AI 中文总结
本文针对一般有界区域，提出带解析激活函数的线性化浅层网络构造的张量积型参数集，实现Sharp Sobolev逼近，其参数集分布固定更便于实际计算。
AI 中文摘要
我们研究了有界区域上的Sobolev逼近问题，所用方法是其内部参数独立于目标函数预设的线性化浅层神经网络。我们的主要步骤是针对解析激活函数开展一维构造。我们证明，对于一类满足泰勒系数定量非消去条件的解析激活函数，具有单变量分辨率m的类切比雪夫参数集可生成达到Sharp H^r到H^s逼近阶m^{-(r-s)}的固定特征空间。将该结果与文献[SIAM J. Math. Anal. 30 (1998), pp. 155-189]中的山脊函数提升定理，以及本工作建立的其对任意拟均匀方向集的扩展相结合，我们构造了张量积型参数集，对所有r>0均满足逼近界||f-f_n||_{L^2(Ω)}≲n^{-r/d}||f||_{H^r(Ω)}（其中f∈H^r(Ω)）。与文献[Neural Comput. 8 (1996), pp. 164-177]中的有限差分构造相比，后者的显式容许条件可能要求参数尺度极小，而本文提出的参数集保持分布在固定区间上，因此更便于实际计算。
英文摘要
We study Sobolev approximation on bounded domains by linearized shallow neural networks whose inner parameters are prescribed independently of the target function. Our main step is a one-dimensional construction for analytic activations. We prove that quasi-Chebyshev parameter sets with univariate resolution $m$ generate fixed feature spaces attaining the sharp $H^r$-to-$H^s$ approximation order $m^{-(r-s)}$ for a class of analytic activations satisfying a quantitative non-cancellation condition on their Taylor coefficients. Combining this result with the ridge-function lifting theorem in [SIAM J. Math. Anal. 30 (1998), pp. 155-189] and its extension to arbitrary quasi-uniform direction sets established in this work, we construct tensor-product-type parameter sets that attain the sharp rate $$\|f-f_n\|_{L^2(Ω)}\lesssim n^{-\frac rd}\|f\|_{H^r(Ω)},\quad f\in H^r(Ω)$$ for all $r>0$. In contrast to the finite-difference construction in [Neural Comput. 8 (1996), pp. 164-177], whose explicit admissibility condition may require an extremely small parameter scale, the proposed parameter sets remain distributed over fixed intervals and are therefore more amenable to practical computation.
Comments32 pages
