FAPA: A Dual-Branch Framework for Time Series Forecasting with Frequency Amplitude Shaping and Harmonic Period Alignment

Time series forecasting supports applications in energy, transportation, and meteorology. Spectral analysis reveals that time series often consist of high-amplitude periodic components and low-amplitude irregular residuals. Fitting structured periodic patterns and irregular residuals jointly with one set of shared weights is harder than modeling them separately, since the two components have very different statistical properties. Predicting the periodic component in a separate branch removes this coupling but raises two problems. First, per-window normalization, the standard remedy for distribution shift, maps windows that differ only in scale to the same input, so the amplitude of the periodic component is inherited from the input window instead of being predicted. Second, the branch has no directly available training target: re-estimating its periods on the ground truth gives periods that are unavailable at inference, while the input-window periods can fall between the frequency bins of the target window, where nearest-bin reconstruction can lose more than half of the component's energy. We propose FAPA, a plug-and-play dual-branch framework that improves existing forecasters. Its periodic branch predicts the periodic component at its original scale and refines the predicted amplitudes with a lightweight frequency Amplitude Shaping Module (AmpShape). This branch is supervised by a dedicated periodic loss whose target is built by Harmonic Period Alignment (HPA), which projects the ground truth onto sinusoids at the input-window periods and is exact for any horizon at no inference cost. A residual branch passes the residual to the backbone. Experiments on eight backbones (e.g., PatchTST, TimesNet) and five datasets show that FAPA lowers the MSE on the five-dataset average for every backbone: it falls by 10.7% overall, and by 19.6% and 18.9% on ECL and Traffic, respectively. A controlled synthetic study further shows that the gain grows monotonically with the amplitude imbalance between the periodic component and the residual.