Solving inverse problems requires knowledge of the forward operator, but accurate models can be computationally expensive and hence cheaper variants are desired that do not compromise reconstruction quality. This chapter reviews reconstruction methods in inverse problems with learned forward operators that follow two different paradigms. The first one is completely agnostic to the forward operator and learns its restriction to the subspace spanned by the training data. The framework of regularisation by projection is then used to find a reconstruction. The second one uses a simplified model of the physics of the measurement process and only relies on the training data to learn a model correction. We present the theory of these two approaches and compare them numerically. A common theme emerges: both methods require, or at least benefit from, training data not only for the forward operator, but also for its adjoint.

翻译：求解反问题需要了解前向算子，但精确模型可能计算成本高昂，因此需要在确保重建质量的前提下寻求更经济的变体。本章综述了基于学习前向算子的反问题重建方法，这些方法遵循两种不同范式。第一种方法完全对前向算子不可知，仅学习其在训练数据张成的子空间上的限制，随后利用投影正则化框架进行重建。第二种方法采用测量过程物理过程的简化模型，仅依赖训练数据学习模型修正。我们给出了这两种方法的理论基础，并进行了数值比较。一个共同主题浮现：两种方法都需要（或至少受益于）训练数据，不仅涉及前向算子，还涉及其伴随算子。