title: Ð¡ÑÐ°Ð²Ð½ÐµÐ½Ð¸Ðµ ÑÐ¸ÑÐ»ÐµÐ½Ð½ÑÑ ÑÐµÑÐµÐ½Ð¸Ð¹ ÐºÑÐ°ÐµÐ²Ð¾Ð¹ Ð·Ð°Ð´Ð°ÑÐ¸ Ð¼ÐµÑÐ¾Ð´Ð°Ð¼Ð¸ 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: Ð¡omparison of numerical solutions of the boundary value problem obtained by machine learning methods
Creators: name: Ð¡ÑÑÐ¾ÑÐ¾Ð², ÐÐ°ÐºÑÐ¸Ð¼
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: ÐºÑÐ°ÐµÐ²Ð°Ñ Ð·Ð°Ð´Ð°ÑÐ°
Ð´Ð¸ÑÑÐµÑÐµÐ½ÑÐ¸Ð°Ð»ÑÐ½ÑÐµÂ­ ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ñ
ÑÐ¸ÑÐ»ÐµÐ½Ð½Ð¾Ð¹ ÑÐµÑÐµÐ½Ð¸Ðµ
Ð¼Ð°ÑÐ¸Ð½Ð½Ð¾Ðµ Ð¾Ð±ÑÑÐµÐ½Ð¸Ðµ
Ð½ÐµÐ¹ÑÐ¾Ð½Ð½ÑÐ¹ ÑÐµÑÐ¸
Â«Ð³Ð»ÑÐ±Ð¾ÐºÐ¸Ð¹Â» Ð¼ÐµÑÐ¾Ð´ Ð Ð¸ÑÑÐ°
Â«Ð³Ð»ÑÐ±Ð¾ÐºÐ¸Ð¹Â» Ð¼ÐµÑÐ¾Ð´ ÐÐ°Ð»ÐµÑÐºÐ¸Ð½Ð°
ÑÐ¸ÑÐ»ÐµÐ½Ð½ÑÐ¹ Ð°Ð½Ð°Ð»Ð¸Ð·
ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ðµ ÑÐ¸ÑÐ»ÐµÐ½Ð½ÑÑ Ð¼ÐµÑÐ¾Ð´Ð¾Ð²
boundary value problem
differential equations
numerical solution
machine learning
neural networks
Deep Ritz method
Deep Galerkin method
numerical analysis
comparison of numerical methods

description: Ð Ð´Ð°Ð½Ð½Ð¾Ð¹ ÑÐ°Ð±Ð¾ÑÐµ Ð¿ÑÐµÐ´ÑÑÐ°Ð²Ð»ÐµÐ½Ð¾ ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ðµ Ð½Ð° Ð¾ÑÐ½Ð¾Ð²Ðµ ÑÐ¸ÑÐ»ÐµÐ½Ð½ÑÑ ÑÐºÑÐ¿ÐµÑÐ¸Ð¼ÐµÐ½Â­ÑÐ¾Ð² ÑÑÑÐµÐºÑÐ¸Ð²Ð½Ð¾ÑÑÐ¸ Ð¼ÐµÑÐ¾Ð´Ð¾Ð² ÑÐµÑÐµÐ½Ð¸Ñ ÐºÑÐ°ÐµÐ²Ð¾Ð¹ Ð·Ð°Ð´Ð°ÑÐ¸ Ð´Ð»Ñ ÐÐÐ£ Ð²ÑÐ¾ÑÐ¾Ð³Ð¾ Ð¿Ð¾ÑÑÐ´ÐºÐ°, Ð¾ÑÐ½Ð¾Ð²Ð°Ð½Ð½ÑÑ Ð½Ð° Ð¸ÑÐ¿Ð¾Ð»ÑÐ·Ð¾Ð²Ð°Ð½Ð¸Ð¸ Ð¸ÑÐºÑÑÑÑÐ²ÐµÐ½Ð½ÑÑ Ð½ÐµÐ¹ÑÐ¾Ð½Ð½ÑÑ ÑÐµÑÐµÐ¹ Ð² ÐºÐ°ÑÐµÑÑÐ²Ðµ ÑÐ¿Ð¾ÑÐ¾Ð±Ð° Ð¿ÑÐ¸Ð±Ð»Ð¸Ð¶ÐµÐ½Ð¸Ñ ÑÑÐ½ÐºÑÐ¸Ð¹ â Deep Ritz Ð¸ Deep Galerkin. ÐÑÐ¸Ð²ÐµÐ´ÐµÐ½Ñ Ð½ÐµÐºÐ¾ÑÐ¾ÑÑÐµ ÑÐµÐ¾ÑÐµÑÐ¸ÑÐµÑÐºÐ¸Ðµ Ð°ÑÐ¿ÐµÐºÑÑ ÑÑÐ¾Ð¹ Ð¿ÑÐµÐ´Ð¼ÐµÑÐ½Ð¾Ð¹ Ð¾Ð±Ð»Ð°ÑÑÐ¸ Ð¸ ÑÑÑÐ»ÐºÐ¸ Ð½Ð° ÐºÐ»ÑÑÐµÐ²ÑÑ Ð»Ð¸ÑÐµÑÐ°ÑÑÑÑ Ð¿Ð¾ Ð½ÐµÐ¹. ÐÐ¾ÐºÐ°Ð·Ð°Ð½Ð¾, ÑÑÐ¾ Ð¿ÑÐ¸ ÑÐ°Ð²ÐµÐ½ÑÑÐ²Ðµ ÐºÐ»ÑÑÐµÐ²ÑÑ Ð¿Ð°ÑÐ°Ð¼ÐµÑÑÐ¾Ð² â ÐºÐ¾Ð»Ð¸ÑÐµÑÑÐ²Ð° ÑÐ¿Ð¾Ñ Ð¾Ð±ÑÑÐµÐ½Ð¸Ñ ÑÐµÑÐ¸, ÑÐ°Ð·Ð¼ÐµÑÐµ Ð¾Ð±ÑÑÐ°ÑÑÐµÐ¹ Ð²ÑÐ±Ð¾ÑÐºÐ¸ Ð¸ Ð°ÑÑÐ¸ÑÐµÐºÑÑÑÐµ Ð½ÐµÐ¹ÑÐ¾Ð½Ð½Ð¾Ð¹ ÑÐµÑÐ¸ â Deep Galerkin Ð¼ÐµÑÐ¾Ð´ Ð¾Ð±ÐµÑÐ¿ÐµÑÐ¸Ð²Ð°ÐµÑ Ð´Ð¾ÑÑÐ¸Ð¶ÐµÐ½Ð¸Ðµ Ð·Ð½Ð°ÑÐ¸ÑÐµÐ»ÑÐ½Ð¾ Ð±Ð¾Ð»ÐµÐµ Ð²ÑÑÐ¾ÐºÐ¾Ð¹ ÑÐ¾ÑÐ½Ð¾ÑÑÐ¸ ÑÐµÑÐµÐ½Ð¸Ñ ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ñ, Ð²Ð¿Ð»Ð¾ÑÑ Ð´Ð¾ Ð´ÐµÑÑÑÐ¸ÐºÑÐ°ÑÐ½Ð¾Ð³Ð¾ Ð¿ÑÐµÐ²ÑÑÐµÐ½Ð¸Ñ ÑÐ¾ÑÐ½Ð¾ÑÑÐ¸ Deep Ritz Ð¼ÐµÑÐ¾Ð´Ð°. ÐÐ¾ÐºÐ°Ð·Ð°Ð½Ð¾ ÑÐ°ÐºÐ¶Ðµ, ÑÑÐ¾ Deep Galerkin Ð¼ÐµÑÐ¾Ð´ Ð±Ð¾Ð»ÐµÐµ ÑÑÑÐµÐºÑÐ¸Ð²ÐµÐ½ Ñ ÑÐ¾ÑÐºÐ¸ Ð·ÑÐµÐ½Ð¸Ñ ÐºÐ¾Ð»Ð¸ÑÐµÂ­ÑÑÐ²Ð° Ð²ÑÑÐ¸ÑÐ»ÐµÐ½Ð¸Ð¹. Ð Ð·Ð°ÐºÐ»ÑÑÐµÐ½Ð¸Ðµ ÑÐ°Ð±Ð¾ÑÑ Ð°ÐºÑÐµÐ½ÑÐ¸ÑÐ¾Ð²Ð°Ð½Ð¾ Ð²Ð½Ð¸Ð¼Ð°Ð½Ð¸Ðµ Ð½Ð° Ð¿ÑÐµÐ¸Ð¼ÑÑÐµÑÑÐ²Ðµ Ð¿Ð¾Ð´Ð¾Ð±Ð½Ð¾Ð³Ð¾ ÑÐ¾Ð´Ð° Ð¼ÐµÑÐ¾Ð´Ð¾Ð² Ð¿ÑÐ¸ ÑÐµÑÐµÐ½Ð¸Ð¸ ÐºÑÐ°ÐµÐ²ÑÑ Ð·Ð°Ð´Ð°Ñ Ð´Ð»Ñ Ð´Ð¸ÑÑÐµÑÐµÐ½ÑÐ¸Ð°Ð»ÑÐ½ÑÑ ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ð¹ Ð²ÑÑÐ¾ÐºÐ¾Ð¹ ÑÐ°Ð·Ð¼ÐµÑÐ½Ð¾ÑÑÐ¸.
description: This paper presents a based on numerical experiments comparison of the effectiveness of methods for solving the boundary value problem for second-order ODEs based on the use of artificial neural networks as a way of approximating the functions â Deep Ritz and Deep Galerkin methods. Some theoretical aspects of this subject area and references to key literature on it are given. It is shown that with the equality of the key parameters - the number of epochs of network training, the size of the training sample and the architecture of the neural network - the Deep Galerkin method achieves significantly higher accuracy of solving the equation, up to ten times the accuracy of the Deep Ritz method. It is also shown that the Deep Galerkin method is more efficient in terms of the number of calculations. In conclusion, attention is focused on the advantage of such methods in solving boundary value problems for high-dimensional differential equations.