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[2509.08759] Fourier Learning Machines: Nonharmonic Fourier-Based Neural Networks for Scientific Machine Learning
Summary
The paper introduces Fourier Learning Machines (FLMs), a novel neural network architecture that utilizes nonharmonic Fourier series for scientific machine learning, demonstrating superior performance on various computational problems.
Why It Matters
This research is significant as it presents a new approach to neural networks that enhances the representation of multidimensional Fourier series, potentially improving the efficiency and accuracy of scientific computations. By addressing limitations in existing models, FLMs could advance applications in fields requiring precise mathematical modeling.
Key Takeaways
- FLMs utilize a feedforward structure with cosine activation functions for learning frequencies and amplitudes.
- The architecture allows for a problem-specific spectral basis adaptable to both periodic and nonperiodic functions.
- FLMs demonstrate a one-to-one correspondence between Fourier coefficients and model parameters.
- Performance evaluations show FLMs outperform traditional architectures like SIREN in scientific computing tasks.
- This research could lead to advancements in solving complex Partial Differential Equations and Optimal Control Problems.
Computer Science > Machine Learning arXiv:2509.08759 (cs) [Submitted on 10 Sep 2025 (v1), last revised 13 Feb 2026 (this version, v2)] Title:Fourier Learning Machines: Nonharmonic Fourier-Based Neural Networks for Scientific Machine Learning Authors:Mominul Rubel, Adam Meyers, Gabriel Nicolosi View a PDF of the paper titled Fourier Learning Machines: Nonharmonic Fourier-Based Neural Networks for Scientific Machine Learning, by Mominul Rubel and 1 other authors View PDF Abstract: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, inc...