title: ÐÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ðµ Ð¼ÐµÑÐ¾Ð´Ð¾Ð² Ð¼Ð°ÑÐ¸Ð½Ð½Ð¾Ð³Ð¾ Ð¾Ð±ÑÑÐµÐ½Ð¸Ñ Ð´Ð»Ñ ÑÐµÑÐµÐ½Ð¸Ñ Ð´Ð²ÑÑ ÑÐ¾ÑÐµÑÐ½Ð¾Ð¹ ÐºÑÐ°ÐµÐ²Ð¾Ð¹ Ð·Ð°Ð´Ð°ÑÐ¸
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: Research of methods of machine learning for the two-point boundary value problem
Creators: name: Ð¢ÑÐ¾ÑÐµÐ½ÐºÐ¾Ð², ÐÐ²Ð°Ð½
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: 2023
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¸
Ð¼ÐµÑÐ¾Ð´ Deep Ritz
ÑÐ¾ÑÐ½Ð¾ÑÑÑ ÑÐµÑÐµÐ½Ð¸Ñ
machine learning
boundary value problems
neural networks
Deep Ritz method
solution accuracy

description: 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¸ 1 ÑÐ¾Ð´Ð° Ð´Ð»Ñ Ð¾Ð±ÑÐºÐ½Ð¾Ð²ÐµÐ½Ð½Ð¾Ð³Ð¾ Ð´Ð¸ÑÑÐµÑÐµÐ½ÑÐ¸Ð°Ð»ÑÐ½Ð¾Ð³Ð¾ ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ñ 2 Ð¿Ð¾ÑÑÐ´ÐºÐ° Ð¸ Ð¼ÐµÑÐ¾Ð´ 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¿Ð¾ÑÑÐ°Ð²Ð»ÐµÐ½Ð½Ð¾Ð¹ Ð·Ð°Ð´Ð°ÑÐ¸. ÐÑÐ½Ð¾Ð²Ð½Ð°Ñ ÑÐµÐ»Ñ ÑÐ°Ð±Ð¾ÑÑ â Ð¿Ð¾ÐºÐ°Ð·Ð°ÑÑ, ÐºÐ°Ðº Ð¼Ð¾Ð¶ÐµÑ Ð¼ÐµÐ½ÑÑÑÑÑ ÑÐ¾ÑÐ½Ð¾ÑÑÑ Ð¿Ð¾Ð»ÑÑÐµÐ½Ð½Ð¾Ð³Ð¾ 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 Ð² ÑÐ¾Ð´Ðµ Ð¸ÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ñ Ð´Ð»Ñ Ð´Ð¾ÑÑÐ¸Ð¶ÐµÐ½Ð¸Ñ ÑÐµÐ»Ð¸ ÑÐ°Ð±Ð¾ÑÑ: â ÐÐ¿Ð¸ÑÐ°Ð½Ð¸Ðµ Ð¼ÐµÑÐ¾Ð´Ð° Ð¼Ð°ÑÐ¸Ð½Ð½Ð¾Ð³Ð¾ Ð¾Ð±ÑÑÐµÐ½Ð¸Ñ (Deep Ritz), Ð¿ÑÐ¸Ð¼ÐµÐ½ÑÐµÐ¼Ð¾Ð³Ð¾ 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Ð¾Ñ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¸ Ð½Ð° Ð¾ÑÐ½Ð¾Ð²Ðµ ÑÐ°Ð³Ð°. â ÐÑÐ±Ð¾Ñ Ð¾Ð´Ð½Ð¾Ð¹ ÑÐµÑÑÐ¾Ð²Ð¾Ð¹ ÐºÑÐ°ÐµÐ²Ð¾Ð¹ Ð·Ð°Ð´Ð°ÑÐ¸, Ð¿ÑÐµÐ´ÑÑÐ°Ð²Ð»ÑÑÑÐµÐ¹ 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¸ Ð¼ÐµÑÐ¾Ð´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° 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´ Ð¾Ð¿ÑÐ¸Ð¼Ð¸Ð·Ð°ÑÐ¸Ð¸. ÐÐ¾ÑÐ»Ðµ Ð¿Ð¾Ð»ÑÑÐµÐ½Ð¸Ñ Ð¿ÑÐ¸Ð±Ð»Ð¸Ð¶ÐµÐ½Ð½Ð¾Ð³Ð¾ ÑÐµÑÐµÐ½Ð¸Ñ Ð¸ Ð½ÐµÐºÐ¾ÑÐ¾ÑÑÑ Ð´ÑÑÐ³Ð¸Ñ ÑÐ°ÑÐ°ÐºÑÐµÑÐ¸ÑÑÐ¸Ðº Ð¼ÐµÑÐ¾Ð´Ð° Ð±ÑÐ» 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 ÑÐºÑÐ¿ÐµÑÐ¸Ð¼ÐµÐ½ÑÐ¾Ð².
description: In this work objects of research are boundary value problems of the first sort for the ordinary differential equation of the second order and method of machine learning, with which it is possible to receive the approximate solution of the set problem. The main purpose of this work is to show how the accuracy of the obtained solution can change depending on the parameters of the neural network. The tasks that were solved during the research to achieve the purpose of the work are: â A description of the machine learning method (Deep Ritz) used to solve the differential equation. â Development of the software implementation of the neural network model. â Using an algorithm to optimize a random objective function based on a step. â Selection of one test boundary value problem of interest for a neural network solution. â Performing an accuracy analysis of the method based on the obtained results. As a result of solving the considered test cases, a feed-forward neural network was implemented. An optimization method was applied in training. After obtaining an approximate solution and some other characteristics of the method, a comparative analysis of the results of numerical experiments was performed.