TimeEmb: A Lightweight Static-Dynamic Disentanglement Framework for Time Series Forecasting
Authors: Mingyuan Xia
Chunxu Zhang
Zijian Zhang
Hao Miao
Qidong Liu
Yuanshao Zhu
Bo Yang
Venue: NeurIPS 2025 poster ()
Keywords: Time series forecasting
Frequency domain analysis
Disentangled representation learning
Primary area: deep_learning

Abstract: Temporal non-stationarity, the phenomenon that time series distributions change over time, poses fundamental challenges to reliable time series forecasting. Intuitively, the complex time series can be decomposed into two factors, i.e., time-invariant and time-varying components, which indicate static and dynamic patterns, respectively. Nonetheless, existing methods often conflate the time-varying and time-invariant components, and jointly learn the combined long-term patterns and short-term fluctuations, leading to suboptimal performance facing distribution shifts. To address this issue, we initiatively propose a lightweight static-dynamic decomposition framework, TimeEmb, for time series forecasting. TimeEmb innovatively separates time series into two complementary components: (1) time-invariant component, captured by a novel global embedding module that learns persistent representations across time series, and (2) time-varying component, processed by an efficient frequency-domain filtering mechanism inspired by full-spectrum analysis in signal processing. Experiments on real-world datasets demonstrate that TimeEmb outperforms state-of-the-art baselines and requires fewer computational resources. We conduct comprehensive quantitative and qualitative analyses to verify the efficacy of static-dynamic disentanglement. This lightweight framework can also improve existing time-series forecasting methods with simple integration. To ease reproducibility, our code is available at https://github.com/showmeon/TimeEmb.

TimeEmb: A Lightweight Static-Dynamic Disentanglement Framework for Time Series Forecasting
Mingyuan Xia*
Chunxu Zhang* Jilin University
Jilin University
xiamy23220mails. j.u. edu.
zhangchunxu@ji edu. cn
Zijian Zhang
Hao Míao
Jilin University
The Hong Kong Polytechnic University hao mww.mpolly....mmmmmm edu. hk
zhangzij jaanejijianjj.. lu edu. cn
Qidong Liu
Yuanshao Zhu
Bo Yang Jilin University ybo@jlu. edu. cn
Xi' am Jiaotong University liugidongoxitu tu. edu, cn
City University of Hong Kong yuanshaolieee. org
Abstract
Temporal non-stationarity, the phenomenon that time series distributions change over time, poses fundamental challenges to reliable time series forecasting. In- mittivlly, the complex time series be decomposed into two factors, i.e., time- invariant and time-varying components, which indicate static and dynamic patterns, respectively. Nonetheless, existing methods often conflate the time-varying and time-invariant components, and jointly learn the combined long-term patterns and short-term fluctuations, leading to suboptimal performance facing distribution shifts. To address this issue, we initiatively propose a lightweight static-yynamic decompo- sition framework, TimeEmb, for time series forecasting. TimeEmb innovatively sep- arates time series into two complementary components: (1) time- iine-invariant compo- nent, captured by a novel global embedding module that learns persistent representa- tions across time series, and (2) time-varying component, processed by an efficient frequency-domain filtering mechanism inspired by full-spectrun analysis in signal processing. Experiments on real-world datasets demonstrate that TimeEmb outper- forms state-of-the-arr baselines and requires fewer computational resources. We conduct comprehensive quantitative and qualitative analyses to verify the efficacy static-dynamic disentanglement. This lightweight framework can also improve existing time-series forecasting methods with simple integration. To ease repro- ducibility, the code available at https: /tte:////tttttt com/ showmeon/ CatimeonnTiimeennn TmmeEmb.
Introduction
The proliferation of edge devices and mobile sensing results in large amount of time series data, enabling various real-world applications [58, 51, 15, 26). this study, we focus on time series forecasting, which plays pivotal role in decision-makinn across critical domains including energy management [11], transportation systems [57, 7], and financial markets [44].
"These authors contributed equally to this work.
I Cooresponding author.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Traditional statistical methods, e.g., ARMA [2], employ moving average techniques model temporal dependencies. With the advances of neural networks, deep learning methods have revolutionized temporal pattern extraction, de- livering superior performance. These methods include recurrent neural networks (RNNs) mod- els [13, 35J that capture sequential dynamics, convolutional neural networks (CNNs) mod- els [8. 18] to obtain hierarchical features, and waanformer-baseee models [46] to learn long- range dependencies with self-attention mech- anisms. Recently, Multi-layer perceptron (MIP) methods [56] have demonstrated their effec- tiveness and superior efficiency compared to transformer-bai based counterparts.
0.26
Bectricity (321 Variables)
Autofornier 12.22M, 66.2s
FITS
0.25
2.5.00000M,2000
20.75
DLinear 0.14M, 17.8s
FEDformer 17,58M, 140.85
Filterfiet 1.0M,
MSE
PatchTST 10.74M, 77.3s
Irrnnformer 5.15M, 23.25
0.22
SOFTS 4.8M, 21.5s
1.0M
5,0M
CycleNot
0.21
0.47M. 18.9s
TimeEmb Emb
MLP-based
0.9 19.8s
Frequency based
Ours
0.20
40
100
120
Training Time (s/opoch)
Figure 1: Efficiency and performance comparison on the Electricity dataset.
Despite the advancements of existing methods, a fundamental challenge persists in modeling complex temporal dependencies, i.e., non-stationarity. Real-world time series often exhibit dynamic distribution shifts due to evolving trends and external interventions, showing high non- non-stationarity [32, 20]. This dynamic distribution shift violates the independent and identical distribution (IID) assumption of most existing forecasting methods [9]. It poses great challenges to robust temporal dependency modeling [38] and calls for a novel method that can handle such non-stationarity and learn comprehensive temporal dependencies.
Intuitively, time series can be considered as combination of two complementary parts: static time-invariant and dynamic time-varying components [21, 36, 6]. Time-invariant component represents stable long-term patterns in time series. For example, traffic flow typically follows a regular pattern, with peaks in the morning and troughs at night. Time-varying component reflects local fluctuations in time series, e.g., abnormal traffic flow caused by extreme weather or accidents. We contend that effective disentanglement of these two components can prevent the model from mistaking short-term noise for long-term patterns. It can explicitly capture stable long-term and dynamic local dependencies, thereby improving the effectiveness of time series forecasting.
However, it non-trivial to develop this kind of model. limitations unsolved for time series disentanglement: (1) Ignorance of long-term invariant pattern modeling. Existing seasonal-trend disentanglement methods often generate the trend component by moving the average kernel, and consider the rest the seasonal component [51, 56, 59, 25). It is performed on local time series by smoothing [37], and can hardly learn the global static patterns in the whole time series. (2) Rigorous assumption. To pursue explicit disentanglemenn, some methods rely on strong assumptions that may not always hold in practice. For example, CycleNet [14] assumes fixed periodic pattern in the dataset and extracts it using learnable recurrent cycle. However, this assumption does not always hold, as the complex periodicities can vary or have diverse lengths. Moreover, relying on pre-defined cycle length leads to limited flexibility and unstable efficacy. It cannot learn periodicity without providing an exact cycle length. (3) High model complexity. The quadratic complexity of the self-attention mechanism hinders practical application [51, 59]. As shown in Figure 1, Transformer-based methods exhibit relatively large model sizes and high training costs. Recent methods based frequency analysis and MLP partially alleviate this with more efficient architectures. However, satisfactory balance between performance and efficiency remains elusive.
general, there remain three major
To address these problems. we propose TimeEmb, a lightweight static-dynamic disentanglement framework. TimeEmb decomposes the original time series into inme-invariant and time-varying components, and processes them accordingly. Specifically, introduce a learnable time-invariant embedding bank to extract static time-invarinn patterns. These embeddings are consistent across all time series segments within the entire dataset, aiming capture long-term and stable temporal patterns. In addition, the embedding bank provides specific embedding for individual timesteps. This enables the model to adapt local data distribution shifts since time-invariant patterns may differ at different timesteps. By separating the time-invariant component from the time series, we obtain the remaining time-varying component, illustrating dynamic disturbance. Frequency analysis describes complex signals using their intensity in the frequency spectrum [1]. which presents clear intrinsic periodicity features. Inspired by this, we design an efficient frequency filter to process the

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vire-ig component through dense weighting. Based on the explicit decomposition and parallel processing of the static and dynamic components, TimeEmb achieves state-of- th-therrrt performance. Meanwhile, due to its lightweight architecture, it requires fewer computational resources. As shown by its optimal position Figure 1, the proposed TimeEmb strikes an excellent balance between performance and efficiency.
Our major contributions are summarized as follows:
For the first time, we propose to leverage learnable embedding bank to capture the global recurrent features while adapting to local distribution shifts.
We propose TimeEmb, which explicitly disentangles the time series and systematically addresses the time-invariant component using learnable embedding bank and time-varying component via frequency filtering.
The proposed TimeEmb can easily and seamlessly serve as plug-in to enhance existing methods with minimum additional computational cost.
Experiments seven benchmark datasets from diverse scenarios demonstrate the superior per- formance of the proposed TimeEmb. TimeEmb efficient in terms of computation and storage compared to existing state-of-the-art basclines.
Related Work
Transformer-based Time Series Forecasting. Transformers have shown strong sequence model- ing capabilities in time series forecasting [51, 58, 22]. PatchTST [28] segments sequences into fixed-length patches for local-global modeling, while iTransformer [22] and Informer [58] reduce attention complexity improve scalability. However, attention-based models still incur considerable computational and memory costs [12, 43], limiting deployment in resource-constraines settings. In contrast, TimeEmb leverages lightweight spectral modules. i.e. including an embedding bank and frequency filter, to achieve strong performance with reduced overhead.
MLP-based Time Series Forecasting. Recently, MLP methods, e.g. TSMixer [4] and TimeMixer [49], have demonstrated competitive forecasting performance with reduced complexity. DLinear [56] further improves efficiency by separating trend and residual components. By contrast, TimeEmb provides an explicit disentanglement framework in the frequency domain, enabling simultancous modeling of time-invariant and time-varying patterns beyond what time-domain MLPs can express.
Frequency-based Time Series Forecasting. Recent work has explored Fourier-based representations to model periodicity and reduce noise sensitivity [52. 59, 29]. While most methods apply global spectral analysis, TimeEmb introduces a fine-grained disentanglement strategy: a time-invariant component is learned across the full spectrum via embedding, while the dynamic part is filtered adaptively by a learnable frequency modulation. This structured spectral design extends the utility of eiqueenc-domaain modeling for complex time series.
mambedding-enhaneee Forecasting. Embedding strategies have been adopted to encode positional, spatial, or temporal context [28, 39, 10]. For instance, STID [39] and D2STGNN [41] use spatiotem- poral embeddings, while SOFTS [10] shares embeddings across channels. Unlike these, TimeEmb establishes a learnable temporal embedding bank that captures global time-invariant patterns across the dataset, with each embedding specializing in a specific time slot to model static structures in a data-driven and frequency- manner.
LLM-based Time Series Forecasting. With the rapid development of Large Language Models, recent research has explored their potential in time series forecasting by treating temporal signals as sequential tokens [19, 16]. LLM- based approaches benefit from strong generalization and transfer capabilities, enabling zero-shot or few-shot forecasting across diverse domains [17, 24, 48). Neverthe- less, these models are typically resource-intensive and require massive pretraining corpora, making them impractical for lightweight or Domain-speciffe applications. Compared with these paradigms, TimeEmb focuses on a compact yet effective disentanglement mechanism that achieves comparable forecasting accuracy with substantially reduced computational cost and training complexity.
3

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Methodology
3.1 Framework Overview
Given historical time series X E RLXD with L timesteps and channels, time series prediction aims to infer the future states of H timesteps, i.e., X RHxD
No F:1
C
C
x dnspg
θ
Embedding bank
E RNKPPD
R
TimeEmb addresses time series forecasting via disentangled representation learning in the frequency domain. The core idea is to decompose the input sequence into time-invariant component X and a time- varying component X
Time series &
Static X,
Dynamic X₃
Static-dynamic disentanglement
Figure 2: TimeEmb framework overview.
Specifically, we first transform the input series X into its frequency representation
X via the Fourier transform. Then, we retrieve X, from a learnable embedding bank E based on the input timestamp, capturing long-term stable patterns. The dynamic part X a is obtained by subtracting X, from X. To model complex dynamics, Hi to X emphasizing informative frequencies and suppressing noise. The filtered dynamic and static components then fused and transformed back to the time domain for final prediction. This trequency-basee decomposition allows TimeEmb to efficiently capture both periodic structures and transient variations in a lightweight and interpretable manner.
apply learnable frequency filter
3.2 Domain Transformation
Viewing time series data from the perspective of the frequency domain offers unique insights into its underlying structure. Unlike the time domain, where patterns may be obscured by noise or nonlinearity, the frequency spectrum reveals the distribution of different periodic components and their relative energy contributions. Transforming time series into the frequency domain decomposes it into distinct frequency components, describing the complex signal as linear combination of sine and cosine waves with varying frequencies and amplitudes. This process helps reveal underlying periodicities and hidden features that are otherwise obscure in the time domain [42].
Given a discrete temporal sequence X RLXD (we consider the univariate case with D - 1 for clarity), we first conduct instance normalization InstNorm() to standardize each instance's distribution at every timestep. Then, its frequency -omaann representation X E CFD can be obtained using real-valued Fast Fourier Transform (rFFT) [27].
X
Σ n=0
x -2ann//
22nk/LL k= 0. 1-
where j = /-1 is the imaginary unit. Due to the conjugate symmetry property of real signals in the Fourier domain, the number of unique frequency components is F [L/2] + allowing for compact representation without redundancy.
3.3 Static Component via Embedding Bank
Existing approaches modeling úime-invariant patterns, such scasonal-tres -renn decomposition [51, 56], typically divide time series into trend and residual components using local smoothing methods. However, this method merely considers locally stable and dynamic parts in the input time series, and fails to uncover the man-sitnding invariant features in the dataset. Recently, CycleNet [14] attempts to address this by learning periodic embedding, but depends on predefined period length from expert knowledge, and slight changes can cause severe performance drops.
To address these limitations, TimeEmb proposes flexible and learnable mechanism to capture long-term, recurrent patterns shared across time series through temporal embedding bank. For example, in traffic forecasting, aim capture recurring daily structures such typical rush-hour

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patterns. Since intra-day patterns also vary over time (e.g., hourly traffic flow fluctuations), we construct embeddings for each timestep.
In specific, we define a learnable embedding bank E E RMXFxD consisting of M embeddings preserve invariant patterns in day. M controls the granularity of intra-day specific patterns. For instance, when M = 24, E assigns an embedding to each hour; when M = 96, it captures common patterns every 15 minutes. To guarantee the embedding learns the general pattern across the time series, we leverage the last timestep of the input X as faast. This index enables us to retrieve embedding from E, i.e., X, Ettt mod M). Then, we separate the embedding X, from time series X and obtain the vurying component Xa as follows,
X
X X,-
In this operation, we subtract the real number X, from the real part of the complex number X, which can reduce the computational and storage cost of the embedding bank. The embedding bank E is optimized across the entire dataset and learns to encode consistent patterns that emerge at the same time across different days. For instance, when M 24, each embedding is tuned capture the average behavior at specific hour of the day (e.g., peaks around 8:00 and lows around 23:00), enabling the model to represent both the global temporal structure and local variations. Importantly, this embedding structure is flexible: while we focus on day-level periodicities, it can be naturally extended to model weekly or other common-sense periods by adjusting M, without relying on domain knowledge. This design enables TimeEmb to learn shared expressive representations of components, which are essential for disentangled modeling and robust generalization.
3.4 Dynamic Component via Frequency Filtering
To effectively model the dynamic component Xa. we apply a learnable spectral filter in the frequency domain. This design motivated by the Convolution Theorem [23], i.e., circular convolution in the time domain is equivalent to element-wis multiplication in the frequency domain. Thus, frequency-domain filtering provides an efficient and expressive way to implement time-invariant linear operations temporal signals.
We introduce complex-valuse valued spectral modulation vector E Cx1,, shared across channels, to selectively reweight different frequency bands. The filtering operation is defined as:
Haw X
[k)
X @ (kk,
where C represents dot product.
This operation can be interpreted as learning the frequency response function of linear time- invariant (LTTT) system [50]. By optimizing w end-to-end, the model can flexibly approximate linear time-invariant transformations of temporal signals. This provides both theoretical generality and practical flexibility for modeling diverse temporal dynamics. Theoretical analysis can be referred to Appendix A. After modulation, the filtered dynamic component is fused with the time-invariant part X, s to recover the full frequency representation:
X (Xa)
3.5 Prediction Layer
We leverage a prediction layer fo to produce the final prediction given representation X. It can be customized to specific requirements. We adopt a two-layer MLP architecture in TimeEmb,
fo(X) = W2(Re X x++b)) + b2,
(5)
where W, E L and We E T xx are projection matrices, H denotes the forecasting horizon, and b, E Rdd by H are the corresponding biases.
To restore the time series to its original scale, we conduct inverse normalization with the instance- specific mean and variance. Consequently, the final prediction X Hx is computed as,
5

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X (FFN
(6)
3.6 Optimization Objective
For model optimization, we employ the Mean Squared Error (MSE) to measure the loss between prediction and ground truth. Inspired by the self-correlation of the values in time series [47], introduce Mean Absolute Error (MAE) loss the frequency domain to alleviate the influence of self-correlation. In summary, our optimization objective function L can be expressed as follows,
((X = oMAAE(FFT(> X), FFT(Y)) (1 -a) MSE X, Y); L
where E [0, 1] is hyper-pate papre-parameter. The workflow of TimeEmb is detailed in Appendix B.
3.7 Computational Efficiency Analysis
We analyze the computational complexity of the core components of our TimeEmb, i.e.. time-invariant embedding bank and frequency filtering.
Time-invariant Embedding. The embedding bank E MxFxD supports two lightweight operations: embedding lookup and frequency-wise subtraction. Given an input time series X E RLXD an embedding X, E Fx is retrieved based on its last timestamp with complexity ((M).
The subtraction step X X X involves O(F D) operations. Thus, the overall complexity is linear, i.e., O(M + Fx x D).
The embedding bank is also parameter-efficient: for example, in ETTh1 with T, = 96, M 24, F = 49, and D 7, the total parameters required are only 24 x 49 x 7 = 8, 232.
Frequency Filtering. The spectral modulation of the dynamic component X & is performed by element-wise multiplication with the learnable filter E C2x1 yielding a complexity of O(F D).
Finally, the dominant cost in TimeEmb arises from the Fourier Transform, which operates at O(D x Llog L). Overall, the computational complexity of key components is linear, making it highly efficient and scalable for long sequences and multivariate inputs.
Experiments
In this section, we conduct extensive experiments with real-world time series benchmarks to suffi- ciently assess the performance of our proposed model, including comparison with SOTA baselines (Section 4.2), compatibility evaluation (Section 4.3), time series disentanglement capability analysis (Section 4.4), and modules' effectiveness verification (Section 4.5).
4.1
Experimental Setup
4.1.1 Datasets and Baselines
Following the mainstream evaluation setup in existing time series prediction studies [51, 58), we conduct experiments seven real-world benchmark datasets, including four ETT datasets (ETTh1, ETTh2, ETTml, ETTm2) [58], Weather [51], Electricity (ECL) [51]. and Traffic [51]. Following prior works [51, 22], we split the ETTs dataset into training, validation, and sets with ratio of 6:2:2, while the other datasets were split in a ratio of 7:1:2.
To comprehensively evaluate the effectiveness, select comprehensive SOTA baselines across three representative frameworks: (1) Frequency-based models: FilterNet [54], FITS [53], and FreTS [55]; (2) MLP-based models: DLinear [56] and CycleNet [14]; and (3) Transformer-baser models: iTTranffomme [22], PStchTST [28), and Fredformer [34]. Detailed introduction of datasets and baselines can be found in Appendix C.
4.1.2 Implementation Details
To ensure fair comparison, we adapt common experimental settings: lookback window lengths E (96, 336, 720} and prediction lengths H {96, 192, 336, 720} for all baselines across datasets [51,

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Table 1: Performance comparison with prediction lengths H (96, 192, 336, 720} and lookback
window length L = 96. The best results are highlighted in bold and the second best are underlined.
Madel
2004
2024
2024
2024
2003
2024
3523
2023
Merric
M55
MARI
MA1
14.5
14527
30AT:
3451
14.1:
14007
MAA
5451
5051
MAABL
14527
MAT
0.364-00011
0.378
5.3M1
0.323
0.362
0.375
0.855
0.361
0.990
0.906
0.306
0.345
0.800
01 1:385
0-100
356
6487250011
6446250001
1364
0450
0.433
0.433
D4476
0.465
0.487
0.453
0.411
0
0675
0.444
0.493
01412
14481
0.429
720
0489200002
046324001
0.661
5.600
06483
1.45
04244
0.400
0.500
0.431
0.467
0479
0,902
0.455
0.553
0,532
0,512
0,516
145
192
0.356:0001
0.37
0.373
0.59
0.371
0.385
0.399
0.555
0.380
0.853
0.376
0.965
0.501
0.386
0.385
0.425
0.477
9.676
336
0.1ECH0002
1.113-0011
0.431
0.453
0.80
0.309
0.220
0.43
0.438
0.433
0.480
0.511
0.066
0.838
0.463
0.867
0.994
0.541
0L38220001
5.04
11355
0.391
0.404
0.383
0.40
0.465
0.399
0.315
90
0339
0200
1.33%
0.36
1021
0.5%
0.344
0.308
0.334
0:365
0:355
0.315
0.335
0.373
1.345
0.572
IUTT22 677al
190
6373300011
0360
0201
1631
L31
0385
0317
11241
01385
0.26
6-60
0.395
11:31
01:00
11:80
0:39
720
048520001
6428:50001
1241
0253
0496
0444
0.491
0.45
0.415
6428
6:80
0.449
0.485
0.465
0.234
0.55
14g
0338800001
6.35220001
0.59
0.394
0.99
0394
0.586
0.407
0.489
0.356
0.49
0445
0.406
0.455
0,415
0.40
0.407
0.802200011
0:34220011
0.39
0222
032
0:53
0.180
0.354
0-34
6.66
0.58
0.366
0.187
0.37
0,93
0.2%2
356
032500001
0.33420001
6335
6340
0001
633
0311
0.341
0.35
1365
8360
0.342
0.343
0.56
0339
0
770
100
0415
5/00
8-90
0.89
0100
8551
0.532
026
019--000
0.14
006
0.60
0.174
0166
0:334
0.38
0.96
Weathe
180
0380
833
633
1700
6700
0.311
0351
6:31
636
671
0.54
0.3
0350
0777
0.96
306
0.29220011
622222001
032
1223
020
0278
0.22
0:13
0-26
020
0204
0.350
0233
0.35
0:32250011
6271
0.345
0.312
0.46
0.258
0.218
0.256
6279
632
0.26
0.250
0.235
0.255
0.317
Tratte Elookitee
95
10226
6219
11547
0.45
0.148
0.240
0.324
0331
6:30
0.218
0.775
0,234
0.597
0.22
195
01333
0.24
0121
11000
0.20
0.162
01333
0.23
0202
0:30
0.30
0.35
013000001
0.31
0.39
720
0388-0091
6207-2011
0312
8.000
0313
0.364
D210
5.0
0.225
0337
0.33
6315
0255
0.33
0.235
0315
0.45
0.353
0.068
0.266
0.26
0.17
0.275
01:99
0.26
0217
0.20
01397
0.239
0222
0.30
192
8444281001
0.457
0.34
1.55
11360
D4422
0.36
0.417
0.276
0.454
060
13337
0.511
0.70
336
155441012
1333-40033
0.70
0.50
0.383
1450
0.366
0.535
0.387
0.450
036
060
0.366
0.333
0.733
720
0.48731023
0.000
0311
0463
0.388
0.23
0.667
0.454
BM6
0.38
0.433
0.253:0001
0.365
Table 2: Performance comparison of average prediction lengths with lookback lengths L E {336, 720]. The best results are highlighted in bold and the second best are in underlined.
Lookhack
L
336
720
Modal
TimeEmb
CycleNet
FilterNet
iTrnnfommr
TimeEmb
CycleNet
SOFTS
DLinger
Metric
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAB
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
ETTbL
0.410
0.423
0.415
0.426
0.423
0.437
0.440
0.447
0.418
0.433
0.430
0.439
0.434
0.455
0.437
0.448
ETTml
0.371
0.355
0.352
0.381
0.365
0.392
0.345
0.376
0.381
0.364
0.396
0
KTTm2
0.247
0,913
0.251
0.309
0.265
0.325
0.286
0.337
0.24N
0.308
0.249
0.312
0.268
0.331
0.261
0.327
Weather
0.221
0.255
0.226
0.266
0.224
0.239
0.236
0.272
0.218
0.257
0.224
0.266
0.230
0.232
0.340
0.292
58, 14J. Forecasting metrics include MSE and MAE, with results averaged over five independent runs. TimeEmb is trained for 30 epochs with early stopping (patience = on the validation set). Batch sizes are 256 for ETTs and the Weather dataset, and 64 for others. Learning rates are selected from {0.0005, 0.001, 0.002, 0.005). with TimeEmb's ' hidden layer size fixed at 512. Experiments use Py PoTor 2.1 [33] on an NVIDIA RTX 4090 24GB GPU, with details in Appendix C.
4.2 Overall Performance
Table 1 presents the comparison results with L, = 96 and H E (96, 192, 336, 720). The baseline results are from the original papers. Several conclusions can be made as follows:
(1) TimeEmb consistently outperforms strong baselines across diverse datasets. Across multiple benchmarks and forecast horizons, TimeEmb achieves a significant reduction in MSE, with relative improvements ranging from 3.0% to 8.7% on average. This highlights the effectiveness of our frequency-based dynamic-static decomposition framework, which explicitly separates and models üime-invrrian and time-varying components.
(2) TimeEmb surpasses disentanglementtbusel baselines by offering more expressive and flexible decomposition. While CycleNet relies on a single long-period embedding and DLinear adopts a local moving average for trend extraction, both approaches struggle to capture long-term temporal patterns effectively. In contrast, TimeEmb leverages global, timestamp-aware embedding bank to learn and represent recurring invariant patterns, enabling more accurate long-range forecasting.
7

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Table 3: Performance of integrating TimeEmb with different backbones Electricity and Weather. The best results are bold. Impr. indicates the performance improvement by equipping TimeEmb.
Licercay
Weater
Herino
W
197
135
770
45
190
175
730
Marki
MSS
MA4
1656
MA4
MSS
MAA
MSS
MAF
MSS
MAF
KIS
MAA
5I55
MAA
MSF
MAR
Linaa lege
1.990
0.779
0.20
1.204
0.397
0.218
0.259
0.366
+1..6
13.26
18.20
+2.05
+729
13440
14.00
$3.9%
+3..5
114.%
06.76
-1195
+3.5%
$1100
.03%
+0.45
MLP
0372
0.266
0.383
0771
0387
01333
01334
0:231
0.80
0:354
0333
0.22
0.338
8,30
11770
-oa tadd
Diinc
0,995
0.276
0.196
0201
0.227
0.339
0.343
0,501
0.355
0.254
0.237
0.205
0.211
0.329
0.341
IL385
-as modd
0.77
0.771
8J881
0.31
0.398
0.359
0,323
8.321
0.888
0
0.236
0.377
0.364
0.386
0.335
0
legss
+2.5%
40075
-8.2%
+33.09
$5.95
-409
44.0%
+5.95
mode..
0,542
0.242
0.31
0.375
0.383
0,362
0,238
0.251
0,296
0.346
0L344
Trege
-7.2%
+1.36
+1.6
-1.5%
+5.8%
-0.0%
16.95
+3.5%
+0.96
+6.9%
+7.19
+3.50
+5.3%
-2.0%
+3.95
+2.35
6.8
0.6
0.5
Analitude
0.4
Ampittud 2
w
0.4
0.2
2
29
25
30
88
40
45
25
30
35
40
45
10
20
30
40
10
20
30
40
Prequency
Prequency
(a) Time series X spectrum (colorful lines) and time- invariant embedding X, spectrum (bold red line)
(b) Time-varying component X spectrum
Figure 3: Disentangled features visualization in frequency spectrum. Frequency components from 20 to 49 are zoomed in at the top right corner.
(3) TimeEmb outperforms frequency-domain models by jointly modeling invariant and dynamic components. FilterNet and FITS adopt fixed filtering approach, which may not effectively handle non-stationary frequency components, as they ignore the global invariant patterns. Conversely, except for the frequency filter for the vary-ng component, the embeddings in TimeEmb can preserve long-term invariant patterns, indicating the structural information of time series.
To evaluate model efficiency, we compare the number of trainable parameters, training time, and MSE on the Electricity dataset against mainstream baselines, as shown in Figure 1. Notably, TimeEmb uses over 5x fewer parameters than the representative Transformer-based model f'rransormeer while simultaneously achieving the best predictive performance. Benefiting from its lightweight design, TimeEmb significantly accelerates training without compromising accuracy, demonstrating an exceptional balance between efficiency and effectiveness.
To further assess the model's ability to capture long-term dependencies, we evaluate TimeEmb under extended lookback windows. Table 2 reports the average performance across all prediction lengths for = 336 and L = 720. Full results are deferred to Appendix D. TimeEmb maintains state-of-the performance under long input horizons, showcasing its strong temporal modeling capacity.
4.3 Compatibility Analysis
To assess the generalizability of our proposed disentanglement mechanism for decoupling time- invariant and time-varying components, we integrate it into several state-of-the-arr time series forecasting models, spanning both MLP-based and Transformer-based architectures. Specifically, our method only replaces the backbone prediction layer with the alternative model in the time domain, enabling seamless integration with different models. As shown in Table 3, incorporating our method consistently improves baseline performance across various prediction horizons, validating its effectiveness as a plug-and-play enhancement for diverse forecasting frameworks. Importantly, this integration incurs minimal computational overhead, enabling seamless adoption without significantly increasing model complexity or training cost. These results highlight the broad applicability of our disentanglement framework and its potential to strengthen existing models with negligible trade-offs.

---

0.5
5.40
5475
0.47
0.2
0.4
5.46
0.46
0.8
0.42
0.52
044
6415
0.2
2.13
68 att Jwern Marng
0.42
5.43
pa0 Hel
(a) Amplitude analysis
(b) Frequency analysis
(c) MSE on ETTh1
(d) MAE on ETThl
Figure Key components contribution analysis.
4.4 Disentangled Features Visualization
To evaluate the disentanglement capability of TimeEmb in separating fime-invariant and time-varying components, we conduct diverse visualization illustrations. We first select the first channel from the ETTm2 dataset and extract a dozen time series ending at 0 o' clock (i.e., time index lastt 0) from different days. In Figure 3()), illustrate the sisuunncy-ommittn representations X of these series as colorful lines, and the corresponding learned time-invarian embedding X, as bold red line. For clarity, zoom in frequency components in the range of 20 49. We can observe that the multiple X from different days exhibit similar spectral structures, and the learned corresponding time-invariant embedding X captures the common pattern to a certain extent. Figure 3 (b) shows the time-varying component X d. which are relatively distinct from one another. The results clearly indicate that the original time series are hard to distinguish, but they become more separable after subtracting the time-invrrian embedding. It shows that TimeEmb successfully captures the shared inverinvariant components across the input sequences, preserving the general structural information.
In addition, we present the distribution of the data before and after disentanglement from high-level perspective.
100
We project the data samples from the Electricity test set onto a two-dimensional space using the T-SNE [45]. To capture week-level time-invariant patterns, we add an em- bedding bank composed of 7 learnable embeddings.
75
25
Day Index
The time series X and the time-varying component X are depicted in Figure 4 (a) and (b), respectively. Time se- ries corresponding to different days of the week are color- coded. For example, the time series for Monday is in dark blue, and that for Sunday is in light yellow. As can be observed from Figure 4 (a), the time series X from differ- ent days of the week tend to be intermixed, suggesting the existence of inherent similar patterns among them. After separating the time-invariant component from X. the time- varying components X for different days of the week earry specific information, as evidenced by their distinct and isolated distribution in Figure (b). This visualization supports our previous finding again: by disentangling the time-invariant component X., the time-narying compo- nent X, becomes more distinguishable compared to the original time series X.
25
550
775
100
30-60-40-20
20
40
60
60
75
50
25.
Day Index
-50
More visualization of TimeEmb can be found in Ap- pendix D.
100
60 17-60-4--00
20
40
60
80
Figure 4: T-SNE visualization results.
4.5
Ablation Study
We conduct comprehensive ablation study to evaluate the contributions of the key components in TimeEmb. The analysis is performed from two main perspectives: (1) the frequency composition of the time-invariant embedding, and (2) the impact of removing altering individual modules.

---

4.5.1 Frequency Spectrum Analysis of X
To examine how different frequency components contribute to the time-invariant embedding X.. we design two controlled perturbation strategies. Amplitude-based masking: For each input X. we preserve only the top-k frequency components of X, with the highest amplitudes, and zero out the rest. Frequency-based filtering: We apply a low-pass filter by retaining only certain proportion of low-frequency components, discarding the High-frequency parts. The results are shown in Figure 5 (a) and (b), with full details provided in Appendix D. As shown in Figure 5(a), model performance improves as more high-amplitude components are retained, indicating that both principal and subordinate frequencies carry useful invariant information [52]. Similarly, Figure 5(b) shows that increasing the proportion importance of capturing both short-term and long-term periodicities in the invariant representation. These findings support the use of the full spectrum in constructing X.
low-frequency components leads to better performance, reflecting the
4.5.2 Component-wise Ablation
To assess the individual impact of key modules, we construct several variants of TimeEmb by altering or removing components: Random: The embedding bank is randomly initialized between training and testing. Zero/Mean: The embedding bank is fixed zeros or the global mean value, respectively. w/o X,: The time-invariant component is entirely removed. w/o Hww The frequency filter is removed from the dynamic processing path. The results in Figure (c) and (d) demonstrate that both the embedding bank and the frequency filter substantially contribute to model performance. In particular, removing either module leads to notable degradation, confirming the importance of jointly modeling the time-invariant and time-varying components. Complete ablation results are reported in Appendix D. Together, these findings validate the effectiveness of our systematic disentanglement framework, in which X, and X d are processed independently via dedicated structures to capture complementary temporal characteristics.
5 Conclusion
In this paper, we tackle the crucial issue of temporal non-stationarity in time series forecasting using well-structured decomposition framework. We introduce TimeEmb, lightweight yet effective architecture that combines global temporal embeddings and spectral filtering. TimeEmb enables separate processing of the disentangled time-variane and time-invan invariant components. Specifically, utilize learnable embeddings to preserve the long-term invariant patterns within time series. Moreover, devise a frequency filter to capture the temporal dependencies of the time- -vryiig component. Extensive experiments confirm that our method not only attains offers interpretable insights into temporal patterns via its dual-path design. It achieves outstanding balance between performance and efficiency. Furthermore, it can be easily integrated with existing methods, thereby enhancing the ability predict time series.
performance but also
Acknowledgements
This work was supported by the China Postdoctoral Science Foundation under Grant No. 2025M771587; the Open Funding Programs of State Key Laboratory of AI Safety (2025-09): National Science and Technology Major Project under Grant No. 202 2021ZD01 12500; the National Natural Science Foundation of China under Grant Nos. U22A2098, 62172185, 62206105, and 62202200; the Major Science and Technology Development Plan of Jilin Province under Grant No.20244011003GG the Major Science and Technology Development Plan of Changchun under Grant N0.2024WX05; the XJTU Research Fund for AI Science (No. 2025YXYC004).
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A In-Depth Analysis of TimeEmb
A.1 Innovation Discussions
TimeEmb vs. Disentanglement Methods. While prior disentanglement approaches [31, 51] focus primarily on separating trend and residual components based local statisties within individual time series, TimeEmb introduces two fundamental advancements, First, instead of local decomposition, our model leverages learnable embedding bank to capture globally consistent and recurrent patterns across the entire dataset, effectively preserving system-level invariants, Second, TimeEmb is the first incorporate learnable requency-domaal filtering into the disentanglement framework, enabling efficient and expressive modeling of dynamic components in the spectral space.
Eib vs. Embeedding-enhnncee Methods. Embeddinn-ennancel models [40, 39] typically use identifier-baset embeddings (e.g., time slot, spatial ID) to encode auxiliary information. In contrast, TimeEmb adopts a decomposition- baaeed design where learnable temporal embedding bank explicitly models the latent periodic patterns without relying on predefined identifiers or external priors.
signal component. This enables data-driven recovery of
A.2 Theoritical Support
In this section, we theoretically analyze the core design of TimeEmb from a frequency-domain perspective. We focus main aspects: the completeness of frequency-done -omaann representation and operations, and the expressiveness of the learnable spectral filtering mechanism for modeling dynamic temporal signals.
A.2.1 Completeness of Frequency-Domain Representation and Operations
TimeEmb operates entirely in the frequency domain by applying the real-valued Fast Fourier Trans- form (rFFT) to input sequences. For real-valued time series X E L its spectral representation is obtained as X FxD, where F L/2] + 1 due the conjugate symmetry of the spectrum. The rFFT is defined as:
X Σ
x [[n] e -22kkmn
k == F 1.
This transformation is invertible via the corresponding inverse real FFT (irFFT), guarantecing that no information is lost the process. Thus, rFFT offers a complete and efficient frequency representation of real-valued signals [30, 5]. Beyond transformation, TimeEmb performs sequence of operations entirely in the frequency domain:
1. Subtraction time-invariant embedding X, from the input spectrum X;
Frequency-wise modulation of the residual X
X X via a learnable filter w;
3. Reconstruction of the final spectrum X output in the time domain.
Xx + X W, followed by an irFFT to recover the
Each of these operations, i.e., subtraction, modulation, and addition, is algebraically well-defined and closed in the frequency domain. Because the rFFT is invertible, the entire transformation chain in TimeEmb is pppresentaaion-complle. all original information is preserved, while allowing structured manipulation the spectral space.
This design offers several important advantages. First, it enables precise modeling of periodicity and oscillatory behavior, which are often hard localize in the time domain. Second, working entirely in the frequency space allows for efficient and interpretable decomposition of long-range temporal patterns. Lastly, the model avoids any information loss due to projection or truncation, ensuring theoretical soundness in its design.
A.2.2 Expressiveness Frequency-Donie -omain Filtering
To model the dynamic (time-varying) component of the input sequence, TimeEmb applies a frequency- domain filter over the residual spectrum. Formally, given the residual X E C x learnable
15

---

modulation vector w E Px1 is applied as:
Haw H. Xa(k] C wik).
(9)
This operation is grounded in the Convolution Theorem: pointwise multiplication in the frequency domain corresponds to convolution in the time domain [30]. Therefore, the frequency filter can be interpreted as learning the impulse response of a Linear Time-Invariant (LTI) system directly in the spectral domain.
This interpretation grants the model several expressive and practical advantages [3]. First, it enables the learning of flexible signal transformations that go beyond local convolutions, e.g., capturing long-range dependencies with global frequency-aware operations. Second, the filtering process is computationally efficient, operating in O( D), and avoids the kernel length constraints inherent in time-domain CNNs. Finally, this formulation provides intuitive control over the model's sensitivity various periodic structures, allowing it emphasize suppress spectral bands depending task-specific dynamics.
In essence, the frequency filter in TimeEmb serves powerful and compact operator that simulates a broad family of spectral responses.
A.3 Convolution Theorem
Frequency filtering modifies signal's frequency content. Given signal z[n] and filter with frequency response H[:], the filtered signal y X k
H in the frequency domain. By the
convolution theorem, the filtered signal ymm] IDFT(Y k (Ir h)) [n] in the time domain. The proof is as follows:
Let ærn] and [r] be length N sequences with DFTs X [kk] and [A:]:
X|
x r/n]e
5
0, 1,
(10)
H Σ h = .N
(11)
The circular convolution of ærn]] and h[n] is defined y/n] (xr & h)[m]] > [<m]][[( m) mod N]. where @ represents the circular convolution operation, and (n m) mod N denotes the modulo N operation of n m.
The DFT y/n]. denoted as [kk]
N Σ
y|k]
ylnle kn
(12)
0=
Σ
Σ
22mmh[(( m) mod N)
e- kn
(13)
Let [ (n m) mod N, and the above equation can be rewritten as:
N-1
N-1
YıK = N
[mm]
hllle - k(1+m)
(14)
T==
1=0
According to the exponential operation rule, we have:
Σ x/m]e km
Σ
hille 37
(15)
Therefore:
(16)
16

---

Algorithm Workflow of Timelimb.
Input: Time series E RLXD
Output: Prediction X E RDxD
1: // Domain transformation
2: IFT (X))
3: // Time series disentanglement
4: Xd - - X (Eq. (2))
5: // Frequency filtering
6: X (X X, (Eq. (4))
7: // Final prediction
8: X = (nnNorm(Jo(IFFT/
(Eq. (6)
9: Return: X
Table 4: Dataset statistics. "Channels" denotes the number of variables in each dataset; "M of each bank denotes the capacity of each embedding bank utilized in TimeEmb. "d" refers to the day-level embedding bank, while "w" indicates the week-level embedding bank.
Datasets
ETThl
ETTh2
ETTml
ETTm2
Flectricity
Weather
Traffic
Channels
7
7
21
862
Timesteps
17420
17420
69680
69680
26304
52696
17544
Frequency
Hourly
Hourly
15min
15min
Hourly
10min
Hourly
Domain
Electricity
Electricity
Electricity
Electricity
Electricity
Weather
Traffic
M of each bank
24
(d)
(d)
24 (d)
(d) + (w)
24 (d)
24 (d) + (w)
The same is true for inverse derivation. We can ultimately infer that:
N
y [n]] IDFT(){() (x h)
Σ
x|m]h[(n- m) mod N].
(17)
In conclusion, we have proved that the DFT of the circular convolution is equal the product of the DFTs, and the IDFT of the product of the DFTs is equal to the circular convolution, which means that frequency filtering (multiplication the DFT domain) equivalent to circular convolution in the time domain.
B Algorithm
We present the pipeline TimeEmb in Algorithm We begin by performing domain transformation for frequency analysis (line 2). Specifically, we apply instance normalization to the input sequence X, followed by a Fast Fourier Transformation (FFT) to obtain the frequency series X. Next, to retrieve the corresponding embedding from the embedding bank,
disentangle the time series, which serves the time-invariant component X s- We then separate it from X extract the time- varying component X a (line 4). Subsequently, we implement frequency filtering with spectral modulation operator w effectively model the dynamic component. After this step, we add back the time-invariant series X, (line 6). The combined series then undergoes the Inverse Fast Fourier Transform (IFFT), followed by a projection layer, and concludes with inverse normalization to generate the final prediction (line 8).
C Experimental Setup
C.1 Datasets
We detail the description the datasets here:
ETT (Electricity Transformer Temperature) contains two subsets of data: ETTh and ETTm. These datasets are based on hourly and 15-minute intervals, collected from electricity transformers between July 2016 and July 2018.
17

---

Weather records 21 weather features, including air temperature and humidity, every ten minutes throughout 2020.
Electricity collects 321 clients' electricity consumption hourly from 2012 to 2014.
Traffic comprises the hourly data of 862 sensors of San Francisco freeways from 2015 to 2016.
Detailed statistics are displayed in Table 4.
C.2 Baselines
We compare TimeEmb with 9 representative and state-of-the-art models to evaluate the performance and effectiveness, including Frequency-based models, MLP-based models, and Transformer-base models. The details of these baselines are as follows:
FilterNet proposes two kinds of learnable filters-Plait shaping filter and Contextual shaping filter-to approximately surrogate the linear and attention mappings widely adopted in time series literature. The detailed implementation is available https: / //i/ github. com/ ai aikunyi/FFF /Fiitterett...
FITS conducts time series analysis using interpolation the complex frequency domain, achieving low cost with 10K parameters. The detailed implementation is available at https://g/ thub. com/ VEWOXIC/FITS.
FreTS presents new approach to utilizing MLPs in the frequency domain, effectively capturing the underlying patterns of time series while benefiting from a global view and energy compaction. The detailed implementation is available at https: : h////gtttt thub. com/ aikunyi/ Cauuuii/Free /FreTS.
DLinear employs a straightforward one-layer linear model capture temporal relationships through season-trend decomposition. The detailed implementation is available at https: //gith///ttt com/ cure-lab/LTSP Linear.
SOFTS introduces an efficient MLP-based model that utilizes a centralized strategy to enhance performance and lessen dependence the quality of each channel. The detailed implementation is available at https: /etp::/gtttttt thub. com/Secclies Lia-Cxy/SOFTS..
CycleNet utilizes RCF technique to separate the inherent periodic patterns within sequences and then performs predictions on the residual components of the modeled cycles. The detailed implementation is available at https: / www/ACAT-SOOE ACAAAAAATTTCCC----/AAAAA hatp//Cyceeeesss
iTransformer uses attention and feed-forward networks on inverted dimensions. embeds time points of individual series into variate tokens for the attention mechanism to capture multivariate correlations. Additionally, the feed-forward network applied each variate token learn non- linear representations. The detailed implementation is available at https:// http//ggttttb/tt com/ con/thum// i ormer.
PatchTST breaks down time series data into subseries-level patches, which helps in extracting local semantic information. The detailed implementation is available https://githu: com/ Citiniee3/PPacch00
Fredformer is Transformer-base based framework that addresses frequency bias by equally learning features across various frequency bands, which ensures the model does not neglect lower amplitude features that are erucial for accurate forecasting. The detailed implementation is available https: //github. com/ chenzRG/ GeanzRR/Freefff Fredf ormer.
C,3 Implementation Details
We implemented TimeEmb with PyTorch and conducted experiments on a single NVIDIA RTX4090 GPU that has 24GB of memory. TimeEmb was trained for 30 epochs, with early stopping implemented and patience level of 5 based on the validation set. The batch size was set to 256 for both the ETT and Weather datasets, while batch size of 64 was used for the remaining datasets. This adjustment was necessary because the latter datasets have larger number of channels, which requires smaller batch size to prevent aut-of-memory issues. The learning rate was chosen from the range 0.0005, 0.001, 0.002, 0.005, based on the performance on the validation set. The size of the hidden layer in TimeEmb was consistently set 512.
18

---

Table 5: Full results with lookback lengths L 336. The best results are in bold and the second best are underlined.
Model
TimeEmb
CycleNet
Filtereet
SOFTS
iTransformer
DI Minear
Metric
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
96
0.367
0.394
0.374
0.396
0.379
0.404
0.386
0.405
0.396
0.415
0.374
0.398
192
0.403
0.414
0.406
0.415
0.417
0.428
0.428
0.432
0.434
0.438
0.430
0.440
336
0.422
0.425
0.431
0.430
0.437
0.443
0.449
0.448
0.452
0.451
0.442
0.445
720
0.446
0,459
0.450
0.464
0.458
0.472
0,460
0.476
0.476
0.485
0.497
0.507
avg
0.410
0.423
0.415
0.426
0.423
0.437
0.431
0.440
0.440
0.447
0.436
0.448
96
0.276
0,333
0.279
0.341
0.302
0,356
0.298
0.356
0.334
0.379
0,281
0.347
ETTh2
192
0.335
0.378
0.342
0.385
0.350
0.393
0.360
0.394
0.413
0.424
0.367
0.404
336
0.370
0.405
0.371
0,413
0.376
0.414
0.385
0.415
0.414
0.432
0.438
0.454
720
0.396
0.433
0.426
0,451
0.414
0,444
0,449
0,463
0,433
0.454
0.598
0.549
avg
0.344
0.387
0.355
0.398
0.361
0.402
0.373
0.407
0.399
0.422
0.421
0.439
96
0.282
0.332
0.299
0.348
0.289
0.344
0.296
0.350
0.303
0.357
0.307
0.350
ETTml
192
0.323
0.361
0.334
0.367
0.331
0.369
0.336
0.374
0.345
0.383
0.340
0,373
336
0.353
0,380
0.368
0,386
0.364
0.389
0.371
0.396
0.375
0.397
0.377
0.397
720
0,403
0.410
0.417
0.414
0.425
0.423
0.433
0.432
0.435
0.432
0.433
0.433
avg
0,340
0.371
0.355
0.379
0.352
0.381
0.359
0,388
0.365
0.392
0,364
0.388
96
0.160
0.243
0.159
0.247
0.177
0.265
0.174
0.259
0,184
0.273
0.165
0.257
ET
192
0,218
0,283
0.214
0.286
0.232
0,304
0.240
0.307
0.262
0.322
0,227
0.307
336
0,265
0,316
0.269
0.322
0.284
0.339
0.295
0.342
0.307
0.351
0.304
0.362
720
0,346
0.370
0.363
0.382
0.367
0,390
0.377
0.396
0,390
0.402
0,431
0.441
avg
0,247
0.303
0.251
0.309
0.265
0.325
0,272
0.326
0.286
0.337
0,282
0.342
96
0.144
0.189
0.148
0.200
0.150
0.183
0,160
0.209
0.163
0.213
0,174
0.235
Weather
192
0.187
0,233
0.190
0.240
0.193
0.221
0,204
0.250
0.203
0.250
0,219
0.281
336
0.238
0.271
0.243
0.283
0.246
0.258
0,249
0.284
0,253
0.288
0,264
0.317
720
0,315
0.326
0.322
0.339
0.308
0.295
0.324
0.335
0.326
0.338
0,324
0.363
avg
0,221
0.255
0.226
0.266
0.224
0.239
0.234
0.270
0.236
0.272
0,245
0.299
96
0,128
0.223
0.128
0.223
0.132
0.224
0.127
0.221
0.133
0.229
0,140
0.237
Electricity
192
0,146
0.240
0,144
0.237
0.143
0,237
0.148
0.242
0.156
0.251
0,153
0.250
336
0,161
0.256
0,160
0.254
0.155
0.253
0,166
0.261
0,172
0.267
0.169
0.267
720
0.198
0.289
0.198
0,287
0.195
0.292
0,202
0.293
0,209
0.304
0,203
0.299
avg
0.158
0,252
0.158
0,250
0.156
0.252
0.161
0.254
0.168
0.263
0.166
0.263
96
0,381
0.263
0.386
0.268
0.398
0.289
0.346
0.246
0.361
0.255
0.410
0.282
Traffic
192
0.398
0.271
0.404
0.276
0.422
0.303
0.373
0.258
0,380
0,268
0.423
0.288
336
0.411
0.278
0.416
0.281
0.437
0.312
0.385
0.265
0.389
0.273
0.436
0.296
720
0.439
0.294
0.445
0.300
0.464
0.325
0.419
0.283
0.415
0,285
0.466
0.315
avg
0.407
0.277
0.413
0,281
0.430
0.307
0.381
0.263
0.386
0,270
0.434
0.295
D Detailed Results
D.1 Full Results with Lookback Window Length L E (336, 720]
To evaluate the performance of TimeEmb in modeling long-term temporal dependencies, we further conduct experiments with extended lookback window lengths of 336 and 720. As shown in Table 5 and Table 6, TimeEmb consistently achieves competitive or superior performance across various forecast horizons under these challenging settings. Unlike many baseline models whose performance degrades significantly as the input length increases, TimeEEm maintains stable accuracy, demonstrating strong temporal generalization.
This performance stems from the architectural design of TimeEmb. The time-invariant embedding bank allows the model to effectively summarize recurring structural patterns, regardless of input length. Meanwhile, the frequency-domale filter adaptively emphasizes relevant dynamic components without being constrained by local receptive fields. Together, these modules enable TimeEmb to capture both long-range dependencies and localized variations efficiently.
Overall, these results indicate that TimeEmb is not only effective under standard settings, but also exhibits strong scalability and resilience when applied to long-context forecasting tasks desirable property for real-world time series applications.
19

---

Table 6: Full results with lookback lengths L 7200. The best results are in bold and the second best are underlined.
Model
TimeEmb
CycleNet
FitterNet
SOFTS
iTransformer
DI Jinear
Metric
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
96
0.372
0.400
0.379
0.403
0.390
0.418
0.384
0.416
0.401
0.430
0.379
0.402
192
0.413
0.427
0.416
0.425
0.424
0.439
0.423
0.442
0.434
0.452
0.419
0.429
336
0.438
0.443
0.447
0.455
0.450
0.461
0.446
0.461
0.468
0.475
0.456
0.456
720
0.449
0,462
0.477
0.483
0.484
0.488
0,481
0.500
0.525
0.520
0.493
0.506
avg
0.418
0.433
0.430
0.439
0.437
0.452
0.434
0.455
0.457
0.469
0.437
0.448
96
0,290
0.348
0.271
0.337
0.297
0.357
0.295
0.357
0.306
0.369
0,309
0.373
ETTh2
192
0.347
0.387
0.332
0.380
0.361
0.400
0.365
0.401
0.372
0.409
0.409
0.433
336
0.376
0.411
0.362
0.408
0.397
0.431
0.398
0.426
0.403
0.434
0.508
0.495
720
0.399
0.439
0,415
0,449
0.435
0,460
0,448
0,473
0,434
0.464
0.851
0.653
avg
0.353
0.396
0.345
0.394
0.373
0.412
0.377
0.414
0.379
0.419
0.519
0.489
96
0.293
0.346
0.307
0.353
0.301
0.358
0.299
0.357
0.317
0.367
0.309
0.353
ETTml
192
0.326
0,366
0.337
0.371
0.340
0.379
0.342
0.381
0.347
0.385
0.345
0.376
336
0.356
0,382
0.364
0,387
0.375
0.398
0.375
0.401
0.377
0.402
0.376
0.398
720
0.405
0.410
0.410
0,411
0.434
0.426
0.441
0.443
0.429
0.431
0.436
0.436
avg
0,345
0.376
0.355
0.381
0.363
0.390
0.364
0.396
0.368
0.396
0.367
0.391
96
0.163
0.251
0.159
0.249
0.180
0.271
0.181
0.272
0.187
0.278
0.163
0.256
ET
192
0,219
0.290
0.214
0.289
0.239
0,313
0.234
0.310
0.251
0.319
0,220
0.300
336
0,265
0.320
0.268
0.326
0.283
0,341
0.284
0.342
0.307
0.355
0,283
0.347
720
0,343
0.372
0.353
0.384
0.361
0,394
0.373
0.398
0.391
0.411
0.376
0.406
avg
0,248
0.308
0.249
0.312
0.266
0..30
0,268
0.331
0.284
0.341
0,261
0.327
96
0.143
0,193
0.149
0.203
0.153
0,208
0,152
0.205
0.168
0.222
0,169
0.227
Weather
192
0.188
0.237
0.192
0.244
0.199
0.250
0.199
0.251
0.209
0.256
0,213
0.271
336
0.236
0,274
0.242
0.283
0.248
0,287
0,248
0.288
0.267
0.302
0,259
0.311
720
0,306
0.325
0,312
0.333
0.313
0,333
0.322
0.343
0.337
0.352
0,319
0.359
avg
0,218
0.257
0,224
0.266
0.228
0.270
0.230
0.272
0.245
0.283
0,240
0.292
Electricity
96
0,129
0.225
0.128
0.223
0.137
0.235
0.137
0.232
0.142
0.243
0.134
0.232
192
0.145
0.241
0.143
0.237
0.160
0.259
0.157
0.252
0.1G0
0.261
0,148
0.245
336
0,161
0.257
0.159
0.254
0.174
0.274
0,172
0.268
0,179
0.281
0,163
0.263
720
0.197
0.289
0.197
0,287
0.212
0.307
0.198
0.291
0,220
0.316
0.198
0.296
avg
0.158
0,253
0.157
0,250
0.171
0.269
0.166
0.261
0.175
0.275
0.161
0.259
96
0.374
0,268
0.374
0.268
0.386
0.285
0.355
0.253
0.358
0.254
0.388
0.275
Traffic
192
0.387
0.268
0.390
0.275
0.401
0.281
0.369
0.261
0.375
0,263
0.399
0.279
336
0.401
0.275
0.405
0.282
0.408
0.288
0.387
0.271
0,387
0.273
0.414
0.291
720
0.434
0.292
0.441
0.302
0.447
0.306
0.409
0.286
0.418
0.292
0.449
0.308
avg.
0.399
0.274
0.403
0,282
0.411
0.290
0.380
0.268
0.385
0.271
0.413
0.288
D.2 Full Results of Efficiency Analysis
To verify the lightweight characteristics of TimeEmb, we conduct several efficiency experiments on it. We first compare the maximum meM)ry(MB), training time, and MSE on the ETTml dataset against mainstream baselines. Subsequently, we calculate the extra parameters from TimeEmb based on the compatibility study, which can also prove the extensibility and efficiency of TimeEmb. The detailed results are displayed in Table 7 and Table 8. It indicates that TimeEmb can operate in resource-consee constrained environment and achieve excellent performance. Moreover, TimeEmb can play the role of plug-in in module in other models, improving their performance at low cost.
D.3
Full Results of Ablation Study
D.3.1 Ablation Results of Embedding Frequency Spectrum Analysis
The complete results of our embedding frequency spectrum ablation are displayed in Table 9 and Table 10. The term "k" in Table 9 represents the number of frequency components of X, selected based on their top amplitudes, while the term ">" in Table 10 refers to the ratio of low-passing filtering applied to the embeddings. The results reveal that leveraging the entire frequency spectrum leads to the best performance. Note that the more frequency components are covered, the better performance TimeEmb can achieve, which indicates that each frequency in the band is important.
20

---

Table 7: Efficiency comparison ETTml dataset shows TimeEmb leads in performance and efficiency.
Model
Training Time(s/epoch)
MSF
Max Memory(MB)
CycleNet
1.77
0.447
91.37
Fredformer
4.46
0.453
1512.32
FilterNer
1.63
0.456
79.51
iTransformer
2.44
0.482
275.12
SOFTS
2,00
0.466
183.95
TimeFmb
1.61
0.435
82.36
Table 8: We equip Fredformer and CycleNet with TimeEmb ETTh2. It brings stable perfor- mance(MSH) gains with trivial extra training costs.
Horizon
96
192
336
720
Fredformer 0.293
0.371
0.382
0.415
+TimeEmb
0.387
Impr.
1.4%
3,5%
5.8%
6.7%
former param
32820131
33465731
9894911
13656959
current param
32830860
33476460
9905640
13667688
extra param
10729 (0.03%
-0.1
CycleNet
0.285
0.373
0.421
0.453
+TimeEmb
0.277
0,351
0.399
0.415
Impr.
2.8%
5.9%
5.2%
8.4%
former param
99080
222200
419192
current
109809
159057
232929
429921
extra param
10729 (2.56
-100883%)
0.45
0.44
0.42
0.28
0.42
0.28
0.25
w0.26
0.22
0.25
0.19
0.23
0.16
ETTh1 ETTm2
Weather
0.16
ETTh1
Weather Electricty
Electricty
ET7m2
0.13
0.15
6
12
24
96
0
0.25
0.5
0.75
1
The number of embedding
Loss Weight
(a) Results of different embedding lengths
(b)
Results of different loss weight
Figure
analysis.
D.3.2 Ablation Results of Key Components in TimeEmb
In this section, we create different versions of the model by altering or removing specific components from TimeEmb. The results are available Table 11. Random: The embedding bank is randomly initialized between training and testing. Zero/Mean: The embedding bank is fixed to zeros or the global mean value, respectively. "w/o X," " represents removing the time-invariant embedding X., "w/o H..." indicates removing frequency filter Ha. "w/o RevIN" refers to removing the reversible instance normalization. It is observed that the time-invariant component embedding contributes most in TimeEmb.
21

---

Table 9: Ablation study results. E contains frequency components with top-k amplitude. The best results are in bold.
5
15
30
40
49(full)
Metric
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
96
0.375
0.393
0.375
0.392
0.374
0.392
0.371
0.390
0.366
0.387
192
0.427
0.422
0.425
0.421
0.424
0.420
0.422
0.419
0.417
0.416
ETTT
336
0.465
0.441
0.463
0.440
0.463
0,440
0.459
0.439
0.457
0.436
720
0,486
0.473
0.472
0.467
0.463
0,463
0.460
0,461
0,459
0,460
avg
0.438
0.432
0.434
0.430
0.431
0.429
0.428
0.427
0.425
0.425
96
0.167
01.447
0.166
0.247
0.166
0.247
0.165
0.244
0.163
0.242
192
0.238
0.295
0.236
0.293
0,230
0,288
0.230
0.287
0.226
0.285
BTTm2
336
0.293
0,,30
0.292
0.328
0,288
0.326
0.287
0.324
0.2866
0.324
720
0,393
0.388
0.393
0.387
0,386
0.384
0.385
0.383
0.383
0.381
avg
0.273
0.315
0.272
0.314
0.268
0.311
0.267
0.310
0.265
0.308
96
0.158
0.202
0.155
0.197
0.154
0.195
0.152
0.192
0.150
0.190
192
0.212
0,250
0.206
0.244
0.205
0.242
0.203
0.240
0.200
0.238
Weather
336
0.268
0.291
0.265
0.288
0.264
0.287
0.261
0.284
0.259
0.282
720
0.350
0.344
0.347
0.342
0.346
0.341
0.344
0.340
0.339
0.336
avg
0.247
0.272
0.243
0.268
0,242
0,266
0.240
0.264
0.237
0.262
96
0.157
0,250
0.151
0.247
0.145
0,241
0.140
0.236
0.136
0.231
192
0.171
01.261
0.168
0.261
0.161
0.156
0.250
0.153
0.246
Electricity
336
0.188
0.278
0.185
0.278
0.179
0,273
0.173
0.267
0.170
0.264
720
0.231
0.315
0.229
0.316
0,220
0.307
0.214
0.302
0.208
0.297
avg
0.187
0.276
0.183
0.276
0.176
0.269
0.171
0.264
0.167
0.260
Table 10: Ablation study results. E contains frequeney components filtered with low low-passing filtering by y. The best results are in bold.
0
0.3
0.6
0.9
Metric
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
96
0.376
0.391
0.370
0.390
0.368
0.389
0.367
0.388
0.366
0.387
192
0.431
0.422
0.420
0.418
0.420
0.417
0.418
0.417
0.417
0.416
ETThl
336
0,470
0.441
0.459
0,437
0.457
0.437
0.457
0,437
0.457
0,436
720
0.487
0.471
0.465
0.463
0.461
0.461
0.459
0.461
0.459
0.460
avg
0.441
0.431
0.429
0.427
0.427
0.426
0.425
0.426
0.425
0.425
0.173
0.252
0.164
0.243
0.164
0.243
0.164
0.243
0.163
0.242
192
0.237
0.294
0.227
0,285
0.228
0,286
0.227
0,285
0.226
0.285
ETTm2
336
0.295
0.332
0.286
0.324
0,287
0.324
0,286
0.324
0.286
0.324
720
0.391
0.389
0.382
0.382
0.383
0.382
0.383
0.382
0.383
0.381
avg
0.274
0.317
0.267
0.309
0,267
0,309
0.266
0.309
0.265
0,308
96
0.182
0.221
0.151
0.191
0.151
0,191
0.150
0.190
0.150
0.190
192
0.228
0.259
0.202
0.240
0,201
0.239
0.201
0.238
0.200
0.238
Weather
336
0,282
0.299
0.261
0.284
0.260
0,283
0.259
0,283
0,259
0,282
720
0.356
0.347
0,344
0.340
0,343
0,339
0.343
0.339
0.339
0.336
avg
0.262
01.282
0.240
0.264
0.239
0.263
0.238
0.263
0.233
0.262
0.178
0.259
0.141
0.236
0.139
0.234
0.138
0.233
0.136
0.231
192
0.184
0.266
0.158
0.251
0.156
0.249
0.154
0.248
0,153
0.246
Electricity
336
0.200
0.282
0.175
0.268
0.1
0.266
0.171
0.265
0.170
0.264
720
0.214
0.301
0,211
0.299
0.210
0.298
0.297
avg
0.201
0.281
0.172
0.264
0.170
0,262
0.168
0,261
0.167
0.260
D.4 Hyper-Parameter Analysis
We conduct experiments to evaluate the impact of essential hyper- of TimeEmb in predic- tion performance, including the number of embeddings M and loss weight a. The dyper-parameter are adjusted individually for each setting based on the performance displayed in Figure 6. Full results can be referred to Table 12 and Table 13.
For the number of embeddings of embedding bank E, we set M E {6, 12, 24, 96), and the results are displayed in Figure 6. From the results, it can be observed that, (1) The variation in M settings affects model performance slightly, highlighting the TimelEm''s robustness and its ability to perform well with minimal manual tuning, making it both user-friendly and easy to deploy. (2) Different datasets exhibit distinct characteristics due to their different periodic patterns. For dataset ETTm2, the best performance is achieved when M = 96, as the shorter time intervals require a finer granularity of
22

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Table 11: Ablation study results of key modules. The best results are in bold.
Dataset
ETTh1
ETTm2
Weather
Electricity
Metric
MSB
MAE
MSE
MAE
MSE
MAE
MSE
MAE
96
0.366
0.387
0.163
0.242
0.150
0.190
0.136
0.231
192
0.417
0.416
0.226
0.285
0.200
0.238
0.153
0.246
TimeEmb
336
0.457
0.436
0.286
0.324
0.259
0.282
0.170
0.264
720
0.459
0.460
0.3883
0.381
0.339
0.336
0.208
0.297
avg
0.425
0.425
0.265
0.308
0.237
0.262
0.167
0.260
96
0,407
0.415
0.240
0.318
0.197
0.237
0.242
0.344
0.453
0.443
0.299
0.249
0.279
0.251
0.354
Random
336
0.499
0.471
0.361
0.386
0.314
0.321
0.291
0.393
720
0,550
0.524
0.452
0.434
0.422
0,384
0.370
0.455
avg
0.477
0.338
0.372
0.296
0.305
0.289
0.387
0.405
0.240
0.198
0.342
192
0,452
0,442
0.299
0.351
0,248
0,279
0.248
0.352
Zero
336
0.497
0.470
0.360
0.385
0.314
0.521
0.289
0.391
720
0,547
0,523
0.450
0.433
0.421
0.383
0.367
0.453
0.475
0,462
0.337
0.372
0.295
0,305
0.286
0.385
96
0.397
0,410
0.204
0.289
0.175
0,221
0.241
0.339
192
0.444
0.439
0.266
0.327
0,220
0.261
0.250
0.349
Mean
336
0.488
0.464
0.328
0,364
0.267
0.29%
0.291
0.387
720
0,537
0,517
0.427
0.418
0,346
0,348
0.374
0.453
avg
0.467
0.458
0.306
0.350
0.252
0.282
0.289
0.382
0,376
0.391
0.173
0.252
0,182
0.221
0.178
0.259
192
0.431
0.422
0.237
0.294
0.228
0.259
0.184
0.266
w/o. X,
336
0.470
0.441
0.295
0.332
0.282
0.299
0.200
0.282
0.487
0.391
0.356
0l347
0.316
avg
0.441
0.431
0.274
0.317
0.262
0,282
0.201
0.281
96
0.366
0.391
0.164
0.244
0.151
0.192
0.137
0.233
192
0.417
0.421
0.229
0.287
0,201
0.239
0.155
0.249
w/o. Fl
0.451
0.286
0.324
0.259
0.283
0.172
0.269
720
0.492
0.474
0.385
0.382
0.341
0.337
0.210
0.299
avg
0.432
0,432
0.266
0.309
0.238
0,263
0.169
0.263
96
0.383
0.403
0.193
0.286
0.147
0.191
0.137
0.235
0.450
0.449
0.283
0.352
0.194
0.238
0.154
0.252
w/o. RevIN
336
0.485
0.463
0.427
0.449
0.248
(.290)
0.171
0.271
720
0.517
0.511
0.516
0.482
0.326
0.346
0.213
0.306
avg
0.459
0.457
0.355
0.392
0.229
0.266
0.169
0.266
embedding to capture temporal patterns. For datasets ETThl and Weather, M = 24 yields optimal results, effectively capturing data complexity while preventing overfitting.
For the loss weight, we conduct experiments with a E {0, 0.25, 0.5, 0.75, 1}. Results in Figure show that an optimal value of a improves TimeEmb's 's performance. Combining both time domain and frequency domain losses outperforms using time domain loss alone (a = 0), highlighting that integrating information from both domains enhances TimeEmb's 's ability to capture diverse patterns in the time series data.
We examine the impact of several key hyperparameters on TimeEmb performance: the number of embeddings in the embedding bank, denoted "M", and the loss weight the optimization objective, denoted as "a". The detailed results are presented in Table 12 and Table 13. It demonstrates that appropriate typee-parameters can enhance the performance of TimeEmb.
D.5 Visualization of the Learned Embeddings
We present the time-invariant component embeddings in Figure 7. The "x" in the term "hour x" represents the hour-index of the last timestep of the input series. Figure 7 depicts the distinct embeddings learned from various datasets and channels. For instance, Figure (a) presents the time-invariant components learned in channel 302 of the electricity dataset, where the hour index is 5. In contrast, Figure (b) shows the time-invariant components in channel 15 of the Weather dataset corresponding to the input series, with the hour index being 2. These embeddings, derived from the global sequence, capture the time-invariant components, offering vital supplementary information to the model and enabling a better understanding of the stable patterns within the time series data.
23

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Table 12: Influence of the number of time inverinvariant embedding M. The best results are in bold.
M
6
12
24
96
Metric
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
96
0.372
0.389
0.370
0.388
0.366
0,387
0.371
0.390
192
0.424
0.419
0.422
0.417
0.417
0.416
0.420
0.417
ETThl
336
0.464
0.437
0.461
0.436
0.457
0.436
0.458
0.437
720
0.481
0.465
0.464
0.462
0.459
0.460
0.460
0.461
avg
0.435
0.428
0.429
0.426
0.425
0.425
0.427
0.426
96
0,168
0.248
0.167
0.246
0.164
0.243
0.163
0.242
192
0.232
0.290
0.231
0.289
0.227
0.285
0.226
0.285
ETTm2
336
0.291
0.328
0.290
0.327
0.285
0.323
0.286
0.324
720
0.387
0.385
0.385
0.383
0.383
0.382
0.383
0.381
avg
0.270
0.313
0.268
0.311
0.265
0.308
0.265
0.308
96
0.159
0.199
0.153
0.193
0.150
0,190
0.157
0.198
192
0,205
0.242
0.202
0.239
0.201
0.238
0.209
0.246
Weather
336
0.262
0.284
0.259
0.282
0.259
0.282
0.266
0,288
720
0.342
0.338
0.339
0.336
0.342
0.339
0.349
0.344
avg
0.242
0.266
0.238
0.263
0.238
0.262
0.245
0.269
96
0,140
0.235
0.138
0.234
0.136
0,231
0.136
0.231
192
0.157
0.249
0,155
0.248
0.153
0.246
0.153
0.247
Electricity
336
0.173
0.266
0.172
0.266
0.170
0.264
0.170
0.264
720
0.212
0.299
0.212
0.300
0.208
0.297
0.208
0.296
avg
0.171
0.262
0.169
0.262
0.167
0.260
0.167
0.260
Table 13: Influence of loss weight a.
The best results are in bold.
0
0.25
0.5
0.75
Metric
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
96
0,382
0.402
0.366
0.389
0,366
0.388
0.366
0.387
0.367
0.387
192
0.423
0.424
0.418
0.419
0.418
0.417
0.417
0.416
0.417
0.416
ETThl
336
0.463
0.444
0.463
0.440
0.460
0.438
0.457
0.437
0.457
0,436
720
0.459
0.460
0.464
0.457
0.468
0.462
0.472
0.464
0.474
0.465
avg
0.432
0.433
0.428
0.428
0.426
0.428
0.426
0.429
0.426
0.170
012511
0.166
0.245
0.165
0.244
0.164
0.243
0.164
0.243
192
0.234
0.293
0.230
0.288
0.228
0,286
0.227
0,285
0.227
0,285
ETTm2
336
0.294
0.333
0.285
0.325
0,286
0.324
0,285
0.323
0.287
0.324
720
0.405
0.398
0.386
0,386
0.386
0.384
0,383
0,382
0.383
0.382
avg
0,276
0.319
0.267
0,311
0,266
0,310
0.265
0.308
0.265
0.309
96
0.154
0.197
0.151
0.193
0.151
0,193
0.150
0.190
0.150
0.190
192
0.203
0.243
0.202
0.240
0,201
0.239
0,201
0.238
0.201
0.238
Weather
336
0.263
0.288
0.260
0.285
0,260
0,283
0,259
0.283
0.259
0.282
720
0.344
0,344
0.342
0.340
0.342
0,340
0.342
0.339
0.342
0.339
avg
0,241
0.268
0.239
0.265
0.239
0.264
0.238
0,263
0.238
0.262
96
0.137
0.234
0.136
0.231
0.136
0.231
0.137
0.231
0.137
0.232
192
0.155
0.250
0.153
0.246
0.153
0.246
0.154
0.247
0,154
0.247
Electricity
336
0.172
0.267
0.170
0.264
0.170
0.264
0.171
0.264
0.171
0.264
720
0.211
0.303
0.208
0.297
0.209
0.297
0.210
0.298
0.210
0.298
avg
0.169
0.264
0.167
0.260
0.167
0,260
0.168
0.260
0.168
0.260
/ May
(a) ECL
(b) Weather
(c) Traffic
M
(d) ETTHl
(0) ETTml
( ETTm2
Figure 7: Visualization of the learned time-invariant embeddings X..
24

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D.6 Visualization of Prediction
We present a prediction showcase on the ETTh2, Electricity, and Traffic datasets, shown in Figure 8. The predictions closely align with the ground truth, demonstrating that TimeEmb is capable of capturing the complex temporal dependencies of these datasets.
Prodetion
(a) Electricity
(b) ETTh2
(c) Traflic
Figure 8: Visualization of TimeEmb prediction and corresponding groundtruth.
Limitation
While TimeEmb demonstrates strong performance and interpretability in frequency-domat time series modeling, several limitations remain. First, the current design adopts a fixed-resolution embedding bank, which limits its ability to adaptively capture stable patterns across multiple temporal granularities (e.g., hourly, daily, weekly). more flexible mechanism for multi-scale lime-invariant representation would further enhance its capacity for learning multi-pe multi-periodic structures. Second, the embedding structure is currently discrete, which may restrict its ability to model continuously evolving periodicity. Extending the embedding formulation continuous kernelized representation could enable smoother generalization across unseen temporal slots.
25

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Reviews:
[{"review_id": "ANQHn3HIJh", "reviewer": "Reviewer_9Nmb", "summary": "This paper proposes a lightweight static-dynamic decomposition framework for time series forecasting. To obtain optimal performance in the face of distribution shifts, the proposed method learns to separate long-term patterns and short-term fluctuations by separating the time series into two complementary components. The authors demonstrate superior performance and storage efficiency on seven diverse benchmark datasets and functions as an efficient plug-in to enhance existing methods with minimal computational cost.", "questions": "Q1) The proposed method captures time-invariant components through an embedding bank and utilizes them to separate time-variant components. However, the method fuses the two separated components just before the model’s output layer.  In contrast, other existing methods, such as Fredformer and DLinear, typically transform each separated feature back into the time domain before performing fusion. Is there a logical or theoretical reason why the proposed method does not adopt a similar design?\nQ2) Could the authors clarify how the parameter *M* is determined in each experiment? Is it selected based on prior knowledge, validation performance, or a fixed heuristic?\nQ3) While the paper claims that the proposed method can be easily extended to existing models, Table 3 shows only four base models being expanded, and only a subset of datasets used in Table 1 are included. Do the authors have additional experimental results (as seen in appendices for other experiments) demonstrating broader compatibility across more datasets and more base models?\nQ4) In Figure 4(b), the time-variant components extracted by the proposed method show clear similarities within the same day of the week and are distinguishable across different days. However, the authors describe the time-variant components as reflecting local fluctuations in the time series, such as anomalies caused by unusual weather or traffic incidents. If such components are indeed day-specific and recurring, wouldn’t it be more appropriate for this structure to emerge in the feature space of the time-invariant components instead? How do the authors reconcile this discrepancy between their definition and the observed clustering pattern?", "limitations": "Yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 3, "originality": 2, "strengths_and_weaknesses": "Strengths\nS1) The proposed method achieves competitive accuracy compared to state-of-the-art approaches, despite its simple architecture that does not rely on mechanisms such as attention.\nS2) The method is quantitatively evaluated against several recent baselines using benchmark datasets, demonstrating improved performance in time series forecasting. Additionally, ablation studies are conducted to validate the contribution of each component of the proposed method.\nS3) The source code is publicly available, allowing for detailed inspection and verification of the implementation.\n\n- Weaknesses\nW1) The rationale for fusing the separated time-invariant and time-variant components in the feature space remains unclear. While Section 4.5 emphasizes the importance of high-amplitude or low-frequency components in future prediction, this does not sufficiently justify why the static time-invariant and dynamic time-varying components should be recombined.\nW2) The authors criticize an existing method (CycleNet) for assuming a fixed periodic pattern in the dataset, stating that this assumption may not hold under complex or variable periodicities. However, the proposed method similarly relies on a user-defined value of *M* to impose periodicity, and thus suffers from the same limitation—namely, the inability to adapt to varying or complex periodic patterns.\nW3) There is a discrepancy between the model described in the paper and the implementation provided in the submitted code. In particular, for the computation described in Section 3.2, the code applies different processing to the real and imaginary parts, yet this distinction is not mentioned in the main text.\nW4) The authors claim that their method can be easily extended to enhance existing approaches. However, the paper lacks concrete explanation or examples of how such extensions would be implemented, leaving open the question of whether the method is truly straightforward to apply in practice.", "was_revised": true, "final_justification": "I have no further concerns. I suggest that the authors provide detailed explanations from the rebuttal in the revised paper for clarity. I'm raising my score to 4.", "rebuttal": "Thanks for your constructive feedback!\n## W1: The rationale for fusing the separated time-invariant and time-variant components in the feature space.\nIn our paper, we decompose complex time series into time-varying and time-invariant components, which are then modeled independently to enable each factor to learn a more precise representation. These two factors are recombined to integrate information to derive the final comprehensive forecasting representation. \n\n## W2: Explanation of TimeEmb's periodicity handling and differences from CycleNet\nFirst, due to the patterns of human activities, time series data such as electricity consumption and traffic flow have stable periodic basic patterns.\nSecond, while existing works have extensively focused on modeling such stable patterns [1], our work diverges by explicitly decoupling time-invariant static patterns (capturing long-term stable periodicities) and time-varying dynamic components (reflecting transient fluctuations). This design choice not only enhances interpretability but also enables more targeted modeling.\nThird, compared to CycleNet, we would like to kindly emphasize that it requires a strong expert knowledge through the RCF algorithm, that is, the length of a stable periodicity. By contrast, we simply model daily patterns, and have achieved optimal results in nearly all experimental settings without the limitation on domain expertise.\nFinally, we fully concur with your emphasis on the importance of modeling complex periodicity in time series analysis. We highlight that the core innovation of this paper lies in proposing an explicitly decoupled framework for complex time series, an efficient, interpretable, and compatible modeling scheme. To address more intricate periodic patterns, we plan to explore adaptive multi-period modeling methods in future work. Thank you again for this insightful feedback.\nFor details on the selection of M, please refer to Q2.\n\n## W3: The clarification for the discrepancy between the model in the paper and the implementation in the code.\nWe kindly ask you to specify the inconsistency. In Section 3.2, we transform the original sequence into the frequency domain via rFFT, which is consistent with the implementation provided in the accompanying code.\n\nRegarding the reviewers’ observation on the differential handling of real and imaginary parts, this likely refers to the subtraction of real components described in Section 3.3. To mitigate computational and storage overhead associated with the embedding bank, we leverage the properties of complex conjugation: specifically, we only use the real part of $\\overline{X}$ for decoupling. Critically, the real and imaginary components are reintegrated before frequency-domain filtering, aligning with the procedures detailed in this paper(Line 180).\n\n## W4: The concrete explanation for the extensions of TimeEmb.\nThanks for pointing it out.\nIn our model, sequences are processed in both the frequency domain and the time domain. Specifically, operations in the frequency domain can be interpreted as reshaping the sequence spectrum within the original feature space, while in the time domain, a projection layer is employed to generate the final prediction. Building on this framework, the time-domain projection process in our original model serves as a backbone component. When extending our approach to other models, integration is streamlined: only replace our backbone (i.e., the projection layer) with the alternative models. Notably, this modification yields significant performance improvements with minimal additional parameter overhead. Thanks again, and we will add this introduction in the next version.\n\n## Q1: The reason for performing fusion before transforming back into the time domain.\nIn contrast to existing methods such as DLinear, where each separated feature is projected into the prediction dimension before fusion, our approach prioritizes feature fusion. This design ensures that decoupling and fusion operations are performed within the original feature space, thereby mitigating the risk of information loss that may arise from premature dimensionality changes.\nThank you for this valuable discussion. We have conducted experiments to validate this approach, and the results confirm that the \"project-first-then-fuse\" strategy leads to significant performance degradation and incurs computational overhead.\n||TimeEmb|Proj-Then-Fuse|\n|-|-|-|\n|Dataset|MSE,MAE|MSE,MAE|\n|ETTh1|**0.425,0.425**|0.442,0.433|\n|ETTh2|**0.362,0.390**|0.366,0.393|\n|ETTm1|**0.368,0.384**|0.374,0.388|\n|ETTm2|**0.265,0.308**|0.269,0.312|\n|Weather|**0.237,0.262**|0.245,0.270|\n|ECL|**0.167,0.260**|0.171,0.264|\n|Traffic|**0.454,0.293**|0.476,0.316|\n\n## Q2: The choice of the embedding bank size M. \nThanks for highlighting this. We use the embedding bank to learn global time-invariant dataset information. The choice of M is based on common sense and requires no additional expert knowledge. Setting $M = 24$ captures daily patterns; $M = 7$ or $12$ gives weekly or half-daily patterns if needed. The setup of $M$ is straightforward, robust, and knowledge-free. \n|Dataset|ETTh1|ETTh2|ETTm1|ETTm2|Weather|Electricity|Traffic|\n|-|-|-|-|-|-|-|-|\n|M|24(daily)|24(daily)|24(daily)|24(daily)|24(daily)|24(daily) + 7(weekly)|24(daily) + 7(weekly)|\n\n## Q3: Additional experimental results for the compatibility analysis.\nTo verify compatibility, we select representative baselines within our study. As per your requirement, we conducted further experiments on the ETTh2 dataset. These consistently demonstrate superior performance when models are equipped with our TimeEmb. Furthermore, the additional trainable parameters introduced by integrating TimeEmb are negligible.\n|Horizon|96|192|336|720|\n|-|-|-|-|-|\n|Metric|MSE,MAE|MSE,MAE|MSE,MAE|MSE,MAE|\n|Informer|1.642,1.063|5.115,1.847|4.560,1.749|3.490,1.592|\n|+TimeEmb|**1.292,0.905**|**2.803,1.288**|**2.344,1.204**|**2.381,1.248**|\n|Transformer|2.484,1.312|7.652,2.419|5.177,1.941|3.748,1.669|\n|+TimeEmb|**2.254,1.218**|**5.147,1.801**|**3.504,1.555**|**2.726,1.370**|\n|Fredformer|0.293,0.342|0.371,0.389|0.382,0.409|0.415,0.434|\n|+TimeEmb|**0.289,0.335**|**0.358,0.380**|**0.360,0.394**|**0.387,0.421**|\n|CycleNet|0.285,0.335|0.373,0.391|0.421,0.433|0.453,0.458|\n|+TimeEmb|**0.277,0.327**|**0.351,0.375**|**0.399,0.415**|**0.415,0.436**|\n|DLinear|0.339,0.392|0.457,0.463|0.576,0.530|0.773,0.632|\n|+TimeEmb|**0.336,0.388**|**0.419,0.431**|**0.467,0.466**|**0.721,0.607**|\n|iTransformer|0.304,0.353|0.380,0.399|0.420,0.431|0.422,0.443|\n|+TimeEmb|**0.302,0.348**|**0.380,0.397**|**0.419,0.430**|**0.420,0.441**|\n|**Extra Params**|**10729**|**10729**|**10729**|**10729**|\n\n## Q4: The explanation for the discrepancy between our definition and the observed clustering pattern.\nThe observed clustering pattern of time-variant components(${X_d}$) in Figure 4(b) and their day-of-week specificity can be reconciled with our definition by clarifying the granularity and nature of the disentanglement:\n\nFirst, the time-invariant component(${X_s}$) captured by our embedding bank is designed to model long-term stable patterns across dataset (e.g., general daily rush hours in traffic or hourly electricity consumption baselines). In Figure 4, we explicitly configure the embedding bank with M=7 to capture week-level invariant patterns (i.e., shared characteristics across the same day of the week, such as typical Monday vs. Sunday behaviors). Thus, the core week-level stability is already encoded in ${X_s}$.\n\nSecond, the time-variant component reflects residual fluctuations that deviate from these stable patterns. While $X_d$ exhibits day-of-week clustering, this does not imply it is \"stable\" in the long term. Instead, these clusters represent subtle, context-dependent variations within each day type (e.g., additional electricity consumption on Monday due to factory start-up, and fluctuations in electricity consumption on Sunday due to increased household activities). These variations are not universal enough to be considered part of the invariant $X_s$, as they lack the persistent, dataset-wide consistency required for the time-invariant component.\n\nIn essence, the day-of-week clustering in $X_d$ arises from the fact that $X_s$ has already subtracted the most dominant week-level invariants, leaving behind structured but non-stationary residuals that retain contextual dependencies. This aligns with our definition: $X_d$ captures local dynamics that, while partially structured, are not stable enough to be included in the time-invariant component.\n\n[1] Zeng, A., et al. (2023, June). Are transformers effective for time series forecasting?. AAAI, 2023."}, {"review_id": "WkFvKqw0ZD", "reviewer": "Reviewer_pNtD", "summary": "This paper introduces TimeEmb, a novel and highly effective framework for time series forecasting that addresses temporal non-stationarity by disentangling static (time-invariant) and dynamic (time-varying) components. The core innovation lies in using a learnable global embedding bank for static patterns and an efficient frequency-domain filter for dynamic fluctuations, all primarily operating within the frequency domain. The experimental results are compelling, demonstrating state-of-the-art performance with significantly reduced computational resources, and a remarkable plug-and-play compatibility with existing models. The paper also provides insightful visualizations supporting its disentanglement claims.", "questions": "1. The authors employ a fixed-resolution embedding bank to represent time-invariant components. Beyond the current approach of manually adjusting the number of basis vectors ($M$), have the authors explored or considered mechanisms for adaptively learning or inferring optimal multi-scale granularities during training? Such an approach could provide a more holistic solution to the acknowledged limitations related to static representation flexibility.\n\n2. Figures 3 and 4 provide qualitative visualizations to illustrate the separation between static and dynamic components, suggesting improved disentanglement. To further substantiate the interpretability claims, could the authors elaborate on the specific types of insights or actionable information these disentangled components yield in real-world time series applications? For instance, how might practitioners leverage the static vs. dynamic decomposition in forecasting, anomaly detection, or downstream decision-making tasks?", "limitations": "See **Weaknesses** and **Questions**.", "rating": 5, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths:**\n\n- Proposes a novel disentanglement framework that separates static and dynamic components, addressing limitations in prior time series models.\n- Introduces a learnable global embedding bank for time-invariant patterns, improving over traditional local trend-season decompositions.\n- Demonstrates strong empirical performance across diverse datasets and baselines, with better efficiency (e.g., significantly fewer parameters than iTransformer).\n\n**Weaknesses:**\n\n- The time embedding bank requires manual tuning of $M$ based on domain knowledge. A fully adaptive, data-driven approach to learning multi-scale structures would improve applicability to real-world time series with nested or irregular periodicities.", "was_revised": false, "final_justification": "", "rebuttal": "Thanks for your constructive feedback!\n## W & Q1: A fully adaptive, data-driven approach to learning multi-scale structures.\nThis is an excellent observation that merits further exploration. The embedding bank mechanism proposed in our paper is specifically designed to model regular periodic patterns—such as daily, weekly, or other recurring temporal structures—enabling it to effectively handle tasks characterized by such regular periodicity. Given the constrained search space of regular periodic patterns, manual parameter adjustment (e.g., setting M as discussed earlier) is more efficient in practical applications than fully adaptive learning mechanisms. Moreover, the regular periodic characteristics of most datasets can be pre-analyzed to guide this process. For datasets with irregular periods and mean drift, such as ETTh2 [1], our model still shows excellent performance. However, extreme irregular periodicity remains a valuable long-term research direction, and we plan to expand our model to conduct in-depth analysis of this issue and develop targeted solutions.\n\n## Q2: Specific types of insights for the disentangled components in real-world time series applications.\nIn this paper, we focus on one of the most critical tasks in long-term time series forecasting. Looking ahead, the time-invariant components modeled in our framework hold significant potential for extension to a broader range of related tasks. For instance, they can be leveraged to enhance performance in scenarios such as forecasting with missing values, data imputation, and other temporal data modeling tasks.\nFrom a practitioner’s perspective, consider the application of our framework in Energy Grid Management:\nStatic (time-invariant) component: This component, learned via the embedding bank, captures consistent daily electricity usage patterns—such as recurring morning and evening demand peaks. For practitioners, this insight enables proactive grid resource allocation: for example, activating backup generators or pre-positioning energy reserves ahead of anticipated peak periods to ensure stable supply.\nDynamic (time-varying) component: By contrast, this component identifies short-term, irregular fluctuations, such as sudden demand spikes triggered by extreme weather events (e.g., heatwaves driving increased air conditioning use). Armed with this real-time information, operators can make agile adjustments to supply—such as ramping up output from renewable energy sources or redistributing power across the grid—to mitigate risks of overloads or blackouts.\n\n[1] Shao, Z., Wang, F., Xu, Y., Wei, W., Yu, C., Zhang, Z., ... & Cheng, X. (2024). Exploring progress in multivariate time series forecasting: Comprehensive benchmarking and heterogeneity analysis. IEEE Transactions on Knowledge and Data Engineering."}, {"review_id": "hFYFNvIZc3", "reviewer": "Reviewer_5idw", "summary": "This paper proposes TimeEmb, a lightweight framework for time series forecasting by disentangling static and dynamic components. It employs a learnable embedding bank for time-invariant patterns and a frequency filter for time-varying components, primarily operating in the frequency domain. The paper highlights its efficiency and effectiveness on various real-world datasets and its plug-and-play capability.", "questions": "Q1. The paper uses ``Instance Normalization’’ at the beginning of the framework. Could the authors elaborate on the impact of this choice compared to other normalization techniques or no normalization, especially in the context of non-stationary time series?\n\nQ2. Please response to the comments in Weaknesses.", "limitations": "Yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 3, "originality": 4, "strengths_and_weaknesses": "Strengths:\n\nS1. This paper proposes a well-motivated disentanglement for time series prediction. The static-dynamic decomposition is a straightforward approach to model long-term patterns separate from short-term fluctuations. Beyond existing research which concatenates learnable time-in-day embeddings, the embedding bank of TimeEmb shares the same embedding space with time series and maintains the common pattern shared across the whole dataset. The well-designed disentanglement representation learning approach yields consistently superior performance beyond diverse state-of-the-art baselines.\n\nS2. The proposed TimeEmb is both effective and efficient. TimeEmb's lightweight architecture is a significant advantage, showcasing competitive performance with fewer parameters and lower training times compared to many strong baselines. \n\nS3. The paper is well-written and generally clear. The figures, especially Figure 1 illustrating performance vs. efficiency, are highly informative and well-designed. The methodology section provides a good overview of the framework and its components. Equations are presented clearly, and the high-level algorithm is provided in Appendix B. The experimental setup and implementation code are detailed, aiding reproducibility. \n\n\\\nWeaknesses:\n\nW1. Absence of statistical significance reporting for main results: The paper presents mean MSE/MAE values but lacks error bars or statistical significance tests in Tables 1 and 2. While the authors specify averaging over five runs, showing the variance (e.g., standard deviation) would greatly strengthen the empirical claims.\n\nW2. The ablation study shows that removing $H_{\\omega}$ leads to notable degradation. However, the paper doesn't delve deeper into how the learnable $\\omega$ adapts across different datasets or non-stationary patterns.\n\nW3. There are many related papers on time series forecasting. More through discussion and comparison with these papers are encouraged to be included in the paper.", "was_revised": true, "final_justification": "Based on the rebuttal discussions, I have adjusted the score accordingly.", "rebuttal": "Thanks for your constructive feedback!\n## W1: Absence of statistical significance reporting for main results.\nThe detailed results are as follows.\n|Dataset|ETTh1|ETTh2|ETTm1|ETTm2|Weather|Electricity|Traffic|\n|-|-|-|-|-|-|-|-|\n|Metric|MSE,MAE|MSE,MAE|MSE,MAE|MSE,MAE|MSE,MAE|MSE,MAE|MSE,MAE|\n|96|0.366±0.001,0.387±0.001|0.277±0.001,0.328±0.001|0.304±0.001,0.343±0.001|0.163±0.001,0.242±0.001|0.150±0.001,0.190±0.001|0.136±0.001,0.231±0.001|0.432±0.002,0.279±0.001|\n|192|0.417±0.001,0.416±0.001|0.356±0.001,0.379±0.001|0.354±0.001,0.373±0.001|0.226±0.001,0.285±0.001|0.200±0.001,0.238±0.001|0.153±0.001,0.246±0.001|0.442±0.001,0.289±0.001|\n|336|0.457±0.001,0.436±0.001|0.400±0.002,0.417±0.001|0.379±0.001,0.393±0.001|0.286±0.001,0.324±0.001|0.259±0.001,0.282±0.001|0.170±0.001,0.264±0.001|0.456±0.002,0.295±0.002|\n|720|0.459±0.002,0.460±0.001|0.416±0.001,0.437±0.002|0.435±0.001,0.428±0.001|0.383±0.001,0.381±0.001|0.339±0.001,0.336±0.001|0.208±0.001,0.297±0.001|0.487±0.003,0.311±0.001|\n|Avg|0.425±0.001,0.425±0.001|0.362±0.001,0.390±0.001|0.368±0.001,0.384±0.001|0.265±0.001,0.308±0.001|0.237±0.001,0.262±0.001|0.167±0.001,0.260±0.001|0.454±0.002,0.293±0.001|\n\n## W2: How the learnable $\\omega$ adapts across different datasets or non-stationary patterns?\nThanks for discussing the valuable point.\nFrom a design perspective, $\\omega$ is a complex-valued vector that selectively weights time-varying frequency bands via element-wise multiplication. This design draws on the convolutional theorem in signal processing, equating frequency-domain filtering to time-domain convolution operations—enabling the model to approximate any linear transformation through end-to-end optimization and adapt flexibly to diverse data. \n\n**Cross-dataset adaptation**: \n- For low-frequency dominant data (e.g., power grids with diurnal patterns), $\\omega$ amplifies low-frequency weights to capture stable fluctuations. \n- For high-frequency rich data (e.g., weather with bursts), it prioritizes high-frequency bands to track transient changes. \n\n**Non-stationarity handling**: \n- In stable phases, $\\omega$ weakens high-frequency noise for smoothness. \n- During anomalies, it boosts high-frequency weights to capture abrupt shifts. \n\nThis adaptive weighting mechanism can be interpreted as learning the frequency response function of a linear time-invariant (LTI) system, enabling the model to robustly handle diverse spectral characteristics and dynamic shifts without the need for manual tuning.\n\n## W3: More discussion and comparison with time series papers.\nThanks for your suggestion, we will add two more categories of related discussions—LSTM-based models and LLM-based models to Related Work in\nthe next release.\n\n## Q1: The impact of Instance Normalization in the context of non-stationary time series.\nIn time series forecasting, Instance Normalization (IN) differs from Batch Normalization (BN) and Layer Normalization (LN) in key ways. BN normalizes using batch-level statistics, which can blur temporal variations in non-stationary data (e.g., mixing peak and off-peak periods in a batch) . LN normalizes across features for a single sample, potentially smoothing critical local fluctuations. IN, by contrast, normalizes each instance independently at every timestep, preserving instance-specific temporal patterns. This suits non-stationary time series, as it avoids distorting dynamic shifts, providing a stable foundation for subsequent frequency decomposition and embedding learning in TimeEmb."}, {"review_id": "Y6rd6QbYJf", "reviewer": "Reviewer_jvWT", "summary": "The paper introduces an efficient and novel time series forecasting framework TimeEmb. It disentangles time series into static and dynamic components, processed by a learnable embedding bank and a frequency filter, respectively. Extensive experimental results suggest improved performance and efficiency. TimeEmb can also enhance existing methods with trivial extra trainable parameters.", "questions": "-\tThe paper focuses on non-stationarity due to changing trends and external interventions. How would TimeEmb perform on time series characterized by highly irregular or chaotic patterns where traditional periodic assumptions might not hold, or where infrequent, large-magnitude events dominate?\n\n-\tCould the authors provide a conceptual diagram or pseudocode illustrating the integration mechanism for TimeEmb when it functions as a ‘plug-in’ module for existing backbone models?", "limitations": "Yes", "rating": 5, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 4, "originality": 4, "strengths_and_weaknesses": "Strengths:\n\n-\tTimeEmb pioneers the use of a learnable global embedding bank to capture recurrent and time-invariant patterns across the entire dataset. This approach avoids the rigid assumptions of pre-defined periodicities found in methods like CycleNet.\n\n-\tThe framework innovatively processes the time-varying component through an efficient frequency-domain filtering mechanism, which is both effective and efficient.\n\n-\tThe model consistently achieves state-of-the-art or highly competitive performance across seven real-world benchmark datasets and various prediction horizons. \n\n-\tTimeEmb exhibits remarkable compatibility, consistently improving the performance of various existing MLP-based and Transformer-based forecasting models when integrated as a plug-in.\n\n\nWeaknesses:\n\n-\tWhile the embedding bank is claimed to capture \"global recurrent features\", the description also notes it \"provides specific embedding for individual timesteps\"  and \"time-invariant patterns may differ at different timesteps\". This creates a conceptual ambiguity about the precise nature of the \"time-invariant\" component being learned, blurring the distinction between a truly global pattern and a collection of localized periodic patterns.\n\n-\tThe paper does not clearly specify how the optimal value of loss weight $\\alpha$ is determined for each dataset or scenario.\n\n-\tIt would enhance the empirical verification to analyze the performance under a distribution-shifting setting.", "was_revised": true, "final_justification": "The author's reply addressed all my concerns and I maintained the positive score.", "rebuttal": "Thanks for your constructive feedback!\n## W1: The clarification for the conceptual ambiguity about the \"time-invariant\" component.\nTo avoid ambiguity, we define the time-invariant component:\nThe \"time-invariant\" component in TimeEmb refers to periodically recurrent, position-specific(decided by the time index) stability across the dataset, not absolute invariance across all timesteps. The embedding bank achieves \"global recurrent features\" by learning these stable patterns for each cycle position, ensuring they generalize across all instances of that position in the time series.\n\n## W2: How the optimal value of loss weight $\\alpha$ is determined for each scenario?\nWe empirically select the loss weight $\\alpha$ from the candidate set {0, 0.25, 0.5, 0.75, 1} and report the optimal results obtained. The detailed values of $\\alpha$ are provided in the accompanying code, and a table listing these $\\alpha$ values will be included in the main text in subsequent revisions.\n\n## W3: Analysis of the performance under a distribution-shifting setting.\nAs noted in related studies [1], certain datasets—ETTh2, for instance—exhibit marked irregular periodicity and distribution shifts. Our model demonstrates strong performance on such datasets, achieving substantial improvements over existing methods specifically designed to address distribution drift.\n\n## Q1: How would TimeEmb perform on time series characterized by highly irregular or chaotic patterns?\nAs noted in the response to W3, TimeEmb already demonstrates strong performance on datasets with irregular patterns such as ETTh2. Theoretically, even in scenarios of extreme irregularity, TimeEmb can still capture sufficient information: Static components, through learnable embedding banks, capture the underlying long-term stable patterns in data, thereby serving as a reference anchor for chaotic sequences. Even in the absence of traditional periodicity, latent regularities may still be uncovered. In contrast, dynamic components are specifically designed to capture irregular fluctuations and transient changes caused by rare events via frequency-domain filtering, enabling adaptation to abrupt disturbances from non-periodic events.\nExtreme irregular periodicity remains a valuable long-term research direction. In future work, we plan to expand our model to conduct in-depth analysis of this issue and develop targeted solutions.\n\n## Q2: A conceptual diagram or pseudocode illustrating the integration mechanism for TimeEmb.\nIn our model, sequences are processed in both the frequency domain and the time domain. Specifically, operations in the frequency domain can be interpreted as reshaping the sequence spectrum within the original feature space, while in the time domain, a projection layer is employed to generate the final prediction. Building on this framework, the time-domain projection process in our original model serves as a backbone component. When extending our approach to other models, integration is streamlined: only replace our backbone (i.e., the projection layer) with the alternative models. Notably, this modification yields significant performance improvements with minimal additional parameter overhead.\n\nThe pseudocode is as follows:\n\n(Choosable) Instance Normalization($X$)\n\n$\\overline{X}$ = $rFFT(X)$\n\n$X_d$ = $\\overline{X}$ − $X_s$\n\n$\\dot{X}$ = $H_ω$ ($X_d$) + $X_s$\n\n$Z$ = $irFFT(\\dot{X})$\n\n$Y$ = $Backbone(Z)$\n\n(Choosable) Reverse Instance Normalization($Y$)\n\nwhere Backbone is referred to the alternative model.\n\nPS: Whether or not to perform Instance Normalization depends on the alternative model.\n\n[1] Shao, Z., Wang, F., Xu, Y., Wei, W., Yu, C., Zhang, Z., ... & Cheng, X. (2024). Exploring progress in multivariate time series forecasting: Comprehensive benchmarking and heterogeneity analysis. IEEE Transactions on Knowledge and Data Engineering."}]

Author rebuttal:

Meta-review: (a) Summarize the scientific claims and findings of the paper based on your own reading and characterizations from the reviewers
This paper proposes TimeEmb, a lightweight framework for time series forecasting that explicitly disentangles static (time-invariant) and dynamic (time-varying) components. The static part is captured via a learnable global embedding bank, while the dynamic part is modeled through an efficient frequency-domain filter inspired by full-spectrum analysis. The model is lightweight, easily integrable into existing forecasting methods, and empirically outperforms strong baselines on multiple benchmark datasets while requiring fewer parameters and computational resources. Extensive ablation studies and qualitative visualizations are provided to validate the disentanglement and interpretability.

(b) Strengths

Novelty & Motivation: Introduces a clear static-dynamic decomposition framework for time series, improving interpretability and robustness under distribution shifts.

Efficiency: Achieves strong results with significantly fewer parameters compared to transformer-based baselines.

Empirical validation: Demonstrates consistent improvements across diverse datasets, with comprehensive ablations and qualitative analyses.

Reproducibility: Publicly available code supports transparency.

Practicality: Plug-and-play capability shows its utility for enhancing existing models with minimal modifications.

(c) Weaknesses / missing elements

Fusion rationale: Some reviewers questioned the theoretical justification for fusing invariant and variant components directly in the feature space rather than following more standard designs.

Periodicity assumption: While positioned as an advance over CycleNet, the method still requires user-specified embedding granularity (M), which may limit adaptability to irregular or nested periodicities.

Paper–code mismatch: One reviewer flagged an inconsistency regarding handling of real vs. imaginary parts in the frequency domain, though authors clarified during rebuttal.

Extension clarity: The claim that TimeEmb can be broadly applied as a plug-in lacked concrete, step-by-step examples in the initial draft (later clarified in rebuttal).
Overall, these weaknesses are incremental rather than fundamental; they do not undermine the technical soundness.

(d) Reasons for final decision
The reviewers converged toward a positive view after rebuttal. The paper is technically solid, empirically validated, and well-motivated. While not a conceptual breakthrough at the level of an oral/spotlight, the method is practical, interpretable, and efficient, which makes it a valuable contribution to the NeurIPS community. The main reasons for acceptance as a poster are:

Clear methodological novelty in disentangling static/dynamic factors.

Strong empirical gains with lightweight design.

Solid rebuttal that resolved initial concerns.
The decision not to elevate to oral/spotlight is because the contributions, while solid and useful, are evolutionary rather than revolutionary: the framework does not introduce fundamentally new theory or modeling paradigms, and its novelty mainly lies in a careful design choice.

(e) Reviewer discussion and rebuttal

R9Nmb raised concerns about fusion design, periodicity assumptions, model–code consistency, and unclear plug-in extensions. The authors responded with clarifications (e.g., fusion before projection reduces information loss, M selection is simple and knowledge-free, code handling is consistent, and integration with baselines is straightforward). Additional experiments were provided to validate compatibility. After rebuttal, R9Nmb acknowledged concerns were addressed and raised the score.

pNtD found the paper strong but noted manual tuning of embedding size and suggested adaptive mechanisms. The rebuttal clarified that manual settings are pragmatic for common datasets, with adaptivity as future work. Authors also provided concrete use cases for disentangled insights (e.g., energy grid management). The reviewer maintained a positive score.

5idw emphasized the well-motivated disentanglement design, efficiency, and superior performance across baselines. Concerns were relatively minor, and the reviewer was consistently positive.

Overall, the rebuttal period was constructive: authors engaged thoroughly, reviewers acknowledged clarifications, and the final consensus shifted toward acceptance.
Decision: Accept (poster)