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.

翻译：本文研究了深度ReLU神经网络（NN）在连续分片多项式函数逼近中的Sobolev范数表达速率与稳定性问题，其中定义域为有界区间$(a,b)$上的任意有限分割$\mathcal{T}$。我们提出了一种新型ReLU NN代理函数构造方法，通过切比雪夫多项式展开系数编码被逼近函数。切比雪夫系数可利用Clenshaw-Curtis点上的函数值通过快速傅里叶逆变换高效计算。相较于文献[Opschoor, Petersen, Schwab, 2020]中基于单项式ReLU NN逼近的构造方案，本文方法在表达速率与稳定性方面展现出更优边界。所有逼近边界均显式依赖于区间（任意）划分、目标逼近精度以及划分各单元内的多项式阶数。针对数值分析中常见的多类函数空间与范数，本文提供了ReLU NN逼近误差估计。特别地，我们证明了对含点奇点的解析函数存在指数级ReLU逼近速率边界，并建立了Chebfun逼近与构造性ReLU NN逼近之间的接口。