Solving inverse problems requires the knowledge of the forward operator, but accurate models can be computationally expensive and hence cheaper variants that do not compromise the reconstruction quality are desired. 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：求解逆问题需要前向算子的知识，但精确模型可能计算成本高昂，因此希望在不牺牲重建质量的前提下采用更经济的变体。本章回顾了基于学习型前向算子的逆问题重建方法，这些方法遵循两种不同范式。第一种方法完全未知前向算子，仅通过学习其在训练数据张成子空间上的限制，然后利用投影正则化框架进行重建。第二种方法采用简化的测量过程物理模型，仅依赖训练数据学习模型修正。我们阐述了这两种方法的理论，并进行了数值对比。一个共性特征浮现：这两种方法不仅需要（或至少受益于）前向算子的训练数据，还需其伴随算子的训练数据。