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

• URL: http://arxiv.org/abs/2509.08759v1
• Date: Wed, 10 Sep 2025 16:49:20 GMT
• Title: Fourier Learning Machines: Nonharmonic Fourier-Based Neural Networks for Scientific Machine Learning
• Authors: Mominul Rubel, Adam Meyers, Gabriel Nicolosi,
• Abstract summary: We introduce the Fourier Learning Machine (FLM), a neural network (NN) architecture designed to represent a multidimensional nonharmonic Fourier series.<n>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.<n> Computational experiments show that the performance of FLMs is comparable, and often superior, to that of established architectures like SIREN and vanilla feedforward NNs.
• Score: 0.5097809301149341
• License: http://creativecommons.org/licenses/by/4.0/
• 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 complete, separable Fourier basis in multiple dimensions 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.

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