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FreDN: Spectral Disentanglement for Time Series Forecasting via Learnable Frequency Decomposition

Zhongde An, Jinhong You, Jiyanglin Li, Yiming Tang, Wen Li, Heming Du, Shouguo Du

TL;DR

FreDN addresses spectral entanglement in frequency-domain forecasting for non-stationary time series by introducing a learnable Frequency Disentangler that separates trend and seasonal components directly in the spectral domain, and a parameter-efficient ReIm Block to model complex spectra with real-valued projections. It provides a theory-grounded loss analysis showing the benefits of frequency-domain MAE and demonstrates substantial empirical gains across seven long-horizon benchmarks, along with notable reductions in parameter count and computation compared with standard complex-valued architectures. The work offers a practical, scalable approach to capture global periodic patterns while maintaining compatibility with real-valued networks, and it supplies theoretical insights into gradient propagation in time- and frequency-domain losses. Overall, FreDN delivers improved forecasting accuracy and efficiency in non-stationary settings, highlighting the value of learnable frequency-domain decomposition for real-world time series tasks.

Abstract

Time series forecasting is essential in a wide range of real world applications. Recently, frequency-domain methods have attracted increasing interest for their ability to capture global dependencies. However, when applied to non-stationary time series, these methods encounter the $\textit{spectral entanglement}$ and the computational burden of complex-valued learning. The $\textit{spectral entanglement}$ refers to the overlap of trends, periodicities, and noise across the spectrum due to $\textit{spectral leakage}$ and the presence of non-stationarity. However, existing decompositions are not suited to resolving spectral entanglement. To address this, we propose the Frequency Decomposition Network (FreDN), which introduces a learnable Frequency Disentangler module to separate trend and periodic components directly in the frequency domain. Furthermore, we propose a theoretically supported ReIm Block to reduce the complexity of complex-valued operations while maintaining performance. We also re-examine the frequency-domain loss function and provide new theoretical insights into its effectiveness. Extensive experiments on seven long-term forecasting benchmarks demonstrate that FreDN outperforms state-of-the-art methods by up to 10\%. Furthermore, compared with standard complex-valued architectures, our real-imaginary shared-parameter design reduces the parameter count and computational cost by at least 50\%.

FreDN: Spectral Disentanglement for Time Series Forecasting via Learnable Frequency Decomposition

TL;DR

FreDN addresses spectral entanglement in frequency-domain forecasting for non-stationary time series by introducing a learnable Frequency Disentangler that separates trend and seasonal components directly in the spectral domain, and a parameter-efficient ReIm Block to model complex spectra with real-valued projections. It provides a theory-grounded loss analysis showing the benefits of frequency-domain MAE and demonstrates substantial empirical gains across seven long-horizon benchmarks, along with notable reductions in parameter count and computation compared with standard complex-valued architectures. The work offers a practical, scalable approach to capture global periodic patterns while maintaining compatibility with real-valued networks, and it supplies theoretical insights into gradient propagation in time- and frequency-domain losses. Overall, FreDN delivers improved forecasting accuracy and efficiency in non-stationary settings, highlighting the value of learnable frequency-domain decomposition for real-world time series tasks.

Abstract

Time series forecasting is essential in a wide range of real world applications. Recently, frequency-domain methods have attracted increasing interest for their ability to capture global dependencies. However, when applied to non-stationary time series, these methods encounter the and the computational burden of complex-valued learning. The refers to the overlap of trends, periodicities, and noise across the spectrum due to and the presence of non-stationarity. However, existing decompositions are not suited to resolving spectral entanglement. To address this, we propose the Frequency Decomposition Network (FreDN), which introduces a learnable Frequency Disentangler module to separate trend and periodic components directly in the frequency domain. Furthermore, we propose a theoretically supported ReIm Block to reduce the complexity of complex-valued operations while maintaining performance. We also re-examine the frequency-domain loss function and provide new theoretical insights into its effectiveness. Extensive experiments on seven long-term forecasting benchmarks demonstrate that FreDN outperforms state-of-the-art methods by up to 10\%. Furthermore, compared with standard complex-valued architectures, our real-imaginary shared-parameter design reduces the parameter count and computational cost by at least 50\%.

Paper Structure

This paper contains 51 sections, 2 theorems, 33 equations, 22 figures, 10 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Time Series Forecasting
  4. Time Series Decomposition
  5. Method
  6. Preliminary
  7. Frequency Disentangler
  8. ReIm Block
  9. Loss Function Analysis
  10. Experiments
  11. Main results
  12. Frequency Disentanglement Analysis
  13. Efficiency of ReIm Block
  14. Conclusion
  15. Datasets Description
  16. ...and 36 more sections

Key Result

Theorem 1

Let $f \in W^{m,2}([0,1])$ be a sobolev-smooth function with square-integrable $m$-th derivative for some $m \geq 1$. $\hat{f}(k)$ is the $k$-th Fourier coefficient of the periodic extension of $f$. Then there exists a constant $C > 0$ such that for all $k \in \mathbb{Z} \setminus \{0\}$,

Figures (22)

  • Figure 1: Spectral component entanglement. Top row: (a) the synthetic sequence with trend generated by B-spline; (b) the frequency spectrum of the real trend; (c) the frequency spectrum of the real periodicity. Bottom row: (d) the frequency-wise proportion of trend, periodic, and noise in the original synthetic sequence; (e) the spectrum of the trend extracted by the moving average in the time domain; (f) the spectrum of the TopK-selected frequency components.
  • Figure 2: Overview of the proposed FreDN architecture. The input is first embedded by adding a redundant dimension, then decomposed by a learnable Frequency Disentangler into trend and seasonal components. The trend is transformed back to the time domain and processed by a TimeMLP. The seasonal real and imaginary parts are separately modeled by a shared MLP under the ReIm Block. The final prediction sums both outputs.
  • Figure 3: (a) Original signal from the ETTh1 with $L=720$; (b) Trend spectrum in our FreDN; (c) Trend spectrum learned by MovDN using moving average, the red line indicating the theoretical sidelobe leakage introduced by the moving average filter.
  • Figure 4: Efficiency of ReIm Block. (a) Parameter count of the ReIm Block under varying input/output lengths; (b) Parameter count of the Complex-Linear structure; (c) Average training time on different datasets.
  • Figure 5: Effect of lookback window length on average MAE across all prediction horizons. FreDN maintains consistently superior accuracy under different lookback settings, with performance improving monotonically as the input length increases.
  • ...and 17 more figures

Theorems & Definitions (3)

  • Theorem 1: Spectral of Sobolev-Smooth Trends
  • Definition 1: Complex Linear Projection
  • Theorem 2: Expressiveness of ReIm Block