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 · College of Computing and Data Science, Nanyang Technological University, Singapore
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
+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
+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
Amp
Phase
FD
MSE
MAE
Amp
Phase
FD
MSE
MAE
ETTm1
×
×
×
0.382
0.395
×
×
×
0.398
0.407
×
✓
✓
0.379
0.397
×
✓
✓
0.392
0.405
✓
✓
×
0.386
0.400
✓
✓
×
0.396
0.405
✓
✓
✓
0.377
0.395
✓
✓
✓
0.391
0.404
Weather
×
×
×
0.263
0.283
×
×
×
0.251
0.276
TABLE IV: Impact of amplitude, phase, and frequency-domain components on forecasting performance. The best results are highlighted in bold .
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.
Yangyou Liu, Zezhi Shao, Xinyu Chen +3
College of Computer Science, Sichuan University, Chengdu, China · Institute of Computing Technology, Chinese Academy of Sciences, Beijing, China · Institute of Artificial Intelligence, University of Central Florida, Orlando, USA +1
Real-world time series forecasting faces the fundamental challenge of non-stationary statistical properties, including shifts in mean and variance over time. While reversible instance normalization (RevIN) has shown promise by stationarizing inputs and denormalizing outputs, it relies on the strong assumption that historical and future distributions remain identical. We observe that in many practical applications, distribution shifts follow cyclical patterns that correlate with periodic positions (e.g., seasonal and holiday volatility). To this end, we propose PAMod, a lightweight yet powerful framework that models cyclical distribution shifts via Phase-Amplitude Modulation in the normalized feature space. PAMod learns periodic embeddings to modulate representations: phase modulation captures mean shifts, while amplitude modulation adapts to variance changes. Crucially, we prove mathematically that modulating in normalized space is equivalent to applying dynamic denormalization, offering an elegant unification of distribution adaptation and representation learning. Extensive experiments on twelve real-world benchmarks demonstrate that PAMod achieves state-of-the-art performance with fewer computational resources. Furthermore, our modulation mechanism, as a novel plug-and-play technique, can improve existing time-series forecasting methods with simple integration.
Yingbo Zhou, Yutong Ye, Shuhao Li +5
Fudan University · Beihang University · Beijing University of Posts and Telecommunications
Instance normalization (IN) is widely used in non-stationary multivariate time series forecasting to reduce distribution shifts and highlight common patterns across samples. However, IN can over-smooth instance-specific structural information that is essential for modeling temporal and cross-channel heterogeneity. While prior methods further suppress distribution discrepancies or attempt to recover temporal specific dependencies, they often ignore a central tension: how to adaptively model common and instance-specific dependency based on each instance's non-stationary structures. To address this dilemma, we propose SeesawNet, a unified architecture that dynamically balances common and instance-specific dependency modeling in both temporal and channel dimensions. At its core is Adaptive Stationary-Nonstationary Attention (ASNA), which captures common dependencies from normalized sequences and specific dependencies from raw sequences, and adaptively fuses them according to instance-level non-stationarity. Built upon ASNA, SeesawNet alternates dedicated temporal and channel relationship modeling to jointly capture long-range and cross-variable dependencies. Extensive experiments on multiple real-world benchmarks demonstrate that SeesawNet consistently outperforms state-of-the-art methods.
Hao Li, Lu Zhang, Liu Chong +3
1Sichuan University · 2Chengdu University of Information Technology · 3McGill University +1