TimeAPN: Adaptive Amplitude-Phase Non-Stationarity Normalization for Time Series Forecasting

Yue Hu; Jialiang Tang; Siwei Yu; Baosheng Yu; Jing Zhang; Dacheng Tao

cs.LGMar 18, 2026

TimeAPN: Adaptive Amplitude-Phase Non-Stationarity Normalization for Time Series Forecasting

Authors: Yue Hu, Jialiang Tang, Siwei Yu, Baosheng Yu, Jing Zhang, Dacheng Tao

Organizations: School of Mathematics, Harbin Institute of Technology, Harbin 150001, China · School of Computer Science and Engineering, Nanjing University of Science and Technology, China · Lee Kong Chian School of Medicine, Nanyang Technological University, Singapore · School of Computer Science, Wuhan University, Wuhan, China ·

Abstract

Non-stationarity is a fundamental challenge in multivariate long-term time series forecasting, often manifested as rapid changes in amplitude and phase. These variations lead to severe distribution shifts and consequently degrade predictive performance. Existing normalization-based methods primarily rely on first- and second-order statistics, implicitly assuming that distributions evolve smoothly and overlooking fine-grained temporal dynamics. To address these limitations, we propose TimeAPN, an Adaptive Amplitude-Phase Non-Stationarity Normalization framework that explicitly models and predicts non-stationary factors from both the time and frequency domains. Specifically, TimeAPN first models the mean sequence jointly in the time and frequency domains, and then forecasts its evolution over future horizons. Meanwhile, phase information is extracted in the frequency domain, and the phase discrepancy between the predicted and ground-truth future sequences is explicitly modeled to capture temporal misalignment. Furthermore, TimeAPN incorporates amplitude information into an adaptive normalization mechanism, enabling the model to effectively account for abrupt fluctuations in signal energy. The predicted non-stationary factors are subsequently integrated with the backbone forecasting outputs through a collaborative de-normalization process to reconstruct the final non-stationary time series. The proposed framework is model-agnostic and can be seamlessly integrated with various forecasting backbones. Extensive experiments on seven real-world multivariate datasets demonstrate that TimeAPN consistently improves long-term forecasting accuracy across multiple prediction horizons and outperforms state-of-the-art reversible normalization methods.

Figures & tables

Fig. 1: Comparison of DDN and TimeAPN in the time and frequency domains. In the time domain (left), TimeAPN achieves a lower prediction error (MSE = 0.075) than DDN (MSE = 0.123). In the frequency domain (middle and right), DDN fails to accurately capture the ground-truth amplitude and phase spectra, whereas TimeAPN produces spectra that align more closely with the true amplitude and phase distributions.

Fig. 2: Overview of the proposed TimeAPN framework. The model first normalizes the input sequence to extract stationary components and non-stationary factors. Stationary sequences are fed into a forecasting model (FM), while non-stationary factors are predicted using the Mean and Phase Prediction Module (MPPM). Predicted non-stationary factors are then combined with FM outputs through de-normalization to reconstruct the future series.

Methods		FEDformer		+TimeAPN		DLinear		+TimeAPN		PatchTST		+TimeAPN		S_Mamba		+TimeAPN
Metric		MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
ETTh1	96	0.375	0.415	0.377	0.399	0.384	0.405	0.371	0.393	0.393	0.408	0.382	0.392	0.388	0.406	0.382	0.396
	192	0.427	0.448	0.424	0.435	0.443	0.450	0.405	0.415	0.445	0.434	0.430	0.420	0.445	0.441	0.439	0.427
	336	0.458	0.465	0.466	0.458	0.447	0.448	0.430	0.431	0.484	0.451	0.472	0.442	0.490	0.465	0.484	0.451
	720	1.093	0.827	0.498	0.499	0.504	0.515	0.435	0.454	0.480	0.471	0.468	0.464	0.506	0.497	0.481	0.473
ETTh2	96	0.341	0.385	0.288	0.343	0.290	0.353	0.272	0.335	0.298	0.347	0.289	0.338	0.297	0.349	0.290	0.340

TABLE I: Multivariate long-term forecasting results on benchmark datasets under prediction lengths of 96, 192, 336, and 720. Results are reported in terms of MSE and MAE. TimeAPN is integrated into four representative backbone models (FEDformer, DLinear, PatchTST, and S_Mamba). The best results are highlighted in bold .

Methods		FEDformer							DLinear
Methods		+TimeAPN	+DDN	+RevIN	+NST	+Dish-TS	+SAN	IMP	+TimeAPN	+DDN	+RevIN	+SAN	IMP
ETTh1	96	0.377	0.389	0.396	0.394	0.390	0.384	-0.5%	0.371	0.372	0.371	0.381	3.4%
	192	0.424	0.425	0.439	0.441	0.441	0.430	0.7%	0.405	0.405	0.406	0.419	8.6%
	336	0.466	0.474	0.484	0.485	0.495	0.474	-1.7%	0.430	0.431	0.442	0.438	3.8%
	720	0.498	0.679	0.515	0.505	0.519	0.504	54.4%	0.435	0.457	0.442	0.446	13.7%
ETTh2	96	0.288	0.299	0.385	0.381	0.806	0.302	15.5%	0.272	0.278	0.278	0.277	6.2%

TABLE II: Comparison of TimeAPN with existing reversible normalization methods under different prediction lengths (96, 192, 336, and 720). Results are reported in terms of MSE using two backbone models, FEDformer and DLinear, across multiple benchmark datasets. IMP denotes the relative improvement achieved by TimeAPN over the original backbone without normalization. The best results for each setting are highlighted in bold .

Methods		PatchTST						S_Mamba
Methods		+TimeAPN		+DDN		+SAN		+TimeAPN		+DDN		+SAN
Metric		MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
ETTh2	96	0.289	0.338	0.291	0.341	0.304	0.358	0.290	0.340	0.291	0.343	0.316	0.368
	192	0.369	0.390	0.378	0.392	0.417	0.434	0.372	0.394	0.383	0.402	0.400	0.416
	336	0.409	0.426	0.418	0.428	0.427	0.446	0.413	0.430	0.429	0.436	0.418	0.434
	720	0.402	0.431	0.430	0.446	0.428	0.455	0.420	0.443	0.428	0.444	0.447	0.463

TABLE III: Comparison of forecasting errors for DDN, SAN, and TimeAPN. The best results are highlighted in bold .

Fig. 3: Comparison of reversible normalization methods on a forecasting sample from the PatchTST model trained on the Electricity dataset. (a) PatchTST without normalization, (b) PatchTST with RevIN, (c) PatchTST with DDN, and (d) PatchTST with the proposed TimeAPN.

Methods	PatchTST					S_Mamba
Methods	Amp	Phase	FD	MSE	MAE	Amp	Phase	FD	MSE	MAE
ETTm1	$\times$	$\times$	$\times$	0.382	0.395	$\times$	$\times$	$\times$	0.398	0.407
	$\times$	$\checkmark$	$\checkmark$	0.379	0.397	$\times$	$\checkmark$	$\checkmark$	0.392	0.405
	$\checkmark$	$\checkmark$	$\times$	0.386	0.400	$\checkmark$	$\checkmark$	$\times$	0.396	0.405
	$\checkmark$	$\checkmark$	$\checkmark$	0.377	0.395	$\checkmark$	$\checkmark$	$\checkmark$	0.391	0.404
Weather	$\times$	$\times$	$\times$	0.263	0.283	$\times$	$\times$	$\times$	0.251	0.276

TABLE IV: Impact of amplitude, phase, and frequency-domain components on forecasting performance. The best results are highlighted in bold .

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PULSE: Generative Phase Evolution for Non-Stationary Time Series Forecasting

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Time series forecasting under non-stationarity faces a fundamental tension between capturing stable representations and adapting to distribution shifts. Existing methods implicitly rely on static historical assumptions, leading to a critical failure mode we term Phase Amnesia, where models become blind to the evolving global context. To resolve this, we formalize non-stationary dynamics through three physical hypotheses: wold decomposition, dynamical phase evolution, and heteroscedastic manifold generation. These principles inspire PULSE, a physics-informed, plug-and-play framework adopting a Disentangle--Evolve--Simulate design philosophy. Specifically, PULSE utilizes phase-anchored disentanglement to resolve optimization interference caused by dominant trends, employs a Phase Router to actively generate future trajectories, and introduces Statistic-Aware Mixup (SAM) to ensure robustness against out-of-distribution volatility. Empirically, PULSE enables a simple MLP backbone to achieve state-of-the-art or highly competitive performance across 12 real-world benchmarks. This validates that a correct physics-informed inductive bias is far more critical than raw architectural complexity for non-stationary forecasting. The code is available at: https://github.com/Gemost/PULSE.

Abstract

Non-Stationary Time Series Forecasting

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