PhaseFormer: From Patches to Phases for Efficient and Effective Time Series Forecasting
Authors: Yiming Niu
Jinliang Deng
Yongxin Tong
Venue: ICLR 2026 Poster (2026)
Keywords: time series forecasting
nonstationary
efficiency
Primary area: learning on time series and dynamical systems
TL;DR: PhaseFormer replaces patch-based inefficiency with a phase-driven approach, achieving efficient and robust time series forecasting.

Abstract: Periodicity is a fundamental characteristic of time series data and has long played a central role in forecasting. Recent deep learning methods strengthen the exploitation of periodicity by treating patches as basic tokens, thereby improving predictive effectiveness. However, their efficiency remains a bottleneck due to large parameter counts and heavy computational costs. This paper provides, for the first time, a clear explanation of why patch-level processing is inherently inefficient, supported by strong evidence from real-world data. To address these limitations, we introduce a phase perspective for modeling periodicity and present an efficient yet effective solution, PhaseFormer. 
PhaseFormer features phase-wise prediction through compact phase embeddings and efficient cross-phase interaction enabled by a lightweight routing mechanism. Extensive experiments demonstrate that PhaseFormer achieves state-of-the-art performance on the evaluated benchmarks with around 1k parameters, consistently across benchmark datasets. Notably, it excels on large-scale and complex datasets, where models with comparable efficiency often struggle. This work marks a significant step toward truly efficient and effective time series forecasting.
Code is available at this repository: https://github.com/neumyor/PhaseFormer_TSL

Published as a conference paper at ICLR 2026
PHASEFORMER: FROM PATCHES TO PHASES FOR EF- FICIENT AND EFFECTIVE TIME SERIES FORECASTING
Yiming Jinliang Deng!a., Yongxin Tongh
State Key Laboratory of Complex & Critical Software Environment, Beihang University
(yimingniu, yxt (oooonnouaaa @ouaa. edu. cn
2 Research Institute of Trustworthy Autonomous Systems, SUSTech com
j jin1 wlllannddnggg5
ABSTRACT
Periodicity is fundamental characteristic of time series data and has long played a central role in forecasting. Recent deep learning methods strengthen the ex- ploitation of periodicity by treating patches as basic tokens, thereby improving predictive effectiivnness. However, their efficiency remains a bottleneck due to large parameter counts and heavy computational costs. This paper provides, for the first time, a clear explanation of why patch-level processing is inherently in- efficient, supported by strong evidence from wal-world data. To address these limitations, we introduce phase perspective for modeling periodicity and present an efficient yet effective solution, PhaseFormer. PhaseFormer features phase-wise prediction through compact phase embeddings and efficient cross-phase interac- tion enabled by lightweight routing mechanism. Extensive experiments demon- strate that PhaseFormer achieves state-of -fate-of-the-art performance on the evaluated benchmarks with around Ik parameters, consistently across benchmark datasets. Notably, it excels on large-scale and complex datasets, where models with com- parable efficiency often struggle. This work marks a significant step toward truly efficient and effective time series forecasting. Code is available this repository: TSL
Intpps:/
com www htw.tt/....co///eeem////////eere
INTRODUCTION
Time series forecasting underpins decision-making across diverse domains such as finance, en- ergy, climate science, and healthcare, playing a pivotal role in tasks including weather forecast-
ing (Qureshi et al. 2025 Wu et al. 2021a), energy consumption planning ([ai et al.] 2018 Alvarez et al. 2010 Cheng et al. 2021). traffic scheduling (Cirstea et al. 2022 2021 Wu et al. 202Tb). In recent years, deep learning has demonstrated promising potential in this field by leveraging end-to-
end modeling and powerful representational capacity to extrapolate from history to future trends.
A central inductive bias in forecasting models is periodicity-the recurring temporal structure in- herent in many real-world time series. Periodicity are ubiquitous in practice, appearing do- mains ranging from urban traffic flow to cloud resource utilization and other workloads, making periodicity-award modeling broadly applicable (Deng et al.] 2024a Xia et al. 2025). Recent ad- vances exploited this property by segmenting sequences into patch tokens, potentially aligned with cycles, prior to processing by the crafted models (Nie et al. 2023 Zhang & Yan 2023 Huang et al.) tokenized time series to capture temporal correlations within and between cycles, while Zhang & Yan (2023)
2025| Tang & Zhang (025b). For instance, NNe et al. (2023) applied Transformer
extended this paradigm by modeling cross-dine dimeenionn dependencies and cross-scale interactions.
Despite their effectiveness, patch-based based approaches struggle to scale efficiently to large and com- plex datasets (Nie et al.) 20033 Zhang & Yan 20233; Tang & Zhang (025b). We attribute this poor scalability to the substantial variability of cycle patterns in real-world scenarios. This variability stems from dynamic external factors, which continuously shift the cycle patterns. For instance,
Yiiiigg Niu and Jinliang Deng contributed equally to this work.
1 "orresponniig author.

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Published as a conference paper at ICLR 2026
Phase Token
0.30
DHELOTS
FITS
20PLOPS
mannsforet
TimeMiber
Paach
eSPLors
SparseTsE
0.15
SPOPFLOPS
TineBasa
0.10
PatchTSTA
0.5
PhasoFormer
1130710PS
Pluse-hased Patch-tased
0.00
MAE
07-04
07:05
07:06
07-07
07-08
07:08
07-10
(a) Patch Token vs. Phase Token.
(h) Model Accuracy and Efficiency Comparison.
Figure 1: Comparison between patch-based and phase-based representations for time-series fore- casting. (a) illustrates the difference in tokenization. (b) provides joint evaluation of model accu- racy, parameter scale, and computational overhead on the Traffic dataset, using an input length of 720 steps and predicting 96 future steps, where the marker size denotes the FLOPs.
traffic flow patterns may evolve new infrastructure is introduced, while electricity demand can change with adjustments in work schedules (Shao et al. 2025b] 2026; [Chenyyang Zhou & Hul 2026). This variability forces models to construct a high-dimensional representation space to faith- fully accommodate the broadened distribution, which inevitably inflates both parameter counts and computational costs (Shao et al. 2025a). Additionally, these methods also struggle to generalize under such varying behavior, resulting in unreliable forecasts for samples beyond training data.
To address this challenge, introduce novel phase-based perspective that focuses values aligned at the same offset across successive cycles. From this perspective, the dynamics of a time series are characterized by the cross-period trends of each phase-captured disregarding the full cyclic behavior. As illustrated in Fig. [a| phase tokens exhibit significantly lower variability than patch tokens, enabling more efficient and generalizable representation, which in turn makes PhaseFormer well suited for resource cowurree-oostrrieeed environments. Importantly, exclud- ing cycle patterns has minimal impact on forecasting effectiveness, since the cyclic behaviors remain locally stable and thus require little effort to predict. We study and verify these properties in depth in Sec. uuiing real-world data, showing the stationarity and compactness the feature space offered by phase tokenization.
phase ookens-while
Building on these insights, we propose Phase-based Routing Transformer, abbreviated as Phase- Former, which reframes time series as a collection of phase tokens and casts step-wise prediction as phase-wise prediction. Specifically, PhaseFormer (i) aligns and extracts phase tokens from the in- put sequence and maps them into shared low- www-dimesiionl latent space, (ii) employs lightweight routing mechanism to enable efficient communication across phases, and (iii) applies a shared pre- dictor project the latent representations into forecasts for each phase. Extensive experiments demonstrate that, compared with PatchTST (Nie et al. (2023) and Crossformer (Zhang & Yan 2023), PhaseFormer achieves delivering consistent improvements in prediction accuracy across all seven benchmark datasets, as illustrated by the Traffic dataset in Fig. Tb Moreover, in contrast to methods with comparable ef- ficiency such as Spa (Lin et al. 2024b) and TimeBase (Huang et al., 2025), PhaseFormer significantly enhances predictive effectiveness, particularly on large and complex datasets. Finally, we conduct a comprehensive analysis of different configurations to reveal the necessity of the con- structed components and the effects of various hyperparameters. Our contributions are as follows:
99.9 % reduction in both parameter count and computational cost, while
1. We introduce phase-based perspective that aligns values across cycles for the characterization of long-term time series, empirically and theoretically demonstrating improved feature station- arity and compactness over the patch-based perspective.
2. We propose PhaseFormer, a lightweight forecasting model that reframes time series as phase tokens, maps them into a shared latent space, and employs low-rank routing mechanism specifically designed around phase tokens to enable efficient phase-wise forecasting.
3. Extensive experiments conducted to showcase that PhaseFormer achieves substantial ef- ficiency gains while consistently improving forecasting accuracy, establishing a superior effi- ciency-effecttiveess trade-off across diverse benchmarks.
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ETTHI
Electricity
Traffic
Explained ratio Cumulative
1.0
30%
Patch Tokens
Top-2 explain > 30% higher than Patch
0.4
Phase
Tokens
0.2
Phase
Patch
0.0
MMD
-85.4%
-86.8% (1.44 = 0.19)
-94.7% (1.14 = 0.06)
(1.51 0.23)
Princippl component
(a) Temporal shift comparison across multiple datasets.
(b) PCA for phase tokens on Traffic.
Figure 2: Visualization of phase tokenization and its advantages.
(a) Phase tokenization yields
more stable representations than patch-based embeddings. dimensionality compared with patch tokens.
(b) Phase tokens exhibit clear low-
RELATED WORKS
Transformer-Base Forecasting Architectures. Early Transformer-based models for long se- quence forecasting often overlooked the periodicity in time series (Zhou et al. [021] Li et al.) 2019). Subsequent research introduced domain-specific priors that better understand recurring tem- poral structures. Autoformer (Wu et al. 2021a) and FEDformer (Zhou et al., 2022) incorporated decomposition strategies and frequency- sonal-trend patterns. Pyraformer (Liu et al., 202I) and Crossformer (Zhang & Yan. 2023) further enriched temporal modeling by embedding multi-scale hierarchies and cross- varialle dependencies, while Liu et al. (2022) explicitly accounted for distributional shifts. More recently, PatchTST (Nie) eet al. 2023) reframed time series as patch sequences to enable more accurate characterization of sequence-level semantics, followed by an extension to jointly consider spatial and temporal correla- tions (Huo et al., 2025). Generally speaking, these models embed progressively stronger temporal biases, though often at the cost of massive parameter counts and heavy computation.
modeling enabling explicit representation of sea-
Efficiency-Orienter Forecasting Models. A growing body of research emphasizes efficiency, aim- ing design lightweight forecasting architectures. Patch-based MLP variants such as xPatch (Stit- syuk & Choi 2025). TimeMixer (Wang et al.] 2024). and PITS (Lee et al. 2024) exploited compact tokenization or hierarchical dependencies to reduce parameter counts while maintaining accuracy. Beyond patches-baat based methods, frequency- based counterparts leverage spectral representations for compression and denoising. FreTS (Yi et al. 2023) applied MIPs in the frequency domain, Yi et al. (224)) learned frequency filters to improve noise robustness, and FITS (Xu et al.) (024) achieved strong accuracy with only 10k parameters. Deng et al. (2024b) demonstrated that selective decom- position can deliver both parsimony and capability. More recently, SparseTSF (Lin et al. 2024b) and TimeBase (Huang et al., 2025) highlighted the importance of cross-period correlation, sharing similar motivation with ours. Despite their impressive computational efficiency, these methods still fall short on forecasting accuracy for large and complex datasets such as Traffic and Electric- ity (Lin al., 2002bb] [Huang al.) (2025). Moreover, they lack systematic analysis to answer the fundamental question: Why can phase tokens serve as an efficient alternative to putch tokens?
MOTIVATIONS
To motivate our approach, we conduct a comparative analysis of the geometric structures of patch and phase tokens across three widely used datasets. As illustrated in Fig. la patch token is com- posed of adjacent observations within local period, whereas a phase token is constructed by ex- tracting values identical offsets across consecutive periods. We gain the following important insights from the thorough analysis.
globally stationary, while patch tokens are locally stationary. To provide an intuitive overview of their geometric structures, we project both types of tokens into ioo-dimensional spaces using t-SNE (van der Maaten & Hinton 2008). As shown in Fig. 2a the
Insight 1: Phase tokens
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distributions of patch tokens drift continuously over time but exhibit local coherence, indicating local stationarity and supporting the minimal impact of excluding cycle patterns from intensive process- ing. In contrast, phase tokens form compact and coherent clusters that remain stable over the long term, reflecting strong global stationarity. To rigorously quantify the long-term drift, we compute the average diserepancy distance between each subsequent week and the initial week. Specifically, we adopt the Maximum Mean Discrepancy (MMD) metric (Ouyang & Key] 2021), statistical measure of distributional divergence:
(M, Q) = Em.a'~pk((z = + By.s/=//ky v')l - 2 Paa-Pa- v)],
(I)
where P' and Q denote tokens collected from two different weeks, respectively, and k(:)) is the RBF kernel function. As two distributions become closer, their MMD value approaches zero. The results the bottom of Fig. Palshow that the average MMD distance of the phase token space is significantly smaller than that of the patch token space. Taken together, both qualitative and quantitative analyses demonstrate that phase tokenization exhibits substantially lower temporal distribution divergence, thereby facilitating better generalization across the time axis.
Insight 2: Phase tokens reside in lower-dimensional subspace than patch tokens. To measure the effective dimensionality of the token space, we perform principal component analysis (PCA) on it. Surprisingly, as illustrated Fig. 2b] two dimensions are already sufficient to explain over 90% of the variance of phase tokens, whereas patch tokens require more than eleven dimensions to achieve the same degree of explanation, owing to their drifting behavior observed in Fig. 2a Consequently, phase information resides in low-dimensional subspace, providing a principled basis for parameter- and ceappuutttio-effcccien modeling.
We further establish, using perturbation theory, that phase tokenization remains stable under per- turbations of cycle patterns, whereas patch tokenization exhibits structural drift. To formalize this result, we represent univariate periodic sequence two-dimensional matrix via delay embedding and analyze the structural properties of this embedded matrix. Due to space limitations, only the core theorem is presented here, while the detailed proof is provided in Sec. A.7
Theorem (Phase Tokenization Stability) Let X = AGT + N E RDxH with rank(A) = Gan = < min(D, D, H), and consider the transformed data
x + R,
where || S ||
||]| S <[ [|N])r,, and let 8uni > denote the minimal
spectral separation. Then there exists universal constant C > O such that:
1. For phase tokenization and corresponding subspace U,, there exists:
((4,())) S CC || + || + [|]]
(3)
8min
with exact invariance in the noiseless case (N R =
2. For patch tokenization and corresponding subspace Vr, there exists:
d(v,(x) > d(Col(G), Col(G) - || +
IN 2 + |
(4)
8min
Takeaways. Phase tokenization is structurally invariant under the eycle pattern change S and only subject to perturbations from noise and small day-to-day mismatches. In contrast, patch tokenization generally suffers from a non-vanishing structural offset. Hence, phase tokenization is more robust and consistent under cycle pattern driftss.
4 METHODOLOGG
Given the focus on periodicity, we adopt the Shannel-independent paradigm (Nic et al. 2023 Zeng et al. 2023) and omit the channel dimension throughout the remainder of this paper. The objective of forecasting is to predict the future trajectory Y E Tou from an input sequence X E tm where Li and Lour denote the input and output lengths, respectively. In the following sections, we describe the data preprocessing procedure, present the proposed network architecture, and finally analyze the computational complexity of the method.
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Phase Tokenization
1yur
y+
phasc
Embedding
Phase-lo-Router Aggregation
Router-to- Phase Distribution
Plase Tukens
Ronters &
Cortestualized
Planse Tokets Z
Cross-Phar Routing Layer
W
W
W
W
W
W
PhaseFormer
Aggregation
K,
X
Phase De Tokenkzation
Distribution
Multi-llend Attention
Multi-lieal Atlentim
pluse
Predictor
Contextuslized Routens H
Enhunced Phase Tokens Zgres
Figure 3: The overview of PhaseFormer.
4.1 DATA PRE-PROCESSING
Normalization and De-Normalization. Following Kim et al. (2021), we normalize inputs with their estimated mean and standard deviation, and de-normalize predictions the original scale.
Phase Tokenization and De- Tokenization. Phase tokenization transforms the one-dimensional in- sequence into two-dimensional phase-periol matrix for the following processing. Conversely, phase de-tokenization reconstructs the predicted phase-period matrix back into a one-dimensional output sequence. Let Lppuse denotes the period length, which can be estimated using frequency analysis as we further illustrated in Sec. [533 and remain fixed during the process.
To ensure that the input sequence length is multiple of Lphase. circularly pad the sequence to Ln Lyhen
length Pin Lybase, where Pin
As illustrated in Fig. 3 the padded sequence X is then
reshaped into phase-period matrix X phase E RR X.P., where each entry X [f corresponds the observation at the th phase the pth period. In the de-tokenization process, the predicted phase-period matrix is mapped back to the temporal domain by reversing the transformation, thereby reconstructing the final miedimensional forecast sequence.
4.2 PHASE-BASED ROUTING TRANSFORMER
The phase-period matrix is fed into our proposed phase-based routing Transformer, termed Phase- Former, to capture and extrapolate temporal dynamics at the phase level in an efficient and effective way. As illustrated in Fig. 3 PhaseFormer first applies an embedding layer to the phase tokens, then refines them through multiple cross-phase routing layers, and finally maps them to the target via shared predictor. Next, we elaborate on the design of these modules detail.
4.2.1 EMBEDDING LAYER
The embedding layer projects the phase tokens Xphase into low-dimensional representation space, allowing the informative components to be extracted from raw observations that are often contam- inated by perturbations. Formally, for each phase index É [l..... Lphase): the corresponding phase token X phusee (.. is mapped into d-dimensional representation through linear function fo. parameterized by A E TR... Paxd;
Z = fe fe (Xphase)) E
IR. Liyres xd
(5)
To better capture the temporal ordering among phases, we introduce set of Iearnable positional embeddings Ep R mm xd distinguish the relative position of each phase, following Liu al. (2023a). These embeddings are added to Z in phase-wise manner, so that each phase representation enriched with its positional information:
Ž Epos
(6)
The resulting is then forwarded to the cross phase routing layers for higher-level feature interac- tion and forecasting.
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Published as a conference paper at ICLR 2026
4.2.2 CROSS-PHASE ROUTING LAYER
Directly modeling full pairwise interactions among phase representations via self-attention is com- putationally expensive. To handle this, we introduce set of learnable routers R E I xx to mediate information exchange across phases. While the design is inspired by prior routing-style design (Jaegle et al. [ & Yan 2023), our contribution lies in leveraging the inherent low- rank structure of phase-aligned lokess. The cross-phase routing substantially reducing the quadratic cost of self-attention while preserving rich cross-phase dependencies.
Cross-phase routing consists of two steps: (i) phase-to-router aggregation, which selectively com- presses information from phase representations into the compact set of routers; and (ii) router-to- phase distribution, which selectively propagates the aggregated cross-phase information from the routers back to the phase representations. Both steps implemented via cross prossaatention allowing the model to scale efficiently while preserving strong representational capacity.
Phase-to to-Router Aggregation. The routers attend to the phase representations to extract contextual information, yielding contextualized router embeddings H R Mxd, Specifically, the routers act queries while the phases provide keys and values. The projection matrices W W W pdxd map the representations into query, key, and value spaces, respectively:
RW
K. = 2w
V. = 2
K
The aggregated router embeddings then obtained via multi-head attention (MHA) with d, heads: H = MH, V.).
Router-to-Phase Distribution. The aggregated information in the routers is subsequently redis- tributed the phase representations, thereby enabling cross-phase information flow. In this step, the phase representations serve queries while the routers provide keys and values, yielding re- fined phase representations Zunn- The projection matrices Wdist Wdist Wdist E Rdxd are used for
Q
this distribution:
Q2 = -Wddist Q K, HW
dist
V HW V K
(9)
Zum (H, K,,, V,).
(10)
This mechanism restores phase-level resolution while simultaneously enforcing coherence across phases through the contextualized routers. Ultimately, each phase representation attends to all others through a two-stage routing pathway.
4.2.3 PREDICTOR
The predictor produces multi-step forecasts length Paut for all phases simultaneously, based on their refined representations. Taking input the refined phase representations Zun E RR Te from the final cross-phase routing layer, the predictor is realized as a linear mapping Id, parameterized by o E Pad:Pa
phuse = (Zuun) E R- x Pon.
(11)
All phases share the same predictor parameters, which enforces consistency across phases and re- duces the number of trainable parameters. This not only improves efficiency but also regularizes learning, thereby enhancing generalization. Finally, the predicted phase-period matrix Y phase is passed through de-tokenization and de-normalization to produce the final forecast Y.
4.3 COMPLEXITY OF PHASEFORMER
For each variable, the overall complexity of PhaseFormer can be summarized follows: the phase embedding layer requires O OL Pid) time and d( memory. The cross- posss- routing layer, which dominates computation, incurs O(Lpphae dZ Lphased) time and O M (Lephuse + MM)) memory. Finally, the predictor costs O (Ipphaee d. doPu time and ( P Pow)) memory. Aggregating across blocks, the end-to-end time complexity is:
ON + Mdd + M Lphaasee) + Lpphase d( Pin + Poux)

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Published as a conference paper at ICLR 2026
Substituting Pmn Lin Lphuse] and Pour
Louu Lphase| into the above expression gives:
o(N ((Lphas + M)d + M Lphnsed) + d(Lin + Looa)):
As investigated in Sec. the phase token space exhibits inherently low-dimensional structure, which allows M and to be chosen as fixed and small numbers. Thus, the computational cost grows in linear manner with both the input length Tin and the output horizon Traa-
4.4 DISCUSSION
Discussion on Periodic Alignment. Although PhaseFormer built around a periodic alignment mechanism, it does not completely fail on weakly periodic partially non-periodic time series. When no strong periodic structure is present, the phase representation naturally degenerates into a coarse-scale subsampling of the sequence. These coarse phase tokens function similarly trend embeddings sampled every k steps, capturing slowly varying components and broad temporal struc- tures. Through the routing mechanism, the model can still aggregate information across these coarse tokens, thereby suppressing high-frequency noise and emphasizing the underlying smooth temporal evolution. This prevents overfitting to short-term stochastic fluctuations and enables the model to retain certain degree of robustness even without clear periodicity.
Discussion Failure Modes. In datasets lacking a clear dominant period or exhibiting only weak periodicity, the automatically estimated phase length may be affected by noise or minor harmonics, resulting in unstable or incorrect phase alignment. Such misalignment constitutes primary failure mode for the model. In multi multi-period settings, if the model aligns to suboptimal harmonic instead of the true dominant component, it can still operate tends deliver inferior predictions, as we later shown in Sec. 5.3
Discussion on Padding. PhaseFormer employs cycle-padding mechanism ensure aligmment along the phase dimension. However, when the chosen phase length is excessively large or small, a substantial amount of padding is introduced. Excessive padding may inject artificial boundary information into the input, creating boundary artifacts that degrade prediction quality. Selecting an appropriate phase length is therefore crucial for minimizing padding and maintaining stable perfor- mance. In practice, using input sequence lengths that are divisible by the phase length is recom- mended whenever possible to avoid unnecessary padding and its associated artifacts.
EXPERIMENTS
5.1 LONG-TERM TIME SERIES FORECASTING
We conduct a joint evaluation of model efficiency and predictive accuracy. The comparative analysis highlights that the proposed PhaseFormer establishes an improved effectiivenes--effiiinnyy tradeoff terms of parameter scale and error metrics. We also provide the code in https://gsthub. com/ ne
TSL
Datasets and Setup. Experiments are performed on seven widely used long-term time series fore- casting datasets: ETTh/, ETTh2, ETTml, E7Tm2] Weather Electricity] B and Truffic Fl
covering a
diverse range of real-world scenarios. The details of the datasets are provided in Sec. Sec. A.2 Following
prior works (Nie et al. 2023 Zhang & Yan 2023 Huang et al. (005). we adopt 6:2:2 split for the ETT datasets and 7:1:2 split for the other datasets. For PhaseFormer, we report the average results over three random seeds, while for the other baselines we follow their official implementations and released code. We evaluate the forecasting accuracy of tested models using mean squared error (MSE) and mean absolute error (MAE), and assess efficiency in terms of floating-pon poitt operations (FLOPs) and the number of parameters (Params).

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Published as a conference paper at ICLR 2026
Table I: Main results for long-term forecasting. The input sequence length Lisput 720, and results are averaged over forecast horizons Lou E (96, 192, 336, 720}. The best results are shown in bold, and the second-bes best in underline.
Dutaee
MSL
Plasekemer
MAE
MSL
ParehTST
NEAL
MSL
Timesfommer
MAA
MSN
Crosdomme
MAN
MSS
FElDeemmrr
MAA
MSG
FimeRuse
MEA
MSL
Sparsc TFF
MSC
FITS
MSE
TimeMFeerr
XAA
ET151
11,033
0.415
134.0
0.439
0.435
0.467
0.117
0.312
0.525
0.533
1.314
0.4.0:
0,303
0.433
0:215
0.435
0.252
0.474
ETTh2
0.346
0.388
0.344
0.390
0.392
0.422
1.468
0.867
0.428
0.469
0.347
0.397
0.345
0.383
0.334
0.382
0.386
0.425
kTTml
0.346
0.374
0.354
0.383
0.370
0.401
0.390
0.417
0.438
0.465
0.256
0.390
0.362
0.383
0.359
0.352
0.383
0.413
ETTe22
0.250
0,313
0.251
0.319
0.278
0.337
0.392
0.426
0.461
0.52
0.250
0.314
0.252
0.316
0.285
0.336
0.314
0.367
Dlctricity
0.160
0.250
0.189
0.255
0.185
0.263
0.180
0.275
0.235
0.348
0.167
0.258
0.168
0.263
0.173
0.270
0.171
0.273
Trattic
0.249
0.266
0.290
0.282
0.400
0.27M
0.250
0.430
0,250
0.248
Weather
0.260
0.764
0.273
0.303
0.
0.385
0.283
0.281
10-
ind
Ponee Facettor
Creet
(a) FLOPs on Traffic
(b) Params on Traffic
(c) FLOPs on Electricity (d) Params on Electricity
Figure 4: Comparison of FLOPs and parameter counts across models on the Traffic and Electricity, using an input length of 720 steps and predicting 96 future steps. basch models are shown in green, thase-based models in blue, and other models in gray.
Baselines and Implementation Details. We evaluate our approach against eight competitive base- lines, following the standard -onteretaa setting where all models are trained and evaluated directly on domain-specifte datasets. The baselines span both state-of-the-art Transformer- -based architectures and recent parameter-efficient forecasting models. We compare our method with Patc)TST(2223) TTransformer(2223b)) Crossformer(022) 2023).
(enfformer(2222))) Time 2025), SparseTSF((0244), FITS(2024). and TimeMixer(20244 Among these, PatchTST, Crossformer, and TimeMixer are patch-based; SparseTSFF is phase-based; TimeBase integrates patch and phase paradigms; FITS and FEDformer are frequency-domain; and iTransformer models the full sequence directly. For all baselines, we adopt the recommended configurations provided in their official im- plementations, keeping hyperparameters aligned with |Huang et al. (2025). The model is optimized using the Adam optimizer with fixed learning rate of 1 10-3 Following the settings from efficiency-oriente works (Huang et al.) 2025 Lin et al., [024b] Xu et al.) (024)), the look-back
length is set to 720 time steps. More implementation details are provided in Sec. A.2
Main Results. We evaluate the predictive accuracy of PhascFormer and the baseline methods on seven datasets. Tab. reports the average prediction errors across four forecasting horizons, with detailed results provided in Sec. IA.3. Overall, PhaseFormer consistently achieves superior perfor- mance on nearly all datasets, with particularly notable gains on complex and dynamic benchmarks such as Weather, Electricity, and Traffic. For example, on the largest dataset, Traffic, PhaseFormer surpasses the second-best method, PatchTST, by 6.3% and outperforms TimeBase by 10.4%, under- scoring its robustness on large-scos scale and heterogeneous data. The only exception is ETTh2, where PhaseFormer ranks second to FITS while still maintaining highly competitive accuracy. closer examination reveals that patch-based baselines, including PStchTSST, Crossformer, and TimeMixer, exhibit performance degradation on the Electricity, likely due to stronger distributional shifts.
Efficiency Comparison. We evaluate the computational overhead of all models, with detailed re- sults in Sec. A.3.2 Fig. shows the FLOPs and the number of parameters of all tested models on the Electricity and Traffic. Overall, phase-based models incur lower overhead than patch-based ones. On the Traffic dataset, PhaseFormer achieves an extraordinary FLOPs reduction of about 99.99% over PStchTST and Crossformer. Beyond patch-based baselines, it also outperforms other phase- based models like SparseTSFF, consistently delivering high efficiency. This stems from the lower variety of phase tokens over time (Sec. (3). making them inherently more efficient to process. Taken together with the previous accuracy evaluations, these results clearly demonstrate that PhaseFormer provides an efficient yet effective solution, delivering superior performance on complex datasets.
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4.34
-0h 49
+0x 142
3318
1344
0259
3356
1211
020
1..
020
3373
9302
330
2200
0150
010
3213
number
(a) Electricity
(b) Traffic
(c) Weather
Figure 5: Effect of varying the number of routers M on forecasting performance on three datasets.
Table 2: Cross-Phase Routing layer ablation. Each cell reports MSE, MAE, and FLOPs. Lower is better for all metrics. FLOPs are reported in millions (MFLOPs). The best results are highlighted with Bold, and the second-best results with Underlined.
PhaseFormer
FullAttention
LinearMixing
Routing
Dataset
MSE
MAE
FLOPS
MSE
MAE
FLOPs
MSE
MAE
FLOPs
MSE
MAE
FLOPs
Weather
0.1503
0.1971
3.119
0.1527
0,2005
3.202
0.1700
0.2226
0.920
0.1907
0.2406
0.783
Electricity
0.1290
0.2209
42.213
0.1295
0.2212
48.951
0.1403
0.2334
14.08
0.1423
0.2365
11.922
Traftic
0,3721
0.2475
113.356
0.3791
0.253
131.452
0,3842
0.2532
37.776
0.3892
0.2584
32.149
5.2 ABLATION STUDIES AND ANALYSIS
Varying the Number of Routers. We systematically evaluate the impact of different number of routers M on model performance, with results summarized in Fig. The experiments indicate that across three datasets, the model's prediction error generally decreases the number of routers M increases, before eventually stabilizing or slightly rising. is worth noting that the best performance usually achieved when M {4, 8), which is much smaller compared the actual number of phase tokens, Lphase = 24. This observation indicates that the phase token spans an inherently low-dimensional space, so only a small number of routers sufficient to effectively capture and represent its underlying structure. More detailed results are provided in Sec. A.3.4
Effectiveness Pross-Phase Routing. To assess the contribution of the cross-phass routing layer, we compare four variants of the model: PhaseFormer, which adopts the original cross cross-phase rout- ing layer; w/ FullAttention, which substitutes the cross- routing layer with full attention mechanism; and w/ LinearMixing. which replaces the cross cross-phase routing layer with linear layer; and w/o Routing, which directly projects each phase into its own future. All other experimental settings are kept identical across these variants.
As summarized in Tab. 2 PhaseFormer consistently outperforms w/ LinearMixing and w/o Rout- ing, indicating that explicit cross-phase routing is crucial for modeling periodic dynamics. More- over, PhaseFormer not only incurs less computational and memory overhead, but also achieves lower prediction error than w/ FullAttention, showing that the routing layer is both efficient and effective. We attribute these gains to operating in a diw-dinnsiioal phase token space, which concentrates informative interactions and reduces cost.
5.3 SENSITIVITY ANALYSIS OF PIIASE LENGTII CHOICES
sysname automatically determines the phase
Phase Length
MSE
MAE
length through frequency-domain analysis (see Sec. A.2). extracting salient periodic components directly from the data. To assess its sensitivity to phase length, we conduct experiments on the Traffic dataset. Its frequency spectrum contains several clear periodic components (see Fig. 6). Based on spectral amplitudes, five candidate phase lengths and run PhaseFormer for forecasting. The results are shown in Tab.
24
0.3619
0.2384
12
0.3960
0.2765
8
0.4032
0.2801
28
0.4063
0.2753
21
0.4184
0.2970
select the top
Table 3: Comparison of forecasting perfor-
mance under different phase lengths on the Traf- fie dataset (lookback=720, horizon=96).

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Published as a conference paper at ICLR 2026
300
220
21
28
100
10
20
40
50
phase length
Figure 6: The spectrum calculated on the Traffic dataset. The top-5 phase length is 24, 12, 8, 28, 21.
0.044
Phose Adl
10
0.130
2
Phose
Phose
Phoaee
Phase
0.125
0.041
0
20
800
Router Axia
Router Auis
Step
(a) Aggregation Weights
(b) Distribution Weights
(c) Visualization of three selected phase tokens
Figure 7: Case study on sample from the Traffic dataset. (a) Attention weight matrix during Phase-to-Route aggregation. (b) Attention weight matrix during Router-to-Phase distribution. Both matrices capture the association between routers and 24 input phases. (c) Visualization of three representative phases (1, 5, and 9), each represents a distinct attentive pattern with routers.
The experiments show that the model performs best when the phase length is 24, corresponding to the dominant spectral component. As the phase length deviates from this main period, forecasting errors increase, consistent with the magnitude patterns in the frequency domain. These findings demonstrate that PhaseFormer effectively captures the dataset's primary periodic structure.
5.4
CASE STUDY
We select one sample from Traffic dataset, comprising an input sequence of length 720 and an output sequence of length 96 (816 time steps in total). The input sequence is fed into PhaseFormer, and we record the attention- weight matrices at the first cross-phase routing layer during both aggregation (Phase-Routee and distribution (Router->Phasee patterns exhibit clear local similarity: adjacent phases tend to be assigned to the same routers and to receive attention from similar routers. This indicates that the routing mechanism captures temporally consistent phase relationships. Meanwhile, the attention weights reveal that certain phases share similar attentive patterns. To analyze this further, focus three phases with distinct attentive patterns and visualized them in Fig. [7c These phases display different temporal behaviors: Phase remains relatively stable over long horizons, whereas Phase 9 and Phase both exhibit pronounced 7-day periodicity but with opposite trends. The differing patterns of these phase tokens suggest that the router structure not only distinguishes among phase tokens but also effectively models their periodicity and trend characteristics.
As shown in Fig. 7a and Fig. [bb both attention
CONCLUSION
This work identifies the inefficiencies of patch-based forecasting and presents PhaseFormer, phase- centrie model that captures periodicity via compact phase representations and lightweight cross- phase routing. Both theoretical analysis and empirical validation converge the same conclusion that phase representations remain both more robust and more efficient than patch-based approaches under eycle pattern shifts. Consequently, PhaseFormer maintains high predictive accuracy while re- maining lightweight compared to patch-based methods. More broadly, these results provide a prac- tical pathway for building lightweight yet powerful forecasting models that retain accuracy without heavy and complex architectures.
However, the approach assumes locally stable periodicity across the input and output horizons; under highly irregular or non-repetitive cycles, phase representations may fail to capture meaning- ful dynamies. Future work will relax this assumption by modeling non-stationarity and complex drifts, aiming to develop more resilient phase representations and further establish PhaseFormer as benchmark for long-term time-series forecasting.
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7 ETHICS STATEMENT
This study focuses on methodological advances in time-scries forecasting and does not involve hu- man subjects, personally identifiable data, sensitive private information. All experiments use publicly available benchmark datasets that are widely adopted the research community, and their complies with the terms of release. We do not employ proprietary confidential data, and no conflicts of interest exist. The contributions are purely technical in nature and do not promote harm- ful applications. All authors affirm adherence to fairness, research integrity, and relevant legal and ethical standards, in line with the ICLR Code of Ethics.
REPRODUCIBILITY STATEMENT
We make substantial efforts to ensure reproducibility. All datasets used in our experiments are publicly accessible, with links provided in Sec. 5 Detailed dataset statistics, preprocessing steps, and partitioning procedures appear in Sec. A.2 Model architectures, hyperparameters, and training procedures (including optimizer choice, learning rate, look-back window length, and router config-
uration) are described Sec. and the Sec. Sec. A.2
For fair comparison, we follow the official implementations of all baseline models and provide ref- erences to their sources. Comprehensive experimental results, including ablation studies, efficiency analyses, and visualizations, appear in Sec. and Sec. A.3 Theoretical analyses supporting design choices also appear in the Sec. A.7
Finally, facilitate independent verification, release anonymized source code and experiment scripts in a public repository at /https: higggthuub.. com/ Pow.sr/rr/aaeeeeeeeeeeeem TSL Collectively, these measures ensure that our reported results are reliably reproducible and exten- sible by the research community.
ACKNOWLEDDGENT
The authors would like to thank all the anonymous reviewers for their insightful comments. This work was partially supported by National Science Foundation of China (NSFC) (Grant Nos. 62425202, 62336003), the Beijing Natural Science Foundation (Z220001), the Fundamental Re- search Funds for the Central Universities No. JK2024-03, the Didi Collaborative Research Pro- gram, the State Key Laboratory of Complex & Critical Software Environment (SKLCCSE), the Open Project Program of State Key Laboratory of Virtual Reality Technology and Systems, Bei- hang University (No. VLLAB222444)))...
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A APPENDIX
A.1 DETAILS ABOUT BASELINES
In experiments, incorporated a diverse of time series forecasting models, with particular emphasis on approaches based on Patch Tokenization and efficient forecasting models. The details of these models are follows:
1. PatchTST - A channel-independent Transformer that treats each variable as an individ- ual channel and segments the time series into patches tokens. This design reduces the complexity of the attention mechanism and enables the utilization of longer historical se- quences, thereby improving long-tenn tergttermm forecasting accuracy.
2. iTransformer - A channel-dependent Transformer that models variables themselves as tokens to capture inter-variable relationships, while simultaneously accounting for nonlin- ear temporal variations within each variable.
3. Crossformer - A multi-scale Transformer that performs patching or segmentation along the temporal dimension and employs two-stage attention mechanism (within-time and cross-variable). This design effectively captures both temporal dependencies and inter- variable correlations, making particularly suitable for datasets characterized by strong inter- variable coupling and mixed long- and short-term patterns.
4. FEDformer - A model that integrates trend-seasonal decomposition with frequency- domain analysis. It extracts small number of significant frequency components to hance periodic forecasting performance while maintaining controlled model complexity in long-term forecasting tasks.
5. Sparse - A lightweight model that reduces temporal complexity through periodic down-sampling or subsequence selection, aiming achieve competitive periodic forecast- ing performance with minimal resource consumption.
6. FITS - lightweight model that leverages frequency-domain features and interpolation operations reconstruct the predicted sequences. With fewer parameters and low compu- tational overhead compared with other models, it demonstrates strong performance on time series with distinct spectral structures.
7. TimeBase - A model that constructs temporal bases (via patching or segmentation strate- gies) represent historical and future variations. Its objective is to maintain satisfactory forecasting accuracy while reducing computational and parameter costs.
8. TimeMixer - An match-based forecasting model fully based on MLPs. It employs Past- Decooppsable--ixxnng to decouple seasonal and trend components across different scales (fine and coarse) and utilizes Multpr--utiireiiitoo-- Miiieiiieto--iiinnggggg to aggregate multi-scale pre- dictions. This design achieves balance of efficiency and accuracy in both short-term and long-term forecasting tasks.
9. Cy02eNet(2024a) A lightweight model explicitly modeling stable periodicity to enhance the performance of models in long-term time series forecasting. introduces the Residual Cycle Forecasting (RCF) technique, which utilizes learnable recurrent cycles model the inherent periodic patterns within sequences, and then performs predictions on the residual components of the modeled cycles.
10. Pat2hMMP((2225)) A Patch-based MLP model. It leverages patching mechanism to enhance sequence locality and replaces complex Transformer architectures with a simple yet highly effective MLP framework.
A.2 IMPLEMENTATION DETAILS
We present detailed statistics the datasets in Tab. The data loading and preprocessing proce- dures follow prior works (Nie et al. 2023 Huang et 2025).
All baseline methods are implemented based on their original papers or official code. For cases where fixed random seeds not specified, each experiment is repeated three times to ensure sta- bility. All experiments are conducted using PyToreh (Paszke et al. (019) on a single NVIDIA A100 24GB GPU.
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Dataset
Var
Length
Freq
Scale
ETTh1
7
14,400
720
96~-220
1hour
0.1M
ETTh2
7
14,400
720
96~~720
1hour
0.IM
ETTml
7
57,600
720
96~~720
15mins
0.4M
ETTm2
7
57,600
720
96~~220
15mins
0,4M
Weather
21
52,696
720
96~~720
10mins
I.IM
Electricity
321
26,304
720
96~~720
Ihour
8.1M
Traffic
862
17,544
720
96~-720
1hour
15.0M
Table Dataset statistics used in experiments.
For model configuration, the primary period obtained from frequency-domain analysis by iden- tifying the dominant spectral component and remains fixed throughout training. The number of routers is selected via grid search on validation set from to 8, and the inner dimension Phase- Former is tuned over {8, 16, 32, 64, 128} using the same procedure. selected from (1,2.3) using the same grid-search procedure, while the number of attention heads tuned over (1) For other hyperparameters such as learning rate, batch size, optimizer set- tings, and early-stopping patience, we adopt commonly used configurations in the current research community, consistent with prior work (Huang al., (025). Please refer to the released code for complete training details at
The depth of the model is
com
__SL_
A.3 FULL RESULTS
A.3.1 THE DETAILED FORECASTING ACCURACY RESULTS
We present detailed forecasting results across all prediction horizons on the test sets in Tab, 6 with the input length fixed 720. PhaseFormer consistently delivers strong and stable performance across most datasets and forecasting lengths. Starting from the general observations, PhaseFormer shows clear advantages
wide range of real-world scenarios. In particular, when compared with CycleNet (an advanced model designed for periodic structures), PhaseFormer achieves 28 Top-1 re- sults versus CycleNet's 15. This substantial difference directly highlights the expressive power and potential of phase representations in time series modeling. Nevertheless, there a few exceptions. For example, on the relatively simple and highly periodic ETTh2 dataset, FITS slightly outperforms PhaseFormer. This suggests that for datasets with simpler structures and more predictable trends, some specialized baseline models may still exhibit better accuracy. It is also worth noting that TimeBase, which adopts phase-based strategy, achieves competitive results the relatively sim- ple ETT datasets. In contrast, PhaseFormer demonstrates its advantage primarily on Traffic and Electricity, which are more complex and challenging datasets. This distinction illustrates that while caase-inspired models may be effective in straightforward settings, PhaseFormer generalizes better and excels in more demanding
contexts.
A.3.2 THE DETAILED FORECASTING EFFICIENCY RESULTS
We further provide the efficiency comparison of PhaseFormer against all baselines in terms of FLOPs and number of parameters, with the input length set 720 and the output length fixed at 96. The results in Fig. reveal that PhaseFormer achieves favorable trade-off between accuracy and efficiency. Despite its stronger predictive performance, PhaseFormer maintains moderate model size and computational cost, often comparable to even lower than other transformer-bases models: On complex datasets such as Traffic, PhaseFormer outperforms large baselines like PatchTST with substantially fewer FLOPs; On simpler datasets, even when specialized models such as TimeBase or FITS show competitive accuracy, their efficiency advantage diminishes when considering scalability to larger, real-wor real-world datasets. These findings underscore that PhaseFormer is not only accurate but also efficient, making more suitable for deployment in resource-constraing or latency-sensitive environments.
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Table 5: Parameters and FLOPS across models for different datasets.
Model
F1.0PS
FLOPS
Params
FLOPS
FLOPS
Partis
FLOPS
Parms L155S
PLOPS
FLORS
PraeFemmer
L1158
13.9
306
0.15
1.55K
5.18
L115K
0.11
L.1588
0.11
0.11
L15KK
0.11
PackTST
199888
248.177.4
1:333M
1364477
1.57MM
16.12.03
58763K
51.30
567 6aK
S1.99
187
51.2
S8T68K
$128
Rondomer
6.51M
11 14
515334
357
5151M
3.917:97
K.12
5-6
7.30
V4
Cheafemmet
13.57M
96.56463
22003
21.3634
13,79.70
2.552.24
5.792M
2734.51
273451
5.5983
2.54.05
5.7834
2.75455
Tmmlita::
214
8.44
214
0.31
216
3.14
234
0.00
214
0.07
704
6.25
TM4
SpeatTPP
17.98K
751.31
4.59KK
5338
4.59KK
76.11
4.99K
1.71
4.90K
1:31
4.90K
1311
4.98K
121
FITS
L054K
1.76
372
0.01
462
0.28
272
0.004
271
2,66K
0.04
2.66KK
0.04
Tinshiier
5,07M
2.036.53
5.63M
5554M
139.4
4.04M
4:004
13591
266
PochMLPP
3,957M
7,311922
2.45MM
105
2650M
20011.5MM
2.409M
31,50M
2.40M
3435M
2.40M
34.0M
2.40M
34.6
Table 6: Full results across datasets and prediction lengths. Each entry reports MAE and MSE. The input length is set to 720. The best results are marked with Bold, and the second-best results are marked with Underlined.
Lh
MFA
SEST
MEAI
VEXI
MAN
MAL
MAL
KINI
15
10:00
0.311
11813
1200
12
11-00
11300
1030
12
203
TEATO
ETT1
306
01:34
13.86
0.533
0.533
1:075
11554
720
0.3
01433
0.455
01.25
6365
0.39
0270
0..35
0.535
0.445
8426
D429
0.55
0.25
0 5.000
18477
0.425
0.548
0.551
192
0.541
0,235
8.342
0.385
0.46
1414
0.830
0-25
0.464
0,399
0.585
0.34
0.377
0.303
8.5
0.53
0.455
0,383
11.33
0.381
5/41
176
01903
01905
13330
0.17
0.30
0.94
0.355
8.306
0.066
01300
0.41
1303
03311
0241
131
035
0331
0354
1030
133
190
8.335
01:61
0.358
0.17
6.3
0.774
0.00
01377
01313
0.05
730
0113
0.818
6.030
01171
0.136
0551
0.15
0.13
0407
411
6.65
8.41
0/136
11:00
10
10303
010
110
11.15
136
0.700
0.13
8.56
111
8/15
610
130
0.50
180
0.506
0170
5.50
720
0,55
0.395
0631
0.49
035
0.30
0.500
0,99
0,353
Waster
3.22
1.10
0.337
115
0.21
0111
6am
0.311
050
5370
0263
0766
0.200
0.261
6.01
276
0.06
0:03
S/x
0131
5,5
35
1150
0.21
5:14
0150
11:10
0140
0333
02255
024
11.9
0:31
01.0
0:55
10:41
Ekcaicky
536
01237
6.172
0.267
0.00
0.55
5100
0.36
5226
0.33
0.099
0.35
0.35
01099
0.10
0171
0.26
0,359
0.254
0.196
5250
720
3201
0.38
0.66
036
0.31
0271
0.71
0.354
022
0.00
0214
3.00
0.00
6515
1L,97
0.221
0.500
Tolt:
196
192
8
8.85
0.311
0,34
0382
0131
6398
0.31
01361
8913
8603
037
0,392
B:17
0.31
0412
0399)
0.11
0.77
0,933
0:32
0-454
6,22
0,934
0.233
0.417
04:3
0:17
2300
A.3.3
THE DETAILED RESULTS OF PCA VISUALIZATION
We present PCA visualization results on the ETTh1, ETTh2, ETTml, ETTm2, Electricity, and Weather datasets in Fig. 0 in addition to Fig. 26) The findings are consistent with those observed on the Traffic dataset: phase tokenization yields significantly more compact space compared to patch tokenization.
A.3.4 THE DETAILED RESULTS OF VARYING ROUTER NUMBERS
We further provide detailed results on the effect of varying the number of routers (1,2,4,8,16) across three datasets: Traffic, Electricity, and Weather. The input window was fixed at 720, and the output length was set to 96.
Our observations show that the number of routers does influence model performance, but the optimal configuration typically involves a relatively small number of routers. Specifically, the best perfor- mance was achieved with 8 routers on the Weather dataset, and with 4 routers on both the Electricity and Traffic datasets. Since routers serve as the foundation for aggregation and distribution in the phase token space, these results provide supporting evidence that the phase token space captures low-dimensional features, allowing strong performance even with fewer routers.
In practice, we recommend selecting the router number through a light-weight grid search, as the optimal value typically small and stable across datasets. Consistent with our empirical findings, we observe that using routers within the range of 2-8 is generally sufficient to achieve strong and robust performance. This aligns with our analysis in Fig. [ and Fig. 2b where the phase representations exhibit a clear low-rank structure-fewe than eight principal components already explain over 90% of the total variance. Consequently, increasing the number of routers beyond this range yields only marginal improvements, and we do not observe any consistent trade-off patterns across datasets. Overall, these results suggest that the model does not require a large number of routers, and a modest choice of M can already provide reliable and efficient configuration in most scenarios.
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Table Impact of router number R on prediction accuracy. Each entry reports MSE and MAE. The input length is set to 720. The best results are marked with Bold, and the second-bes, results are marked with Underlined.
R=
R=2
R=4
R=
R=16
Dataset
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
Weather
0.162
0.210
0.153
0.199
0.151
0.199
0.148
0.195
0.149
0.198
Traffic
0.372
0.249
0.367
0.243
0.361
0.238
0.364
0.242
0.368
0.243
Electricity
0.133
0.228
0.132
0.226
0.129
0.221
0.130
0.223
0.130
0.222
Exlaned eto Cumulativ
Expleied Cww./uulliiiee
Dglened ratio Cumullttive
20%
100
0.6
0.6
0.6
0.4
0.4
0.4
0.2
Huse Patch
Phove
Phate Parch:
Parch
11
11
Phinppal
Prncipall
Phnttppll component
(a) PCA for tokens ETThl.
(b) PCA for tokens on E7Th2.
(c) PCA for tokens ETTml.
Explanedd rato Cumutshie
1.0
Explenedd Cumuuletive
Expl retoo Cumuletive
0.0
8.8
0.6
066
0.6
6.2
Phase
Phose
Phase
0.0
0.0
0.0
compoment
component
component
(d) PCA for tokens on ETTm2.
(c) PCA for tokens on Electricity.
(1) PCA for tokens on Weather.
Figure 8: Visualization of phase tokenization across six datasets: ETThl, ETTh2, ETTml, ETTm2, Electricity, and Weather.
A.3.5 COMPARISON WITH TIME SERIES FOUNDATION MODELS
To contextualize the performance of approach, compare PhaseFormer against several repre- sentative foundation models drawn from the GIFT-Eval benchmark suite (Aksu et al. 2024). Specif- ically, we include Chronos (Ansari et al. 2024). Moirai (Woo et 2024). and the recently released Moirai-2 Liu et al. 2025) as baseline models.
Due to space limitations, report results the Web/DevOps subsets of the benchmark Tab. These Web/DevOps datasets are not well-studied by many existing works (Nie et al., 2023 Zhang Yan 2023 Huang al.) 2025 Xu et al., 2024 Lin et al.] 2024b). Their heterogeneous and highly non-stationary behaviors make them particularly challenging, and thus provide realistic environment to evaluate the generalization capability of large time series foundation models.
Beyond demonstrating strong overall competitiveness, the empirical results clearly highlight the unique strengths phase-based representation handling high-variance, noise-heavy workload patterns. In particular, PhaseFormer consistently outperforms all foundation mondal baselines on the bitbrains na.rnddddddddd..d. and bitbrains wtttbrains.fsttssttetetssss fast_storage datasets- -two challenging subsets in GIFT-Eval due frequent spikes and abrupt intensity shifts. These results indicate that Phase- Former's ' phase-aware design provides substantially improved robustness and stability under highly volatile real-world workload traces, where traditional architecture tend degrade.
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Table Comparison of MSE (Mean) Across Four Forecasting Models on Web/CloudOps Datasets
provided in GIFT-Eval.
Dataset
PhaseFormer
Moirai-2
Moiral-base
Chronos-base
Best Model
39.4347
20.6240
22.10000
24.1000
Moirai-2
bizitobs 12c/H/short
99.9582
67.8501
272.0000
243.0000
Moirai-2
bitbrains mnd///short
1,970,330
2,235,573
4,020,000
2,210,000
PhaseFormer
bithrains mmdS5//short
1,606,034
1,660,532
180,000,000
1,800,000
PhaseFormer
buritobs_applicanee Siteaatiioo/OS/horttt 10S/short
4,653,132
621,852
6,360,000
6,030,000
Moirai-2
bithrains fast storage/H/short
2,585,180
2,942,651
600,000,000
2,620,000
PhaseFormer
bitbrains fast storage/ST/short
1,887,398
2,142,125
25,400,000.000
1,920,000
PhaseFormer
bizitobs, service/1OS/shen
26.523
4,798
51,900
37,100
Moirui-2
Table 9: Comparison PhaseFormer variants. Each cell reports MSE, MAE, and FLOPs. The input length is fixed as 720 steps and the output length is fixed as 96 steps. The best results are marked with Bold, and the second-best results are marked with Underlined.
PhameFormer-I.78
Formerrk
Former333
Dataset
MSE
MAE
FLOPs
MSE
MAE
FLOPs
MSE
MAE
FLOPs
Electricity
0.129
0.220
9,41M
0.129
0.221
31.97M
0.131
0.223
221.05M
Traffic
0.361
0.241
25.27M
0.366
0.243
85,84M
0.360
0.236
593.61M
Weather
0.150
0.199
0.62M
0.151
0.194
2.09M
0.174
0.217
14.46M
A.4 ADDITIONAL HYPER PARAMETERS ANALYSIS
A.4.1 IMPACT OF MODEL PARAMETER SCALE ON PERFORMANCE
We conducted comparative experiments on three variants of the PhaseFormer model with different parameter scales across the Electricity, Traffic, and Weather datasets. The three configurations are: a single-layer model with latent dimension
1.2K parameters), a single-layer model with la- tent dimension 16 (R55.48K parameters), and a two-layer model with latent dimension 32 (R:37.1K parameters). Fig. summarizes the results in terms MSE, MAE and FLOPs.
Overall, the effect of model scale on performance is not consistent. On the Traffic dataset, larger models yield slight improvements, whereas on the Electricity and Weather datasets, the smaller and medium-sizen models perform better. These findings indicate that PhaseFormer achieves a favor- able balance between computational efficiency and predictive accuracy at relatively small parameter scales, and increasing model size does not lead to uniform gains across all tasks.
A.4.2 IMPACT OF INPUT LENGTH ON PERFORMANCE
We examine how the input window size affects the prediction accuracy and computational cost of PhaseFormer. Throughout this section, L denotes using the most recent L time steps as model input. The output length is fixed 96 steps. As summarized in Fig. increasing L reduces MSE and MAE across datasets, indicating that PhaseFormer benefits from longer historical context for modeling long-range temporal dependencies.
In terms of efficiency, the parameter count and FLOPs forward pass remain nearly constant as increases, with only modest growth (see Fig, 10) attributable primarily to the embedding stage. This behavior arises because the sequence length processed by the core encoder/decoder is governed by the number of phases, which depends on the data's learned periodic structure rather than by the raw input length. Consequently, scaling L mainly affects the embedding computations, whose cost is relatively small compared to the phase-basee modules.
A.5 ANALYSIS OF FINE-GRAINED TEMPORAL INFORMATION RETENTION
To examine whether Phase Tokenization preserves essential intra-cycle dynamics after compressing periodic structures, conduct reconstruction experiment the Traffic dataset. Given an input sequence of length 720, PhaseFormer is trained to reconstruct sequences of the same length, remov- ing forecasting uncertainty and isolating the representational capacity of the phase embeddings.
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MS2
0.24
0.4
0.22
1 038
0.18
014
66
302
136
700
95
150
306
720
86
190
305
720
(a) Electricity
(h) Traffic
(c) Weather
Figure 9: Prediction error test results across datasets under different input lengths.
Aaaan. HC)
1.50
530
356
122
95
332
138
120
332
136
Vesit Lergth
kpett longth
Inect Longts
(a) Electricity
(b) Traffic
(c) Weather
Figure 10: Efficiency evaluation of PhaseFormer across datasets with varying input lengths.
Orign
0.30
0:35
0.55
Mawaoh MMMb A
-2
300
602
200
ppentodogrami
16.
10
0.0
Figure 11: Time-domain and power-spectrum visualizations of the reconstruction residuals when PhaseFormer reconstructs 720 input time steps on the Traffic dataset.
Setup. We evaluate the retention fine fine-grained information by computing the reconstruction resid- ual and analyzing it in both the time and frequency domains. In particular, we estimate the power spectral density (PSD) of the residual to detect whether certain frequency components, especially those associated with short-term or fieg-frryyency dynamics, systematically diminished.
Frequency Analysis of the Residual. The PSD analysis in Fig. II shows that residual energy evenly distributed across the spectrum, with noticeable concentration suppression in any frequency band. This indicates that Phase Tokenization does not introduce drequency-deependen information loss. Despite compressing each cycle into compact phase representation, the model retains the ability to capture fine -raiied intra-cycle dynamics, supporting the effectiveness of phase- based temporal abstraction for long-sequence forecasting.
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Hosure sell
60-
3.0
10-
Fredition Time
Predition Time
(a) Phaserormer
(h) PatchTST
Coorrd Fradicsonn
RUP 3762
Tiratase fredition
-..
Peditiin Thre Stepe
Prodictoon Time Stops
(c) FITS
(d) TimeBase
Visualization of forecasting results on Traffic dataset. The black lines stand for the ground truth and the blue lines stand for forecasting results.
Figure 12:
A.6 SHOWCASES
To provide clearer comparison of predictive performance across different models, present the results of PhaseFormer, PatchTST, FITS, and TimeBase on the Traffic dataset. PhaseFormer demon- strates strong predictive performance, as reflected by both the shape of its forecasts and the actual prediction errors.
Compared with PhaseFormer, PatchTST produces lower peak values within each cycle, failing to fully match the true curve. This discrepancy likely due to phase shifts the periodic pattern that reduce peak amplitudes. FITS, which performs prediction in the frequency domain with frequency band partitioning, tends to overlook high- figgufryyy information. As a result, its predictions deviate from the ground truth, though the forecasts still preserve periodic structure. TimeBase aligns well with the general cyclical pattern but fails to capture true variations across cycles, a limitation stemming from its patch-based basis construction mechanism.
A.7 THEORETICAL ANALYSIS FOR PHASE TOKENIZATION
Under the classical finite-order linear system assumption, the structural dependencies a time series (especially periodic quasi-periodic components) are reflected in the rank properties of suitable delay-embedding matrices.
Consider univariate time series
22........ TT.
Fix collection of delays (lags) .I,..., and set of reference indices 81, such that 8p + TD for all h. The corresponding H delay-embedding matrix is
Teitn
Teatn
Tsstn Een +72
Destra
Ts1-TD
TegtTD
Tay +Tp_
The classical Hankel matrix is recovered as the special case with unit delay and consecutive indices, e.g. Ta d and 8д h, but analysis does rely this specific structure.
the series is dominated by periodic or harmonic components, can often be generated (or well approximated) by low-dimensional linear dynamical system. In this case, the delay-embedding
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matrix X exhibits a natural low-rank structure: its columns lie approximately in low-dimensional subspace determined by the underlying system dynamics. Only when the series resembles unstruc- tured white noise does the delay-embe matrix approach full rank.
For a univariate periodic (or quasi-periodic) series, such delay-embedding admits the factorization
= AG,
where and G encode, respectively, the delay-dependent responses and the coefficients associated with different reference indices (e.g., days windows). The parameters D, the choice of delays {Ta}, and the number of columns H jointly determine the shape of the embedding.
Thus, model the data matrix
= AG +N E R
where A E RDxr G E RHxr are column full rank with rank(A) rank(G) r, and N is noise. The true signal is M = AGT. We assume r < min(D, H).
Patch tokenization corresponds to the row space Row(X) (X) (the right singular r-subspace), while phase tokenization corresponds to the column space Col(X) (the left singular r-subspace).
shared transformation applies IR H*H on the hourly dimension:
X' XS = A(SG) + +
In our formulation, the set S represents a family of linear transformations applied to the input se- quence. Such transformations frequently arise in practical systems due to sensor delays, timing jitters, or preprocessing procedures including smoothing or temporal alignment.
For matrix Y, define the spectral separation
scp,
min lo:(Y) a())) isr.j>r
When rank(Y) r, this equals o, (Y). In partieular,
== o, 8' = (MMM)) 8ain = min666,))
Assume S is invertible on Col(G), i.. rank(SG)) = r. Define
k = min i (S) (S)ca(())) 0.
Moreover, since (MM) 2 or (A)o, and o, (MS 2 ko, (A)o, we have the useful bound 8min 2 min(1, x) o, (G)
For two r-dimensional subspaces U1 their distance is
d(24, V) || Pylla sin O (M, V),
where Pu is the orthogonal projector onto 44 This metric satisfies the triangle inequality.
Lemma (Column space preservation) Let M AG Ifrank(SG) then Col(MS = Col(M) = Col(A).
(aan < r, then the column space shrinks.
Lemma 2 (Row space change) For M AGT,
Row = Col(G),
Row(MS Col(SG).
As a result:
d(Col(G), Col(SG))
S(Col(G)) # Col(G))
Lemma (Wedin's sin θ theorem) Let + and 8 = sep, (M) > 0. Then
|E'lz
(M) <
8
where U4 (M) denotes the leading left singular r-subspace of M (the right case is analogous). Here C is absolute constant (often C E 22). The condition 8 > 0 necessary.
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Theorem Assume rank(SG) and 8, 8 > 0. Then:
1. For phase tokenization,
((4,(x)) & c( + :) &
c 1l ||
Smin
In the noiseless case, Lemma I ensures exact invariance, so the distance is 0.
2. For patch tokenization,
(V,())) 2 do -c(
[ (
IN'll2
where do == d(Col(G), Col(SG)). If S(Col(G)) + Col(G), then do > 0.
Proof For phase tokenization, apply the triangle inequality: (MT, + du,,(Ms (.
S d4,, (X)1)
By Lemma the middle term vanishes, and the two boundary terms are bounded by Wedin's 's theorem, yielding the stated inequality.
For patch tokenization, have
d(V, dXV. (X')) 2 ((V.
d(), - (X'))
By Lemma 2 the first term equals do. and the other two are controlled by Wedin't 's theorem, proving the bound.
In real-world scenarios, slight variations in timing conditions occur from day to day, the daily
transformations are not exactly identical. We model this systematic inconsistency by introducing a small perturbation Ad, which captures the mismatch between the ideal linear transformation S and the actual data-gend gentraenera process. Suppose each day's 's transform is Sa = S + Aa with ||aall I c. Then
X' = XST + R
Rd = Xa.
Bounding row by row gives | 2|2 S SeIla,,112. hence
||| P S = x|
|R]|a S <<(I + [[N]))).
The stability tokenization-induce subspaces measures whether the model able to maintain consistent internal representations of key periodic trend components when subjected to these perturbations. Correspondingly. the distances between iikenization-inducee subspaces provide quantitative assessment of how much the representation changes, with smaller distances indicating more stable and reliable token features.
Phase-based tokenization exhibits near-invariance under these transformations, offering inductive bias that is particularly beneficial for real-world forecasting scenarios.
Theorem (Stability under day-wise perturbations) Under the relaxed model, each day uses Sa S Aa with ||al| S E, that
X' XST +R,
Rd, AT
Let M+N with M AGT, rank(A) rank(G) r, and assume rank(SG) = r so that 8 (, > 0 and 8' = (MS > O. Define 8uin = min(,, 8') and do = di (C), Col(SG)).
Then there exists absolute constant C E [2, (222 such that:
I. Phase tokenization (left r-subspace):
||
|
|Rll2
(x()())) S C
8
8
<(
(II + [[
|N + |
C
8 8min
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2. Patch tokenization (right r-subspace):
||
|
||/2|12
((v(()))Vx(x)) 2 do C
+
8'
8'
<(ll - IN - ||
IN
do C
Ómin
In particular, if ((C)(GG)) = Col(G) then do = 0 and patch tokenization is also preserved up to the same perturbation scale.
Moreover, using 8min 2 min(1, (,) with k Tmin (S) (S1Co)())) signal strength explicit.
0 makes the role of
Proof By row-wise control, RRd S =|| X|:la, hence
| <|||X||= S [[| [[]] [|R||2 S S S
Insert the chain
X MS + MS +N' - X MST + N' +
For phase subspace Ur. according to the triangle inequality,
(X)) & (A,, + (M), + (, + N*) + d(U + (x))
The middle term vanishes by Column space preservation (Lemma 1). Applying Wedin's 's sin θ the- orem (Lemma to the remaining three perturbations C'|/''lla/8' R 8'. Use 8min S 5, 8' and Step I to obtain Item 1.
{N, N', R} yields C/ +
For patch subspace Vr, we use the reverse triangle inequality:
(.)((x)...X) 2 d(V.(M), (MS - (M))
- -d(V.(MST) V.MSS +*)) - d(V "N',,,(X")
The first term equals do by Row space change (Lemma 2). Apply Wedin's theorem to the other three terms obtain the final results.
Therefore, we can conclude that our Phase Tokenization performs better than Patch Tokenization when facing periodic perturbations and is more tolerant of mild periodic fluctuations. The above theoretical analysis can be applied real-world time-serie series data that have stable primary period but exhibit small variations.
A.8
THE USE OF LARGE LANGUAGE MODELS
In this work, large language models (specifically ChatGPT-5) are used solely for polishing the writ- ing, identifying grammatical issues, and performing proofreading.
24


Reviews:
[{"review_id": "vynr8fUDLa", "reviewer": "Reviewer_bPyT", "summary": "This paper continues a line of research on highly parameter-efficient univariate long-term forecasting for time series with a dominant periodicity. It establishes a new best performance among these parameter-efficient models on standard baselines. Additional analysis is provided on the role of different approaches for reshaping time series into tokens.", "questions": "- The code shows significantly different hyperparameters being used for different datasets and horizons, but hyperparameter tuning is not discussed in the paper. How was hyperparameter tuning done?\n- Were hyperparameters tuned for the baseline models? If not, were dataset-specific hyperparameters used or were they kept the same for all datasets and horizons?\n- Are different model sizes used for different horizons? This seems to be the case in the code but the horizon is not given for Figure 1.b) and Figure 4.\n- Can you explain how the terms used in Theorem 1 specifically relate to a time series signal and time series operations on it?", "limitations": "", "rating": 2, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "- The model appears to improve forecasting performance relative to models of comparable size in the given domain.\n- The architecture used is substantially novel and demonstrates useful approaches for minimizing parameter usage in these kinds of models.", "weaknesses": "- The comparison to parameter-efficient models is up to date but the general comparison to time series forecasting models is not, with the most recent evaluated large models being published in May 2024. (See [1] for examples of more recent models.) While I don't think this comparison is crucial for the paper, claims relating to the state of the art should be amended to limit the scope to parameter-efficient models.\n- It's not clear how the statistical model in Theorem 1 relates to periodic signal forecasting. For instance, the data is modelled as a low rank matrix with additive noise, and it's not explained how that relates to a univariate time series with a dominant periodicity.\n- The practical relevance of the approach seems limited, since it relates to univariate forecasting on signals with a known dominant periodicity where extremely low parameter counts are needed. Prior works establishing that this was possible with reasonable performance were an interesting result, but that's well-established now, and it's not clear to me that incremental improvements to performance in this regime are impactful. The reliance on a small number of very well-studied benchmarks also raises the possibility of overfitting.\n- Important details on hyperparameter tuning and model variants used for experiments are not given, making it difficult to evaluate the soundness of experiments - see Questions.\n\n[1] GIFT-Eval: A Benchmark For General Time Series Forecasting Model Evaluation https://huggingface.co/spaces/Salesforce/GIFT-Eval", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Ma5cC23Afh", "reviewer": "Reviewer_Nubb", "summary": "This paper proposes PhaseFormer, a highly efficient Transformer architecture for long-term time series forecasting.\nInstead of modeling local patch tokens, the authors introduce a novel phase tokenization mechanism that reorganizes the input sequence into a phase across period structure, capturing cross-cycle consistency.\nA lightweight Cross-Phase Routing Layer with two submodules—Phase to Router (aggregation) and Router to Phase (distribution)—enables global information exchange with only a few learnable routers.\nThe authors also provide a theoretical justification via the Phase Tokenization Stability Theorem, arguing that phase representations are more stable and low-dimensional under periodic perturbations.\nExperiments on seven benchmarks show that PhaseFormer achieves comparable or better accuracy than SOTA models while reducing parameters and FLOPs by orders of magnitude.", "questions": "1. The paper mentions that phase tokenization depends on the cycle length $L_{phase}$, yet the main text only briefly states that it is detected via autocorrelation. Could the authors clarify how $L_{phase}$ is determined for each dataset and whether it is fixed or adaptive during training?\n2. Table 5 shows the number of routers $M$ affects performance. Could the authors provide a short discussion or guideline on selecting $M$? For example, is there a trade-off pattern across datasets?\n3. In the Phase Tokenization Stability Theorem, the perturbation analysis is central. Could the authors clarify whether this theorem assumes a fixed period or allows mild variations? This would help readers understand the scope of applicability.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "1. The idea of aligning data across cycles (phase tokenization) directly targets the instability of patch-based representations. Both theoretical and empirical analyses (PCA, MMD, t-SNE) support the claim that phase tokens form a more stable, low-dimensional subspace.\n2. PhaseFormer drastically reduces parameters and FLOPs (up to 99.9% less than PatchTST) while maintaining or even improving forecasting accuracy across multiple datasets.\n3. The Cross-Phase Routing Layer elegantly combines efficiency and expressiveness. Router attention visualizations reveal interpretable phase dependencies.\n4. Extensive comparisons and ablations confirm the effectiveness of both phase tokenization and the router mechanism.", "weaknesses": "1. The theory and architecture depend on locally stable periodicity. The method’s behavior on weakly periodic or nonperiodic series is not fully tested, though acknowledged as a limitation.\n2. The paper mentions autocorrelation-based detection but does not fully describe how multiple or unstable periods are handled. Automatic phase-length estimation and failure cases should be detailed.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "xE8k9wQbtB", "reviewer": "Reviewer_ZUVB", "summary": "This paper proposes PhaseFormer, a phase-centric framework for long-term time-series forecasting. Instead of patch tokens, the method aligns values at identical offsets across cycles to form phase tokens, argues (empirically + theoretically) that phase tokens are more stationary and lower-dimensional than patch tokens, and introduces a lightweight cross-phase routing mechanism (phase→router aggregation, router→phase distribution) plus a shared predictor. Experiments on 7 benchmarks report strong accuracy with ~1k parameters and very large FLOPs reductions versus patch-based Transformers, with an explicit complexity analysis and ablations.", "questions": "As in weaknesses.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper motivates phase tokens via t-SNE/MMD/PCA analyses showing lower drift and dimensionality than patch tokens, and formalizes stability under cycle perturbations (Theorem 1). This gives a principled basis for efficiency and generalization.  The cross-phase routing layer (few routers, shared predictor) is easy to implement and yields large efficiency gains; on Traffic, FLOPs are reduced by ≈99.99% vs PatchTST/Crossformer while improving error.", "weaknesses": "1. While TimeMixer is included, PatchMLP  is not in the main tables. Given the paper’s efficiency narrative, a direct comparison would strengthen claims. Please report PatchMLP with the same look-back (720), horizons, and official settings. \n2. The main table shows FITS slightly outperforming PhaseFormer on ETTh2 (e.g., 0.334/0.382 vs 0.346/0.388 for MSE/MAE), and PatchTST essentially ties on ETTm2. Discuss failure modes and whether non-strictly periodic signals undermine the phase assumption. \n3. L_phase is chosen via frequency-domain/autocorrelation analysis and inputs are circularly padded to multiples of L_phase. Please provide a sensitivity study to period mis-specification and show robustness when cycles drift or are multi-periodic (e.g., Traffic/Electricity), as well as clarify any risk that circular padding induces boundary artifacts for long horizons.\n4. The method follows a channel-independent paradigm (treating each variable separately). For multivariate datasets with strong cross-variable dependencies, can PhaseFormer leverage cross-variable structure beyond phase-wise routing? A small study contrasting channel-independent vs channel-dependent variants would help.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "VtcuH3WnmY", "reviewer": "Reviewer_X2Dk", "summary": "This paper proposes PhaseFormer, a lightweight Transformer architecture for time series forecasting that replaces conventional patch-based tokenization with a novel phase-based tokenization paradigm. Instead of segmenting sequences into contiguous time windows, the model aligns values at identical offsets across cycles (phases) to form compact and stable phase tokens. A lightweight phase-based routing Transformer is introduced to enable efficient cross-phase interaction via a two-stage \"phase-to-router\" and \"router-to-phase\" attention mechanism. Experiments on seven benchmark datasets (ETTh1/2, ETTm1/2, Weather, Electricity, Traffic) show that PhaseFormer achieves comparable or superior accuracy to existing patch-based and frequency-based models.", "questions": "1. The paper does not include a comparison against CycleNet[1], which also explicitly models periodic patterns to enhance forecasting performance.\n  2. How sensitive is PhaseFormer to inaccurate or variable period lengths? Could adaptive or learned phase estimation improve robustness on datasets with weak periodicity?\n  3. Have the authors examined whether phase tokenization retains sufficient intra-cycle information? For instance, reconstruction experiments or mutual information analysis could verify that the compressed phase embeddings still capture critical fine-grained temporal cues.\n\n[1] Lin S, Lin W, Hu X, et al. Cyclenet: Enhancing time series forecasting through modeling periodic patterns[J]. Advances in Neural Information Processing Systems, 2024, 37: 106315-106345", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. Conceptual novelty: introduces a new phase-based view for modeling periodic time series, complementing the patch and frequency paradigms.\n  2. Efficiency and scalability: achieves over 99.9% parameter and FLOP reduction without losing accuracy.\n  3. Clarity of presentation: writing, diagrams, and methodology are clear and logically connected.", "weaknesses": "1. Representational limitation: phase tokenization compresses intra-cycle information; it remains unclear whether this sacrifices fine-grained dynamics.\n  2. Incremental architecture: routing attention is inspired by Perceiver-style cross-attention; architectural novelty mainly lies in its adaptation to phase-level modeling.\n  3. Strong periodicity assumption: the model assumes a fixed, known period; performance on irregular or multi-periodic data is not analyzed.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)