Physics-informed multi-grid neural operator: theory and an application to porous flow simulation

Suihong Song
Traditional PDE solvers are computationally expensive and slow, making inverse and optimization problems challenging. We propose a physics-informed multi-grid finite neural operator as an efficient alternative for PDE solvers. The training of the operator involves using the physics-informed method to train a neural network on the original fine 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 fine grid. The operator theory is validated with three porous flow examples of varying dimensions: its predictions closely match results obtained from a numerical solver while executing 10 to 100 times faster.
 

​Figure: Comparison of results from neural operator and traditional solver (Eclipse) in porous flow simulations