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.

翻译：在科学与工程领域，经常需要反复求解大规模、高分辨率的偏微分方程（PDEs）。神经算子是一种新型模型，能够映射函数空间，使训练后的模型可以模拟PDE的解算子。本文提出了一种具有多网格架构的新型傅里叶神经算子（MgFNO）。MgFNO将深度神经网络（DNNs）的频率原理与求解线性系统的多网格思想相结合。为了加速FNO的训练过程，采用了一种三层V循环多网格架构。该架构涉及在粗网格上多次训练模型，然后将其迁移到细网格上以加速模型训练。基于DNN的求解器从低频到高频学习解，而多网格方法则从高频到低频获取解。值得注意的是，FNO是一种分辨率不变的解算子，因此相应的计算得以大幅简化。最后，在Burgers方程、Darcy流和Navier-Stokes方程上进行了实验。结果表明，所提出的MgFNO优于传统的傅里叶神经算子。