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.

翻译：传统的神经时间序列分析主要集中于在时域中对数据进行建模，部分方法会将等效的傅里叶域表示作为辅助的频谱特征。在本研究中，我们将主要焦点转移到频域表示上，在傅里叶域中完全且直接地对时间序列数据进行建模。我们提出了神经傅里叶建模（NFM），一种用于时间序列分析的紧凑而强大的解决方案。NFM基于傅里叶变换（FT）的两个关键特性：（i）能够将有限长度的时间序列建模为傅里叶域中的函数，将其视为函数空间中的连续时间元素；（ii）在傅里叶域内进行数据操作（如重采样和时间跨度扩展）的能力。我们将傅里叶域的数据操作重新解释为频率外推与内插，并将其作为NFM的核心学习机制，适用于各种任务。为了支持结合频谱先验的灵活频率扩展以及对频率表示的有效调制，我们提出了两个学习模块：可学习频率令牌（LFT）和隐式神经傅里叶滤波器（INFF）。这些模块实现了在傅里叶域中进行紧凑且富有表现力的建模。大量实验表明，NFM在广泛的任务（预测、异常检测和分类）上达到了最先进的性能，包括在测试时遇到先前未见过的采样率这一具有挑战性的时间序列场景。此外，NFM高度紧凑，在时间序列长度从100到16K不等的每个任务中，所需参数少于40K。