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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2509.08759 |
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| _version_ | 1866908898663858176 |
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| author | Rubel, Mominul Meyers, Adam Nicolosi, Gabriel |
| author_facet | Rubel, Mominul Meyers, Adam Nicolosi, Gabriel |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_08759 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fourier Learning Machines: Nonharmonic Fourier-Based Neural Networks for Scientific Machine Learning Rubel, Mominul Meyers, Adam Nicolosi, Gabriel Machine Learning Optimization and Control 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. |
| title | Fourier Learning Machines: Nonharmonic Fourier-Based Neural Networks for Scientific Machine Learning |
| topic | Machine Learning Optimization and Control |
| url | https://arxiv.org/abs/2509.08759 |