kaz29Вестник Казахстанско-Британского технического университетаHerald of the Kazakh-British Technical University1998-66882959-8109Казахстанско-Британский Технический Университет10.55452/1998-6688-2026-23-3-33-46kaz29-3173Research ArticleМАТЕМАТИЧЕСКИЕ НАУКИMATHEMATICAL SCIENCESПРИМЕНЕНИЕ ФИЗИКО-ОРИЕНТИРОВАННЫХ НЕЙРОННЫХ СЕТЕЙ ДЛЯ РЕШЕНИЯ ЗАДАЧ ЛИНЕЙНЫХ УРАВНЕНИЙ В ЧАСТНЫХ ПРОИЗВОДНЫХ С НЕОДНОРОДНЫМИ СМЕШАННЫМИ ГРАНИЧНЫМИ УСЛОВИЯМИAPPLICATION OF PHYSICALLY-ORIENTED NEURAL NETWORKS TO SOLVING PROBLEMS OF LINEAR PARTIAL DERIVATIVE EQUATIONS WITH INHOMOGENEOUS MIXED BOUNDARY CONDITIONShttps://orcid.org/0000-0002-1937-8615ИсаховА. А.IssakhovA. A.

Профессор

Алматы

Professor

Almaty

alibek.issakhov@gmail.comhttps://orcid.org/0009-0004-5395-4569БейсембековМ. Е.

Бакалавр

Алматы

m_beisembekov@kbtu.kzКазахстанско-Британский технический университетКазахстанKazakh-British Technical UniversityKazakhstan2026240920262333346Copyright © Исахов А.А., Бейсембеков М.Е., 20262026Исахов А.А., Бейсембеков М.Е.Issakhov A.A., Бейсембеков М.Е.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/3173

В работе рассматривается применение Physics-Informed Neural Networks (PINN) для решения двумер­ного уравнения Пуассона с неоднородными смешанными граничными условиями. Метод PINN объединяет нейросетевые аппроксимации и физические законы, включая невязку уравнения и ошибки на границе в единую функцию потерь. Модель реализована в среде PyTorch и обучалась на выборке, сгенерированной ме­тодом латинской гиперкубической выборки. Для проверки точности проведено сравнение с аналитическим решением, а также с традиционными численными методами. Систематически исследовано влияние ключе­вых гиперпараметров: числа обучающих точек, весов в функции потерь, архитектуры сети и шага обучения. Установлено, что увеличение веса на граничные условия улучшает сходимость и точность, а оптимальные архитектуры с 4-5 скрытыми слоями и 64 нейронами обеспечивают относительную ошибку менее 0.1-0.2%. В отличие от численных схем, требующих сеточной дискретизации, PINN продемонстрировал универсаль­ность и высокую точность при работе с разреженными данными и сложными условиями, что делает его пер­спективным инструментом для задач вычислительной математики и моделирования физических процессов в условиях сложной геометрии и неполной информации.

The paper considers the application of Physics-Informed Neural Networks (PINN) to solving a two-dimensional Poisson equation with inhomogeneous mixed boundary conditions. The PINN method combines neural network approximations and physical laws, including the residual of the equation and errors at the boundary, into a single loss function. The model is implemented in the PyTorch environment and trained on a sample generated by the Latin hypercubic sampling method. To verify the accuracy, a comparison is made with an analytical solution, as well as with traditional numerical methods. The influence of key hyperparameters is systematically studied: the number of training points, weights in the loss function, network architecture, and training step. It is found that increasing the weight on the boundary conditions improves convergence and accuracy, and optimal architectures with 4-5 hidden layers and 64 neurons provide a relative error of less than 0.1-0.2%. Unlike numerical schemes requiring grid discretization, PINN has demonstrated versatility and high accuracy when dealing with sparse data and complex conditions, making it a promising tool for computational mathematics and modeling of physical processes in conditions of complex geometry and incomplete information.

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