# Fourier Learning Machines: Nonharmonic Fourier-Based Neural Networks for Scientific Machine Learning

arXiv: https://arxiv.org/abs/2509.08759 · HTML: https://mlsanity.com/paper/2509.08759/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 40 references that are in the map.
- First posted on arXiv 10 Sep 2025.

## Builds on (most cited first, 15 of 15)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [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)
- [Autoformer: Decomposition Transformers with Auto-Correlation for Long-Term Series Forecasting](https://mlsanity.com/paper/2106.13008/) (2021)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [SAITS: Self-Attention-based Imputation for Time Series](https://mlsanity.com/paper/2202.08516/) (2022)
- [Estimating Missing Data in Temporal Data Streams Using Multi-directional Recurrent Neural Networks](https://mlsanity.com/paper/1711.08742/) (2017)
- [Seeing Implicit Neural Representations as Fourier Series](https://mlsanity.com/paper/2109.00249/) (2021)
- [TSI-Bench: Benchmarking Time Series Imputation](https://mlsanity.com/paper/2406.12747/) (2024)
- [Fourier Neural Networks: A Comparative Study](https://mlsanity.com/paper/1902.03011/) (2019)
- [Control Problems with Vanishing Lie Bracket Arising from Complete Odd Circulant Evolutionary Games](https://mlsanity.com/paper/1710.09000/) (2017)
- [Approximation of Optimal Control Surfaces for 2× 2 Skew-Symmetric Evolutionary Game Dynamics](https://mlsanity.com/paper/2207.07167/) (2022)
- [Approximation of Optimal Control Surfaces for the Bass Model with Stochastic Dynamics](https://mlsanity.com/paper/2304.12265/) (2023)
- [An Initial Condition-Dependent Neural Network Approach for Optimal Control Problems](https://mlsanity.com/paper/2502.08607/) (2025)

## Abstract (arXiv)

We introduce the Fourier Learning Machine (FLM), a neural network (NN) architecture designed to represent a multidimensional nonharmonic Fourier series. The FLM uses a simple feedforward structure with cosine activation functions to learn the frequencies, amplitudes, and phase shifts of the series as trainable parameters. This design allows the model to create a problem-specific spectral basis adaptable to both periodic and nonperiodic functions. Unlike previous Fourier-inspired NN models, the FLM is the first architecture able to represent a multidimensional Fourier series with a complete set of basis functions in separable form, doing so by using a standard Multilayer Perceptron-like architecture. A one-to-one correspondence between the Fourier coefficients and amplitudes and phase-shifts is demonstrated, allowing for the translation between a full, separable basis form and the cosine phase-shifted one. Additionally, we evaluate the performance of FLMs on several scientific computing problems, including benchmark Partial Differential Equations (PDEs) and a family of Optimal Control Problems (OCPs). Computational experiments show that the performance of FLMs is comparable, and often superior, to that of established architectures like SIREN and vanilla feedforward NNs.