Let $\Omega\subset \mathbb{R}^d$ be a bounded domain. We consider the problem of how efficiently shallow neural networks with the ReLU$^k$ activation function can approximate functions from Sobolev spaces $W^s(L_p(\Omega))$ with error measured in the $L_q(\Omega)$-norm. Utilizing the Radon transform and recent results from discrepancy theory, we provide a simple proof of nearly optimal approximation rates in a variety of cases, including when $q\leq p$, $p\geq 2$, and $s \leq k + (d+1)/2$. The rates we derive are optimal up to logarithmic factors, and significantly generalize existing results. An interesting consequence is that the adaptivity of shallow ReLU$^k$ neural networks enables them to obtain optimal approximation rates for smoothness up to order $s = k + (d+1)/2$, even though they represent piecewise polynomials of fixed degree $k$.

本文解决了浅层ReLU$^k$神经网络在Sobolev空间中如何高效近似的关键问题。作者利用拉东变换和差异理论的最新成果，提供了一种简单的证明，展示了在多种情况下几乎最佳的近似率，显示出这种网络在平滑性达到$s = k + (d+1)/2$时仍能获得最佳近似率，其结果显著推广了现有研究。 

通过拉东变换对Sobolev空间中浅层ReLU$^k$神经网络的近似率研究