title: ÐÐ¿ÑÑ Ð¿ÑÐ¸Ð¼ÐµÐ½ÐµÐ½Ð¸Ñ Ð½ÐµÐ¹ÑÐ¾Ð½Ð½ÑÑ ÑÐµÑÐµÐ¹ Ð´Ð»Ñ ÑÐ°ÑÑÐµÑÐ° Ð´Ð²ÑÐ¼ÐµÑÐ½ÑÑ ÑÐµÑÐµÐ½Ð¸Ð¹ Ð²ÑÐ·ÐºÐ¾Ð¹ Ð¶Ð¸Ð´ÐºÐ¾Ñ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: Experience in using neural networks for calculating two-dimensional flows of viscous fluid
Creators: name: Ð¢ÑÐ¾ÑÐ¸Ð½, ÐÐ¸ÑÐ°Ð¸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: 2023
Subjects: Ð¼Ð°ÑÐ¸Ð½Ð½Ð¾Ðµ Ð¾Ð±ÑÑÐµÐ½Ð¸Ðµ
ÑÐ¸Ð·Ð¸ÐºÐ¾-Ð¸Ð½ÑÐ¾ÑÐ¼Ð¸ÑÐ¾Ð²Ð°Ð½Ð½Ð°Ñ Ð½ÐµÐ¹ÑÐ¾Ð½Ð½Ð°Ñ ÑÐµÑÑ
Ð»Ð°Ð¼Ð¸Ð½Ð°ÑÐ½Ð¾Ðµ ÑÐµÑÐµÐ½Ð¸Ðµ
ÐºÐ°Ð½Ð°Ð» Ñ Ð²Ð½ÐµÐ·Ð°Ð¿Ð½ÑÐ¼ ÑÑÐ¶ÐµÐ½Ð¸ÐµÐ¼
PINN
Tensorflow
Pytorch
machine learning
physics-informed neural network
laminar flow
sudden constriction channel

description: ÐÐ°Ð½Ð½Ð°Ñ ÑÐ°Ð±Ð¾ÑÐ° Ð¿Ð¾ÑÐ²ÑÑÐµÐ½Ð° Ð¸ÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ñ Ð²Ð¾Ð·Ð¼Ð¾Ð¶Ð½Ð¾ÑÑÐµÐ¹ Ð¿ÑÐ¸Ð¼ÐµÐ½ÐµÐ½Ð¸Ñ Ð½ÐµÐ¹ÑÐ¾Ð½Ð½ÑÑ ÑÐµÑÐµÐ¹ Ð² Ð¾Ð±Ð»Ð°ÑÑÐ¸ Ð³Ð¸Ð´ÑÐ¾Ð´Ð¸Ð½Ð°Ð¼Ð¸ÐºÐ¸ Ñ ÑÐµÐ»ÑÑ Ð¿Ð¾Ð»ÑÑÐµÐ½Ð¸Ñ Ð´Ð¾ÑÑÐ¾Ð²ÐµÑÐ½ÑÑ Ð³Ð¸Ð´ÑÐ¾Ð´Ð¸Ð½Ð°Ð¼Ð¸ÑÐµÑÐºÐ¸Ñ ÑÐ°ÑÐ°ÐºÑÐµÑÐ¸ÑÑÐ¸Ðº. 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Ð±Ð·Ð¾Ñ Ð¸ Ð°Ð½Ð°Ð»Ð¸Ð· ÑÑÑÐµÑÑÐ²ÑÑÑÐµÐ¹ Ð»Ð¸ÑÐµÑÐ°ÑÑÑÑ, Ð¿Ð¾ÑÐ²ÑÑÐµÐ½Ð½Ð¾Ð¹ Ð¿ÑÐ¸Ð¼ÐµÐ½ÐµÐ½Ð¸Ñ Ð½ÐµÐ¹ÑÐ¾Ð½Ð½ÑÑ ÑÐµÑÐµÐ¹ Ð² Ð³Ð¸Ð´ÑÐ¾Ð´Ð¸Ð½Ð°Ð¼Ð¸ÐºÐµ. 2. ÐÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ðµ Ð¸ Ð²ÑÐ±Ð¾Ñ Ð¾Ð¿ÑÐ¸Ð¼Ð°Ð»ÑÐ½Ð¾Ð³Ð¾ Ð¿Ð¾Ð´ÑÐ¾Ð´Ð° Ðº Ð¿ÑÐ¾Ð³Ð½Ð¾Ð·Ð¸ÑÐ¾Ð²Ð°Ð½Ð¸Ñ. 3. Ð ÐµÐ°Ð»Ð¸Ð·Ð°ÑÐ¸Ñ Ð¿ÑÐ¾Ð³ÑÐ°Ð¼Ð¼Ñ Ð´Ð»Ñ Ð¼Ð¾Ð´ÐµÐ»Ð¸ÑÐ¾Ð²Ð°Ð½Ð¸Ñ Ð»Ð°Ð¼Ð¸Ð½Ð°ÑÐ½Ð¾Ð³Ð¾ ÑÐµÑÐµÐ½Ð¸Ñ Ð² ÐºÐ°Ð½Ð°Ð»Ðµ Ñ ÑÐµÐ·ÐºÐ¸Ð¼ ÑÑÐ¶ÐµÐ½Ð¸ÐµÐ¼. 4. ÐÑÐ¾Ð²ÐµÐ´ÐµÐ½Ð¸Ðµ ÑÐµÑÐ¸Ð¸ Ð²ÑÑÐ¸ÑÐ»Ð¸ÑÐµÐ»ÑÐ½ÑÑ ÑÐºÑÐ¿ÐµÑÐ¸Ð¼ÐµÐ½ÑÐ¾Ð² Ð½Ð° Ð¾ÑÐ½Ð¾Ð²Ðµ ÑÐ°Ð·ÑÐ°Ð±Ð¾ÑÐ°Ð½Ð½Ð¾Ð¹ Ð¿ÑÐ¾Ð³ÑÐ°Ð¼Ð¼Ñ Ð´Ð»Ñ ÑÐ°Ð·Ð»Ð¸ÑÐ½ÑÑ ÑÐ¸ÑÐµÐ» Ð ÐµÐ¹Ð½Ð¾Ð»ÑÐ´ÑÐ°. 5. Ð Ð°Ð·ÑÐ°Ð±Ð¾ÑÐºÐ° Ð½ÐµÐ¹ÑÐ¾Ð½Ð½Ð¾Ð¹ ÑÐµÑÐ¸ Ñ Ð¾Ð±ÑÐ°ÑÐ½ÑÐ¼ Ð¿ÑÐ¾Ð³Ð½Ð¾Ð·Ð¾Ð¼. Ð Ð´Ð°Ð½Ð½Ð¾Ð¼ Ð¸ÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ð¸ Ð±ÑÐ»Ð¾ Ð²ÑÐ±ÑÐ°Ð½Ð¾ Ð¿ÑÐ¸Ð¼ÐµÐ½ÐµÐ½Ð¸Ðµ ÑÐ¸Ð·Ð¸ÐºÐ¾-Ð¸Ð½ÑÐ¾ÑÐ¼Ð¸ÑÐ¾Ð²Ð°Ð½Ð½Ð¾Ð¹ Ð½ÐµÐ¹ÑÐ¾Ð½Ð½Ð¾Ð¹ ÑÐµÑÐ¸, Ð²Ð²Ð¸Ð´Ñ Ð¸Ñ ÑÑÑÐµÐºÑÐ¸Ð²Ð½Ð¾ÑÑÐ¸ Ð² ÑÑÐµÐ½Ð°ÑÐ¸ÑÑ Ð¿ÑÐ¾Ð³Ð½Ð¾Ð·Ð¸ÑÐ¾Ð²Ð°Ð½Ð¸Ñ, Ð¾ÑÐ½Ð¾Ð²Ð°Ð½Ð½ÑÑ Ð½Ð° ÑÐ¸Ð·Ð¸ÑÐµÑÐºÐ¸Ñ Ð·Ð°ÐºÐ¾Ð½Ð°Ñ Ð¸ Ð¾Ð³ÑÐ°Ð½Ð¸ÑÐµÐ½Ð¸ÑÑ. ÐÑÐ»Ð¸ÑÐ¸ÑÐµÐ»ÑÐ½Ð°Ñ Ð¾ÑÐ¾Ð±ÐµÐ½Ð½Ð¾ÑÑÑ Ð´Ð°Ð½Ð½ÑÑ ÑÐµÑÐµÐ¹ Ð·Ð°ÐºÐ»ÑÑÐ°ÐµÑÑÑ Ð² Ð²Ð¾Ð·Ð¼Ð¾Ð¶Ð½Ð¾ÑÑÐ¸ ÑÐ½Ð¸Ð¶ÐµÐ½Ð¸Ñ Ð·Ð°Ð²Ð¸ÑÐ¸Ð¼Ð¾ÑÑÐ¸ Ð¾Ñ Ð¸ÑÑÐ¾Ð´Ð½Ð¾Ð³Ð¾ Ð¾Ð±ÑÐµÐ¼Ð° Ð´Ð°Ð½Ð½ÑÑ Ð¸Ð»Ð¸ Ð´Ð°Ð¶Ðµ ÐµÐ³Ð¾ Ð¿Ð¾Ð»Ð½Ð¾Ð³Ð¾ Ð¸ÑÐºÐ»ÑÑÐµÐ½Ð¸Ñ Ð² ÑÐ¾Ð´Ðµ Ð¾Ð±ÑÑÐµÐ½Ð¸Ñ Ð½ÐµÐ¹ÑÐ¾Ð½Ð½Ð¾Ð¹ ÑÐµÑÐ¸. ÐÐµÑÑ Ð¿ÑÐ¾ÑÐµÑÑ Ð¿ÑÐ¾ÐµÐºÑÐ¸ÑÐ¾Ð²Ð°Ð½Ð¸Ñ Ð¸ ÑÐ°Ð·ÑÐ°Ð±Ð¾ÑÐºÐ¸ Ð¿ÑÐ¾Ð²Ð¾Ð´Ð¸Ð»ÑÑ Ñ Ð¿ÑÐ¸Ð¼ÐµÐ½ÐµÐ½Ð¸ÐµÐ¼ Ð¾ÑÐºÑÑÑÑÑ Ð±Ð¸Ð±Ð»Ð¸Ð¾ÑÐµÐº Ð¸ ÑÐ·ÑÐºÐ° Ð¿ÑÐ¾Ð³ÑÐ°Ð¼Ð¼Ð¸ÑÐ¾Ð²Ð°Ð½Ð¸Ñ Python. Ð ÑÐµÐ·ÑÐ»ÑÑÐ°ÑÐµ Ð¿ÑÐ¾Ð²ÐµÐ´ÐµÐ½Ð½Ð¾Ð¹ ÑÐ°Ð±Ð¾ÑÑ Ð¿Ð¾Ð»ÑÑÐµÐ½Ñ ÑÐµÐ½Ð½ÑÐµ Ð½Ð°Ð²ÑÐºÐ¸ Ð¸ Ð·Ð½Ð°Ð½Ð¸Ñ Ð² Ð¾Ð±Ð»Ð°ÑÑÐ¸ Ð¿ÑÐ¸Ð¼ÐµÐ½ÐµÐ½Ð¸Ñ Ð½ÐµÐ¹ÑÐ¾ÑÐµÑÐµÐ²ÑÑ Ð¼ÐµÑÐ¾Ð´Ð¾Ð² Ð² Ð³Ð¸Ð´ÑÐ¾Ð´Ð¸Ð½Ð°Ð¼Ð¸ÐºÐµ. ÐÑÐ»Ð¸ Ð¸Ð·ÑÑÐµÐ½Ñ Ð¸ Ð¿ÑÐ¸Ð¼ÐµÐ½ÐµÐ½Ñ Ð°ÐºÑÑÐ°Ð»ÑÐ½ÑÐµ Ð¸Ð½ÑÑÑÑÐ¼ÐµÐ½ÑÑ Ð´Ð»Ñ ÑÐ°Ð·ÑÐ°Ð±Ð¾ÑÐºÐ¸ Ð¸ Ð¿ÑÐ¾ÐµÐºÑÐ¸ÑÐ¾Ð²Ð°Ð½Ð¸Ñ, Ð° ÑÐ°ÐºÐ¶Ðµ Ð¿Ð¾Ð»ÑÑÐµÐ½Ñ ÑÐµÐ·ÑÐ»ÑÑÐ°ÑÑ, Ð¾ÑÑÐ°Ð¶Ð°ÑÑÐ¸Ðµ ÑÐ¾Ð²ÑÐµÐ¼ÐµÐ½Ð½ÑÐµ Ð´Ð¾ÑÑÐ¸Ð¶ÐµÐ½Ð¸Ñ Ð² Ð¾Ð±Ð»Ð°ÑÑÐ¸ Ð¸ÑÐ¿Ð¾Ð»ÑÐ·Ð¾Ð²Ð°Ð½Ð¸Ñ ÑÐµÑÐ½Ð¾Ð»Ð¾Ð³Ð¸Ð¸ Ð½ÐµÐ¹ÑÐ¾Ð½Ð½ÑÑ ÑÐµÑÐµÐ¹ Ð² Ð²ÑÑÐ¸ÑÐ»Ð¸ÑÐµÐ»ÑÐ½Ð¾Ð¹ Ð³Ð¸Ð´ÑÐ¾Ð´Ð¸Ð½Ð°Ð¼Ð¸ÐºÐµ.
description: This work is dedicated to exploring the possibilities of applying neural networks in the field of hydrodynamics to obtain reliable hydrodynamic characteristics. The tasks that were solved in the course of the research are: 1. Review and analysis of existing literature dedicated to the application of neural networks in hydrodynamics. 2. Exploration and selection of the optimal approach to prediction. 3. Implementation of a program for modeling laminar flow in a channel with a sharp narrowing. 4. Conducting a series of computational experiments based on the developed program for different Reynolds numbers. 5. Implementation of an Inverse neural network. In this study, the use of physics-informed neural networks was chosen due to their effectiveness in prediction scenarios based on physical laws and constraints. A distinctive feature of these networks is the possibility of reducing dependence on the initial volume of data or even its complete exclusion during the training of the neural network. The entire design and development process was carried out using open libraries and the Python programming language. As a result of the work carried out, valuable skills and knowledge were gained in the field of application of neural network methods in hydrodynamics. Actual tools for development and design were studied and applied, and results were obtained that reflect modern achievements in the field of using neural network technology in computational hydrodynamics.