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.

通过两种不同范例，学习正演算子在逆问题中的重建方法：完全对正演算子不可知的范例通过投影正则化找到重建，而基于测量过程的简化模型则依赖于训练数据来学习模型修正。这两种方法都需要训练数据不仅仅是正演算子，也是其伴随算子。