发表机构
Nanjing University（南京大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文构造了具有固定高斯和tanh差激活的浅层神经网络，通过确定性规则独立于目标设置参数，实现最优Sobolev逼近速率，并验证了二维和三维数值实验。
AI 中文摘要
我们在有界Lipschitz域$\Omega\subset\mathbb{R}^d$（$d\ge2$）上构造具有固定高斯和tanh差激活函数的浅层神经网络。对于每个$s>0$，显式的确定性规则独立于目标$u\in H^s(\Omega)$规定所有隐藏参数。在频率分辨率$M$下，$N_M\asymp M^d$个规定的特征张成线性空间$V_M$。对于每个足够大的整数$N$，我们选择$M\asymp N^{1/d}$使得$N_M\le N$，并构造一个有界线性算子$A_N:H^s(\Omega)\to V_M$，使得对于每个$u\in H^s(\Omega)$和每个整数$0\le m<s$，有\\[ \\|u-A_Nu\\|_{H^m(\Omega)} \le C N^{-(s-m)/d}\\|u\\|_{H^s(\Omega)} \\]，其中$C$独立于$u$、$N$和$m$。Sobolev宽度下界表明，在独立于目标选择的至多$N$维线性空间中，该速率是最优的。我们使用显式的球面cubature节点用于方向，以及Gauss-Legendre节点或等距中点用于偏移。我们证明外部偏移余量可以随着$M\to\infty$缩小到零，同时保持逼近速率。对于高斯激活，超出$\sup_{x\in\Omega}|x|$的$M^{-1}\log M$阶余量，对于tanh差激活，$M^{-1}(\log M)^2$阶余量，足以截断脊积分并移除外部中点节点。在二维和三维中的数值实验检验了$L^2$和$H^1$逼近。
英文摘要
We construct shallow neural networks with fixed Gaussian and tanh-difference activations on bounded Lipschitz domains $Ω\subset\mathbb{R}^d$, $d\ge2$. For each $s>0$, explicit deterministic rules prescribe all hidden parameters independently of the target $u\in H^s(Ω)$. At frequency resolution $M$, the $N_M\asymp M^d$ prescribed features span a linear space $V_M$. For every sufficiently large integer $N$, we choose $M\asymp N^{1/d}$ with $N_M\le N$ and construct a bounded linear operator $A_N:H^s(Ω)\to V_M$ such that \[ \|u-A_Nu\|_{H^m(Ω)} \le C N^{-(s-m)/d}\|u\|_{H^s(Ω)} \] for every $u\in H^s(Ω)$ and every integer $0\le m<s$, with $C$ independent of $u$, $N$, and $m$. Sobolev width lower bounds show that this rate is optimal among linear spaces of dimension at most $N$ chosen independently of the target. We use explicit spherical cubature nodes for the directions and Gauss-Legendre nodes or equispaced midpoints for the offsets. We show that the exterior offset margin can shrink to zero as $M\to\infty$ while preserving the approximation rate. A margin beyond $\sup_{x\inΩ}|x|$ of order $M^{-1}\log M$ for the Gaussian and $M^{-1}(\log M)^2$ for the tanh difference suffices to truncate the ridge integral and remove exterior midpoint nodes. Numerical experiments in two and three dimensions examine $L^2$ and $H^1$ approximation.
