发表机构
School of Mathematics, Sichuan University（四川大学数学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出新型傅里叶特征网络（FENs），其无需对输入变量做仿射变换即可获高精度解，经实验验证FENs在求解偏微分方程时精度始终高于极限学习机（ELMs）。
AI 中文摘要
基于单隐层神经网络的基础，本文提出了傅里叶特征网络（Fourier Feature Networks，FENs），该网络利用余弦函数（$\boldsymbol{\text{cos}}$）、正弦函数（$\boldsymbol{\text{sin}}$）或两者的组合来融入傅里叶特征。与极限学习机（Extreme Learning Machines，ELMs）类似，FENs采用单隐层架构生成一组基函数，随后将目标函数近似为这些基函数的线性组合，其系数通过最小二乘法确定。然而，ELMs通常依赖仿射变换来提升表示能力，而FENs无需对输入变量进行此类变换即可获得高精度解。为评估这些网络的表示能力，我们在预定义范围内为随机初始化且固定的权重与偏置搜索最优缩放因子，通过调整该缩放因子，确保FENs与ELMs在使用各类激活函数（如$\boldsymbol{\text{sigmoid}}$、$\boldsymbol{\text{tanh}}$、$\boldsymbol{\text{swish}}$）时的公平比较。数值实验表明，FENs的精度始终高于ELMs。
英文摘要
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.
DOI：10.1016/j.cnsns.2025.109274
