this thesis aims to explore the potential of statistical concepts, specifically the vapnik-chervonenkis dimension (vcd)[33], in optimizing neural networks. with the increasing use of neural networks in replacing human labor, ensuring the safety and reliability of these systems is a critical concern. the thesis delves into the question of how to test the safety of neural networks and optimize them through accessible statistical concepts. the thesis presents two case studies to demonstrate the effectiveness of using vcd in optimizing neural networks. the first case study focuses on optimizing the autoencoder, a neural network with both encoding and decoding functions, through the calculation of the vcd. the conclusion suggests that optimizing the activation function can improve the accuracy of the autoencoder at the mathematical level. the second case study explores the optimization of the vgg16 neural network by comparing it to vgg19 in terms of their ability to process high-density data. by adding three hidden layers, vgg19 outperforms vgg16 in learning ability, suggesting that adjusting the number of neural network layers can be an effective way to analyze the capacity of neural networks. overall, this thesis proposes that statistical concepts such as vcd can provide a promising avenue for analyzing neural networks, thus contributing to the development of more reliable and efficient machine learning systems. the final vision is to allocate the mathematical model reasonably to machine learning and establish an idealized neural network establishment, allowing for safe and effective use of neural networks in various industries.