We propose a physics-informed multi-grid finite neural operator as an efficient alternative for PDE solvers. The operator aims to map a random parameter function to the corresponding solution function of a linear PDE, both distributed on a fine grid (called original grid). The training of the operator involves using the physics-informed method to train a neural network on the original grid to approximate the PDE solution. The grid is then coarsened, and a new neural network is trained on the coarse grid to approximate the prediction error of the previous network. This process is iteratively repeated until the final grid's neural network achieves adequate accuracy. The combination of all neural networks from coarse to fine grids forms the multi-grid neural operator capable of mapping random parameters to solutions on the original grid. The above process involving fine-coarse-fine grid is called one V cycle (following the traditional multi-grid PDE literature), and multiple V cycles can be included into the operator to enhance its performance. Additionally, the operator integrates the idea of using coarse-grid predictions as initial guesses for fine-grid predictions. Validation is performed using three porous flow examples of varying dimensions. The operator's prediction accuracy improves with more V cycles included, and its predictions closely match results obtained from a numerical solver while executing 10 to 100 times faster.