TY - JOUR
T1 - Improved Analysis of PINNs: Alleviate the CoD for Compositional Solutions
AU - Jiao , Yuling
AU - Lu , Xiliang
AU - Yang , Jerry Zhijian
AU - Yuan , Cheng
AU - Zhang , Pingwen
JO - Annals of Applied Mathematics
VL - 3
SP - 239
EP - 263
PY - 2023
DA - 2023/09
SN - 39
DO - http://doi.org/ 10.4208/aam.OA-2023-0021
UR - https://global-sci.org/intro/article_detail/aam/21993.html
KW - Composition function, deep neural network, approximation.
AB - <p style="text-align: justify;">In this paper, we present an improved analysis of the Physics Informed Neural Networks (PINNs) method for solving second-order elliptic equations. By assuming an intrinsic sparse structure in the underlying solution,
we provide a convergence rate analysis that can overcome the curse of dimensionality (CoD). Specifically, using some approximation theory in Sobolev space
together with the multivariate Faa di Bruno formula, we first derive the approximation error for composition functions with a small degree of freedom in
each compositional layer. Furthermore, by integrating several results on the
statistical error of neural networks, we obtain a refined convergence rate analysis for PINNs in solving elliptic equations with compositional solutions. We
also demonstrate the benefits of the intrinsic sparse structure with two simple
numerical examples.</p>