TStruct: Learning Shared Temporal Structures for Long-Term Time-Series Forecasting

Yangze Li; Zhengyang Zhou; Qihe Huang; Pengkun Wang; Binwu Wang; Yang Wang

cs.LGOct 13, 2026

TStruct: Learning Shared Temporal Structures for Long-Term Time-Series Forecasting

Authors: Yangze Li, Zhengyang Zhou, Qihe Huang, Pengkun Wang, Binwu Wang, Yang Wang

Organizations: University of Science and Technology of China (USTC), Hefei, China · Suzhou Institute for Advanced Research, USTC, Suzhou, China

Abstract

Long-term time series forecasting requires temporal dependency modeling that remains stable and generalizable under temporal distribution shifts. However, real-world time series are often noisy and non-stationary, making deep learning models prone to fitting incidental temporal interactions in the training data. A key question is whether there exist temporal relationship structures that recur across different temporal positions and variables. We analyze temporal relationship matrices on commonly used time series datasets and find that samples from different temporal positions and different variables exhibit shared dominant temporal relationship structures. This indicates that temporal relationship structures contain patterns that recur stably across samples and variables. To learn such shared structures, we propose Temporal Structure Network (TStruct), a simple yet effective network for directly learning shared and generalizable temporal dependencies. Specifically, TStruct directly parameterizes a set of shared temporal dependency matrices and generates normalized combination weights using dynamic sample information and static variable priors. These weights represent a soft allocation of each sample-variable pair over shared temporal dependency structures, enabling the model to generate adaptive temporal dependency matrices under shared structural constraints, rather than fitting incidental interactions in a fully unconstrained manner. Notably, this simple structure-aware design consistently outperforms several strong baselines across all datasets, achieving superior accuracy while maintaining high computational efficiency.

Figures & tables

Figure 1: Patch-level temporal relationship structures. (a) Construction of patch-level relationship matrices and qualitative comparison between attention-based and TStruct-based temporal dependency modeling. (b,c) Concentrated singular value spectra across samples and variables. (d) Higher spectral concentration in real data than in patch-shuffled and IID Gaussian controls supports reusable temporal structures.

Figure 2: Overall architecture of TStruct. (a) The Temporal Structure Mixer (TSM) generates sample- and variable-specific temporal dependencies by adaptively combining shared temporal dependency structures with dynamic sample information and static variable priors. (b) The PCAHead handles variable heterogeneity by generating variable-aware prediction weights in a parameter-efficient manner.

Methods	TStruct (ours)		DUET [ 2025 ]		PhaseFormer [ 2026 ]		GTR [ 2026 ]		TimeEmb [ 2025 ]		TQNet [ 2025 ]		CycleNet [ 2024a ]		PatchTST [ 2023 ]		DLinear [ 2023 ]
Datasets	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
ETTh1	0.390	0.415	0.398	0.419	0.403	0.415	0.415	0.419	0.418	0.433	0.420	0.428	0.415	0.426	0.417	0.431	0.425	0.439
ETTh2	0.331	0.381	0.334	0.383	0.347	0.389	0.346	0.394	0.353	0.396	0.358	0.399	0.345	0.394	0.351	0.395	0.470	0.468
ETTm1	0.337	0.367	0.338	0.369	0.347	0.374	0.345	0.377	0.345	0.376	0.353	0.388	0.352	0.378	0.349	0.381	0.356	0.378
ETTm2	0.243	0.304	0.248	0.307	0.251	0.314	0.255	0.315	0.248	0.308	0.258	0.318	0.249	0.311	0.256	0.314	0.259	0.324
Weather	0.213	0.249	0.218	0.252	0.223	0.261	0.227	0.264	0.218	0.257	0.230	0.266	0.237	0.277	0.224	0.262	0.242	0.293

Table 1: Average multivariate time series forecasting results over all prediction horizons. The best results are marked in bold red , and the second-best results are marked in blue underline .

Dataset	ETTh1		ETTh2		ETTm1		ETTm2		Weather		Electricity		Traffic
Metric	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
①	0.425	0.444	0.350	0.395	0.376	0.391	0.259	0.318	0.229	0.265	0.162	0.254	0.403	0.270
②	0.403	0.424	0.341	0.384	0.347	0.372	0.248	0.308	0.218	0.254	0.153	0.246	0.382	0.253
③	0.394	0.417	0.339	0.386	0.341	0.368	0.248	0.307	0.215	0.251	0.152	0.245	0.382	0.253
TStruct	0.390	0.416	0.331	0.381	0.337	0.367	0.243	0.303	0.213	0.249	0.151	0.244	0.381	0.252

Table 2: Ablation study on the main components of TStruct. Results are averaged over all prediction horizons. Detailed results for each prediction horizon are provided in Table 11 .

Panel of Figure 3 — Figure 3: Generalization analysis on ETTh1. (a) Models are trained with the earliest timestamp-ordered training samples, evaluating temporal distribution shift. (b) Models are trained with randomly sampled training instances, evaluating sample efficiency.

Figure 4: Effect of look-back length on ETTh1.

Figure 5: Interpretability analysis on the Weather dataset. Different variables exhibit different ACF patterns and assign different weights to the learned shared temporal structures through $Q_{\mathrm{static}}$ . The representative learned structures show distinct patch-level dependency patterns, including broader recurrent relations and local diagonal-dominant continuity.

Appendix figures & tables17 assets

Supplementary material from the paper’s appendix.

Appendix

Dataset	Variables	Timestamps	Split	Prediction Length	Sampling	Information
ETTh1	7	14400	6:2:2	{96,192,336,720}	Hourly	Electricity
ETTh2	7	14400	6:2:2	{96,192,336,720}	Hourly	Electricity
ETTm1	7	57600	6:2:2	{96,192,336,720}	15min	Electricity
ETTm2	7	57600	6:2:2	{96,192,336,720}	15min	Electricity
Weather	21	52696	7:1:2	{96,192,336,720}	10min	Weather
ECL	321	26304	7:1:2	{96,192,336,720}	Hourly	Electricity

Appendix

Table 3: Detailed descriptions of datasets.

(a) Sample side: across temporal windows
Dataset	Patch size $P$	Patch number $N$	Variables $C$	Real	Shuffle	IID	Real/Shuffle	Real/IID
ETTh1	24	30	7	5.59	2.80	2.61	$2.00\times$	$2.14\times$
ETTh2	24	30	7	5.73	2.68	2.61	$2.14\times$	$2.20\times$
ETTm1	24	30	7	4.74	2.41	2.61	$1.96\times$	$1.82\times$
ETTm2	24	30	7	4.27	2.43	2.61	$1.76\times$	$1.64\times$
Weather	24	30	21	5.92	2.44	2.61	$2.43\times$	$2.27\times$

Appendix

Table 4: Shuffle denotes independent patch permutations and IID denotes independent Gaussian windows. Results are averaged over repeated runs of each null control.

Seed		2024		2025		2026		2027		2028		Mean		Std
Dataset	Metric	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
ETTh1	96	0.345	0.380	0.344	0.380	0.345	0.381	0.344	0.390	0.345	0.380	0.345	0.382	0.001	0.004
	192	0.382	0.404	0.383	0.406	0.383	0.405	0.384	0.406	0.384	0.406	0.383	0.405	0.001	0.001
	336	0.407	0.421	0.405	0.421	0.405	0.420	0.407	0.422	0.407	0.421	0.406	0.421	0.001	0.001
	720	0.424	0.453	0.426	0.453	0.431	0.456	0.428	0.456	0.424	0.450	0.427	0.454	0.003	0.003
	Avg.	0.390	0.415	0.390	0.415	0.391	0.416	0.391	0.419	0.390	0.414	0.390	0.416	0.001	0.002

Appendix

Table 5: Stability analysis under different random seeds.

Initialization		Random		Orthogonal		Xavier		Kaiming
Dataset	Metric	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
ETTh1	96	0.366	0.398	0.345	0.380	0.345	0.380	0.345	0.380
	192	0.421	0.439	0.383	0.405	0.383	0.405	0.383	0.405
	336	0.426	0.442	0.407	0.420	0.407	0.420	0.407	0.420
	720	0.497	0.503	0.433	0.456	0.434	0.459	0.430	0.458
	Avg.	0.428	0.446	0.392	0.415	0.392	0.416	0.391	0.416

Appendix

Table 6: Sensitivity analysis under different initialization strategies.

Loss	Huber+FFT						Plain MSE
Method	TStruct		DUET		PhaseFormer		TStruct		DUET		PhaseFormer
Datasets	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
ETTh1	0.390	0.415	0.430	0.450	0.417	0.425	0.410	0.432	0.425	0.447	0.431	0.437
ETTh2	0.331	0.381	0.364	0.410	0.374	0.401	0.349	0.400	0.358	0.405	0.358	0.402
ETTm1	0.337	0.367	0.432	0.426	0.351	0.378	0.345	0.373	0.382	0.409	0.358	0.384
ETTm2	0.243	0.304	0.282	0.342	0.253	0.314	0.246	0.311	0.279	0.338	0.266	0.329

Appendix

Table 7: Forecasting performance under two training objectives with

L=720

for all methods. The lowest MSE and MAE for each dataset within each objective are highlighted in bold.

Setting	Varying noise ratio						Setting	Varying noise strength
Method	TSM		MHA		$\Delta$		Method	TSM		MHA		$\Delta$
Ratio	MSE	MAE	MSE	MAE	MSE	MAE	Strength	MSE	MAE	MSE	MAE	MSE	MAE
0.05	0.392	0.417	0.434	0.449	0.042	0.033	2	0.393	0.417	0.437	0.452	0.044	0.035
0.10	0.394	0.418	0.441	0.455	0.047	0.036	3	0.394	0.418	0.441	0.455	0.047	0.036
0.20	0.397	0.425	0.443	0.457	0.046	0.033	5	0.396	0.420	0.434	0.449	0.038	0.029
0.30	0.400	0.427	0.442	0.458	0.042	0.031	8	0.404	0.426	0.479	0.473	0.075	0.047

Appendix

Table 8: Noise injection robustness on ETTh1 under varying noise ratios (left) and strengths (right).

\Delta

denotes the error reduction of TSM relative to MHA.

Model	MSE	Params (M)	MACs (G)	Memory (GB)	Train Time (s)	Infer Time (ms)
Crossformer	0.171	36.936	47.543	22.932	0.184	75.474
PatchTST	0.171	5.496	23.134	12.848	0.269	326.133
TStruct	0.151	3.324	11.779	7.143	0.059	56.059
Improvement ( $\downarrow$ )	1.13 $\times$	1.65 $\times$	1.96 $\times$	1.80 $\times$	4.56 $\times$	1.35 $\times$

Appendix

Table 9: Efficiency comparison on the Electricity dataset. Crossformer [ Zhang and Yan, 2023 ] and PatchTST [ Nie et al., 2023 ] are included as representative patch-based Transformer baselines. All models are evaluated under the same setting. We report MSE, parameters, MACs, GPU memory, training time per step, and inference time per batch. Bold indicates the best result, and underline indicates the second-best result.

Figure 6: Hyperparameter sensitivity analysis for the number of shared temporal structures

K

on the ETT benchmarks. Bar plots denote MSE and line plots denote MAE under different prediction horizons.

Figure 8: Learned Temporal Structures on ETTh1.

Figure 9: Learned Temporal Structures on ETTh2.

Figure 10: Learned Temporal Structures on ETTm1.

Figure 11: Learned Temporal Structures on ETTm2.

Figure 12: Learned Temporal Structures on Weather.

Figure 13: Learned Temporal Structures on Electricity.

Figure 14: Learned Temporal Structures on Traffic.

Methods		TStruct (ours)		DUET [ 2025 ]		PhaseFormer [ 2026 ]		GTR [ 2026 ]		TimeEmb [ 2025 ]		TQNet [ 2025 ]		CycleNet [ 2024a ]		PatchTST [ 2023 ]		DLinear [ 2023 ]
Datasets	Metric	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
ETTh1	96	0.345	0.380	0.352	0.384	0.359	0.382	0.367	0.391	0.372	0.400	0.371	0.393	0.374	0.396	0.375	0.399	0.379	0.403
	192	0.383	0.405	0.398	0.409	0.397	0.404	0.409	0.420	0.413	0.427	0.410	0.418	0.406	0.415	0.414	0.421	0.408	0.419
	336	0.407	0.421	0.414	0.426	0.425	0.424	0.432	0.434	0.438	0.443	0.432	0.433	0.431	0.430	0.431	0.436	0.440	0.440
	720	0.427	0.454	0.429	0.455	0.431	0.450	0.451	0.463	0.449	0.462	0.466	0.468	0.450	0.464	0.449	0.466	0.471	0.493
	Avg.	0.390	0.415	0.398	0.419	0.403	0.415	0.415	0.419	0.418	0.433	0.420	0.428	0.415	0.426	0.417	0.431	0.425	0.439

Appendix

Table 10: Full multivariate time series forecasting results. The best results are marked in bold red , and the second-best results are marked in blue underline .

Datasets		ETTh1		ETTh2		ETTm1		ETTm2		Weather		Electricity		Traffic
Method	Metric	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
TSM $\rightarrow$ MHA	96	0.367	0.399	0.285	0.350	0.298	0.340	0.191	0.272	0.143	0.189	0.131	0.225	0.377	0.255
	192	0.421	0.440	0.360	0.395	0.344	0.369	0.222	0.293	0.188	0.232	0.148	0.240	0.390	0.262
	336	0.434	0.447	0.360	0.402	0.424	0.426	0.266	0.322	0.260	0.295	0.167	0.258	0.405	0.272
	720	0.478	0.491	0.393	0.433	0.437	0.429	0.356	0.383	0.326	0.344	0.203	0.291	0.439	0.289
	Avg.	0.425	0.444	0.350	0.395	0.376	0.391	0.259	0.318	0.229	0.265	0.162	0.254	0.403	0.270

Appendix

Table 11: Complete ablation results of TStruct under different prediction horizons. TSM

\rightarrow

MHA replaces the Temporal Structure Mixer with multi-head attention, TSM

\rightarrow

Linear replaces it with a shared linear temporal mixer, and PCAHead

\rightarrow

Linear replaces the proposed prediction head with a linear head.

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Three-Stage Learning Unlocks Strong Performance in Simple Models for Long-Term Time Series Forecasting

Zhenan Yu, Guangxin Jiang, Jin Yang

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Multivariate Time Series Forecasting Long-Term Time Series Forecasting Multilayer Perceptrons

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Recent studies on long-term time series forecasting have shown that simple linear models and MLP-based predictors can achieve strong performance without increasingly complex architectures. However, many competitive baselines still rely on structural priors such as frequency-domain modeling, explicit decomposition, multi-scale mixing, or sophisticated cross-variable interaction modules, while paying less attention to how simple temporal mappings should be trained and organized. In this paper, we propose STAIR, short for Stagewise Temporal Adaptation via Individualization and Residual Learning, a training paradigm for long-term time series forecasting that aims to unlock the capacity of simple temporal mapping models without introducing complex architectural modules. STAIR decomposes forecasting ability into three progressive stages: it first learns common temporal dynamics across variables through a shared temporal mapping, then adapts the shared model to each variable via channel-wise fine-tuning to capture variable-specific patterns, and finally complements the backbone with cross-variable information through residual learning. We further introduce Shared-to-Individual Fine-tuning and alpha-RevIN to mitigate the limitations of strict channel independence and the overly strong normalization prior induced by standard RevIN. This design gradually increases modeling flexibility while keeping the core temporal predictor as a shallow MLP in the main experiments, with linear variants analyzed separately. Experiments on nine long-term forecasting benchmarks show that STAIR matches or outperforms recent strong baselines while preserving a simple temporal backbone, providing a concise and effective modeling perspective for long-term time series forecasting.

Abstract

Long-Term Time Series Forecasting

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