@Article{EAJAM-14-841,
author = {Du , TingHuang , Zhongyi and Li , Ye},
title = {Approximation and Generalization of DeepONets for Learning Operators Arising from a Class of Singularly Perturbed Problems},
journal = {East Asian Journal on Applied Mathematics},
year = {2024},
volume = {14},
number = {4},
pages = {841--873},
abstract = {<p style="text-align: justify;">Singularly perturbed problems present inherent difficulty due to the presence of thin layers in their solutions. To overcome this difficulty, we propose using deep
operator networks (DeepONets), a method previously shown to be effective in approximating nonlinear operators between infinite-dimensional Banach spaces. In this paper,
we demonstrate for the first time the application of DeepONets to one-dimensional singularly perturbed problems, achieving promising results that suggest their potential as
a robust tool for solving this class of problems. We consider the convergence rate of the
approximation error incurred by the operator networks in approximating the solution
operator, and examine the generalization gap and empirical risk, all of which are shown
to converge uniformly with respect to the perturbation parameter. By utilizing Shishkin
mesh points as locations of the loss function, we conduct several numerical experiments
that provide further support for the effectiveness of operator networks in capturing the
singular layer behavior.</p>},
issn = {2079-7370},
doi = {https://doi.org/10.4208/eajam.2023-128.051023},
url = {http://global-sci.org/intro/article_detail/eajam/23440.html}
}