In this paper, we focus on approximating a natural class of functions that are compositions of smooth functions. Unlike the low-dimensional support assumption on the covariate, we demonstrate that composition functions have an intrinsic sparse structure if we assume each layer in the composition has a small degree of freedom. This fact can alleviate the curse of dimensionality in approximation errors by neural networks. Specifically, by using mathematical induction and the multivariate Faa di Bruno formula, we extend the approximation theory of deep neural networks to the composition functions case. Furthermore, combining recent results on the statistical error of deep learning, we provide a general convergence rate analysis for the PINNs method in solving elliptic equations with compositional solutions. We also present two simple illustrative numerical examples to demonstrate the effect of the intrinsic sparse structure in regression and solving PDEs.

翻译：本文聚焦于一类自然函数——光滑函数复合而成的函数的逼近问题。不同于对协变量低维支撑的假设，我们证明若复合函数的每一层具有较小的自由度，则此类函数具有内在稀疏结构。该事实可缓解神经网络逼近误差中的维数诅咒问题。具体而言，通过数学归纳法与多元Faa di Bruno公式，我们将深度神经网络的逼近理论推广至复合函数情形。进一步结合深度学习统计误差的最新成果，我们为求解含复合解的椭圆方程的PINNs方法提供了通用收敛率分析。文中还给出两个简单数值示例，以直观展示回归与偏微分方程求解中内在稀疏结构的效用。