  Regularization plays a pivotal role in integrating prior information into
inverse problems. While many deep learning methods have been proposed to solve
inverse problems, determining where to apply regularization remains a crucial
consideration. Typical methods regularize neural networks via architecture,
wherein neural network functions parametrize the parameter of interest or the
regularization term. We introduce a novel approach, denoted as the
"data-regularized operator learning" (DaROL) method, designed to address PDE
inverse problems. The DaROL method trains a neural network on data, regularized
through common techniques such as Tikhonov variational methods and Bayesian
inference. The DaROL method offers flexibility across different frameworks,
faster inverse problem-solving, and a simpler structure that separates
regularization and neural network training. We demonstrate that training a
neural network on the regularized data is equivalent to supervised learning for
a regularized inverse map. Furthermore, we provide sufficient conditions for
the smoothness of such a regularized inverse map and estimate the learning
error in terms of neural network size and the number of training samples.
