title: ÐÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ðµ ÑÐ¾ÑÐ½Ð¾ÑÑÐ¸ ÑÐ¸ÑÐ»ÐµÐ½Ð½Ð¾Ð³Ð¾ ÑÐµÑÐµÐ½Ð¸Ñ ÐºÑÐ°ÐµÐ²Ð¾Ð¹ Ð·Ð°Ð´Ð°ÑÐ¸ Ð¼ÐµÑÐ¾Ð´Ð¾Ð¼ Ð¼Ð°ÑÐ¸Ð½Ð½Ð¾Ð³Ð¾ Ð¾Ð±ÑÑÐµÐ½Ð¸Ñ
title: Ð²ÑÐ¿ÑÑÐºÐ½Ð°Ñ ÐºÐ²Ð°Ð»Ð¸ÑÐ¸ÐºÐ°ÑÐ¸Ð¾Ð½Ð½Ð°Ñ ÑÐ°Ð±Ð¾ÑÐ° Ð±Ð°ÐºÐ°Ð»Ð°Ð²ÑÐ°
titleType: Subtitle
title: Investigation of the accuracy of the numerical solution of a boundary value problem by machine learning.
Creators: name: Ð¢Ð¸Ð¼Ð¾ÑÐµÐµÐ²Ð°, ÐÐ½Ð½Ð°
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: 2022
Subjects: Ð¼Ð°ÑÐ¸Ð½Ð½Ð¾Ðµ Ð¾Ð±ÑÑÐµÐ½Ð¸Ðµ
ÐºÑÐ°ÐµÐ²ÑÐµ Ð·Ð°Ð´Ð°ÑÐ¸
Ð½ÐµÐ¹ÑÐ¾Ð½Ð½ÑÐµ ÑÐµÑÐ¸
Ð³Ð»ÑÐ±Ð¾ÐºÐ¾Ðµ Ð¾Ð±ÑÑÐµÐ½Ð¸Ðµ
ÑÐ¾ÑÐ½Ð¾ÑÑÑ ÑÐµÑÐµÐ½Ð¸Ñ
machine learning
boundary value problems
neural networks
deep learning
solution accuracy

description: Ð¢ÐµÐ¼Ð° Ð²ÑÐ¿ÑÑÐºÐ½Ð¾Ð¹ ÐºÐ²Ð°Ð»Ð¸ÑÐ¸ÐºÐ°ÑÐ¸Ð¾Ð½Ð½Ð¾Ð¹ ÑÐ°Ð±Ð¾ÑÑ: Â«ÐÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ðµ ÑÐ¾ÑÐ½Ð¾ÑÑÐ¸ ÑÐ¸ÑÐ»ÐµÐ½Ð½Ð¾Ð³Ð¾ ÑÐµÑÐµÐ½Ð¸Ñ ÐºÑÐ°ÐµÐ²Ð¾Ð¹ Ð·Ð°Ð´Ð°ÑÐ¸ Ð¼ÐµÑÐ¾Ð´Ð¾Ð¼ Ð¼Ð°ÑÐ¸Ð½Ð½Ð¾Ð³Ð¾ Ð¾Ð±ÑÑÐµÐ½Ð¸ÑÂ». Ð Ð´Ð°Ð½Ð½Ð¾Ð¹ ÑÐ°Ð±Ð¾ÑÐµ Ð¾Ð±ÑÐµÐºÑÐ°Ð¼Ð¸ Ð¸ÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ñ ÑÐ²Ð»ÑÑÑÑÑ ÐºÑÐ°ÐµÐ²ÑÐµ Ð·Ð°Ð´Ð°ÑÐ¸ Ð¸ Ð¼ÐµÑÐ¾Ð´Â Ð¼Ð°ÑÐ¸Ð½Ð½Ð¾Ð³Ð¾ Ð¾Ð±ÑÑÐµÐ½Ð¸Ñ, Ñ Ð¿Ð¾Ð¼Ð¾ÑÑÑ ÐºÐ¾ÑÐ¾ÑÐ¾Ð³Ð¾ Ð¼Ð¾Ð¶Ð½Ð¾ Ð¿Ð¾Ð»ÑÑÐ¸ÑÑ Ð¿ÑÐ¸Ð±Ð»Ð¸Ð¶ÐµÐ½Ð½Ð¾Ðµ ÑÐµÑÐµÐ½Ð¸Ðµ. ÐÑÐ½Ð¾Ð²Ð½Ð°Ñ ÑÐµÐ»Ñ ÑÐ°Ð±Ð¾ÑÑ â Ð¿Ð¾ÑÐ¼Ð¾ÑÑÐµÑÑ, ÐºÐ°Ðº Ð¼Ð¾Ð¶ÐµÑ Ð¼ÐµÐ½ÑÑÑÑÑ ÑÐ¾ÑÐ½Ð¾ÑÑÑ Ð¿Ð¾Ð»ÑÑÐµÐ½Ð½Ð¾Ð³Ð¾Â ÑÐµÑÐµÐ½Ð¸Ñ Ð² Ð·Ð°Ð²Ð¸ÑÐ¸Ð¼Ð¾ÑÑÐ¸ Ð¾Ñ Ð¿Ð°ÑÐ°Ð¼ÐµÑÑÐ¾Ð² Ð½ÐµÐ¹ÑÐ¾Ð½Ð½Ð¾Ð¹ ÑÐµÑÐ¸. ÐÐ°Ð´Ð°ÑÐ¸, ÐºÐ¾ÑÐ¾ÑÑÐµ ÑÐµÑÐ°Ð»Ð¸ÑÑÂ Ð² ÑÐ¾Ð´Ðµ Ð¸ÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ñ: â ÐÐ¿Ð¸ÑÐ°Ð½Ð¸Ðµ Ð¼ÐµÑÐ¾Ð´Ð° Ð³Ð»ÑÐ±Ð¾ÐºÐ¾Ð³Ð¾ Ð¾Ð±ÑÑÐµÐ½Ð¸Ñ, Ð¿ÑÐ¸Ð¼ÐµÐ½ÑÐµÐ¼ÑÐ¹ Ð´Ð»Ñ ÑÐµÑÐµÐ½Ð¸Ñ Ð´Ð¸ÑÑÐµÑÐµÐ½ÑÐ¸Ð°Ð»ÑÐ½Ð¾Ð³Ð¾ ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ñ. â Ð Ð°Ð·ÑÐ°Ð±Ð¾ÑÐºÐ° Ð¿ÑÐ¾Ð³ÑÐ°Ð¼Ð¼Ð½Ð¾Ð¹ ÑÐµÐ°Ð»Ð¸Ð·Ð°ÑÐ¸Ð¸ Ð¼Ð¾Ð´ÐµÐ»Ð¸ Ð½ÐµÐ¹ÑÐ¾Ð½Ð½Ð¾Ð¹ ÑÐµÑÐ¸. â ÐÑÐ¿Ð¾Ð»ÑÐ·Ð¾Ð²Ð°Ð½Ð¸Ðµ Ð°Ð»Ð³Ð¾ÑÐ¸ÑÐ¼Ð° Ð´Ð»Ñ Ð¾Ð¿ÑÐ¸Ð¼Ð¸Ð·Ð°ÑÐ¸Ð¸ ÑÐ»ÑÑÐ°Ð¹Ð½Ð¾Ð¹ ÑÐµÐ»ÐµÐ²Ð¾Ð¹ ÑÑÐ½ÐºÑÐ¸Ð¸Â Ð½Ð° Ð¾ÑÐ½Ð¾Ð²Ðµ ÑÐ°Ð³Ð°. â ÐÑÐ±Ð¾Ñ Ð´Ð²ÑÑ ÑÐµÑÑÐ¾Ð²ÑÑ ÐºÑÐ°ÐµÐ²ÑÑ Ð·Ð°Ð´Ð°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Ð¾Ð½Ð½Ð¾Ð¹ ÑÐµÑÐ¸. â ÐÑÐ¾Ð²ÐµÐ´ÐµÐ½Ð¸Ðµ Ð°Ð½Ð°Ð»Ð¸Ð·Ð° ÑÐ¾ÑÐ½Ð¾ÑÑÐ¸ Ð½Ð° Ð¾ÑÐ½Ð¾Ð²Ðµ Ð¿Ð¾Ð»ÑÑÐµÐ½Ð½ÑÑ ÑÐµÐ·ÑÐ»ÑÑÐ°ÑÐ¾Ð². 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 ÑÐµÑÑÐ¾Ð²ÑÑ ÑÐ»ÑÑÐ°ÐµÐ², Ð±ÑÐ»Ð° ÑÐµÐ°Ð»Ð¸Ð·Ð¾Ð²Ð°Ð½Ð°Â Ð½ÐµÐ¹ÑÐ¾Ð½Ð½Ð°Ñ ÑÐµÑÑ Ð¿ÑÑÐ¼Ð¾Ð³Ð¾ ÑÐ°ÑÐ¿ÑÐ¾ÑÑÑÐ°Ð½ÐµÐ½Ð¸Ñ. ÐÑÐ¸ Ð¾Ð±ÑÑÐµÐ½Ð¸Ð¸ Ð±ÑÐ» Ð¿ÑÐ¸Ð¼ÐµÐ½ÐµÐ½ Ð¼ÐµÑÐ¾Ð´Â Ð¾Ð¿ÑÐ¸Ð¼Ð¸Ð·Ð°ÑÐ¸Ð¸. ÐÐ¾ÑÐ»Ðµ ÑÐµÐ³Ð¾, Ð¿ÑÐ¾Ð²ÐµÐ´ÐµÐ½ 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Â ÑÐºÑÐ¿ÐµÑÐ¸Ð¼ÐµÐ½ÑÐ¾Ð².
description: The subject of the graduate qualification work is Â«Investigation of the accuracy ofÂ the numerical solution of a boundary value problem by machine learningÂ». In the given work the objects of study are boundary value problems and a machineÂ learning method, with which you can get an approximate solution. The main goal of theÂ work is to see how the accuracy of the obtained solution can change depending on theÂ parameters of the neural network. The research set the following tasks: â A description of a deep learning method used to solve a differential equation. â Development of a software implementation of a neural network model. â Using an algorithm to optimize a random objective function based on a step. â Selection of two test boundary value problems of interest for obtaining a solutionÂ using a neural network. â Conducting an accuracy analysis based on the results obtained. As a result of solving the considered test cases, a feed -forward neural network wasÂ implemented. During training, the optimization method was applied. After that, aÂ comparative analysis of the results of numerical experiments was carried out.