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

let data talk: data-regularized operator learning theory for inverse problems