An Enhanced V-cycle MgNet Model for Operator Learning in Numerical Partial Differential Equations

TL;DR

This paper introduces an improved MgNet model with a low-frequency correction structure for operator learning in PDEs, achieving better accuracy and robustness than existing methods.

Contribution

The paper proposes a novel low-frequency correction in MgNet, enhancing its ability to learn operators for PDEs more effectively than standard models.

Findings

Outperforms state-of-the-art methods in standard operator learning tasks.

More robust with low- and high-resolution data during training and testing.

Captures low-frequency features of solutions more effectively.

Abstract

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.