UFO: Learning Universal Frequency Representation with Neural Operator for Time Series Forecasting

Frequency representations play a vital role in time series forecasting, with recent deep learning approaches increasingly modeling temporal dependencies in the Fourier basis. In parallel, operator learning frameworks such as the Fourier Neural Operator (FNO) demonstrate that complicated operators can be parameterized via linear transforms in frequency space coupled with local non-linearities. However, existing methods remain constrained by discretization errors and full-frequency mode representation, which inherently bottleneck their prediction capacity. To mitigate this limitation, we replace Fourier bases with adaptive basis functions and determine their rotation coefficients. Time translation is then converted into these coefficients acting on the basis functions. Mathematically, we prove that both the basis functions and the rotation coefficients can be universally approximated by neural networks. Grounded in this theoretical framework, we propose the Universal Frequency Operator (UFO) for long-term time-series forecasting (LSTF). Instead of relying on fixed Fourier or wavelet bases, UFO learns adaptive bases and the corresponding rotation coefficients, capturing temporal dynamics by combining them through complex multiplication. Extensive experiments on 8 real-world datasets demonstrate that UFO consistently outperforms state-of-the-art baselines. To demonstrate that UFO captures core temporal dynamics in the basis functions, we ablate the history input's high-frequency mode and observe that UFO consistently outperforms other methods across all high-frequency cutoff levels. Code is available at this repository: https://anonymous.4open.science/status/UFO-3D30.