title: ÐÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ðµ Ð²Ð»Ð¸ÑÐ½Ð¸Ñ Ð¿Ð°ÑÐ°Ð¼ÐµÑÑÐ¾Ð² Ð½ÐµÐ¹ÑÐ¾Ð½Ð½Ð¾Ð¹ ÑÐµÑÐ¸ Ð½Ð° ÑÐ¸ÑÐ»ÐµÐ½Ð½Ð¾Ðµ Ð½Ð° ÑÐ¸ÑÐ»ÐµÐ½Ð½Ð¾Ðµ ÑÐµÑÐµÐ½Ð¸Ðµ Ð´Ð²ÑÑ ÑÐ¾ÑÐµÑÐ½Ð¾Ð¹ ÐºÑÐ°ÐµÐ²Ð¾Ð¹ Ð·Ð°Ð´Ð°ÑÐ¸ Ð¼ÐµÑÐ¾Ð´Ð¾Ð¼ Deep Ritz
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 Neural Network Parameters Influence on Numerical Solution of Two-Point Boundary Problem using Deep Ritz Method
Creators: name: ÐÐ¼Ð¾Ð½Ð¾Ð²Ð°, Ð¡Ð¸ÑÐ¾ÑÐ°
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: 2024
Subjects: Ð´Ð²ÑÑ ÑÐ¾ÑÐµÑÐ½Ð°Ñ ÐºÑÐ°ÐµÐ²Ð°Ñ Ð·Ð°Ð´Ð°ÑÐ°
Ð¼ÐµÑÐ¾Ð´ deep ritz
Ð½ÐµÐ¹ÑÐ¾Ð½Ð½ÑÐµ ÑÐµÑÐ¸
ÑÐ¸ÑÐ»ÐµÐ½Ð½Ð¾Ðµ ÑÐµÑÐµÐ½Ð¸Ðµ
Ð¾Ð¿ÑÐ¸Ð¼Ð¸Ð·Ð°ÑÐ¸Ñ Ð¿Ð°ÑÐ°Ð¼ÐµÑÑÐ¾Ð²
Ð³ÑÐ°Ð´Ð¸ÐµÐ½ÑÐ½ÑÐ¹ ÑÐ¿ÑÑÐº
ÑÐ¾ÑÐ½Ð¾ÑÑÑ ÑÐµÑÐµÐ½Ð¸Ð¹
Ð³Ð»ÑÐ±Ð¾ÐºÐ¾Ðµ Ð¾Ð±ÑÑÐµÐ½Ð¸Ðµ
two-point boundary problem
deep ritz method
neural networks
numerical solution
parameter optimization
gradient descent
solution accuracy
deep learning

description: Ð Ð´Ð°Ð½Ð½Ð¾Ð¹ Ð´Ð¸Ð¿Ð»Ð¾Ð¼Ð½Ð¾Ð¹ ÑÐ°Ð±Ð¾ÑÐµ Ð¿ÑÐ¾Ð²ÐµÐ´ÐµÐ½Ð¾ Ð¸ÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ðµ Ð²Ð»Ð¸ÑÐ½Ð¸Ñ Ð¿Ð°ÑÐ°Ð¼ÐµÑÑÐ¾Ð² Ð½ÐµÐ¹ÑÐ¾Ð½Ð½Ð¾Ð¹ ÑÐµÑÐ¸ Ð½Ð° ÑÐ¸ÑÐ»ÐµÐ½Ð½Ð¾Ðµ ÑÐµÑÐµÐ½Ð¸Ðµ Ð´Ð²ÑÑÑÐ¾ÑÐµÑÐ½Ð¾Ð¹ ÐºÑÐ°ÐµÐ²Ð¾Ð¹ Ð·Ð°Ð´Ð°ÑÐ¸ Ñ Ð¸ÑÐ¿Ð¾Ð»ÑÐ·Ð¾Ð²Ð°Ð½Ð¸ÐµÐ¼ Ð¼ÐµÑÐ¾Ð´Ð° Deep Ritz. ÐÑÐµÐ´Ð¼ÐµÑÐ¾Ð¼ Ð¸ÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ñ ÑÐ²Ð»ÑÐµÑÑÑ Ð¼ÐµÑÐ¾Ð´ Deep Ritz, Ð¿ÑÐ¸Ð¼ÐµÐ½ÑÐµÐ¼ÑÐ¹ Ð´Ð»Ñ ÑÐµÑÐµÐ½Ð¸Ñ ÐºÑÐ°ÐµÐ²ÑÑ Ð·Ð°Ð´Ð°Ñ, Ð° ÑÐµÐ¼Ð° ÑÐ°Ð±Ð¾ÑÑ â Ð°Ð½Ð°Ð»Ð¸Ð· ÑÐ¾ÑÐ½Ð¾ÑÑÐ¸ Ð¸ ÑÑÑÐµÐºÑÐ¸Ð²Ð½Ð¾ÑÑÐ¸ ÑÑÐ¾Ð³Ð¾ Ð¼ÐµÑÐ¾Ð´Ð° Ð¿ÑÐ¸ ÑÐ°Ð·Ð»Ð¸ÑÐ½ÑÑ Ð¿Ð°ÑÐ°Ð¼ÐµÑÑÐ°Ñ Ð½ÐµÐ¹ÑÐ¾Ð½Ð½Ð¾Ð¹ ÑÐµÑÐ¸. Ð¦ÐµÐ»Ñ ÑÐ°Ð±Ð¾ÑÑ Ð·Ð°ÐºÐ»ÑÑÐ°ÐµÑÑÑ Ð² Ð¾Ð¿ÑÐµÐ´ÐµÐ»ÐµÐ½Ð¸Ð¸ Ð¾Ð¿ÑÐ¸Ð¼Ð°Ð»ÑÐ½ÑÑ Ð¿Ð°ÑÐ°Ð¼ÐµÑÑÐ¾Ð² Ð½ÐµÐ¹ÑÐ¾Ð½Ð½Ð¾Ð¹ ÑÐµÑÐ¸, ÑÐ°ÐºÐ¸Ñ ÐºÐ°Ðº ÐºÐ¾Ð»Ð¸ÑÐµÑÑÐ²Ð¾ ÑÐ»Ð¾ÐµÐ², ÐºÐ¾Ð»Ð¸ÑÐµÑÑÐ²Ð¾ Ð½ÐµÐ¹ÑÐ¾Ð½Ð¾Ð², ÑÐ¸Ð¿ Ð°ÐºÑÐ¸Ð²Ð°ÑÐ¸Ð¾Ð½Ð½Ð¾Ð¹ ÑÑÐ½ÐºÑÐ¸Ð¸ Ð¸ Ð¿Ð°ÑÐ°Ð¼ÐµÑÑÑ Ð¾Ð¿ÑÐ¸Ð¼Ð¸Ð·Ð°ÑÐ¸Ð¸, Ð´Ð»Ñ Ð¿Ð¾Ð²ÑÑÐµÐ½Ð¸Ñ ÑÐ¾ÑÐ½Ð¾ÑÑÐ¸ Ð¸ ÑÑÑÐµÐºÑÐ¸Ð²Ð½Ð¾ÑÑÐ¸ ÑÐ¸ÑÐ»ÐµÐ½Ð½Ð¾Ð³Ð¾ ÑÐµÑÐµÐ½Ð¸Ñ ÐºÑÐ°ÐµÐ²ÑÑ Ð·Ð°Ð´Ð°Ñ. ÐÐµÑÐ¾Ð´Ð¾Ð»Ð¾Ð³Ð¸Ñ Ð¸ÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ñ Ð²ÐºÐ»ÑÑÐ°Ð»Ð° ÑÐ¸ÑÐ»ÐµÐ½Ð½Ð¾Ðµ ÑÐµÑÐµÐ½Ð¸Ðµ Ð´Ð²ÑÑ Ð·Ð°Ð´Ð°Ñ Ñ Ð·Ð°Ð´Ð°Ð½Ð½ÑÐ¼Ð¸ ÑÑÐ½ÐºÑÐ¸ÑÐ¼Ð¸ \( f = \sin(x) \) Ð¸ ÐºÑÑÐ¾ÑÐ½Ð¾Ð¹ ÑÑÐ½ÐºÑÐ¸ÐµÐ¹ Ð¿ÑÐ¸ Ð¿Ð¾Ð¼Ð¾ÑÐ¸ Ð¼ÐµÑÐ¾Ð´Ð° Deep Ritz Ð¸ Ð½ÐµÐ¹ÑÐ¾Ð½Ð½Ð¾Ð¹ ÑÐµÑÐ¸. Ð¡ÑÐ°Ð²Ð½ÐµÐ½Ð¸Ðµ ÑÐ¾ÑÐ½Ð¾Ð³Ð¾ Ð¸ Ð¿ÑÐµÐ´ÑÐºÐ°Ð·Ð°Ð½Ð½Ð¾Ð³Ð¾ ÑÐµÑÐµÐ½Ð¸Ð¹ Ð¿ÑÐ¾Ð²Ð¾Ð´Ð¸Ð»Ð¾ÑÑ Ñ Ð¿Ð¾ÑÑÑÐ¾ÐµÐ½Ð¸ÐµÐ¼ ÑÑÐµÑÐ¼ÐµÑÐ½ÑÑ Ð³ÑÐ°ÑÐ¸ÐºÐ¾Ð² Ð·Ð°Ð²Ð¸ÑÐ¸Ð¼Ð¾ÑÑÐ¸ Ð³Ð»ÑÐ±Ð¸Ð½Ñ ÑÐµÑÐ¸, ÐºÐ¾Ð»Ð¸ÑÐµÑÑÐ²Ð° Ð¸ÑÐµÑÐ°ÑÐ¸Ð¹ Ð¸ Ð¾ÑÐ½Ð¾ÑÐ¸ÑÐµÐ»ÑÐ½Ð¾Ð¹ Ð¿Ð¾Ð³ÑÐµÑÐ½Ð¾ÑÑÐ¸. ÐÐ° Ð¾ÑÐ½Ð¾Ð²Ðµ ÑÑÐ¸Ñ Ð³ÑÐ°ÑÐ¸ÐºÐ¾Ð² Ð±ÑÐ»Ð¸ Ð½Ð°Ð¹Ð´ÐµÐ½Ñ Ð¾Ð¿ÑÐ¸Ð¼Ð°Ð»ÑÐ½ÑÐµ Ð·Ð½Ð°ÑÐµÐ½Ð¸Ñ Ð¿Ð°ÑÐ°Ð¼ÐµÑÑÐ¾Ð², Ð¿Ð¾ÑÐ»Ðµ ÑÐµÐ³Ð¾ Ð¿ÑÐ¾Ð²ÐµÐ´ÐµÐ½Ñ Ð¿Ð¾Ð²ÑÐ¾ÑÐ½ÑÐµ ÑÐºÑÐ¿ÐµÑÐ¸Ð¼ÐµÐ½ÑÑ Ð´Ð»Ñ Ð¾ÑÐµÐ½ÐºÐ¸ ÑÑÐ°Ð±Ð¸Ð»ÑÐ½Ð¾ÑÑÐ¸ ÑÐµÐ·ÑÐ»ÑÑÐ°ÑÐ¾Ð². Ð ÐµÐ·ÑÐ»ÑÑÐ°ÑÑ Ð¿Ð¾ÐºÐ°Ð·Ð°Ð»Ð¸, ÑÑÐ¾ Ð¼ÐµÑÐ¾Ð´ Deep Ritz Ð¿Ð¾Ð·Ð²Ð¾Ð»ÑÐµÑ Ð¿Ð¾Ð»ÑÑÐ°ÑÑ ÑÐµÑÐµÐ½Ð¸Ñ Ñ Ð²ÑÑÐ¾ÐºÐ¾Ð¹ ÑÐ¾ÑÐ½Ð¾ÑÑÑÑ, Ð±Ð»Ð¸Ð·ÐºÐ¾Ð¹ Ðº ÑÐ¾ÑÐ½ÑÐ¼ ÑÐµÑÐµÐ½Ð¸ÑÐ¼. ÐÐ¿ÑÐ¸Ð¼Ð°Ð»ÑÐ½ÑÐµ Ð¿Ð°ÑÐ°Ð¼ÐµÑÑÑ ÑÐµÑÐ¸ Ð·Ð½Ð°ÑÐ¸ÑÐµÐ»ÑÐ½Ð¾ ÑÐ»ÑÑÑÐ°ÑÑ ÑÐºÐ¾ÑÐ¾ÑÑÑ ÑÑÐ¾Ð´Ð¸Ð¼Ð¾ÑÑÐ¸ Ð¸ ÑÐ¾ÑÐ½Ð¾ÑÑÑ Ð¼ÐµÑÐ¾Ð´Ð°. ÐÐ¾Ð²ÑÐ¾ÑÐ½ÑÐµ ÑÐºÑÐ¿ÐµÑÐ¸Ð¼ÐµÐ½ÑÑ Ð¿Ð¾Ð´ÑÐ²ÐµÑÐ´Ð¸Ð»Ð¸ Ð½ÐµÐ·Ð½Ð°ÑÐ¸ÑÐµÐ»ÑÐ½ÑÐµ Ð¸Ð·Ð¼ÐµÐ½ÐµÐ½Ð¸Ñ Ð² ÑÐµÑÐµÐ½Ð¸ÑÑ, ÑÑÐ¾ ÑÐ²Ð¸Ð´ÐµÑÐµÐ»ÑÑÑÐ²ÑÐµÑ Ð¾ Ð½Ð°Ð´ÐµÐ¶Ð½Ð¾ÑÑÐ¸ Ð¼ÐµÑÐ¾Ð´Ð°. ÐÐ±Ð»Ð°ÑÑÑ Ð¿ÑÐ¸Ð¼ÐµÐ½ÐµÐ½Ð¸Ñ ÑÐµÐ·ÑÐ»ÑÑÐ°ÑÐ¾Ð² Ð´Ð°Ð½Ð½Ð¾Ð¹ ÑÐ°Ð±Ð¾ÑÑ Ð²ÐºÐ»ÑÑÐ°ÐµÑ ÑÐ°Ð·ÑÐ°Ð±Ð¾ÑÐºÑ Ð±Ð¾Ð»ÐµÐµ ÑÑÑÐµÐºÑÐ¸Ð²Ð½ÑÑ Ð°Ð»Ð³Ð¾ÑÐ¸ÑÐ¼Ð¾Ð² ÑÐ¸ÑÐ»ÐµÐ½Ð½Ð¾Ð³Ð¾ ÑÐµÑÐµÐ½Ð¸Ñ ÐºÑÐ°ÐµÐ²ÑÑ Ð·Ð°Ð´Ð°Ñ Ñ Ð¸ÑÐ¿Ð¾Ð»ÑÐ·Ð¾Ð²Ð°Ð½Ð¸ÐµÐ¼ Ð³Ð»ÑÐ±Ð¾ÐºÐ¾Ð³Ð¾ Ð¾Ð±ÑÑÐµÐ½Ð¸Ñ. ÐÑÐ²Ð¾Ð´Ñ ÑÐ°Ð±Ð¾ÑÑ Ð¿Ð¾Ð´ÑÐ²ÐµÑÐ¶Ð´Ð°ÑÑ, ÑÑÐ¾ Ð¼ÐµÑÐ¾Ð´ Deep Ritz Ð¼Ð¾Ð¶ÐµÑ ÑÑÑÐµÑÑÐ²ÐµÐ½Ð½Ð¾ Ð¾Ð±Ð»ÐµÐ³ÑÐ¸ÑÑ ÑÐµÑÐµÐ½Ð¸Ðµ Ð¿Ð¾Ð´Ð¾Ð±Ð½ÑÑ Ð·Ð°Ð´Ð°Ñ, Ð¾Ð±ÐµÑÐ¿ÐµÑÐ¸Ð²Ð°Ñ Ð²ÑÑÐ¾ÐºÑÑ ÑÐ¾ÑÐ½Ð¾ÑÑÑ Ð¸ ÑÑÑÐµÐºÑÐ¸Ð²Ð½Ð¾ÑÑÑ.
description: This thesis explores the impact of neural network parameters on the numeric- al solution of a two-point boundary value problem using the Deep Ritz method. The subject of the study is the Deep Ritz method applied to boundary value problems, and the topic is analyzing the accuracy and efficiency of this method under various neural network parameters. The goal of the work is to determine the optimal parameters of the neural network, such as the number of layers, number of neurons, type of activation function, and optimization parameters, to improve the accuracy and efficiency of the numerical solution of boundary value problems. The research methodology included numerically solving two problems with given functions \( f = \sin(x) \) and a piecewise function using the Deep Ritz method and a neural network. Comparison of the exact and predicted solutions was conducted by constructing three-dimensional graphs of the dependence of network depth, number of iterations, and relative error. Based on these graphs, optimal parameter values were found, followed by repeated experiments to assess the stability of the results. The results showed that the Deep Ritz method provides solutions with high accuracy, close to the exact solutions. Optimal network parameters significantly improve the convergence speed and accuracy of the method. Repeated experim- ents confirmed insignificant changes in solutions, indicating the reliability of the method. The application area of the results of this work includes the development of more efficient algorithms for the numerical solution of boundary value problems using deep learning. The conclusions of the work confirm that the Deep Ritz method can significantly simplify the solution of such problems, ensuring high accuracy and efficiency.