AI 中文总结
该研究解决了一般维度下浅层Sigmoidal网络的最优Sobolev逼近问题，构造确定性与随机方向-偏移字典并通过实验验证了其最优逼近速率。
AI 中文摘要
具有预设或随机采样隐藏参数的浅层网络被广泛用作数值试验空间，但在一般维度下，其采用标准光滑Sigmoidal激活函数时的最优Sobolev逼近能力仍未得到解决。我们针对一类具有Schwartz类导数衰减的光滑Sigmoidal激活函数（包括tanh、逻辑Sigmoid和误差函数erf）建立了对应的最优速率。我们首先构造具有M个特征的确定性方向-偏移字典，使得对所有0≤m≤k，每个u∈H^k(Ω)都能在H^m(Ω)中达到阶为M^(-(k-m)/d)的逼近误差，该速率在Sobolev球的Kolmogorov宽度意义下是最优的。我们进一步证明，从任意远离零的有界密度中独立采样参数得到的字典，在对数过采样的情况下，以高概率达到相同的逼近指数。该分析发展了Sigmoidal脊表示，并将其与方向-偏移空间中的确定性或概率求积相结合，同时保持输出系数的多项式控制。在广泛的维度、目标正则性和Sobolev误差范数范围内的数值实验，验证了确定性和随机特征字典的预测代数速率。
英文摘要
Shallow networks with prescribed or randomly sampled hidden parameters are widely used as numerical trial spaces, yet their optimal Sobolev approximation power with standard smooth sigmoidal activations in general dimension remains unresolved. We establish the corresponding optimal rates for a class of smooth sigmoidal activations with Schwartz-class derivative decay, including $\tanh$, the logistic sigmoid, and the error function $erf$. We first construct deterministic direction--offset dictionaries with $M$ features such that every $u\in H^k(Ω)$ can be approximated with error of order $M^{-(k-m)/d}$ in $H^m(Ω)$ for all $0\le m\le k$. This rate is optimal in the sense of Kolmogorov widths for Sobolev balls. We further prove that dictionaries obtained by independent parameter sampling from any prescribed density bounded away from zero attain the same approximation exponent with high probability, up to logarithmic oversampling. The analysis develops a sigmoidal ridge representation and combines it with deterministic or probabilistic quadrature in direction--offset space while retaining polynomial control of the output coefficients. Numerical experiments across a broad range of dimensions, target regularities, and Sobolev error norms recover the predicted algebraic rates for both deterministic and random feature dictionaries.
