural re- parameterization
we can achieve the fusion of two individual neural networks by fusing each of their layers at corresponding positions.
And specifically for time series community, most of the complicated models are built upon linear layers. For example, the embedding layer, the prediction head and the backbone in MLP-based models are all made up of linear layers. Similarly, the backbone in convolution-based models made up of convolution layers. For more complicated wransformer-based models, their learnable parameters are also within linear layers (e.g., QKV and output projections in the attention block and those in the FFN block). Thus, by applying Seasonal-Trent Re-parameterization to fuse each corresponding linear convolution layer, we can realize the fusion of two complicated expert models. And in this fusion process, À factors are learned independently for each layer.
In this paper. our fusion mechanism relies strict luyer-wise alignment and the prerequisite for our fusion process is that the two expert models must have the same model architecture. And this prerequisite definitely holds true in this paper, since the two expert models in our fusion process two copies of the same original model, ensuring that they definitely have the same architecture. will be our future work to further study how handle the architectural mismatches during fusion. Please see Appendix Q for more details.
Experiments
In this section, we conduct extensive experiments to evaluate the effectiveness of our DecompNet. In particular, we investigate the enhancement capability of DecompNet by integrating it with various DecompNet with other enhance- ment frameworks in Section 4.22 and compare our idea of implicit
advanced time series backbone models in Section [ compare decomposition with the previous and
idea of explicit decomposition setups. Please also refer to Appendix S for visualization showcases and forecasting examples.
Section 4.3 See Appendix A and C details of experimental
4.1 Main Results
Setups We conduct long-term forecasting experiments on 8 popular real-world benchmarks, in- cluding Weather [40), Traffic [33], ECL (36). Solar-Energy (2] and 4 ETT datasets [48]. Following the previous settings, we set prediction lengths as {96, 192, 336, 720} and fix the input length as 384. We calculate the mean squared error (MSE) and mean absolute error (MAB) of multivariate time series forecasting metrics. In all experiments, we set the moving average window size as 25 for time series decomposition during the decoupled pretraining stage. And we provide the parameter sensitivity study in Section 4.5 Our method is trained with the L2 loss, using the ADAM optimizer with an initial learning rate in {10 3,5 x 10 10 10-4). $). The default training process is 30 epochs with proper early stopping (i.e., we set the maximum epoch 30 and set the early stop patience 3). More datasets details and implementation details are in Appendix and C
Backbone Models DecompNet is a model-agnostic enhancement framework and can applied to any time series forecasters. To better prove our enhancement capability, adopt following

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representative forecasters as strong backbone models, including the Transformer-basel models: PatchTST iTransformmr [8];; the MLP-based models: RLinear, RMLP [4] and the Convolution-
based model: ModernTCN 23 The selection criterion of backbone models are in Appendix C
Results As shown in Table DecompNet consistently enhances the performance of all backbone models by large margin. For instance, on Solar, ETTh1, and Traffic datasets, the average MSE performance improvements are rather substantial: 8.0%, 6.4% and 6. % respectively. Note that these performance promotions are brought by time series decomposition. proves that our DecompNet can really inherit the performance promotion obtained from time series decomposition even though not actually decomposing the input time series, verifying the feasibility and soundness of our idea of implicit decomposition.
It is worth noting that the selected backbone models have already achieved the previous state-of- the-art performance by themselves. And our framework can still further improve their performance, which can truly reflect strong enhancement capability. Meanwhile, the backbone models cover the mainstream types of time series forecasters and our DecompNet brings consistent performance enhancement to them, which can firmly prove our framework generality. Moreover, by improving the performance of these state-of-the-arl models, this paper greatly pushes the performance limit of time series forecasting, setting a new standard for future researches.
Table 1:
Performance promotion obtained by our DecompNet framework in long-term forecasting tasks. We adopt five mainstream state-of ofttr--rttt-tttt time series forecasters as backbone models. A lower MSE or MAE indicates a better performance and the best results are in bold. Results are averaged from four prediction lengths T E {96, 192, 336, 720}. ). See Table 12 in Appendix R. for full results. And we also report the full error bar in Appendix which can venfy that the improvement in our main result is meaningful and is not due to chance.
Models
PadiTST
Maassforeer
RLiseer
XMLP
ModeanCCN
Settings
Original
[0
iDecouspbit
(Ours)
Orginal
[+]
DecoonpNei
(Curs)
Original
14
DDcoompheet
(Ours)
Original
[4]
:DeommpNee
(Ours)
Original
H
Decommpeer
(Ours)
Metric
MMSE MSE MAE MAE MSB MAE MAEIMSE MAB [MSB MAE
MAE MSE
ETThi
10.18 0433/0400 0.421 0.461 0.464 0.417 9.45 10.15 0445/04400
0.422 (0452 0:469
0.225
0.400 J1403 0419)6.387
0.411
ETTh2
10.44
(,30]](0.46 0,387
0,344 0.359
0.300
0,383 00354 0.401
0,390 J0.328 0.382/0.312
ETTml
[0,35 0.38||0.445 0,399 0,312 0.399 0,358 0.365 10.364
0,361 (1.39 0.377 (0.31 0.394 (0,55 0,383 [0,365 0.396J0.284
BTTm2
[0200
0315|ju257 0.314 0.274 0.332 0.262 0.321 [0,000
0/310/6.255
0.312 11271 0.326 9.255
0,318 [1.26] 0,301|uu251
IL314
Weuther [0227
0.264)12224 0.261 0.275 0.272 0.229 0.268 10.300 0.781 (6255 11275 10,244 0.279 (:231 0,275 [0,233 0,271JU1231
11266
Solar- Eneegy/ 0:317/11199 0,257 0.212 272 0,204 0.259
0:32|ju227 0.274 (02311 0.254 19.214 0,249 [1:218 O 0,300/LL211
11258
[0:66] 0.208001111 0.260
0.168 0.264
0220 0.259 ja172 0.165
[0: 0,264)0.199
10.10 0.28|Jo2387 0.368 0.428 0:300 0,395 0.280 Joues 0311//1425
0.286 10:43
0.308
0.413 0,285 [0.412 0.200/0.194
0.275
4.2
Compared with Other Enhancement Frameworks
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Figure 4: Comparison of our DecompNet with other enhancement frameworks. (a) and (b) Perfor- mance comparison. A lower MSE (a shorter bar) means better performance. Results are averaged from datasets. See Appendix R.3 for full results. (c) Efficiency comparison. We conduct experi- ments in Traffic dataset under 3p-r-384--rediit-96 setting. A smaller memory usage and lower inference time indicate better efficiency. (d) Performance under different input lengths. We conduct experiments in Traffic dataset and fix the prediction length as 96.
Setups We compare our DecompNet with other time series enhancement frameworks, including the
popular normalization-bar based frameworks: Dish-TS [9] SAN (20). FAN [43] and the latest prior-based
frameworks: SparseTSS TSF [66) CycleNet [15]. And we adopt RLinear 14 and PatchTST [30] as backbone models for this comparison.
7

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Performance Comparison Figure (a) and (b) show the excellent performance superiority of DecompNet. Concretely, our DecompNet achieves the best performance among existing enhancement frameworks and brings the best enhancement effects to all backbone models. In detail, DecompNet the first decomposition- based method that outperforms the well-recognized normalization-bant methods, which is great breakthrough in this direction. And DecompNet also surpasses the latest prior-based methods. For instance, when using PStchTST as the backbone model, DecompNet achieves the best performance with an average MSE of 0.290, obtaining an average of 6.3% MSE reduction than the latest prior-based methods in all datasets. Besides, as an advantage over these prior- bosod methods, our method can perform well without the need of any prior information about the dataset, thus being suitable for a wider range of application scenarios.
Framework Generality Comparison Although normalization- baaeiiiatatto-b-dddddd enhancement frameworks can work well on the simple backbone model like RLinear [14), they fail to enhance the performance of more advanced backbone model like PatchTST [30]. As shown in Figure (b), combining normalization-basee
frameworks with PStchTST brings negative effects, making their performance even worse than the original model's. s. By contrast, our DecompNet can bring consistent performance promotion to both kinds of backbone models, showing better framework generality. And we'd d like to highlight that our method is currently the only one that can improve the performance of more advanced backbone models, which is unique advantage that sets us apart from other competitors.
According to previous studies 30 23,, the more advanced backbone models can learn to handle the statistical informaition by themselves. As result, the normaliztion- based enhancement framework, which also works from the perspective of statistical informaition, is not necessary for these advanced backbone models. By contrast, our results indicate that existing backbone models still fail to obtain the decomposition knowledge on their owns. Therefore, the external assistance from our decomposition- enhancement framework is still needed.
Efficiency Comparison We comprehensively compare the forecasting performance and inference costs (i.c., inference speed and memory usage) of the selected enhancement frameworks. And the results are shown in Figure (c). When bringing the best enhancement effect to the backbone model, our method has totally the same inference speed and memory usage as the original model, proving that our method can greatly improve the forecasting performance and bring completely no additional inference costs. Our efficiency superiority can be credited to the idea of implicit decomposition. Since we don't 't need to explicitly decompose the input series for inference, it can greatly improve our efficiency. As a comparison, other enhancement frameworks need to explicitly extract the periodic, statistical or frequency information of the input series for inference, which inevitably brings more memory usage and inference time, making them less efficient.
Robustness to Vary Input Length As shown in Figure (d), all enhancement frameworks gain continuous performance improvement with the increasing input length, validating their effectiveness in extracting useful information from longer history. Overall, our method achieves the best performance under most input lengths, demonstrating our great adaptability to vary input lengths and highlighting our performance superiority.
Conclusion In conclusion, DecompNet achieves the best in both performance and efficiency. And our method also enjoys a wider range of applicability since it doesn't 't require any prior information of the dataset. These results validate that our DecompNet is ideal enhancement framework to solve the non-stationary issue in time series data. Besides, the success of DecompNet makes the decomposition-basee enhancement frameworks surpass the well-recognized normalization-base methods for the first time. This breakthrough reveals the great potential of decomposition-basee methods, which can encourage further studies in this direction.
4.3 Compared with Explicit Decomposition
Setups Our DecompNet is designed based on the idea of implicit decomposition, which is brand new idea to apply decomposition in time series forecasting. And our implicit decomposition holds very different opinions from the previous idea of explicit decomposition. To provide an adequate comparison of these two design ideas, we compare our DecompNet with some latest and representative explicit decomposition methods, i.e., MICN (37 and Leddam 444 Please refer to Appendix F33 for a detailed introduction of their model structures and data processing pipelines.
Overall Comparison As shown in Figure 5 (a), our implicit decomposition outperforms the previ- ous explicit decomposition by large margin all datasets, indicating that our implicit decomposition
is a better solution to apply decomposition in time series forecasting.
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womponent-wise Analysis To clearly demon- strate where our performance superiority comes from, we also gradually modify these explicit decomposition methods with our implicit de- composition designs. The results are in Figure (b). And we have some interesting findings that challenge the commonly held belief in pre- vious decomposition- methods, which are follows:
a.00 Jine
me
Madiiiiiii Explast Drcorppsite
(1) What contributes to better performance? Previous explicit decomposition pays more at- tention on data processing, while our implicit decomposition focuses more on model training. To provide a comparison, we remove the so- phisticated data processing in previous explicit decomposition methods and apply our decou- pled training strategy to them. We observe great performance improvement after this modifica- tion (14.7% promotion in MICN and 6.3% in Leddam), which can prove the importance of model training, highlighting that a better training strategy is the key to fully unleash the performance potential of decomposition-bent based methods, not the more sophisticated data processing.
01 Performane Comparkee da Datsom
Figure 5: (a) Performance (average MSE results) comparison with explicit decomposition methods on 8 datasets. (b) Modifications on explicit de- composition methods with our implicit decomposi- tion designs. Results are further averaged from datasets. Full results are in Appendix R.2
(2) Do the two expert models have to use different model structures? The two expert models in our implicit decomposition have the same model structure, since they are actually two copies of the same original model. But previous explicit decomposition advocates to use different model structures two expert models and consider it as a prerequisite to handle the different nature in seasonal and trend components. We also re-examine this advocacy and surprisingly find that using the same model structure does not result in performance degradation. We attribute this finding to decoupled training strategy, since it can better train the two expert models and fully realize the strong representation capabilities within neural networks, making them able to capture the meaningful seasonal and trend representations even with same model structure. This finding verifies the feasibility to use same model structure in two expert models, paving the way to improve the efficiency by our Seasonal- -rend Re- parameterization.
Conclusion In this section, we conduct comprehensive comparisons and prove that our implicit decomposition is better idea to apply decomposition in time series forecasting. Meanwhile, we re-examine many well-established practices in existing decomposition-boe -baeed methods and reveal that they are not optimal choices, making the research on time series decomposition still an open question. We hope these new findings can prompt people to rethink this classic research topic and design more innovative methods for time series decomposition.
4.4 A Ablaioon Study
Inférane
Inférvace
Original Model
Oriigiaal Medsl
Deecepped Truning Stritegy
Decouppel Traisin Stritgr
0,433
Comparban: Jahnt Traintng Strategy
Compurban: John Training Stratgg
Seessal-Trel Repeetanttriiainoon
E.S07
Scavsnal Reperaantecintion
8:455
Por NwfoomaareetSt (MSE))
4.355
0L345
Parfomaaer (MSF))
8.365
4,345
(a) Ablation results in Traffic dataset using PT as backbone model
(b) Ablation results in Traffic dataset using RLinear as backbone model
Figure 6: Ablation study. lower MSE and inference time (a shorter blue bar and a green star closer to the vertical axis) indicate better performance and efficiency. More results are in Appendix G
To validate the effectiveness of our designs, we start from an original model and gradually enhance it by adding our designs step by step. And we provide the trajectory going from the original model to its enhanced version in Figure 6
Decoupled Training Strategy We first obtain two copies of the original model and train them to be the expert models for seasonal and trend prediction tasks. And we directly deploy these two expert models for inference in an explicit decomposition manner without fusion. We observe great

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performance improvement when using our decoupled training strategy, while using joint training strategy results in less performance promotion. This result proves that our decoupled training strategy is a better strategy to fully unleash the performance potential of decomposition- methods.
Seasonal-Trend Re-parametteriaatin Secondly, the fusion by our Seasonal-Trend Re- parameterization can greatly improve the efficiency. And it causes no performance degradation, indicating that our Seasonal- Trend Re- Se-paraaeetrrizaton can assist the fused model to successfully inherit the decomposition related knowledge. Meanwhile, we observe that Geasonal-Tremt Re- parameterization can bring further performance promotion in some datasets, since the trainable fusion factor À helps to fuse the seasonal and trend knowledge more appropriate proportion than the direct summation in explicit decomposition.
After equipped with all our designs, the final model can be seen as the enhanced version of the original model, enjoying better performance and having totally the same inference costs (e.g., in traffic dataset, it can boost the performance from 0.410 to 0.387 under the same inference time 322ms), which verifies the necessity and effectiveness of our decomposition-basel enhancement framework.
4.5 Parameter Sensitivity
Moving average window size is the only tunable parameter for our framework, which is used in time series decomposition during our decoupled pretraining stage. We perform its parameter sensitivity study in Figure The results show that our method is robust to the choice of moving average window size. And using the default value of 25 can provide ideal performance across various datasets.
0.40
MSS.
ETTSS
MST
EITYS
0.25
Taffic
Wouther
Waster
0.15
Moving Average Window Sizee
Moving Average Window Sazes
Backhone Model: PatchTST
Backhone Model: RLincar
Figure 7: Parameter sensitivity. We conduct experiments under input-384-predicte settings, adopt- ing PatchTST [30 and RLinear [1] as backbone models.
4.6 Training Efficiency
In addition to the excellent inference efficiency, our method also enjoys nice training efficiency. As stated in Section 3.3 and 3.4, the decoupled pretraining process for expert models and the extra training process for fusion can converge quickly. making the total training cost of our method comparable the original model. Please refer to Appendix E.] for detailed experimental evidence.
Conclusion and Future Work
In this paper, we pioneer the idea of implicit decomposition and propose a powerful decomposition- based enhancement framework named DecompNet. Our method enable time series model to inherit the decomppoitiion-rlltee knowledge, even though this model is not actually decomposing the input time series. Therefore, our DecompNet can greatly enhance the forecasting performance of various time series models and bring no additional inference costs. Experimentally, DecompNet demonstrates excellent enhancement capability and framework generality, greatly pushing the perfor- mance limit of time series forecasting. Meanwhile, DecompNet also shows compelling performance and efficiency superiority, making the decomposition based enhancement framework surpass the well-recogee rell-recognized normalization sraaal frameworks for the first time. We hope our exploration on implicit decomposition can provide some fresh perspectives and facilitate the future researches in both time series decomposition and time series enhancement frameworks.
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Acknowledgment
This work was supported by the Hebei Innovation Plan (20540301D).
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Appendix C and the Reference. The baselines mostly using Apache-2.0 license and MIT license. The datasets we used are all extensively utilized for benchmarking and publicly available. We provide the URL for the datasets in Appendix B And we are trying to reach out to the dataset's creators for the license information.
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A Details of Experimental Setup
Datasets We conduct long-term forecasting experiments on popular real-world benchmarks, including Weather 40), Traffic [33], ECL [36], Solar-Energy (29 and 4 ETT datasets [48] More
details of these datasets are provided in Appendix B
Backbone Models DecompNet is a model-agnostic enhancement framework and can be applied to any time series forecasters. To better prove our enhancement capability, adopt following representative forecasters as strong backbone models, including the Transformer-base
models: PatchTST [30), i'Transformer (18); the MI.P-based models: RLinear, RMLP [14 and the Convolution- based model: ModemnTCN 23 The selection criterion for above backbone models are introduced in
Appendix C
Comprehensive Comparison To better prove our comprehensive superiority, we compare our DecompNet with other time series enhancement frameworks in Section 4.2 including the popular
normalization-base frameworks: Dish-TS [9]. SAN 20) FAN 43 and the latest prior-based
frameworks: Sparse'TSF [16], CycleNet
15
And we also compare our idea of implicit decomposition with the previous idea of explicit de-
composition in Section 4.3 including the latest and representative explicit decomposition methods:
MICN 37 and Leddam 4
More information of these methods adopted for comparisons are provided in Appendix F3 and
F.4
Evaluation We conduct long-term forecasting experiments for evaluation. Following the previous settings, we set prediction lengths as (96, 192, 336, 720} and calculate the mean squared error (MSE) and mean absolute error (MAE) of multivariate time series forecasting as metries.
Meanwhile, to truly reflect the enhancement effects, we should ensure that each backbone model has achieved its best performance. And according to previous findings 45 14 30 21 the time series models achieve their best performance when the input length longer than 300. Thus fix the input length as 384 to truly unleash the potential of backbone models and obtain strong baselines, which can make our results more persuasive. More implementation details are in Appendix
23 22). most of
B Datasets
Table 2: Detailed descriptions of multivariate datasets. The Dataset Size denotes the total number of time points in (Train, Validation, Test) split respectively.
Dataset
Variates
Prediction Length
Dataset Size
Frequency
Information
ETTh1
7
196, 192, 336, 720)
(8545, 2881, 2881)
Hourly
Electricity
ETTh2
7
(96, 192, 336, 720)
(8545, 2881, 2881)
Hourly
Electricity
ETTml
7
196, 192, 336, 720)
(34465, 11521, 1 11521)
15min
Electricity
ETTm2
(96, 192, 336, 720)
(34465, 11521, 11521)
15min
Electricity
Weather
196, 192, 336, 720)
(36792, 5271, 10540)
10min
Weather
Solar- Entery
137
(96, 192, 336, 720)
(36601,516)) 10417)
10min
Energy
ECL
321
(96, 192, 336, 720)
(18317, 5261)
Hourly
Electricity
Traffic
862
196, 192, 336, 720)
(12185, 1757, 3509)
Hourly
Transportation
We evaluate the multivariate long-term forecasting performance on popular world datasets, including Weather, Traffic, ECL, Solar-energy and 4 ETT datasets (ETThI, ETTh2, ETTm1, ETTm2). These datasets have been extensively utilized for benchmarking and cover many aspects of life.
The variate number, dataset size and sampling frequency of each dataset are summarized in Table We follow standard protocol [48] and split all datasets into training, validation and test set in chronological order by the ratio of 6:2:2 for the ETT datasets and 7:1:2 for the other datasets. And
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training, validation and test sets are zero-mean normalized with the mean and standard deviation of training set. Each of above datasets only contains one continuous long time series, and obtain samples by sliding window.
More introduction of the datasets follows:
1) Weather contains 21 meteorological indicators of Germany in 2020.
2) Traffic contains the road occupancy rates measured by 862 different sensors on San Francisco Bay area freeways in 2 years.
3) ECL(Clectricity contains hourly electricity consumption of 321 clients from 2012 to 2014.
D
4) ETT(Electriciyy Transformer Temperature contains the data collected from two different elec- tricity transformers (labeled 1 and 2) with two different resolutions (denoted as for 15 minutes and for 1 hour) by 7 sensors. As a result, in total we have ETT datasets: ETTh1, ETTh2, ETTm1, ETTm2.
5) Solar(Solar-Fnergy contains 137 time series representing the solar power production in Alabama state 2006.
C Experiment Details
Implementation Details Our method is trained with the L2 loss, using the ADAM [12 optimizer
with an initial learning rate in {10-3,3 10-3,3 5 10 10-4,10-$ 10-4). The default training process is 30 epochs with proper early stopping (i.c., we set the maximum epoch as 30 and set the early stop patience as 3). The mean square error (MSE) and mean absolute error (MAE) are used as metrics. All the experiments are repeated 5 times with different seeds and the means of the metrics are reported as the final results. All the deep learning networks are implemented PyTorch [32 and conducted on NVIDIA A100 40GB GPU.
Selection Criterion for Backbone Models DecompNet is model-agnostic framework and can be applied any time series forecasters. To better prove our effectiveness, we adopt following representa- tive forecasters as strong backbone models, including the Transformer-basee models: PatchTST (30), iTransformer [18; the MLP-based models: RLinear and RMLP [4] and the Convolution-base model: TCNernTNN (23).
We choose above backbone models for following reasons: (i) the selected backbone models cover the mainstream types of time series forecasters, and the capability to enhance the performance of various types of models can prove our framework generality; (ii) the selected backbone models represent the ff-te-of-the-art art level in time series forecasting, and the capability to further enhance the performance of these state-of-the-art models can truly reflect our strong enhancement effects.
Parameter Selection for Backbone Models For all backbone models, we follow the official codes in their original papers and search their model parameters from following searching space: number of layers L from {1,2, 3), model dimension from {16, 32, 64, 128, 256] and FFN expansion a from 21, 4,88, 8}. which can fully unleash the performance potential of backbone models and provide persuasive benchmark to truly reflect the enhancement effects of all enhancement frameworks.
We only search model parameters for the baseline backbone models. After the model parameters of a specific baseline backbone model have been determined, we directly apply our DecompNet framework into this baseline backbone model without additionally modifying its model parameters. Therefore, the model parameters in our method are same as those in the baseline, which can ensure fair comparision.
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Parameter Selection for Enhancement Frameworks By default, our DecompNet sets the moving average window size as 25 for time series decomposition during the decoupled pretraining stage, And we provide the parameter sensitivity study in Section 4.5 For other enhancement frameworks adopted for comparison, we follow their official codes with their recommended framework parameters the original papers.
Metric We adopt the mean square error (MSE) and mean absolute error (MAE) of multivariate time series forecasting metrics.
T
MSE
Σ
(Y,-Y,)
T
1=0
MAE
1 T
Σ i=U
Y
Y,
where Y, Y E Tx are the M variates prediction results of length I" and corresponding ground
truth. Y, means the i-th time point in the prediction result.
D More About Our Methods
Advantages Decoupled Training Strategy Our decoupled training strategy has following advantages: (i) Better training results. The goal of decomposittion-assed methods is to train two expert models specialized in the trend prediction task and seasonal prediction task, respectively. And our strategy directly trains the expert models these two tasks, which can provide clearer supervision signals and bring better training results. For comparison, the previous joint training strategy trains the expert models messily entangled supervision signals, limiting their performance (Figure [ (b)). As a result, our strategy is more in line with the purpose of time series decomposition, therefore can fully unleash the performance potential of decomposition- -based methods. (ii) Faster convergence. Since the ground truth Y is raw time series entangled with various patterns, directly predicting the whole Y is more difficult 43 20 And our strategy can unravel Y into more predictable components and make the training process easier, contributing faster convergence.
indirect whole-series prediction task, which can only provide
Please refer to Appendix
E.1 and S.I for more experimental verifications.
E
More About Our Training Phase
E.1
Training Efficiency
In this paper, we mainly focus on the inference efficiency for following reasons. Compared with the training phase which is usually performed on devices with substantial GPU resources, the inference phase is more likely to be conducted with limited computational budgets (e.g., deployed on resource- limited user devices). Therefore, focusing and optimizing the inference cost can enable our method to be deployed in a wider range of real-world application scenarios (especially in resource-constrain environments) and thus increase the utility of our method. Under such consideration, we emphasize our excellent inference efficiency in this paper and highlight our inference-cost-fre property as key advantage that sets us apart from other competitors.
In addition to the excellent inference efficiency, our method also enjoys nice training efficiency. As stated in Section 3.3 and 3.4, the decoupled pretraining process for two expert models and the extra training process for fusion can converge quickly, making the total training cost of our method comparable to the original model.
We conduct experiments to verify our nice training efficiency. The experimental setups are as follows:
We use PatchTST 30 96 setting. For all training processes, the training setting totally the same, where we set the maximum epoch as 30 and set the early stop patience 3, And based on our observation,
the backbone model. Results are recorded under 1appu--84-predicee
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the models can converge quickly in practice. So the actual training epochs consumed by model are often less than the maximum number 30.
For original model, we train it from scratch and report its total training epochs.
For our method, we first pretrain the trend model and seasonal model respectively, then we conduct an extra training to fuse these two expert models. So the epochs of +Ours are recorded as (trend + seasonal fuse = total).
Due to the same structure of each model, the per epoch cost similar. So the total training epochs can reflect the overall training cost.
We conduct experiments and report the training cost and model performance in Table The results prove that:
Although we pretrain two expert models during the decoupled pretraining process, the total training cost is not doubled. It is comparable or less than the original model.
Although our method has an extra training process for fusion, can converge quickly and bring very little training cost.
As result, the total training epochs consumed by our method are comparable to the original model.
Considering the comparable training cost, the totally same inference cost and great performance improvement, it can be proven that our method performs much better than the original model.
Table 3: Training cost comparison and model performance comparison. We use MSE to reflect the model performance and we use the total training epochs to reflect the overall training cost. A lower MSE and less training cost indicate better forecasting performance and training efficiency. The best results are bold.
ETTh1
ETTml
Solar
Traffic
Dataset
Metric
Fpoch
MSE
Epoch
MSE
Epoch
MSE
Epoch
MSE
Original Backbone Model|
20
0.376
15
0.292
26
0.197
0.385
+Ours
(1+10+1=12) 0.361
(4+ (4+12+3=19) 0.283 (8+ 18+12+1=21 0.172
(10+18+3=31) 0.350
E.2
Ablation Study on Decoupled Pretraining Stage
We conduct ablation study verify the necessity of our decoupled pretraining stage. As shown in T the removal of pretraining leads to performance degradation, validating the necessity of our decoupled pretraining stage. And this result is consistent with intuitive judgment: since the decoupled pretraining stage is key process to inject the decomposition-relatee knowledge into models, it can not be removed.
Table 4: Ablation study on our decoupled pretraining stage. Ours indicates that W, and W; are firstly pretrained with decomposition-rrlateed knowledge before fusion. And Without Pretraining indicates that W, and W, are not pretrained with decomposition-relate knowledge before fusion. They are just randomly initialized and directly used for fusion. Results are averaged from four prediction lengths T E (96, 192, 336, 720), using PatchTST (30 as the backbone model. A lower MSE MAE indicates better performance. The best results are in bold.
ETThl
ETTm1
Solar
Traffic
Dataset
Metric
MSE MAE
MSE MAE
MSE MAE
MSE MAE
Ours
0.400 0.421
0.345 0.380
0.199 0.257
0.387 0.268 0.412 0.287
Without Pretraining
0.423 0.435
0.356 0.386
0.217 0.315
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E.3 Ablation Study on Learnable Fusion Factor X
We conduct ablation study verify the importance of our learnable fusion factor As shown in Ta the removal of À degrades the performance, suggesting that it is vital to learn an optimal proportion À to fuse the seasonal and trend knowledge.
Table 5: Ablation study learnable fusion factor x. Ours indicates that we fuse W. s and W under the help of the learnable fusion factor A, letting W = W, + A W, And Fusion Without indicates that we fuse W, and W, without the help of the learnable fusion factor X, only letting
W = W, + W.. Results are averaged from four prediction lengths I" E {96, 192, 336, 720]. using PatchTST [30] as the backbone model. A lower MSE or MAE indicates better performance. The
best results are in bold.
ETTh1
ETTml
Solar
Traffic
Dataset
Metric
MSE MAE
MSE MAE
MSE MAE
MSE MAE
Ours
0.403 0.421 0.345 0.380 0.199 0.257 0.387 0.268
Fusion Without A 0.419 0.436
0.352 0.382
0.208 0.268
0.399 0.282
F More Related Works
F.1
Other Procedure for Time Series Decomposition
As introduced in Section 2.1 the mainstream procedure of previous decomposition- -based methods to decompose the time series into seasonal and trend components and process them with two individual expert models respectively.
Apart from above mainstream procedure, there is also other type of procedure for decomposition-
based methods, which is exemplified by Autoformer [41] and FEDformer [49]. They take time
series decomposition as an inner block of the model structures and they decompose the time series at
each layer, instead of at the beginning of the model. But this more sophisticated procedure is less straightforward and less performing, making not the mainstream choice.
F.2 Drawbacks of Explicit Decomposition
We denote the previous well-established procedures for decomposition- based methods as explicit decomposition, which meets several drawbacks:
(i) Previous methods need to explicitly decompose the input time series and use several expert models for inference, which brings heavier inference costs.
(ii) Previous methods advocate that the seasonal model and trend model should use different model structures, which makes it impossible to improve the efficiency by our Seasonal- -renn Re-parameterization.
(iii) Previous methods pay more attention on data processing, instead of model training. They prefer to design more sophisticated decomposition process (e.g., multi-kernel decomposition (37] learnable decomposition (44) to better decompose the input time series. But they ignore the importance of how to better train the two expert models. They simply use the sub-optimal joint training strategy to train the seasonal and trend models, making their performance potential not fully unleashed.
Due to above drawbacks, these previous explicit decomposition methods suffer from the performance and efficiency issues. And our implicit decomposition can handle these challenges from a brand new perspective, providing better solution to apply decomposition in time series forecasting.
E33
Details of Explicit Decomposition Baselines in Section 4
MICN 37 adopts a sophisticated decomposition process (i.e., multi-kernel moving average decom- position) to better decompose the input time series. And it uses different model structures for two
26

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expert models: Linear layer for trend part and multi-scale isometric convolution for seasonal part. And is trained based on joint training strategy.
Leddam adopts a sophisticated decomposition process (i.e., learnable decomposition) to better decompose the input time series. And it uses different model structures for expert models: Linear layer for trend part and Dual Attention Module for seasonal part. And it is trained based on joint training strategy.
In Section we modify MICN and Leddam with our implicit decomposition designs. In the first step, compare the importance between data processing and model training. We replace their sophisticated decomposition processes (i.e., multi-kernel moving average decomposition in MICN and learnable decomposition in Leddam) with simple single-kernel moving average decomposition. And we apply our decoupled training strategy to replace their original joint training strategy. In the second step, re-examine the feasibility of using same model structure for expert models. We adopt multi-scale isometric convolution for two expert models in MICN and adopt Dual Attention Module for two expert models Leddam.
F.4 Details of Enhancement Frameworks in Section 4.2
Normalization-basee enhancement frameworks are developed based on RevIN RevIN proposes to first normalize the input time series with zero mean and unit standard deviation, and then add the mean and deviation back to the predicted outputs for de-normalization. RevIN puts an assumption that the input time series and the predicted outputs share the same statistical properties. Since this assumption may be at odds with reality, Dish-TS ( further introduces an additional coefficient network to specifically learn the statistical properties for output series. As further improvement, SAN 20 explores statistical properties at a finer granularity (e.g., the sliced level) and FAN [4] conducts normalization in frequency domain. Meanwhile, they also adopt multi-stage and multi- target training strategy to unleash their performance potential, which is similar to our decoupled training strategy.
Prior-based enhancement frameworks can also improve the model performance, provided that the periodicity prior of a dataset is known in advance. Given the priori periodic length P of dataset, SparseTSFF [16] proposes a cross-period sparse forecasting technique, which will downsample the original series into several subsequences based on the value of P and apply model with shared parameters to these subsequences for prediction. And CycleNet [I5] utilizes an additional length-P learnable matrix specifically model the shared periodic patterns of the given dataset.
G More Ablation Results
More ablation results are provided in Figure and which are using PatchTST [30 and RLinear 14 as the original backbone models, respectively.
H Error Bar
We report the standard deviation of DecompNet performance under five runs with different random seeds in T using PatchTST 30 as the original backbone model. The results exhibit that the performance of DecompNet stable.
I Full Error Bar on More Backbone Models
The full results with full error bars of performance promotion obtained by our DecompNet framework are provided in Table 7 It is shown that the average performance improvement is roughly an order of magnitude larger than the random fluctuation from multiple runs (i.e., 0.0x vs 0.00x). Thus, the improvement in our main result (Table is meaningful and not due to chance.
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(b) Abbation roults in ETTml dateeat asing Patch TST backhone model
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1.19
(Dunn)
8.307
(c) Ablation reselts in Satar datuset using PST hackhone model
Ablation rovals in Traffie dataset using Patch TST backbone mode
Figure 8: More ablation study results using PatchTST 30 as the original backbone model. From the top the bottom, each row means design add the original model to enhance it. We report the averaged MSE of four prediction lengths. The inference time is recorded under input-384- --e-predict- 96 setting per iteration. A lower MSE (a shorter blue bar) and smaller inference time (a green star closer to the vertical axis) indicate better performance and efficiency.
Orighel
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(a) Ablation results in ETTh1 doineet using RLinear as backbune model
(b) Ablation resalts in ETTml dataset using RLisear as backbone model
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6.331
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(c) Ablation rocsles in Solar datzoeel ming RI iinsr as buckhone model
(4) Aldation resultts in Traffic dataert teing RLinar as buckhone model
Figure 9: More ablation study results using RLinear [4 as the original backbone model. From the top to the bottom, each row means one design we add on the original model to enhance it. We report the averaged MSE of four prediction lengths. The inference time is recorded under input-384-prediee 96 setting per iteration. A lower MSE (a shorter blue bar) and smaller inference time (a green star closer to the vertical axis) indicate better performance and efficiency.
J How Robust is DecompNet to Non-stationary Test-time Inputs?
DecompNet can robustly handle non-stationary test-time inputs. And we compare our DecompNet with the popular RevIN 111 from the perspectives of mechanism guarantees and empirical evidence:
Table 6: Error bar.
Dataset
ETTbl
ETTh2
ETTml
ETTm2
Horizon
MSE
MAB
MSE
MAE
MSB
MAE
MSE
MAE
0..0110.000
0.39010.001
0.277
0.337::0.001
0.283.10.001
0.337:10.001
0.166110.000
0.25610.000
192
0.391:0.001
0.100100001
0.351=200022
0.386:=0.001
0.328:40.01
0.368:00.001
0.222:0.001
0.293:40.001
336
0.18+00003
0.28+000022
0.371+40.022
0.999=0.002
0.365-H0.022
0,392:+0.002
0.272+00001
0.325+00011
720
0442:0000
0455:000003
0,386-2.003
0,42==0.002
0.405-00.022
0,421=20.022
0. 0.062000022
0.385+00002
Dataset
Weather
Solar-Energy
ECL
Traffic
Horizan
MSE
MAE
MSF
MAF
MSR
MAE
MSF
MAE
96
0.146:000000
0.19240.000
0,172-50.100
0.229::0.000
0.133:40.000
0.233:20.001
0.350:00001
0.347:40001
192
0.190+0.001
0.238+00000
0.197=50.001
0.254=0..001
0.149::0.001
0.250=0.001
0.374:#00002
0.263++0.001
336
0.243:00000
0.2794:00011
0.211:20.001
0.270::0.000
0.163:20.001
0.265:20.00
0.268:00001
720
0.317:0.001
0.33210.001
0.216:2.002
0,2711:00002
0.199:10.001
0,293-20.001
0.135:10.003
0.293:10.022
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Table 7: Full results with full eπror bars of performance promotion obtained by our DecompNet framework. We conduct experiments in long-term forecasting tasks and adopt the mainstream state- of-the-art time series models as backbone models. The input length is fixed as 384 and the prediction lengths are in T E (96, 192, 336, 720). Avg means the average results from all four prediction lengths. A lower MSE or MAE indicates better performance and the best results are in bold.
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J.1 Mechanism Guarantees
DecompNet has mechanism to handle non-stationary test-time inputs, but the mechanism is different from RevIN's:
RevIN is a normalization- based method. It alleviates the non-stationary issue from the perspective of statistical measures.
DecompNet is a decomposition-basee based method. It handles the non-stationary issue based on time series decomposition, since decomposition can unravel the entangled temporal patterns in raw time series and extract the stationary components [0
J.2 Empirical Evidence
To prove our superiority in handling non-stationary issues, we compare our performance with RevIN in the top-3 highly non-stationary datasets, which are ETTh2, Exchange and ILI [13 2 48 19).
For ILI, the input length is 96 and the results are averaged from prediction lengths f24, 36, 48, 60). For other datasets, the input length is 384 and the results are averaged from 4 prediction lengths (96, 192, 336,737777 720).
As shown in Table R DecompNet outperforms RevIN in all these highly non-stationary datasets, indicating its better ability in handling non-stationary issues.
J.3
Conclusion
As verified on the currently available and publicly accessible datasets in Section 4.2 and in Appendix DecompNet performs the best in handling non-stationary issues and is even better than RevIN and other normalizationnbasse enhancement frameworks. These results indicate that DecompNet can robustly handle non-stationary test-time inputs.
J.2
29

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Table 8: Performance comparison with RevIN in the top-3 highly non-stationary datasets. The Original model equips with RevIN by default. And Ours replaces the RevIN with our DecompNet framework. So these results can provide fair comparison between our DecompNet and RevIN in handling non-stationary issues.
Models
PatchTST
iormnffommer
RLinear
RMLP
ModeraTCN
Settings
Original
Decomppett
Original
DecompNNel
Original
DecompNee
Original
+ Decomppet
Original
+DecompNet (Ours)
(Ours)
18
(Ours)
14
(Ours)
(Ours)
23
Metric
MSE MAE
MSE MAE
MSE MAEJ
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
ETTh2
0.391 (0.366 0.387
(0.377 0.44
0.389
0.354
0.338
|0.364
0.401
0.390
00.328 0.382 (0.312 0.373
Exchange (0 3.787 0.419/0.300 0.404
(o:378 0.418/0.317
0.291
|0:378
0.41
(0.314
(0.385
0:414
(0.320
00.376
(0.319
2,096 0.969/ |1.04 0.886 |,2331 1.020|1.955
1,959 0.934
|3.686 1.366/2.305
1,042 |2.796 1.172 |2.056 0.972 11.44 1.198
1.198|1.738 (1733 0.874
We are well aware that there are non-stationary issues that cannot be represented by the current datasets. In future work, we will keep monitoring the development time series community and promptly validate the effectiveness of our methods on the latest and more challenging non-stationary datasets.
K DecompNet's Generalization to Other Forms of Decomposition like STL and Scalability to More Than Two Decomposition Components
K.1 Preliminary Exploration on the Scalability and Generalization of DecompNet
We conduct a preliminary exploration on the scalability and generalization of DecompNet. The experiment settings are follows:
We integrate our DecompNet with STL and decompose the time series into three finer components (i.e., seasonal, trend and residual components).
We introduce more fusion factors to fuse three expert models, letting W = W, + A,W, - W2, where W,, W,, W, are the weight matrices from seasonal, trend and residual expert models respectively.
We use Riinear
14 and PStchTST as the backhone models.
As shown in Table 9 after integrating DecompNet with more advanced decomposition methods and scaling into more and finer decomposed components, we observe continuous performance improvement, which can prove the scalability and generalization of DecompNet.
Table Preliminary exploration on the scalability and generalization of DecompNet.
ETTh1
ETTml
Solar
Traffic
Dataset Metric
MSE MAE
MSE MAE
MSE MAE
MSE MAE
PatchTST + 2 Components 0.403 0.421
0.345 0.380
0.199 0.257
0.387 0.268
PatchTST + 3 Components 0.394 0.407
0.338 0.370
0.193 0.247
0.380 0.261
RLinear + 2 Components
/0.410 0.422
0.359
0.377
0.227 0.274
0..425 0.286
RLinear + Components
0.399 0.410
0.349 0.367
0.213 0.265
0.412 0.277
K.2 Discussion on Potential Technical Challenges during Scaling
We discuss some potential technical challenges follows, including increased fusion complexity, parameter growth, or optimization difficulty.
Based preliminary exploration, we find that:
30

---

Fusion complexity and optimization difficulty are not increased. Through directly general- izing the equation from W W, AW, W W, + λ W; + X2 Wr, we can achieve the fusion of three expert models. And the training process is still stable and can converge quickly.
Parameter growth: The number of parameters is only increased during training. When scaling to more decomposition components, we need to train more expert models and introduces more fusion factors. This will introduce more parameters, but will also bring better performance. During inference, all expert models are fused as one single model, so there is no additional inference costs.
K.3 Future Work
Above preliminary experiments demonstrate the scalability and generalization of DecompNet. In future work, will conduct more in-depth researches to further unleash its scaling potential and report more insights on optimization.
L Comparison with Other Prominent Decomposition-based Models
We compare method with DLinear 45 NHiTss
As shown in Table TO method surpasses these prominent decomposition based models, achieving an average MSE reduction of 11.1% compared DLinear, 16.7% compared NHiTs and 18.6% compared N-BEATS, thus verifying our performance superiority.
3 and N-BEATS 31.
Table 10: Comparison with other prominent decomposition- based models like DLinear, NHiTs and N-BEATS.
ETTh1
ETTml
Solar
Traffic
Datasct
Metric
MSE MAE
MSE MAE
MSE MAE
MSE MAE
DecompNet + PatchTST
0.403 0.421
0.345
0.380
0.199
0.257
0.387 0.268
DecompNet RLinear
0.410 0.422
0.359 0.377
0.227 0.274
0.425 0.286
DLinear
0.428 0.444
0.365 0.385
0.255
0.338
0..45 0.318
NHiTs
0.486 0.477
0.372
0.397
0.265
0.348
0.479 0.327
N-BEATS
0.500 0.486 0.385 0.404
0.266 0.345
0.487 0.331
M Enhancing the Performance of Classic Models (e.g., Standard RNN)
We utilize our DecompNet to enhance the performance of classic models (e.g., standard RNN). As shown in Table [I our method can also work on the classic model and greatly improve its performance, making catch up with the state-of-the-art level in time series forecasting.
Table 11: Enhancing the Performance of Classic Models (e.g., Standard RNN)
ETTh1
ETTml
Solar
Traffic
Dataset
Metric
MSE MAE
MSE MAE
MSE MAE
MSE MAE
RNN
1.114 0.794
1.130 0.779/0.913 0.498
1.043 0.486 0.474 0.309
RNN + Ours 0.448 0.447
0.372 0.404
0.246 0.288
N
Reproducibility Statement
31

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N.1 Code Release
Our method is introduced detail with equations and figures in the main text. All the implementation details are included in the Appendix, including dataset descriptions, evaluation metrics, model configurations and experiment settings. Code available this repository: https://gi com/ Ttodhhh/Decompleet
N.2 Code Related to the Implementation Details of Model Fusion Process
The implementation of our fusion process are provided in the following two files within our repository (i.e., attil//sasoonal_reeee trend_reparant and quils/seasonal_rees Ltt___trrrerraarnnte Sarraamm_o___aarrppappp Please refer these files for implementation details of our fusion process.
More Discussion on the Choice of Moving Average Window Size
In this paper, we use fixed moving average window size of 25 for following reasons:
Experimental Fairness: The primary motivation for adopting fixed window size of 25 is to ensure strict compliance with mainstream protocols. Starting from the famous decomposition-based methods like Autoformer and DLinear, becomes a widely adopted protocol to use fixed moving average window size of 25 for the decomposition imple- mentations. Consequently, our main experiments adhere to this convention ensure fair experimental setting.
Simplicity: In our humble opinion, minimizing the numbers of tunable hyperparameter is critical for real-world applicability, it reduces the implementation complexity and enhances the adaptability across various real-world scenarios. Since the window size of 25 can already provide sufficient performance while aligning with mainstream conventions, we prioritize simplicity and directly use the fixed moving average window size of 25 without further tuning this hyperparameter.
To better choose the appropriate moving average window size:
For the setting in experimental scenarios: We advocate using fixed moving average window size of 25 for the consideration of simplicity and experimental fairness.
For users seeking to maximize performance in their own real-world application scenarios: Although the window size of 25 can already provide sufficient performance, we are well aware that other window sizes may bring even better performance. To fully push the performance limit in specific real-world applications, we encourage the users to explore other moving average window sizes. And the powerful technique like automatic period detection can serve as an effective guidance for their further parameter adjustments.
P Ethies Statement and Broader Impact
P.1 Ethics Statement
Our work only focuses on the time series forecasting problem, so there no potential ethical risk.
P.2 Impact on Future Research
In this paper, we pioneer the idea of implicit decomposition, which is a brand new idea to apply decomposition in time series forecasting and provides guidance upgrade the time series decomposi- tion into a agnosicc enhancement framework. Based on this idea, we establish a new category of decomposition based enhancement framework and propose DecompNet as its representative, which demonstrates compelling enhancement capability. framework generality, performance superiority and efficiency superiority, making decomposition -baed enhancement framework surpass well-recognized normalization-baal frameworks for the first time. We hope our exploration on implicit decomposition can provide some fresh perspectives and facilitate the future researches in both time series decomposition and time series enhancement frameworks.
32

---

P.3 Broader Impact on Real-world Applications
As an advanced -adeos-ggnostic time series enhancement framework, our method can enhance the forecasting performance of the latest and state-of-the time series models, greatly pushing the performance limit of time series forecasting. Therefore, our proposed method makes promising to tackle real-world forecasting problem, helping our society make better decisions and prevent risks in advance.
Meanwhile, our DecompNet can enhance the model performance without introducing any additional inference costs, enjoying totally the same inference costs the original backbone model. This inference-cost-free property allows for wider application of our method, especially in some real- world scenarios that with limited computational budgets.
Our paper mainly focuses on scientific research and has no obvious negative social impact.
Limitations and Future Work
In this paper, we mainly implement our idea of implicit decomposition based the mainstream moving average decomposition and we only decompose the time series into seasonal and trend compo- nents. Combining implicit decomposition with more advanced decomposition implementations (e.g.. Seasonal-Trew decomposition using Loess (STL) [5] and frequency domain decomposition (38) and decomposing the time series into finer components (e.g., seasonal, trend and residual components) can further improve the performance. It will be our future work to conduct these further explorations and further push the performance limit of our method.
In this paper, our fusion mechanism relies on strict layer-wise alignment. The prerequisite for the fusion process is that the expert models must have the same model architecture. And it will be our future work to further study on how to handle the architectural mismatches during fusion and study the fusion of models with different architectures.
In this paper, we mainly verify the effectiveness of our method through extensive empirical results and comprehensive experimental evidence. It will be future work conduct deeper analysis model fusion and provide better theoretical explanation.
R
Full Results
Due the space limitation of the main text, place the full results of all experiments in the following subsections. And we also provide the showcases in Appendix
R.1
Performance Promotion by DecompNet Framework
The full results of performance promotion obtained by DecompNet framework are provided in Table 12
R.2
Compared with Explicit Decomposition
The full results of performance comparison between our implicit decomposition and the previous explicit decomposition are provided in Table
R.3 Compared with Other Enhancement Frameworks
The full results of performance comparison between our DecompNet and other enhancement frame- works are provided Table 14 and which adopt RLinear [14 and PatchTST 30 backbone models, respectively.
33

---

Table 12: Full results of performance promotion obtained by our DecompNet framework. We conduct experiments in long-term forecasting tasks and adopt the mainstream state-of- of-the-art time series models as backbone models. The input length is fixed as 384 and the prediction lengths are in T E (96, 192, 336, 720). Avg means the average results from all four prediction lengths. A lower MSE or MAE indicates a better performance and the best results are in bold.
Models
PatchTST
iTrnnfoommer
RLineur
RMLP
ModemnTN
Settings
Original 30
+DecompNett (Ours)
Original 18
+DccompNet (Ours)
Original 14
+DecompNet (Ours)
Original 13
+DecompNed (Ours)
Original 23
+DecompNet (Ours)
Metric
MSE MAE
MSE MAE
MSE
MAE
MSE
MAE
ISSE
MAB
MSE
MAB
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
96
0.376
0.401
0.361
0.390
00.413
0.427
0.379
0.404
0.332
0.397
0.369
0,392
0.339
0.414
0.375
0.399
0,368
0.355
0.382
0.421
0.411
0:421
0.403
0.412
0.439
0.399
0.410
0.385
0.404
336
0.434
0.439
0.418
0.428
0.455
0.458
0.429
0.441
0.425
0.429
0.427
0.427
0.499
0.479
0.437
0.435
0.394
0.415
0.376
0.407
720
0.451
0.471
0.442
0.455
005528
0.521 0.446
0.469
0.440
0.465
(0.441
0.456
0.595
0.541
0.469
0.467
0.445
0.461
0..43
0.450
Avg
0.418
0.433
00.403
0.421
[0.461
0.464
0.417
0.435
[0.415
0.428
[0.410
0.422
|0.482
0.469
00.423
0.430
|0.403
0.419|0.387
0.411
96
0.275
0.337
0,277
0.337
(0.306
0.363 0.279
0.340
0.2778
0.338
(0.272
0.334
10,302
0.358
0.283
0.341
|0.263
0.332 0..555
0.329
E17h2
0.388
0.386
00.366
0.402 0.347
0.356
0.393
00.337
0.376
0.350
0.387
0.321
0.320
0.378
336
0.368
01403
0.371
0.399
0.400
0,427
0.356
0.404
0.364
0.408
0.356
0.399
00.387
0.416
0.367
0.405
0.3100
0.374
00,290
0.163
720
0.398
0.435
0.386
0.425
0.:45
0,454
0.392
0.427
00:419
0.447
(0.36
0.426
(0.416
0.443
0.396
0.433
00,417
0.447
0.382
0,422
Avg
0.349
0.391|
00.346
0.387
(0.377
0.412 0.344
0.389
[0.354
0.397
|0.338
0.383
|0.364
0.401
(o.347
0.390
|0.328
0.382
(0.312
0.373
96
0,292
0.344
0.283
0.337
(0.314
0.366 0.299
0.354
00.316
0.355
00.303
0.344
10,309
0.356
(0.297
0.351
|0,303
0.35|
0..297
0.343
0.340
0.374
0.368
00353
0.389 0.336
0.337
0.366
0.336
0.364
00
0.380
0.337
0.352
0.379
0.337
0.367
336
0,367
0.394
0,365
0.,92
003883
0,405 0.372
0,392
0.375
0,389
0.370
0.384
00.383
0.403
0.364
0.388
0.375
0.393
00.372
0.390
720
0.422
0:427 0.405
0.421
00:437
0.435 0.424
0.422
0.4126
0.415
(0.426
0.415
00:447
0.437
0.421
0.415
00:429
0.421 00.410
0.416
Avg.
0,355
0.385
01.445
0.380
(0.372
0.399
0,358
0.385
[0.364
0,381
[0.359
0,377
|0.371
0.394
|0.35
0.383
|0.365
0.386)0.354
0.379
96
0,166
0.255
0.166
0.254
17.179
0,271 0.172
0.265
00.164
0.25.
(0.164
0.253
10.180
0.26770.177
10.171
0.260
(0.109
0.259 00.170
0.257
ETTm2
192
0.228
0.299
0.222
0.293
0.243
0.313 0.227
0.298
0.227
0.297
0..18
0.289
0.2247
0.308
0.228
0.299
0.223
0.296 0..222
0.292
336
0,280
0.331
0,272
0.325
0.2995
0.345
0.278
0.331
0.275
0.329
(.222
0.324
0.289
0.341
0.279
0.332
0.288
0.339 0..668
0.324
720
0.373
0.389
0.366
0.385
(0.377
0.397
0.371
0.388
00.368
0.386
(0.365
0.382
0.366
0.387
0.358
0.382
0.365
0.386 (0.357
0.383
Avg
0,262
0.319
0,257
0.314
[0.274
0.352
0.262
0,321
(0,259
0.316
(0.255
0,312
|0.271
0.326
(0.259
0.318
|0.261
0.320
0.314
96
0.150
0.203 0.146
0.194
00.160
0.230 0.155
0.207
10.174
0.227
[0.162
0.217
10.169
0.220
0.214
[0.15]
0.205
0..147
0.199
Weather
192
0.194
0.244
0.190
0.238
0.203
0.251
0.198
0.247
0.216
0.261
00.206
0.256
0.221
0.259
0.207
0.254
0.197
0.248
0.195
0.243
336
0,243
0.279
0,243
0.279
0.255
0.287
0.245
0.284
0.263
0.294
0..253
0.290
0.260
0.293
0.258
0.293
0.251
0.287 0.253
0.286
720
0.319
0.333
0.317
0.332
(0.355
0.339
0.317
0.333
[.339
0.340
00.20
0.336
0.333
0.345
0.327
0.340
0.332
0.340
00.128
0.337
Avp.
0,227
0.264
0.224
0.261
10:235
0,277 0.229
0.268
[0,246
0,281
[0.235
0,275
|0.244
0.279
00.39
0.275
|0.233
0.270(0.23)
0.266
96
0.197
0.291 0.172
0.229
18.
0.246 0.185
0.245
[0.223
0.297
(0.201
0.257
10.215
0.269
0.196
0.234
[0.194
0.261
00.177
0.232
Kolar-Energy
192
0.214
0.319
0.197
0.254
00.209
0.269 0.209
0.270
0.251
0.312
0..31
0.274
0.230
0.291
0.220
0.252
0,216
0.317
0.19M
0.251
336
0.235
0.328
0.211
0.270
(0.222
0.281
0.207
0.258
0.265
0.338
0.238
0.282
0.238
0.304
0.231
0.254
0.230
0.321
0.215
0.275
720
0.225
0.329
0.216
0.274
(0:231
0.291
0.214
0.263
(0:284
0.345
0.238
0,282
0.223
0. 0.306
0.220
0.256
0.233
0.323
00.313
0.272
Avg
0,218
0.317
0.199
0.257
[0.212
0.272
0.204
0.259
(0,256
0.323
(0.227
0.274
|0.231
0.294
00.214
0.249
|0.218
0.306
0.200
0.258
96
0,133
0.232
0,133
0:233
00.140000
0,240
0.133
0.230
101151
0:251
(0.140
0.233
10.144
0.239/00.134
0..34
0.229
(0.137
0.234
00.133
0.233
192
0.148
0.246
0.1-9
0.250
0.1
0.258
0.153
0.250
0.163
0.262
0.153
0.245
0.1558
0.251
0.151
0.245
0.156
0.254
0.144
0.248
0.169
0.265
0.163
0.265
175 0.275
0.165
0.264
0.9
0.277
0.170
0.262
0.114
0.270
0.167
0.261
0.168
0.265
0.164
0.266
720
0,202
0.295
0.199
0.293
(0.215
0.310
0.219
0,313
0.2188
0.308
0.208
0.295
00.213
0.501
0.207
0.295
0.200
0.303
0..19
0.285
Avg
0.163
0.260
(0.61
0.260
[0.177
0.271
0..168
0.264
[0.178
0.275
(0.168
0.259
10.172
0.265
[0.165
0.258
|o.168
0.264
00.159
0.258
96
0,385
0.277
0.350
0.247
(0.393
0.298
0.357
0.261
10.433
0.305
(0.403
0.277
10.404
0.294
00.89
0.275
[0.39]
0.283
(0.363
0.255
Traffi
192
0.401
0,283
0.374
0.263
00.414
0.312
0.387
0.276
0.441
0.307
0.416
0,280
0.423
0.302
0.403
0.280
0.407
0.287
0.380
0.267
0.287
0.268
00.432
0.397
0,279
0.453
0.313
0.424
0.285
0.4439
0.285
0,416
0.293
0,393
0.271
720
0.443
0,304
0.435
0.293
00.472
0.347
0.437
0.305
(0:483
0.331
00.456
0,341
0.417
0.327
0.444
0.301
0.415
0,309
0.438
0.308
Avg
0.430
0.283]
0.387
0.268
[0.428
0.320
0.395
0.280
[0.453
0.314
(0.425
0.286
|0.436
0.308
00.413
0.285
[0.417
0.293
(0.39A 0.275
34

---

Table 13: Full performance comparison between our implicit decomposition and the previous explicit decomposition. We conduct experiments in long-term forecasting tasks. The input length is fixed as 384 and the prediction lengths are in T E {96, 192, 336, 720). And we also gradually modify the explicit decomposition methods with our implicit decomposition designs. Avg means the average results from all four prediction lengths. A lower MSE or MAE indicates a better performance. And Total Arg means the further average results from all eight datasets.
Type
Implicit Decorppoaiion [Ours]]
Explicit Decomposition
Frameworks
DecompNict (Ours)
MICN 37
Leddam 1
Settings
DecompNaal
DecompNet
Original
Use Decoupled Training Use Same Simutture
Original t+
Use Decoupted Triiiig
Use Sane Stractue
*PaichTST
iRLincar
+Our Design
+Our Design
+Our Design
+Our Design
Metric
MSE MAF
MSF
MAR
MSF
MAF MSE
MAF
MSE
MAF
MSF
MAF
MSF
MAF
MSE
MAF
96
0.361 0.399
0,392
0.408 0.434 0.396
0.417
00,400
0.420
0,381 0,400
0.575
0.396
0.373
0.394
192
0.403
0442
0.421 0.441 0412
0.431
0.413
0.429
0.499 0.427 0.411
0.413
336
0.418
0.428 0.428
0.427
0.427
0.461 0:470 0.440
0.447
0.445
0.450
0.439
0:445 0.429
0.436
0.432
0.438
720
0.412
0.155 0.111
0.456
0.695 0.629 0.515
0.489
0.514
0.485
0.463
0.474
0.440
0:163
0.461
0.461
Avg
0.400
0.121 |0.410
0.422
0.999 0.199 0.441
0,446
00.44
0.416
0.426 0.437 0.414
0.430
00:455
0.430
96
0:277 0.337/0.272
0.334
0.306 0.377 0.281
0.343
0.285
0.346
0.308 0.358 0.275
0.336
0.278
0.340
E1
192
0.351
0.333
0.376
0.423 0.447 0.351
0,390
0.351
0.391
0.370 0.398 0.337
0.376
0.336
0.378
336
0.399 0.356
0.397
0.555 0.532 0.365
0.405 0.436 0.355
0.356
720
0.386
0.425 0.386
0.426
0.903 0.738 0.393
0.432
0.394
0.433
0.446 0.461
0.382
0.423
0.383
0.424
Avg
0.346
0.337
0.383
0.562 0.524 0.348
0.393
0.349
0.393
0.3382 0:411 0.337
0.383
0.338
96
0.283
0.337 0.303
0,344
0.321 0.368 0.304
0.350
0.305
0.352
0,317 0.362 00301
0.347
0.298
0.345
E17ml
192
0.336
0.364
0,300 0.292 0.344
0.346
0.357
0.300
0.370
0.336
336
0:365 0.392 0.370
0.370
0.384
0.429 4:42 0.379
0.395
0.376
0.393
0.389 0:407 0.371
0.388
0.372
0.300
720
0.405 0.421 0.426
0.415
0.474 0.459 0.431
0.423
0.434
0:427
0.442 0:435 0.424
0.418
0.420
0.414
Avg
0.345
0, 0.380
0,377
|0.396 0.413 0.365
0,386
10.365
0.386
10.376 0.398 (..39
0.381
|0.357
0.380
96
o 0.166 0.254
[0.187 0.77
0.255
10.168
10.185 0:267 66166
10.170
0.257
B1Tm2
192
0.222
0.293 0.218
0.289
0.265 0.341 0.222
0.291
0.221
0.293
0.251
0.200
0.291
0.223
0.290
336
0.272 0.325 0.272
0,324
0.333 0.383 0.275
0.278
0,317
0.353 0.272
0.272
0.324
720
0.366
o. 385 p.365
0.382
0.466 4666 0.368
0.384
0.367
0.382
0.379 0.394 0.364
0.381
0.369
Aug
0:257 0.314j
0.312
[0.313 0.367 (0:258
0.315
|0.259
0.315
10,283 0.331 (0.255
0.313
|0.259
0.314
96
0.146 0. 19400 162
0,217
0.177
0.246 0.173
0.226
0.199
0.215
0.156
0.206 0.52
0.205
00,149
0.301
Weather
192
0.238 0.206
0.256
0.228 0.292 0.218
0.263
0.209
0.199
0.247 0.196
0.241
0:191
336
0:243 0.279 0.25
0.290
0.319 0.372 0.264
0.296
0.265
0.297
0.249
0.286 0.242
0.279
0.258
0.2775
720
0.317 0.332 0.320
0.336
0.577 408 0.332
0,344
0.336
0.345
0.333 0.339 0.330
0.332
0,311
0.304
Avg
0.224 0.261|0.235
0.275
|0.276 o 330 ( 1.47
0.282
10.342
0.279
J0,234 0.270/p.238
0.264
|0,222
0.260
Solar-Fnees EnneEy
96
0.172
0.229
0.257
002118 0.366
0.257
00118
10.201 0.267 08184
10.176
192
0.197 0.254
0.274
0.250 0 299 0.232
0.273
0.234
0.278
0.222
0.280 0.206
0.260
0.196
0.246
336
0.211
0.270 0.238
0,282
0.267 0.312 0.220
0,283
0.250
0.288
0.253
0.290 0.215
0.279
0.212
0.271
720
0.216 0.274 0.27400.33
0.238
0.282
0.208 0.313 0.255
0.281
0.255
0.281
0.245
0.302 0.215
0.275
0.217
Avg
0.399 0.277
0.214
00251 0.298 0.234
0.274
102332
0.273
102225 0.286
0.265
00.000
0.255
96
0.133 0,233/0, 140
0.233
001662 0 275 0.1
0.250
1.156
0.257
0.137
0.236 1316
0.231
001131
0.228
BCL
192
0.
0.245
0.177 0.291
0.160
0.253 0.151
0.150
336
0.103 0.260 0.10 0.170
0,262
0.195 0.305 0.183
0.286
0.183
0.288
0.169 0.200 0.166
0.264
0.165
0.266
720
0.399 0,293 o
0.295
0.215 0.326 0.210
0,312
0.215
0.319
0.205 0.500 0.198
0.291
0.200
0.297
Avg
0.361 0.261)o.166
0.259
[0.187 0.300 0.179
0.282
00118
0.285
001168 0.264 163
0.259
|0.162
0.259
96
0.277
[.499 0.318
0.294
0.466
0.291
103991
0.286 0.365
10.367
Traflic
192
0.374 0.263 0.263..11 0.416
0.280
0.500 (1480
0.302
0.478
0.300
0.421
0.207 0.396
0.227
0.400
0.281
336
0.389 0 0.124
0.285
0.517 0.321 0.490
0.307
0.487
0.305
0.432 0.302 0.405
0.279
0.402
0.276
720
022930.450
0,303
0.549 0.332 0.521
0.319
0.516
0.317
0.464
0.314 0.440
0.302
0.446
Avg
0.387 0.268j0.425
0.286
|0.516 0.322 0.489
0.305
|0.487
0.303
[0.427 0.300j0.402
0.280
|0.404
0.283
Tocal Avg
0.319/0030
0.324
[0.375 0.381 (0.320
|0.320
|0.315 0.337|0.295
|0.295
35

---

Table 14: Full performance comparison between DecompNet and other enhancement frameworks. We conduct experiments in long-term forecasting tasks and adopt RLinear [14] as the backbone model. The original backbone model equips with RevIN [ by default. And we replace RevIN with other time series enhancement frameworks for comparisons. The input length is fixed as 384 and the prediction lengths are in T E {96, 192, 336, 720). Avg means the average results from all four prediction lengths. A lower MSE or MAE indicates a better performance. The best results are in bold and the second best are underlined. And Total Avg means the further average results from all eight datasets.
Backbone Model: RLincar
Time Series Enhancement
Decomposition-busee based
Prior-based
Normalization-buser
Original Backbone Model
Frameworks
+DecompNet (Ours)
+CycleNet SSpraseTFFF
+FAN
+SAN 20
+Dish TS
+RevIN I
0
Metric
MSE
MAE
MSE
MAB
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
96
0.369
0.392
0.374 0.39 0,402 0.417
0.395
0.419 (0.381 0.398 0.419
0.433
0.372
0.397
E7Th
0.403
0.412
0.428 0.434
0.416 0.439
0.455 (0.416 0.419
0.447
0.411
0.421
336
0.427
0.427
0.448
0.448
0.422
0.420
0.490
0.485 (0.432 0,432
0.473 0.473
0.425
0.429
720
0.441
0.456
0.462
0.474
00.432
0.431
0,560
0,544 (0.448 0.461
00.512 0.522
0.450
0.465
Avg
00.410
0.422
0.4428
0.438
(0.418 0.427
00.473
0.476 (o.419 0.428
(0:463
0.470
[0.415
0.428
96
0.272
0.334
0.299
0.355 00.303
0.351
00.118
0.382 (0.282 0.342
0..343
0.398
0..278
0.338
E7
0.376
01.379
0.397
0.434 (0.343 0
0.449
0.462
0.356
0.393
336
0,356
0.397
0.369
0.411
0,381
0.410
0.454
0.465
(0.366
0.406
0.564
0.526
0.364
0.408
720
0.386
0.426
0.405
0.439 0.390
0.424
0.623
0.572 (0.394 0:435
0.7748
0.619
0.449
0.447
Avg
|0.338
0,383
(0.356
0.399
(0.363
0.396
00.4488
0.463
(0.346
0.393
00.526
0.501
|o.354
0.397
96
10.303
0.344
0.312
0.35410.311
0.354
00.311 0.353
00.301
0.350
00.297
0.347
00.312
0.354
00.316
0.355
ElTml
0.364
013446
0.373
0.341
0.375 0..33
0.366
0.345
0.366
336
0.370
0,384
0.378
0.392
0.377
0.391
0,376
0.397
00.367
0.38.
0.379
0.395
0.375
0.389
720
0.426
0.415
0.433 0:421
0.424
0.416
0.438
0.436 (0.415
0.419
0.4441 0.436
(0.426
0,415
Avg
|0.359
0.377
00.367
0.385
(0.365
0.383
(0.364
0.390 (0.353 0.379 (0.369
0.390
(o.364
0.381
0.164
0.253
0.5
0.260/0.
1800 0.268
10.
0.266 (0.177 0.272
0.1178 0.275
00.164
0.253
ETTm2
0.218
0.289
0.223
0.29
0.231
0.301
0.276
0.344 (0,224 0.301
0.250
0.331
0.227
0.297
336
0.272
0.324
0.277
0.330
0,288 0.335
0.324
0.371 (0.276
0.337
0.308
0.385
0.275
0.329
720
0.365
0,382
0.372 0.391 0.337 0.386
0.420
0.434 (0.362 0.391
0.445
0.456
0.368
0.386
Avg
|0.255
0.312
0.2262
0.319 002668 0.325
(0,299
0.354 (0.260
0.325
00.302
0.362
(0.259
0.316
00162
0,217
0.1666
0.220/0.
174 0.225
1.6
0.209
10.1 0.2.11
00.176
0.235
00.174
0.227
Weather
192
0.206
0,256
0.213
0.260
0.216 0.260
0.202
0.259 (0.196
0.253
0.217
0.273
0.216
0.261
336
0.253
0.290
0.259 0.292
0,265 0.297
0.255
0.297
(0.246
0.294
0.262
0.31
0.263
0.294
720
0.320
0.336
0.326 0.339
0.332
0.343
00.320
0,352
0.321 0.346
0.3225
0.362
0..329
0,340
Avg
|0.235
0.275
0.2241 0.278
(0.247
0.281
00,233
0.279
(0.229
0.276|0.2445
0.276
(0.245
0.295
(0.246
0.281
Solar- Solar-Energy Energy
10.201
0.257
0.212
0.267 [0.227 0.267
10.197
0.248
19.898
0.253
0.2224
0.294 (0.223
0.297
192
0.231
0.274
0.240
0.293 0.259 0.287
0.229
0.270
0.2334
0.273
0.270
0.312 0.251
0.312
336
0.238
0.282
0.255
0. 302 0.276 0.297
0.240
0.280
(0.243
0.287
0.264
0.323
0.265
0.338
720
0.238
0.282
0.257 0.301 00.299 0.293
0.233
0.277
0.241
0.281
0.266
0.326 00.284
0.345
Avg
0.222
0.274
002241
0.291 |0266
0.286
(0.226
0.269
(0.229
0.274
0.256
0.314 |0.256
0.323
1.144
0.233
0.233
147
0.241
140
0.239
1.0
0.2334
0..
160
0.266
10.151
0.251
ECL
192
0.153
0.245
0.154
0.247
0.157
0.250
0.16
0.254
0.1555
0.247
0.172 0.276
0.163
0.262
336
0.170
0.262
0.170
0.264
0.174
0.268
0.171
0.271
(73
0.267
0.7
0.291
0..19
0.277
720
0.208
0.295
0.208
0.297
0..21
0.298
0.207
0.305
(0.202
0.296
0.221 0.323
0..218
0.308
Avg
[0.168
0.259
0..168
0.260
0.264
00.169
0.267 (0.168 0.261
10.185 0.289
10.178
0.275
10.403
0.277
0.413
0.287
(0.420
0.283
(0.416
0.290
(0.414 0.288
00.438
0. 0.308
(0.433
0.305
Traffic
192
0.416
0,280
0.421
0.290
0..36
0.292
0.430
0.304 00.427
0.298
0.450
0.31
0.44
0.307
336
0.424
0.285
0.432
0.297
0.465
0.310
0.443
0 (0.441 0.305
0.462
0.317
0.453
0.313
720
0.456
0.303
0.459
0.30 3m
0.497
0.330
0.448
0.336 (0.476 0. 0.320
0.489
0.338
0.483
0.331
Avg
00.425
0.286
0.443
205
00.456
0.304
[0.444
1,
(0.440 0. 303
0.460
0.319 00.453
0.314
1d Count
23
29
2
2
Total Avg
|0.302
0.324
|0.312
0,33//0.319
0.333
J0.332 0.351 (0.306 0.351 0.368 00336
0.339
36

---

Table 15: Full performance comparison between DecompNet and other enhancement frameworks. We conduct experiments in long-term forecasting tasks and adopt PatchTST (30] as the backbone model. The original backbone model equips with RevIN [ by default. And we replace RevIN with other time series enhancement frameworks for comparisons. The input length is fixed as 384 and the prediction lengths are in T E {96, 192, 336, 720). Avg means the average results from all four prediction lengths. A lower MSE or MAE indicates a better performance. The best results are in bold and the second best are underlined. And Total Avg means the further average results from all eight datasets.
Backbone Model: ParchTST
Time Series Enhancement
Decomposition-busee bused
Prior-based
Normalization--used
Original Backbone Model
Frameworks
+DecompNet (Ours)
+CycleNet +SpraseTSF
+FAN
+SAN 20
+Dish TS
+RevIN 1
0
Metric
MSE
MAE
MSE
MAB
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
96
0.361
0.390
0.382 0.407
0.368
0.393
0.418
0.430 (0.410 0:418
0.405
0.422
0.376
0.401
E7Th
0.391
0.410
0.419
0.399
0.450 (0.473 0.452 0.447
0.410
0.421
336
0,418
0.428
0.453
0.456
0.434
0.437
0,490
0.476 (0.568 0.502
0:481
0.481
0.434
0.439
720
0.442
0.455
0.468 0.482
00.487
0.486
0.557
0.530 (0.578 0.530
00.579 0.535
0.451
0.471
Avg
|0.403
0.421
00.431
0.443/0..42
0.432
00.478
0.472 (0.507 0.476/0.478
0.474
[0.418
0.433
96
0.277
0.337
0.277
0.348
00.311 0.357
0..300
0.365 (0.319 0.366 0.388 0.426
(0.275
0.337
E7
0.386
01.374
0.405
0.423 (0.396 0,427 0.496
00.354
0
336
0.371
0.399
0.381
0.411
0,375
0.409
0.453
0,507 (0.414 0,436
0.507 0.496
0,368
0.403
720
0.386
0.425
0.396
0.431
00.435
0.471
0,578
0.529 (0.4511 0.475 0.7706
0.576
00.398
0.435
Avg
00.346
0.387
0.335
0.393
00.374
0.411
[0.430
0.456
(0.395
0,426
00.525
0.492
(0.349
0.391
96
10.283
0.337
0.289
0.342
0..309
0.363
00.303
0.351 (0,289
0.342
0..305
0.360
00.292
0.344
ElTml
0.328
0.368
0.371
0.344
0.379
0.341
0.374
(0.334
0.370
0.351
0.383
00,340
0.374
336
0.365
0.392
0.365
0.389
0,395
0.411
0.385
0,400
(0.368
0.395
0.385
0.410
0.367
0.394
720
0.405
0.421
0.428
0.427
0.449
0.434
0.4400
0.436 (0:424
0.433
00:431
0.437
0.422
0.422
Avg
00.345
0.380
00.356
0. 0.382 0.368
0.397
(0.367
0.390 (0.354
0.385
00.368
0.39%
(0.355
0.385
96
00166
0.254
0.177
0.265
087 0.280
00.171 0.266 10.170 0.265 0..172 0.264
(0.166
0.255
ETTm2
192
0.222
0.293
0.234
0.307
0.244
0.316
0.238
0.316 (0.239 0.314 0.236
0.312
0.2228
0.299
336
0.272
0.325
0.279
0.332
0.291
0.341
0,305
0,366 (0.377 0.390 0.304
0.352
0,280
0. 0.331
720
0.366
0,385
0636770
0.395
0.367
0.387
0.414
0.435 (0.550 0.475
0.442
0.437
0.373
0.389
Avg
|0.257
0.314
0.2664
0.325
(0.272
0.331
[0,282
0.344 (0.334 0.361
(0.283
0.341
(0,262
0.319
1.6
0.194
0.145
199
(0.
166
0.226
0.4
0.211 10.150 0.208
0.115
0.207
10.
0.201
Weather
192
0.190
0.238
0.188
0.240
0.208
0.261
0.199
0.254
10.194 0.251
0.195
0.256
0.194
0.244
336
0.243
0.279
0.247
0.286
0,258
0.299
0,249
0.295 (0.24.
0.293
0.248
0.304
0..43
0.279
720
0.317
0,332
0.316
0.333
0,331
0.353
0.317
0,346 (0.322 0.344
0.324
0.370
0.319
0.333
Avg
00.224
0.261
0.224
0.265
(0.241 0.285
[0.230
0.277 [o.227 0.274
22.22/
0.284
(0.227
0.264
Solar- Solar-Energy Energy
00.12
0.229
0.1998
0.266 10. 196 0.273
187
0.243 18. 0.240/0.1
080
0.241
10.197
0.291
192
0.197
0.254
0.271
0.320 0.211 0.286
0.201
0.273 (0.207
0.265
0.202
0.256
0.214
0.319
336
0.211
0.270
0.243
0.306 0,233 0.298
0.,233
0.288 (0.223 0.278
0.218
0.276
0.235
0.328
720
0.216
0,274
0.240 0. 0.223 0.296
0.2255 0.290 (0.232 0.285
0.219
0.276
0.225
0.329
Avg
00.199
0.257
0.233 0.296 (0.220 0.288 [0.209
0.274 (0.211 0.267
|0205
0.262
(0.218
0.317
0..33
0.233
0.139
0.239 138 0.235
137
0.235
10. 0.231
0.6
0.234
10.133
0.232
ECL
192
0.149
0.250
0.157
0.256 0.161 0.264
0.252
0.250
04.148
0.248
0.151
0.246
0.148
0.246
336
0.163
0.265
0.1 0.267
0.171 0.272
168
0.268
0.166
0.267
0.166
0.266
0.169
0.265
720
0.99
0.293
0.210 0. 0.300
0.210 0.309
0.200
0.300
19.
0.294
0. 199
0.295
0.202
0.295
Avg
|0.161
0.260
00.169 0.266
0.270 165 0.263
(0.161
0.260
0.163
0.260
00.163
0.260
00.350
0.247
00.411
0.295 (0.39 (0.96
0.274
[0.405
0.284 (0.408 0.282
00.394
0.277
10.385
0.277
Traffic
192
0.374
0.263
0.422
0.301 0..40 0.281
0.423
0.294 (0.417
0.287
0.408
0.282
0.401
0.283
336
0.389
0.268
0.432
0.309 0..21 0.287
0..337 0.302 (0.436 0.298
0.423
0.291
0.410
0.287
720
0.435
0.293
0.458 0.319 0.442 0.303
0..476 0.321
[ (0.464
0.31
0.452
0.309
0.443
0. 0.304
Avg
|0.387
0.268
|0.431
0.306/0.417
0.306
00.417
0.286
(0.435 0,300
(0.431 0.295
00.419 0. 0.290 00.40
0 288
1d Count
33
36
2
5
Total Avg
|0.290
0.319
|0.308
0.335j0.311 0.338
J0.325 0.347 (0.328 0.343 00.334 0.350 000000
0.332
37

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S Showcases
S.1 Intuitive Comparison between Decoupled Training Strategy and Joint Training Strategy
We provide the intuitive comparison between our decoupled training strategy and previous joint training strategy, We visualize the performance of two expert models seasonal prediction task and trend prediction task, respectively. The results are in Figure To and [T When trained with our decoupled training strategy. the trend expert model can perform much better in trend prediction task. And the seasonal expert model can also better handle the details in seasonal prediction task.
Trend Prediction Task
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Precccioon
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(a) Decoupled Training Strategy (Ours)
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Groandhuth
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(b) Joint Training Strategy (Previous Method)
Figure 10: Intuitive comparison between our decoupled training strategy and previous joint training strategy. We visualize the performance of two expert models in seasonal prediction task and trend prediction task, respectively. We conduct experiment in ECL dataset under amput-344prredctt 192 setting, using RLinear [4] as the backbone model. The blue lines stand for the ground truth and the orange lines stand for predicted values.
38

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Trend Prediction Task
Scasonal Prediction Task
Aedistion
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(a) Decoupled Training Strategy (Ours)
Trend Prediction Task
Seasonal Prediction Task
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(b) Joint Training Strategy (Previous Method)
Figure 11: Intuitive comparison between our decoupled training strategy and previous joint training strategy. We visualize the performance of two expert models in seasonal prediction task and trend prediction task, respectively. We conduct experiment Traffic dataset under 111-338--prrdicct setting, using RL.inaar [14] as the backhone model. The blue lines stand for the ground truth and the orange lines stand for predicted values.
39

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S.2 Intuitive Comparison among Different Enhancement Frameworks
To provide an intuitive comparison among different enhancement frameworks, we provide showeases to the long-term forecasting tasks. The results are in Figure Among the various enhancement frameworks, our DecompNet predicts the most precise future series variations and exhibits superior performance.
-12
-131
NecompNet (Ours)
m
m
mmm
-.34
-35
-139
-.79
CycleNet
SSa
FAN
-30
410
4123
Dish-TS
Original Backbone
Figure 12: Visualization of ETTm2 predictions by different enhancement frameworks under the 1ouut-384-preddee 192 setting. We adopt PatchTST 30 the backbone model for this visualization. The blue lines stand for the ground truth and the orange lines stand for predicted values.
40


Reviews:
[{"review_id": "CRc0x44Skx", "reviewer": "Reviewer_USxs", "summary": "This paper introduces a framework called DecompNet that implements implicit decomposition to improve time-series forecasting performance without added inference cost. The key idea is to train two copies of a base forecasting model on the decomposed seasonal and trend components of the data (so each becomes an expert in one component) and then fuse them into a single model via a novel Seasonal-Trend re-parameterization technique. During this fusion, the two expert networks’ parameters are merged (with a learned scaling factor) into one set of weights, yielding an enhanced model that can directly ingest raw time series and still leverage decomposition-related knowledge.", "questions": "I would consider increasing my score if most of the following points are clearly addressed. Please also refer to the weaknesses section.\n\n- How sensitive is DecompNet’s performance to the choice of decomposition hyperparameters? For instance, the moving average window size (25) was fixed. Did you try other window lengths or methods like STL, and if so, how did they affect results?\n- In the fusion stage with the learnable factor $\\lambda$, do you allow the weights $W_s$ and $W_t$ of the expert models to update as well, or is only $\\lambda$ learned? How stable is this training process? Does $\\lambda$ reliably converge to a meaningful value (e.g. close to 1 or something data-dependent), and have you observed any instances where the fused model underperformed the explicit two-model ensemble?\n- Implicit decomposition is demonstrated with two components (trend and seasonal). How would the framework extend to decomposing into more than two components (say seasonal/trend/residual)? Would you envision introducing multiple fusion factors (e.g. $\\lambda_1, \\lambda_2$) to fuse three expert models, or some iterative fusion process?\n- In existing explicit decomposition approaches, researchers sometimes choose heterogeneous model types for seasonal vs trend components (anticipating different dynamics). Your approach instead uses the same base architecture for both, which has the huge benefit of enabling weight fusion. Did you experiment with using different architectures for the two expert models without fusion (just to see performance)? If not fused, would, say, a CNN for seasonality and an RNN for trend perform better, or did you find that using the same architecture with decoupled training already captures both aspects well?\n- Could you provide more details on the training overhead introduced by DecompNet? For example, how much longer (in terms of epochs or wall-clock time) does it take to train a model with DecompNet compared to training the base model normally?", "limitations": "The authors do address some limitations of their work, albeit briefly in the appendix. They acknowledge that their implementation of implicit decomposition is currently tied to the classic moving-average decomposition with a fixed seasonal/trend split. They note that extending the framework to use more advanced decomposition techniques (such as STL or frequency-domain methods ) and to decompose time series into finer components (e.g., adding a residual/noise component beyond just trend and seasonal) could further improve performance.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n- The paper introduces a new approach to time-series forecasting by injecting decomposition knowledge via training, rather than performing decomposition at inference. This idea of implicit decomposition is highly novel and addresses a gap, as it transforms decomposition into a model-agnostic enhancement that incurs no runtime cost.\n- DecompNet delivers consistent t forecasting improvements across a wide range of models and datasets.\n- The paper is well-written and logically organized.\n\nWeaknesses:\n\n- DecompNet’s benefits come at the cost of a more complex training pipeline. The method requires training two separate model instances on decomposed data and then performing an extra fusion fine-tuning stage. This means the total training time and computational resources are higher than for training a single model once. The authors do argue that the extra stages converge quickly, but the paper does not quantify the overhead.\n- The framework relies on a specific decomposition technique (moving average with fixed window, yielding one seasonal and one trend series). This choice might not be optimal for all data – for example, a window of 25 might not align with the true seasonality in certain datasets, or some time series might have multiple seasonalities or a large residual component. The current method does not adapt the decomposition to data characteristics (e.g., no automatic period detection).\n-  While the paper’s approach is empirically validated, it lacks a deeper theoretical explanation for why the fused model can perform as well as two separate models, especially given the non-linear nature of deep networks. The derivation in Equation (10) is a helpful intuition for linear layers, but it doesn’t formally guarantee performance preservation for complex networks – instead the authors rely on a training-based solution. The success of the fusion is demonstrated experimentally, but the paper does not analyze scenarios where it might fail or whether any bounds/guarantees exist.\n- The study focuses on one particular way of decomposing (seasonal/trend via moving average). It does not explore whether the implicit decomposition idea generalizes to other forms of decomposition or other non-stationarity issues (e.g., piecewise trends, irregular seasonal patterns).\n-", "was_revised": true, "final_justification": "After reviewing the rebuttal and discussion, my overall evaluation (4) remains unchanged. The authors provided useful clarifications on training overhead, robustness to decomposition parameters, and generalization to alternative decomposition methods. However, several of my key questions, particularly those concerning the principled justification of design choices and a deeper theoretical understanding of the fusion mechanism, were not fully addressed. While the approach is technically solid and has practical relevance, the limited resolution of these points prevents me from raising my score. However, I appreciate the overall novelty and scope of the work and the effort that the authors put into the rebuttal. I therefore maintain my positive rating.", "rebuttal": "Many thanks to Reviewer USxs for the insightful review.\n\n> W1, Q5: Quantify the training costs (in terms of epochs).\n+ Thanks for highlighting this. We have reported the quantitative comparison results in **Table 3 in Appendix E.1**. The results show that **the total training epochs consumed by our method are comparable or sometimes less than the original model**.\n+ To ensure visibility and highlight our nice training efficiency, we will move **Appendix E.1** to the main text in our final version.\n\n> W2-1, Q1-1: Parameter sensitivity to the choice of moving average window size.\n+ As shown in our parameter sensitivity study in **Figure 7 in Appendix G**, our method is very robust to the choice of moving average window size:\n    + Our method maintains consistent performance across window sizes spanning an order of magnitude (7 to 169).\n    + Using the default value of 25 can provide consistently strong performance across various datasets.\n+ To ensure visibility and highlight our robustness to moving average window size, we will move **Appendix G** to the main text in our final version.\n\n> W2-2: Why don't we adapt the decomposition to data characteristics?\n+ As mentioned above, our method is very robust to window size and can achieve consistently strong performance across various datasets just using the default value of 25. This property **eliminates the need to spend extra effort analyzing the dataset in advance or adjusting hyperparameters based on the characteristics of the dataset**.\n+ Since we aim to propose a **prior-free** enhancement framework (i.e., a method that does not require any prior analysis of a dataset), we do not analyze the characteristics of the dataset in advance to determine the hyperparameters.\n\n> W3-1: Analysis on model fusion\n\n>> Theoretical explanation:\n+ We sincerely thank the reviewer for highlighting the importance of theoretical explanation. However, rigorous theoretical explanation for model fusion remains an open challenge beyond current methodological capabilities—even leading works in this nascent field (e.g., [1-4]) rely primarily on empirical validation.\n+ In this paper, we mainly verify the effectiveness of our method through **extensive empirical results and comprehensive experimental evidence**, which is consistent with state-of-the-art practice.\n+ It will be our future work to conduct deeper analysis on model fusion and provide better theoretical explanation.\n\n>> Guarantees.\n+ The extensive experimental results in our paper can firmly verify the effectiveness of our fusion process.\n+ Similar researches on model fusion in the fields of Natural Language Processing (NLP) [1-3] and Computer Vision (CV) [4] can also help to guarantees the feasibility of such fusion process.\n\n>> Bounds:\n+ Since deriving rigorous theoretical bounds remains challenging, we expand our experimental analysis to address this concern empirically. \n+ We consider the following boundary cases to test the performance bounds of our method:\n    + Enhancing the performance of state-of-the-art models. The results are provided in **Table 1 in Section 4.1**.\n    + Enhancing the performance of classic models (e.g., standard RNN). The results are provided in **following table**.\n    + Handling highly non-stationary data. Please refer to the **Response to W3, Q3 of Reviewer f7pT** for detailed results.\n+ Above boundary cases cover the extreme scenarios in time series domain. And the results show that our method can work well for each case, which can ensure the reliability of our method in most application scenarios.\n\n|MSE/MAE|RNN|+Ours|\n|:---:|:---:|:---:|\n|ETTh1|1.114/0.794|0.448/0.447|\n|ETTm1|1.130/0.779|0.372/0.404|\n|Solar|0.913/0.498|0.246/0.288|\n|Traffic|1.043/0.486|0.474/0.309|\n\n> W4-1, Q1-2, Q3, Limitation: Can our method generalize to other decomposition method like STL? Can our method extent to more and finer decomposition components?\n+ Following your valuable suggestion, we integrate our DecompNet with STL and decompose the time series into three finer components (i.e., seasonal, trend and residual components). The experiment settings are as follows.\n    + We introduce more fusion factors to fuse three expert models (i.e., $W = W_{s}+\\lambda_{1} W_{t}+\\lambda_{2} W_{r}$).\n    + We use RLinear and PatchTST as the backbone models.\n+ As shown in following table, after integrating DecompNet with more advanced decomposition methods and decomposing the time series into more components, we observe continuous performance improvement, which can prove our ability to further generalize and extend.\n+ It will be our future work to further explore the potential of DecompNet in generalization and extension.\n \n|MSE/MAE|PatchTST + 2 Components|PatchTST + 3 Components|RLinear + 2 Components|RLinear + 3 Components|\n|:---:|:---:|:---:|:---:|:---:|\n|ETTh1|0.403/0.421|0.394/0.407|0.410/0.422|0.399/0.410|\n|ETTm1|0.345/0.380|0.338/0.370|0.359/0.377|0.349/0.367|\n|Solar|0.199/0.257|0.193/0.247|0.227/0.274|0.213/0.265|\n|Traffic|0.387/0.268|0.380/0.261|0.425/0.286|0.412/0.277| \n \n> W4-2: Can our method generalize to other non-stationary issues?\n+ In this paper, we compare our DecompNet with other enhancement frameworks on many public non-stationary datasets. These results verify that DecompNet performs the best in handling non-stationary issues and outperforms all other enhancement frameworks, which demonstrates the strong capabilities of our method in handling non-stationary issues.\n+ We are well aware that there are non-stationary issues that cannot be represented by the current datasets. In future work, we will keep monitoring the development in time series community and promptly validate the effectiveness of our method on the latest and more challenging non-stationary datasets.\n\n> Q2-1: Do you allow the weights $W_s$ and $W_t$ to update as well?\n+ Yes. As stated in **line 202-203**, $W_{s}$ and $W_{t}$ are also updated with $\\lambda$.\n+ Thanks for highlighting this. We will bold this sentence in our final version to ensure visibility.\n\n> Q2-2: How stable is the training process for $\\lambda$?\n+ As shown in **Appendix E.1**, the training process for $\\lambda$ is very stable and it can converge quickly with in 3 epochs in most cases.\n\n> Q2-3: What value do $\\lambda$ converge to?\n+ $\\lambda$ can reliably converge to a meaningful value. This value is data-dependent and this value is usually not equal to 1. \n+ As shown in **Appendix E.3**, the fusion of $W_{s}+\\lambda W_{t}$ consistently performs better than $W_{s}+W_{t}$, which indicates that the final learned $\\lambda$ is usually not equal to 1. Instead, it is learned to be an optimal value data-dependently. \n\n> Q2-4, W3-2: Scenarios where the fusion process might fail. Can our fusion process guarantee the performance preservation for complex networks? Have you observed any instances where the fused model underperformed the explicit two-model ensemble?\n+ As shown in our ablation study in **Section 4.4 and Appendix H**, the fused model outperforms the explicit two-model ensemble in most cases.\n+ The only exception case is in ETTh1, where the MSE sightly increase after fusion. When using PatchTST as the backbone, the MSE increases by 0.001 (0.402-->0.403) (**Figure 8 (a) in Appendix H**). And when using RLinear as the backbone, the MSE increases by 0.003 (0.407-->0.410) (**Figure 9 (a) in Appendix H**). In this exceptional case, although the fused model exhibits inferior performance, **the performance difference is very small**.\n+ These results empirically validate that our fusion process can guarantee performance preservation for both complex networks (e.g., PathcTST) and simple networks (e.g., RLinear).\n\n> Q4: \"using different architectures for two expert models\" or \"using the same architecture for two expert models\"\n\n>> Did you experiment with using different architectures for the two expert models without fusion (just to see performance after decoupled training)? \n+ Yes. In **Section 4.3**, we conduct experiments with two popular heterogeneous models, i.e., MICN and Leddam. MICN uses linear layers for trend part and uses CNN for seasonal part. Leddam uses linear layers for trend part and uses attention for seasonal part.\n+ Our decoupled training strategy can also enhance the performance of heterogeneous models. Compared with previous joint training strategy, our strategy reduces the average MSE of MICN from 0.375 to 0.320, and reduces the average MSE of Leddam from 0.315 to 0.295 (**Figure 5 (b) in Section 4.3**).\n\n>> Did you find that using the same architecture with decoupled training already captures both aspects well?\n+ Yes. After using our decoupled training strategy, we find that adopting the same architecture for two expert models can achieve the same performance as heterogeneous models (**Figure 5 (b) in Section 4.3**).\n+ We attribute this finding to our decoupled training strategy, since it can better train the two expert models and fully realize the strong representation capabilities within neural networks, making them able to capture the meaningful seasonal and trend representations even with same model structure (**line 337-339 in Section 4.3**).\n\n>> Experimental evidence for Q4. This table is a copy of the result from **Figure 5 (b) in Section 4.3**\n\n|Average MSE in eight datasets|heterogeneous models with joint training|heterogeneous models with our decoupled training|same model structure with our decoupled training|\n|:---:|:---:|:---:|:---:|\n|MICN|0.375|0.320|0.320|\n|Leddam|0.315|0.295|0.295|\n\n> Concern about figure size\n\nThanks for your reminder. We will increase the size of key figures to ensure they can be read easily.\n\n> Reference\n\n[1] Qwen3 Embedding: Advancing Text Embedding and Reranking Through Foundation Models\n\n[2] Language Models are Super Mario: Absorbing Abilities from Homologous Models as a Free Lunch\n\n[3] Model Merging in Pre-training of Large Language Models\n\n[4] Multimodal Pathway: Improve Transformers with Irrelevant Data from Other Modalities"}, {"review_id": "67jJk4JXQs", "reviewer": "Reviewer_f7pT", "summary": "This paper proposes DecompNet, a model-agnostic framework that enhances time series forecasting models via implicit decomposition. During training, two expert models are trained separately on the seasonal and trend components of the input series. These expert models are later fused into a single model using a learnable fusion factor, enabling inference on raw time series without requiring explicit decomposition at test time. Extensive experiments across 8 benchmark datasets and multiple backbone models (e.g., PatchTST, RLinear) show consistent improvements in accuracy. The authors also claim that DecompNet outperforms prior decomposition methods and normalization-based frameworks like RevIN, without increasing inference cost. The main contributions are: (1) introducing implicit decomposition, (2) proposing a decoupled-and-fusion training scheme, and (3) demonstrating consistent gains across model types and datasets", "questions": "1.\tAre the fusion weights (λ factors) shared across all layers, or learned independently per layer?\nClarifying this would help assess the flexibility and generalization capability of the fusion strategy. A more detailed explanation—ideally supported by a schematic or pseudocode—would improve transparency and reproducibility.\n2.\tHow does DecompNet handle architectural mismatches during fusion? \nIf the expert models have different depths or contain non-matching layers (e.g., normalization, residual connections), how is the fusion performed? Does the fusion mechanism rely on strict layer-wise alignment, and how are such inconsistencies resolved?\n3.\tHow robust is DecompNet to non-stationary test-time inputs?\nWithout any mechanism to adjust for distribution shifts at inference time, how does the model perform when the input time series exhibits level shifts, changing trends, or variance fluctuations not seen during training? Is there any empirical evidence comparing its performance with methods like RevIN that explicitly account for such shifts?\n4.\tHow do the authors plan to address the scalability of DecompNet to more than two decomposition components?\nThe paper acknowledges as a limitation that the current implementation only supports seasonal and trend components and suggests future integration with methods like STL or frequency-domain decomposition. Could the authors elaborate on what specific technical challenges they foresee in extending DecompNet to handle additional components (e.g., residuals or noise)? For example, how might fusion complexity, training stability, or parameter efficiency be affected?", "limitations": "Yes, the authors acknowledge a key limitation of their current implementation: it only supports decomposition into two components—seasonal and trend—and does not yet incorporate more advanced methods such as STL or frequency-domain decomposition. This limitation is clearly stated in Appendix L, along with a plan for future work. However, it would strengthen the paper if the authors also discussed the potential technical challenges involved in extending the framework to handle more than two components (e.g., residuals or noise), such as increased fusion complexity, parameter growth, or optimization difficulty. Addressing these considerations would provide a clearer picture of the method’s scalability.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 4, "originality": 3, "strengths_and_weaknesses": "Strengths:\n- Practical Innovation: Injects decomposition-related knowledge via a learnable fusion step, eliminating the inference overhead of prior decomposition-based frameworks.\n- Model-Agnostic Compatibility: Enhances diverse forecasting architectures (e.g., Transformer, MLP, CNN) without modifying their design.\n- Strong Empirical Rigor: Evaluates across 8 datasets and multiple baselines, with ablations and multiple seeds confirming consistency.\n- Efficiency + Effectiveness: Achieves notable performance gains without any added inference cost—which is rare among enhancement frameworks. The decomposition is done implicitly during training, so the final model runs as fast and light as the original, making it highly practical for real-world deployment.\n\nWeaknesses:\n- Limited Technical Novelty: The framework builds on existing ideas such as structural re-parameterization and training expert models on decomposed components. As a result, it may be seen more as a practical engineering extension rather than a fundamentally novel algorithmic breakthrough. The core idea – train separate models for trend and seasonal components and then merge – is clever but not entirely unprecedented.\n- Unclear Fusion Implementation: Although the paper outlines a high-level strategy for fusing expert models by aligning corresponding linear or convolutional layers, it does not clarify how architectural mismatches are handled, such as differences in model depth, or the presence of normalization layers and residual connections. Furthermore, it remains unclear whether the fusion weights (e.g., λ factors) are shared across all layers or learned independently for each layer. These omissions limit clarity and may hinder reproducibility.\n- Lack of Robustness to Test-Time Shifts: The method assumes stationary test distributions and currently has no mechanism to adapt to distribution shifts at inference (such as sudden level changes or variance shifts in the input data). Unlike normalization-based methods like RevIN or more adaptive frameworks, DecompNet does not explicitly address these forms of non-stationarity. This could make the approach vulnerable when the training and testing data distributions differ significantly.", "was_revised": true, "final_justification": "The rebuttal provided clear answers to my concerns regarding the fusion mechanism, including the prerequisite of identical architectures and the independent learning of fusion weights across layers. Additional experiments comparing DecompNet with RevIN on highly non-stationary datasets, as well as the preliminary extension to three-component decomposition using STL, convincingly support the robustness and scalability of the method. While the method primarily builds on existing concepts, the integration of implicit decomposition with structural re-parameterization is practical and novel within the time series forecasting context. Some limitations remain, such as the reliance on identical architectures for fusion, but these are acknowledged by the authors and clearly outlined as directions for future work. In light of these clarifications and additions, I find the contribution solid and impactful, and have accordingly increased my score.", "rebuttal": "Many thanks to Reviewer f7pT for the insightful comments.\n\n> W1: Technical Novelty\n\nWe'd like to highlight our difference from previous works as follows: \n+ Although the idea of training expert models on decomposed components already exists, **we innovatively propose a better training strategy**, which can fully unleash the performance potential of decomposition-based methods.\n    + Previous methods mainly adopt a joint training strategy, which can only provide messily entangled supervision signals and limit the performance of expert models.\n    + In this paper, we propose a novel decoupled training strategy. By decoupling the whole-series prediction task into a seasonal prediction task and a trend prediction task, our strategy can provide more direct and clearer supervision signals for expert models, fully unleashing their performance potential.\n+ Although the technique of structural re-parameterization has been utilized in other domain, we **innovatively integrate it with time series decomposition**.\n    + Our Seasonal-Trend Re-parameterization is **a novel variant** of structural re-parameterization **specially designed for implicit decomposition**, helping to achieve **inferece-cost-free** forecasting performance enhancement.\n+ With the help of the above two innovations, we **successfully pioneer and verify the idea of implicit decomposition**, which can provide some **novel insights** to time series community:\n    + We are **the first** to propose a decomposition-based enhancement framework that surpasses the well-recognized normalization-based frameworks. And we are **the first** to propose a inference-cost-free enhancement framework. **These unprecedented findings can encourage further exploration on more types and more efficient enhancement frameworks**. \n    + As a **brand new idea**, our implicit decomposition can **bring some fresh perspectives** and provide a better solution to the classic research topic of **time series decomposition**.\n+ We hope the above clarification helps to distinguish our method from previous works and clarify our contributions, thus highlighting our novelty.\n\n> W2-1, Q2: How does DecompNet handle architectural mismatches during fusion? \n\n>> Does the fusion mechanism rely on strict layer-wise alignment?\n+ Yes, it relies on strict layer-wise alignment. The prerequisite for the fusion process is that **the two expert models must have the same model architecture**. \n\n>> We will not encounter the situation of architectural mismatches in this paper.\n+ In this paper, **the two expert models in our fusion process definitely have the same architecture**, since they are two copies of the same original model.\n+ As a result, **we will not encounter the situation of architectural mismatches during fusion** and we do not need to handle this situation specifically.\n\n>> Future work\n+ We fully acknowledge the reviewer's interest in handling the architectural mismatches during model fusion. However, such topic remains an open challenge beyond current methodological capabilities—even leading works in this nascent field (e.g., [1-4]) mainly focus on the fusion of models with same architecture.\n+ Aligning with state-of-the-art practice, our method mainly focuses on the fusion of models with same architecture. And our method can avoid the situation of architectural mismatches during fusion.\n+ It will be our future work to further study on how to handle the architectural mismatches during fusion.\n\n[1] Qwen3 Embedding: Advancing Text Embedding and Reranking Through Foundation Models\n\n[2] Language Models are Super Mario: Absorbing Abilities from Homologous Models as a Free Lunch\n\n[3] Model Merging in Pre-training of Large Language Models\n\n[4] Multimodal Pathway: Improve Transformers with Irrelevant Data from Other Modalities\n\n> W2-2, Q1-1: Whether λ factors are shared across all layers or learned independently for each layer.\n+ λ factors are learned independently for each layer.\n\n> W2-3, Q1-2: About clarity and reproducibility\n+ Thanks for the suggestion. \n+ To enhance clarity, we will emphasize the same model architecture as a prerequisite for model fusion in **Section 3.4**. And we will also clarify that λ factors are learned independently for each layer. \n+ To ensure reproducibility, we guarantee to make the code public upon paper acceptance. And we will add a pseudocode related to model fusion process in the appendix for the final version. \n \n> W3, Q3: How robust is DecompNet to non-stationary test-time inputs? About mechanism guarantees. Empirical evidence comparing DecompNet with RevIN in non-stationary issues.\n+ DecompNet can robustly handle non-stationary test-time inputs. And based on our experimental results, it performs better than RevIN. \n+ Mechanism Guarantees: DecompNet **has mechanism** to handle non-stationary test-time inputs, but the mechanism is different from RevIN's:\n    + RevIN is a normalization-based method. It alleviates the non-stationary issue **from the perspective of statistical measures**.\n    + DecompNet is a decomposition-based method. It handles the non-stationary issue **based on time series decomposition**, since decomposition can unravel the entangled temporal patterns in raw time series and and **extract the stationary components** [5-7].\n+ Empirical Evidence: To prove our superiority in handling non-stationary issues, we compare our performance with RevIN in the top-3 highly non-stationary datasets, which are ETTh2, Exchange and ILI[8].\n    + For ILI, the input length is 96 and the results are averaged from 4 prediction lengths {24,36,48,60}. For other datasets, the input length is 384 and the results are averaged from 4 prediction lengths {96,192,336,720}.\n    + In following results, the *Original* model equips with RevIN by default. And *Ours* replaces the RevIN with our DecompNet framework. **So these results can provide a fair comparison between our DecompNet and RevIN in handling non-stationary issues**.\n    + It is shown that DecompNet outperforms RevIN in all these highly non-stationary datasets, indicating its better ability in handling non-stationary issues.\n+ Conclusion:\n    + As verified on the currently available and publicly accessible datasets, DecompNet performs the best in handling non-stationary issues and is even better than RevIN. \n    + We are well aware that there are non-stationary issues that cannot be represented by the current datasets. In future work, we will keep monitoring the development in time series community and promptly validate the effectiveness of our methods on the latest and more challenging non-stationary datasets.\n\n|ETTh2(MSE/MAE)|Original|Ours|\n|:---:|:---:|:---:|\n|PatchTST|0.349/0.391|0.346/0.387|\n|iTransformer|0.377/0.412|0.344/0.389|\n|RLinear|0.354/0.397|0.338/0.383|\n|RMLP|0.364/0.401|0.347/0.390|\n|ModernTCN|0.328/0.382|0.312/0.373|\n\n|Exchange(MSE/MAE)|Original|Ours|\n|:---:|:---:|:---:|\n|PatchTST|0.387/0.419|0.330/0.404|\n|iTransformer|0.378/0.418|0.317/0.391|\n|RLinear|0.378/0.411|0.314/0.386|\n|RMLP|0.385/0.414|0.320/0.385|\n|ModernTCN|0.376/0.407|0.319/0.376|\n\n|ILI(MSE/MAE)|Original|Ours|\n|:---:|:---:|:---:|\n|PatchTST|2.096/0.969|1.904/0.886|\n|iTransformer|2.231/1.020|1.959/0.934|\n|RLinear|3.686/1.366|2.305/1.042|\n|RMLP|2.796/1.172|2.056/0.972|\n|ModernTCN|1.944/1.198|1.738/0.874|\n\n[5] Forecasting: principles and practice\n\n[6] Forecasting: theory and practice.\n\n[7] Autoformer: Decomposition Transformers with Auto-Correlation for Long-Term Series Forecasting\n\n[8] Non-stationary Transformers: Exploring the Stationarity in Time Series Forecasting\n\n> Q4, Limitation: The scalability of DecompNet to more than two decomposition components. Integration with STL.\n\n>> Experimental results from our preliminary exploration\n+ Following your valuable suggestion, we conduct a preliminary exploration on the scalability of DecompNet. \n+ The experiment settings are as follows.\n    + We integrate our DecompNet with STL and decompose the time series into three components (i.e., seasonal, trend and residual components).\n    + We introduce more fusion factors to fuse three expert models, letting $W = W_{s} + \\lambda_{1} W_{t} + \\lambda_{2} W_{r}$, where $W_{s}, W_{t}, W_{r}$ are the weight matrices from seasonal, trend and residual expert models respectively.\n    + We use RLinear and PatchTST as the backbone models.\n+ As shown in the following table, we observe continuous performance improvement after scaling into more and finer components, which can prove the scalability of DecompNet.\n\n|MSE/MAE|PatchTST + 2 Components|PatchTST + 3 Components|RLinear + 2 Components|RLinear + 3 Components|\n|:---:|:---:|:---:|:---:|:---:|\n|ETTh1|0.403/0.421|0.394/0.407|0.410/0.422|0.399/0.410|\n|ETTm1|0.345/0.380|0.338/0.370|0.359/0.377|0.349/0.367|\n|Solar|0.199/0.257|0.193/0.247|0.227/0.274|0.213/0.265|\n|Traffic|0.387/0.268|0.380/0.261|0.425/0.286|0.412/0.277|\n\n>> About potential technical challenges like increased fusion complexity, parameter growth, or optimization difficulty.\n\nBased on our preliminary exploration, we find that:\n+ Fusion complexity and optimization difficulty are not increased. Through directly generalizing the equation from $W = W_{s} + \\lambda W_{t}$ to $W = W_{s} + \\lambda_{1} W_{t} + \\lambda_{2} W_{r}$, we can achieve the fusion of three expert models. And the training process is still stable and can converge quickly.\n+ Parameter growth: \n    + The number of parameters is only increased during training. When scaling to more decomposition components, we need to train more expert models and introduces more fusion factors. This will introduce more parameters, but will also bring better performance.\n    + During inference, all expert models are fused as one single model, so there is no additional inference costs.\n\n>> Future work\n\nAbove preliminary experiments demonstrate the scalability of DecompNet. In future work, we will conduct more in-depth researches to further unleash its scaling potential and report more insights on optimization."}, {"review_id": "X6RzHWZp89", "reviewer": "Reviewer_5VM5", "summary": "The authors propose training the seasonal module and trend module separately, and then fusing them into any time series model. However, the details of the model are not clearly explained, and treating different time series models merely as simple linear combinations is not appropriate.", "questions": "see weaknesses", "limitations": "yes", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths\n\nThe authors  pretrain two copies of it on the decomposed seasonal and trend data, helping them grasp  the decomposition-related knowledge and making them to be the seasonal and trend expert models.  Then we fuse these two expert models back as one single model, merging all of their decomposition related knowledge into the fused model. In this way, author make it to inject the decomposition related knowledge into a single model (i.e., the fused model).\n\nweaknesses:\n1. What are the network architectures of Model X and Model Y? \n2. Wt and Ws cannot be fused using Equation 10. On one hand, Xs and Xt are derived under a non-stationary mode. On the other hand, for Equation 10, the second equality does not hold—there are no parameters $\\lambda$ that can make the two sides equal. \n3. In Fusion of complicated models module, the explanation lacks a detailed discussion on the fusion of the two, particularly  for models like patchTST and iTransformer, which do not fall under the category of simple linear models. \n4. In the introduction, the authors mention the implementation of inference costs; however, in the experiments, there is no detailed analysis of the inference costs after incorporating DecompNet.\n5. The DecompNet section lacks an overall model architecture diagram. \n6. Including specific experimental results in the introduction and abstract would further enhance the clarity and impact.", "was_revised": true, "final_justification": "The author’s explanation effectively addresses w1, w3, and w4, and I truly appreciate the clarity in their responses. However, while I acknowledge the author's perspective on w1, I still maintain my own viewpoint regarding it. As for w5, although the model framework diagram is helpful, it could benefit from further optimization as it currently appears to be somewhat simplistic.", "rebuttal": "Many thanks to Reviewer 5VM5 for the detailed review. However, it seems that some comments in the \"Summary\" part of the review **stem from a misunderstanding of our work**.\n\nTo eliminate potential misunderstanding, we'd like to clarify and explain our work as follows:\n+ DecompNet is a model-agnostic enhancement framework and it can be applied to any time series models. When using our DecompNet to enhance a given time series model, it has following steps:\n    + When given a model to be enhanced, we first pretrain two copies of it on the decomposed seasonal and trend data, making these two copies to become the seasonal and trend expert models. \n    + Then we fuse these two expert models (i.e., the seasonal expert model and the trend expert model) back as one single model, merging all of their decomposition-related knowledge into the fused model. In this way, we make it to inject the decomposition-related knowledge into a single model (i.e., the fused model).\n    + And this fused model can be seen as an enhanced version of the original given model.\n\n> Responses to some comments in the \"Summary\" part\n+ Response to comment \"The authors propose training the seasonal module and trend module separately, and then fusing them into any time series model\":\n    + Our method is not \"fuse two separate seasonal module and trend module into any time series model so as to enhance that time series model\". \n    + Our method is \"when given any time series model to be enhanced, we first pretrain two copies of the given model to be the seasonal and trend expert models. Then we fuse these two expert models as one single model. And this fused model can be seen as an enhanced version of the original given model\".\n+ Response to comment \"the details of the model are not clearly explained\":\n    + We'd like to clarify that our proposed DecompNet is not a specific model. It is a model-agnostic enhancement framework. \n    + And the details of DecompNet are clearly introduced in **Section 3** and summarized in **Section 1 line 50-59**.\n+ Response to comment \"treating different time series models merely as simple linear combinations\":\n    + We do not treat different time series models merely as simple linear combinations.\n    + To ensure we address your concern most effectively, could you please specify which sentences in our paper lead to this comment?\n\n> W1: What are the network architectures of Model X and Model Y?\n+ We suppose this question may be \"What are the network architectures of seasonal model and trend model\", because we do not mention \"Model X\" and \"Model Y\" in this paper.\n+ Seasonal model and trend model are **two copies of the original model to be enhanced**. So they have the same network architecture as the original model.\n+ For \"what is the network architecture of the original model\": Since DecompNet is a model-agnostic enhancement framework, the original model can be any time series models. And specifically in our main experiments, we use PatchTST, iTransformer, RLinear, RMLP and ModernTCN as the original models.\n\n> W2-1: About fusion process: Can $W_{s}$ and $W_{t}$ be fused using Equation 10?\n\nYes. We'd like to highlight that **$W_{s}$ and $W_{t}$ can be fused**. And **we have verified this through adequate experimental evidence**.\n+ The consistent forecasting improvements across a wide range of models and datasets in **Section 4.1** and our ablations in **Section 4.4** can verify the effectiveness of our fusion process.\n+ Similar researches on model fusion in other fields of Natural Language Processing (NLP) [1-3] and Computer Vision (CV) [4] can also help to guarantees the feasibility of such fusion process.\n\n[1] Qwen3 Embedding: Advancing Text Embedding and Reranking Through Foundation Models\n\n[2] Language Models are Super Mario: Absorbing Abilities from Homologous Models as a Free Lunch\n\n[3] Model Merging in Pre-training of Large Language Models\n\n[4] Multimodal Pathway: Improve Transformers with Irrelevant Data from Other Modalities\n\n> W2-2: About Equation 10: $X_{s}$ and $X_{t}$ are derived under a non-stationary mode.\n\nAs defined and mentioned in **line 121, 157, 181**, $X_{s}$ and $X_{t}$ are two mathematical symbols to denote the decomposed seasonal and trend components. **They are not derived**.\n\n> W2-3: About Equation 10: Can we make Equation 10 hold? Is there a fusion factor $\\lambda$ that can make the two sides equal?\n\nYes. As mentioned in **line 197-198**, we introduce an extra training process to learn the optimal fusion factors $\\lambda$ to make Equation 10 hold. And **our experimental results in Section 4.1 and our ablations in Section 4.4, Appendix E.3 can verify that such optimal fusion factors $\\lambda$ really exist**.\n\n> W3: Lack of discussion on the fusion of the models like patchTST and iTransformer.\n+ Thanks for highlighting this. PatchTST and iTransformer fall under the category of Transformer-based models. And the fusion of Transformer-based models is explicitly addressed in **line 219-222 (Section 3.4 Paragraph \"Fusion of complicated models\")**, where we can achieve the fusion of two structurally identical Transformer-based models by fusing each of the corresponding linear layers.\n\n> W4: Lack of analysis of the inference costs.\n+ Thanks for highlighting this. We have provided a detailed analysis of the inference costs in **line 279-288 (Section 4.2 Paragraph \"Efficiency Comparison\")** and provided the experimental evidence in **Figure 4 (c)**.\n+ The results show that our method has totally the same inference speed and memory usage as the original model, demonstrating that our method can greatly improve the forecasting performance and bring completely no additional inference costs.\n\n> W5: The DecompNet section (Section 3) lacks an overall model architecture diagram.\n+ Thanks for highlight this. We have provided the diagram to demonstrate the overall workflow of DecompNet in **Figure 1 (b)**. And we have included the sentence \"Figure 1 (b) shows the overall workflow of DecompNet\" in **Section 3 line 126** as a reference to this diagram.\n+ And we'd like to claim that DecompNet is not a specific model architecture but a model-agnostic enhancement framework.\n\n> W6: Including specific experimental results in the introduction and abstract would further enhance the clarity and impact.\n+ We very much appreciate your great suggestions about the paper writing. But considering the the page limitation of the main text, we hope we can maintain to arrange the presentation of experimental results as follows:\n    + In **Abstract (line 9-15) and Introduction (line 59-63, 67-72)**, we **briefly summarize our experimental performance** to give readers a preliminary understanding of our performance superiority.\n    + In **Section 4**, we provide **specific experimental results and detailed performance analysis** to further enhance the clarity and impact. \n---\n**Due to space limitation, we place some responses to Reviewer CJUx here.**\n> W4 of Reviewer CJUx: Convey the training cost details (Appendix E.1) in the main text.\n\nThanks for your suggestion. We will move **Appendix E.1** to the main text in our final version and highlight our nice training efficiency.\n\n> W5 of Reviewer CJUx: Include a summary sentence before redirecting readers to the appendix.\n+ For the two baselines MICN and Leddam, we will rephrase **line 309-311** as: \"We compare our DecompNet with some latest and representative explicit decomposition methods, i.e., MICN  and Leddam. Please refer to **Appendix F.3** for a detailed introduction of their model structures and data processing pipelines\". \n+ For moving average window size, we will add a new sentence in **line 234**: \"We set the moving average window size as 25 for time series decomposition during the decoupled pretraining stage. And we provide the parameter sensitivity study in **Appendix G**\".\n+ For training details, we will add a new sentence in **line 234**: \"Our method is trained with L2 loss, using ADAM optimizer with an initial learning rate in {$10^{-3}, 5\\times 10^{-4}, 10^{-4}$}. The default training process is 30 epochs with early stopping. More implementation details are in **Appendix C**\".\n\n> Other2 of Reviewer CJUx: Comparison with other prominent decomposition-based models like DLinear, NHiTs and N-BEATS\n+ Following your valuable suggestion, we conduct these comparisons. As shown in following table, our method surpasses these prominent decomposition-based models, achieving an average MSE reduction of 11.1% compared to DLinear, 16.7% compared to NHiTs and 18.6% compared to N-BEATS, thus verifying our performance superiority.\n+ Please also see **Section 4.3** for our comparison with two more competitive decomposition-based models (i.e., MICN and Leddam).\n\n|MSE/MAE|DecompNet+PatchTST|DecompNet+RLinear|DLinear|NHiTs|N-BEATS|\n|:---:|:---:|:---:|:---:|:---:|:---:|\n|ETTh1|0.403/0.421|0.410/0.422|0.428/0.444|0.486/0.477|0.500/0.486|\n|ETTm1|0.345/0.380|0.359/0.377|0.365/0.385|0.372/0.397|0.385/0.404|\n|Solar|0.199/0.257|0.227/0.274|0.255/0.338|0.265/0.348|0.266/0.345|\n|Traffic|0.387/0.268|0.425/0.286|0.453/0.318|0.479/0.327|0.487/0.331|\n\n> Other3 of Reviewer CJUx: Appendix G could be referenced in the main text. \n\nThanks for your reminder. We will move **Appendix G** to the main text in our final version and highlight our robustness to moving average window size.\n\n> Other4 of Reviewer CJUx: Include some forecasting examples in the main text.\n+ Considering the large space occupation of visualization figures and the page limitation of the main text, we place the forecasting examples in **Appendix N**.\n+ We will add a sentence in **Section 4 line 228** to direct readers to **Appendix N** for forecasting examples.\n\n> Q1 of Reviewer CJUx: Is $\\lambda$ learned during training, or is it tuned as a hyperparameter?\n+ Thanks for highlighting this. As mentioned in **line 198**, $\\lambda$ is learned during training. \n+ To ensure visibility, we will bold this sentence in our final version."}, {"review_id": "ibOs1Qn2C2", "reviewer": "Reviewer_CJUx", "summary": "This paper proposes a novel time series forecasting method centered on trend-seasonality re-parametrization, introducing a new concept referred to as \"implicit decomposition.\" The proposed approach demonstrates consistent improvements across various baselines.", "questions": "1. Is the lambda fusion parameter learned end-to-end during model training as a network weight, or is it tuned as a hyperparameter? If the latter, what is the range of values considered?", "limitations": "Yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 2, "strengths_and_weaknesses": "Strengths:\n- The proposed model shows strong performance gains across a wide range of datasets and baselines, as reflected in the presented tables and figures. Table 7 in the appendix nicely summarizes the model's improvements.\n\n- The figures are clear and convey the findings well. Figure 4 provides an effective visual comparison of benchmarks for multiple factors, including loss, inference time, and varying input lengths.\n\n- Forecast examples included in Appendix N.1 and N.2 help showcase the performance improvement.\n\n- An ablation study on each component of the method is provided to showcase the combined improvement of the proposed techniques.\n\nWeaknesses:\n1. The reference to trend-seasonality re-parameterization as “implicit decomposition” is potentially misleading. In practice, the model performs explicit decomposition during training using a moving average filter. Implicit decomposition is assumed to occur during inference through the re-parameterized model; however, there is no clear evidence or analysis confirming that this actually happens. This behavior seems to be inferred from the empirical forecasting results.\n\n2. Many of the claims throughout the paper use subjective or vague language where quantitative evidence is not explicitly provided to verify them. The reader must try to parse the claims from the provided figures and tables. I recommend that the authors revise these statements and explicitly tie each claim to specific metrics results. Some examples include:\n   - Line 241: “by a large margin”\n   - Lines 261–262: “DecompNet also significantly surpasses...”\n   - Line 366: “enjoying better performance and having totally the same inference costs”\n\n3. The authors report the standard deviation of DecompNet performance under five runs with different random seeds in Table 6 in the Appendix using PatchTST, but do not do the same for the other benchmarks. The average results along with standard deviations are not included in the main tables. With results across random seeds for both the baseline and the proposed method (for each model), one can also consider adding statistical significance tests to confirm whether improvements are meaningful or due to chance.\n\n4. I understand the focus on inference cost for deployment practicality. However, including training cost information is still valuable to ensure the method incurs a reasonable expense. Training cost details are not explicitly conveyed in the main paper, though metrics like the number of epochs are used in the appendix. Including a couple of sentences on this in the main paper with explicit metrics would be helpful. Also, metrics like FLOPs and wall-clock time are generally used to convey computational cost and training time, respectively.\n\n5. For many key details, the reader is directed to refer to the appendix. For example, explanations of benchmarks like MICN and Leddam, details on the average window size for the proposed method, and references to model training details in Appendix C. Including a summary sentence on these topics before redirecting readers to the appendix would be helpful.\n\nOther:\n1. The trend-seasonality re-parameterization, which appears to be one of the main contributions of the paper, is not introduced until page 3. This core idea should be presented earlier, as it seems to be the main contribution and may be both clearer and more familiar to readers than the concept of implicit decomposition. The notion of implicit decomposition could then be introduced in relation to this concept.\n\n2. It would strengthen the paper to compare the proposed model and its variants against other prominent decomposition-based models such as DLinear, NHiTs, and N-BEATS. This comparison would better contextualize the method’s performance and practical relevance.\n\n3. Section G is helpful and addresses the effect of moving window size. This could be referenced in the main text to assure readers that this aspect has been considered and studied.\n\n4. Including some forecast examples in the main paper would be very helpful to demonstrate the performance improvement of the proposed approach.", "was_revised": true, "final_justification": "I recommend increasing my score to a 4 (borderline accept) based on the authors response to my comments. \n\nI am happy that the authors have added error bars for experiments across random seeds. However, I would like to see their statement, \"The average performance improvement is roughly an order of magnitude larger than the random fluctuation from multiple runs\", supported with statistical tests and to valid the use of the term “significant” when referring to performance improvements. Additionally, I would like to see metrics like FLOPs and wall-clock time to convey computational cost and training time, respectively in addition to current metric (training epochs).", "rebuttal": "Many thanks to Reviewer CJUx for the detailed comments.\n\n> W1-1: About the relationship between implicit decomposition and Seasonal-Trend Re-parameterization\n\nWe'd like to clarify that we do not refer Seasonal-Trend Re-parameterization as implicit decomposition.\n+ Implicit decomposition is **a brand new idea** we pioneered in this paper, which aims to improve a model's performance by decomposition-related knowledge during inference but without actually decomposing the input time series.\n+ Seasonal-Trend Re-parameterization (along with decoupled training strategy) are **two key designs we proposed to achieve the goal of implicit decomposition**.\n\n> W1-2: About the naming of implicit decomposition\n\nTo eliminate potential misleading, we provide the reasons for the naming of implicit decomposition as follows:\n+ Previous decomposition-based methods hold a belief that: if we want to utilize the decomposition-related knowledge to improve a model's performance, we **must explicitly and actually** decompose the input time series during inference.\n+ While in this paper, we find it possible to utilize the decomposition-related knowledge to improve a model's performance **without actually** decomposing the input time series during inference.\n+ This is a brand-new finding that challenges the commonly held belief in previous decomposition-based methods. To distinguish it from the previous methods, we name it as **implicit decomposition**.\n\n> W1-3: The model performs explicit decomposition during training\n+ Thanks for highlighting this. The concept of explicit decomposition and implicit decomposition in this paper are used to distinguish whether a method needs to actually decompose the input time series **during inference**.\n+ Both previous explicit decomposition method and our implicit decomposition method need to explicitly decompose the data during training. **But the main difference lies in the inference stage**, our implicit decomposition can avoid explicitly decomposing the input time series during inference, achieving inference-cost-free performance enhancement.\n\n> W1-4: Evidence and analysis\n+ Experimental evidence: Implicit decomposition aims to improve a model's performance by decomposition-related knowledge during inference but without actually decomposing the input time series. And as shown in our main results in **Table 1 in Section 4.1**, this kind of performance improvement really happens, which can serve as the experimental evidence for implicit decomposition.\n+ Theoretical analysis: Since rigorous theoretical analysis for a model-fusion-based method like implicit decomposition remains an open challenge beyond current methodological capabilities—even leading works in this nascent field (e.g., [1-4]) rely primarily on empirical validation, we provide **comprehensive experimental evidence** to demonstrate our efficacy and will leave the theoretical explanation in future work.\n\n[1] Qwen3 Embedding: Advancing Text Embedding and Reranking Through Foundation Models\n\n[2] Language Models are Super Mario: Absorbing Abilities from Homologous Models as a Free Lunch\n\n[3] Model Merging in Pre-training of Large Language Models\n\n[4] Multimodal Pathway: Improve Transformers with Irrelevant Data from Other Modalities\n\n> Other1-1: Seasonal-Trend Re-parameterization should be presented earlier.\n\nThanks for the recognition of our Seasonal-Trend Re-parameterization. Following your valuable suggestion, we will rephrase the sentence in **line 66** as \"we propose a powerful decomposition-based enhancement framework called DecompNet, which features two key designs: Seasonal-Trend Re-parameterization and decoupled training strategy\" in our final version.\n\n> Other1-2: About contributions.\n\nBeyond the proposal a novel technique like Seasonal-Trend Re-parameterization, **the introduction of the brand-new idea of implicit decomposition is also an important contribution** of our paper, which can provide some **novel insights** to time series community:\n+ Based on the idea of implicit decomposition, we are **the first** to propose a decomposition-based enhancement framework that surpasses the well-recognized normalization-based frameworks. And we are **the first** to propose a inference-cost-free enhancement framework. **These unprecedented findings can encourage further exploration on more types and more efficient enhancement frameworks**. (Quantitative evidence is provided in **Section 4.1 and 4.2**.)\n+ Implicit decomposition re-examines and challenges the commonly held belief in previous decomposition-based methods, which can **bring some fresh perspectives** and provide a better solution to the classic research topic of **time series decomposition**. (Quantitative evidence is provided in **Section 4.3**.)\n\n> Other1-3: Clarification on why we introduce implicit decomposition at first.\n\nThanks for your great suggestions about the paper writing. While we do improve the other parts, we hope we can maintain to introduce implicit decomposition at first for following reasons:\n+ Implicit decomposition represents the core theme of this paper—improving a model's performance by decomposition-related knowledge but without actually decomposing the input time series during inference. Therefore, introducing implicit decomposition at first can help readers quickly grasp the research objectives and the key contributions of this paper.\n+ The idea of implicit decomposition can provide novel insights to time series community. Thus, we introduce implicit decomposition and share these insights with readers at first, which can highlight our contributions and novelty.\n+ From a technical perspective, implicit decomposition represents the overall design idea, while Seasonal-Trend Re-parameterization is a specific technical detail. In our humble opinion, introducing the overall design idea first, followed by the specific technical details, will help to better clarify our method.\n\n> W2: About the claims and quantitative evidence\n+ We'd like to clarify that our claims are not using subjective or vague language. All of our claims can be verified by the experimental results in our figures and tables.\n+ Following your valuable suggestion, we will revise these statements in our final version and explicitly tie each claim to specific metrics results as follows:\n    + Line 241: DecompNet consistently enhances the performance of all backbone models by a large margin. For instance, on Solar, ETTh1, and Traffic datasets, the average MSE performance improvements are rather significant: 8.0%, 6.4% and 6.1% respectively.\n    + Line 261–262: DecompNet also significantly surpasses the latest prior-based methods. For instance, when using PatchTST as the backbone model, DecompNet achieves the best performance with an average MSE of 0.290, obtaining an average of 6.3% MSE reduction than the latest prior-based methods in all datasets.\n    + Line 366: enjoying better performance and having totally the same inference costs (e.g., in traffic dataset, it can boost the performance from 0.410 to 0.387 under the same inference time 322ms.)\n\n> W3: Main result (Table 1) with full error bar\n\nWe report the **Table 1** with full error bar as follow. The average performance improvement is **roughly an order of magnitude larger** than the random fluctuation from multiple runs (0.0x vs 0.00x). Thus, the improvement in our main result is meaningful and not due to chance.\n\n|MSE/MAE|PatchTST|PatchTST+Ours|iTransformer|iTransformer+Ours|RLinear|RLinear+Ours|RMLP|RMLP+Ours|ModernTCN|ModernTCN+Ours|\n|:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:|\n|ETTh1|0.418±0.002/0.433±0.001|0.403±0.002/0.421±0.002|0.461±0.001/0.464±0.001|0.417±0.001/0.435±0.001|0.415±0.001/0.428±0.001|0.410±0.001/0.422±0.001|0.482±0.002/0.469±0.002|0.423±0.002/0.430±0.001|0.403±0.002/0.419±0.002|0.387±0.002/0.411±0.001|\n|ETTh2|0.349±0.001/0.391±0.001|0.346±0.002/0.387±0.002|0.377±0.001/0.412±0.001|0.344±0.002/0.389±0.002|0.354±0.001/0.397±0.001|0.338±0.001/0.383±0.001|0.364±0.002/0.401±0.001|0.347±0.002/0.390±0.002|0.328±0.002/0.382±0.001|0.312±0.002/0.373±0.001|\n|ETTm1|0.355±0.002/0.385±0.002|0.345±0.002/0.380±0.002|0.372±0.001/0.399±0.001|0.358±0.001/0.385±0.001|0.364±0.001/0.381±0.001|0.359±0.001/0.377±0.000|0.371±0.001/0.394±0.001|0.355±0.001/0.383±0.001|0.365±0.002/0.386±0.001|0.354±0.001/0.379±0.001|\n|ETTm2|0.262±0.001/0.319±0.001|0.257±0.001/0.314±0.001|0.274±0.001/0.332±0.001|0.262±0.001/0.321±0.000|0.259±0.000/0.316±0.001|0.255±0.001/0.312±0.001|0.271±0.002/0.326±0.001|0.259±0.001/0.318±0.001|0.261±0.002/0.320±0.001|0.254±0.001/0.314±0.001|\n|Weather|0.227±0.000/0.264±0.000|0.224±0.001/0.261±0.001|0.235±0.001/0.272±0.001|0.229±0.001/0.268±0.001|0.246±0.000/0.281±0.001|0.235±0.000/0.275±0.000|0.244±0.001/0.279±0.000|0.239±0.000/0.275±0.000|0.233±0.001/0.270±0.001|0.231±0.000/0.266±0.000|\n|Solar|0.218±0.001/0.317±0.001|0.199±0.001/0.257±0.001|0.212±0.002/0.272±0.001|0.204±0.002/0.259±0.001|0.256±0.002/0.323±0.002|0.227±0.001/0.274±0.001|0.231±0.003/0.294±0.003|0.214±0.002/0.249±0.002|0.218±0.001/0.306±0.001|0.200±0.001/0.258±0.001|\n|ECL|0.163±0.001/0.260±0.001|0.161±0.001/0.260±0.001|0.172±0.001/0.271±0.001|0.168±0.001/0.264±0.001|0.178±0.001/0.275±0.001|0.168±0.001/0.259±0.001|0.172±0.001/0.265±0.001|0.165±0.000/0.258±0.000|0.168±0.002/0.264±0.001|0.159±0.001/0.258±0.001|\n|Traffic|0.410±0.002/0.288±0.001|0.387±0.002/0.268±0.001|0.428±0.002/0.320±0.001|0.395±0.002/0.280±0.002|0.453±0.001/0.314±0.002|0.425±0.001/0.286±0.001|0.436±0.001/0.308±0.001|0.413±0.001/0.285±0.001|0.417±0.003/0.293±0.002|0.394±0.002/0.275±0.001|\n|Average|0.300±0.001/0.332±0.001|0.290±0.002/0.319±0.001|0.316±0.001/0.343±0.001|0.297±0.001/0.325±0.001|0.316±0.001/0.339±0.001|0.302±0.001/0.324±0.001|0.321±0.002/0.342±0.001|0.302±0.001/0.324±0.001|0.299±0.002/0.330±0.001|0.286±0.001/0.317±0.001|\n\n> Due to space limitation, the responses to W4, W5, Other2, Other3, Other4, Q1 are placed **at the bottom of the Response to Reviewer 5VM5**"}]

Author rebuttal:

Meta-review: This paper introduces DecompNet, which performs implicit time series decomposition to enhance time-series forecasting without additional inference costs. In the training phase, two expert models are trained on the seasonal and trend components of the input time series, respectively. The two experts are fused into a single model through seasonal-trend re-parameterization with a learnable scaling factor.

Initially, this paper received mixed recommendations. Major concerns raised by reviewers include the additional costs for training two experts, lack of evaluation on the inference costs, the dependency on a specific decomposition technique, lack of discussion or theoretical explanation on the success of the fusion and the implicit decomposition during inference, the generalization to other forms of decomposition and non-stationarity, the validation of some technical choices, limited technical novelty, unclear fusion implementation, lack of quantitative evidence regarding some claims, lack of standard deviations or statistical significance tests, lack of comparisons with other decomposition-based models, and presentation issues. 

The rebuttal addressed most of these concerns, but several limitations regarding the principled justification of design choices, a deeper theoretical understanding of the fusion mechanism, the reliance on identical architectures for fusion, the lack of statistical tests, and the lack of metrics such as FLOPs and wall-clock time on computational cost remain. The reviewers have provided very detailed feedback. The authors are encouraged to follow their comments closely and include all additional experiments in the camera-ready version. Especially, the authors should (1) include error bars, (2) not use the term “significant” if there is no statistical test conducted, as suggested by the reviewer CJUx.
Decision: Accept (poster)