Mathematical and Computational Understanding of Neural Networks: From Representation to Learning and From Shallow to Deep Abstract.

In this talk, I will present some understanding of a few basic mathematical and computational questions for neural networks, as a particular form of nonlinear representation, and show how the network structure, activation function, and parameter initialization can affect its approximation properties and the learning process. In particular, we propose structured and balanced multi-component and multi-layer neural networks (MMNN) using sine as the activation function with an initialization scaling strategy.  At the end, I will discuss a few issues and challenges when using neural networks to solve partial differential equations