PhaseFormer: From Patches to Phases for Efficient and Effective Time Series Forecasting

TL;DR

PhaseFormer reframes long-horizon time-series forecasting by substituting patch-based tokens with phase tokens, producing phase-wise predictions through a lightweight cross-phase routing Transformer. The method combines data-driven phase extraction with a two-stage routing mechanism and a shared predictor, achieving state-of-the-art accuracy with roughly $\,\approx 10^3$ parameters while dramatically reducing FLOPs, especially on large-scale datasets like Traffic and Electricity. The authors theoretically justify phase-token stability under cycle-pattern drifts and provide extensive empirical evidence across seven benchmarks, including ablations and case studies, to demonstrate both robustness and efficiency. This work offers a practical path toward truly efficient and effective forecasting, and the accompanying code is available for reproducibility.

Abstract

Periodicity is a fundamental characteristic of time series data and has long played a central role in forecasting. Recent deep learning methods strengthen the exploitation of periodicity by treating patches as basic tokens, thereby improving predictive effectiveness. However, their efficiency remains a bottleneck due to large parameter counts and heavy computational costs. This paper provides, for the first time, a clear explanation of why patch-level processing is inherently inefficient, supported by strong evidence from real-world data. To address these limitations, we introduce a phase perspective for modeling periodicity and present an efficient yet effective solution, PhaseFormer. PhaseFormer features phase-wise prediction through compact phase embeddings and efficient cross-phase interaction enabled by a lightweight routing mechanism. Extensive experiments demonstrate that PhaseFormer achieves state-of-the-art performance with around 1k parameters, consistently across benchmark datasets. Notably, it excels on large-scale and complex datasets, where models with comparable efficiency often struggle. This work marks a significant step toward truly efficient and effective time series forecasting. Code is available at this repository: https://github.com/neumyor/PhaseFormer_TSL

Figures (10)

• Figure 1: Comparison between patch-based and phase-based representations for time-series forecasting. (a) illustrates the difference in tokenization. (b) jointly evaluates model accuracy, parameter scale, and computational overhead on the Traffic dataset, where marker size indicates FLOPS.
• Figure 2: Visualization of phase tokenization and its advantages. (a) Phase tokenization yields more stable representations than patch-based embeddings. (b) Phase tokens exhibit clear low-dimensionality compared with patch tokens.
• Figure 3: The overview of PhaseFormer.
• Figure 4: Comparison of FLOPs and parameter counts across models on the Traffic and Electricity. Patch-based models are shown in green, phase-based models in blue, and other models in gray.
• Figure 5: Effect of varying the number of routers $M$ on forecasting performance on three datasets.

Theorems & Definitions (8)

• Theorem 1: Phase Tokenization Stability
• Lemma 1: Column space preservation
• Lemma 2: Row space change
• Lemma 3: Wedin’s $\sin\Theta$ theorem
• Theorem 2
• Proof 1
• Theorem 3: Stability under day-wise perturbations
• Proof 2