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This paper introduces Fourier Feature Networks (FENs), a single-hidden-layer architecture designed specifically to solve partial differential equations with high precision without relying on affine transformations. By leveraging trigonometric basis functions and optimizing scaling factors, the method outperforms standard Extreme Learning Machines in numerical experiments involving various activation functions. The work advances machine learning techniques for scientific computing by improving representational capacity through spectral feature engineering rather than complex network depth or input warping.
A Novel Fourier Feature Network for Solving Partial Differential Equations
arXiv:2608.14733v1 Announce Type: new Abstract: Building on the foundation of single-hidden-layer neural networks, Fourier Feature Networks (FENs) are proposed, which incorporate Fourier features using $\cos$, $\sin$, or a combination of both. Similar to Extreme Learning Machines (ELMs), FENs employ a single-hidden-layer architecture to generate a set of basis functions. The target function is then approximated as a linear combination of these basis functions, with the coefficients determined using the least squares method. However, unlike ELMs, which often rely on affine transformations to improve representational power, FENs can achieve high-precision solutions without requiring such transformations on the input variables. To evaluate the representational capacity of these networks, we search for an optimal scaling factor within a predefined range for the randomly initialized and fixed weights and biases. By adjusting this scaling factor, we ensure a fair comparison between FENs and ELMs using various activation functions, such as $\text{sigmoid}$, $\tanh$, and $\text{swish}$. Our numerical experiments demonstrate that FENs consistently achieve higher accuracy than ELMs.
Abstract:Building on the foundation of single-hidden-layer neural networks, Fourier Feature Networks (FENs) are proposed, which incorporate Fourier features using $\cos$, $\sin$, or a combination of both. Similar to Extreme Learning Machines (ELMs), FENs employ a single-hidden-layer architecture to generate a set of basis functions. The target function is then approximated as a linear combination of these basis functions, with the coefficients determined using the least squares method. However, unlike ELMs, which often rely on affine transformations to improve representational power, FENs can achieve high-precision solutions without requiring such transformations on the input variables. To evaluate the representational capacity of these networks, we search for an optimal scaling factor within a predefined range for the randomly initialized and fixed weights and biases. By adjusting this scaling factor, we ensure a fair comparison between FENs and ELMs using various activation functions, such as $\text{sigmoid}$, $\tanh$, and $\text{swish}$. Our numerical experiments demonstrate that FENs consistently achieve higher accuracy than ELMs.
Machine Learning (cs.LG); Artificial Intelligence (cs.AI)
arXiv:2608.14733 [cs.LG]
(or arXiv:2608.14733v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2608.14733 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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