This study used a multigrid-based convolutional neural network architecture known as MgNet in operator learning to solve numerical partial differential equations (PDEs). Given the property of smoothing iterations in multigrid methods where low-frequency errors decay slowly, we introduced a low-frequency correction structure for residuals to enhance the standard V-cycle MgNet. The enhanced MgNet model can capture the low-frequency features of solutions considerably better than the standard V-cycle MgNet. The numerical results obtained using some standard operator learning tasks are better than those obtained using many state-of-the-art methods, demonstrating the efficiency of our model.Moreover, numerically, our new model is more robust in case of low- and high-resolution data during training and testing, respectively.

翻译：本研究采用基于多重网格的卷积神经网络架构MgNet进行算子学习，以求解数值偏微分方程（PDEs）。鉴于多重网格方法中光滑迭代具有低频误差衰减缓慢的特性，我们引入了一种残差低频校正结构来增强标准V循环MgNet。增强后的MgNet模型在捕捉解的低频特征方面显著优于标准V循环MgNet。通过若干标准算子学习任务获得的数值结果优于许多最先进方法，验证了模型的有效性。此外，数值实验表明，在训练低分辨率数据与测试高分辨率数据时，新模型具有更强的鲁棒性。