Neural Fourier Modelling: A Highly Compact Approach to Time-Series Analysis

TL;DR

Neural Fourier Modelling (NFM) reframes time-series analysis by modeling directly in the Fourier domain, exploiting the Fourier transform’s function-space view to enable frequency interpolation and extrapolation. It introduces Learnable Frequency Tokens (LFT) to inject spectral priors and Implicit Neural Fourier Filters (INFF) to provide a compact, global Fourier-processing operator, forming a backbone that maps input sequences to target spectral representations. Across forecasting, anomaly detection, and classification, NFM achieves competitive or state-of-the-art results with under 40K parameters, including scenarios with unseen test-time discretizations, and demonstrates robustness to changes in sampling rates. The frequency-domain approach offers a compact, continuous-time representation suitable for on-device or low-resource settings, while remaining extendable to multivariate signals; irregular-time series remain a future challenge.

Abstract

Neural time-series analysis has traditionally focused on modeling data in the time domain, often with some approaches incorporating equivalent Fourier domain representations as auxiliary spectral features. In this work, we shift the main focus to frequency representations, modeling time-series data fully and directly in the Fourier domain. We introduce Neural Fourier Modelling (NFM), a compact yet powerful solution for time-series analysis. NFM is grounded in two key properties of the Fourier transform (FT): (i) the ability to model finite-length time series as functions in the Fourier domain, treating them as continuous-time elements in function space, and (ii) the capacity for data manipulation (such as resampling and timespan extension) within the Fourier domain. We reinterpret Fourier-domain data manipulation as frequency extrapolation and interpolation, incorporating this as a core learning mechanism in NFM, applicable across various tasks. To support flexible frequency extension with spectral priors and effective modulation of frequency representations, we propose two learning modules: Learnable Frequency Tokens (LFT) and Implicit Neural Fourier Filters (INFF). These modules enable compact and expressive modeling in the Fourier domain. Extensive experiments demonstrate that NFM achieves state-of-the-art performance on a wide range of tasks (forecasting, anomaly detection, and classification), including challenging time-series scenarios with previously unseen sampling rates at test time. Moreover, NFM is highly compact, requiring fewer than 40K parameters in each task, with time-series lengths ranging from 100 to 16K.

Figures (13)

• Figure 1: Illustration of Fourier-domain manipulations (left), including zero-padding/truncation (top) and zero-interleaving (bottom), and equivalent effects in the time domain (right).
• Figure 2: Overall workflow (forecasting scenario is exemplified) of the proposed NFM which deals with discrete signals as continuous-time elements in the compact function space through Fourier lens. NFM finds an interpolation/extrapolation from discrete input to target directly in the Fourier domain.
• Figure 3: Illustration of NFM architecture consisting of three main learning modules: 1) LFT block to allow flexible frequency extension and provides effective spectral priors, 2) a plain MLP for channel mixing, and 3) INFF module for effective token mixing with global convolution operation. The M in LFT block denotes frequency extension operation.
• Figure 4: Comparison of NFM with different ablation cases on SpeechCommand dataset. PMU is peak memory usage during inference time.
• Figure 5: Visualization of INFF on synthetic data. The top-left figure shows the frequencies of input sequence, bottom-left the frequencies of filtered sequence, bottom-right the learned INFF's coefficients, and top-right the coefficients averaged over hidden dimension.