  Singularly perturbed problems present inherent difficulty due to the presence
of a thin boundary layer in its solution. 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 boundary layer behavior.
