Recovering Lost Details: Multi-Scale Frequency Compensation for Long-Term Time Series Forecasting

Runmin Zou; Siyi Xie; Yaohui Huang; Yun Wang

cs.AISep 21, 2026

Recovering Lost Details: Multi-Scale Frequency Compensation for Long-Term Time Series Forecasting

Authors: Runmin Zou, Siyi Xie, Yaohui Huang, Yun Wang

Organizations: Central South University Changsha, Hunan, China

Abstract

Long-term time series forecasting has made significant progress by leveraging multi-scale information to capture hierarchical temporal patterns and model long-range dependencies. However, temporal downsampling in existing multi-scale methods inevitably smooths detailed temporal fluctuations, and this information loss is further aggravated by their emphasis on dominant trends across scales, resulting in insufficiently expressive representations. To address this, we propose a Multi-Scale Wavelet Mixing (MWMixer) model, which incorporates a Bidirectional Frequency-Bands Mixing strategy to recover lost temporal details across scales, enabling complementary cross-scale information interactions. Then, a Dynamic Scale-Adaptive Fusion module learns time-varying weights for each scale to fuse multi-scale forecasts into the final prediction, enhancing the flexibility of multi-scale aggregation. In addition, a cross-scale consistency loss aligns each coarse-scale prediction with the interval-averaged fine-scale outputs, while a multi-scale supervision loss enforces prediction accuracy at each scale, promoting consistent learning across scales. Extensive experiments on seven real-world datasets demonstrate that MWMixer achieves competitive performance in long-term forecasting.

Figures & tables

Figure 1. Motivation of the proposed framework. (a) Downsampling preserves global trends from fine to coarse scale but erodes fine-grained local details. (b) Unlike Discrete Wavelet Transform (DWT) sequence reduction, the Maximal Overlap DWT (MODWT) maintains full temporal resolution. (c) Energy maps reveal that frequency-domain compensation Mixing successfully restores information lost during downsampling compared to the pre-compensation state w/o Mixing. A three-part motivation diagram. Panel (a) depicts a fine-scale time series becoming progressively smoother after downsampling to coarser scales, illustrating the loss of local fluctuations. Panel (b) contrasts DWT, whose coefficients become shorter after decomposition, with MODWT, whose coefficient sequences retain the original temporal resolution. Panel (c) compares coefficient-energy heatmaps before and after frequency-band mixing and shows the recovery of energy patterns weakened by downsampling.

Figure 2. Overview of the proposed model architecture. The model processes multi-scale time-series inputs and decomposes them in the frequency domain via MODWT, enabling explicit multi-band representations. The BFBM module captures cross-scale interactions from coarse to fine and vice versa, followed by DSAF to integrate multi-scale predictions. A left-to-right architecture diagram of MWMixer. The input sequence is downsampled into multiple temporal scales, and each scale is decomposed by MODWT into approximation and detail coefficients. Bidirectional Frequency-Bands Mixing exchanges information between adjacent scales in both coarse-to-fine and fine-to-coarse directions. The reconstructed scale-specific representations are processed by predictors, and Dynamic Scale-Adaptive Fusion combines their forecasts using time-varying scale weights to produce the final prediction.

Method		MWMixer (ours)		CASA (2025)		DUET (2025)		AMD (2025)		TimeMixer (2024)		SparseTSF (2024)		iTransformer (2024)		FilterNet (2024)		FITS (2024)		MICN (2023)
Metric		MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
ETTh1	96	0.368	0.389	0.382	0.404	0.400	0.405	0.383	0.393	0.372	0.398	0.385	0.390	0.385	0.404	0.382	0.402	0.384	0.392	0.421	0.431
	192	0.416	0.417	0.427	0.428	0.430	0.425	0.435	0.422	0.437	0.427	0.414	0.420	0.439	0.434	0.430	0.429	0.434	0.421	0.474	0.487
	336	0.450	0.436	0.474	0.451	0.479	0.452	0.475	0.443	0.489	0.457	0.476	0.439	0.492	0.463	0.472	0.451	0.475	0.442	0.569	0.551
	720	0.440	0.449	0.484	0.475	0.527	0.494	0.479	0.464	0.479	0.471	0.461	0.454	0.509	0.493	0.481	0.473	0.462	0.459	0.770	0.672
	Avg	0.418	0.422	0.441	0.439	0.459	0.444	0.443	0.430	0.444	0.438	0.434	0.425	0.456	0.448	0.441	0.438	0.438	0.428	0.558	0.535

Table 1. Multivariate long-term forecasting results (MSE/MAE) under prediction horizons

H\in\{96,192,336,720\}

with a fixed look-back length

L=96

. “Avg” denotes the average performance across all horizons. Lower values indicate better performance. The best results are highlighted in bold , and the second-best results are underlined . The “1st count” row reports the number of best results achieved by each method in this table.

Dataset	Timesteps	Sample Frequency	Dimension
ETTh1	17,420	1 hour	7
ETTh2	17,420	1 hour	7
ETTm1	69,680	15 min	7
ETTm2	69,680	15 min	7
Weather	52,696	10 min	21
Electricity	26,304	1 hour	321

Table 2. Dataset statistics.

Method		Ours		w/o BFBM		w/o DetailMix		w/o ApproxMix		w/o Scale Loss		w/o Supervision		w/o Consistency
Metric		MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
ETTh1	96	0.368	0.389	0.372	0.387	0.369	0.389	0.372	0.387	0.367	0.388	0.368	0.388	0.368	0.389
	192	0.416	0.417	0.424	0.417	0.415	0.416	0.428	0.420	0.415	0.419	0.424	0.417	0.416	0.417
	336	0.450	0.436	0.479	0.445	0.474	0.444	0.482	0.444	0.466	0.441	0.479	0.442	0.450	0.436
	720	0.440	0.449	0.495	0.471	0.442	0.449	0.491	0.468	0.460	0.460	0.479	0.468	0.460	0.460
	Avg	0.418	0.422	0.442	0.430	0.425	0.424	0.443	0.429	0.427	0.427	0.437	0.428	0.423	0.425

Table 3. Ablation study of BFBM, the Scale Loss, and their internal branches. The best results are highlighted in bold .

Figure 3. Ablation study on DSAF and learned scale weighting using average results. A comparison of average forecasting errors for the complete model and variants using different scale-fusion strategies. The plot contrasts Dynamic Scale-Adaptive Fusion with alternatives that remove adaptive fusion or replace learned time-varying scale weights. Lower MSE and MAE values indicate better forecasting performance.

Method	Ours		Direct Add		Residual Add		Concat + Linear		Concat + MLP		Nearest		MLP Ups.		DWT		Fourier
Metric	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
ETTh1	0.418	0.422	0.441	0.431	0.443	0.430	0.438	0.428	0.432	0.430	0.417	0.422	0.432	0.430	0.596	0.518	0.440	0.433
ETTm1	0.373	0.390	0.377	0.392	0.378	0.391	0.380	0.392	0.379	0.396	0.373	0.390	0.374	0.391	0.430	0.420	0.377	0.392
Exchange	0.340	0.391	0.383	0.411	0.373	0.407	0.364	0.402	0.380	0.412	0.348	0.397	0.373	0.406	0.422	0.444	0.357	0.398

Table 4. Average results of cross-scale mixing and upsampling/decomposition strategies over four prediction horizons

H\in\{96,192,336,720\}

. Full results are provided in Table 5 in Appendix B. The best results are highlighted in bold .

Figure 4. Hyperparameter sensitivity analysis on ETTh1 and Exchange. Each subplot reports MSE under different values of one hyperparameter, while the other two hyperparameters are fixed. A set of sensitivity plots for ETTh1 and Exchange. In each subplot, the horizontal axis varies one model hyperparameter while the remaining hyperparameters are fixed, and the vertical axis reports mean squared error. The plots show how forecasting performance changes across the tested settings and identify ranges in which the model remains stable.

Figure 5. Visualization of adaptive weight allocation by the DSAF module on the ETTh1 dataset for (a) Variable 5 and (b) Variable 1. Top panels display the comparison between predictions and ground truth, while bottom panels show the multi-scale prediction sequences and the time-varying fusion weights learned by the model. Two groups of time-series plots for variables 5 and 1 of ETTh1. The upper plot in each group overlays the final prediction and ground truth over the forecast horizon. The lower plots display predictions from individual temporal scales together with the fusion weights assigned to those scales at each time step, illustrating that DSAF changes its scale allocation over time and across variables.

Panel of Figure 8 — Figure 8. Scalability analysis on Weather. The length analysis fixes the number of variables to 21, while the variable analysis fixes the lookback window to 96. Scalability plots on the Weather dataset. One analysis varies the sequence length while holding the number of variables at 21, and the other varies the number of variables while holding the lookback window at 96. The plotted resource or runtime measurements compare how MWMixer and baseline models scale as each input dimension increases.

Method		Ours		Direct Add		Residual Add		Concat + Linear		Concat + MLP		Nearest		MLP Ups.		DWT		Fourier
Metric		MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
ETTh1	96	0.368	0.389	0.376	0.392	0.375	0.390	0.371	0.388	0.378	0.397	0.367	0.389	0.368	0.389	0.569	0.490	0.370	0.391
	192	0.416	0.417	0.427	0.419	0.429	0.420	0.426	0.418	0.416	0.417	0.415	0.417	0.416	0.417	0.582	0.502	0.426	0.424
	336	0.450	0.436	0.481	0.445	0.484	0.445	0.473	0.441	0.481	0.448	0.449	0.436	0.453	0.438	0.616	0.526	0.482	0.448
	720	0.440	0.449	0.481	0.468	0.484	0.465	0.480	0.465	0.453	0.456	0.439	0.449	0.492	0.479	0.620	0.556	0.484	0.471
ETTm1	96	0.310	0.351	0.311	0.353	0.312	0.352	0.314	0.353	0.323	0.362	0.310	0.351	0.313	0.353	0.375	0.390	0.314	0.354

Table 5. Full results of cross-scale mixing, upsampling, and decomposition strategies. MSE/MAE are reported under four prediction horizons

H\in\{96,192,336,720\}

. The best results are highlighted in bold .

Panel of Figure 9 — Figure 9. Prediction Results of Our Method (i.e. MWMixer) and Baseline Models on the ETTm2 Dataset. Four aligned time-series prediction plots on ETTm2 compare MWMixer, CASA, AMD, and TimeMixer over the same forecast interval. Each panel overlays a model prediction with the ground-truth sequence, enabling comparison of how closely each method follows the temporal level, fluctuations, and turning points of the target series.

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Method	Ours		w/o DSAF		Learned Scale Weight
Metric	MSE	MAE	MSE	MAE	MSE	MAE
ETTh1	0.418	0.422	0.431	0.427	0.422	0.424
ETTm1	0.373	0.390	0.375	0.390	0.375	0.391
Exchange	0.340	0.391	0.380	0.412	0.359	0.404

Appendix

Table 6. Average ablation results for DSAF and learned scale weighting.

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SEMixer: Semantics Enhanced MLP-Mixer for Multiscale Mixing and Long-term Time Series Forecasting

Xu Zhang, Qitong Wang, Peng Wang, Wei Wang

Shanghai Key Laboratory of Data Science, College of Computer Science and Artificial Intelligence, Fudan University, Shanghai, China · Harvard University, Cambridge, Massachusetts, United States

Long-Term Time Series Forecasting Multi-Scale Feature Fusion Multiscale Time-Series Modeling

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Modeling multiscale patterns is crucial for long-term time series forecasting (TSF). However, redundancy and noise in time series, together with semantic gaps between non-adjacent scales, make the efficient alignment and integration of multi-scale temporal dependencies challenging. To address this, we propose SEMixer, a lightweight multiscale model designed for long-term TSF. SEMixer features two key components: a Random Attention Mechanism (RAM) and a Multiscale Progressive Mixing Chain (MPMC). RAM captures diverse time-patch interactions during training and aggregates them via dropout ensemble at inference, enhancing patch-level semantics and enabling MLP-Mixer to better model multi-scale dependencies. MPMC further stacks RAM and MLP-Mixer in a memory-efficient manner, achieving more effective temporal mixing. It addresses semantic gaps across scales and facilitates better multiscale modeling and forecasting performance. We not only validate the effectiveness of SEMixer on 10 public datasets, but also on the \textit{2025 CCF AlOps Challenge} based on 21GB real wireless network data, where SEMixer achieves third place. The code is available at the link https://github.com/Meteor-Stars/SEMixer.

Abstract

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