The solution of inverse problems is central to a wide range of applications including medicine, biology, and engineering. These problems require finding a desired solution in the presence of noisy observations. A key feature of inverse problems is their ill-posedness, which leads to unstable behavior under noise when standard solution methods are used. For this reason, regularization methods have been developed that compromise between data fitting and prior structure. Recently, data-driven variational regularization methods have been introduced, where the prior in the form of a regularizer is derived from provided ground truth data. However, these methods have mainly been analyzed for Tikhonov regularization, referred to as Network Tikhonov Regularization (NETT). In this paper, we propose and analyze Morozov regularization in combination with a learned regularizer. The regularizers, which can be adapted to the training data, are defined by neural networks and are therefore non-convex. We give a convergence analysis in the non-convex setting allowing noise-dependent regularizers, and propose a possible training strategy. We present numerical results for attenuation correction in the context of photoacoustic tomography.

翻译：逆问题的求解是医学、生物学和工程学等多个应用领域的核心问题。此类问题需要在存在噪声观测的情况下寻找期望的解。逆问题的关键特征在于其不适定性，这导致使用标准求解方法时会在噪声影响下出现不稳定性。为此，研究者开发了在数据拟合与先验结构之间进行权衡的正则化方法。近年来，数据驱动变分正则化方法被提出，其中以正则化形式呈现的先验信息来源于提供的真实数据。然而，这些方法主要针对Tikhonov正则化（即网络Tikhonov正则化，简称NETT）进行分析。本文提出并分析了将莫罗佐夫正则化与学习型正则化器相结合的方法。这些正则化器可通过神经网络定义并适应训练数据，因此具有非凸性。我们给出了非凸框架下的收敛性分析，允许正则化器依赖噪声水平，并提出了一种可行的训练策略。最后，我们展示了光声层析成像中衰减校正的数值实验结果。