Title: PMLF: A Physics-Guided Multiscale Loss Framework for Structurally Heterogeneous Time Series
Venue: NeurIPS 2025
Accepted: yes
Keywords: Time Series Forecasting

Physics-Guided Learning

Energy-Based Loss Design

Seasonal-Trend Decomposition
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

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.

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 highfrequency 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 [29], seasonal-trend decomposition[39, 38], and frequency-aware representations [45, 41, 5, 28]. While these approaches improve the model's ability to represent multiscale signals, they apply a uniform loss functions during optimization, with limited attention paid to how different temporal structures are supervised [3, 15, 10].

Figure 1 illustrates the divergent error behaviors of seasonal and trend components. Seasonal components are highly sensitive to minor phase shifts or amplitude distortions, which rapidly accumulate over time and significantly degrade predictive accuracy. In contrast, trend components

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 [36, 33]. However, most existing approaches adopt a single loss, such as MSE or MAE, applied indiscriminately across the entire sequence [13, 17, 2]. This uniform treatment implicitly assumes that all temporal components contribute equally to forecasting error. As a 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.

To resolve the mismatch in optimization objectives between seasonal and trend components in multiscale time series, we draw inspiration from structural response mechanisms in molecular systems [37]. Specifically, we design 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 in 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 seasonal and trend components in time series.

Motivated by this analogy, we propose a Physics-guided Multiscale Loss Framework (PMLF) that supervises each component using a loss function grounded in its corresponding physical dynamics. As illustrated in Figure 2, 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 a structurally consistent objective. The seasonal branch is optimized using a 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-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 to harmonic oscillations and irreversible structural drift in physical systems.

Based on this insight, we formulate a physics-guided loss framework that assigns quadratic penalties to seasonal errors and logarithmic potentials to trend deviations, reflecting their distinct structural behaviors. We further integrate a softmax-based 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.

2 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 decomposition, and hierarchical temporal abstraction. Explicit decomposition methods, such as Autoformer [38], 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-domain approaches model signals in the spectral domain to identify dominant periodicities and multiscale frequency structures, as

Baselines

Physics-Guided MultiscaleLossFramework

Dynamic Loss Balancing

美E

MLP

Prediction

P-Seasonal

P-Trend

eβx

MMMMn

Decompose

imMM

e

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e

eo

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Input X

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min（Ls,L)

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IDecompose

ConvldxN

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Transformer

Concat

Self Attention

C

Scaled Dot-Product Attention

=Ns*Ls+*Lt

demonstrated by FEDformer [45], FreTS[42], and WPMixer[21]. Hierarchical models [17, 18, 8] instead organize inputs into layered or patch-wise representations to capture dependencies across temporal scales. PatchTST[22], for example, achieves strong performance by applying self-attention to fixed-length patches, enabling joint modeling of short- and long-term dynamics. Although these models incorporate multiscale structures at the architectural level, they typically adopt uniform loss functions across all components. This mismatch between representation and supervision may limit the model's ability to 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 to improve sequence-level accuracy. These approaches can be broadly categorized into alignmentbased and structure-aware objectives. Alignment-based losses, including DTW [1], Soft-DTW [3], DILATE [15], 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 [11] 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 physics-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 a loss function aligned with its temporal dynamics. This structure-aware supervision allows the model to optimize multiscale representations more effectively, leading to improved longrange forecasting performance.

Molecular

Time Series

Loss Design

Response Mode

Representation

MolecularVibration

SeasonalPattern

HarmonicEnergyLoss

(High-frequency)

Loss

Value

t

Strict

Constraint

error

External Influences

Trend Pattern

RelaxationEnergyLoss

(Low-frequency)

+

Extemal Field

Value

Loss

→

Drift

Tolerant

Constraint

t

error

3.2 Problem Formulation

Given a multivariate time series X = { x 1 , x 2 , ..., x T } ∈ R T × C , the model f θ aims to predict the next H time steps conditioned on a historical window of length L :

To capture multiscale temporal structure, the prediction and ground truth sequences are decomposed into two components via seasonal-trend decomposition:

where Y τ and Y s denote the trend and seasonal component, respectively. This decomposition is 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, as it lacks structured patterns and thus cannot be effectively supervised.

3.3 Physics-Inspired Structural Supervision

In molecular systems, physical responses to perturbations generally fall into two categories: intrinsic vibrations and externally induced structural drift [37, 23, 9]. As shown in Figure 3, the former corresponds to 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 or mechanical stress, leading to 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 a structure-consistent manner, we replace uniform loss penalties with energy-based objectives whose growth profiles align with the dynamic behavior of each component.

Energy-Based Loss for Seasonal Oscillation. In molecular systems, small displacements from equilibrium are governed by harmonic motion, where the potential energy increases quadratically with deviation:

where x is the system state, x 0 the equilibrium position, and k the stiffness constant. This response structure reflects the behavior of seasonal time series components, which exhibit high-frequency, reversible oscillations and are sensitive to phase or amplitude shifts.

Given predicted and target seasonal values ˆ s ( t ) and s ( t ) , the potential energy difference is:

By expanding the square and reorganizing terms, the energy difference can be rewritten as:

Due to the symmetric oscillation of s ( t ) around its equilibrium, the coupling term ( s ( t ) -x 0 )(ˆ s ( t ) -s ( t )) has an expected value close to zero across a complete seasonal cycle:

Consequently, the expected energy difference simplifies to a dominant quadratic penality, which defines the seasonal loss as:

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, we introduce a diminishing structural force model:

which captures the saturation of structural resistance under persistent displacement.

The integral of this response yields the effective structural energy potential:

which reflects the sublinear accumulation of internal structural energy associated with irreversible drift. Linear and constant terms are omitted as they yield non-informative or structurally insensitive gradients. We retain only the logarithmic penalty, which preserves structural sensitivity and yields the final trend loss:

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 two components, causing the trend branch to dominate gradient updates and suppressing high-frequency corrections. To address this, we adopt a softmax-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-order effects.

The total loss is defined as a weighted sum:

where the adaptive weights λ s and λ τ are computed as:

Here, L † = SG ( L ) denotes the gradient-detached loss to avoid backpropagating through the weighting mechanism. The scalar m = max( L † s , L † τ ) ensures numerical stability, and the parameter β = 1 unless otherwise specified.

4 Experiments

4.1 Setup

Datasets To comprehensively evaluate the effectiveness of PMLF on real datasets, we conducted experiments on eight multivariate time series datasets, including ETT (4 sub-datasets) [44], Weather, Electricity, Traffic, and Solar-Energy [12]. Detailed dataset descriptions are provided in Appendix.

Baselines To evaluate the ability of PMLF on different baselines, we extensively selected Transformer, Mamba, MLP, CNN, and their hybrid methods, including Amplifier [5], TimeXer [31], S-Mamba [32], iTransformer [19], TimeMixer [29], PatchTST [22], and TimesNet [39].

MSEPMLF

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0.600

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-2.84%

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TimesNet

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 a fixed moving average filter with kernel size 25, and we also support an optional learnable version implemented as a plug-in module. Our loss framework is model-agnostic and can be seamlessly integrated into a wide range of forecasting backbones, including MLPs [5], Transformers [26], and Mamba [6, 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 PyTorch with four NVIDIA RTX 4090 GPUs.

4.2 Main Results

We evaluate the proposed PMLF loss across a set of state-of-the-art forecasting models on eight benchmark datasets covering diverse application domains. The standard MSE loss is used as the baseline for comparison. As shown in Table 1, the proposed model-agnostic loss framework yields consistent improvements over the standard MSE baseline across five recent state-of-the-art models. This demonstrates that PMLF effectively guides the optimization of both short-term seasonal patterns and long-term trends, which frequently coexist in real-world time series. Especially, the performance improvement of the model on the MAE index is often higher than that on the MSE index, mainly due to the long-term control of trend. For example, among a total of 64 comparisons in iTransformer, the PMLF improvement item accounted for 93.75% (60/64). In addition, the improvement in the 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-Energy, PMLF achieves significant improvements in both MSE and MAE. Additional results covering all models and datasets are provided in the appendix.

4.3 Comparison with Other Loss Functions

We compare PMLF with traditional loss function and specially designed objectives proposed for time series forecasting. These approaches typically enhance standard objectives by introducing additional

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(c) ETTh2 Trend

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(e) ECL Seasonal

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PMLF

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6

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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-domain comparisons. In contrast, PMLF [27] supervises decomposed seasonal and trend components individually at the loss level, aiming to guide structural learning rather than refining a unified temporal loss. As shown in Table 2, 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 can 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 6, 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-frequency oscillations as well as long-term drift.

4.5 Ablation Studies

As shown in Table 3, we conduct ablation studies on the ETTh2, ETTm2, and Weather datasets using Amplifier as the backbone. The results show that removing either the seasonal or trend component from the loss formulation leads to a 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 fixed-kernel version, suggesting that the proposed loss is compatible with both static and adaptive decomposition strategies. Finally, as illustrated in Figure 7, we assess the robustness of the hyperparameter β on the ETTh1 and Weather datasets using the TimeXer model. The results indicate that the performance remains stable across a wide range of β values, from 0.2 to 5, demonstrating that our loss design is insensitive to this choice.

5 Conclusion

Weintroduce a Physics-Guided 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, a 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 is to explore adaptive decomposition mechanisms or hybrid supervision strategies that enhance the framework's applicability to structure-invariant or single-scale time series.

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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 (Electricity Transformer Temperature): The ETT dataset contains 7 variables of electricity transformer temperature from July 2016 to july 2018. There are 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 to 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-test set split protocol used in TimesNet, where the train, validation, and test datasets are strictly divided according to chronological order to make sure there are no data leakage issues. As for the forecasting settings, we fix the length of the lookback series as 96 and the prediction length varies in {96, 192, 336, 720}.

A.2 Implementation Details

All the experiments are implemented in PyTorch 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 is decomposed into seasonal and trend components using the moving average method. The kernel size refers to the setting in Autoformer, which is 25. But in order to avoid fixed kernel sizes affecting time series with different sampling frequencies, we designed a hybrid expert decomposition mechanism that uses a 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.

B Baselines

To evaluate the general applicability of PMLF, we compare it with a varied collection of leading timeseries forecasters that span the principal architectural families: state-space (S-Mamba), Transformer (iTransformer, TimeXer, PatchTST), multilayer perceptron (Amplifier, TimeMixer), and convolutional neural network (TimesNet). The core ideas of these baselines are outlined below.

Amplifier: An energy-amplification block heightens weak spectral bands, the spectrum is renormalised, and parallel seasonal and trend heads with a lightweight channel-interaction module enhance performance on low-signal datasets.

TimeXer: Targets mixed endogenous and exogenous forecasting. Patch-wise self-attention models the target series, variate-wise cross-attention injects exogenous cues, and global endogenous tokens integrate the two streams, improving robustness to abrupt external shocks.

S-Mamba: Represents each time step with a single per-channel token and applies a 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.

iTransformer: Reassigns Transformer roles by applying attention across variables to learn crosschannel 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 a parallel head simultaneously outputs all future steps. The absence of attention provides GPU-friendly speed without sacrificing accuracy.

PatchTST: Divides long sequences into fixed-length temporal patches that serve as Transformer tokens and shares encoder parameters among channels. This strategy reduces attention complexity from O ( L 2 ) to O (( L/P ) 2 ) while retaining local semantics.

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 a single modelling philosophy but instead reflect a broad enhancement of time-series learning.

C More Experimental Results

C.1 Robustness Assessment

To examine the robustness of our framework, the Amplifier baseline was trained five times with independent random seeds. Table 5 reports the mean performance together with the corresponding standard deviations. The consistently small variances confirm that the Amplifier yields repeatable results, underscoring the robustness of the proposed approach.

C.2 Parameter Sensitivity

To examine how the dynamic-weighting coefficient β influences forecasting accuracy, we conducted a grid search over β ∈ { 0 . 2 , 0 . 4 , 0 . 6 , 0 . 8 , 1 , 1 . 5 , 3 , 5 } for two representative networks: TimeXer and TimeMixer. Figure 7 reports the resulting MSE and MAE on the ETTh2 and Weather datasets. For TimeXer (top row) and TimeMixer (bottom row), the error curves remain nearly flat across the entire range, and the optimal β 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 β . These results confirm the robustness of our framework with respect to this hyper-parameter.

(e) TimeMixer:ETTh1

(f) TimeMixer:ETTh1

(g) TimeMixer:Weather

(h) TimeMixer:Weather

0.44

H=96

Z6T=H --

-H=336

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96=H

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- H=336

0Z/=H

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= H=336

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β

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-- H=192

±- H=336

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-- H=192

☆- H=336

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-I-H=192

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+

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C.3 Additional Evaluation on Classical Forecasting Networks

Additionally, we assesses the architecture-independent effectiveness of PMLF on three widely used forecasting models: the Transformer-based PatchTST, the selective state-space model S Mamba, and the convolutional network TimesNet. For each architecture, the original MSE objective was replaced 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 error metrics (MAE, MSE), confirming that PMLF improves forecasting accuracy and robustness even in classical network settings that are not covered by the main set of state-of-the-art architectures.

D Visualization

As shown in Figure 9 and 10, we provide a visual comparison between PMLF and MSE on six benchmark datasets: ETTh2, ETTm2, Weather, ECL, 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

(a) ETTh1:MSE

(b) ETTh2:MSE

(c) ETTm1:MSE

(d) ETTh1:MAE

(e) ETTh2:MAE

(f) ETTm1:MAE

(g) ETTm2:MSE

(h) ECL:MSE

(i) Solar:MSE

(j) ETTm2:MAE

(k) ECL:MAE

(l) Solar:MAE

MSEPMLF

MSEPMLF

MSEPMLF

0.600

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%19-

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MSE1PMLF

MSEPMLF

MSEPMLF

-2.59%

-10.1%

0.400

-10.0%

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0%

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-4.53%

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TimesNet

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TimesNet

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0.400

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0.400

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-4.28%

-15.99%

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-4.94%

2.40%

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-11.74%

MAE

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0.000

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PatchTST

TimesNet

S-Mamba

PatchTST

TimesNet

S-Mamba

PatchTST

TimesNet

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.

(a) ETTh2 Prediction

(b) ETTh2 Seasonal

(c) ETTh2 Trend

(d) ETTm2 Prediction

(e) ETTm2 Seasonal

(f) ETTm2 Trend

(g) Weather Prediction

(h) Weather Seasonal

(i) Weather Trend

Prediction

Seasonal

Trend

0.75

MSE

1.00

MSE

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MSE

PMLF

PMLF

PMLF

0.50

gt

0.75

6

gt

0.25

0.50

0.0

0.00

anen

0.25

0.00

0.50

0.25

0.4

0.75

1.00

0.50

0.6

20

40

60

80

0.75

20

40

60

80

20

40

60

80

Time Steps

Time Steps

Time Steps

Prediction

Seasonal

Trend

0.6

PMLF

MSE

PMLF

MSE

0.15

PMLF

MSE

0.4

gt

0.10

6

0.4

0.2

0.2

S00

Value

Value

anjeA

0.00

00

0.0

0.2

0.2

0.05

0.10

0.4

20

40

Time Steps

60

80

20

40

Time Steps

60

80

20

40

is a

60

80

Prediction

Seasonal

Trend

0.08

3SW

MSE

0.010

MSE

0.07

PMLF

gt

0.07

gt

PMLF

PMLF

0.06

0.06

0.005

0.05

N

0.05

0o.:4-

anen

0.000

0.03

-E00

0.005

0.02

0.02

100

0.01

0.010

0.00

20

40

60

80

20

40

60

80

20

40

60

80

Time Steps

Time Steps

Time Steps

(a) ECL Prediction

(b) ECL Seasonal

(c) ECL Trend

(d) Traffic Prediction

(e) Traffic Seasonal

(f) Traffic Trend

(g) Solar Prediction

(h) Solar Seasonal

(i) Solar Trend

Prediction

Seasonal

Trend

MSE

1.5

MSE

0.50

MSE

1.0

PMLF

PMLF

PMLF

gt

6

0.25

6

1.0

0.5

0.00

0.5

anjea

00

anjeA

0.25

0.0

0.5

0.50

1.0

0.5

0.75

1.5

1.0

1.00

20

40

Time Steps

60

80

20

40

Time Steps

60

80

1.25

0

20

40

Time Steps

60

80

Prediction

Seasonal

Trend

MSE

MSE

MSE

1.0

PMLF

gt

1.0

PMLF

6

0.4

PMLF

gt

0.5

0.5

0.2

0.0

0.0

anjeA

0.0

0.2

0.5

0.5

0.4

1.0

1.0

20

40

60

80

20

40

60

80

0.6

20

40

60

80

Time Steps

Time Steps

Time Steps

Prediction

Seasonal

Trend

1.5

MSE

PMLF

1.5

MSE

PMLF

0.4

PMLF

MSE

6

6

gt

1.0

1.0

0.2

0.0

0.5

anieA

0.5

0.0

0.0

0.4

0.5

0.5

0.6

0.8

20

40

60

80

20

40

60

80

20

40

60

80

Time Steps

Time Steps

Time Steps

Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis 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.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\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.\n\nQUESTIONS:\nPlease refer to the weakness.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 5, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-04"}

comments: 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.

STRENGTHS AND WEAKNESSES:
Strengths: 
1.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

2.Extensive experiments across multiple datasets and diverse baselines demonstrate consistent improvements in both MSE and MAE. 

Weaknesses:
1. The role of the balancing parameter (β) is briefly discussed, but its impact on different datasets/models could be analyzed further.

2. computational complexity is not adequately addressed, as the decomposition operations and dynamic weighting mechanism may introduce non-negligible overhead.

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
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-06-25

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis 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.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\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.\n\nQUESTIONS:\n1. 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?\n\nLIMITATIONS:\n1. 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).", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis 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\n\nSTRENGTHS AND WEAKNESSES:\n1. **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.\n\nQUESTIONS:\n- 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?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-06"}