  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.
