(untitled) (Communications in Computational Physics, 2024)
The success of deep learning in various applications has generated a growing interest in understanding its theoretical foundations. This paper presents a theoretical framework that explains why over-parameterized neural networks can perform well. Our analysis begins from the perspective of approximation theory and argues that over-parameterized deep neural networks with bounded norms can effectively approximate the target. Additionally, we demonstrate that the metric entropy of such networks is independent of the number of network parameters. We utilize these findings to derive consistency results for over-parameterized deep regression and the deep Ritz method, respectively. Furthermore, we prove convergence rates when the target has higher regularity, which, to our knowledge, represents the first convergence rate for over-parameterized deep learning.