Frequency-Constrained Learning for Long-Term Forecasting

TL;DR

This paper introduces a frequency-constrained learning method that enhances long-term time series forecasting by explicitly modeling periodicity through spectral initialization and frequency-aware optimization, improving accuracy and interpretability.

Contribution

It proposes a spectral initialization and frequency-constrained optimization approach that can be integrated into existing models to better capture periodic patterns in long-term forecasting.

Findings

Consistent performance improvements on real-world benchmarks.

Effective recovery of true frequencies on synthetic data.

Enhanced long-range forecasting accuracy, especially at long horizons.

Abstract

Many real-world time series exhibit strong periodic structures arising from physical laws, human routines, or seasonal cycles. However, modern deep forecasting models often fail to capture these recurring patterns due to spectral bias and a lack of frequency-aware inductive priors. Motivated by this gap, we propose a simple yet effective method that enhances long-term forecasting by explicitly modeling periodicity through spectral initialization and frequency-constrained optimization. Specifically, we extract dominant low-frequency components via Fast Fourier Transform (FFT)-guided coordinate descent, initialize sinusoidal embeddings with these components, and employ a two-speed learning schedule to preserve meaningful frequency structure during training. Our approach is model-agnostic and integrates seamlessly into existing Transformer-based architectures. Extensive experiments across diverse real-world benchmarks demonstrate consistent performance gains--particularly at long horizons--highlighting the benefits of injecting spectral priors into deep temporal models for robust and interpretable long-range forecasting. Moreover, on synthetic data, our method accurately recovers ground-truth frequencies, further validating its interpretability and effectiveness in capturing latent periodic patterns.