🤖 machine learning

POEM: Phase-Aware $\mathrm{SO}(2)$ Feature Rotation for Time Series Forecasting Under Periodicity Drift

This paper introduces POEM, a phase-aware time series forecasting framework that addresses periodicity drift by learning a phase-correction coordinate and applying invertible $\mathrm{SO}(2)$ rotations to latent features, enhanced by a Directional Phase Increment Attention mechanism for future phase extrapolation.

Original authors: Jiawen Zhu, Shuhan Liu, Shengxuan Li, Qiming Shi, Di Weng

Published 2026-08-05

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CC BY 4.0

Original authors: Jiawen Zhu, Shuhan Liu, Shengxuan Li, Qiming Shi, Di Weng

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to predict the future by looking at a repeating pattern, like the tides rolling onto a beach or the daily rush of cars through a city intersection. For a long time, scientists and computers have been great at spotting these loops. They assume that if a wave crashes every 12 hours, it will keep crashing at exactly the same time forever. But in the real world, life is messy. Sometimes the tide comes in a little early because of a storm; sometimes the traffic jam starts later because of a holiday. This "drifting" of the schedule—where the rhythm stays the same but the timing gets wobbly—is a huge headache for computers trying to forecast what happens next. This field, known as time series forecasting, is crucial for everything from managing electricity grids to predicting heart rates. The big question is: how do we teach a computer to understand that a pattern can stretch, shrink, and shift its timing without losing its shape?

Enter POEM, a new method developed by researchers at Zhejiang University that acts like a time-traveling dance instructor for data. The paper suggests that instead of forcing these wobbly rhythms to fit a rigid, unchanging clock, we should let the computer "rotate" the data to smooth out the timing glitches before making a prediction. Think of it like watching a spinning record player. If the record is slightly warped, the music sounds like it's speeding up and slowing down (that's the "periodicity drift"). POEM doesn't just listen to the warped music; it calculates exactly how the record is warping and digitally "rotates" the sound waves to make them perfectly circular again. Once the data is straightened out, a simple, lightweight predictor can easily see the pattern and guess what comes next. After the guess is made, the system rotates the prediction back to its original, wobbly shape so it matches reality.

The researchers found that this approach works surprisingly well. By using a mathematical tool called SO(2) rotation—which is just a fancy way of saying "spinning a pair of numbers around a circle"—POEM can untangle complex time shifts. They tested this on nine different real-world datasets, ranging from weather patterns and stock market exchanges to heart rate monitors and birth rates. The results suggest that POEM is a strong contender, often beating other top methods in accuracy. For instance, on a dataset tracking US births, the new method reduced the error significantly compared to previous attempts. The team also showed that by "rotating" the data, the hidden patterns inside the computer's brain become much smoother and more organized, almost like turning a tangled ball of yarn into a neat spool.

However, the paper is careful not to claim this solves every problem in the universe. The method works best when the data has a clear, repeating cycle that just happens to drift in time. It suggests that while we can't stop the world from changing its rhythm, we can build tools that dance along with it. The researchers also noted that while their method is slightly slower than the fastest existing models, it is still incredibly quick (taking less than half a millisecond per sample) and much faster than some other complex approaches. Ultimately, POEM offers a playful yet powerful new way to handle the fact that time, in our messy world, rarely ticks with perfect uniformity.

Technical Summary: POEM – Phase-Aware SO(2) Feature Rotation for Time Series Forecasting Under Periodicity Drift

1. Problem Definition: Periodicity Drift

Time series forecasting relies heavily on identifying cyclic regularities in historical data to extrapolate future states. However, real-world systems rarely adhere to strict periodicity. The authors identify periodicity drift as a critical challenge where cycle timing and phase vary over time due to changing operating conditions. Unlike fixed-period models that assume cycles recur at constant rates, periodicity drift involves non-uniform phase progression. This is characterized by a time-varying phase velocity (governing how fast a cycle advances) and a phase offset (governing where the cycle is anchored).

The paper argues that existing methods, including template-based approaches and phase-aware models like AlignTime or PhaseFormer, often rely on fixed-length templates or global/discretized alignment. These approaches fail to explicitly construct a channel-wise phase trajectory from local evidence, leaving a gap in modeling the continuous, local evolution of phase shifts.

2. Methodology: The POEM Framework

To address periodicity drift, the authors propose POEM, a framework that mitigates phase-related variability by learning a phase-correction coordinate and applying invertible rotations using the special orthogonal group in two dimensions, SO(2). The framework consists of five sequential modules:

A. Local Phase-Correction Estimator (LPCE)
This module estimates variable-specific phase trajectories from the look-back window.

Local Dynamics: The input sequence is divided into overlapping windows. A local encoder (convolutional mixing, self-attention, temporal aggregation) extracts features for each window.
Parameterization: For each window and variable, the model predicts:
1. Phase Offset ( $\theta$ ): An angular anchor constrained to $(-\pi, \pi)$ to respect cyclic nature.
2. Phase Velocity ( $v$ ): A positive scalar representing the speed of cycle progression, parameterized via a clipped MLP and exponentiation to ensure positivity.
Integration: These local estimates are aggregated into a continuous, channel-wise phase correction trajectory $\Phi_c(t)$ using a circular mean and temporal unwrapping.

B. Phase-Guided SO(2) Feature Rotation
Before forecasting, the model transforms the latent features to a "phase-corrected" space.

Feature Encoding: Observations are encoded into $D$ -dimensional vectors.
Rotation: The feature vectors are split into pairs to form 2D subspaces. An SO(2) rotation matrix $R(-\Phi_c(t))$ is applied to these pairs using the negative of the estimated phase correction angle.
Effect: This operation compensates for estimated phase shifts, projecting physical latent states onto a more regular periodic orbit, thereby simplifying the subsequent forecasting task.

C. Phase-Corrected Temporal Predictor
A lightweight Multi-Layer Perceptron (MLP) operates on the rotated, phase-corrected features. Because the phase-induced irregularities have been removed, this predictor only needs to model the remaining temporal dependencies to forecast future states in the corrected space.

D. Directional Phase Increment Attention (DPIA)
Since periodicity drift continues into the forecast horizon, the model must predict how the phase will evolve.

Mechanism: DPIA retrieves historical phase increments observed under similar temporal contexts. It uses normalized timestamp encodings as queries (future) and keys (historical).
Attention: A learnable scaled cosine attention mechanism computes weights based on the similarity between future and historical timestamp contexts.
Integration: Historical phase increments are aggregated via these weights to construct the future phase-correction trajectory.

E. Phase Restoration and Decoding
The final step reverses the initial transformation.

Restoration: The predicted future features (in the corrected space) are rotated back using the predicted future phase angles $R(\hat{\Phi}_c(L+h))$ .
Decoding: A linear decoder maps these restored features back to the original data scale to produce the final forecast.

3. Key Contributions

The paper outlines three primary contributions:

Problem Formulation: It formally defines periodicity drift as a coupling between intrinsic system dynamics and a time-varying phase that distorts the observed time axis.
POEM Framework: It introduces a method that constructs a continuous phase correction trajectory and applies SO(2) transformations to features. This creates a more regular representation for a lightweight temporal predictor, addressing the limitations of fixed-template or global-alignment methods.
Empirical Validation: Extensive experiments on nine real-world benchmarks demonstrate competitive performance. Ablation studies and visualizations confirm that the learned phase-aware transformations regularize latent trajectories and that the SO(2) rotation is superior to alternative transformations (e.g., frequency-domain correction or shear).

4. Experimental Results

The authors evaluated POEM on nine datasets spanning energy (ETT), finance (Exchange), weather, and other domains (NN5, US Birth, ZafNoo).

Performance: POEM achieved the best or second-best results in 12 out of 18 comparisons (across 9 datasets and 2 metrics: MSE and MAE). It reduced the average MSE by 11.9% compared to the leading baseline, PMDformer, outperforming it on 7 of 9 datasets.
Ablation Studies:
- Removing the SO(2) Rotation significantly increased error, confirming the necessity of phase correction.
- Removing the LPCE (replacing it with a simple linear head) caused the largest performance degradation, validating the need for structured local estimation of offsets and velocities.
- Replacing DPIA with linear projections or standard attention resulted in higher errors, supporting the efficacy of retrieving historical increments via scaled cosine attention.
Alternative Transformations: SO(2) rotation outperformed Frequency-Domain Phase Correction, Phase-Conditioned FiLM, and Phase-Conditioned Shear, suggesting that norm-preserving rotation of paired latent features is the most effective mechanism for this task.
Efficiency: POEM maintains low inference latency (0.201–0.441 ms per sample), comparable to PMDformer and significantly faster (2.5–4.3x) than SRSNet.

5. Significance and Claims

The paper claims that POEM offers a geometrically grounded solution to periodicity drift. By treating phase drift as a temporally evolving process and correcting it via SO(2) rotations, the framework transforms a complex, non-stationary forecasting problem into a more regular one that a lightweight predictor can handle.

The authors emphasize that their approach does not attempt to recover the complete underlying dynamical system but rather models the specific effect of phase progression. Qualitative visualizations (PCA projections) show that post-rotation feature trajectories are smoother and form more coherent cycles, directly supporting the claim that the method successfully reduces phase-related variability. The work concludes that this phase-aware geometric approach is a robust strategy for improving time series forecasting in the presence of non-uniform periodicity.