MgFNO: Multi-grid Architecture Fourier Neural Operator for Parametric Partial Differential Equations

In science and engineering, there is often a need to repeatedly solve large-scale and high-resolution partial differential equations (PDEs). Neural operators are a new type of models that can map between function spaces, allowing trained models to emulate the solution operators of PDEs. This paper introduces a novel Fourier neural operator with a multigrid architecture (MgFNO). The MgFNO combines the frequency principle of deep neural networks (DNNs) with the multigrid idea for solving linear systems. To speed up the training process of the FNO, a three-layer V-cycle multigrid architecture is used. This architecture involves training the model multiple times on a coarse grid and then transferring it to a fine grid to accelerate the training of the model. The DNN-based solver learns the solution from low to high frequency, while the multigrid method acquires the solution from high to low frequency. Note that the FNO is a resolution-invariant solution operator, therefore the corresponding calculations are greatly simplified. Finally, experiments are conducted on Burgers' equation, Darcy flow, and Navier-Stokes equation. The results demonstrate that the proposed MgFNO outperforms the traditional Fourier neural operator.