Deep learning has become a powerful tool for approximating complex functions, with significant implications for mathematics and physics. In this talk, we examine neural networks from the perspective of function approximation, discussing their theoretical foundations and practical applications. We begin by framing deep learning as a method for learning mappings between input and output spaces, highlighting the roles of activation functions and gradient-based optimization in the training process.

We then discuss universal approximation theorems, which establish the theoretical expressive power of neural networks, and address common challenges such as vanishing gradients. To overcome these challenges, we explore architectural innovations like residual networks (ResNets) and their application to solving ordinary differential equations (ODEs).

Next, we connect mathematical convolution kernels to convolutional neural networks (CNNs), demonstrating their utility in processing structured data. Turning to partial differential equations (PDEs), we review modern approaches, including physics-informed neural networks (PINNs) and collocation methods, which enable PDE solutions without traditional discretization techniques.

To illustrate these concepts, we present a case study: solving the 2D Poisson equation using PINNs. This example highlights the effectiveness of neural networks in computational physics and demonstrates their potential for addressing scientific problems.

This talk aims to provide mathematicians and physicists with an understanding of the theoretical principles underlying deep learning while showcasing its practical applications in solving differential equations.