Expression rates and stability in Sobolev norms of deep ReLU neural networks (NNs) in terms of the number of parameters defining the NN for continuous, piecewise polynomial functions, on arbitrary, finite partitions $\mathcal{T}$ of a bounded interval $(a,b)$ are addressed. Novel constructions of ReLU NN surrogates encoding the approximated functions in terms of Chebyshev polynomial expansion coefficients are developed. Chebyshev coefficients can be computed easily from the values of the function in the Clenshaw--Curtis points using the inverse fast Fourier transform. Bounds on expression rates and stability that are superior to those of constructions based on ReLU NN emulations of monomials considered in [Opschoor, Petersen, Schwab, 2020] are obtained. All emulation bounds are explicit in terms of the (arbitrary) partition of the interval, the target emulation accuracy and the polynomial degree in each element of the partition. ReLU NN emulation error estimates are provided for various classes of functions and norms, commonly encountered in numerical analysis. In particular, we show exponential ReLU emulation rate bounds for analytic functions with point singularities and develop an interface between Chebfun approximations and constructive ReLU NN emulations.

对于具有指定参数定义的深度ReLU神经网络（NN），本文讨论了其在Sobolev范数中的表达速率和稳定性，针对有限分区上的连续的分段多项式函数。我们通过Chebyshev多项式展开系数独特构建了ReLU NN的代理，这些系数可以通过在Clenshaw-Curtis点上函数的值使用逆快速傅里叶变换轻松计算得到。与基于ReLU NN模拟多项式的构造方法相比，本文获得了更优的表达速率和稳定性界限。所有模拟界限均明确以区间的分割、目标模拟精度和每个分割元素的多项式次数来表示。本文还针对在数值分析中常见的各种函数和范数提供了ReLU NN模拟误差估计，特别是对于具有点奇异性的解析函数展示了指数级的ReLU模拟速率界限，并发展了Chebfun逼近和构造性ReLU NN模拟之间的接口。

深度ReLU网络和高阶有限元方法 II：切比雪夫模拟