We consider ill-posed inverse problems where the forward operator $T$ is unknown, and instead we have access to training data consisting of functions $f_i$ and their noisy images $Tf_i$. This is a practically relevant and challenging problem which current methods are able to solve only under strong assumptions on the training set. Here we propose a new method that requires minimal assumptions on the data, and prove reconstruction rates that depend on the number of training points and the noise level. We show that, in the regime of "many" training data, the method is minimax optimal. The proposed method employs a type of convolutional neural networks (U-nets) and empirical risk minimization in order to "fit" the unknown operator. In a nutshell, our approach is based on two ideas: the first is to relate U-nets to multiscale decompositions such as wavelets, thereby linking them to the existing theory, and the second is to use the hierarchical structure of U-nets and the low number of parameters of convolutional neural nets to prove entropy bounds that are practically useful. A significant difference with the existing works on neural networks in nonparametric statistics is that we use them to approximate operators and not functions, which we argue is mathematically more natural and technically more convenient.

翻译：我们考虑不适定逆问题，其中前向算子$T$未知，但可获取由函数$f_i$及其含噪图像$Tf_i$组成的训练数据。这是一个具有实际意义且富有挑战性的问题，现有方法仅能在对训练集施加强假设的条件下解决。本文提出一种新方法，该方法对数据要求极低，并证明重建误差率取决于训练样本数与噪声水平。研究表明，在"大样本"训练数据场景下，该方法达到极小极大最优。该方法采用卷积神经网络（U-net）与经验风险最小化技术来"拟合"未知算子。简言之，我们的方法基于两个核心思想：首先，将U-net与小波等多尺度分解建立联系，从而将其纳入现有理论框架；其次，利用U-net的层级结构和卷积神经网络的低参数量特性，推导具有实用价值的熵界。与非参数统计中现有的神经网络研究不同，本文的关键突破在于将神经网络用于算子逼近而非函数逼近，这在数学上更为自然且技术上更具便利性。