Patch-wise Structural Loss for Time Series Forecasting
Authors: Dilfira Kudrat
Zongxia Xie
Yanru Sun
Tianyu Jia
Qinghua Hu
Venue: ICML 2025 poster (2025)
Keywords: Time Series Forecasting
Loss Function
Deep Learning
Primary area: deep_learning->sequential_models_time_series
TL;DR: We propose PS loss, a novel loss function that measures local structural similarity of time series for more accurate forecasting.

Abstract: Time-series forecasting has gained significant attention in machine learning due to its crucial role in various domains. However, most existing forecasting models rely heavily on point-wise loss functions like Mean Squared Error, which treat each time step independently and neglect the structural dependencies inherent in time series data, making it challenging to capture complex temporal patterns accurately. To address these challenges, we propose a novel **P**atch-wise **S**tructural (**PS**) loss, designed to enhance structural alignment by comparing time series at the patch level. Through leveraging local statistical properties, such as correlation, variance, and mean, PS loss captures nuanced structural discrepancies overlooked by traditional point-wise losses. Furthermore, it integrates seamlessly with point-wise loss, simultaneously addressing local structural inconsistencies and individual time-step errors. PS loss establishes a novel benchmark for accurately modeling complex time series data and provides a new perspective on time series loss function design. Extensive experiments demonstrate that PS loss significantly improves the performance of state-of-the-art models across diverse real-world datasets. The data and code are publicly available at: \url{https://github.com/Dilfiraa/PS_Loss}.

Patch-wise Structural Loss for Time Series Forecasting
Dilfira Kudrat' Zongxia Xie
Yanru Sun
Tianyu Jia
Qinghua Hu
Abstract
MSE 0.83
MSC 00.03
MSE -588
Time-series forecasting has gained significant at- tention in machine learning due to its crucial role in various domains. However, most exist- ing forecasting models rely heavily on point-wise loss functions like Mean Squared Error, which treat each time step independently and neglect the structural dependencies inherent in time se- ries data, making it challenging to capture com- plex temporal patterns accurately. To address these challenges, we propose a novel Patch-wise Structural (PS) loss, designed to enhance struc- tural alignment by comparing time series at the patch level. Through leveraging local statisti- cal properties, such as correlation, variance, and mean, PS loss captures nuanced structural dis- crepancies overlooked by traditional point-wise losses. Furthermore, it integrates seamlessly with point-wise loss, simultaneously addressing lo- cal structural inconsistencies and individual time- step errors. PS loss establishes a novel bench- mark for accurately modeling complex time series data and provides new perspective on time se- ries loss function design. Extensive experiments demonstrate that PS loss significantly improves the performance of state-of-the-art models across diverse real-world datasets. The data and code are publicly available at: /ttps:///gthhun com/Dilfiraa/Pe Loss.
Pe 4p-1-113 Ap-10
oy - 06 Per
0,-0,6
(a) Poor correlation (b) Variance mismatch
(c) Mean offset
Figure I. Limitations of MSE loss in capturing structural differ- ences. Predictions with similar MSE but different statistical prop- erties: (a) Poor linear correlation leads to misalignment in direc- tionality and pattern; (b) Variance mismatch distorts the fluctuation dynamics; (c) Mcan offset causes shift in the overall level of the forecast.
that capture complex dependencies and intricate patterns in time series data, significantly improving forecasting accu- racy (Zhou et al., 2021; 2022; Sun et al., 2024a; Yu et al., 2024b; Sun et al., 2024b; Yu et al., 2024a; Qiu et al., 2025b; Sun et al., 2025). These models aim to produce multi-step predictions based on historical data that closely follow the trajectory future series.
Despite these advancements, most current forecasting mod- els rely on Mean Squared Error (MSE) the primary loss function. MSE evaluates predictions by averaging point- wise distances, which overlooks important structural charac- teristics inherent in time series data, leading to suboptimal forecasts that fail to accurately capture the underlying tem- poral dynamics (Le Guen & Thome, 2019; Lee et al., 2022: Cuturi & Blondel, 2017).
1. Introduction
To illustrate the limitation of MSE, we consider three pre- dictions with similar MSE values but different forecasting quality, as shown in Figure 1. The prediction in Figure la exhibits poor correlation with the ground truth, failing to capture the overall directionality and pattern. In contrast, the prediction Figure 1b aligns with the general shape of the ground truth but fails match its variability, as indicated by the variance mismatch, leading to distorted fluctuation dynamics. On the other hand, the prediction in Figure Ic maintains the shape but exhibits mean offset, causing a consistent shift in the overall forecast level. These exam- ples highlight that correlation, variance, and mean capture distinct yet complementary aspects of structural similarity,
Time series forecasting plays a crucial role across various real-world domains, including traffic (Kong et al., 2024; Long et al., 2024), weather (Lam al., 2023; Wu et al., 2023b), and finance (Huang et al., 2024a), where accurate forecasting is essential for informed decision-makin Re- cent advancements in deep learning have developed models
'College of Intelligence and Computing, Tianjin Uni- versity, China. Correspondence to: Zongxia Xie <ead- diexie heaii@@@otmail.com...
Proceedings of the 42nd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Patch-wise Structural Loss for Time Series Forecasting
providing a more comprehensive assessment of the align- ment between prediction and ground truth.
models excel modeling local temporal relationships (Wu et al., 2023a; Luo & Wang, 2024; Liu al., 2022). In con- trast, Transformer-based models have gained significant attention for their ability to model long-range dependencies through self-attention mechanisms (Nie et al., 2023; Liu et al., 2024a; Chen et al., 2024b; Lin et al., 2024b; Zhou et al., 2022; Yu et al., 2023). Additionally, MLP-based models offer efficient alternatives that balance simplicity with performance (Zeng et al., 2023; Wang et al., 2024; Lin et al., 2024a; Ekambaram et al., 2023; Jia et al., 2025), Recently, LLM-based models have emerged as promising direction, leveraging the capabilities of large language mod- els to effectively capture complex temporal dependencies (Zhou et al., 2023; Liu et al., 2024b; Jin et al., 2023; Niu et al., 2025). These models use point-wise MSE as their loss function, treating each time step independently, which limits their ability to capture the structural nuances of time series data and may hinder forecasting performance.
However, measuring correlation, mean, and variance at a global level is insufficient, as global metrics often fail to capture local variations in statistical properties, especially in non-stationary time series where these properties evolve over time. This limitation highlights the need for a localized approach that can adapt to these variations and provide more precise comparison of time series.
To address these challenges, we propose Patch-wise Structural (PS) loss that improves time series forecasting by introducing structural similarity metrics at the patch level. Our method comprises three key components. First, we introduce the Fourier-based Adaptive Patching (FAP) mod- ule, which adaptively segments the time series into patches, enabling a finer- finer-grained assessment of structural alignment. Second, we propose the Patch-wise Structural (PS) loss, which integrates correlation, variance, and mean losses to capture local structural similarities and account for varia- tions across different regions of the data. Third, we employ a Gradient- Dynamic Weighting (GDW) strategy to adaptively balance the contributions of each loss term based on their gradients, ensuring that all structural aspects are adequately accounted for during training. PS loss can be seamlessly combined with MSE loss, resulting in more robust and accurate forecasting. Our contributions are as follows:
2.2. Loss Functions for Time Series Forecasting
Recent efforts to address the limitations of MSE have given rise to alternative loss functions (Le Guen & Thhome, 2019; Cuturi & Blondel, 2017; Lee et al., 2022; Wang et al., 2025), which can be broadly categorized into shape-focused losses losses. Shape-focused losses aim
and dependency-foeure to capture structural similarities between the ground truth and predictions by explicitly addressing shape misalign- ment. For instance, Dynamic Time Warping (DTW)-based methods such as Soft-DTW (Cuturi & Blondel, 2017) and DILATE (Le Guen & Thome, 2019) align sequences under temporal distortions. While effective at improving shape alignment, their high computational complexity limits seal- ability. TILDE-Q (Lee et al., 2022), on the other hand, introduces transformation invariance, making it robust to amplitude shifts, phase shifts, and scaling differences, thus focusing shape similarity. Dependency-fouuue focusee losses focus on capturing temporal dependencies within the fore- cast window. FreDF (Wang et al., 2025) bypasses label correlation complexity by learning to forecast in the fre- quency domain. Despite these advancements, most prior methods rely on global comparisons of entire time series, overlooking crucial localized structural details. In contrast, our proposed approach incorporates patch-wise statistical properties into the loss function, enabling more granular structural measurement of the data and providing a funda- mentally new perspective on time series loss design.
We introduce novel structural similarity measure that combines correlation, variance, and mean, offering more comprehensive understanding of structural pat- terns and improving alignment between predictions and ground truth.
We propose a patch-wise comparison method that en- ables localized similarity measures and accounts for changing statistical properties within the series, ensur- ing accurate structural alignment.
Extensive experiments on five state-of-the the archi- tectures and seven real-world datasets confirm the ef- fectiveness of our approach, demonstrating consistent performance gains and enhanced forecasting accuracy.
2. Related Work
2.1. Time Series Forecasting Models
In recent years, numerous deep forecasting models have been proposed to capture complex dependencies in time series data by leveraging the powerful representation capa- bilities of neural networks (Kim et al., 2025; Shao et al., 2024; Qiu et al., 2024; 2025a). Based on their architec- ture, these models can be categorized into CNN, Trans- former, MIP, and L--e- approaches. CNN-based
3. Methodology
3.1. Preliminaries
Given historical time series X { F1,..., #L-1) RCx where C. is the number of channels and L is the length of the lookback window. The goal of time series
2

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Patch-wise Structural Loss for Time Series Forecasting
Prediction
Fourier-based
Patch-wise Structural Loss
Gradient-base -based Dynamic Weighting
Adaptive Patching
Ly
mile
B
Eyteon
Leore
Lyar aogllly magar
Lutan
Corr
Lagenn
Bockoon
PARAAN
Before weighting
After weighting
Carr
Eyar
y Extoan
Ground Truth Y
FFT
Lps
MSE
Figure 2. Overview of the proposed PS loss. The method consists of three main components: (1) Fourier-based Adaptive Patching, where the ground truth Y and predicted series Y are adaptively segmented into patches: (2) Patch-wise Structural Loss, which measures local similarity between patches by integrating correlation (CCorr), variance (Lvar), and mean losses (CMees)) ); and (3) Gradicnt-based Dynamic Weighting, which dynamically adjusts the weights of these loss components (a, 5 and 7) based on their gradient magnitudes to ensure balanced optimization. The PS loss (Crs) seamlessly integrated to the MSE loss (CMSE) to improve forecasting accuracy.
forecasting is to learn mapping g RCXL - RCT that generates predictions Ý {in, ,1, E RCxT approximating the ground truth = (vo, , -r E RCxT, where T' is the forecasting length. For simplicity of notation, we focus on the univariate case (C = 1) in the following discussions. MSE one of the most common loss functions for evaluating prediction accuracy, and it is defined as:
By seamlessly combining PS loss with MSE, we handle both global point-wise errors and local structural discrepancies. This integration enhances forecasting accuracy, demonstrat- ing the advantages of incorporating patch-wise structural analysis into time series loss functions.
3.3. Fourier-based Adaptive Patching
To enable localized structural similarity measures between time series, we propose patch-wise comparison approach that divides the prediction Y E and the ground truth Y RT into patches. By comparing these patches, better account for the structural patterns within the series.
T LMSE
1
(y: ==0
2:)2
T
3.2. Overview
In order to adaptively determine the patch length, we intro- duce the Fourier- Adaptive Patching (FAP) technique. By analyzing the amplitude spectrum of the ground truth via the Fast Fourier Transform (FFT), we identify its dominant periodicity:
Our method addresses the structural alignment limitations of MSE loss by focusing localized pattern differences. To achieve this, incorporate three key components: Fourier- based Adaptive Patching (FAP), Patch-wiss wise Structural (PS) Loss, and Gradient-Basee Dynamic Weighting (GDW), which collectively measure the patch-wise structural simi- larity between prediction and ground truth, as illustrated in Figure 2.
A Amp (FFT(Y)), f /=
arg max f.E(1.....IT/2 [T/2]]
(A); (2)
where, Amp(-) calculates the amplitude values and f de- notes the dominant frequency. The period p = [T/f] serves an initial patch length candidate, ensuring the patches reflect recurring structural patterns. However, dominant pe- riodicity alone may produce overly large patches, missing finer structure nuances. To balance granularity and coher- ence, the final patch length is determined as:
We first generate the prediction Y using backhone model, forming the basis for subsequent patch-level comparisons. Following this, FAP segments both the ground truth and prediction into patches based on the dominant frequency of the ground truth, ensuring that the patches capture re- curring patterns and coherent structural information. Once the series is segmented, the PS loss evaluates the similarity between patches by integrating correlation, variance, and mean losses, enabling a comprehensive assessment of local structural alignment across the time series. To further re- fine the optimization process, GDW dynamically adjusts the weights of the three losses based on their gradients, ensuring balanced contributions from each component and leading to more effective optimization and consistent model training.
P= min (L)
T f
p p=
(3)
where 5 is predefined threshold. This adjustment ensures patches remain focused on localized structural patterns, even when the dominant periodicity is unclear or too large.
With patch length P and stride S LP/2], J. the series is seg- mented into patches Y, E TRRPx and Y, R PxN where
3

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Patch-wise Structural Loss for Time Series Forecasting
N ( L[|S] is the total number of patches. The (6)
where ob.) is the softmax function. Due to the shift- invariance property of softmax, i.e., ((,p ( - =l)) =
i-th patch of Y is defined as y {wo (8) (2) ..... 4p-1h th where u/) refers to the VixS+j in the unpatched time series.
o o(pp))) the variance loss formulation simplifies to:
3.4. Patch-wise Structural Loss
N W-1 Σ KL (x 1 (x²²))
Cvar
(6)
Building on the adaptive patching framework, we propose the Patch-wise Structural (PS) loss to enhance time series forecasting by capturing and aligning localized statistical properties within each patch. PS loss focuses on three key metrics, namely correlation, variance, and mean, offer deeper structural comparison.
By aligning the relative dispersion within patches, the vari- ance loss encourages the model to capture the underlying fluctuation patterns of the time series, leading to more struc- turally coherent forecasts.
Correlation quantifies the degree of directional and pattern consistency between the ground truth and predicted patches. We define the correlation loss based on the Pearson Correlation Coefficient (PCC) (Cohen et al., 2009), as follows:
Correlation Loss.
Mean Loss. The mean value of patch provides clear indicator of its central tendency. While correlation and vari- ance losses effectively capture directionality patterns and localized variability. they may overlook systematic shifts or biases in the predictions. To address this, we introduce a mean loss term that directly measures the consistency of average values.
1 N-1
Loor
Σ
p(x, ,2,00
N
Specifically, the mean loss is defined as the Mean Absolute Error (MAE) between the mean values of the ground truth and predicted patches:
0=
1 N-1
Σ E
(0) 9 10,
i(())
Σ
N
o(00(0
(4)
N-1 (ett)
MM
(2(t)
where ((()) and (1) represent the mean and standard devi- ation of the i-th patch, respectively, and -(::-) denotes the PCC. Minimizing Loor encourages the model to align its predictions with the ground truth's directionality and trends. This alignment helps the model capture underlying pattern relationships, which are critical for time series forecasting tasks that require more than just point-wise accuracy.
where (((() and pa(() represent the mean values of the i-th patch in the ground truth and predictions, respectively. By aligning the mean values, the mean loss corrects prediction biases and ensures the model maintains accurate overall levels and central tendencies, enabling it to adapt to dynamic shifts in the time series.
Variance Loss. Variance quantifies the degree of fluctuation within a patch, providing a localized measure of variability. To ensure that predictions represent these localized dynam- ics, we introduce a variance loss that compares the relative dispersion between ground truth and predicted patches.
PS Loss. Finally, we integrate the correlation, variance, and mean losses into a unified Patch-wise Structural (PS) loss, which formulated as the weighted sum of the three individual loss components:
Rather than directly comparing variance values, our ap- proach focuses on softly aligning the relative dispersion of each time step within patches. Specifically, we compute the deviations, y, ( and (,(") f))0, and normalize them into probability distributions using the softmax function. This transformation highlights the relative magnitude of the deviations, allowing to focus how the values within each patch are dispersed relative to one another (Shu al., 2021). The similarity between these distributions is mea- sured via the Kullback-Leine (KL) divergence, leading to the following variance loss:
LPs = aLcorr 3L + MLseans
(8)
where a, 3, and y balance the contributions of the individual loss components. This integration provides a comprehen- sive structural alignment by leveraging the complementary strengths of each loss.
3.5. Gradient-based Dynamic Weighting
PS loss combines three loss components to measure patch- level similarity. To ensure effective training, it is crucial to balance these components, preventing any single loss term from dominating the optimization process (Chen et al., 2018). In multi-task learning, a common approach to bal- ancing multiple losses is to weight them based on their gradient magnitudes, which is robust to differences in loss
1 N-1 Σ KL (o (y,0 -2(0) 1 (y )())) Lvar= N 1=0
(5)

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Patch-wise Structural Loss for Time Series Forecasting
scales and directly reflects each loss's influence the opti- mization process (Sener & Koltun, 2018; Lin et al., 2019; Liu et al.). Inspired by this, we propose Gradient-based Dynamic Weighting (GDW) strategy, which adaptively ad- justs loss weights based on gradients to achieve balanced optimization.
4. Experiments
4.1. Experimental Setup
Datasets. We conduct our experiments on seven real-world multivariate time series datasets, including ETT (ETThI, ETTh2, ETTml, ETTm2), Weather, ECL,, and Exchange. follow the standard protocol
At training step t, let W denote the model's output layer parameters, We compute the L2-norm of each loss compo- nent's gradient with respect to W:
For data preprocessing. used by backbone studies. The detailed description of the datasets is provided in Appendix A.
Backbones. We select five state-of-the-arr (SOTA) time series forecasting models with diverse architectures to com- prehensively evaluate PS loss function. Specifically, we include Transformer-based models: iTransformer (Liu et al., 2024a), PatchTST (Nie et al., 2023); MLP- models: DLinear (Zeng et al., 2023), TimeMixer (Wang et al., 2024); CNN-baser model: TimesNet (Wu et al., 2023a).
Corr = V| (t) Corrlle
Var = V1 W Varllzz 12:
Mean = V W. W.anlllllz
where G(t) Cory is the gradient magnitude of the correlation loss, and G G(t) Mean are defined analogously. Using these gradient magnitudes, we compute the average gradient magnitude:
Implementation Details. We use the official implementa- tions of each backbone from their respective repositories for our experiments. For fair evaluation, when integrating PS loss enhance the backbone model's performance, we (10) follow their original experimental and hyperparameter set- tings, while only tuning the PS loss weight À and the patch length threshold 6. The experiments are implemented using Py PyTorch and executed on NVIDIA RTX 3090 24GB GPU.
(C()) Garr + GVvar + G!t) GMean
3
The loss weights at iteration / are then set as:
(t))
(())
(1)
8(0
700 = c.
4.2. Main Results
(
(
Corr
GVar
Mean (11)
We present the MSE and MAE on seven real-world datasets for five SOTA long-term multivariate forecasting models in Table 1. Notably, PS loss achieves remarkable im- provement DLinear, with average reductions of 5.22% in MSE and 4.39% in MAE. Similar performance gains are observed across other backbone models and PS loss consis- tently enhances the forecasting performance significantly outperforming MSE loss in 134 out of 140 cases, further validating the robustness and broad applicability of the pro- posed loss function. These improvements highlight the unique strength of PS loss in capturing localized structural patterns within time series data.
To further refine the mean loss contribution, we introduce two scaling factors, e and v, which adjust the weight of the mean loss based on the model's performance in correlation and variance. These factors are defined as:
c= 2 1 -1+ 3
Or8 )r Qy0g"
20 of + +0²
(12)
where Tyy is the covariance between ground truth and predictions, and oy and oy are their standard deviations. Both e and v lie in the range [0, 1), increasing as the predic- tions align more elosely with the ground truth in terms of correlation and variance. As aligmment improves, the rela- tive weight of the mean loss increases, ensuring it plays greater role when correlation and variance are well- matched, thereby enhancing the refinement of predictions.
4.3. Results on LLM-basse Models
To further evaluate the generalizability and effectiveness of the proposed PS loss, we conduct experiments on two LLM-based time series forecasting models: OFA (Zhou et al., 2023), built on GPT-2 (Radford et al.) and Autotimes (Liu al., 2024b). which utilizes LLaMA (Touvron et al., 2023).
Finally, the total loss function for optimizing the time series forecasting model is formulated as the sum of the standard MSE loss and the PS loss:
The results in Table 2 confirm that incorporating PS loss produces consistent performance improvements over the standard MSE loss. These findings highlight the potential of PS loss to serve as a universal enhancement for LLM-based time series models, further solidifying its value as a powerful tool for advancing multivariate time series forecasting.
= CMSE ALPs.
(13)
where À is hyperparameter that controls the contribution of PS loss.
5

---

Patch-wise Structural Loss for Time Series Forecasting
Table Long- term multivariate forecasting results. The table reports MSE and MAE for different forecasting lengths T' (96, 192, 336, 7203. The input sequence length is consistent with the backbone setting. The better results are highlighted in bold.
Models
iTransformer (20244)
PT (2025)
TimeMixer (2024)
Di ineur (2023)
TimseNer (2023a)
Functions
MSE
MSE
MSE
PS
MSE
PPS
MSE
Metric
MSE
MAB
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
96
0.387
0,405
0,379
0
0.375
0.394
0.385
0.402
0.366
0.392
0.384
0.405
0.367
0.389
0.384
0.405
0.382
0.406
192
01.411
0,436
014228
11444
01414
0.421
0.407
0.415
0.443
0.430
0.421
0.421
0.445
0.450
0.411
0.411
0.443
0.450
0.434
0.432
ETTM
336
0.491
0.462
0.474
0.453
0.432
0.436
0,420
0.426
0.512
0.470
0.489
0.453
0.447
0.448
0.435
0.435
0.494
0.471
0.457
0.448
720
0.509
0.494
0.480
0.478
0.450
0.466
0.425
0.449
0.497
0.476
0.474
0.464
0.504
0.515
0.463
0.484
0.504
0.515
0.477
0.477
Avg
0.457
0.449
0.440
0.438
0.418
0.431
0.405
0.421
0.460
0.444
0.437
0.432
0.445
0.454
0.417
0.430
0.457
0.460
0.435
0.441
96
0,301
0.350
0.289
0.340
0.274
0.336
0.272
0.332
0.291
0.342
0.284
0.330
0.299)
0.353
0.280
0.341
0.330
0.367
0.316
0.354
192
0.380
0.399
0..773
0.391
0.339
0.380
0.336
0.375
0.376
0.396
0.355
0.381
0.388
0.422
0.358
0.396
0.405
0.415
0.387
0.39
ETT52
336
0.424
0,432
0,416
0.331
0.324
0.377
0.437
0.439
0.412
0.423
0.463
0,473
0.435
0.450
0:454
0.451
0.428
0.431
720
0./10
0.117
0,421
0.441
0.378
0.421
0.376
0.417
0.16-1
0.16-1
0.423
0,440
0.733
0.606
0.597
0.540
0.434
0.448
0.138
0.118
Avg
0.381
0.407
0.351
0.379
0.327
0.375
0.392
0.410
0.369
0.394
0.169
0.463
0.417
0.432
0:406
0,120
0.392
0.407
96
0.342
0.377
0.326
0.360
0.288
0.342
0.284
0.334
0.328
0.364
0.316
0.350
0.301
0.345
0.296
0.339
0.334
0.375
0.339
0.369
192
0.383
0.396
0.374
0.384
0.334
0.372
0.328
0.364
0.364
0,382
0.357
0,373
0.336
0.366
0.333
0.362
0.406
0.413
0.388
0.400
ETTml
336
0.118
0.118
0,410
0.406
0.367
0.393
0.357
0.383
0.387
0. 0.102
0.385
0.394
0,572
0.389
0.365
0.380
0:115
0/122
0.394
0.406
720
0,447
0.457
0.472
01.440
0.417
0.422
0.406
0.412
0.472
0.449
0,444
0.432
0.427
0.423
0.419
0,413
0.511
0.472
00,669
0,444
Avg
0..40%
0,412
0.396
0.399
0.352
0.382
0.344
0.373
0.388
0.359
0.375
0.387
0.359
0.381
0.353
0.374
00:47
01:421
0.395
0.405
96
0.186
0.277
0.175
00.254
0.164
0.253
0.160
0.247
0.175
0.257
0.171
0.252
0.172
0.267
0.163
0.251
0.189
0.266
0,180
0.257
192
0.254
0.314
0,243
0,299
0.291
0.216
0.287
0.240
0.302
0.234
0.295
0.237
0.314
0.222
0.296
0.263
0.312
0.246
0.298
ETTm2
720
336
0.414
0.316
0.407
0.351
0.401
0.304
0.394
0.338
0.365
0.277
0.384
0.329
0.357
0.267
0.378
0.321
0.396
0.303
0.400
0.343
0.386
0.291
0.389
0.331
0:427
0.295
0.439
0.359
0.277
0.377
0.397
0.332
0.418
0.326
0.405
0.354
0,303
0.418
0.402
0.334
Avg
| 0.299
0.336
0.281
0.321
0.257
0.314
0.250
0.308
0.278
0.326
0.270
0.317
0.283
0.345
0.260
0.319
0.299
0.334
0,286
0.323
96
0.176
0.216
0.167
0.203
0.151
0.198
0.149
0.190
0.165
0.212
0.161
0.201
0.174
0.283
0.171
0.222
0.172
0.220
0.170
0.219
192
0.227
0.260
0.219
0.249
0.196
0.242
0.193
0.235
0.211
0.254
0,207
0.244
0.218
0.278
0.212
0.259
0.225
0.264
0.225
0,263
Weather
336
0.282
0.299
0,274
0,291
0,218
0,282
0.245
0.276
0,263
0,293
0,263
0,285
0.263
0.314
0.258
0.300
0.281
0.301
0.281
0.303
720
0.357
0.348
0.353
0.343
0.319
0.335
0.318
0.330
0.343
0.345
0.340
0.337
0.332
0.374
0.323
0.356
0.359
0.354
0.358
0.352
Avg
0,261
0.281
0.253
0.272
0.228
0.264
0.226
0.258
0.245
0.276
0.243
0.267
0.247
0,300
0.241
0.284
0.259
0.285
0,259
0.284
96
0.148
0.239
0.223
0.220
0.153
0.245
0.152
0.245
0.140
0.237
0.140
0.236
0.168
0.272
0.164
0.266
192
0.167
0,258
0.161
0.250
0.119
0.240
0.145
0.235
0.166
0.257
0.165
0.256
0.154
0.250
0.153
0.348
0,186
0,289
0,177
0,277
ECL
336
0.178
0.271
0.165
0.185
0.275
0,182
0.272
0.169
0.268
0.169
0.266
0.19%
0.207
0.192
0.292
720
0.211
0.300
0.208
0.297
0.208
0.296
0.196
0.282
0.224
0.312
0.223
0.312
0.204
0.500
0.203
0.297
0.235
0.329
0,220
0.314
Avg
0.176
0,267
0,172
0,262
0.163
0.254
0.158
0.247
0,182
0,272
0,180
0,271
0.167
0.264
0.166
0.262
0.196
0.297
0,188
0.287
96
0,086
0.206
0.087
0.206
0.093
0.214
0.089
0.208
0.086
0,201
0.084
0.202
0.085
0.209
0.078
0.199
0.105
0.235
0.107
0,236
192
0,181
0,304
0,180)
0,303
0.194
0.315
0.186
0.317
0.187
0.306
0.186
0.306
0.162
0.296
0.156
0.290
0.232
0.351
112215
0.337
Exchange
336
0.338
0.336
0.430
0.436
0.427
0.386
0.454
0.332
0.417
0,441
0.325
0.432
0.393
0.462
0.377
0.450
720
0.853
0.696
0.860
0.700
0.903
0.712
0.904
0.713
0.928
0.727
0.917
0.719
0.898
0.725
0.811
0.690
1.011
0.768
1.010
0.768
Avg
0,364
0,407
0.366
0.407
0.386
0.414
0.397
0.423
0.380
0.411
0.381
0.420
0.373
0.412
0.435
0.454
0.427
0.448
Table 2. LLMd long-term multivariate forecasting results. The table reports MSB and MAE for different forecasting lengths T E (96, 192, 336, 720]. The input sequence length is consistent with the backbone setting. The better results are highlighted in bold.
Loss
Dataset
ETTbl
ETTh2
ETTml
ETTm2
Models
Functions
Pred Len
96
192
336
720
96
192
336
720
96
192
336
720
96
192
336
720
MSE
0.382
0.424
0.444
0.445
0.292
0.359
0.379
0.418
0.298
0.336
0.368
0.419
0.170
0.230
0.285
0.364
OFA (2023)
MSE
MAE
0.402
0.425
0.437
0.458
0.348
0.398
0.414
0.454
0.344
0.375
0,392
0.425
0.263
0.306
0.345
0.392
MSF
0.371
0.413
0.434
0.422
0.283
0.355
0.372
0.398
0.290
0.331
0.364
0.416
0.162
0.218
0.275
0.355
+ PPS
MAE
0.393
0.416
0.430
0.445
0.338
0.381
0.402
0.431
0.343
0.367
0.388
0.418
0.248
0.289
0.329
0.375
MSE
0.360
0.388
0,401
0.405
0.295
0.359
0.392
0.465
0.289
0.337
0,373
0.430
0.181
0,244
0.299
0.372
AutoTmies
MSE
MAE
0.400
0.419
0.429
0,440
0.355
0.400
0.431
0.483
0.345
0.374
0.396
0.432
0.269
0.311
0.349
0.398
(2024b)
MSE
0.358
0.386
0.398
0.404
0.287
0.348
0.370
0.422
0.280
0.327
0.360
0.416
0.171
0.231
0.284
0.366
MAE
0.398
0.415
0.425
0.438
0.347
0.387
0.410
0.450
0.337
0,366
0.387
0.421
0,258
0.299
0.335
0,387
4.4. Comparison with Other Loss Functions
localized structural similarity by evaluating correlation, vari-
ance, and mean at the patch level, which enables a more precise alignment between the ground truth and predictions.
We evaluate the proposed PS loss against advanced loss functions. TILDE-Q (Lee et al., 2022) emphasizes shape similarity using cetafoommationnivvriaaa loss terms, while FreDF (Wang et al., 2025) bypasses label correlation by comparing time series in the frequency domain. The results are shown in Table 3. PS loss achieves the lowest MSE and MAE in most cases across various datasets and forecasting horizons. This is due to its ability to effectively measure
4.5. Ablation Study
To evaluate the contribution of each component within the proposed PS loss, we conducted ablation experiments on the ETTh and weather datasets using -inee as the backbone models. The results are presented in Table 4, with additional
6

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Patch-wise Structural Loss for Time Series Forecasting
Table Comparison between the proposed Fatch-wise Structural loss and other loss functions. The model is iTrrasformmer and report the result of three highlighted in underline.
ETTml, and Weather. The best results are highlighted in bold, and the second-bes best results are
Dutaset
ETTh1
ETTml
Weather
Forecast length
192
336
720
192
336
720
192
336
720
MSE
0.387
0.441
0.491
0.509
0.342
0.383
0.418
0.487
0.176
0.227
0.282
0.357
MSE
MAE
0.405
0.436
0.462
0.494
0.377
0.396
0.418
0.457
0.216
0.260
0.299
0.348
TILDE-Q
MSE
0.386
0.434
0.477
0,521
0.336
0,382
0.418
0.483
0.172
0.221
0.279
0.356
(2022)
MAE
0.401
0.429
0.454
0,502
0.366
0.392
0.416
0.453
0.208
0.252
0.296
0.346
FreDF
MSF
0.381
0.433
0.471
0.487
0.334
0.380
0.416
0.486
0.170
0.222
0.279
0.357
(2025)
MAE
0.396
0.426
0.446
0.481
0.367
0.393
0.414
0.451
0.210
0.254
0.297
0.348
MSE
0.379
0.428
0.474
0.480
0.326
0.374
0.410
0.472
0.167
0.219
0.274
0,353
+ PP
MAE
0.396
0.424
0.453
0.478
0.360
0.384
0.406
0.440
0.203
0.249
0.291
0.343
Table 4. Ablation study of the components of PS loss on the ETThl and Weather datasets using DLinear as a backbone.
Method
Dlinear
PS
w/o Lourr
w/o Lvar
w/o Lxtan
w/o Patching
w/o Weighting
Metric
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
96
0.367
0.389
0,381
0.402
0.380
0.399
0.372
0.395
0.380
0.401
0,384
0.404
192
0.402
0,411
0.439
0.446
0.440
0.444
0.404
0.414
0.436
0.441
0.403
0.411
ETTH1
336
0.435
0.435
0.443
0.445
0.439
0.486
0.440
0.443
0.442
0.442
0.439
0.440
720
0.463
0.484
0.510
0.519
0.509
0,517
0.474
0.493
0.522
0.528
0.528
0.532
96
0,171
0.222
0.174
0.233
0.172
0.223
0.172
0.223
0.171
0.224
0.172
0.226
192
0.212
0.259
0.222
0.286
0.217
0.271
0.217
0.271
0.220
0,280
0.214
0.268
Weather
336
0.258
0.300
0.261
0.311
0.259
0.303
0.259
0.303
0.263
0.314
0.261
0.307
720
0.323
0.356
0.329
0.369
0.328
0.365
0.328
0.365
0.331
0.372
0.332
0.370
results provided in Appendix I.
The results in Table 5, measured on target datasets with a forecasting length of 192, highlight the consistent advan- tages of PS loss. It outperforms MSE loss in 34 out of 36 scenarios, demonstrating substantial improvements in gen- eralization across diverse datasets and granularities. These gains arise from PS loss's ability to analyze time series at the patch level, incorporating local statistical properties to achieve better structural alignment and enabling models to adapt more effectively to unseen data distributions.
Loss components. Removing any one of the three loss com- ponents leads to noticeable performance degradation across all horizons. This highlights the synergistic relationship between these components in capturing localized structural similarities.
Adaptive patching. To evaluate the importance of patching, we conducted a direct comparison by processing the entire time series without segmentation. This approach signifi- cantly worsens performance, especially for longer horizons, underscoring the value of the patching mechanism in cap- turing fine-grained patterns.
4.7. Forecasting Visualization
We visualize samples from the Weather dataset using iTrans- former as the backbone model. As shown in Figure 3, PS loss significantly improves alignment with the ground truth compared to MSE loss: (a) its predictions closely follow the trajectory of the ground truth, preserving overall trends and patterns. (b) it effectively captures amplitude fluctuations, reflecting the localized variability. (c) reduces biases at the overall level, ensuring better alignment with the central tendency of the ground truth. These results highlight the strengths of PS loss in capturing localized structural patterns. More visualizations are provided in Appendix H.
Dynamic weighting, To evaluate the importance of dy- namic weighting, we replaced the adaptive weighting with a fixed weight of 1.0 for all three loss terms. This static weighting strategy results in performance degradation, demonstrating that dynamic weighting effectively balances each loss term and improves optimization and forecasting.
4.6. Zero-shot Forecasting
To evaluate how PS loss improves generalization to unseen datasets, we conducted zero-shot forecasting experiments. Following prior works (Jin et al., 2024; Chen et al., 2024a), we sequentially use each of ETTh1, ETTh2, ETTml, and ETTm2 as the source dataset, treating the remaining datasets as targets.
4.8. Hyperparameter Sensitivity
Our method introduces two key hyperparameters: PS loss weight À and patch length threshold 8. We evaluate their impact using iTransformer model.
7

---

Patch-wise Structural Loss for Time Series Forecasting
Table 5. Zero-shot forecasting results on ETT datasets. The table reports MSE and MAE values for backbone models trained with and without PS for zero-shot forecasting. The forecasting length is 192.
Models
iTransformer
DLinear
TimeMixer
Loss Function
MSE
+ PS
MSE
+ PS
MSE
+ PS
Source
Target
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
ETTh2
0.379
0.395
0,381
0.394
0.464
0.465
0,356
0.394
0.378
0.391
0.378
0.390
ETTh1
ETTml
0,963
0.611
0.798
0,575
0.737
0,571
0.717
0,550
1.015
0.633
0.923
0.606
ETTm2
0.299
0.353
0.280
0.338
0.383
0.436
0.273
0.353
0.290
0.348
0.285
0.343
ETTh1
0.617
0.538
0.541
0.498
0,446
0,446
0,436
0.437
0.667
0,552
0.645
0.541
ETTh2
ETTml
0.924
0.607
0.915
0.599
0.713
0.551
0.732
0.559
1.081
0.635
0.991
0.619
ETTm2
0.292
0.350
0.291
0.349
0.307
0.388
0.268
0.346
0.296
0.353
0.296
0.352
ETTh
0.718
0.564
0.601
0,516
0.479
0.459
0.472
0.453
0.774
0.576
0.628
0.524
ETTml
ETTb2
0.443
0,436
0.435
0.427
0.375
0.411
0.371
0,407
0,492
0.459
0,466
0,443
RTTm2
0.260
0.314
0,260
0.310
0.244
0.322
0.239
0.315
0.262
0.315
0.257
0.307
ETTh1
0,914
0.637
0.613
0,523
0,542
0.498
0.498
0.470
0.622
0,530
0.590
0,514
ETTm2
ETTh2
0.447
0.445
0.426
0.421
0,385
0.410
0.366
0.397
0.415
0.424
0,412
0.417
ETTml
0.674
0.525
0.562
0.476
0.417
0.420
0.375
0.394
0.659
0.514
0.523
0.460
Tru
MSB
P8
True
MSE
PS
MSC
1.75
120
Fonccaring Valar
1.10
Forcaring Valae
1.50
Porconsting Value
-0.50
1.00
-1.00
0.90
1.00
-150
0.55
0
25
50
75
25
S01
75
25
30
Time Steps
Time Steps
Time Stops
(a) Better Correlation
(b) Better Variance Match
Better Mcan Match
Figure 3. Forecasting visualization comparing PS loss and MSE loss as objective functions. Predictions with PS loss align more closely with the ground truth's statistical properties, yielding results with higher structural similarity.
96
192
336
720
ETTbl
ETTml
limits the maximum patch length to maintain the granular-
0.48
ity of comparison. We evaluate ō E {24, 36, 48, 60} and present the results in Figure 4b. For ETTh1, where the Fourier-basse patch length is smaller than 5, increasing 8 does not affect performance. However, for ETTm1, where the Fourier-based patch length exceeds 6, large 6 results in diminished performance, indicating that overly large patches blur fine-grate -reigraided structural patterns, and proper patch length is crucial for structural alignment.
0.45
MSB
0.47
0.40
0.12
0.41
10.0
-00
2
(a) Robustness of parameter X on ETTml.
(b) Effectiveness of the ō on ETThl and ETTml.
Figure 4. The impact of the hyperparameter on ETThl and ETTml based irransformer. More results on parameter X are shown in Appendix F.
5. Conclusion
In this study, we addressed the limitations of traditional point-wise loss functions by introducing the Patch-wise Structural (PS) loss, a novel method that leverages local statistical properties for more precise structural alignment. By integrating patch-level analysis and dynamic weight- ing strategies, PS loss improves forecasting accuracy and strengthens generalization across multivariate time series models. This innovation establishes new benchmark for accurately modeling complex time series data. In future work, we will focus on developing a new loss function and model that accounts for both local and global structural dynamics within the prediction.
PS loss weight A. The weight À determines the relative im- portance of PS loss in combination with the MSE. We eval- uate a range of E {0.1, 0.3, 0.5, 0.7, 1.0, 3.0, 5.0, 10.0). as depicted in Figure 4a. The results demonstrate that per- formance improves as À increases, peaking at À 3.0. Notably, the performance remains stable across broad range of X, indicating that PS loss is robust and can be easily applied to various datasets and models.
Patch length threshold 5. The patch length threshold
8

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Patch-wise Structural Loss for Time Series Forecasting
Acknowledgments
Huang, H., Chen, M., and Qiao, X. Generative learning for financial time series with irregular and scale-invariant patterns. In The Twelfth International Conference on Learning Representations, 2024a.
This work was supported by the National Natural Sci- ence Foundation of China under Grants 62376194, 61925602, U23B2049 and China Scholarship Council Grant 202406250137.
Huang. Q. Shen, L.,, Zhang, R., Cheng, J., Ding, S., Zhou, Z., and Wang, Y. Hdmixer: Hierarchical dependency with extendable patch for multivariate time series forecasting. In Proceedings of the AAAI Conference Artificial In- telligence, volume 38, pp. 1 12608-12616, 2024b.
Impact Statement
This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
Jia, T., Xie, Z., Sun, Y., Kudrat, D., and Hu, Q- Lightweight channel-wise dynamic fusion model: Non-stationary time series forecasting via entropy analysis. ArXiv, abs/2503.02609, 2025. URL https:
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Patch-wise Structural Loss for Time Series Forecasting
A. Datasets
Our study leverages a diverse set of forecasting datasets to evaluate the effectiveness of our loss function across various domains.
ETT (Electricity Transformer Temperature): The ETT (Zhou et al., 2021) dataset contains seven features of electricity transformer data collected from two separate counties between July 2016 and July 2018. These datasets vary in granularity, with "h" indicating hourly data and "m" indicating 15-minute intervals. The suffixes "I" and "2" refer to two different regions from which the datasets originated.
Weather: The Weather (Zhou et al., 2021) dataset includes 21 meteorological factors recorded every 10 minutes in 2020 at the Max Planck Biogeochemistry Institute's Weather Station.
ECL (Flectricity Consuming Load): The ECL (Li et al., 2019) dataset contains hourly electricity consumption data of 321 clients from 2012 to 2014.
Exchange: The Exchange (Lai al., 2018) dataset consists of daily exchange rates from 1990 to 2016 for eight countries: Australia, the United Kingdom, Canada, Switzerland, China, Japan, New Zealand, and Singapore.
We follow the same data processing and train-validation-test split protocol as previous works (Zhou et al., 2021; Wu et al., 2023a; Liu et al., 2024a). The ETT dataset is divided into training, validation, and testing sets in 12/4/4-monts ratio, while other datasets are split in a 7:1:2 ratio. The details of the datasets are provided in Table 6.
Table Descriptions of the forecasting datasets. "Channel" shows the variate number of cach dataset. "Dataset Size" indicates the size of the (Train, Validation, and Test) split respectively. "Frequency" denotes the sampling interval of time points.
Dataset
Channel
Length
Datset Size
Frequency hour hour
Information
ETThl
7
17420
(8545, 2881, 2881)
Electricity
ETTh2
7
17420
(8545, 2881, 2881)
Electricity
ETTml
7
69680
(34465, 11521, 11521)
15 min
Electricity
ETTm2
7
69680
(34465, 11521, 11521)
15 min
Electricity
Weather
21
52696
(36792, 5271, 10540)
10 min
Weather Electricity
ECL
321
26304
(18317, 2633, 5261)
hour 1 day
Exchange
8
7588
(5120, 665, 1422)
Exchange Rates
B. Backbone Models
We select seven state-of-the-art (SOTA) models from different architectures as backbones. These include Transformer-based models: iTransformer (Liu et al., 2024a) and PatchTST (Nie et al., 2023); MLP-based models: DLinear (Zeng et al., 2023) and TimeMixer (Wang et al., 2024);
model: TimesNet (Wu et al., 2023a); LLM-based models: OFA (Zhou al., 2023) and AutoTimms (Liu et al., 2024b). We use the official implementations of each backbone from their respective repositories and follow their optimal haaeaallyypppaaaa hyperparameter settings for our experiments.
iTransformer*: It embeds the entire series as a single token and applies a Transformer along the channel dimension to explicitly capture channel dependencies.
PatchTST²: It segments time series into patches and applies Transformer along the time dimension to capture long-term temporal dependencies.
DLinear"; It simplifies forecasting by decomposing time series into trend and seasonal components, modeling each separately with linear layers.
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12

---

Patch-wise Structural Loss for Time Series Forecasting
TimeMixer": It captures multi-scale information by decomposing time series into different scales and blending them through linear layers.
TimesNet': It transforms ID time series into 2D tensors to account for multi-periodicity and employs CNNs to capture both inter- and intra-period dependencies.
It embeds time series patches into tokens and leverages pre-trained language models for general time series analysis tasks.
OFA::
: It projects time series into the embedding space of language tokens and autoregressively generates future predictions with arbitrary lengths.
C. Related Work
Patching Mechanism in Time Series Forecasting. In recent years, patch-based approaches have gained significant traction in time series forecasting, offering substantial improvements in prediction performance (Zhang & Yan, 2023; Chen et al., 2024b). By segmenting the input time series into patches, these methods enhance locality and capture comprehensive semantic information that point-level techniques often overlook (Nie et al., 2023; Zhang & Yan, 2023).
For instance, PatchTST (Nie et al., 2023) employs series patching combined with channel-independent (CI) modeling strategy to capture long-term temporal dependencies. Crossformer (Zhang & Yan, 2023) partitions time series into patches and utilizes self-attention mechanisms to model dependencies across both temporal and variable dimensions. TimesNet (Wu et al., 2023a) and PDF (Dai et al., 2024) apply Fast Fourier Transform (FFT) to determine the dominant frequency of the input series, segmenting the data accordingly to ensure patches capture clear periodic patterns. HDMixer (Huang et al., 2024b) adaptively expands patch lengths to enrich boundary information and mitigate semantic incoherence within the series.
While these methods employ patching to extract localized semantic information from input time series, we take novel approach by introducing patching to the prediction series. This enables a localized comparison between prediction and ground truth, providing a more fine-grained assessment of structural similarity.
D. Computational Cost
To evaluate the computational cost introduced by the proposed PS loss, we conducted a theoretical complexity analysis and measured the actual runtime overhead during training.
D.1. Time Complexity Analysis
We analyze the complexity of our proposed PS loss with respect to the forecasting length T, channel number C, and the last hidden layer's dimension d. The overall time complexity comes from three main components:
Fourier-based Adaptive Patching (FAP). The time complexity for this component is dominated by the Fast Fourier Transform (FFT) operation, which has complexity of O(T log 7') for a time series of length T. Since the FFT is applied independently to each of the channels, the total time complexity is O(C T).
Patch-wise Structural Loss (PS). The series is divided into N = - 1 F patches, where P denotes the patch length and the stride S is set - Computing the correlation, variance, and mean for each patch requires P operations. With a total of C N patches, the time complexity of this component is O N P) O T))
Gradient-based Dynamic Weighting (GDW). The gradient computation for each loss term with respect to the output layer weights, which have shape d x T. requires O(d T) operations. Considering all three loss components, the total time complexity of GDW is O(3 d d = (dd T).
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13

---

Patch-wise Structural Loss for Time Series Forecasting
Therefore, the overall time complexity of the PS loss is O(C TlogT + C C C.T++..T)
D.2. Actual Runtime Overhead
To assess the runtime overhead introduced by the PS loss, we measure the training time per epoch (seconds/epoch) for both the standard MSE loss and the proposed PS loss, using the iTransformer model as the backbone. We select three datasets of different scales for the experiment: ETThl (small), Weather (medium), and ECL (large). As shown in Table while PS loss introduces additional computational cost, the increase in training time remains modest across datasets of different sizes. This overhead is justified by the consistent improvements in forecasting accuracy achieved with the PS loss.
Table Runtime Overhead of PS loss. The table reports the running time per epoch (seconds/epoch) for iTransformer model using MSE loss and PS loss the three datasets: ETThI, Weather, and ECL. "Time Inc." represents the additional running time introduced by PS loss.
Datasets
ETTh1
Weather
ECL
Train Time (seppoch)
MSE Time
PS Time
Time Inc.
MSE Time
PS Time
Time Inc.
MSE Time
PS Time
Time Inc.
96
1.99
2.64
0.65
10.25
13.27
3,03
24.03
27.36
3.34
192
1.98
2.73
0.74
10.62
14.17
3.55
24.35
28.23
3.89
336
1.91
2.64
0.73
10.71
14.52
3.81
25.15
30.13
4.98
iTransformer
720
1.95
2.65
0.70
10.96
14.19
3.23
26.55
35.07
8.52
Avg.
1.96
2.66
0.71
10.63
14.04
3.40
25.02
30.20
5.18
E. Additional Evaluation Metrics for PS Loss Performance
To provide a more comprchensive evaluation of the PS loss, we incorporate additional metrics, including Dynamic Time Warping (DTW) (Müller, 2007), Time Distortion Index (TDI) (Frias-Paredess et al., 2016), and Pearson Correlation Coefficient (PCC) (Cohen et al., 2009), to assess the quality of the forecasting results.
E.1. Metric Definitions
DTW computes the distance between two time series by identifying an optimal warping path, allowing non-linear alignments along the time axis emphasize shape similarity (Müller, 2007). Given two time series Y (yo,
Dynamic Time Warping.
E IR and Y {ūo, ú1. ......, . E RT, the DTW distance is defined as:
DTW(Y,8) = min AEA(Y))) (AJ)E
Σ
0,) = Σ d(vi,
(L))A
where .(:.)) denotes a distance measure, typically the squared Euclidean distance. A a warping path consisting of K index pairs ((aa. jo), (f1, (1x-1jk--)) where < iR. T A(Y. Y) denotes the set of all admissible warping paths, and A A(Y,Y) is the optimal path that yields the minimum total distance between aligned time steps. warping path A is considered admissible if it satisfies the following conditions:
Boundary condition: (No, jo) = (0, 0) and (ix-1, jK 1) (T 1,7-11
Monotonicity condition: ix+1 2 is. and ju+1 2 ju. for all k E {0,
Step size condition: (i++1 -x+ - j&) E (),,0), (0, 1), (1, for all E KO -
Time Distortion Index. TDI quantifies the degree of temporal distortion between two sequences (Frías-Paredess et al., 2016). was originally defined as the area between the optimal DTW warping path A and the identity path I = {(1,1)) [-) In our work, we adopt simplified version of TDI proposed in (Le Guen & Thome, 2019). Given two time series Y e R and Y RT and their optimal warping path A", the TDI is calculated as:
(i
TDI(Y,
Σ
7
(DJEAA
14

---

Patch-wise Structural Loss for Time Series Forecasting
Pearson Correlation Coefficient. PCC measures the degree of linear correlation between time series. Given Y (yo, :::--: E TR? and y {ūo: .1.-........ DT } E R², the PCC is calculated as:
PCC(Y,Y) Y)= = ΣT- (ye 7) Ú)
- (y: 0) Vr= (0t D)
where ij and i denote the mean values of Y and Y, respectively.
E.2. Evaluation Results
We evaluated the forecasting performance of the iTransformer model trained with MSE and PS loss using DTW, TDI, and PCC. The results are presented in Table 8. PS loss consistently improves all three shape-aware metrics, indicating better structural alignment. On the ETTh2 dataset, the iTransformer trained with MSE achieves lower DTW scores, reflecting smaller numerical differences after optimal alignment. However, the higher TDI in this case suggests that the alignment involves more extensive temporal warping, indicating greater structural distortion compared to the forecasts generated using PS loss.
Table Additional evaluation metrics for assessing the performance of PS loss. This table reports the performance of the iTransformer model trained with MSE and PS loss, evaluated using three shape-aware metries: DTW. TDI, and PCC.
Model
iTransformer
Metrics
MSR
DTW
TDI
PCC
Loss Functions
MSE
+ PS
MSE
+ PS
MSE
+ PS
MSE
+ PS
96
0.387
0.379
3.743
3.725
2.475
2.278
0.562
0.572
192
0.441
0.428
5.728
5.685
4.549
4.479
0.530
0.539
336
0.491
0.474
8,055
8.042
9.356
8.393
0.496
0.519
ETTh
720
0.509
0.480
11.896
11.844
15.171
12.686
0.469
0.488
Avg
0.457
0.440
7,355
7.324
7.888
6.959
0.514
0.530
96
0.301
0.289
3.256
3.213
5.739
5.207
0.355
0.408
192
0,380
0.373
5,227
5.321
14.124
13.937
0.311
0.353
336
0.424
0.416
7.494
7.644
27.298
24.994
0.286
0.325
ETTh2
720
0.430
0.421
11,586
11,886
51,733
46.683
0.244
0.283
Avg
0.384
0.375
6.891
7.016
24.723
22.705
0.299
0.342
96
0.342
0.326
3.275
3.167
6.199
5.569
0.575
0.604
192
0.383
0.374
4,887
4.796
8.221
7.561
0.556
0.573
ETTml
336
0.418
0.410
6.917
6.815
11.863
10.831
0.530
0.545
720
0.487
0.472
11.195
10,960
23.522
21.562
0.490
0.505
Avg
0.408
0.396
6.568
6.435
12.451
11,381
0.538
0.557
96
0.186
0.175
2.575
2.358
6.338
5.544
0.364
0.425
192
0.254
0.242
4.141
3.906
11.684
10.748
0.337
0.403
ETTm2
336
0.316
0.304
6.167
5.878
25.611
21.580
0.318
0.377
720
0.414
0.401
10.770
10.300
64.242
52.106
0.281
0.341
Avg
0.292
0.281
5.913
5.611
26.969
22.495
0.325
0.387
96
0.176
0.167
2.174
2.107
11,990
11.166
0.344
0.381
192
0.227
0.219
3.725
3.703
22.741
21.400
0.339
0.371
336
0.282
0.274
5.768
5.772
42.628
41.029
0.319
0.344
Weather
720
0.357
0.353
9.975
10.053
88.401
87.777
0.296
0.313
Avg
0.261
0.253
5.410
5.409
41.440
40.343
0.324
0.352
F. More Results on Hyperparameter Sensitivity
We investigate the impact of hyperparameter PS loss weight À on ETTm2 and Weather datasets using iTransformer and DL Dinnear as the backbone models. We set À E (0.1, 0.3, 0.5, 0.7, 10,33.00 5.0, 10.0) for the experiment.
15

---

Patch-wise Structural Loss for Time Series Forecasting
The results show that À increases, the performance of the model improves, with À 5.0 achieving the relatively best results. Moreover, the performance remains robust across wide range of À values tested, which simplifies PS loss application to different datasets and models.
96
336
720
96
336
720
192
336
96
192
336
720
0.3
MSB
0.3
MSB
MSE
MSB
0.25
0.2
0.2
0.20
5.0
10.0
0.0
3.0
5.0
105
3.0
10.
10.0
2
(a) Transformmr as backhone. (left: ETTm2, right: Weather)
(b) iinin as backhone. (left: ETTm2, right: Weather)
Figure 5. Influence PS loss weight on ITransffrmer and DI Dlincar backbone. We plot the change MSE with respect to on the FTTm2 and Weather datasets, using I'Transfommer and DI iinea the backhone model. The performance is robust across wide range of À values.
G. Impact of PS Loss on Generalization
To examine how PS loss affects training dynamics and generalization, we visualize the MSE on both the training and test datasets across all epochs, using MSE loss and PS loss objective functions, as shown in Figure 6. The experiments were conducted on the ETTh1, ETTh2, ETTm1, and ETTm2 datasets, with iTransformer as the backbone model.
Training Error
Tosting Error
Training
Testing
MEE
0.42
MSL
PS
0.42
0.313
0.40
20
MNl
0.2973
0.660
0.513
0.40
0.34
0. 300
0.00
0.200
Epodic
(a) Results on E dataset.
(b) Results on ETTh2 dataset.
Training Error
Testing Earor
Training Ervor
Testing Frror
MBE
MSE
MRE
MBE
0:34
XPNI
0.300
0.34
0.182
0.23
0.100
0.315
0.20
0.178
0.28
0.21
0.176
10
Epodio
Epocka
Epoclo
Epodt
(c) Results on ETTml dataset.
(d) Results on ETTm2 dataset.
Figure 6. Training and testing MSE loss curves across all training epochs for the irransformer model trained with MSE loss and PS loss on the ETT datasets (left: Training errors, right: Testing errors). The model trained with PS loss shows higher training errors but achieves lower testing errors, highlighting its effectiveness enhancing generalization and mitigating overfitting.
From Figure 6, we observe a consistent trend across both datasets. During training, models optimized solely with MSE loss achieve lower training error per epoch compared to those optimized with PS loss. However, testing data, models trained with MSE loss exhibit higher testing error per epoch compared to those trained with PS loss. These observations suggest that models trained with MSE loss focus solely on the point-wise error, resulting in smaller training losses, but fail to generalize effectively to the testing data. In contrast, PS loss mitigates overfitting by encouraging the model to focus on local structural similarity in addition to pise accuracy, highlighting its effectiveness in improving generalization and forecasting accuracy.
16

---

Patch-wise Structural Loss for Time Series Forecasting
H. Forecasting Visualization
To evaluate the quality predictions using MSE loss and our proposed PS loss, we visualize samples from the ETTml, ETTm2, and Weather datasets with iTransformer the backbone model. Predictions generated with PS loss exhibit enhanced alignment with the ground truth, accurately capturing directionality, fluctuation dynamics, and average levels while preserving localized patterns and structural details.
Trse
MSE
PS
Tioe
MSE
PS
Time
MSE
1.00
0.50
Flrncertig Vabe
0.10
Ftrcesstios Valee
1.00
Freeastia Vile
1.00
0.50
0.00
0.00
0.00
0.00
1..00
Time
Time
(a) Results on ETTml dataset.
Troe
PS
Tire
Trme MSSI
Tree
MSE
0.25
poreciing Vahue
0.25
Forecosting Valae
0.50
0.00
0.00
0.00
Famouting
0.25
0.25
0.25
0.00
0.50
-2.50
20.25
30
23
30
Time Staps
Time Steps
Time Steps
Time Steps
(b) Results on ET dataset.
Tree
MSS
PS
Thu
MSE
1rne
MIS
adagy Aagghh Callhheerrn my M
0.20
Porecattngg Vahe
Forecatting Valee
1.00
-1.00
1.000
Poncueting
-3.10
0.00
0..00
-3.30
0.90
-0.20
Mana Arl
www
-1.30
00.00
23
30
75
0.50
25
30
30 Time Stegs
Time Steps
Time Staps
Timse Steps
(c) Results on Weather datasct.
Figure 7. Forccasting visualization comparing PS loss and MSE loss objective functions.
I. More Results on Ablation Study
We conduct additional ablation studies on the DLinear and iTransformer backbones using the ETTh1, ETTh2, ETTml, and ETTm2 datasets further demonstrate the necessity of each component, shown in Table 9 and Table 10.
Loss components. Omitting any of the three loss components results in decline in performance, underscoring the unique and complementary roles each loss plays in capturing localized structural similarities.
Adaptive Patching. Comparing time series globally leads to performance degradation, especially for longer forecasting horizons. This highlights the importance point-wit comparison in capturing nuanced local variations in statistical properties.
Dynamic weighting. Replacing the GDW strategy with fixed weighting leads to performance degradation, highlighting the importance of GDW in ensuring the balanced utilization of each loss term for effective model training.
17

---

Patch-wise Structural Loss for Time Series Forecasting
Table 9. Ablation study of the components of PS loss the ETT datasets using DLinear hackhone.
Method
Dlinear
PS
w/o Corr
w/o Lva
w/o Laann
w/o Patching
w/o Weighting
Metric
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
96
0.367
0.389
0.381
0.402
0,380
0.399
0.372
0.395
0,380
0.401
0.384
0.404
192
0.402
0.411
0.439
0.446
0.440
0.444
0.404
0.414
0.436
0.441
0.403
0.411
336
0.435
0.435
0.443
0.445
0.439
0.486
0.440
0.443
0.442
0.442
0.439
0.440
ETThl
720
0.463
0.484
0.510
0.519
0.509
0.517
0.474
0.493
0.522
0.528
0.528
0.532
Avg
0.417
0.430
0.443
0.453
0.442
0.462
0.423
0.436
0.445
0.453
0.439
0,447
96
0.280
0.341
0.289
0.351
0.281
0.342
0.287
0.351
0.280
0.341
0.283
0.345
192
0.358
0.396
0.381
0.415
0.359
0.396
0.397
0.428
0.386
0.399
0.371
0.404
336
0.435
0.450
0.455
0.466
0.457
0.462
0.477
0.477
0.438
0.450
0.439
0.452
ETTh2
720
0.597
0.540
0.720
0.599
0.673
0.574
0.747
0.612
0.663
0.583
0.679
0.580
Avg
0.417
0.432
0.461
0.458
0.443
0.443
0.477
0.467
0.442
0.443
0.443
0.445
96
0.296
0.339
0.300
0.343
0.296
0.338
0.301
0.345
0.304
0.347
0.297
0.340
192
0.333
0.362
0.334
0.364
0.332
0.359
0.335
0.366
0.341
0.373
0.332
0.361
336
0.365
0,380
0.370
0.387
0.368
0.381
0.369
0.386
0.371
0.387
0.367
0.381
ETTml
720
0.419
0.413
0.424
0.420
0.422
0.415
0.423
0.420
0.423
0.419
0.422
0.415
Avg
0.353
0.374
0.357
0.378
0.355
0.373
0.357
0.379
0.360
0.381
0.355
0.374
96
0.163
0.251
0.170
0.265
0.163
0.252
0.171
0.268
0.164
0.254
0.164
0.255
192
0.222
0.296
0,234
0.308
0.222
0.296
0.238
0.320
0.223
0.299
0.225
0.301
336
0.277
0.332
0.282
0.342
0.283
0.342
0.295
0.353
0.286
0.347
0.289
0.350
ETTm2
720
0.377
0.397
0.414
0.432
0.398
0.418
0.451
0.453
0.393
0.415
0.410
0.426
Avg
0.260
0.319
0.275
0.337
0.267
0.327
0.289
0.348
0.266
0.329
0.272
0.333
Table 10. Ablation study of the components of PS loss on the ETT datasets using iTransformer as backbone. "iTranss" refers to the
iTransformer model.
Method
iTrans
PS
w/o Loorr
w/o Lvar
w/o Litenn
w/o Patching
w/o Weighting
Metric
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
96
0.379
0.396
0.381
0.399
0.385
0.400
0.382
0.397
0.383
0.400
0.380
0.396
192
0.428
0.424
0.441
0.433
0.436
0.429
0.431
0.426
0.435
0.429
0.430
0.425
336
0.474
0.453
0.486
0.458
0.499
0.466
0.471
0.446
0.483
0.455
0.485
0.458
E
720
0.480
0.478
0.515
0.496
0.521
0.498
0.477
0.471
0.511
0.492
0.532
0.505
Avg
0.440
0.438
0.456
0.447
0.460
0.448
0.440
0.435
0.453
0.444
0.457
0,446
96
0.289
0.340
0.294
0.344
0.296
0.343
0.292
0.343
0.290
0.340
0.292
0.341
192
0.373
0.391
0.374
0.393
0.375
0.392
0.379
0.395
0.376
0.393
0.374
0.392
336
0.416
0.428
0.423
0.431
0.421
0.429
0.424
0.431
0.419
0.428
0.419
0.428
ETTh2
720
0.421
0.441
0.439
0.453
0.425
0.442
0.426
0.443
0.423
0.441
0.423
0.441
Avg
0.375
0.400
0.383
0.405
0.379
0.402
0.380
0.403
0.377
0.401
0.377
0.401
96
0.326
0.360
0.334
0.365
0.328
0.360
0.350
0.375
0.343
0.374
0.328
0.361
192
0.374
0.384
0.382
0.387
0.374
0.381
0.384
0.390
0.385
0.394
0.375
0.383
336
0.410
0.406
0.419
0.412
0.412
0.406
0.417
0.412
0.420
0.416
0.411
0.407
ETTml
720
0.472
0.440
0.482
0.446
0.479
0.443
0.483
0.449
0.488
0.455
0.477
0.444
Avg
0.396
0.397
0.404
0.402
0.398
0.397
0.408
0.407
0.409
0.410
0.398
0.399
96
0.175
0.254
0.179
0.259
0.177
0.256
0.183
0.266
0.182
0.265
0.177
0.258
192
0.242
0.299
0.248
0.306
0.244
0.300
0.247
0.306
0.248
0.306
0.245
0.302
336
0.304
0.338
0.309
0.343
0.308
0.340
0.308
0.344
0.310
0.344
0.309
0.343
ETTm2
720
0.401
0.394
0.409
0.400
0.405
0.397
0.410
0.401
0.410
0.400
0.410
0.399
Avg
0.281
0.321
0.286
0.327
0.283
0.323
0.287
0.329
0.287
0.329
0.285
0.325
18

---

Patch-wise Structural Loss for Time Series Forecasting
J. More Experiments on Zero-shot Forecasting
Different Forecasting Lengths. We conducted zero-shot forecasting experiments on the iTransformer model with forecasting lengths of 96, 336, and 720. As results shown in Table 11, PS loss consistently enhances forecasting accuracy across all horizons, yielding improvements in 33 out of 36 settings. These findings underscore the effectiveness of PS loss in improving zero-shot forecasting performance under varying prediction lengths.
Table 11. PS loss zero-shot performance on different forecasting lengths using iTransformer as the backhone model. The forecasting lengths are (96, 336, 720).
Forecasting Length
96
336
720
Loss Function
MSE
PS
MSE
PS
MSE
PS
Source
Target
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
ETTh2
0.296
0.344
0.296
0.342
0.424
0.431
0.422
0.425
0.429
0.444
0.432
0.445
ETThl
ETTml
0,963
0.611
0.821
0.570
0.845
0.597
0.949
0.616
0.866
0.610
0.828
0.601
ETTm2
0.238
0.321
0.222
0.307
0.342
0.375
0.353
0.382
0,442
0.428
0.436
0.424
ETTh1 0.575
0,515
0.465
0.455
0.656
0,559
0.607
0.534
0.708
0.603
0.626
0,563
BTTh2
ETTml
1,066
0.641
0.927
0.597
0.918
0.615
0.887
0.603
0.906
0.629
0.869
0.614
ETTm2
0.237
0.321
0.228
0.313
0.349
0.381
0.347
0.380
0.445
0.430
0.441
0.428
ETThl 0.708
0.556
0.617
0.521
0.731
0.576
0.626
0.532
0.743
0.599
0.633
0.559
ETTml
ETTh2
0.351
0.390
0.342
0.380
0.490
0.469
0.473
0.457
0.486
0.476
0.467
0.463
ETTm2
0,202
0.279
0.198
0.272
0.322
0.352
0.319
0.347
0.422
0,407
0.419
0.403
ETTh1 0.833
0.603
0.568
0.497
1.119
0.716
0.642
0.539
1.202
0.756
0.623
0.544
ETTm2
ETTh2 0,353
0.395
0.341
0.373
0.517
0,491
0,461
0.451
0.528
0.50S
0.456
0.460
ETTml
0.679
0.520
0.465
0.425
0.740
0.557
0.520
0.464
0.880
0.615
0.584
0.500
LLM-based Models. To further assess the generalization ability of the proposed PS loss, we extend our zero-shot forecasting experiments to LLM-bbseed models, including OFA (Zhou et al., 2023), AutoTTmmss (Liu et al., 2024b). and Time-LLM (Jin et al., 2024), with the forecasting length set to 96. As shown in Table 12, incorporating PS loss consistently improves forecasting performance in most settings. These results demonstrate the compatibility of PS loss with LLM-based models and highlight its ability to enhance generalization in zero-shot forecasting.
Table 12, PS loss performance M.M--ddeed models for the zero-shot forecasting task. The backbones are OFA and Autotimes, with forecasting length of 96.
Models
OFA
Auto Times
TiMe-LLM
Loss Function
MSE loss
PS loss
MSE loss
PS loss
MSE loss
PS loss
Source
Target
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
ETTh2
0.289
0.347
0.292
0.343
0.303
0.362
0.305
0.363
0.287
0.349
0.286
0.346
ETTh1
ETTml
0.723
0.532
0.742
0.534
0.728
0.552
0.724
0.549
0.747
0.566
0.706
0.538
ETTm2
0.219
0.311
0.216
0.303
0.231
0.323
0.233
0.325
0.225
0.312
0.224
0.310
6TTh1
0.413
0.419
0.412
0.419
0.469
0.470
0.418
0.432
0.487
0.466
0.422
0.425
ETTh2
ETTml
0.762
0.540
0.742
0.534
0.987
0.618
0.842
0.573
0.814
0.574
0.706
0.533
ElTm2
0.209
0.302
0.208
0.299
0.248
0.327
0.236
0.322
0.216
0.308
0.222
0.308
6TTh1
0.495
0..74
0.493
0.472
0.541
0.497
0.527
0.488
0.530
0.484
0.502
0.468
ETTml
ETTh2
0.331
0.376
0.333
0.377
0.347
0.395
0.324
0.378
0.320
0.372
0.316
0.368
H17m2
0.179
0.262
0.180
0.262
0.190
0.276
0.185
0.271
0.195
0.276
0.197
0.277
ETTh1
0.511
0).455
0.469
0.460
0.544
0.508
0.500
0.477
0.549
0.497
0.471
0.453
ETTm2
ETTh2
0.306
0.364
0.297
0.351
0.298
0.362
0.294
0.354
0.311
0.369
0.294
0.352
ElTml
0.411
0.408
0.359
0.380
0.445
0.429
0.373
0.390
0.411
0.416
0.357
0.378
19


Reviews:
[{"review_id": "cld4U7LNmJ", "reviewer": "Reviewer_K3Dy", "summary": "The authors propose a novel patch-wise structural (ps) loss, which is designed to enhance structural alignment by comparing time series at the patch level. By leveraging local statistical properties, e.g., correlation, variance, and mean, PS loss captures nuanced structural discrepancies overlooked by traditional point-wise loss. Experiments demonstrate that the PS loss can improve the performance of the state-of-the-art models across diverse real-world datasets.\n\n## update after rebuttal\nI support for accept.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you very much for your valuable feedback. Below are our responses to your concerns and suggestions.\n# [Q1] PS loss on ultra-long-term and short-term forecasting \nWe evaluated PS loss on **ultra-long-term (T = {1080, 1440, 1800, 2160})** and **short-term (T = {12, 24, 48})** forecasting tasks using iTransformer and DLinear. We report averaged MSE results. Please refer to [Table 1](https://anonymous.4open.science/r/PS_Re/T1.pdf) and [Table 2](https://anonymous.4open.science/r/PS_Re/T2.pdf) for full results.\n\n- **Ultra-long-term**: MSE reduced by **7.38% (iTransformer)** and **11.01% (DLinear)**.\n\n|Model|iTransformer||Dlinear||\n|-|-|-|-|-\n|Loss Function|MSE|+PS|MSE|+PS\n|ETTh1|0.753|**0.693**|0.696|**0.628**\n|ETTh2|0.545|**0.494**|1.241|**1.127**\n|ETTm1|0.577|**0.536**|0.487|**0.474**\n|ETTm2|0.480|**0.466**|0.557|**0.463**\n|Imp.|-|**7.38%**|-|**11.01%**\n\n- **Short-term**: MSE reduced by **3.43% (iTransformer)** and **1.60% (DLinear)**.\n\n|Model|iTransformer||Dlinear||\n|-|-|-|-|-\n|Loss Function|MSE|+PS|MSE|+PS\n|PEMS03|0.110|**0.107**|0.239|**0.235**\n|PEMS04|0.105|**0.101**|0.283|**0.279**\n|Imp.|-|**3.43%**|-|**1.60%**\n\nThese results demonstrate the effectiveness of PS loss across both **short-term** and **ultra-long-term** forecasting tasks.\n\n# [Q2] Comparison with Ada-MSHyper and FAN\nThe contributions of Ada-MSHyper and FAN differ from our PS loss, as they address distinct challenges:\n- **Ada-MSHyper** introduces a **hypergraph transformer** with a **graph constraint loss** to enhance multi-scale interaction modeling through hypergraph learning.\n- **FAN** proposes a frequency-based adaptive **normalization method** to address both trend and seasonal non-stationary patterns.\n- **PS (Ours)** presents a novel **loss function** that enhances structural alignment between predictions and ground truth via patch-wise statistical metrics.\n\nWe also combined PS loss with both methods. Please refer to [Table 3](https://anonymous.4open.science/r/PS_Re/T3.pdf) and [Table 4](https://anonymous.4open.science/r/PS_Re/T4.pdf) for the full results.\n- **Ada-MSHyper + PS.** PS loss improves the average performance by **8.28% (MSE) and 4.68% (MAE)**.\n\n|Method|Ada-MSHyper||Ada-MSHyper+PS||\n|-|-|-|-|-\n|Dataset|MSE|MAE|MSE|MAE\n|ETTh1|0.137|0.262|**0.132**|**0.254**\n|ETTh2|0.107|0.231|**0.105**|**0.227**\n|Imp.|-|-|**8.28%**|**4.68%**\n\n- **FAN + PS.** When using DLinear as the backbone, PS loss further improves the average performance of FAN by **2.07% (MSE) and 2.31% (MAE)**.\n\n|Method|Dlinear+FAN||Dlinear+FAN+PS||\n|-|-|-|-|-|\n|Dataset|MSE|MAE|MSE|MAE|\n|ETTh1|0.444|0.485|**0.439**|**0.479**\n|ETTh2|0.137|0.262|**0.132**|**0.254**\n|Imp.|-|-|**2.07%**|**2.31%**\n\nThese results demonstrate that while **FAN and Ada-MSHyper focus on different aspects, PS loss can still further improve their performance** by enhancing the structural alignment of the forecasted series.\n\n# [Q3] PS loss on LLM-based models for zero-shot forecasting\nWe conducted zero-shot forecasting experiments with LLM-based models: OFA, AutoTimes, and Time-LLM. PS loss improved forecasting accuracy with average MSE reductions of **2.07% (OFA)**, **6.33% (AutoTimes)**, and **7.29% (Time-LLM)**. Please refer to [Table 5](https://anonymous.4open.science/r/PS_Re/T5.pdf) for the full results.\n|Model|OFA||AutoTimes||Time-LLM||\n|-|-|-|-|-|-|-\n|Loss Function|+MSE|+PS|+MSE|+PS|+MSE|+PS\n|ETTh1→ETTh2/m1/m2|0.410|**0.417**|0.421|**0.421**|0.420|**0.405**\n|ETTh2→ETTh1/m1/m2|0.461|**0.454**|0.568|**0.499**|0.506|**0.450**\n|ETTm1→ETTh1/h2/m2|0.335|**0.336**|0.359|**0.346**|0.349|**0.338**\n|ETTm2→ETTh1/h2/m1|0.411|**0.359**|0.445|**0.373**|0.424|**0.374**\n|Imp|-|**2.07%**|-|**6.33%**|-|**7.29%**\n\n\n\n# [Q4] Time complexity analysis of PS loss\nWe analyze the time complexity of PS loss with respect to the forecast length $T$, number of channels $C$, and hidden dimension $d$. The overall complexity arises from three main components:\n- **Fourier-based Adaptive Patching (FAP):** The complexity of this component is dominated by the Fast Fourier Transform (FFT), which is $O(T\\log T)$ per channel. Since FFT is applied to each of the $C$ channels, the total time complexity is $O(C\\cdot T \\log T)$.\n- **Patch-wise Structural Loss (PS):** The series is split into $N \\approx \\frac{2T}{P}$ patches, where $P$ is the patch length. Calculating correlation, variance, and mean over each patch requires $O(P)$ operations. Given $C \\cdot N$ patches, the total complexity becomes $O(C \\cdot N \\cdot P) = O(C\\cdot T)$.\n- **Gradient-based Dynamic Weighting (GDW):** The gradient computation for each loss component w.r.t. the model output has shape $d \\cdot T$, leading to a complexity of $O(d \\cdot T)$.\n\nTherefore, the overall time complexity of PS loss is **$O(C \\cdot T \\log T + C \\cdot T + d \\cdot T)$**.\n# [Q5] Performance gains in terms of percentage improvements\nWe now report percentage improvements throughout the paper and summarize them in [updated Table](https://anonymous.4open.science/r/PS_Re/T7.pdf).", "claims_and_evidence": "The effectiveness of different module designs has been validated by the experimental results.", "methods_and_evaluation_criteria": "The proposed methods and evaluation criteria are appropriate for time series forecasting.", "theoretical_claims": "I have checked the correctness of the proofs for the theoretical claims.", "experimental_designs_or_analyses": "The manuscript has some weaknesses in the experiments. Please see Questions for details.", "relation_to_broader_scientific_literature": "The introduce of patch-wise structural loss is beneficial for time series forecasting.", "essential_references_not_discussed": "Some important baselines  e.g., Ada-MSHyper [1], and FAN [3], need to be compared. Please see Questions 1 and 2 for details.\n\n[1] Shang Z, Chen L, Wu B, et al. Ada-MSHyper: adaptive multi-scale hypergraph transformer for time series forecasting. NeurIPS, 2024.\n\n[3] Ye W, Deng S, Zou Q, et al. Frequency Adaptive Normalization For Non-stationary Time Series Forecasting. NeurIPS, 2024.", "other_strengths_and_weaknesses": "1.The paper presents a notable innovation by exploring the integration of patch-wise structural loss into time series forecasting, a direction scarcely addressed by existing methodologies.\n\n2.The organization of this paper is clear and the paper is well written.", "other_comments_or_suggestions": "No", "questions_for_authors": "1. As the proposed loss function is specifically designed for time series forecasting, the comparative experiments should not only focus on long-term time series forecasting but also encompass other experimental settings, e.g., short-term time series forecasting and ultra-long-term time series forecasting, as mentioned by existing methods [1, 2].\n\n2. Since some latest methods [1, 3] also improve model performance by introducing constraints or loss functions, as a loss specifically designed for time series, the authors should elaborate on the differences between their proposed loss function and these existing designs in the related work Section. In addition, more comparative experiments should be conducted to validate the effectiveness of the proposed loss functions against these advanced loss functions.\n\n3. In Section 4.6, the authors demonstrate that the PS loss can improve generalization to unseen datasets. Given that LLMs have also been proven to exhibit strong generalization capabilities under zero-shot settings [4, 5, 6], it is suggested to study whether the PS loss can further enhance the performance of LLMs under zero-shot settings. The authors should validate this through additional experiments.\n\n4.The design of the PS loss appears to be somewhat complex. The authors are suggested to include time complexity analysis.\n\n5. To provide an intuitive understanding of the performance improvements brought by the proposed loss, the authors should explicitly list the performance gains in terms of percentage improvements.\n[1]Shang Z, Chen L, Wu B, et al. Ada-MSHyper: adaptive multi-scale hypergraph transformer for time series forecasting. NeurIPS, 2024.\n[2]Jia Y, Lin Y, Hao X, et al. WITRAN: Water-wave information transmission and recurrent acceleration network for long-range time series forecasting. NeurIPS, 2023.\n[3]Ye W, Deng S, Zou Q, et al. Frequency Adaptive Normalization For Non-stationary Time Series Forecasting. NeurIPS, 2024.\n[4]Zhou T, Niu P, Sun L, et al. One fits all: Power general time series analysis by pretrained LM. NeurIPS, 2024.\n[5]Liu Y, Qin G, Huang X, et al. Autotimes: Autoregressive time series forecasters via large language models. NeurIPS, 2024.\n[6]Jin M, Wang S, Ma L, et al. Time-LLM: Time Series Forecasting by Reprogramming Large Language Models. ICLR, 2024.", "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": "p2PO6SSM4q", "reviewer": "Reviewer_YfmW", "summary": "Most previous time series forecasting models use MSE as the loss function, which treats each time step independently and neglect the structural dependency among steps. To fill the gap, this work proposes Patch-wise Structural (PS) Loss. PS Loss first splits target series into patches with patch size determined by FFT. Then correlation, variance and mean losses are computed within each patch and averaged. A gradient-based dynamic weighting mechanism is used to balance the weights of three losses during training. Experiments on real-world datasets show that PS loss can boost forecasting accuracy of both traditional and LLM-based models.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you very much for your valuable feedback. Below are our responses to your concerns and suggestions.\n# [M1] PS loss performance on non-periodic targets\nWhen there is no clear periodic pattern in the target series, the dominant frequency—i.e., the one with the highest amplitude in the FFT spectrum—does not necessarily correspond to a true periodic component in the data. Instead, it typically falls into one of two categories:\n- **Short period ($p<\\delta$)**: This often results from high-frequency components such as local fluctuations or noise. In this case, Equation (3) yields a short patch length, which still allows the model to focus on finer-grained local structure. \n- **Long period ($p>\\delta$)**: This typically corresponds to low-frequency background components or weak global trends. In this case, the patch length will be capped at $\\delta$  to prevent excessively large patches that could hinder fine-grained comparisons.\n\nThis design allows PS loss to adapt to both periodic and non-periodic series, using frequency content to guide patch granularity, while $\\delta$ prevents overly large patches. On the Exchange dataset, which lacks a clear periodic pattern, PS loss still improved MSE by **6.43%** on DLinear, demonstrating its effectiveness.\n# [M2] Purpose of mean loss refinement\nThe purpose behind this design is to **first focus on aligning the shape of the series, and then gradually increase attention to the value offset**. As correlation and variance alignment improve during training (indicated by increasing $c$, $v$), the model allocates more weight to $L_{mean}$ to refine value-level offsets. Ablation on iTransformer and TimeMixer (ETTh1) confirms its effectiveness.\n|Method|iTrans*+PS||W/o c&v||TimeM*+PS||W/o c&v||\n|-|-|-|-|-|-|-|-|-\n|Metric|MSE|MAE|MSE|MAE|MSE|MAE|MSE|MAE\n|96|**0.379**|**0.396**|0.380|0.396|**0.366**|**0.392**|0.368|0.391\n|192|**0.428**|**0.424**|0.429|0.425|0.421|0.421|**0.418**|**0.420**\n|336|**0.474**|**0.453**|0.480|0.458|**0.489**|**0.453**|0.498|0.457\n|720|**0.480**|**0.478**|0.505|0.492|**0.474**|**0.463**|0.480|0.464\n|Avg|**0.440**|**0.438**|0.448|0.443|**0.438**|**0.432**|0.441|0.433\n# [M3] Additional metrics for structural evaluation\nBeyond MSE/MAE, we include additional shape-aware metrics: **DTW, TDI, and PCC**, using iTransformer for evaluation (Please see Reviewer Pd9N [W1] for metric details). **PS loss consistently improves all three metrics**, indicating better structural alignment ([Table 6](https://anonymous.4open.science/r/PS_Re/T6.pdf)).\n|Metrics|DTW||TDI||PCC||\n|-|-|-|-|-|-|-\n|Loss|MSE|+PS|MSE|+PS|MSE|+PS\n|ETTh1|7.355|**7.324**|7.888|**6.959**|0.514|**0.530**\n|ETTh2|**6.891**|7.016|24.723|**22.705**|0.299|**0.342**\n|ETTm1|6.568|**6.435**|12.451|**11.381**|0.538|**0.557**\n|ETTm2|5.913|**5.611**|26.969|**22.495**|0.325|**0.387**\n|Weather|5.410|**5.409**|41.440|**40.343**|0.324|**0.352**\n# [E4] Time complexity analysis of PS loss\n### **1. Theoretical time complexity analysis**\nWe analyze the time complexity of PS loss with respect to the forecast length $T$, number of channels $C$, and hidden dimension $d$. The overall complexity arises from three main components:\n- **Fourier-based Adaptive Patching (FAP):** The complexity of this component is dominated by the Fast Fourier Transform (FFT), which is $O(T\\log T)$ per channel. Since FFT is applied to each of the $C$ channels, the total time complexity is $O(C\\cdot T\\log T)$.\n- **Patch-wise Structural Loss (PS):** The series is split into $N \\approx \\frac{2T}{P}$ patches, where $P$ is the patch length. Calculating correlation, variance, and mean over each patch requires $O(P)$ operations. Given $C\\cdot N$ patches, the total complexity becomes $O(C\\cdot N\\cdot P) = O(C\\cdot T)$.\n- **Gradient-based Dynamic Weighting (GDW):** The gradient computation for each loss component w.r.t. the model output has shape $d\\cdot T$, leading to a complexity of $O(d\\cdot T)$.\n\nTherefore, the overall time complexity of PS loss is **$O(C\\cdot T\\log T+C\\cdot T+d\\cdot T)$**.\n### **2. Actual run time overhead**\nWe report the empirical runtime cost by measuring **seconds per epoch** using **iTransformer** across three datasets: **ETTh1 (small), Weather (medium), and ECL (large)**:\n|Dataset|MSE|PS|Time Increase\n|-|-|-|-\n|ETTh1|1.96|2.66|0.71\n|Weather|10.63|14.04|3.40\n|ECL|25.02|30.20|5.18\n\nDespite the added cost, **the runtime increase is modest and justified by performance gains**.\n# [E5] PS loss performance without MSE\nResults show that PS loss alone yields comparable accuracy to MSE+PS, demonstrating its effectiveness as a **standalone optimization objective** ([Table 8](https://anonymous.4open.science/r/PS_Re/T8.pdf)).\n|Model|iTransformer||TimeMixer||\n|-|-|-|-|-|\n|Loss|MSE+PS|PS Only|MSE+PS|PS Only|\n|ETTh1|0.440|**0.439**|0.437|**0.429**\n|ETTh2|**0.375**|0.380|0.369|**0.364**\n|ETTm1|**0.396**|**0.396**|**0.375**|0.377\n|ETTm2|**0.281**|0.282|**0.270**|0.274\n|Weather|**0.253**|**0.253**|0.243|**0.242**", "claims_and_evidence": "Claims are well supported.", "methods_and_evaluation_criteria": "Overall, the methods and evaluation are convincing. I have a few minor questions or concerns as follows:\n\n1. PS loss utilizes FFT to detect the period in the target series for patching. What if there is no periodic pattern in the target? How does PS loss perform on such datasets (considering most datasets used in experiments have obviously daily periods)?\n2. In gradient-based dynamic weighting, why does mean loss require further adjustment by Equation 12 among three losses? What is the purpose behind this design, and has there been an ablation study conducted on it?\n3. PS loss focuses on structural dependency, but the evaluation metrics are still point-wise MSE and MAE, which may not effectively measure structural consistency. The authors should not be criticized for this, as they are standard metrics. However, I would still like to inquire whether there are other candidate metrics (like DTW) that could better reflect structural information.", "theoretical_claims": "Not applicable, no new theoretical claims is proposed.", "experimental_designs_or_analyses": "1. Considering that FFT brings additional computation cost, how is the efficiency of PS loss w.r.t different forecasting lengths and channel numbers?\n2. Can PS loss along without MSE be used as the loss function? How does it perform?", "relation_to_broader_scientific_literature": "The primary objective of this paper is to enhance time series forecasting, with outcomes that can be effectively applied across diverse downstream domains", "essential_references_not_discussed": "Not applicable, references are generally comprehensive.", "other_strengths_and_weaknesses": "Overall, this is a commendable work. The topic is significant, as most recent studies have primarily focused on backbone designs, leaving loss functions relatively underexplored. The presentation is clear and easy to follow. Should the authors adequately address my concerns, I would be happy to raise the score.", "other_comments_or_suggestions": "See my comments and suggestions above.", "questions_for_authors": "See my questions above.", "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": "KqgHLLXcts", "reviewer": "Reviewer_jewV", "summary": "This paper proposes the Patch-wise Structural (PS) loss function for time series forecasting. The PS loss improves the alignment of local statistical properties (correlation, variance, and mean), addressing the limitations of traditional point-wise loss functions like MSE. By incorporating patch-level analysis, PS loss enhances the ability to model complex temporal structures. Extensive experiments on 7 real-world time series datasets demonstrate that PS loss significantly outperforms traditional methods, improving forecasting accuracy across various models, including LLM-based forecasting models.\n\n## update after rebuttal\nI support for accept", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you very much for your valuable feedback. Below are our responses to your concerns and suggestions.\n\n# [E1] Comparison between GDW and grid-search\nTo evaluate GDW against a traditional grid search for selecting loss weights, we conducted experiments on the ETTh1 dataset using iTransformer. Both methods use the same overall PS loss weight $\\lambda$. For grid search, coefficients $\\alpha$, $\\beta$, and $\\gamma$ were chosen from {0.3, 0.5, 0.7, 1.0}, totaling **64 runs per prediction length**. We report the best and average performance from grid search for comparison:\n\n|Method|GDW||Grid Search (Best)||Grid Search (Average)||\n|-|-|-|-|-|-|-|\n|Metric|MSE|MAE|MSE|MAE|MSE|MAE|\n|96|**0.379**|**0.396**|0.380|**0.396**|0.385|0.398|\n|192|**0.428**|**0.424**|**0.428**|**0.424**|0.432|0.426|\n|336|0.474|0.453|**0.473**|**0.450**|0.483|0.456|\n|720|**0.480**|**0.478**|0.483|0.479|0.513|0.494|\n|Avg|**0.440**|0.438|0.441|**0.437**|0.453|0.444|\n\n**GDW achieves performance comparable to the best grid-searched results, while avoiding exhaustive tuning**. Moreover, it reflects the intuition that the weights of correlation, variance, and mean should evolve dynamically during training to maintain balanced attention across all three loss terms, which static coefficients cannot assure.\n\n# [E2] Visualization of loss weights generated by GDW\nWe visualize the evolution of weights generated by GDW in [Figure 1](https://anonymous.4open.science/r/PS_Re/F1.pdf) and made the following observations:\n\n- **Weight Range:** The weights for correlation, variance, and mean have different ranges, reflecting the inherent variation in their gradient magnitudes, which highlights the need for adaptive balancing.\n- **Weight Evolution:** Correlation weight tends to decrease, while variance and mean weights increase. This does not imply shifting focus, but rather ensures equilibrium among components, allowing structural alignment to be preserved during optimization. \n\nThis confirms that GDW adaptively balances multiple objectives throughout training, improving stability and convergence.\n\n# [W1] Hyperparameter settings\nThe PS loss weight $\\lambda$ is selected from {0.1, 0.3, 0.5, 0.7, 1.0, 2.0, 3.0, 5.0, 10.0}. The patch length threshold $\\delta$ is chosen from {24, 48}.\n\n# [Q1] PS loss as a standalone objective\nResults show that PS loss alone yields comparable accuracy to MSE+PS, demonstrating its effectiveness as a **standalone optimization objective** ([Table 8](https://anonymous.4open.science/r/PS_Re/T8.pdf)).\n|Model|iTransformer||TimeMixer||\n|-|-|-|-|-|\n|Loss|MSE + PS|PS Only|MSE+PS|PS Only|\n|ETTh1|0.440|**0.439**|0.437|**0.429**\n|ETTh2|**0.375**|0.380|0.369|**0.364**\n|ETTm1|**0.396**|**0.396**|**0.375**|0.377\n|ETTm2|**0.281**|0.282|**0.270**|0.274\n|Weather|**0.253**|**0.253**|0.243|**0.242**\n\n# [Q2] PS loss complexity on large datasets\nWe report the empirical runtime cost by measuring the average **seconds per epoch** during training using **iTransformer** across three datasets: **ETTh1 (small), Weather (medium), and ECL (large)**:\n|Dataset|MSE|PS|Time Increase\n|-|-|-|-\n|ETTh1|1.96|2.66|0.71\n|Weather|10.63|14.04|3.40\n|ECL|25.02|30.20|5.18\n\nDespite the added cost, **the runtime increase is modest and justified by performance gains**.\n\n# [Q3] Zero-shot performance across forecasting lengths\n\nWe extended zero-shot experiments based on iTransformer to forecast lengths of **96, 336, and 720**, in addition to 192 (reported in the paper). PS loss improved accuracy in **33 out of 36** settings, confirming its robustness across both short- and long-term horizons. See [Table 9](https://anonymous.4open.science/r/PS_Re/T9.pdf) for details.\n\n|Model|96||336||720||\n|-|-|-|-|-|-|-\n|Loss Function|MSE|+PS|MSE|+PS|MSE|+PS\n|ETTh1→ETTh2/m1/m2|0.499|**0.446**|0.537|**0.575**|0.579|**0.566**\n|ETTh2→ETTh1/m1/m2|0.626|**0.540**|0.641|**0.614**|0.686|**0.645**\n|ETTm1→ETTh1/h2/m2|0.420|**0.385**|0.514|**0.472**|0.550|**0.506**\n|ETTm2→ETTh1/h2/m1|0.622|**0.485**|0.792|**0.541**|0.870|**0.554**\n|Imp|-|**15.58%**|-|**11.39%**|-|**15.42%**\n\n# [Q4] Combination of PS loss and FreDF loss\nFreDF focuses on frequency-domain alignment to **mitigate label autocorrelation**, while PS loss emphasizes **patch-wise structural alignment** in the time domain. Their goals are complementary. We evaluated MSE+FreDF, MSE+PS, and MSE+PS+FreDF using iTransformer as backbone.\n\n|Loss|MSE+FreDF||MSE+PS||MSE+PS+FreDF||\n|-|-|-|-|-|-|-\n|Dataset|MSE|MAE|MSE|MAE|MSE|MAE\n|ETTh1|0.443|0.437|0.440|0.438|**0.436**|**0.433**\n|ETTm1|0.404|0.406|**0.396**|**0.397**|**0.396**|**0.397**\n\nResults show that combining the two losses yields **either improved or comparable performance**, supporting their compatibility. [Table 10](https://anonymous.4open.science/r/PS_Re/T10.pdf) provides full results.", "claims_and_evidence": "The claims made in the paper are well-supported by experimental evidence. The experiments in the submission and appendix demonstrate that the proposed method is effective and robust across different models and architectures. The source code and detailed experimental procedures further enhance the study's reproducibility.", "methods_and_evaluation_criteria": "The methods and evaluation criteria used in this paper are well-suited for the problem of time series forecasting. The authors choose relevant benchmark datasets and employ suitable evaluation metrics to assess model performance. The integration of PS loss with MSE is clearly explained, and the experimental setup is sound. The use of both quantitative and qualitative results strengthens the validity of the conclusions.", "theoretical_claims": "I have checked the proofs and the theoretical claims behind PS loss, which are robust, and do not find any issues.", "experimental_designs_or_analyses": "The experimental design is solid and thorough, and the results support the claims made in the paper. However, there are several issues that need to be discussed:\n1.  The paper does not provide a detailed comparison between the GDW strategy and grid search for loss coefficient selection. It would be helpful if the authors could offer a deeper analysis of the GDW strategy’s effectiveness. \n2. The authors should provide visualizations or analyses showing how the different loss weights evolve during training, to better understand how the GDW strategy influences model optimization.", "relation_to_broader_scientific_literature": "The proposed PS loss builds on existing time series forecasting loss functions but introduces a more flexible and localized approach. The introduction of patch-wise structural alignment is a novel contribution that distinguishes this work from existing methods. This innovation positions PS loss as a valuable contribution to the field of time series forecasting and makes it relevant to both academic research and practical applications.", "essential_references_not_discussed": "The authors have effectively cited and discussed relevant work in time series forecasting, loss functions, and the patching mechanism in time series forecasting in the paper.", "other_strengths_and_weaknesses": "Strengths:\n1. The paper presents a clear and novel contribution to time series forecasting by addressing the limitations of traditional loss functions through the integrate seamless with PS loss, which provides a more precise method for structural alignment and achieves more practical predictions.\n2. The gradient-based dynamic weighting strategy is a novel contribution that enhances the effectiveness of PS loss by adjusting the weight of each component based on gradient magnitudes, improving robustness without the computational cost of grid search.\n\nWeaknesses:\n1. The paper includes a sensitivity analysis of the hyperparameters \\lamba and \\delta, but it lacks a detailed explanation of how they vary in different scenarios. It is recommended that the authors clarify the parameter settings.", "other_comments_or_suggestions": "Minor inconsistency in notation: In Figure 2, \\alpha is used for L_Corr, \\beta for L_Mean, and \\gamma for L_Var, but these notations are inconsistent with the rest of the manuscript and formula. It would be beneficial to unify the notation.", "questions_for_authors": "1. About the ablation study, Could the authors explore whether PS loss can replace MSE loss entirely while achieving similar results in terms of forecasting accuracy?\n2. Since PS loss involves local analysis, how does it affect training time, especially for large-scale datasets such as ECL? Would training time significantly increase for large datasets? \n3. In the zero-shot forecasting experiment, do longer forecasting horizons benefit more from PS loss, or is its impact more significant in short-term forecasting? \n4. How does PS loss compare with other loss functions such as FreDF (Learning to Forecast in Frequency Domain)? Does the combination of PS loss with these other functions enhance forecasting performance?", "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": "kzqU1dzDeR", "reviewer": "Reviewer_Pd9N", "summary": "Traditional loss functions, such as Mean Squared Error, often miss structural dependencies in time series forecasting. This paper proposes a Patch-wise Structural Loss to improve accuracy by focusing on patch-level structural alignment. It uses Fourier-based Adaptive Patching to divide the series and incorporates local statistical features—correlation, variance, and mean—with dynamic gradient weighting. Testing shows enhanced forecasting performance across multiple datasets and models.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you very much for your valuable feedback. Below are our responses to your concerns and suggestions.\n# [W1] Additional quantitative metrics for evaluating PS loss performance\nTo provide a more comprehensive evaluation, we incorporated additional metrics: **Dynamic Time Warping (DTW)**, **Time Distortion Index (TDI)**, and **Pearson Correlation Coefficient (PCC)**, to assess the performance of PS loss. A detailed explanation of these metrics is provided below:\n\n| Metric | Definition | Interpretation |\n|--|--|--|\n| Dynamic Time Warping (DTW) | Measures the minimum cumulative distance between two sequences after applying an optimal non-linear alignment. | Lower DTW indicates that the prediction closely matches the ground truth after optimal alignment. |\n| Time Distortion Index (TDI) | Quantifies the amount of temporal warping or distortion required to achieve the optimal alignment obtained by DTW. | Lower TDI indicates less temporal adjustments for optimal alignment, while a higher TDI signifies greater distortion.|\n| Pearson Correlation Coefficient (PCC) | Measures the linear relationship between two sequences. | Higher PCC indicates better preservation of the sequence's overall trend and structure. |\n\nOn the iTransformer model, **PS loss consistently improves all three shape-aware metrics**, indicating better structural alignment. On the ETTh2 dataset, the iTransformer trained with MSE achieves lower DTW scores, reflecting smaller numerical differences after optimal alignment. However, the higher TDI in this case suggests that the alignment requires more extensive temporal warping, which indicates greater structural distortion compared to the forecasts generated using PS loss. \n\n|Metrics|DTW||TDI||PCC||\n|-|-|-|-|-|-|-\n|Loss|MSE|**+PS**|MSE|**+PS**|MSE|**+PS**\n|ETTh1|7.355|**7.324**|7.888|**6.959**|0.514|**0.530**\n|ETTh2|**6.891**|7.016|24.723|**22.705**|0.299|**0.342**\n|ETTm1|6.568|**6.435**|12.451|**11.381**|0.538|**0.557**\n|ETTm2|5.913|**5.611**|26.969|**22.495**|0.325|**0.387**\n|Weather|5.410|**5.409**|41.440|**40.343**|0.324|**0.352**\n\nDue to space limitations, we report the average metric values across all forecasting lengths. **Please refer to ([Table 6](https://anonymous.4open.science/r/PS_Re/T6.pdf)) for full results.**\n\n# [Q1] Drawbacks of fixed-length patching\n\nFixed-length patching requires grid search over a predefined set of patch lengths, which introduces **computational overhead and lacks adaptability across datasets**.\n\n|Fixed Patch length|3|6|12|24|48|96|\n|-|-|-|-|-|-|-|\n|96|0.379|**0.378**|0.379|0.383|0.383|0.386|\n|192|0.430|0.429|**0.428**|0.431|0.431|0.433|\n|336|0.473|**0.473**|0.474|0.480|0.480|0.483|\n|720|0.496|0.499|**0.480**|0.493|0.493|0.508|\n|Avg.|0.444|0.445|**0.440**|0.446|0.450|0.453|\n\nIn our ETTh1 experiments, the best-performing fixed patch size was found to be $P = 12$, which matches the patch length estimated by our adaptive patching strategy using the dominant period $p$. This demonstrates that our method can **automatically identify an appropriate patch length** without manual tuning.", "claims_and_evidence": "There are no obvious problematic claims in the paper.", "methods_and_evaluation_criteria": "The design and presentation of the method both make sense. The evaluation of the method also aligns with the established standards in the field.", "theoretical_claims": "No question about theoretical claims.", "experimental_designs_or_analyses": "The design of the main experiments is comprehensive, as the PSloss is applied to various model architectures and achieves relatively good results. The ablation experiments are thorough, and their design and analysis help me better understand the method.", "relation_to_broader_scientific_literature": "1. Most current time series works, such as PatchTST[1] and iTransformer[2], use MSE loss as the optimization objective. This paper, however, highlights the limitations of using MSE loss for optimization.\n2. The paper employs the Pearson Correlation Coefficient[3] to characterize correlation loss and the Kullback–Leibler (KL) divergence to characterize variance loss.\n3. Inspired by previous works on balancing multi-task losses, such as [4-5], the paper proposes Gradient-based Dynamic Weighting to achieve balanced optimization.\n\nReferences:\n- [1] A Time Series is Worth 64 Words: Long-term Forecasting with Transformers\n- [2]  iTransformer: Inverted Transformers Are Effective for Time Series Forecasting\n- [3] Pearson correlation coefficient\n- [4] Multi-task learning as multiobjective optimization.\n- [5] Sparsetsf: Modeling long-term time series forecasting with* 1k* parameters.", "essential_references_not_discussed": "No essential reference not discussed.", "other_strengths_and_weaknesses": "Strengths:\n1. The paper is well written and easy to understand.\n2. The paper includes comprehensive evaluations. The diverse ablation study helps to understand the proposed approach.\n\nWeaknesses:\n1. The shortcomings of MSE presented in the paper are not only its limitations as an optimization objective but also as a metric. However, to align with the work in the field, the paper still uses MSE and MAE as main metrics. Figure 3 demonstrates how PSloss contributes to the final prediction results, but I believe the authors could provide more quantitative metrics to further illustrate this point.", "other_comments_or_suggestions": "No more comments or suggestions.", "questions_for_authors": "Question 1: Why use adaptive patching? What are the drawbacks of fixed-length patching?", "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:

Meta-review: All reviewers support acceptance, highlighting significant methodological novelty, comprehensive validation, and clear improvements, while recommending minor revisions and clarifications for final submission.
Decision: Accept (poster)