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

arXiv: https://arxiv.org/abs/2410.04703 · HTML: https://mlsanity.com/paper/2410.04703/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 51 of its 61 references that are in the map.
- First posted on arXiv 7 Oct 2024.

## Builds on (most cited first, 30 of 51)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Efficiently Modeling Long Sequences with Structured State Spaces](https://mlsanity.com/paper/2111.00396/) (2021)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [Informer: Beyond Efficient Transformer for Long Sequence Time-Series Forecasting](https://mlsanity.com/paper/2012.07436/) (2020)
- [A Time Series is Worth 64 Words: Long-term Forecasting with Transformers](https://mlsanity.com/paper/2211.14730/) (2022)
- [WaveNet: A Generative Model for Raw Audio](https://mlsanity.com/paper/1609.03499/) (2016)
- [Autoformer: Decomposition Transformers with Auto-Correlation for Long-Term Series Forecasting](https://mlsanity.com/paper/2106.13008/) (2021)
- [Are Transformers Effective for Time Series Forecasting?](https://mlsanity.com/paper/2205.13504/) (2022)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [FEDformer: Frequency Enhanced Decomposed Transformer for Long-term Series Forecasting](https://mlsanity.com/paper/2201.12740/) (2022)
- [TimesNet: Temporal 2D-Variation Modeling for General Time Series Analysis](https://mlsanity.com/paper/2210.02186/) (2022)
- [HyperNetworks](https://mlsanity.com/paper/1609.09106/) (2016)
- [iTransformer: Inverted Transformers Are Effective for Time Series Forecasting](https://mlsanity.com/paper/2310.06625/) (2023)
- [Scene Representation Networks: Continuous 3D-Structure-Aware Neural Scene Representations](https://mlsanity.com/paper/1906.01618/) (2019)
- [Speech Commands: A Dataset for Limited-Vocabulary Speech Recognition](https://mlsanity.com/paper/1804.03209/) (2018)
- [Learning Continuous Image Representation with Local Implicit Image Function](https://mlsanity.com/paper/2012.09161/) (2020)
- [Neural Controlled Differential Equations for Irregular Time Series](https://mlsanity.com/paper/2005.08926/) (2020)
- [Anomaly Transformer: Time Series Anomaly Detection with Association Discrepancy](https://mlsanity.com/paper/2110.02642/) (2021)
- [FNet: Mixing Tokens with Fourier Transforms](https://mlsanity.com/paper/2105.03824/) (2021)
- [A Fourier-based Framework for Domain Generalization](https://mlsanity.com/paper/2105.11120/) (2021)
- [Global Filter Networks for Image Classification](https://mlsanity.com/paper/2107.00645/) (2021)
- [Self-Supervised Contrastive Pre-Training For Time Series via Time-Frequency Consistency](https://mlsanity.com/paper/2206.08496/) (2022)
- [Adaptive Fourier Neural Operators: Efficient Token Mixers for Transformers](https://mlsanity.com/paper/2111.13587/) (2021)
- [Mobile-Former: Bridging MobileNet and Transformer](https://mlsanity.com/paper/2108.05895/) (2021)
- [GRU-ODE-Bayes: Continuous modeling of sporadically-observed time series](https://mlsanity.com/paper/1905.12374/) (2019)
- [Deep Complex Networks](https://mlsanity.com/paper/1705.09792/) (2017)

## Built on by (most cited first, 3 of 3)

- [Test-Time Efficient Pretrained Model Portfolios for Time Series Forecasting](https://mlsanity.com/paper/2510.06419/) (2025)
- [AR-KAN: Autoregressive-Weight-Enhanced Kolmogorov-Arnold Network for Time Series Forecasting](https://mlsanity.com/paper/2509.02967/) (2025)
- [Neural Optimal Transport in Hilbert Spaces: Characterizing Spurious Solutions and Gaussian Smoothing](https://mlsanity.com/paper/2602.14086/) (2026)

## Abstract (arXiv)

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.