Krylov iterative methods, such as generalized minimal residual (GMRES), are widely employed in the numerical solution of partial differential equations (PDEs). However, the convergence rate of these methods tends to deteriorate as the condition number of the discrete linear system increases: a phenomenon associated with higher contrast in the system matrix A and with spatial mesh refinement [1]. To mitigate this issue, several classical preconditioners can be found in the literature, including Jacobi, incomplete factorizations (ILU), and algebraic multigrid (AMG). In particular, AMG is regarded as one of the most efficient preconditioners and is often classified as optimal for a broad class of problems, such as elliptic PDEs[1]. Recently, the hybridization of Krylov methods with machine learning approaches has attracted increasing attention in the literature [1,2]. In this context, neural networks have emerged as promising alternatives for constructing preconditioners capable of capturing complex features of the discrete operator. In this work, we investigate the use of neural network–based preconditioners with the goal of accelerating the convergence of Krylov methods. It is emphasized that the focus of this study is not on designing the best possible preconditioner for a specific problem, but rather on analyzing the convergence properties and the potential of machine learning–based approaches when employed as preconditioners. The numerical experiments demonstrate that neural preconditioners outperform traditional methods, such as Jacobi and ILU, and achieve competitive performance compared to AMG. The evaluations were conducted on an elliptic Darcy problem using datasets with varying contrasts and resolutions, highlighting the robustness and generalization capability of the neural preconditioners. Acknowledgements. We would like to thank PETROBRAS (Petróleo Brasileiro S.A.) and CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico) for the financial support for this research. REFERENCES [1] Kopaničáková, Alena, and George Em Karniadakis., Deeponet based preconditioning strategies for solving parametric linear systems of equations, SIAM Journal on Scientific Computing, 47(1), C151-C181, 2025. [2] Giraud, Luc, et al., Neural network preconditioning: a case study for the solution of the parametric Helmholtz equation, Doctoral dissertation, Inria Centre at the University of Bordeaux, France, 2025.