{"ID":2872698,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2509.08759","arxiv_id":"2509.08759","title":"Fourier Learning Machines: Nonharmonic Fourier-Based Neural Networks for Scientific Machine Learning","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, 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.","short_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 paramete...","url_abs":"https://arxiv.org/abs/2509.08759","url_pdf":"https://arxiv.org/pdf/2509.08759v3","authors":"[\"Mominul Rubel\",\"Adam Meyers\",\"Gabriel Nicolosi\"]","published":"2025-09-10T16:49:20Z","proceeding":"cs.LG","tasks":"[\"cs.LG\",\"math.OC\"]","methods":"[]","has_code":false}