Efficient solution of large-scale partial differential equations (PDEs) remains a central challenge in computational science and engineering, particularly in matrix-free and high-resolution settings where classical solver components become difficult to construct or scale. Iterative methods are widely used for these problems, but their convergence critically depends on effective preconditioning. While classical smoothers (e.g., Jacobi, ILU, FSAI) efficiently damp high-frequency error components, the treatment of low-frequency modes remains a major bottleneck in complex geometries, heterogeneous media, and matrix-free formulations. Recent advances in Scientific Machine Learning (SciML), and in particular operator learning techniques such as Deep Operator Networks (DeepONet), have demonstrated strong capabilities in approximating solution operators of PDEs. However, most existing approaches focus on replacing entire solvers with data-driven surrogates, raising concerns about scalability, robustness, and generalization in large-scale applications. In this work, we propose a hybrid SciML approach in which DeepONet is used not as a full solver surrogate, but as a learned coarse solver that targets the low-frequency components of the inverse operator. The DeepONet is trained in a supervised manner on carefully constructed low-frequency input vectors. Once trained, the network provides a non-intrusive, matrix-free approximation of coarse-grid correction that can be seamlessly integrated into Krylov methods alongside classical smoothers. Numerical experiments on linear elliptic PDEs demonstrate that the learned operator effectively captures the dominant smooth error modes and significantly accelerates convergence when combined with an FSAI preconditioner, reducing iteration counts by an order of magnitude in representative 2D test cases. These results highlight the potential of operator learning as a scalable and flexible tool for enhancing classical solvers in large-scale SciML applications, offering a data-driven alternative in challenging computational settings.