Neural differential equations: when deep learning meets mathematical modeling

This article discusses neural differential equations when deep learning meets mathematical modeling. The relevance of the topic is due to the fact that modern machine learning technologies, in particular deep learning, are at the peak of their development and find application in a wide variety of fields of science and technology. However, many features of their use and justification remain insufficiently studied, which makes it difficult to optimize and expand the scope of their application. The purpose of this study is to develop a mathematical model of neural differential equations for analyzing and predicting the dynamics of neural networks over time. To achieve this goal, it is necessary to solve problems aimed at studying existing approaches to mathematical modeling of neural networks and identifying their relationship to differential equations, as well as applying numerical methods to solve the resulting model by studying the behavior of a neural network with various input signals. Mathematical modeling, numerical methods, and comparative analysis are used as methods. The theoretical significance lies in expanding the understanding of mathematical modeling of neural networks, as well as in the development of new mathematical models that take into account the complexity of dynamic processes occurring in deep learning networks. The practical significance lies in the development of a mathematical model of neural differential equations and the description of the dynamics of neural networks over time. The developed models can be used to improve the quality of predictions, optimize learning processes and the operation of neural networks in real conditions.