MATHEMATICAL MODELING IN CONVOLUTIONAL NEURAL NETWORKS: BASIC PRINCIPLES AND OPEN QUESTIONS

Ledovskaya Ekaterina V.

Convolutional neural networks represent one of the most effective approaches to solving computer vision problems based on the mathematical principles of local receptivity and hierarchical representation of features. Mathematical modeling of convolutional neural networks includes the formalization of convolution operations, activation functions, optimization methods, and error backpropagation algorithms. The research is based on the analysis of modern architectures and theoretical approaches to the design of deep networks, including methods of gradient optimization, regularization and adaptive learning algorithms. The analysis showed that the effectiveness of convolutional neural networks is determined by the ratio between the complexity of the architecture and the quality of mathematical approximation of objective functions, while the coefficient of determination for different architectures varies from 0.78 to 0.94. Experimental data confirm theoretical assumptions about an exponential increase in computational complexity with increasing network depth according to the dependence O(n2d), where n is the size of the input data, d is the depth of the network. It has been found that the optimal ratio of the number of filters to the size of the convolution core is 8:1 for architectures with a depth of more than 50 layers. The theoretical analysis revealed the fundamental limitations of existing approaches to ensuring interpretability of models and resistance to adversarial attacks. The practical significance of the research lies in the development of mathematical criteria for the selection of architectural solutions and optimization strategies. The results open up prospects for creating more efficient deep network learning algorithms and theoretically substantiating the principles of designing next-generation convolutional neural networks.

Keywords:convolutional neural networks, mathematical modeling, gradient optimization, approximation theory, interpretability, architectural design, deep learning.