• Wasserstein distance
• neural networks
• numerical experiments
• Gaussian Processes
• ANN
• DNN
• machine learning
• deep learning
• deep neural networks
• optimal transport

The thesis regards the quantitative convergence of randomly initialized fully connected deep neural networks towards a suitable Gaussian limit. In particular we estimate the 2-Wasserstein distance between the joint distribution of the outputs of a neural network and a suitable Gaussian Process, the "Neural Network Gaussian Process" (NNGP).
We make our estimates with techniques of optimal transport and techniques linked to Gaussian processes, and we obtain that such rate of convergence can be estimated with the inverse of the square root of the numbers of neurons in the layers.
We provide some numerical simulation to show that our estimates are not sharp, so further studies are needed to find the true rate of convergence and to study other properties of this convergence.