kaz29Вестник Казахстанско-Британского технического университетаHerald of the Kazakh-British Technical University1998-66882959-8109Казахстанско-Британский Технический Университет10.55452/1998-6688-2025-22-4-143-154kaz29-2290Research ArticleКОМПЬЮТЕРНЫЕ НАУКИCOMPUTER SCIENCEМЕТОД ФИЗИЧЕСКИ ОБОСНОВАННОЙ НЕЙРОННОЙ СЕТИ (PINN), ОСНОВАННЫЙ НА САМОПОДОБНЫХ РЕШЕНИЯХPHYSICS-INFORMED NEURAL NETWORK (PINN) METHOD BASED ON SELF-SIMILAR SOLUTIONShttps://orcid.org/0000-0003-4351-0185НуртасМ.NurtasM.

PhD, ассоцированный профессор

г. Алматы

PhD, Associate Professor

Almaty

m.nurtas@iitu.edu.kzhttps://orcid.org/0000-0003-0131-4469АбдикаликоваЗ. Т.AbdikalikovaZ. T.

PhD, ассоцированный профессор

г. Алматы

PhD, Associate Professor

Almaty

z.abdikalikova@iitu.edu.kzhttps://orcid.org/0009-0006-2969-1695АдиатЛ.AdiatL.

магистр, лектор

г. Алматы

MSc, Lecturer

Almaty

l.adiat@iitu.edu.kzМеждународный университет информационных технологийКазахстанInternational University of Information TechnologiesKazakhstan202523122025224143154Copyright © Нуртас М., Абдикаликова З.Т., Адиат Л., 20252025Нуртас М., Абдикаликова З.Т., Адиат Л.Nurtas M., Abdikalikova Z.T., Adiat L.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://vestnik.kbtu.edu.kz/jour/article/view/2290

При численном решении дифференциальных уравнений в частных производных, описывающих сложные физические процессы, такие как теплопроводность и газовая динамика, зачастую требуется значительная вычислительная мощность. Для решения этих сложностей в последние годы особое внимание науки и техники привлекают физически информированные нейронные сети (Physics-Informed Neural Networks, PINN). В данной статье рассматривается задача нахождения решений уравнений теплопроводности и газовой динамики с помощью метода PINN. В отличие от традиционных численных методов, метод физически информированных нейронных сетей позволяет решать задачи, внедряя физические законы в структуру нейронной сети. То есть решение подчиняется не только данным, но и самому уравнению. В статье описываются архитектура метода PINN, структура функций потерь и их связь с уравнением теплопроводности и уравнениями Эйлера на конкретных примерах. Кроме того, анализируются механизмы введения начальных и граничных условий, а также факторы, влияющие на устойчивость и точность решений. Полученные результаты демонстрируют эффективность PINN и возможность их применения в будущем для решения сложных многомерных и многофазных задач. Также предложены исследования, направленные на ускорение вычислительного процесса и повышение стабильности PINN.

In the numerical solution of partial differential equations that describe complex physical processes such as heat conduction and gas dynamics, substantial computational resources are often required. To address these challenges, Physics-Informed Neural Networks (PINNs) have gained increasing attention in recent years within the fields of science and engineering. This paper investigates the application of the PINN methodology to obtain solutions to the heat conduction and gas dynamics equations. Unlike traditional numerical approaches, the physics-informed neural network framework incorporates governing physical laws directly into the neural network architecture. Consequently, the solution is constrained not only by data but also by the underlying differential equations. The paper presents the architecture of the PINN framework and details the structure of loss functions, demonstrating their relationship with the heat equation and the Euler equations using specific examples. Furthermore, the implementation of initial and boundary conditions is discussed, along with an analysis of factors influencing the stability and accuracy of the obtained solutions. The results highlight the efficiency of PINNs and demonstrate their potential for solving complex multiphase and high-dimensional problems in the future. Additionally, current research directions aimed at accelerating the computational process and enhancing the robustness of PINNs are outlined.

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The authors declare that there are no conflicts of interest present.