МЕТОД ПАРАМЕТРИЗАЦИИ И ФИЗИЧЕСКИ ИНФОРМИРОВАННЫЕ НЕЙРОННЫЕ СЕТИ ДЛЯ УРАВНЕНИЯ ДУФФИНГА
DOI:
https://doi.org/10.66941/3106-9037/2026/iss3pp7-16Аннотация
Вступление. Физически информированные нейронные сети (PINN) все чаще предлагаются как универсальные решатели дифференциальных уравнений, однако их надежность для нелинейных осцилляторов с несколькими периодическими решениями изучена недостаточно. Цели и задачи. Сравнить метод параметризации Д.С. Джумабаева с PINN при нахождении периодических решений уравнения Дуффинга по точности, затратам, устойчивости и контролю над сосуществующими решениями. Результаты. Метод параметризации достиг погрешности 10⁻⁸-10⁻⁷ за сотые доли секунды; разбиение на восемь частей при сильной мягкой нелинейности повысило долю сходящихся начальных приближений с 0-9% до 61-100%; в сочетании с продолжением по длине дуги и мультипликаторами Флоке метод воспроизвел полный гистерезис амплитудно-частотной характеристики. PINN достигли погрешности 10⁻⁶ 10⁻² за 20-40 с и в ряде режимов сходились к другому периодическому решению при столь же малой функции потерь. Заключение. Решения PINN нуждаются в верификации, которую обеспечивает метод параметризации.
Ключевые слова:
уравнение Дуффинга, периодическая краевая задача, метод параметризации, физически информированные нейронные сети, метод Ньютона, амплитудно-частотная характеристика.Библиографические ссылки
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