  We develop a unified model, known as MgNet, that simultaneously recovers some
convolutional neural networks (CNN) for image classification and multigrid (MG)
methods for solving discretized partial differential equations (PDEs). This
model is based on close connections that we have observed and uncovered between
the CNN and MG methodologies. For example, pooling operation and feature
extraction in CNN correspond directly to restriction operation and iterative
smoothers in MG, respectively. As the solution space is often the dual of the
data space in PDEs, the analogous concept of feature space and data space
(which are dual to each other) is introduced in CNN. With such connections and
new concept in the unified model, the function of various convolution
operations and pooling used in CNN can be better understood. As a result,
modified CNN models (with fewer weights and hyper parameters) are developed
that exhibit competitive and sometimes better performance in comparison with
existing CNN models when applied to both CIFAR-10 and CIFAR-100 data sets.
