From Theory to Application: A Practical Introduction to Neural Operators in Scientific Computing is out in Mathematics (vol. 14, no. 13).
The review takes the position that a neural operator is only useful insofar as you know when to trust it. It walks from the operator-learning formulation through the main architectures to the places the accuracy actually matters, inside an optimization loop or an inference loop, and to the error-control machinery that makes a surrogate substitutable for the solver rather than merely close to it on a test set.