Abstract: 

In this talk, we will present recent results on the theories, algorithms, and applications of deep neural networks (DNNs) from a numerical analysis perspective. First, we will illustrate the connections between linear finite elements and ReLU DNNs, as well as spectral methods and ReLU^k DNNs. Second, we will show our latest findings regarding the open question of whether DNNs can precisely recover piecewise polynomials of arbitrary order on any simplicial mesh in any dimension. Then, inspired by the multigrid structure in numerical PDEs, we will discuss a unified framework for convolutional neural networks (CNNs) and multigrid methods, known as MgNet. Additionally, we will showcase recent advancements in the theories and applications of MgNet, particularly the first approximation result for CNNs with 2D inputs and an efficient operator learning framework, MgNO.