In the recent years, research on scientific machine learning based on deep learning solvers has been increasingly applied to scientific computing and computational engineering. However, even though these newly data-driven deep learning solvers could be very effective as soon as they have been properly trained, they can (in general) provide us with a solution of limited accuracy. Furthermore, the computational cost in the training phase can be extremely expensive. In this talk, we presents some ways of hybridizing the newly emerging deep learning solvers and the more traditional numerical linear algebra techniques to let them benefit from each other. In the context of solving a heterogeneous Helmholtz equation, we first focus on introducing some mathematical ingredients from classical iterative solver into the training phase of a recently proposed deep neural network solver. The main benefit is a significant improvement in the training phase that is more robust and faster, which turns out to be applicable to the testing process as well. Furthermore, once the network solvers have been properly trained, their inferences can be applied as a nonlinear preconditioner in the traditional flexible GMRES and flexible FOM methods. This part demonstrates that these hybrid variants have clear advantages over both the newly emerging deep neural network approach and the classical iterative Krylov solver in terms of both computational cost and accuracy of the computed solution.