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.

翻译：奇异摄动问题因其解中存在薄边界层而具有内在难度。为克服这一困难，我们提出使用深度算子网络（DeepONets）——一种已被证明能有效逼近无限维巴拿赫空间之间非线性算子的方法。本文首次展示了DeepONets在一维奇异摄动问题中的应用，取得了令人鼓舞的结果，表明其作为解决此类问题的稳健工具的潜力。我们研究了算子网络在逼近解算子时逼近误差的收敛速率，并考察了泛化差距与经验风险，所有这些指标均关于摄动参数一致收敛。通过采用Shishkin网格点作为损失函数的位置，我们开展了多项数值实验，进一步验证了算子网络捕捉奇异边界层行为的有效性。