A Novel Fourier Feature Network for Solving Partial Differential Equations

2026-08-13

AI Summary

This study addresses the limited representational capacity of single-hidden-layer networks and the challenge of high-accuracy partial differential equation (PDE) solving by proposing the Fourier Feature Network. Integrating trigonometric basis functions with the least squares method, this approach achieves high-precision approximation without requiring affine transformations of inputs, while optimizing scaling factors to significantly enhance network expressivity. Numerical experiments demonstrate that the proposed model consistently outperforms Extreme Learning Machines in solution accuracy. Consequently, this work provides both theoretical foundations and a practical paradigm for the efficient and accurate resolution of PDEs, overcoming key limitations inherent in traditional single-layer architectures.

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.

Problem

• Partial Differential Equations
• Fourier Feature Networks
• Numerical Solution
• High-precision Approximation

Innovation

• Fourier Feature Networks
• Partial Differential Equations
• Extreme Learning Machines
• Least Squares Method
• Scaling Factor Optimization