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求解中发挥着重要作用。尽管已提出许多深度学习方法来求解反问题，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，即神经网络函数参数化感兴趣参数或正则化项。我们提出一种名为"数据正则化算子学习"（DaROL）的新方法，专门用于解决偏微分方程反问题。该方法通过Tikhonov变分方法和贝叶斯推断等通用技术对数据训练神经网络进行正则化。DaROL方法在不同框架下均具有灵活性，能够更快求解反问题，且其将正则化与神经网络训练分离的简化结构优势明显。我们证明：对正则化数据训练神经网络等价于对正则化逆映射进行监督学习。此外，我们给出了此类正则化逆映射光滑性的充分条件，并从神经网络规模与训练样本数量两个维度估计了学习误差。