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.

翻译：正则化在将先验信息融入反问题中发挥着关键作用。尽管已提出许多深度学习方法来解决反问题，但确定在何处应用正则化仍是关键考量。典型方法通过架构对神经网络进行正则化，即神经网络函数对感兴趣参数或正则化项进行参数化。我们提出一种新颖方法，称为"数据正则化算子学习" (DaROL) 方法，旨在解决偏微分方程反问题。DaROL方法通过Tikhonov变分方法和贝叶斯推断等常见技术对数据进行正则化，并在正则化数据上训练神经网络。DaROL方法在不同框架下具有灵活性，能更快地求解反问题，并具有将正则化与神经网络训练分离的更简单结构。我们证明，在正则化数据上训练神经网络等价于对正则化逆映射进行监督学习。此外，我们为该类正则化逆映射的光滑性提供了充分条件，并依据神经网络规模和训练样本数量估计了学习误差。