This work exclusively focuses on using convolution neural operator learning for accelerating the solution of some heterogenous PDEs (including Poisson equations, Darcy flow, Diffusion-Advection equations) using the flexible GMRES [1] method. We use operator learning with U-Net [2] neural network architecture. For the sake of learning general information, the neural operator is trained with randomly generated datasets using an unsupervised approach. The trained neural operator exhibits significant generalization features with respect to different aspects. That includes the ability to address varying source terms, diffusivity terms, velocity field for advection, and varying boundary conditions for these heterogeneous equations. Furthermore, it also shows promising results for addressing wider range of the advection dominant situations, which is the challenging case for the advection-diffusion equations. Overall, this work demonstrates the efficiency of applying the neural operator learning as felxible preconditioner for subspace iterative linear solvers.