Special SIAM online seminar

In this talk, I will present recent results on applying multigrid structures to both general and specific operator learning problems in numerical PDEs. First, we will illustrate MgNet as a unified framework for convolutional neural networks and multigrid methods with some preliminary theories and applications. Then, we will discuss some basic background on operator learning, including the problem setup, a uniform framework, and a general universal approximation result. Motivated by the general definition of neural operators and the MgNet structure, we propose MgNO, which utilizes multigrid structures to parameterize these linear operators between neurons, offering a new and concise architecture in operator learning. This approach provides both mathematical rigor and practical expressivity, with many interesting numerical properties and observations. Finally, I will offer some remarks on our most recent work in using multigrid structures to approximate and learn a specific operator in numerical PDEs.

These results are based on joint work with Prof. Jinchao Xu and Dr. Xinliang Liu.