PMLF: A Physics-Guided Multiscale Loss Framework for Structurally Heterogeneous Time Series
Authors: Xinghong Chen
Weilin Wu
Kunping Yang
Guannan Chen
Venue: NeurIPS 2025 poster ()
Keywords: Time Series Forecasting
Physics-Guided Learning
Energy-Based Loss Design
Seasonal-Trend Decomposition
Primary area: deep_learning

Abstract: Forecasting real-world time series requires modeling both short-term fluctuations and long-term evolutions, as these signals typically exhibit multiscale temporal structures. A core challenge lies in reconciling such dynamics: high-frequency seasonality demands local precision, while low-frequency trends require global robustness. However, most existing methods adopt a unified loss function across all temporal components, overlooking their structural differences. This misalignment often causes overfitting to seasonal noise or underfitting of long-term trends, leading to suboptimal forecasting performance. To address this issue, we propose a Physics-guided Multiscale Loss Framework (PMLF) that decomposes time series into seasonal and trend components and assigns component-specific objectives grounded in the distinct energy responses of oscillatory and drift dynamics.  Specifically, we assign a quadratic loss to seasonal components, reflecting the quadratic potential energy profile of molecular vibration, while a logarithmic loss is used for trend components to capture the sublinear energy profile of molecular drift under sustained external forces. Furthermore, we introduce a softmax-based strategy that adaptively balances the unequal energetic responses of these two physical processes. Experiments on different public benchmarks show that PMLF improves the performance of diverse baselines, demonstrating the effectiveness of physics-guided loss design in modeling structural heterogeneity in time series forecasting.

PMLF: A Physics-Guided Multiscale Loss Framework for Structurally Heterogeneous Time Series
Xinghong Chen, Weilin Wu
Kungping Yang, Guannan Chen
Key Laboratory of OptoElectronic Science and Technology for Medicine of Ministry of Education
Fujian Normal university
gbx20220087dyjj.. fjn edu. en; weilin iatpm_ww199011600 .om; kungpingyangüf jnu puw.unuuuc.mm edu. edado@f jnu, edu. cn
Abstract
Forecasting real-world time series requires modeling both short-term fluctuations and long-term evolutions, as these signals typically exhibit multiscale temporal structures. A core challenge lies in reconciling such dynamics: high-frequency seasonality demands local precision, while low-frequency trends require global robustness. However, most existing methods adopt unified loss function across all temporal components, overlooking their structural differences. This misalign- ment often causes overfitting to seasonal noise or underfitting of long-term trends, leading to suboptimal forecasting performance. To address this issuc, we propose a Physics-guided Multiscale Loss Framework (PMLF) that decomposes time series into seasonal and trend components and assigns component-speciin grounded in the distinct energy responses of oscillatory and drift dynamics. Specif- ically, we assign quadratic loss to seasonal components, reflecting the quadratic potential energy profile of molecular vibration, while a logarithmic loss used for trend components to capture the sublinear energy profile of molecular drift under sustained external forces. Furthermore, we introduce adaptively balances the unequal energetic responses of these two physical processes. Experiments on different public benchmarks show that PMLF improves the perfor- mance of diverse baselines, demonstrating the effectiveness of physics-guidee loss design in modeling structural heterogeneity in time scries forecasting.
objectives
boftmax-based strategy that
1
Introduction
Time series forecasting is widely used across a range of scientific and industrial domains [34], including energy consumption [35). climate change [40, 43), financial exchange [24), and traffic flow 20) Real-world time series inherently exhibit complex multiscale behavior, combining high- frequency seasonal variation with low-frequency trend dynamics. This multiscale characteristic introduces conflicting demands on forecasting models, as short-term and long-term patterns evolve at different temporal resolutions, making accurate forecasting particularly challenging [30]. Recent methods address multiscale forecasting primarily through architectural-level innovations, such as hierarchical attention 22) multi-resolution convolutions 229]. seasonal-trend decomposition (39] 38
5 28) While these approaches improve the model's ability to represent multiscale signals, they apply uniform loss functions during optimization, with limited attention paid to how different temporal structures are supervised [5]
and frequency-awar aware representations [45
41
Figure I illustrates the divergent error behaviors of seasonal and trend components. Seasonal components are highly sensitive minor phase shifts or amplitude distortions, which rapidly accumulate over time and significantly degrade predictive accuracy. In contrast, trend components
"Corresponding Author: Guannan Chen, Weilin Wu
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

---

are more tolerant to local errors but may suffer from long-term drift when trained under overly strict constraints. These differences indicate that seasonal and trend components respond to supervision in fundamentally different ways and thus require structurally distinct loss formulations 3d 33
However, most existing approaches adopt a single loss, such as MSE or MAE, applied indiscriminately This uniform treatment implicitly assumes that all temporal components contribute equally to forecasting error. As result, the model fails to distinguish between structurally different error patterns, which limits its ability to adapt supervision to each component and ultimately impairs forecasting accuracy and long-term stability.
across the entire sequence [13 17. 2
To resolve the mismatch in optimization ob- jectives between seasonal and trend compo- nents in multiscale time series, we draw inspi- ration from structural response mechanisms in molecular systems (37 Specifically, we de- sign loss functions based on the energy profiles characteristic of each component. Molecular vibrations follow harmonic dynamics around equilibrium, where small displacements lead to rapid increases potential energy [7]. In contrast, molecular structures under sustained external influence exhibit nonlinear relaxation, with weakening restoring forces and sublinear energy accumulation [25]. These two physical responses closely parallel the behaviors of sea- sonal and trend components in time series.
Seasonal True Sessonal Prediction
phase
trand
Thend True frand Predition
amplltude distortions
Figure 1: Illustration of distinct error patterns aris- ing from different structural components in time series. Seasonal components are particularly sen- sitive to phase shifts and amplitude distortions, whereas trend components are more susceptible to long-term drift.
Motivated by this analogy, we propose Physics-guides Multiscale Loss Framework (PMLF) that supervises each component using loss function grounded in its corresponding physical dynamics. As illustrated in Figure once the model produces the output sequence, both the prediction and the ground truth are decomposed during loss computation into seasonal and trend components, each supervised by structurally consistent objective. The seasonal branch is optimized using quadratic loss, consistent with harmonic energy, while the trend branch adopts a logarithmic loss that reflects sublinear stress accumulation under long-term drift. To ensure balanced training across heterogeneous error profiles, we further incorporate a softmax-basee based weighting mechanism that dynamically adjusts the relative contributions of seasonal and trend losses. This loss design yields the following contributions:
We reveal that seasonal and trend components exhibit fundamentally heterogeneous structural characteristics, analogous harmonic oscillations and irreversible structural drift in physical systems.
Based on this insight, we formulate a physics gusided loss framework that assigns quadratic penalties seasonal errors and logarithmic potentials trend deviations, reflecting their distinct structural behaviors. We further integrate softmax-basce weighting mechanism to dynamically balance their learning contributions during training.
Experiments on different public benchmarks show that PMLF improves the performance of diverse baselines, demonstrating the effectiveness of physics-guided loss design in modeling structural heterogeneity in time series forecasting.
Related Work
2.1
Multiscale Modeling in Time Series Forecasting
Due to the inherent decomposability of time series, many recent forecasting models have taken multiscale modeling strategies, including explicit temporal decomposition. frequency-domain decom- position, and hierarchical temporal abstraction. Explicit decomposition methods, such as Autoformer [8) extract trend signals via moving-average filtering and treat seasonal variations as residual components. TimeMixer [29 extends this idea by operating over multiple temporal resolutions to better capture fine-grained temporal structures. Frequency-doman approaches model signals in the spectral domain to identify dominant periodicities and multiscale frequency structures, as

---

Physics-Guided Guidedd Multiscale Loss Framework
Dynamic Loss Balancing
Prodiction
P-Seasonal
P-Trend
Decompose
wonts mark
Co
Input
y
m
Decompose
MMMM
4
4
Liow
Ground Truth
T-Seasonal
T-Trend
L by +
Figure 2: Overview of the proposed Physical-Guided Multiscale Loss Framework. Given an input time series, the model predicts future values, and both predictions and ground truths are decomposed into seasonal and trend components. Each component is supervised by physically grounded loss aligned with its structural behavior. Finally, softmax taaseax-based dynamic weighting mechanism then balances their contributions to form the final optimization objective.
demonstrated by FEDformer 45
FreTS [2], and WPMixer Hierarchical models 117 18
instead organize inputs into layered or patch-wise representations to capture dependencies across temporal scales. PatchTST [22) for example, achieves strong performance by applying elff-attenioon to fixed-length patches, enabling joint modeling of short- and long-term dynamics. Although these models incorporate multiscale structures the architectural level, they typically adopt uniform loss functions across all components. This mismatch between representation and supervision may limit the model's ability learn the distinct dynamics of trend and seasonal signals.
2.2 Loss Function Design in Time Series Forecasting
Beyond point-wise error minimization, recent work has explored shape-aware loss functions designed improve sequence-level accuracy. These approaches can be broadly categorized into alignment- based and structure-awarg objectives. Alignment-bast based losses, including DTW [I] Soft-DTW B. DILATE [[S]. and TIDLE-Q [16], enable elastic matching between predictions and targets, improving robustness to temporal misalignment and phase variation. Structure-aware losses, in contrast, aim to preserve internal signal characteristics such as frequency, locality, and temporal coherence. For example, FreDF 27 compares signals in the frequency domain to preserve spectral energy distributions. Patch-wise structural loss [[] learns local correlation, variance, and mean patterns by decomposing sequences into overlapping patches, enabling finer-grained structural supervision.
3 Methodology
3.1 Overview and Motivation
While recent forecasting models incorporate structural decomposition to isolate trend and seasonal components, their training objectives typically remain uniform, applying loss functions such as MSE over the full output sequence. However, this uniform supervision fails to reflect the divergent dynamic sensitivities of multiscale components. Localized seasonal oscillations require fine-grained alignment, whereas long-term trends benefit from stable, drift-aware objectives.
To address this mismatch between representation and supervision, we propose a mhysics-guided multiscale loss framework. As illustrated in Figure 2 both predictions and ground truths are decomposed into seasonal and trend components using a shared operator. Each component is then supervised by loss function aligned with its temporal dynamics. This structure-aware supervision allows the model optimize multiscale representations more effectively, leading to improved long- range forecasting performance.

---

Molecular Response Mode
Time Series
Loss Design
Representation
Molecular Vibration
Seasonal Pattern
Harmonic Energy Loss
(High-trequency)
Value
Constraint
External Influences (Low-rrquuency)
Trend Pattern
Energy
Value
Loss
Tolerant Constraint
error
Figure Conceptual illustration our physics-guides loss design. Seasonal components are analogous to high-frequency molecular vibrations and are penalized using harmonic energy loss, which imposes strict local alignment. Trend components resemble structural drift under external fields and optimized via relaxation energy loss that enforces tolerant, sublinear correction.
3.2 Problem Formulation
Given a multivariate time series X [21, IT} RTxC the model fo aims to predict the next H time steps conditioned on a historical window of length L:
Ý (Úr+1,.... DT- =o [o( PT)
(1)
To capture multiscale temporal structure, the prediction and ground truth sequences are decomposed into components via seasonal-tres trrnn decomposition:
Y= Y= +, ÿ Y-
where Y, and Y, denote the trend and seasonal component, respectively. This decomposition performed using a shared low-pass filter 'D to extract the trend, while the seasonal component is defined as the residual. We omit the stochastic noise term in this work, lacks structured patterns and thus cannot be effectively supervised.
3.3 Physics-Inspired Structural Supervision
In molecular systems, physical responses perturbations generally fall into two categories: intrinsic vibrations and externally induced structural drift 33 23 99. As shown Figure 3 the former corresponds reversible, high-frequency oscillations of atomic bonds around equilibrium, exhibiting strong sensitivity to small displacements. The latter arises from persistent external forces such as thermal gradients mechanical stress, leading irreversible, low-frequency drift with gradually diminishing resistance. These two physical response modes naturally correspond to the structural decomposition of time series. Seasonal components exhibit periodic fluctuations and are highly sensitive to phase and amplitude deviations, similar to molecular vibrations. In contrast, trend components evolve gradually under long-term influence and resemble structural drift observed in dissipative systems. To supervise these components in structure conssurr-onnssteent manner, we replace uniform loss penalties with energy- objectives whose growth profiles align with the dynamic behavior of each component.
Energy-Bate -Base Loss for Seasonal Oscillation. cquilibrium are governed by harmonic motion, where the potential energy increases quadratically with deviation:
In molecular systems, small displacements from
U (n)
k(x 1xx 10a-xpp =00)²
(3)

---

where z is the system state, the equilibrium position, and & the stiffness constant. This response structure reflects the behavior of seasonal time series components, which exhibit high-frees nigh-frequency. reversible oscillations and are sensitive to phase or amplitude shifts.
Given predicted and target seasonal values s(t) and s(t), the potential energy difference is:
k [st) s(t)² 2 s(t))
Uss Uew((((t) (tt)
(4)
2
By expanding the square and reorganizing terms, the energy difference can be rewritten
aU (f) =
2 [(a(t) - (()(t)=(tt)) - 2 -
- s(t))]
(5)
Due the symmetric oscillation of stt) around its equilibrium, the coupling term (s(1) - 20 (8(() s(t)) has an expected value close to zero across a complete seasonal cycle:
Eg [(((t) - ()(( s())]
(6)
Consequently, the expected energy difference simplifies to a dominant quadratic penality, which defines the seasonal loss as:
L. = CH
Σ c.
sCc) T+h
T+h
Relaxation Energy Loss for Trend Components. In contrast to the symmetric and reversible oscillations of seasonal dynamics, trend components reflect irreversible structural drift induced by persistent external influences. Similar dynamics occur in dissipative physical systems under sustained forces, such as thermal gradients, where deformation accumulates progressively and the system's internal opposition to change gradually saturates. To capture this non-restorative response, introduce a diminishing structural force model:
ke
r(e)
where e =
(8)
1 + ae'
which captures the saturation of structural resistance under persistent displacement. The integral of this response yields the effective structural energy potential:
(ra(C) = a
(1 +1+ ae log(1 + ace)
which reflects the sublinear accumulation of internal structural energy associated with irreversible drift. Linear and constant terms are omitted they yield non-informative or structurally insensitive
gradients. We retain only the logarithmic penalty, which preserves structural sensitivity and yields the final trend loss:
C,
Σ c.
log [1+
((e) T+hh -T74h
(10)
CH
3.4 Dynamic Loss Balancing
Seasonal and trend components originate from distinct physical mechanisms: harmonic oscillation and structural relaxation, respectively. When supervised jointly, their associated energy responses evolve at different rates. As training proceeds, the trend-related loss often grows several orders of magnitude larger than the seasonal loss due to its slow but persistent accumulation. This numerical imbalance distorts the force equilibrium between the components, causing the trend branch dominate gradient updates and suppressing high-frequency corrections. To address this, we adopt boftmax-based dynamic weighting scheme that adjusts the influence of each component based on its current energy level. Instead of static weights, we compute adaptive coefficients from the exponentially scaled differences between the detached losses. This strategy maintains structural balance during optimization and avoids second-orde effects.
The total loss is defined as weighted sum:
L -= Xr
(II)
5

---

Table 1: Multivariate long-term term forecasting results across diverse real-world datasets. The table reports MSE and MAE across four prediction horizon He (96, 192, 336, 720), with a fixed input sequence length of 96. The better results for each setting are highlighted in bold.
Metied
A
TincMiun
Low Flacion
MSE
PSLLF
XSE
PMILF
MSE
PMLE
MSE
PMLEF
MSE
PMLF
Marros
MSF:
MAR
MATI
M5n
MAB
MSI
MAR
MSS:
MAR:
MSI
MAT:
MSB
MAR.
MSn
MAT
MSR
MAR
MST
MAR:
0.315
0.375
0.304
0.390
0,402
0.369
0.482
0.406
0.407
0.367
0.403
0.385
0.365
0.392
192
11.433
01.64
0.423
0.425
0,443
11433
0.424
0.430
0.415
0.441
0.437
0.43
1.448
0.441
0.438
0.334
0.435
0.425
0.416
0.423
136
11.044
18.65
0.439
0.661
0.475
11453
0.463
0.413
0:455
0,470
01477
0.453
0.485
0.459
0.479
0.453
0.09
0.455
0.464
0.415
11:59
0.489
0.484
0.467
1L479
0.450
0.496
11:79
0.477
0.484
0.486
0.178
0.513
0.483
0,171
0.464
0.447
0.443
0.436
0.442
0.437
0,439
0.455
0.453
0.454
0.445
0.448
0.431
96
03.003
0.355
0.253
0.332
0.355
0.336
0.243
0.333
0.234
0.346
0.224
0.337
0.300
0.351
0.296
0.340
0.297
0.345
0.297
0.337
0.397
0.353
0,379
0.317
0.399
0.361
0.418
0.385
0.434
0.453
0.475
0.455
0,403
0.423
0.441
0.415
0.431
0:431
0.451
11:122
0.424
0.422
0.441
0,412
0.4M0
0.411
0.445
0.402
0.417
AVR
0,383
0.409
0.356
0.365
0.374
0,399
0.363
0.368
0.381
0.407
0.367
0.391
0.378
0.404
0.374
0.395
0,383
0,404
0.363
0.367
11364
0.385
6:311
0.372
0.376
0,599
0.369
6.373
0.384
6:308
0.382
0.993
6.378
0.378
0.368
0.384
0.267
0,218
336
11.966
0.405
0.309
0.395
0,400
11411
0.415
0.399
0.45
0.455
0.421
0,111
B419
0.417
0.413
0.460
0.398
0,405
0.395
0.401
IL460
0.415
0.481
0.436
0.467
0.447
0,463
0,412
01553
0.418
IL471
0,441
01487
0.435
0.417
0.439
0,482
0.451
0.4%6
0.315
0.300
0.386
0.401
0.398
0.368
0.436
0..00
0.408
0.413
0.318
0.393
0.388
0.399
0.391
96
11
11000
0.763
0,172
0.25
0.312
10:55
0.165
0.246
0.195
0.378
11.001
0.240
11188
0.724
0.115
0.352
0.176
0.249
0.170
6.249
01.441
0.301
6.257
0.298
0.237
0.500
0.221
0.290
0.235
6.314
0.341
0.299
0.152
0.313
0.245
0.29
0.238
0.200
0.236
0.298
01.999
0.399
0.391
0.387
0.396
0.314
0.368
0.415
0.407
0.11
0.405
0.482
0.303
0.393
0.390
0,368
Ang
01:251
0.317
0,375
0.316
0.315
01:53
0.271
0.315
0.2000
0.310
11.292
0,525
0.300
0.315
0.261
0.319
0.213
0.314
01.275
6.116
0.209
0.156
0,205
0.156
0.398
0.213
0.253
8.309
0.343
0.304
0.247
0.201
6.237
0.214
6:352
0.31
0.221
0.254
6.219
0.347
0.286
0.25
0.266
0.242
336
0.368
0.292
0.362
0.283
0.251
0.290
0.269
0.260
0.274
0.297
0.363
0.264
0.278
0.296
0.213
0.280
0.251
0.247
0.253
0.254
130
01344
0.342
0.339
0.354
0.340
0.541
0.336
0.333
0.350
0.345
0.342
0.335
0.358
0.347
0.351
0.341
0.339
0.341
0.339
0.336
0.274
96
0.152
0.245
0.145
0.342
0.340
11242
0.135
0.236
0.139
0.255
0.137
0.232
11148
0.240
0.146
0.336
0.159
0.251
0.155
0.245
192
0,162
0.256
6.161
0.343
0.157
01256
0.155
0.251
0.159
0.255
0.161
0.354
0.362
0.253
0.163
0.352
0.171
0.269
0.170
0,259
11114
0.769
6.171
0.267
0.275
0.171
0.269
0.312
IL175
0.769
0.173
0.364
0.187
6,31X
AvA
0.172
0.267
0.170
0.291
0.371
0.270
0.166
0.263
0.170
0.265
0.369
0.262
0.178
0.270
0.173
0.361
0.186
0.217
0.185
0.272
0.245
0.451
0.290
0.271
0.429
0.268
0.261
0,305
0.265
0.393
0.297
0.316
0.467
0,382
0.453
0.207
0117
0.276
6.112
0.491
0.904
0.479
0.316
0.172
0.314
6.473
0.288
0:481
6:246
0.417
0:376
0.11
0.274
8459
0.781
6.428
0.271
0.445
0.246
0.936
0,314
730
11:23
0.335
0.511
0.351
8.516
11:07
0.521
0.385
0.460
0.300
0.215
D4467
0.303
0.461
0.386
11.508
01511
0.531
0.318
0.315
0.475
0,300
0,287
0.472
0.254
0.414
0.276
0.282
0.424
0.31
0.212
6.188
0.214
0.389
0.226
0.187
6.341
0:305
0.344
0:303
0:231
6.199
0.318
0.189
0.2
6,212
192
01:255
0.255
6.221
0.231
0.210
0.205
0.201
0:257
0:231
0.200
0.235
0.239
0.233
0.261
0:236
0.344
0.223
0.263
0.224
6.254
136
0.345
0.265
0.245
0.348
0.215
0.299
0.213
0.267
0.255
0.258
0.341
0.255
0.241
0.273
0.258
0.250
0.231
0.292
0.243
0.266
0.275
0.220
6.218
Arg
0.228
0.255
0.327
0.23
6:211
0.296
0.215
0.239
0.340
0.375
0.234
0.241
0.333
0.363
0.218
0.344
0.206
0.280
0.218
0.256
where the adaptive weights X, and A are computed as:
exp(B(ct m))
exp(B(Ct - m)) exp(B(cI- ( + (xp(3((t-m)
A,
(12)
m)) + (xc( -m))
Here, L = SGC denotes the gradient-detached loss to avoid backpropagating through the weighting mechanism. The scalar m max(C), Ct) ensures numerical stability, and the parameter 3 1 unless otherwise specified.
4 Experiments
4.1 Setup
Datasets To comprehensively evaluate the effectiveness of PMLF on real datasets, we conducted experiments eight multivariate time series datasets, including ETT (4 sub-datasets) [44), Weather, Electricity, Traffic, and Solar-Energy
Detailed dataset descriptions are provided in Appendix.
Baselines To evaluate the ability of PMLF on different baselines, we extensively selected Trans- former, Mamba, MIP, CNN, and their hybrid methods, including Amplifier [5), TimeXer [3] S-Mamba (32), iTTrnsformer [19]. TimeMixer 29). PatchTST (22] and TimesNet [39].

---

0,600
MSE PMLF
MSE PMLF
MSE PMLF
2.84%
-.8.%%
22.
-1.4%
0.400
0.400
4,50%
-1.0%
UI
4.50%
0,100
0.200
0.200
0.000
0.000
0.000
S-Mamba
PatchTST
TimesNet
S-Mamba
PatchTST
TimesNet
S-Mamba
PatchTST
TimesNet
(a)
(b)
(C)
0-400
0.400
22.8%
-0.1%
0.400
10.0%
-.30%
-1 1.23P5
-11.7%%
MAE
S 0.200
0.200
0.200
0.000
S-Mamba
PatchTST
TimesNet
S-Mambba
PatchTST
TimesNet
9-Mamba
PatchTST
TimesNot
(d)
(e)
(f)
Figure 4: Forecasting performance of three backbone models (S (S-Mamba, PatchTST, and TimesNet) on the ETTh1, ECL, and Solar datasets. Subplots (a-c) show MSE and (d-f) show MAE, averaged four forecasting horizons He (96, 192, 336. 720). Each pair of bars compares standard MSE loss with the proposed PMLF.
Implementation Detail All models were trained from scratch using PMLF as the loss function, ensuring a consistent evaluation of its effectiveness. During training, seasonal and trend components are extracted via fixed moving average filter with kernel size 25, and also support an optional learnable version implemented as a plug-in module. Our loss framework is model -adel--gmostic and can be seamlessly integrated into a wide range of forecasting backbones, including MLPs [5], Transformers (26). and Mamba 66, 4), thereby supporting universal applicability to multiscale prediction tasks. To ensure fair comparison, we ensure the consistency of the parameters of the comparison loss, only by adjusting the learning rate to adapt to the gradient dynamics caused by our loss design. All experiments are conducted using Py PyTorch with four NVIDIA RTX 4090 GPUs.
4.2
Main Results
We evaluate the proposed PMLF loss across a set of state-of-the-ar forecasting models on eight benchmark datasets covering diverse application domains. The standard MSE loss used as the baseline for comparison. As shown in Table [ the proposed model-agnostic loss framework yields consistent improvements over the standard MSE baseline across five recent state-of offtthe-rtt models. This demonstrates that PMLF effectively guides the optimization of both short-term seasonal patterns and long-tery term-term trends, which frequently coexist in real-world time series. Especially, the performance improvement of the model the MAE index is often higher than that the MSE index, mainly due comparisons TTranfformer,
to the long-term control of trend. For example, among a total the PMLF improvement item accounted for 93.75 75% (60/64). In addition, the improvement in MAE metric is often more substantial than that in MSE, suggesting that the trend-aware loss contributes significantly to long-term stability. To further assess the generalizability of our framework. Figure 4 reports the average forecasting performance of three representative architectures: S-Mamba, PatchTST, and TimesNet, across multiple prediction horizons. On typical datasets including ETTh1, ECL. and Solar-Enen Eelery PMLF achieves significant improvements both MSE and MAE. Additional results covering all models and datasets provided in the appendix.
4.3 Comparison with Other Loss Functions
compare PMLF with traditional loss function and specially designed objectives proposed for time series forecasting. These approaches typically enhance standard objectives by introducing additional

---

FOW
FOR
1000
(a)
(b)
(c)
(d)
Figure 5: Robustness analysis of the dynamic weighting parameter 3 on the ETTh2 (a-b) and Weather (c-d) datasets based on TemeXer.
Predecte
Suesced
kred
8.0
4.54
6.0
4.00
Tinssops
(a) ETTh2 Overall Prediction
ETTh2 Seasonal
(c) ETTh2 Trend
60
30
0.2
-10
-45
188
(d) ECL Overall Prediction
(e) ECL Seasonal
(1) ECL Trend
Figure 6: Overall and Component-wil Prediction Visualization on ETTh2 and ECL. Comparison between MSE and PMLF losses on ETTh2 (())-c)) and ECL ((d)(f)) Each row presents the overall prediction, trend component, and seasonal component. PMLF exhibits closer alignment with the ground truth in both local seasonal and global trends.
terms to emphasize specific learning characteristics. For instance, TILDE-Q [16] introduces penalties on phase and amplitude shifts to improve temporal alignment, while FreDF strengthens label-wise consistency through frequency-don -omain comparisons. In contrast, PMLF 27 supervises decomposed seasonal and trend components individually at the loss level, aiming to guide structural learning rather than refining unified temporal loss. As shown in Table PMLF achieves better overall performance on three datasets in both MSE and MAE metrics. These results demonstrate that structural supervision introduced solely through the loss function effectively improve forecasting accuracy.
4.4 Visualization
To assess the structural impact of our loss function, we visualize predictions on ETTh2 and ECL using Amplifier as the backbone model. As shown in Figure compared to the standard MSE loss, our proposed PMLF loss produces forecasts that are more closely aligned with the ground truth. To further analyze the effect of structural supervision, we decompose both predicted and true sequences into seasonal and trend components. In both dimensions, the PMLF-based predictions exhibit improved alignment with their respective ground truth counterparts, demonstrating enhanced fidelity in modeling high-frequeee oscillations as well as long-term drift.

---

Table Comparison of PMLF with standard and structure- aware loss functions on ETTh2, ETTm2,
and Weather datasets using Amplifier as the backbone model.
Dataset
ETTh2
ETTm2
Weather
horizon
96
192
336
720
96
192
336
720
96
192
336
720
MSE
0.303
0.369
0.412
0.446
0.180
0.241
0.303
0.399
0.164
0.213
0.268
0.344
MSE
MAE
0.355
0.397
0.428
0.455
0.262
0.301
0.344
0.399
0.209
0.268
0.292
0.342
Soft-DTW
MSE
0.292
0.371
0.413
0.436
0.177
0.242
0.302
0.407
0.164
0.214
0.268
0.344
MAE
0.346
0.396
0.429
0.450
0.259
0,303
0.344
0,406
0.209
0.252
0.291
0.341
MSE
0.298
0.360
0.399
0.432
0.174
0.238
0.299
0.397
0.159
0.209
0.265
0.342
Huber
MAE
0.345
0.388
0.420
0.444
0.255
0.299
0.339
0.396
0.201
0.246
0.287
0.338
MSE
0.286
0.364
0.394
0.420
0.177
0.236
0.296
0.390
0.169
0.214
0.267
0.345
TILDE-Q 16
MAE
0.334
0.384
0.419
0.438
0.254
0.293
0.333
0.390
0.213
0.250
0.291
0.341
MSE 0.285
0.355
0.392
0.422
0.173
0.235
0.296
0.388
0.167
0.212
0.269
0.345
FreDF hi
MAE
0.334
0.381
0.412
0.436
0.253
0.2994
0.334
0.388
0.210
0.254
0.297
0.348
MSE
0.283
0.353
0.39
0.403
0.172
0.237
0.299
0.391
0.156
0.209
0.262
0.339
PMLF
MAE
0.332
0.379
0.417
0.423
0.251
0.293
0.332
0.387
0.193
0.243
0.283
0.334
Table 3: Ablation study of the components of PMLF loss on the ETTh2, ETTm2 and Weather datasets using Amplifier as a backbone.
Method
PMLF
w/o L
w/o Cr
w/o Balancing
w/ LMA
Metire
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
96
0.283
0.332
0.330
0.361
0.375
0.391
0.292
0.335
0.289
0.332
ETTh2
192
0.353
0.379
0.487
0.468
0.428
0,423
0.357
0.382
0.357
0.379
336
0.385
0.405
0.670
0.588
0.421
0.429
0.392
0.409
0.391
0.406
720
0.403
0.423
0.592
0.548
0.434
0.446
0.406
0.425
0.406
0.425
96
0.172
0,251
0.206
0.277
0.208
0,282
0.175
0.256
0.171
0.250
ETTm2
192
0.237
0.293
0.301
0.345
0.249
0.303
0.239
0.297
0.239
0.295
336
0.299
0.332
0.348
0.369
0.308
0.340
0.301
0.338
0.296
0.331
720
0.391
0.387
0.433
0.414
0.405
0.396
0.400
0.392
0.393
0.390
96
0.156
0.193
0.21
0.262
0.180
0.223
0.158
0.197
0.155
0.194
Weather
192
0.209
0.243
0.259
0.292
0.217
0.254
0.212
0.249
0.209
0.243
336
0.262
0.283
0.306
0.323
0.271
0.292
0.265
0.286
0.264
0.283
720
0.339
0.334
0.367
0.360
0.346
0.341
0.343
0.335
0.342
0.334
4.5
Ablation Studies
As shown in Table 3] we conduct ablation studies the ETTh2, ETTm2, and Weather datasets using Amplifier as the backbone. 'he results show that removing either the seasonal or trend component from the loss formulation leads to significant drop in forecasting accuracy, confirming the necessity of jointly modeling both temporal structures. Additionally, eliminating the dynamic weighting mechanism by fixing the loss coefficients also degrades performance, highlighting the importance of adaptive balancing between the two structural modes. Furthermore, we implement a learnable moving average (LMA) module with trainable kernels. Its performance remains comparable to the fxeedkernel version, suggesting that the proposed loss is compatible with both static and adaptive decomposition strategies. Finally, as illustrated in Figure we assess the robustness of the hyperparameter 3 on the ETTh1 and Weather datasets using the TimeXer model. The results indicate that the performance remains stable across a wide range of 3 values, from 0.2 to demonstrating that our loss design is insensitive to this choice.

---

Conclusion
We introduce Physics-Guided Guiddd Multiscale Loss Framework (PMLF) for time series forecasting, which decouples seasonal and trend supervision by grounding each in distinct physical dynamics. Inspired by molecular systems, the framework models short-term oscillations through harmonic potentials and long-term drift via relaxation energy. Furthermore, softmax-based dynamic weighting mechanism balances the contributions of each component during training, enabling adaptive optimization across structural error profiles. Experimental results across multiple datasets and model backbones show that PMLF consistently improves forecasting accuracy.
Limitation and Future Works. However, PMLF assumes that multiscale structures are present and separable. Although this assumption holds for many real-world signals, the benefits of structural decoupling may diminish in settings dominated by abrupt transitions or highly regular periodic signals, such as regime shifts or physiological rhythms like heartbeat and respiration. These signals lack the multi scale heterogeneity that PMLF is designed to exploit. A promising direction for future work to explore adaptive decomposition mechanisms or hybrid supervision strategies that enhance the framework's applicability to structure-invariant or single-scale time series.
Acknowledgements
This study was supported by the Natural Science Foundation of Fujian Province under Grant 2024J01063 and Grant 2024Y4017.
References
[1] Donald J Berndt and James Clifford. Using dynamic time warping to find patterns in time series. In Proceedings of the 3rd international conference on knowledge discovery and data mining, pages 359-370, 1994.
[2] Peng Chen, Yingying Zhang, Yunyao Cheng, Yang Shu, Yihang Wang, Qingsong Wen, Bin Yang, and Chenjuan Guo. Multi-scale transformers with adaptive pathways for time series forecasting. In International Conference on Learning Representations, 2024.
[3] Marco Cuturi and Mathieu Blondel. Soft-dtw: a differentiable loss function for time-series. In International conference machine learning, pages 894-903. PMLR, 2017.
[4] Tri Dao and Albert Gu. Transformers are SSMs: Generalized models and efficient algorithms through structured state space duality. In International Conference on Machine Learning (ICML), 2024.
[5] Jingru Fei, Kun Yi, Wei Fan, Qi Zhang, and Zhendong Niu. Amplifier: Bringing attention to neglected low-energy components time series forecasting. 2025.
[6] Albert Gu and Tri Dao. Mamba: Linear-time sequence modeling with selective state spaces. arXiv preprint arXiv:2312.00752. 2023.
[7] Justin R Gullingsrud, Rosemary Braun, and Klaus Schulten. Reconstructing potentials of mean force through time series analysis of steered molecular dynamics simulations. Journal of Computational Physics, 150(11110-221, 1999. ISSN 0021-9991.
[8] Qihe Huang, Lei Shen, Ruixin Zhang, Jiahuan Cheng, Shouhong Ding, Zhengyang Zhou, and Yang Wang. Hdmixer: Hierarchical dependency with extendable patch for multivariate time series forecasting. In Proceedings of the AAAI conference on artificial intelligence, volume 38, pages 22608-12616, 2024.
[9] Michael E Kellman and Vivian Tyng. The dance of molecules: New dynamical perspectives highly excited molecular vibrations. Accounts of chemical research, 40(4):243-250, 2007.
[10] Taesung Kim, Jinhee Kim, Yunwon Tae, Cheonbok Park, Jang-Ho Choi, and Jaegul Choo. Reversible instance normalization for accurate time-series forecasting against distribution shift. In International Conference on Learning Representations, 2022.
10

---

[11] Dilfira Kudrat, Zongxia Xie, Yanru Sun, Tianyu Jia, and Qinghua Hu. Patch-wise structural loss for time series forecasting, arXiv preprint arXiv:2503. 00877, 2025.
[12] Guokun Lai, Wei-Cheng Chang, Yiming Yang, and Hanxiao Liu. Modeling long-and short-term temporal patterns with deep neural networks. In The st international ACM SIGIR conference on research development in information retrieval, pages 95-104, 2018.
[13] Henning Lange, Steven L Brunton, and J Nathan Kutz. From fourier to koopman: Spectral methods for long-term time series prediction. Journal of Machine Learning Research, 22((1):
1-38, 2021.
[14] Vincent Le Guen and Nicolas Thome. Shape and time distortion loss for training deep time series forecasting models. Advances in neural information processing systems, 32, 2019.
[15] Vincent Le Guen and Nicolas Thome. Shape and time distortion loss for training deep time series forecasting models. In Advances in Neural Information Processing Systems, pages 4191-4200 2019.
[16] Hyunwook Lee, Chunggi Lee, Hongkyu Lim, and Sungahn Ko. Tilde-q: transformation invariant loss function for time-series forecasting. arXiv preprint 0rxiv:2210.113 50050, 2022.
[17] Minhao Liu, Ailing Zeng. Muxi Chen, Zhijian Xu, Qiuxia Lai, Lingna Ma. and Qiang Xu. Scinet: Time series modeling and forecasting with sample convolution and interaction. Advances in Neural Information Processing Systems, 35:5816-5888 2022.
[18] Shizhan Liu, Hang Yu, Cong Liao, Jianguo Li, Weiyao Lin, Alex X. Liu, and Schahram Dustdar. Pyraformer: Low-complexity pyramidal attention for long-range time series modeling and forecasting. In International Conference on Learning Representations, 2022.
[19] Yong Liu, Tengge Hu, Haoran Zhang, Haixu Wu, Shiyu Wang, Lintao Ma, and Mingsheng Long. itransformer: Inverted transformers are effective for time series forecasting. In The Twelfth International Conference on Learning Representations, 2024.
[20] Yisheng Lv, Yanjie Duan, Wenwen Kang, Zhengxi Li, and Fei-Yue Yue Wang. Traffic flow prediction with big data: A deep learning approach. leee transactions on intelligent transportation systems, 16(2):865-88) 873,, 2014.
[21] Md Mahmuddun Nabi Murad, Mehmet Aktukmak, and Yasin Yilmaz. Wpmixer: Efficient multi- resolution mixing for long-tens termttermmne time series forecasting. Proceedings of the AAAI Conference on Artificial Intelligence. 19518)11958,119588, Apr. 2025.
[22] Yuqi Nie, Nam H. Nguyen, Phanwadee Sinthong, and Jayant Kalagnanam. A time series is worth 64 words: Long-term forecasting with transformers. In International Conference on Learning Representations, 2023.
[23] Tamar Schlick and Charles S Peskin. Can classical equations simulate quantum-meccannict behavior? a molecular dynamics investigation of diatomic molecule with a morse potential. Communications pure and applied mathematics, 42(8):1 141-1163, 1989.
[24] Omer Berat Sezer, Mehmet Ugur Gudelek, and Ahmet Murat Ozbayoglu. Financial time series forecasting with deep learning: A systematic literature review: 2005-2019. Applied Soft Computing, 90:106181 2020. ISSN 1568-4946. doi:
106181.
[25] Yuichi Togashi and Alexander S Mikhailov. Nonlinear relaxation dynamics in elastic networks and design principles of molecular machines. Proceedings of the National Academy of Sciences, 104((1):8697-877) 2007.
[26] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Lukasz Kaiser, and Illia Polosukhin. Attention all you need. Advances in neural information processing systems, 30, 2017.
[27] Hao Wang, Licheng Pan, Zhichao Chen, Degui Yang, Sen Zhang, Yifei Yang, Xinggao Liu, Haoxuan Li, and Dacheng Tao. Fredf: Learning to forecast the frequency domain. In ICLR, 2025.
11

---

[28] Jingyuan Wang, Ze Wang, Jianfeng Li, and Junjie Wu. Multilevel wavelet decomposition network for interpretable time series analysis. In Proceedings of the 24th ACM SICKDD international conference knowledge discovery & data mining, pages 2437-2446, 2018
[29] Shiyu Wang, Haixu Wu, Xiaoming Shi, Tengge Hu, Huakun Luo, Lintao Ma, James Y Zhang, and JUN ZHOU. Timemixer: Decomposable multiscale mixing for time series forecasting. In International Conference on Learning Representations (ICLR), 2024.
[30] Yuxuan Wang, Haixu Wu, Jiaxiang Dong, Yong Liu, Mingsheng Long, and Jianmin Wang. Deep time series models: A comprehensive survey and benchmark. 2024.
[31] Yuxuan Wang, Haixu Wu, Jiaxiang Dong, Yong Liu, Yunzhong Qiu, Haoran Zhang, Jianmin Wang, and Mingsheng Long. Timexer: Empowering transformers for time series forecasting with exogenous variables. Advances in Neural Information Processing Systems, 2024.
[32] Zihan Wang, Fanheng Kong. Shi Feng, Ming Wang, Xiaoeui Yang, Han Zhao, Daling Wang, and Yifei Zhang. Is mamba effective for time series forecasting? Neurocomputing, page 129178, 2024.
[33] Qingsong Wen, Jingkun Gao, Xiaomin Song, Liang Sun, Huan Xu, and Shenghuo Zhu. Robust- stl: A robust seasonal-trent decomposition algorithm for long time series. In Proceedings of the AAAI conference artificial intelligence, volume 33, pages 5409-5416, 2019.
[34] Qingsong Wen, Linxiao Yang. Tian Zhou, and Liang Sun. Robust time series analysis and applications: An industrial perspective. In Proceedings of the 28th ACM SIGKDD conference on knowledge discovery and data mining, pages 4836-4837, 2022.
[35] Rafal Weron. Electricity price forecasting: A review of the state-of-the-ar with a look into the future. Internationnl of forecasting, 30(4):1030-1081 2014.
[36] Mike West. Time series decomposition. Biometrika, 84(2):899494, 1997.
[37] Jordan Winetrout, Krishan Kanhaiya, Joshua Kemppainen, Pieter J in 't Veld, Geeta Sachdeva, Ravindra Pandey, Behzad Damirchi, Adri van Duin, Gregory M Odegard, and Hendrik Heinz. Implementing reactivity in molecular dynamics simulations with harmonic force fields. Nature communications, (51): 2024.
[38] Haixu Wu, Jiehui Xu, Jianmin Wang, and Mingsheng Long, Autoformer: Decomposition trans- formers with auto-correlation for long-term series forecasting. Advances in neural information processing systems, 34:22419-22430, 2021.
[39] Haixu Wu, Tengge Hu, Yong Liu, Hang Zhou, Jianmin Wang, and Mingsheng Long. Timesnet: Temporal 2d-variation modeling for general time series analysis. In International Conference on Learning Representations, 2023.
[40] Haixu WI, Hang Zhou, Mingsheng Long, and Jianmin Wang. Interpretable weather forecasting for worldwide stations with a unified deep model. Nature Machine Intelligence, 5(6)6602-61,, 2023.
[41] Kun Yi, Qi Zhang. Wei Fan, Shoujin Wang, Pengyang Wang, Hui He, Ning An, Defu Lian, Longbing Cao, and Zhendong Niu. Frequency-domain mips are more effective learners in time series forecasting. Advances in Neural Information Processing Systems, 36.76656-76679, 2023.
[42] Kun Yi, Qi Zhang, Wei Fan, Shoujin Wang, Pengyang Wang, Hui He, Ning An, Defu Lian, Longbing Cao, and Zhendong Niu. Frequency-domain MLPs more effective learners in time series forecasting. In Thirty-seventh Conference on Neural Information Processing Systems, 2023.
[43] Yuchen Zhang, Mingsheng Long, Kaiyuan Chen, Lanxiang Xing, Ronghua Jin, Michael Jordan, and Jianmin Wang. Skilful nowcasting of extreme precipitation with noweastnet. Nature, 619 (970):526-532.222 2023.
12

---

[44] Haoyi Zhou, Shanghang Zhang, Jieqi Peng, Shuai Zhang, Jianxin Li, Hui Xiong, and Wancai Zhang. Informer: Beyond efficient transformer for long sequence time-series forecasting. In The Thirly-Fifth AAAI Conference on Artificial Intelligence, AAAI 2021, Virtual Conference, volume 35, pages 11 106-11115, AAAI Press, 2021.
[45] Tian Zhou, Ziqing Ma, Qingsong Wen, Xue Wang, Liang Sun, and Rong Jin. FEDformer: Frequency enhanced decomposed transformer for long-term series forecasting. In Proc. 39th International Conference on Machine Learning (ICML 2022), 2022.
13

---

NeurIPS Paper Checklist
1. Claims
Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Answer: [Yes]
Justification:
The abstract and section h clearly claimed the contributions made in the paper.
Guidelines:
The answer NA means that the abstract and introduction do not include the claims made in the paper.
The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.
The claims made should match theoretical and experimental results, and reflect how much the results can expected to generalize to other settings.
It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper.
2. Limitations
Question: Does the paper discuss the limitations the work performed by the authors?
Answer: [Yes]
Justification: please refer the limitation discussion part 5
Guidelines:
The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those
discussed the paper.
The authors are encouraged create a separate "Limitations" section in their paper.
The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect how these assumptions might be violated in practice and what the implications would be.
The authors should reflect on the scope of the claims made, c.g., if the approach was only tested on few datasets or with few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.
The authors should reflect on the factors that influence the performance of the approach. For example, a facial recognition algorithm may perform poorly when image resolution is low images taken in low lighting. Or speech-to-text system might not be used reliably to provide closed captions for online lectures because fails to handle technical jargon.
The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.
If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and faimesss.
While the authors might fear that complete honesty about limitations might be used by reviewers grounds for rejection, worse outcome might be that reviewers discover limitations that aren't it acknowledged judgment and recognize that individual actions in favor of transparency play important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.
the paper. The authors should use their best
3. Theory assumptions and proofs
Question: For each theoretical result, does the paper provide the full set of assumptions and complete (and correct) proof?
Answer: [Yes]
14

---

Justification:
Please refer to Section 3
Guidelines:
The answer NA means that the paper does not include theoretical results.
All the theorems, formulas, and proofs in the paper should numbered and cross- referenced.
All assumptions should be clearly stated referenced in the statement of any theorems.
The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition.
Inversely, any informal proof provided in the core of the paper should complemented by formal proofs provided in appendix or supplemental material.
Theorems and Lemmas that the proof relies upon should be properly referenced.
4. Experimental result reproducibility
Question: Does the paper fully disclose all the information needed to reproduce the main ex- perimental results of the paper to the extent that affects the main claims and/or conclusions of the paper (regardless whether the code and data are provided or not)?
Answer: [Yes]
Justification: The loss designed have been described clearly in main text
Guidelines:
The answer NA means that the paper does not include experiments.
If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.
If the contribution is dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable.
Depending on the contribution, reproducibility can be accomplished in various ways. For example, the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary either make possible for others replicate the model with the same dataset, or provide access to the model. In general. releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to hosted model (e.g., in the case of large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed.
While NeurIPS does not require releasing code, the conference does require all submis- sions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example
(a) If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm.
(b) If the contribution is primarily a new model architecture, the paper should describe the architecture clearly and fully.
(c) If the contribution is new model (e.g., large language model), then there should either be a way to access this model for reproducing the results or a way to reproduce the model (e.g., with open-source dataset or instructions for how to construct the dataset).
(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility. In the case closed-source models, it may be that access the model limited some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results.
5. Open access to data and code
Question: Does the paper provide open access to the data and code, with sufficient instruc- tions to faithfully reproduce the main experimental results, as described in supplemental material?
15

---

Answer: [Yes]
Justification: Please refer to the implementation
details listed the
appendix of the supplementary materials.
The code be found
in
supplementary materials.
Guidelines:
The answer NA means that paper does include experiments requiring code.
Please see the NeurIPS code and data submission guidelines (https://nips.c cc/ public/guides/ Public/guides//odeSuuiiie
for more details.
While we encourage the release of code and data, we understand that this might not possible, so "No" an acceptable answer. Papers cannot rejected simply for not including code, unless this is central to the contribution (e.g., for new open-source benchmark).
The instructions should contain the exact command and environment needed to run to reproduce the results. See the NeurIPS code and data submission guidelines (https
ssionPolicyy for more details.
The authors should provide instructions on data access and preparation, including how to access the raw data, preprocessed data, intermediate data, and generated data, etc.
The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. If only subset of experiments are reproducible, they should state which ones are omitted from the script and why.
At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).
Providing as much information as possible in supplemental material (appended the paper) is recommended, but including URLs to data and code is permitted.
6. Experimental setting/details
Question: Does the paper specify all the training and test details (e.g., data splits, hyper- parameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
Answer: [Yes]
Justification:
All implement detail please refer to Appendix in the supplementary materials.
Guidelines:
The answer NA means that the paper does not include experiments.
The experimental setting should be presented the core of the paper to level of detail that is necessary to appreciate the results and make sense of them.
The full details can be provided either with the code, in appendix, or as supplemental material.
7. Experiment statistical significance
Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
Question:
Answer:
[Yes]
Justification: please refer Appendix in the supplementary materials.
Guidelines:
The answer NA means that the paper does not include experiments.
The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.
The factors of variability that the error bars capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall with given experimental conditions).
The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.)
The assumptions made should be given (c.g., Normally distributed errors).
16

---

It should be clear whether the error bar is the standard deviation or the standard error of the mean.
It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report 2-sigma error bar than state that they have a 96% CI, if the hypothesis of Normality of errors is not verified.
For asymmetric distributions, the authors should be careful not show in tables figures symmetric error bars that would yield results that are out of range (e.g. negative error rates).
If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.
8. Experiments compute resources
For each experiment, does the paper provide sufficient information on the com- puter resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
Question:
Answer: [Yes]
Justification:
All implement detail please refer to Appendix in the supplementary materials.
Guidelines:
The answer NA means that the paper does not include experiments.
The paper should indicate the type of compute workers CPU GPU, internal cluster, cloud provider, including relevant memory and storage.
The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute.
The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g.. preliminary or failed experiments that didn't make it into the paper).
9. Code of ethics
Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics http://urrr...c.pp........ccctaa
Answer: [Yes]
Justification:
We have reviewed and the research conforms with the NeurIPS Code of Ethics.
Guidelines:
The answer NA means that the authors have reviewed the NeurIPS Code of Ethics.
If the authors answer No, they should explain the special circumstances that require deviation from the Code of Ethics.
The authors should make sure to preserve anonymity (e.g., there is special considera- tion due to laws regulations in their jurisdiction).
10. Broader impacts
Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
Answer: [Yes]
Justification: Please refer to the case study in the Section 4.4
4.4
Guidelines:
The answer NA means that there is no societal impact of the work performed.
If the authors answer NA or No, they should explain why their work has no socictal impact or why the paper does not address socictal impact.
Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.
17

---

The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.
The authors should consider possible harms that could arise when the technology being used as intended and functioning correctly, harms that could arise when the technology is being used intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.
If there negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses addition to attacks, mechanisms for monitoring misuse, mechanisms monitor how system learns from feedhack over time, improving the efficiency and accessibility of MI.).
11. Safeguards
Question:
Does the paper describe safeguards that have been put in place for responsible release of data models that have high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
Answer: [NA]
Justification:
This paper does not constitute such risk.
Guidelines:
The answer NA means that the paper poses such risks.
Released models that have high risk for misuse dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions access the model or implementing safety filters.
Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.
We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors take this into account and make best faith effort.
12. Lieenses for existing assets
Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper. properly credited and are the license and terms of use explicitly mentioned and properly respected?
Answer: [Yes]
Justification: All creators of datasets properly credited by citations.
Guidelines:
The answer NA means that the paper does not existing assets.
The authors should cite the original paper that produced the code package or dataset.
The authors should state which version of the asset is used and, if possible, include URL.
The name of the license (e.g., CC-BY 4.0) should be included for each asset.
For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided.
If assets are released, the license, copyright information, and terms of use in the pack- age should be provided. For popular datasets, paperswi thcode. com /mmatttsstts has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset.
For existing datasets that are re-pa re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided.
18

---

If this information is not available online, the authors are encouraged to reach out to the asset's creators.
13. New assets
Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
Answer: [NA]
Justification: The paper does not release new assets. The code and training details are provided in supplementary materials.
Guidelines:
The answer NA means that the paper does not release new assets.
Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc.
The paper should discuss whether and how consent obtained from people whose asset is used.
At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.
14. Crowdsourcing and research with human subjects
Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well details about compensation (if any)?
Answer: [NA]
Justification:
The paper does not involve crowdsourcing nor research with human subjects.
Guidelines:
The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.
Including this information the supplemental material fine, but if the main contri- bution of the paper involves human subjects, then as much detail possible should be included in the main paper.
According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage the country of the data collector.
15. Institutional review board (IRB) approvals or equivalent for research with human subjects
Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or equivalent approval/review based the requirements of your country institution) were obtained?
Answer: [NA]
Justification: The paper does not involve crowdsourcing nor research with human subjects.
Guidelines:
The answer NA means that the paper does not involve crowdsourcing research with human subjects.
Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. you obtained IRB approval, you should clearly state this in the paper.
We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution.
For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.
19

---

for 16. Declaration of LLM usage
Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology. scientific rigorousness, originality of the research, declaration is not required.
Answer: [NA]
Justification: The article uses the LLM for grammar checking and polishing.
Guidelines:
The answer NA means that the core method development this research does not involve LLMs as any important, original, or non-standard components.
Please refer LLM policy http:///eeeerippsc/////ooeeeeeee what should or should not be described.
20

---

A Datasets and Implementation
A.1 Datasets
We conducted experiments on 8 real-world datasets to evaluate the effectiveness of the our proposed PMLF loss functions across various domains. The detailed dateset information are depicted in Figure
4
ETT(Ellctricity Transformer Temperature): The ETT dataset contains 7 variables of electricity transformer temperature from July 2016 to july 2018. There 4 sub datasets: ETTh1, ETTh2, ETTm2, ETTm2, where ETTh recorded hourly and ETTm recorded every 15 minutes.
Weather: Weather contains 21 meteorological variables collected every 10 minutes from the Weather Station of the Max Planck Biogeochemistry Institute in 2020.
ECL: ECL records the hourly electricity consumption data of 321 clients from 2012 2014.
Traffic: Traffic collects hourly road occupancy rates measured by 862 sensors of San Francisco Bay area freeways from January 2015 to December 2016.
Solar-Energy: Solar records the solar power production of 137 PV plants in 2006, which are sampled every 10 minutes.
We follow the same data processing and train-validation-tes set split protocol used in TimesNet, where the train, validation, and test datasets are strictly divided according chronological order to make sure there are no data leakage issues. As for the forecasting settings, we fix the length of the lookback series 96 and the prediction length varies in (96, 192, 336, 720).
Table 4: Detailed Dataset Descriptions. Dim denotes the variable number of each dataset. Prediction Length denotes the future time steps be predicted and four prediction setting are included in each dataset. Dataset Size denotes the total number of time steps in (Train, Validation, Test) split respectively. Frequency denotes the sampling interval of time steps.
Dataset
Dim
Prediction Length
Dataset Size
Frequency
Domain
ETThl
(96, 192, 336, 720)
(8545, 2881, 2881)
hour hour
Electricity
ETTh2
(96, 192, 336, 720)
(8545, 2881, 2881)
Electricity
ETTml
(96, 192, 336, 720)
(34465, 11521, 11521)
15 min
Electricity
ETTm2
(96, 192, 336, 720)
(34465, 11521, 11521)
15 min
Electricity
Weather
21
(96, 192, 336, 720)
(36792, 5271, 10540)
10 min
Weather
ECL
321
(96, 192, 336, 720)
(18317, 2633, 5261)
hour
Electricity
Traffic
862
(96, 192, 336, 720)
(12185, 1757, 3509)
hour 10 min
Transportation
Solar-Energy
137
(96, 192, 336, 720)
(36601, 5161, 10417)
Energy
A.2 Implementation Details
All the experiments are implemented in Py Torch and conducted on four NVIDIA 4090 24GB GPU. For fair comparison, we set the input size for all models to be uniform, with the batchsize for ETT and Weather datasets set to 64 and the ECL, Traffic, and Solar datasets set to 16. Except for changing the learning rate to fully learn new structures, do not change other parameters related to the model.
Before calculating the loss, the time series decomposed into seasonal and trend components using the moving average method. The kernel size refers to the setting in Autoformer, which is 25. But order to avoid fixed kernel sizes affecting time series with different sampling frequencies, we designed a hybrid expert decomposition mechanism that uses set of average pooling layers to extract trends and combines them with learnable weights. The kernel sizes (7, 13, 15, 25, 49) represent the corresponding periods at different frequencies.
21

---

B Baselines
To evaluate the general applicability of PMLF, we compare it with varied collection of leading time- series forecasters that span the principal architectural families: state-space (S- (S-Mamba), Transformer (iTransformer, TimeXer, PatchTST), multilayer perceptron (Amplifier, Time.ixxr), and convolutional neural network (TimesNet). The core ideas of these baselines are outlined below.
Amplifier: An energy- amplifiaiiio block heightens weak spectral bands, the spectrum is re- normalised, and parallel seasonal and trend heads with lightweight channel-interaction module enhance performance on low-signal datasets.
Targets mixed endogenous and exogenous forecasting, Patch-wise wise self-attention models the target series, variate-wise Pross-attention injects exogenous cues, and global endogenous tokens integrate the two streams, improving robustness to abrupt external shocks.
TimeXer:
S-Mamba: Represents each time step with single per-channel token and applies bidirectional Mamba state-space layer along the channel axis to capture inter-variable dependencies. This near-linear-time design surpasses Transformer baselines while greatly reducing computational cost.
Traanfformer; Reassigns Transformer roles by applying attention across variables to learn cross- channel links, while the feed-forward block operates along time to model nonlinear temporal dynamics. This rearrangement lowers memory usage for long horizons and yields interpretable variable-level attention.
TimeMixer: A pure-MLP predictor. Depth-wise convolutions extract multi-scale bands, linear layers mix these components, and parallel head simultaneously outputs all future steps. The absence of attention provides GPU-friendly speed without sacrificing accuracy.
PSt Divides long sequences into fixed-length temporal patches that serve Transformer tokens and shares encoder parameters among channels. This strategy reduces attention complexity from ((L) to ((L/)) while retaining local semanties.
TimesNet: Converts a one-dimensional series into a two-dimensional time-period grid and employs heterogeneous CNN kernels to capture intra-period seasonality as well as inter-period trends. This representation enables direct transfer from vision backbones and delivers strong results in forecasting, anomaly detection, and classification.
These heterogeneous baselines ensure that any improvements attributed to PMLF are not limited to single modelling philosophy but instead reflect broad enhancement of time-series learning.
C More Experimental Results
C.1 Robustness Assessment
To examine the robustness of framework, the Amplifier baseline was trained five times with independent random seeds. T the mean performance together with the corresponding standard deviations. The consistently small variances confirm that the Amplifier yields repeatable results, underscoring the robustness the proposed approach.
C.2 Parameter Sensitivity
To examine how the lynamic-weighteee coefficient influences forecasting accuracy, we conducted a grid search over 3 E {0.2, 0.4, 0.6, 0.8, 1, 1.5. 3. 5} for two representative networks: TimeXer and TimeMixer. Figure reporrt the resulting MSE and MAE on the ETTh2 and Weather datasets. For TimeXer (top row) and TimeMixxr (bottom row), the error curves remain nearly flat across the entire range, and the optimal values cluster around 1.0 on both datasets. The maximum deviation from the best MSE and MAE is below 2.5%, indicating that the proposed dynamic weighting scheme is insensitive to the precise choice of B. These results confirm the robustness of our framework with respect to this
22

---

Table Roubustness of PMLF performance. The results are obtained from five random seeds using the Amplifier as backbone.
Dataset
ETTh1
ETTh2
Weather
Horizon
MSE
MAE
MSE
MAE
MSE
MAE
0.005=10.002
0.394:0.0001
0.203=00000
0.332:E0.000
0.156=
0.001
192
0.423=00.003
0.425+0.0022
0.353=0.002
0.379=000001
0.209
0.243=0.004
336
0.490-10.003
0.456:10.002
0.385+100000
0.405=10.001
0.262:100.02
0.283
720
0.484=10.003
0.467=+00002
0.403=0.002
0.423=10.001
0.339:+0.002
0.334=0.004
Dataset
ETTml
ETTm2
ECL
Horizon
MSE
MAE
MSE
MAE
MSE
MAE
96
0..01
0.342:10.002
0.172:10.002
0.250=
0. 0.140
0.242:10.000
192
0.371=0.004
0.372=00.003
0.237=0.001
0.293-10.002
0.161#0.001
0.243=0.000
336
0.309
0.395=0.003
0.299=0.002
0.332=20.002
0.170=
0.263=0.001
720
0.001
0.436:20.004
0.001
0.387:10.004
0.199:000001
0.280
330
(a) TimeXer:ETThl
(b) TimeXer.ETThl
(c) TimeXer:
(d) TimeXer: Weather
(e) TiTe
(I) EimeMMxer.ETThl
(g) TimeMixer:
(h) TimeMixer: Weather
Figure 7: Sensitivity of the dynamic-weighting coefficient 3. The first row shows TimeXer perfor- mance on ETTh2 (a. b) and Weather (c. d) varies, and the second row gives the corresponding results for TimeMixer.
C.3 Additional Evaluation Classical Forecasting Networks
Additionally, we assesses the setreccture-independeeee effectiveness of PMLF on three widely used forecasting models: the Transformer-basee PatchTST, the selective state-space model S Mamba, and the convolutional network TimesNet. For each architecture, the original MSE objective with PMLF and the models were evaluated on the ETT, Weather, ECL, and Solar datasets under four prediction horizons (96, 192, 336, 720]. As shown in Figure 8 across all datasets and horizons, the substitution of MSE with PMLF consistently reduced the eиror metrics (MAE, MSE), confirming that PMLF improves forecasting accuracy and robustness even in classical network settings that not covered by the main set of sate-of-the-at architectures.
replaced
D Visualization
As shown in Figure and 10 we provide a visual comparison between PMLF and MSE on benchmark datasets: ETTH2, ETTm2, Weather, ECI., Traffic, and Solar. Each row corresponds to one dataset, where the first column presents the overall forecast, and the second and third columns show the decomposed seasonal and trend components, respectively. Across all datasets, the forecasts
23

---

0.600
MSE PMLF
MBE PMLF
MBE PMLF
0.400
2.84%
-.82%%
22..%%
0.400
2.0%
415
-741%
-..4%
UI
0.200
0.200
0.200
0.000
0.000
0.000
S-Mamba
PatchTST
TimesNet
S-Mamba
PatchTST
TimesNet
S-Mamba
PatchTST
TimesNet
(a) ETTh MSE
(b) ETTH2:MSE
(c) ETTmI:MSE
0.400
2.58%
10.1%
10.0%
0.400
..2%%
4.27%
0.400
4.48%
-.35%
MAE
0.200
0.200
0.200
0.000
0.000
S-Mamba
PatchTST
TimesNet
S-Mamba
PatchTST
TimesNet
9-Mamba
PatchTST
TimesNot
(d) ETTHI:MAE
(e) ETTh2:MAE
(f) ETTmI:MAE
MSE PMIFF
MSE PMLF
MSE PMLF
0.400
60%
-1.7%%
a 58%
-1.8%%
.31 8%
4.50%
MSE
2,.0%
0.100
0,200
0.000
0.000
0.000
S-Mamba
PatchTST
Times
S-Mamba
PatchTST
TimesNet
S-Mamba
PatchTST
TimesNet
ETTm2:MSE
(h) ECL:MSE
ii Solar:MSE
0.400
0.400
30.78%
0.400
-.2%%
11 29%
-11.74%
S 0.200
0.200
0,200
S-Mamba
PatchTST
TimesNet
S-Marnba
PatchTST
TimesNet
S-Mamba
PatchTST
TimesNet
(j) ETTm2:MAF
(k) ECL:MAE
()) Solar:MAF
Figure 8: Average forecasting performance of PatchTST, S-Mamba, and TimesNet under four prediction horizons (96, 192, 336, 720) on the ETT (4 subsets), Weather, ECL, and Solar datasets. Models trained with PMLF are contrasted with their original MSE counterparts. Across all horizons and datasets, PMLF consistently reduces MAE, MSE, and RMSE, illustrating its effectiveness on these classical architectures.
generated using PMLF exhibit closer alignment with the ground-truth signals, capturing long-term trends and periodic patterns with higher fidelity compared to those produced with MSE.
24

---

Veed
1.00
4.54
6.10
6.0
-..3
-.08
(a) ETTh2 Prediction
(b) ETTh2 Seasonal
(c) ETTh2 Trend
Pondccion
6--
2000
(d) ETTm2 Prediction
(c) ETTm2 Scasonal
ETTm2 Trend
1.5
6:20
100
(g) Weather Prediction
(h) Weather Seasonal
Weather Trend
Figure 9: Visual comparison of Amplifier forecasts trained with PMLF and MSE across six benchmark datasets. Each corresponds one dataset (ETTh2, ETTm2, Weather), with three columns showing the overall prediction (left), seasonal component (middle), and trend component (right).
25

---

0.5
0.5
50-
-1.0
(a) ECL Prediction
(b) ECL Seasonal
(c) ECL Trend
mediccnn
W
c.-
(d) Traffic Prediction
(c) Traffic Scasonal
(f) Traffic Trend
Predecion
Seaven
Toed
3.
10
(g) Solar Prediction
(h) Solar Seasonal
(i) Solar Trend
Figure 10: Visual comparison of Amplifier forecasts trained with PMLF and MSE across six benchmark datasets. Each row corresponds to one dataset (ECL, Traffic, and Solar), with three columns showing the overall prediction (left), seasonal component (middle), and trend component (right).
26


Reviews:
[{"review_id": "xxDRYCOXhK", "reviewer": "Reviewer_6tYu", "summary": "This work assumes that real-world time series can be decomposed into seasonal and trend components and proposes using different loss functions tailored to each. Seasonal components, which represent high-frequency oscillations, are modeled with a quadratic loss, capturing their need for local precision. Trend components, reflecting long-term, low-frequency changes, are modeled with a logarithmic loss to ensure global robustness. A softmax-based balancing strategy is introduced to adaptively weight these losses based on their energetic behavior. The proposed method outperform compared approach on multiple benchmarks", "questions": "- Can the authors provide empirical evidence (e.g., using toy or real-world datasets) to support the claim that unified loss functions fail to capture both trend and seasonal components effectively?\n- Have the authors considered including ablation studies to evaluate whether the performance gains come from the decomposition itself or the choice of distinct loss functions?\n- Have different combinations of loss functions (e.g., quadratic for both, logarithmic for both, or swapped) been tested to justify the specific assignment proposed?\n- How does the proposed adaptive weighting compare against fixed or heuristic alternatives in terms of performance? Can a baseline be included for comparison?", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "1. **Unrealistic Decomposition Assumption**  \n   The method assumes that all time series can be neatly decomposed into seasonal and trend components. This is a strong and often invalid assumption for many real-world signals that include abrupt changes, irregularities, or structural breaks. The paper does not discuss how the method handles such scenarios, raising concerns about robustness and generalization.\n\n2. **Misleading \"Physics-Guided\" Terminology**  \n   While the paper uses analogies from physics to motivate the choice of loss functions, it does not incorporate any actual physical modeling or constraints. Referring to the method as \"physics-guided\" is misleading and may confuse readers, especially in the context of Physics-Informed Neural Networks (PINNs), which are grounded in physical laws.\n\n3. **Lack of Justification for Key Claims**  \n   - The claim that a unified loss function cannot effectively handle both seasonal and trend patterns is asserted without theoretical or empirical support. This should be backed by experiments (e.g., on toy datasets) comparing standard losses such as MSE, MAE, and DTW to illustrate their limitations.  \n   - The specific choice of quadratic loss for seasonality and logarithmic loss for trend appears ad hoc. There is no ablation study to verify whether these choices are optimal, or whether the performance gains come mainly from the decomposition itself.  \n   - The effectiveness of the proposed adaptive loss weighting strategy is not clearly demonstrated. A comparison with fixed or heuristic weighting schemes is necessary to justify its inclusion.", "was_revised": false, "final_justification": "", "rebuttal": "### **[Q1]:provide empirical evidence to support the claim that unified loss functions fail to capture both trend and seasonal components effectively.**\n\n**Response**. Thank you for your insightful and constructive suggestion. We fully agree that empirical evidence is crucial to support our claim.\n\nTo support our claim, we conducted a controlled experiment on the well-known AirPassengers dataset, which exhibits both a clear seasonal pattern and a long-term upward trend. We trained a simple LSTM model under two settings: (1) using a unified MSE loss, and (2) using a decomposed loss. All other training configurations were held constant.\n\nResults showed that the decomposed MSE setup achieved better forecasting performance (final MSE = 0.02065vs. 0.01267). This confirms that a structure-aware loss provides a more effective supervisory signal than its unified counterpart, even when using the same loss form. We believe this demonstrates the practical importance of respecting structural heterogeneity in loss design.\n\n---\n\n### **[Q2, Q3]: Is the performance gain of PMLF primarily due to the decomposition process, or the specific assignment of quadratic and logarithmic loss functions? Have alternative loss combinations been tested to justify the proposed design?**\n\n**Response**. We appreciate the reviewer’s insightful questions. We agree that distinguishing the contribution of decomposition and loss design is crucial to validate our method’s effectiveness.\n\nTo disentangle these factors, we conducted an ablation study on five benchmark datasets (ETTh1/2, ETTm1/2, Weather). We kept the backbone model fixed and only varied the loss strategies, comparing: \n\n- **Unified loss baselines**: Directly applying MSE, MAE, Huber, and DTW directly applied to raw series.\n\n- **Structure-aware variants**: applying decomposition with various seasonal-trend loss assignments, including:quadratic for both(Q+Q), logarithmic for both (L+L), or swapped (L + Q) and Ours(Q+L)\n\nDue to space constraints, we report average forecasting MSE/MAE across four prediction horizons (96/192/336/720) based on Amplifier. A subset of results is summarized below:\n\n| Loss|PMLF||MSE||Huber||DTW||MAE||Q+Q||L+L||L+Q||\n|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|\n|Metric|MSE|MAE|MSE|MAE|MSE|MAE|MSE|MAE|MSE|MAE|MSE|MAE|MSE|MAE|MSE|MAE|\n|ETTh1|**0.443**|**0.434**|0.464|0.447|0.475|0.445|0.463|0.443|0.459|0.447|0.480|0.453|0.455|0.436|0.481|0.451|\n|ETTh2|**0.356**|**0.385**|0.383|0.409|0.372|0.399|0.378|0.405|0.369|0.394|0.377|0.404|0.363|0.387|0.387|0.386|\n|ETTm1|0.391|**0.386**|**0.384**|0.398|0.388|0.392|0.389|0.399|0.394|**0.386**|0.390|0.399|0.396|0.387|0.389|0.398|\n|ETTm2|**0.275**|**0.316**|0.281|0.327|0.277|0.322|0.282|0.328|0.276|0.318|0.313|0.347|0.277|0.322|0.279|0.325|\n|Weather|**0.242**|**0.263**|0.247|0.274|0.244|0.268|0.245|0.273|0.243|**0.263**|0.248|0.274|0.245|0.268|0.248|0.274|\n\nBased on the above results, we draw the following conclusions:\n- The performance gain is not solely attributed to decomposition, as the decomposition-based methods （e.g Q+Q, L+L） alone do not consistently outperform unified loss functions across all datasets;\n- Our proposed loss combination (quadratic for seasonal and logarithmic for trend) consistently outperforms all other combination  variants, including Q+Q, L+L, and L+Q;\n- Therefore,  the observed performance gains arise from the synergy between a meaningful decomposition and a component-aware loss design. \n\nWe hope this empirical analysis and the above insights sufficiently address your concern. Your thoughtful comments were highly valuable and helped us improve the clarity and rigor of our work.\n\n---\n\n### **[Q4]: How does the proposed adaptive weighting compare against fixed or heuristic alternatives?**\n\n**Response**. We appreciate the reviewer’s suggestion to evaluate the benefits of our adaptive weighting mechanism through direct comparison with fixed or heuristic strategies.\n\nTo investigate this, we conducted ablation experiments on two representative datasets (ETTh2 and ETTm2), using the same model backbone (Amplifier). We compared our dynamic strategy (PMLF) with heuristic loss weight settings, where the seasonal-to-trend ratio ranged from 1:9 to 9:1.\n\n|loss strategy||dynamic strategy ||1:9||2:8||3:7||4:6||5:5||6:4||7:3||8:2||9:1||\n|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|\nDataset|horizon|MSE|MAE|MSE|MAE|MSE|MAE|MSE|MAE|MSE|MAE|MSE|MAE|MSE|MAE|MSE|MAE|MSE|MAE|MSE|MAE|\nETTh2|96|**0.283**|**0.332**|0.292|0.335|0.293|0.335|0.293|0.335|0.292|0.334|0.290|0.333|0.290|0.332|0.289|**0.332**|0.290|0.334|0.294|0.337\n||192|**0.353**|0.379|0.358|0.379|0.357|0.379|0.358|0.379|0.356|0.378|0.354|**0.377**|0.356|0.378|0.359|0.380|0.363|0.381|0.370|0.384|\n||336|**0.385**|0.405|0.390|0.407|0.386|**0.403**|0.392|0.408|0.388|0.405|0.400|0.409|0.387|0.403|0.392|0.406|0.394|0.408|0.400|0.411|\n||720|**0.403**|**0.423**|0.407|0.426|0.420|0.434|0.420|0.433|0.405|0.500|0.424|0.425|0.405|0.425|0.406|0.425|0.409|0.427|0.413|0.429|\n||Avg|**0.356**|**0.385**|0.362|0.387|0.364|0.388|0.366|0.389|0.360|0.404|0.367|0.386|0.360|0.385|0.362|0.386|0.364|0.388|0.369|0.390\n||\nETTm2|96|**0.172**|0.251|0.173|0.251|0.173|0.250|0.173|0.250|0.171|0.250|0.172|0.250|**0.171**|**0.249**|0.172|0.250|0.172|0.250|0.173|0.252|\n||192|**0.237**|**0.293**|0.239|0.294|0.238|0.294|0.238|0.294|0.239|0.294|0.239|0.294|0.240|0.295|0.238|0.294|0.239|0.295|0.239|0.295|\n||336|**0.299**|**0.332**|0.300|0.333|0.300|0.332|0.300|0.334|0.300|0.333|0.302|0.334|**0.299**|0.333|**0.299**|**0.332**|**0.299**|**0.332**|0.301|0.334|\n||720|**0.391**|**0.387**|0.397|0.392|0.394|0.392|0.396|0.389|0.399|0.391|0.398|0.391|0.400|0.399|0.398|0.392|0.397|0.391|0.398|0.392|\n||Avg|**0.275**|**0.316**|0.277|0.318|0.276|0.317|0.277|0.317|0.277|0.317|0.278|0.317|0.278|0.319|0.277|0.317|0.276|0.317|0.278|0.318|\n\nThe results show that while certain fixed ratios (e.g., 7:3 on ETTm2) may yield comparable performance in specific horizons, no single fixed setting consistently outperforms the adaptive approach. In contrast, our method achieves the best or near-best results across all settings without manual tuning.\n\nWhile fixed heuristics can perform well in some cases, they lack adaptability and require manual tuning. In contrast, our softmax-based strategy dynamically learns to balance seasonal and trend losses in a data-driven manner, offering robust and generalizable performance across diverse scenarios. We appreciate the reviewer’s suggestion, which prompted us to conduct this comparative study and reinforced the value of our design.\n\n---\n\n### **[W1] The dilemma of decomposing assumptions**\nWe thank the reviewer for highlighting this important point. Indeed, PMLF operates under the assumption that the time series can be approximately decomposed into a low-frequency trend and a high-frequency seasonal component. We explicitly acknowledge this limitation in Section 5 of the paper.\n\nHowever, the real-world datasets used in our experiments, like Weather and ETTm2, exhibit clear signs of imperfect separability, including irregular fluctuations, external shocks, and nonstationary behaviors. Despite this, PMLF still demonstrates strong performance across all benchmarks, which suggests practical robustness even under imperfect separability. To further mitigate the rigidity of decomposition,  we introduced a learnable moving average (LMA) mechanism, as shown in the last column of Table 3. Unlike fixed-kernel smoothing, LMA allows the decomposition boundary to be **learned** rather than fixed, enabling the model to dynamically adapt to structural irregularities and regime shifts in a data-driven manner.\n\nWhile we agree that perfect decomposition is not always feasible, the learnable decomposition strategy helps PMLF remain robust and adaptable in practice. We appreciate the reviewer for prompting us to emphasize this point more clearly.\n\n---\n\n### **[W2] Misleading \"Physics-Guided\" Terminology**\n\nWe appreciate the reviewer’s concern regarding the terminology. In our work, the term “physics-guided” is used in a qualitative and analogical sense, rather than implying strict enforcement of physical laws or differential equations, as done in physics-informed neural networks (PINNs). Specifically, our seasonal loss adopts a quadratic form motivated by the potential energy of a harmonic oscillator (Eq. 3–7), while the trend loss uses a logarithmic form inspired by the diminishing structural response seen in irreversible relaxation systems (Eq. 8–10). These formulations are intended to provide interpretability and design intuition, rather than enforce hard physical constraints.\n\n We understand how the term \"physics-guided\" may lead to unintended interpretations. To prevent confusion, we will revise the terminology throughout the paper and use “physics-inspired” instead, which more accurately reflects the nature and intent of our design.\n\n---\n\n### W3.3 Please refer to the answer Q4\n- **W3.1** Please refer to the response to Q1.\n- **W3.2** Please refer to the responses to Q2 and Q3.\n- **W3.3** Please refer to the response to Q4."}, {"review_id": "92XiIH61ap", "reviewer": "Reviewer_eskT", "summary": "This paper proposes PMLF, a novel loss framework tailored for time series with structurally heterogeneous components (i.e., seasonal and trend). Drawing inspiration from physical systems, the authors design two specialized loss functions: a quadratic form for high-frequency seasonal components (analogous to harmonic oscillations), and a logarithmic form for low-frequency trend components (analogous to structural relaxation). A softmax-based dynamic weighting mechanism is further introduced to balance the optimization across these two losses. Empirical results on a wide range of datasets and backbones suggest that PMLF improves over standard MSE losses and structure-aware baselines.", "questions": "1. How critical is the choice of quadratic and logarithmic loss functions? Have you evaluated alternative forms (e.g., Huber, MAE, or frequency-weighted losses) to assess the necessity of the proposed physical analogies?\n2. How does the dynamic weighting mechanism evolve during training? Can you provide a plot of the trend vs. seasonal weights over epochs, and clarify whether this weighting stabilizes or oscillates?\n3. Have you tested the learnable decomposition strategy in a controlled way? It’s briefly mentioned but not quantitatively compared with the fixed-kernel approach. Does it improve performance on nonstationary or high-frequency data?\n4. Can PMLF be extended to handle more than two components? For time series that include noise, regime shifts, or multiple periodicities, can the loss framework be generalized to supervise more than just trend and seasonal parts?\n5. What are the computational implications of the proposed loss? Does the log-based trend loss introduce optimization difficulties (e.g., vanishing gradients), or slow down convergence compared to MSE?", "limitations": "1. The effectiveness of PMLF relies on the assumption that the input time series can be cleanly decomposed into distinguishable trend and seasonal components. In real-world applications such as medical signals, high-frequency financial data, or irregular event streams, this assumption may not hold, limiting the applicability of the method.\n\n2. The current design focuses exclusively on a two-part decomposition (trend and seasonal). It is unclear whether the framework can be extended to time series with more complex or hierarchical structures (e.g., noise, abrupt shifts, multi-scale events).", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths\n1. The paper rightly identifies a limitation in current time series forecasting models that apply a uniform loss across structurally different components.\n2. The physical analogies provide an interesting perspective and yield two loss forms that align well with different structural sensitivities in time series.\n3. PMLF is model-agnostic and integrates cleanly into a wide variety of forecasting architectures, as shown in the experiments.\n4. The method demonstrates performance improvements across several benchmarks and baselines, and includes ablation studies and robustness analysis.\nWeaknesses\n1. The proposed use of quadratic and logarithmic losses is based on analogies to molecular dynamics (vibrations and structural drift). While intuitive, this physical metaphor lacks formal statistical or optimization-theoretic grounding. There is no analysis comparing the proposed loss forms to standard alternatives (e.g., MAE, Huber, or frequency-aware metrics), nor a discussion on why these loss functions are optimal or necessary for the intended structural components.\n2. The softmax-based weighting scheme introduces a dynamic balance between trend and seasonal losses during training. However, the paper does not analyze how this mechanism behaves over time, whether it converges, or if it may cause instability due to loss scale mismatch. The lack of gradient norm or loss trajectory analysis limits confidence in its robustness.\n3. The trend-seasonal separation relies on fixed-kernel moving average filters, which may not be sufficient for data with nonstationary, overlapping, or nonlinear structures. Although a learnable decomposition is mentioned, it is not systematically evaluated or compared, leaving open the question of how critical the decomposition quality is to overall performance.", "was_revised": true, "final_justification": "Basically, most questions are answered in the rebuttal, while some points may go beyond this current work. I would like to keep my current rating.", "rebuttal": "### **[Q1]How critical is the choice of quadratic and logarithmic loss functions? Have you evaluated alternative forms (e.g., Huber, MAE, or frequency-weighted losses) to assess the necessity of the proposed physical analogies?**\n\n**Response.** Thank you for the thoughtful question. We address the reviewer’s concern from two perspectives: (1) the empirical necessity of using the proposed physically-motivated losses, and (2) the theoretical motivation for pairing quadratic and logarithmic forms in a structured way.\n\nOur design of PMLF stems from the assumption that seasonal and trend components exhibit distinct error characteristics:\n- Seasonal parts behave like bounded oscillations, where symmetric deviations around the mean are natural—thus, quadratic penalty (convex and sensitive to amplitude) is appropriate.\n- Trend parts correspond to slow structural drifts—susceptible to outliers but insensitive to local noise—where logarithmic loss ensures smoother convergence and robustness to scale shifts.\n\nTo assess whether our proposed PMLF design is indispensable or replaceable by alternative loss functions, we conducted two classes of ablation:\n - **Unified Loss Baselines:** MSE, MAE, Huber, and DTW Applied directly to the time series.\n - **Component-wise Structured Losses:** Pairings applied to seasonal/trend components, including: Q+Q (Quadratic for both components), L+L (Logarithmic for both) and MSE+MAE, Huber+MSE: Non-physics-aligned pairings.\n\nWe report the average results across four prediction lengths on five benchmark datasets using the Amplifier backbone. Due to space limits, we show average results of four horizons below:\n\n| Dataset |ETTh1| ETTh2| ETTm1| ETTm2| Weather |\n|-|-|-|-|-|-|\n|Loss Combination|MSE/MAE|MSE/MAE|MSE/MAE|MSE/MAE|MSE/MAE|\n| **PMLF (Q+L)**|**0.443 / 0.434**|**0.356 / 0.385**| 0.391 / **0.386**|**0.275 / 0.316** | **0.242 / 0.263** |\n|Q+Q|0.480 / 0.453|0.377 / 0.404| 0.390 / 0.399| 0.313 / 0.347| 0.248 / 0.274|\n|L+L|0.455 / 0.436| 0.363 / 0.387| 0.396 / 0.387| 0.277 / 0.322| 0.245 / 0.268|\n|MSE+MAE|0.453 / 0.439|0.366 / 0.390|0.392 / 0.390|**0.275** / 0.317|0.243 / 0.264|\n|Huber+MSE|0.480 / 0.453|0.383 / 0.408|0.391 / 0.400|0.281 / 0.327 |0.248 / 0.274|\nUnified MSE|0.478 / 0.451|0.383 / 0.409|**0.384** / 0.398|0.281 / 0.327|0.247 / 0.274|\nUnified MAE|0.459 / 0.447|0.369 / 0.394|0.394 / **0.386**|0.276 / 0.318|0.243 / **0.263**|\nUnified Huber|0.475 / 0.445|0.372 / 0.399|0.388 / 0.392|0.277 / 0.322|0.244 / 0.268\n|Unified DTW|0.463 / 0.443|0.378 / 0.405|0.389 / 0.399|0.282 / 0.328|0.245 / 0.273|\n\nOur results show that:\n- PMLF consistently ranks first or second across all datasets and metrics, outperforming both standard losses and structured pairings.\n- While certain combinations like MSE+MAE achieve competitive results on specific datasets (e.g., ETTm2), they fail to generalize broadly.\n- Unified baselines such as DTW or Huber underperform compared to component-wise modeling, confirming the benefit of decomposition.\n- Symmetric pairings (Q+Q, L+L) lack the asymmetry needed to capture seasonal vs. trend divergence and often suffer from training instability.\n\nWe conclude that the proposed Q+L formulation is not only empirically effective but also grounded in meaningful physical structure. The consistent generalization across settings underscores its necessity beyond heuristic loss combinations. We hope this analysis addresses the reviewer’s concerns and demonstrates the necessity of our approach.\n\n---\n\n### **[Q2]:How does the dynamic weighting mechanism evolve during training? Can you provide a plot of the trend vs. seasonal weights over epochs, and clarify whether this weighting stabilizes or oscillates?**\n\n**Response.** Thank you for your insightful question. The stability of the dynamic weighting mechanism is crucial to ensuring balanced learning between seasonal and trend components.\n\nWhile submission constraints prevent us from including figures, we provide the following numerical evidence to support the convergence behavior. We monitored the evolution of seasonal and trend weights across four prediction lengths on ETTh1 using the Amplifier backbone. In all cases, the weights converge within a few dozen iterations. Once stabilized, the weights exhibit low variance and do not oscillate unpredictably throughout training. The table below summarizes the **mean ± standard deviation** of the weights after stabilization:\n\n|Horizon|Seasonal Weight| Trend Weight|\n|-|-|-|\n|96 | 0.4810 ± 0.0039| 0.5190 ± 0.0039|\n|192| 0.4779 ± 0.0040| 0.5221 ± 0.0040|\n|336| 0.4854 ± 0.0037| 0.5146 ± 0.0037|\n|720| 0.4679 ± 0.0045| 0.5321 ± 0.0045|\n\nThe low standard deviations (<0.005) across all horizons demonstrate that the learned weights remain highly stable once convergence is reached. These results suggest that the model learns to assign interpretable and horizon-aware importance to seasonal and trend components in a consistent and non-oscillatory manner. We will include full training curves in the final version to visualize the smooth convergence process. We hope this addresses your concern.\n\n---\n\n### **[Q3]Effectiveness of learnable decomposition compared to fixed-kernel smoothing.**\n\n**Response.** Thank you for raising this insightful question. We fully agree that a controlled and quantitative comparison is necessary to assess the value of our learnable decomposition strategy, particularly against traditional fixed-kernel smoothing.\n\nWe conducted such a comparison in Table 3 (last column: “w/ LMA”) of the main paper. However, we understand it might have been overlooked due to its compact format. To clarify, we extract the relevant entries below, highlighting the difference between:\n- **PMLF:** Our proposed model using fixed-kernel decomposition (smoothed trend).\n- **PMLF with LMA**: A variant that replaces the smoothing operation with a learnable mask aggregation (LMA), allowing data-driven decomposition under the same training configuration.\n\nThese results are measured across three representative datasets:\n\n|Dataset|Horizon|PMLF|PMLF with LMA|\n|-|-|-|-|\n|ETTh2|96|**0.283/0.332**|0.289/**0.332**|\n||192|**0.353/0.379**|0.357/0.379|\n||336|**0.385/0.405**|0.391/0.406|\n||720|**0.403/0.423**|0.406/0.425|\n|ETTm2|96|0.172/0.251|**0.171/0.250**|\n||192|**0.237/0.293**|0.239/0.295|\n||336|0.299/0.332|**0.296/0.331**|\n||720|**0.391/0.387**|0.393/0.390|\n|Weather|96|0.156/**0.193**|**0.155**/0.194|\n||192|**0.209/0.243**|**0.209/0.243**|\n||336|**0.262/0.283**|0.264/**0.283**|\n||720|**0.339/0.334**|0.342/**0.334**|\n\nThese results show that LMA achieves comparable or better performance across the majority of settings, particularly on datasets with more nonstationary or high-frequency characteristics (e.g., ETTm2 and Weather). Even without access to fixed filtering kernels, the model is able to autonomously discover effective decomposition patterns, suggesting that LMA enhances adaptability while maintaining stability.\n\n---\n\n### **[Q4]: Can PMLF be extended beyond two components (trend + seasonal) to model noise, regime shifts, or multiple periodicities?**\n**Response.** We appreciate the reviewer’s insightful question. Indeed, the current formulation of PMLF focuses on a two-component decomposition (seasonal and trend), which aligns well with many standard forecasting benchmarks. However, the underlying philosophy of PMLF is structurally extensible. If the time series is decomposed into $n$ components (e.g., trend, multiple periodicities, noise, regime shifts), the framework can be readily generalized by assigning a dedicated loss to each component:\n\n$$\\mathcal{{L}_{total}}=\\sum_i^n \\lambda_i \\mathcal{L_i}$$\n\nThis formulation remains fully compatible with our dynamic weighting strategy, enabling adaptive focus across components. Moreover, our use of physically guided loss design can be generalized, for instance, using entropy-like losses for noise components or change-point-aware losses for regime shifts.\n\n### **[Q5]: Does the proposed log-based loss for trend modeling incur optimization issues (e.g., vanishing gradients) or slow convergence compared to MSE?**\n\n**Response:** We thank the reviewer for raising this important concern. To ensure training stability and computational efficiency, we carefully designed the logarithmic loss function to avoid numerical pitfalls and empirically evaluated its behavior during optimization.\n\nThe trend loss $\\mathcal{L}_\\tau$ adopts a logarithmic form:\n$$\\mathcal{L}\\_\\tau = \\frac{1}{CH} \\sum\\_{c=1}^C\\sum\\_{h=1}^H \\log(1 + | \\hat{\\tau}^{(c)}\\_{T+h} - \\tau^{(c)}\\_{T+h} | )$$\n\nwith gradient:\n\n$$\n\\frac{d \\mathcal{L}_{\\tau}}{de} = \\frac{1}{1 + e} \\in (0,1]\n$$\n\nAlthough $\\mathcal{L}_\\tau$ is non-convex, it is Lipschitz-continuous and differentiable. Thus, the gradient vanishes gradually with growing error, which provides a soft constraint on large deviations while preserving stability during training. Therefore, the logarithmic trend term loss will not encounter optimization difficulties.\n\nDue to the inability to upload images, we provided the loss variation of each epoch validation set trained with different loss functions on ETTh2 using Amlfier\n\n|Epoch|1|2|3|4|5|6|7|8|9|10\n|-|-|-|-|-|-|-|-|-|-|-|\n|MSE|0.2183|0.2160|0.2150|0.2127|0.2146|0.2138|0.2139|0.2139|0.2140|0.2141\n|PMLF|0.1382|0.1364|0.1357|0.1352|0.1352|0.1351|0.1351|0.1351|0.1351|0.1351\n\nIn summary, the log-based trend loss does not suffer from vanishing gradients or convergence slowdown, and instead provides smoother convergence, improved robustness, and more stable optimization. \n\nWe sincerely thank the reviewers for their thorough and constructive feedback. We have carefully addressed all concerns regarding loss design, weight stability, learnable decomposition, and optimization behavior through both controlled experiments and theoretical analysis. Additional materials and clarifications will be provided in the final version to further enhance transparency and reproducibility."}, {"review_id": "ISkiRWnHTh", "reviewer": "Reviewer_mT8C", "summary": "This paper proposes a Physics-guided Multiscale Loss Framework (PMLF) that decomposes time series into seasonal and trend components, then designs physics-inspired loss functions for each (quadratic loss for seasonal components mimicking molecular vibration energy, logarithmic loss for trend components simulating molecular drift relaxation energy). By incorporating dynamic weighting to balance their optimization, PMLF aims to address the limitations of traditional uniform loss functions in simultaneously capturing short-term fluctuations and long-term trends. Experimental results demonstrate PMLF's ability to improve prediction accuracy across multiple baseline models.", "questions": "1. Could you provide formal analysis on the convergence properties of your proposed loss functions? Specifically, under what conditions does the dynamic weighting mechanism guarantee balanced optimization between seasonal and trend components?", "limitations": "NA", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths: \n1.The paper provides a well-motivated theoretical foundation, drawing clear analogies between molecular dynamics and time series components. The loss functions are derived from physical principles, and the dynamic weighting mechanism is carefully justified\n\n2.Extensive experiments across multiple datasets and diverse baselines demonstrate consistent improvements in both MSE and MAE. \n\nWeaknesses:\n1. The role of the balancing parameter (β) is briefly discussed, but its impact on different datasets/models could be analyzed further.\n\n2. computational complexity is not adequately addressed, as the decomposition operations and dynamic weighting mechanism may introduce non-negligible overhead.", "was_revised": true, "final_justification": "The revisions have addressed my concerns regarding the details. Based on my original positive score, I will maintain my score.", "rebuttal": "Thank you very much for your positive evaluation and appreciation of our universality, and empirical performance. Here are our responses to specific questions and concerns.\n\n### **[Q]: Could you provide formal analysis on the convergence properties of your proposed loss functions? Specifically, under what conditions does the dynamic weighting mechanism guarantee balanced optimization between seasonal and trend components?**\n\n**Response.** We appreciate the reviewer’s insightful question. We address this from two perspectives: (1) the convergence behavior of the proposed loss functions, and (2) the conditions under which the dynamic weighting mechanism ensures balanced optimization between seasonal and trend components.\n\n- **Convergence Properties.** Our total loss is defined as:\n\n$$\n\\mathcal{L}\\_{\\text{total}} = \\lambda\\_s \\mathcal{L}_s + \\lambda\\_{\\tau} \\mathcal{L}\\_{\\tau}\n$$\nThe seasonal loss $\\mathcal{L}_s$ is defined as a convex quadratic penalty:\n\n$$\n\\mathcal{L}\\_s = \\frac{1}{CH} \\sum\\_{c=1}^C \\sum\\_{h=1}^H( \\hat{s}^{(c)}\\_{T+h} - s^{(c)}\\_{T+h})^2\n$$\n\nwhich ensures stable convergence under gradient-based optimization. The trend loss $\\mathcal{L}_\\tau$ adopts a logarithmic form:\n\n$$\n\\mathcal{L}\\_\\tau = \\frac{1}{CH} \\sum\\_{c=1}^C\\sum\\_{h=1}^H \\log(1 + | \\hat{\\tau}^{(c)}\\_{T+h} - \\tau^{(c)}\\_{T+h} | )\n$$\n\nwith gradient:\n\n$$\n\\frac{d \\mathcal{L}_{\\tau}}{de} = \\frac{1}{1 + e} \\in (0,1]\n$$\n\nAlthough $\\mathcal{L}\\tau$ is non-convex, it is Lipschitz-continuous and differentiable, ensuring that the total loss $\\mathcal{L}{\\text{total}}$ remains smooth. Empirically, we observe consistent monotonic decrease of $\\mathcal{L}_{\\text{total}}$ across all tasks, validating its practical convergence behavior. Due to submission restrictions, we will supplement the convergence curve in the future.\n\n- **Balanced Optimization via Dynamic Weighting.** To ensure balanced optimization between the seasonal and trend components, we design a softmax-based dynamic weighting strategy:\n\n$$ \\lambda\\_s = \\frac{\\exp(\\beta (\\mathcal{L}\\_s^\\dagger - m))}{\\exp(\\beta (\\mathcal{L}\\_s^\\dagger - m)) + \\exp(\\beta (\\mathcal{L}\\_\\tau^\\dagger - m))}, \\quad\n    \\lambda\\_\\tau = \\frac{\\exp(\\beta (\\mathcal{L}\\_\\tau^\\dagger - m))}{\\exp(\\beta (\\mathcal{L}\\_s^\\dagger - m)) + \\exp(\\beta (\\mathcal{L}\\_\\tau^\\dagger - m))} $$\n\nwhere $\\mathcal{L}s^\\dagger$ and $\\mathcal{L}\\tau^\\dagger$ denote the detached seasonal and trend losses (no gradient flow),$m = \\max(\\mathcal{L}s^\\dagger, \\mathcal{L}\\tau^\\dagger)$ ensures numerical stability. This mechanism possesses several important properties:\n\n- **Responsiveness to imbalance.** If one component (e.g., trend) exhibits significantly larger loss, its corresponding weight $\\lambda_\\tau$ will increase, prompting the model to focus more on the underperforming part. This behavior aligns with the principle of adaptive error compensation.\n- **Stability through max-normalization.** The use of $m = \\max(\\cdot)$ shifts both logits before exponentiation, avoiding disproportionate weight assignment. For example, if one loss is much larger than the other, direct softmax could lead to almost binary weights (e.g., [1, 0]); the max-shifting avoids this by anchoring the maximum at zero and reducing the gap between logits, thus ensuring neither component is neglected.\n- **Temperature control with $\\beta$.** The sharpness of this rebalancing is governed by $\\beta$: A higher $\\beta$ emphasizes the relative difference between the two losses, leading to sharper focus on the harder task. A lower $\\beta$ produces smoother balancing, allowing more gradual joint optimization.\n\nIn our experiments, we set $\\beta=1$ for stability, but as shown in Figure 5, the results remain robust within a wide range of $\\beta \\in [0.2, 5]$, highlighting the insensitivity of the mechanism to this hyperparameter.\n\n- **Empirical Evidence.** We further monitor the evolution of dynamic weights across four prediction lengths on ETTh1 using Amplifier as the backbone. Although minor fluctuations are observed in the early iterations due to initial loss disparity, the weights consistently converge to a stable state within a few dozen steps. The table below reports the mean±std after convergence, confirming that the dynamic weighting mechanism promotes a balanced optimization between seasonal and trend losses without oscillation or dominance.\n\n|Horizon|Seasonal Weight| Trend Weight|\n|-|-|-|\n|96 | 0.4810 ± 0.0039| 0.5190 ± 0.0039|\n|192| 0.4779 ± 0.0040| 0.5221 ± 0.0040|\n|336| 0.4854 ± 0.0037| 0.5146 ± 0.0037|\n|720| 0.4679 ± 0.0045| 0.5321 ± 0.0045|\n\n---\n\n### **[W1] The role of the balancing parameter (β) impact on different datasets/models.**\n\n**Response.** We thank the reviewer for the insightful question regarding the balancing coefficient $\\beta$, which controls the relative weight between the seasonal and trend loss components. To assess its impact, we conduct controlled experiments across three backbone models (TimeMixer, Amplifier, TimeXer) and two benchmark datasets (ETTh2 and Weather). Results are summarized below.\n\nPerformance across different models (ETTh2)\n|method|TimeMixer||Amplifier||TimeXer||\n|----|---|---|---|---|---|---|\n|$\\beta$|MSE|MAE|MSE|MAE|MSE|MAE|\n|0.2|0.364|0.389|**0.359**|0.385|**0.361**|0.389|\n|0.4|**0.363**|**0.388**|0.360|**0.384**|0.362|0.389|\n|0.6|**0.363**|**0.388**|**0.359**|**0.384**|0.363|**0.388**|\n|0.8|0.365|0.389|0.362|0.386|0.362|**0.388**|\n|1.0|0.369|0.391|0.363|0.387|0.364|0.389|\n|2.0|0.372|0.392|0.363|0.388|0.365|0.390|\n|3.0|0.370|0.392|0.371|0.393|0.367|0.392|\n|5.0|0.372|0.395|0.374|0.396|0.375|0.398|\n\nAcross all models, best performance consistently occurs within $\\beta$ ∈ [0.4, 0.8]. When $\\beta$ is too large, model performance degrades significantly. This suggests that overly emphasizing one component can impair learning balance and generalization.\n\nDataset|ETTh2||Weather||\n|-|-|-|-|-\n$\\beta$|MSE|MAE|MSE|MAE|\n0.2|**0.361**|0.389|0.242|0.264|\n0.4|0.362|0.389|0.242|0.264|\n0.6|0.363|**0.388**|0.240|0.262|\n0.8|0.362|**0.388**|0.240|0.262|\n1.0|0.364|0.389|**0.239**|**0.261**|\n2.0|0.365|0.390|0.241|0.263|\n3.0|0.367|0.392|0.241|0.263|\n5.0|0.375|0.398|0.244|0.267|\n\nOn both ETTh2 and Weather, the optimal $β$ also lies between 0.4 and 1.0, with Weather favoring a slightly larger $β$. Notably, performance drops when $β$ becomes too large, consistent with our prior observations.\n\n**Conclusion.** Overall, the balancing coefficient \\$\\beta\\$ performs best within the range of **0.4 to 1.0** across models and datasets. When \\$\\beta\\$ becomes excessively large, model performance consistently deteriorates. This confirms that overly skewing the loss weights disrupts the balance between seasonal and trend learning, leading to suboptimal results.\n\n---\n\n### **[W2] computational complexity is not adequately addressed, as the decomposition operations and dynamic weighting mechanism may introduce non-negligible overhead.**\n\nWe appreciate the reviewer’s concern regarding potential computational overhead. We would like to clarify that both the decomposition and dynamic weighting mechanism are implemented in a lightweight and efficient manner, **incurring negligible additional cost relative to the backbone network.**\n\n- **computational complexity.** We adopt a simple moving average filter for time series decomposition, with a computational complexity of $O(T)$, where $T$ is the sequence length. This decomposition method is memory-efficient, parallelizable, and incurs minimal time cost in practice.\n\n- **Dynamic Weighting Cost.** The softmax-based dynamic weighting is applied once per batch and involves only two scalar loss $\\mathcal{L}\\_s$ and $\\mathcal{L}\\_{\\tau}$. The additional computation amounts to two exponentials and one division, making the complexity constant-time $O(1)$ and negligible compared to the deep learning models.\n\nTo quantify its practical cost, we report the average runtime introduced by our loss function (including decomposition and weighting) on the ETTh1 dataset with a batch size of 64, under different prediction horizons:\n\nHorizon|Cost Time(ms)|\n-|-|\n96|0.6186±0.0294\n192|0.6223±0.0325\n336|0.6474±0.0234\n720|0.6599±0.0235\n\nEven for long-horizon forecasts, the additional cost remains below 0.7 ms per batch, which is negligible compared to the total training time.\n\n---\n\nIn summary, we provide both theoretical grounding (Lipschitz continuity, descent lemma) and empirical validation (convergence curves, β-sensitivity, runtime profiling) to guarantee balanced optimization and negligible overhead.  .Additional materials and clarifications will be provided in the final version to further enhance transparency and reproducibility."}, {"review_id": "tAuuzCZosf", "reviewer": "Reviewer_jE1W", "summary": "This paper introduces a Physics-guided Multiscale Loss Framework (PMLF) for time series forecasting, targeting the structural differences between high-frequency seasonal fluctuations and low-frequency trend dynamics. Unlike existing methods that use a unified loss, PMLF decomposes time series into seasonal and trend components, applying a quadratic loss to the former and a logarithmic loss to the latter. A softmax-based weighting mechanism dynamically balances their contributions. By aligning supervision with the underlying physical behavior of components, PMLF improves forecasting performance across multiple benchmarks, effectively addressing structural heterogeneity in multiscale time series data.", "questions": "Please refer to the weakness.", "limitations": "yes", "rating": 3, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths:\n1. The paper introduces a physics-inspired, component-specific loss formulation that aligns well with the distinct behaviors of seasonal and trend components, offering a novel perspective on supervision in time series forecasting.\n\n2. PMLF can be seamlessly applied across diverse backbone architectures, demonstrating strong generalization and consistent performance improvements on multiple benchmarks.\n\n\nWeaknesses:\n1. The definitions of s(t) and \\tau are not provided. Please clarify their relationship to Y^s  and Y^t, respectively. \n\n2. Equation (7), labeled as the “seasonal loss,” appears mathematically equivalent to the standard MSE applied to the seasonal component. This raises concerns about whether the proposed energy-based loss introduces a genuinely new formulation or merely a reinterpretation of MSE through a physical analogy. The manuscript should clarify whether the energy-based perspective yields any theoretical or empirical advantage beyond reinterpretation. Additionally, it remains unclear how this formulation addresses phase or amplitude shifts. Including a concrete example could significantly aid comprehension.\n\n3. The relaxation energy loss, expressed via a logarithmic penalty (Equation 10), seems to introduce robustness to outliers due to its dampened gradient for large deviations. However, the manuscript lacks a clear explanation of its practical implication. A discussion on its theoretical advantage and empirical behavior in presence of temporal drift or outliers would be beneficial.\n\n4. Since DTW is designed to handle phase shift, a comparison or discussion is warranted. What are the advantages of the proposed loss functions relative to DTW?\n\n5. The trend prediction shown in Figure 1 exhibits unexpected oscillatory behavior. This is counterintuitive given the assumed smoothness of trend components. Providing quantitative evidence or theoretical justification for this phenomenon would strengthen the paper’s claims. Furthermore, the concept of \"temporal drift\" requires clearer definition. How is it distinct from standard deviation?\n\n6. The paper currently lacks a comprehensive comparison against structure-aware baselines across all seven datasets. Including such experiments would provide a more convincing argument for the generalizability and effectiveness of the proposed method.", "was_revised": false, "final_justification": "", "rebuttal": "**[W1] The definitions of s(t) and \\tau and their relationship to Y^s and Y^t, respectively.**\n\n**Response.** In our framework, the multi variate time series $Y(t) \\in \\mathcal{R}^{C \\times T}$ is decomposed into two components via seasonal-trend decomposition:\n\n$Y=Y_s+Y_\\tau$\n\nwhere $Y_s$ represents the seasonal component and $Y_\\tau$ denotes the trend component. For notational clarity, **$s(t)$ and $\\tau(t)$ respectively refer to one individual channel of the seasonal $Y_s$ and trend components $Y_\\tau$**. We sincerely apologize for not explaining this more clearly in the original manuscript, and will revise the notation and provide explicit definitions in the updated version to ensure better readability.\n\n---\n\n**[W2] concerns about whether the proposed energy-based loss introduces a genuinely new formulation or merely a reinterpretation of MSE through a physical analogy. how this formulation addresses phase or amplitude shifts.**\n\n\n**Response.** Thank you for raising this critical and insightful question. We would like to clarify that although Equation (7) takes the form of a quadratic penalty, it is not a heuristic reinterpretation of MSE. It is derived from Potential energy of harmonic oscillator：$U_{vib}(x)=k/2(x-x_0)^2$ and the seasonal loss is defined as the difference of potential energies $\\Delta U_{vib}$.\n\n**Empirical Validation.** To verify whether this physically inspired design yields real benefits, we compared several alternative combinations of loss functions for seasonal/trend components:Quadratic + Quadratic (Q+Q), Logarithmic + Logarithmic (L+L), Swapped (L+Q) and Heuristic combinations including MSE+MAE, MAE+Huber, and Huber+MSE. Due to character limitations, we report average forecasting results across four prediction lengths (96, 192, 336, 720) using Amplifier as the backbone:\n\n|Loss|PMLF||Q+Q||L+L||L+Q||MSE+MAE||MAE+Huber||Huber+MSE||\n|---|---|---|---|---|---|---|---|---|---|---|---|---|-|-|\n|Metric|MSE|MAE|MSE|MAE|MSE|MAE|MSE|MAE|MSE|MAE|MSE|MAE|MSE|MAE|\n|ETTh1|**0.443**|**0.434**|0.480|0.453|0.455|0.436|0.481|0.451|0.453|0.439|0.462|0.440|0.480|0.453|\n|ETTh2|**0.356**|**0.385**|0.377|0.404|0.363|0.387|0.387|0.386|0.366|0.390|0.369|0.396|0.383|0.408|\n|ETTm1|0.391|**0.386**|0.390|0.399|0.396|0.387|0.389|0.398|0.392|0.390|0.387|0.393|0.391|0.400|\n|ETTm2|0.275|0.316|0.313|0.347|0.277|0.322|0.279|0.325|0.275|0.317|**0.274**|0.316|0.281|**0.252**|\n|Weather|**0.242**|**0.263**|0.248|0.274|0.245|0.268|0.248|0.274|0.243|0.264|0.244|0.270|0.245|0.274|\n\nThese results suggest that PMLF is not only theoretically motivated but consistently outperforms all other configurations, including heuristic and purely statistical losses.\n\n**Sensitivity to Phase and Amplitude Shifts.** Seasonal losses mainly come from two types: amplitude shift and phase shift.\n- **Amplitude Shift.** A baseline offset in amplitude causes consistent deviations over time, leading to cumulative penalties. The larger the offset, the stronger the loss, encouraging the model to align the predicted baseline with the ground truth.\n\n- **Phase Shift.** Even slight phase misalignments can significantly amplify the structural loss, as the predicted peaks and troughs become temporally misaligned with the ground truth. This sharp sensitivity to phase errors causes the loss to escalate rapidly, encouraging the model to correct temporal misalignment even when the overall waveform shape is preserved. \n\nTo illustrate this, we use a sine function $y=sin(2\\pi x)$, and compute mse loss variations under controlled shifts:\n\n|$\\Delta$ amplitude |$\\Delta loss_a$||$\\Delta$ phase|$\\Delta loss_p$|\n|---|---|---|---|---|\n|0.1|0.005||$5^{\\circ}$|0.004\n|0.2|0.02||$10^{\\circ}$|0.015\n|0.5|0.125||$30^{\\circ}$|0.132\n\nThese results confirm that even slight amplitude or phase deviations lead to noticeable increases in loss, demonstrating that our formulation effectively penalizes these structural mismatches.\n\n---\n\n**[W3] Clarification on the  theoretical advantage and empirical behavior of the relaxation energy loss**\n\n**Response.** Thank you for your professional and constructive feedback. While the manuscript outlines the relaxation energy loss design, we agree its robustness implications deserve further clarification.\n\n\n- **Theoretical Advantage.** The relaxation loss adopts a logarithmic form $\\log(1 + | \\hat{\\tau} - \\tau |)$, which imposes moderate penalities on large deviations. Unlike MSE or MAE, this design yields smoother gradients and enhances robustness against outliers and temporal drift, which is critical for stable long-term trend modeling.\n- **Empirical behavior.** The real-world datasets used in our study naturally exhibit temporal drift and contain substantial outliers due to noise and irregular trends. We compare the relaxation loss with commonly used robust losses such as MAE and Huber. As shown below, our method consistently achieves superior performance under such noisy conditions, validating its effectiveness in suppressing the influence of anomalous deviations while preserving structural trend signals.\n\n|Dataset|PMLF(MSE/MAE)|Huber(MSE/MAE)|MAE(MSE/MAE)\n|-|-|-|-\n|ETTh1|**0.443**/**0.434**|0.475/0.445|0.459/0.447\n|ETTh2|**0.356**/**0.385**|0.372/0.399|0.369/0.394\n|ETTm1|0.391/**0.386**|**0.388**/0.392|0.394/0.386\n|ETTm2|**0.275**/**0.316**|0.277/0.322|0.276/0.318\n|Weather|**0.242**/**0.263**|0.244/0.268|0.243/0.263\n\n---\n\n**[W4] What are the advantages of the proposed loss functions relative to DTW**\n\n**Response.** Thanks your professional and insightful opinion. Introducing more loss functions, including DTW, for comparison will further demonstrate the advantages of our loss function design. we provide a comprehensive comparison highlighting PMLF’s theoretical and empirical advantages.\n\n- **Differentiability** DTW is inherently non-differentiable and thus unsuitable as a loss for gradient-based optimization. Although Soft-DTW enables differentiability via smoothing, it introduces approximation bias. In contrast, PMLF is fully differentiable, enabling precise and stable gradient updates for both seasonal and trend components.\n- **Efficiency.** Soft-DTW has $O(T^2)$ time and memory complexity, which scales poorly for long sequences. PMLF operates in linear time $O(T)$, enabling fast training and deployment.\n- **Structural Decomposition.** DTW aligns sequences holistically based on overall shape but lacks the ability to disentangle different structural sources of error. PMLF explicitly decomposes the prediction into seasonal and trend components, allowing the model to receive targeted supervision for each structural mode.\n- **Loss Effectiveness.** In experiments across five datasets （below table）, PMLF consistently achieves lower MSE and MAE compared to Soft-DTW, demonstrating its superior effectiveness in guiding model training. \n\n|Dataset|**PMLF (MSE/MAE)**|Soft-DTW (MSE/MAE)|\n|---|---|---|\n|ETTh1|**0.443 / 0.434**|0.463 / 0.443|\n|ETTh2|**0.356 / 0.385**| 0.378 / 0.405|\n|ETTm1|0.391 / **0.386**| **0.389** / 0.399|\n|ETTm2|**0.275 / 0.316**| 0.282 / 0.328|\n|Weather|**0.242 / 0.263**| 0.245 / 0.273|\n\n**[W5]Clarification on oscillatory trend prediction and definition of temporal drift.**\n\n**Response.**\n-  **Oscillatory Trend Prediction.** The mild oscillations observed in the predicted trend result from the scale-dependent nature of trend and seasonality decomposition. Patterns that appear as trends over short time windows, such as hours or days, may correspond to low-frequency seasonal variations when viewed across longer time frames. This ambiguity is common in real-world time series. Our relaxation loss could accommodate gradual structural drift while suppressing rapid fluctuations, allowing the model to capture stable yet realistic trend dynamics without excessive smoothing.\n  \n- **Definition of Temporal Drift.** In time series, temporal drift refers to the gradual, cumulative shift in the baseline level of a time series over time. It is distinct from standard deviation, which measures short-term variability around the local mean. To illustrate this, we refer to the well-known AirPassengers dataset (recorded monthly). As shown below, while the within-year variance (±std) remains relatively stable, the month;y mean increases significantly from 126.67 in 1949 to 197.00 in 1952:\n\n|year|1949|1950|1951|1952\n|-|-|-|-|-\n|mean±std|126.67±13.72|139.67±19.07|170.17±18.44|197.00±22.97\n---\n**[W6] The paper currently lacks a comprehensive comparison against structure-aware baselines across all seven datasets. Including such experiments would provide a more convincing argument for the generalizability and effectiveness of the proposed method**\n\n**Response.**  Thank you for your comprehensive and professional advice. We have expanded the comparison of structural losses to seven datasets. The following is a comparison of the mean values of their four different prediction lengths:\n\n| Loss|PMLF||FreDF||TIDE-Q||\n|---|---|---|---|---|---|---|\n|Metric|MSE|MAE|MSE|MAE|MSE|MAE|\n|ETTh1|**0.443**|**0.434**|0.450|0.435|0.478|0.449||\n|ETTh2|**0.356**|**0.385**|0.364|0.391|0.366|0.394|\n|ETTm1|**0.391**|**0.386**|0.393|0.400|**0.391**|0.395|\n|ETTm2|0.275|**0.316**|**0.273**|0.317|0.275|0.318|\n|Weather|**0.242**|**0.263**|0.248|0.273|0.249|0.274|\n|ECL|**0.170**|**0.259**|**0.170**|0.262|0.176|0.270|\n|Traffic|**0.475**|**0.310**|0.481|0.303|0.482|0.316|\n\nThe above results indicate that although FreDF and TIDE-Q loss have some competitiveness on a very small number of datasets, our proposed PMLF is applicable to all datasets, which demonstrates the universality and effectiveness of our method.\n\n---\n\nWe appreciate the reviewers’ thoughtful comments. In response, we have carefully examined and addressed all aspects of loss design, ablation comparison, conceptual grounding, and training behavior, supported by experiments and analysis. The final version will include additional resources to improve reproducibility and understanding."}]

Author rebuttal:

Meta-review: The paper introduces a Physics-guided Multiscale Loss Framework (PMLF) for time series forecasting. The approach decomposes time series into seasonal and trend components and applies distinct loss functions: a quadratic loss for seasonal fluctuations and a logarithmic loss for long-term trends, drawing inspiration from analogies in molecular vibration and drift energies. An adaptive softmax-based weighting balances these two losses dynamically. The framework is designed to be model-agnostic, and experiments across multiple benchmarks show consistent improvements in forecasting accuracy.

Strengths:  This paper addresses an interesting issue in time series forecasting --- unified losses often fail to capture heterogeneous time series dynamics. The proposed framework is lightweight, model-agnostic, and easy to integrate into existing forecasting methods. Empirical evaluations span several datasets and backbone architectures, with improvements that are often consistent and practically relevant.

Weaknesses:  A major concern about the proposed loss function modeling is that the physics analogy is metaphorical rather than strictly grounded, which reduces the conceptual novelty. In addition, some experiments (e.g., oscillations in trend predictions) are not fully explained, and additional comparisons with structure-aware baselines would strengthen the evaluation. The adaptive weighting mechanism remains under-analyzed, lacking discussion of stability or convergence properties.

Rebuttal: The authors have provided extensive experiments and detailed clarification to the concerns. Ablation studies on the choices of loss functions and other heuristic alternatives for the weighting of loss functions. A better interpretation of the physical meaning the designs is also provided. 

In summary, this paper is a borderline paper. Reviewers raised concerns about the limited novelty, the metaphorical nature of the physics framing, and unclear benefits of the adaptive weighting. In the rebuttal, the authors provided further ablation studies, clarifying implementation choices and showing that the adaptive scheme contributes positively. While some skepticism remains, especially regarding the theoretical grounding, the additional evidence helped mitigate the initial concerns. On balance, I find the empirical strengths and practical utility outweigh the limitations, leading to a positive but cautious recommendation.
Decision: Accept (poster)