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：本文重点关注一类自然函数——即光滑函数的复合函数——的逼近问题。与协变量具有低维支撑的假设不同，我们证明若复合函数中各层具有较小的自由度，则此类复合函数本身存在内在稀疏结构。这一特性可缓解神经网络逼近误差中的维数灾难问题。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研究成果，本文为求解含复合结构解的椭圆型方程的PINNs方法提供了泛化收敛率分析。最后通过两个简单数值算例，验证了内在稀疏结构在回归与偏微分方程求解中的有效性。