title: Ð¡ÑÐ°Ð²Ð½Ð¸ÑÐµÐ»ÑÐ½Ð¾Ðµ ÑÐµÑÑÐ¸ÑÐ¾Ð²Ð°Ð½Ð¸Ðµ Ð°Ð»Ð³Ð¾ÑÐ¸ÑÐ¼Ð¾Ð² Ð¾Ð¿ÑÐ¸Ð¼Ð¸Ð·Ð°ÑÐ¸Ð¸ Ð¿ÑÐ¸ Ð¾Ð±ÑÑÐµÐ½Ð¸Ð¸ Ð½ÐµÐ¹ÑÐ¾Ð½Ð½ÑÑ ÑÐµÑÐµÐ¹ Ð² Ð·Ð°Ð´Ð°ÑÐ°Ñ Ð¼Ð°ÑÐµÐ¼Ð°ÑÐ¸ÑÐµÑÐºÐ¾Ð³Ð¾ Ð¼Ð¾Ð´ÐµÐ»Ð¸ÑÐ¾Ð²Ð°Ð½Ð¸Ñ
title: Ð²ÑÐ¿ÑÑÐºÐ½Ð°Ñ ÐºÐ²Ð°Ð»Ð¸ÑÐ¸ÐºÐ°ÑÐ¸Ð¾Ð½Ð½Ð°Ñ ÑÐ°Ð±Ð¾ÑÐ° Ð¼Ð°Ð³Ð¸ÑÑÑÐ°
titleType: Subtitle
title: Comparative testing of optimization algorithms for training neural networks in mathematical modeling problems
Creators: name: ÐÐ°Ð»ÑÑ, ÐÐ°Ð²ÐµÐ»
Publisher: name: Ð¡Ð°Ð½ÐºÑ-ÐÐµÑÐµÑÐ±ÑÑÐ³ÑÐºÐ¸Ð¹ Ð¿Ð¾Ð»Ð¸ÑÐµÑÐ½Ð¸ÑÐµÑÐºÐ¸Ð¹ ÑÐ½Ð¸Ð²ÐµÑÑÐ¸ÑÐµÑ ÐÐµÑÑÐ° ÐÐµÐ»Ð¸ÐºÐ¾Ð³Ð¾
Year: 2020
Subjects: Ð½ÐµÐ¹ÑÐ¾Ð½Ð½ÑÐµ ÑÐµÑÐ¸
Ð´Ð¸ÑÑÐµÑÐµÐ½ÑÐ¸Ð°Ð»ÑÐ½ÑÐµ ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ñ
Ð¼Ð°ÑÐµÐ¼Ð°ÑÐ¸ÑÐµÑÐºÐ¾Ðµ Ð¼Ð¾Ð´ÐµÐ»Ð¸ÑÐ¾Ð²Ð°Ð½Ð¸Ðµ
Ð¼ÐµÑÐ¾Ð´Ñ Ð¾Ð¿ÑÐ¸Ð¼Ð¸Ð·Ð°ÑÐ¸Ð¸
ÑÐµÑÐµÐ½Ð¸Ðµ Ð¿ÐµÑÐµÐ¾Ð¿ÑÐµÐ´ÐµÐ»ÑÐ½Ð½ÑÑ Ð´Ð¸ÑÑÐµÑÐµÐ½ÑÐ¸Ð°Ð»ÑÐ½ÑÑ Ð·Ð°Ð´Ð°Ñ
neural network
differential equation
mathematical modeling
optimization method
the solution of the overdetermined differential problems

description: ÐÐ°Ð½Ð½Ð°Ñ ÑÐ°Ð±Ð¾ÑÐ° Ð¿Ð¾ÑÐ²ÑÑÐµÐ½Ð° ÑÑÐ°Ð²Ð½Ð¸ÑÐµÐ»ÑÐ½Ð¾Ð¼Ñ ÑÐµÑÑÐ¸ÑÐ¾Ð²Ð°Ð½Ð¸Ñ Ð°Ð»Ð³Ð¾ÑÐ¸ÑÐ¼Ð¾Ð² Ð¾Ð¿ÑÐ¸Ð¼Ð¸Ð·Ð°ÑÐ¸Ð¸ Ð¿ÑÐ¸ Ð¾Ð±ÑÑÐµÐ½Ð¸Ð¸ Ð½ÐµÐ¹ÑÐ¾Ð½Ð½ÑÑ ÑÐµÑÐµÐ¹, Ð¸ÑÐ¿Ð¾Ð»ÑÐ·ÑÐµÐ¼ÑÑ Ð´Ð»Ñ Ð¿Ð¾ÑÑÑÐ¾ÐµÐ½Ð¸Ñ ÑÐµÑÐµÐ½Ð¸Ð¹ Ð·Ð°Ð´Ð°Ñ Ð¼Ð°ÑÐµÐ¼Ð°ÑÐ¸ÑÐµÑÐºÐ¾Ð³Ð¾ Ð¼Ð¾Ð´ÐµÐ»Ð¸ÑÐ¾Ð²Ð°Ð½Ð¸Ñ. ÐÑÐ½Ð¾Ð²Ð½Ð°Ñ ÑÐµÐ»Ñ ÑÐ°Ð±Ð¾ÑÑ â Ð²ÑÑÐ²Ð»ÐµÐ½Ð¸Ðµ Ð½Ð°Ð¸Ð±Ð¾Ð»ÐµÐµ ÑÑÑÐµÐºÑÐ¸Ð²Ð½ÑÑ Ð¼ÐµÑÐ¾Ð´Ð¾Ð² Ð¾Ð¿ÑÐ¸Ð¼Ð¸Ð·Ð°ÑÐ¸Ð¸, Ð¸ÑÐ¿Ð¾Ð»ÑÐ·ÑÐµÐ¼ÑÑ Ð´Ð»Ñ Ð¿Ð¾ÑÑÑÐ¾ÐµÐ½Ð¸Ñ ÑÐµÑÐµÐ½Ð¸Ð¹ ÐºÐ¾Ð½ÐºÑÐµÑÐ½ÑÑ Ð¼Ð°ÑÐµÐ¼Ð°ÑÐ¸ÑÐµÑÐºÐ¸Ñ Ð·Ð°Ð´Ð°Ñ Ð¸ Ð¸Ñ ÑÐµÐ°Ð»Ð¸Ð·Ð°ÑÐ¸Ñ. ÐÐ°Ð´Ð°ÑÐ¸, ÐºÐ¾ÑÐ¾ÑÑÐµ ÑÐµÑÐ°Ð»Ð¸ÑÑ Ð² ÑÐ¾Ð´Ðµ Ð¸ÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ñ: 1. ÐÑÐ±Ð¾Ñ Ð·Ð°Ð´Ð°Ñ Ð¼Ð°ÑÐµÐ¼Ð°ÑÐ¸ÑÐµÑÐºÐ¾Ð³Ð¾ Ð¼Ð¾Ð´ÐµÐ»Ð¸ÑÐ¾Ð²Ð°Ð½Ð¸Ñ, Ð¿ÑÐµÐ´ÑÑÐ°Ð²Ð»ÑÑÑÐ¸Ñ Ð¸Ð½ÑÐµÑÐµÑ Ð´Ð»Ñ Ð¿Ð¾Ð»ÑÑÐµÐ½Ð¸Ñ ÑÐµÑÐµÐ½Ð¸Ñ Ñ Ð¿Ð¾Ð¼Ð¾ÑÑÑ Ð½ÐµÐ¹ÑÐ¾Ð½Ð½ÑÑ ÑÐµÑÐµÐ¹. 2. ÐÑÐ±Ð¾Ñ ÑÐ¾Ð¿Ð¾Ð»Ð¾Ð³Ð¸Ð¸ Ð½ÐµÐ¹ÑÐ¾Ð½Ð½ÑÑ ÑÐµÑÐµÐ¹. 3. ÐÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ðµ ÑÑÑÐµÐºÑÐ¸Ð²Ð½ÑÑ Ð¼ÐµÑÐ¾Ð´Ð¾Ð² Ð¾Ð¿ÑÐ¸Ð¼Ð¸Ð·Ð°ÑÐ¸Ð¸, Ð¿ÑÐ¸Ð¼ÐµÐ½ÑÐµÐ¼ÑÑ Ð´Ð»Ñ Ð¾Ð±ÑÑÐµÐ½Ð¸Ñ Ð½ÐµÐ¹ÑÐ¾Ð½Ð½ÑÑ ÑÐµÑÐµÐ¹. 4. ÐÑÐ¾Ð²ÐµÐ´ÐµÐ½Ð¸Ðµ ÑÑÐ°Ð²Ð½Ð¸ÑÐµÐ»ÑÐ½Ð¾Ð³Ð¾ Ð°Ð½Ð°Ð»Ð¸Ð·Ð° Ð¿Ð¾Ð»ÑÑÐµÐ½Ð½ÑÑ ÑÐµÐ·ÑÐ»ÑÑÐ°ÑÐ¾Ð². ÐÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ñ Ð¿ÑÐ¾Ð²Ð¾Ð´Ð¸Ð»Ð¸ÑÑ Ð½Ð° Ð¾ÑÐ½Ð¾Ð²Ðµ Ð²ÑÐ±ÑÐ°Ð½Ð½ÑÑ ÑÐ¸Ð¿Ð¾Ð² Ð½ÐµÐ¹ÑÐ¾Ð½Ð½ÑÑ ÑÐµÑÐµÐ¹. ÐÐ»Ñ ÑÑÐ¾Ð³Ð¾ Ð½ÐµÐ¹ÑÐ¾Ð½Ð½ÑÐµ ÑÐµÑÐ¸ Ð±ÑÐ»Ð¸ ÑÐµÐ°Ð»Ð¸Ð·Ð¾Ð²Ð°Ð½Ñ ÐºÐ°Ðº Ð¿Ð¾Ð»ÑÐ·Ð¾Ð²Ð°ÑÐµÐ»ÑÑÐºÐ¸Ðµ ÐºÐ»Ð°ÑÑÑ Ð¿Ð¾ÑÑÐµÐ´ÑÑÐ²Ð¾Ð¼ Ð¿ÑÐ¸Ð¼ÐµÐ½ÐµÐ½Ð¸Ñ ÑÐ·ÑÐºÐ° Ð¿ÑÐ¾Ð³ÑÐ°Ð¼Ð¼Ð¸ÑÐ¾Ð²Ð°Ð½Ð¸Ñ Matlab. ÐÑÐ»Ð¸ Ð¿ÑÐ¾Ð²ÐµÐ´ÐµÐ½Ñ ÑÐ°ÑÑÑÑÑ, ÐºÐ¾ÑÐ¾ÑÑÐµ Ð½Ð°Ð³Ð»ÑÐ´Ð½Ð¾ Ð¿Ð¾ÐºÐ°Ð·ÑÐ²Ð°Ð»Ð¸ ÑÑÑÐµÐºÑÐ¸Ð²Ð½Ð¾ÑÑÑ ÑÐµÑÑÐ¸ÑÑÐµÐ¼ÑÑ Ð¼ÐµÑÐ¾Ð´Ð¾Ð² Ð¾Ð¿ÑÐ¸Ð¼Ð¸Ð·Ð°ÑÐ¸Ð¸. Ð ÑÐµÐ·ÑÐ»ÑÑÐ°ÑÐµ ÑÐµÑÐµÐ½Ð¸Ñ ÑÐ°ÑÑÐ¼Ð°ÑÑÐ¸Ð²Ð°ÐµÐ¼ÑÑ Ð·Ð°Ð´Ð°Ñ Ð¼Ð°ÑÐµÐ¼Ð°ÑÐ¸ÑÐµÑÐºÐ¾Ð³Ð¾ Ð¼Ð¾Ð´ÐµÐ»Ð¸ÑÐ¾Ð²Ð°Ð½Ð¸Ñ, Ð±ÑÐ»Ð¸ Ð¿ÑÐ¾Ð³ÑÐ°Ð¼Ð¼Ð½Ð¾ ÑÐµÐ°Ð»Ð¸Ð·Ð¾Ð²Ð°Ð½Ñ ÑÐµÑÐ¸, Ð¿ÑÐ¸ Ð¾Ð±ÑÑÐµÐ½Ð¸Ð¸ ÐºÐ¾ÑÐ¾ÑÑÑ ÑÐµÑÑÐ¸ÑÐ¾Ð²Ð°Ð»Ð¸ÑÑ Ð¼ÐµÑÐ¾Ð´Ñ Ð¾Ð¿ÑÐ¸Ð¼Ð¸Ð·Ð°ÑÐ¸Ð¸. ÐÐ¾ÑÐ»Ðµ ÑÐµÐ³Ð¾, Ð¿ÑÐ¾Ð²ÐµÐ´ÑÐ½ ÑÑÐ°Ð²Ð½Ð¸ÑÐµÐ»ÑÐ½ÑÐ¹ Ð°Ð½Ð°Ð»Ð¸Ð· ÑÐµÐ·ÑÐ»ÑÑÐ°ÑÐ¾Ð² ÑÐ¸ÑÐ»ÐµÐ½Ð½ÑÑ ÑÐºÑÐ¿ÐµÑÐ¸Ð¼ÐµÐ½ÑÐ¾Ð². ÐÐ½Ð°Ð»Ð¸Ð· ÑÐµÐ·ÑÐ»ÑÑÐ°ÑÐ¾Ð² Ð¿Ð¾Ð·Ð²Ð¾Ð»Ð¸Ð» Ð²ÑÑÐ²Ð¸ÑÑ Ð½Ð°Ð¸Ð±Ð¾Ð»ÐµÐµ ÑÑÑÐµÐºÑÐ¸Ð²Ð½ÑÐµ Ð¼ÐµÑÐ¾Ð´Ñ Ð¾Ð¿ÑÐ¸Ð¼Ð¸Ð·Ð°ÑÐ¸Ð¸ ÑÐµÑÐ°ÐµÐ¼ÑÑ Ð·Ð°Ð´Ð°Ñ.
description: The given work is devoted to comparative testing of optimization algorithms for training neural networks used to build solutions to mathematical modeling problems. Main goal this research is identification of the most effective optimization methods used to build solutions to specific mathematical problems and their implementation. The research set the following tasks: 1. The choice of mathematical modeling problems that are of interest for obtaining solutions using neural networks. 2. The choice of the neural network topology. 3. The research of effective optimization methods used for training neural networks. 4. Carrying out a comparative analysis of the results obtained. The research was based on the selected types of neural networks. For this purpose, neural networks were implemented as user classes by the Matlab programming language. Calculations were carried out that clearly demonstrated the effectiveness of the tested optimization methods. As a result of solving the considered problems of mathematical modeling, networks were programmatically implemented, and optimization methods were tested during their training. After that, a comparative analysis of the results of numerical experiments was performed. The analysis of the results allowed to identify the most effective methods for optimizing the tasks to be solved.