FractalLoss: Fractal Structure of the overlooked Non-Dominant Components in Time Series Forecasting

Despite rapid progress in time series forecasting (TSF), model training still largely relies on mean squared error (MSE), which often produces over-smoothed predictions thereby misses fine-grained temporal dynamics. We investigate this limitation from a spectral perspective and show that MSE is energy-preferential: it emphasizes high-energy frequency components while overlooking low-energy ones. Consequently, dominant spectral peaks are well fitted, whereas numerous low-energy background components remain underfitted. More importantly, extensive empirical analysis reveals that these individually weak yet collectively strong background components are not random residuals, but instead exhibit a clear fractal structure characterized by strong linearity in the log-frequency log-energy spectrum. Motivated by these observations, we propose FractalLoss, a model-agnostic regularizer that encourages the recovery of fractal structure for more faithful forecasting. We further theoretically show that FractalLoss provides energy-independent guidance, helping mitigate underfitting across background components irrespective of their spectral energy. Experiments on real-world benchmarks demonstrate forecasting gains and substantially improved fractal structure preservation.