An Enhanced V-cycle MgNet Model for Operator Learning in Numerical Partial Differential Equations
This work addresses operator learning for PDEs, offering incremental improvements in accuracy and robustness for computational science applications.
The study tackled solving numerical partial differential equations by enhancing a multigrid-based neural network (MgNet) with a low-frequency correction structure, resulting in better performance than state-of-the-art methods and improved robustness across data resolutions.
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.