vmaitComputational Mathematics and Information TechnologiesComputational Mathematics and Information Technologies2587-8999Донской государственный технический университет10.23947/2587-8999-2025-9-4-10-21vmait-209Research ArticleМАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕMATHEMATICAL MODELLINGГибридное моделирование экстремальных штормовых процессов и рисков судоходства в Азовском море на основе трёхмерной гидродинамики и методов машинного обученияHybrid Modelling of Extreme Storm Processes and Navigation Risks in the Azov Sea Based on Three-Dimensional Hydrodynamics and Machine Learning Methodshttps://orcid.org/0000-0002-5875-1523СухиновА. И.SukhinovA. I.

Александр Иванович Сухинов, член-корреспондент РАН, доктор физико-математических наук, профессор, директор НИИ Математического моделирования и прогнозирования сложных систем 

344003, г. Ростов-на-Дону, пл. Гагарина, 1

Alexander I. Sukhinov, Corresponding Member of the Russian Academy of Sciences, Doctor of Physical and Mathematical Sciences, Professor, Director of the Research Institute of Mathematical Modeling and Forecasting of Complex Systems

1, Gagarin Sq., Rostov-on-Don, 344003

sukhinov@gmail.comПроценкоС. В.ProtsenkoS. V.

Софья Владимировна Проценко, кандидат физико-математических наук, доцент кафедры математики, научный сотрудник 

344003, г. Ростов-на-Дону, пл. Гагарина, 1

347936, г. Таганрог, ул. Инициативная, 48

Sofia V. Protsenko, Candidate of Physical and Mathematical Sciences, Associate Professor of the Department of Mathematics, Research Fellow

1, Gagarin Sq., Rostov-on-Don, 344003

48, Initiative St., Taganrog, 347936

rab5555@rambler.ruhttps://orcid.org/0000-0001-7911-3558ПроценкоЕ. А.ProtsenkoE. A.

Елена Анатольевна Проценко, кандидат физико-математических наук, доцент кафедры математики, ведущий научный сотрудник 

347936, г. Таганрог, ул. Инициативная, 48

Elena A. Protsenko, Candidate of Physical and Mathematical Sciences, Associate Professor of the Department of Mathematics, Leading Research Fellow

48, Initiative St., Taganrog, 347936

eapros@rambler.ruhttps://orcid.org/0000-0002-9037-5556ПанасенкоН. Д.PanasenkoN. D.

Наталья Дмитриевна Панасенко, кандидат технических наук, доцент кафедры «Математика и информатика», доцент кафедры «Информационная безопасность в вычислительных системах и сетях»

344003, г. Ростов-на-Дону, пл. Гагарина, 1

Natalia D. Panasenko, Candidate of Technical Sciences, Associate Professor of the Department of Mathematics and Computer Science, Associate Professor of the Department of Information Security in Computing Systems and Networks

1, Gagarin Sq., Rostov-on-Don, 344003

natalija93_93@mail.ruДонской государственный технический университетРоссияDon State Technical UniversityRussian FederationДонской государственный технический университет; Таганрогский институт имени А.П. Чехова (филиал) РГЭУ (РИНХ), г. Таганрог, Российская ФедерацияРоссияDon State Technical University; Taganrog Institute named after A.P. Chekhov (branch) of RSUERussian FederationТаганрогский институт имени А.П. Чехова (филиал) РГЭУ (РИНХ), г. Таганрог, Российская ФедерацияРоссияTaganrog Institute named after A.P. Chekhov (branch) of RSUERussian Federation202517012026941021Copyright © Сухинов А.И., Проценко С.В., Проценко Е.А., Панасенко Н.Д., 20252025Сухинов А.И., Проценко С.В., Проценко Е.А., Панасенко Н.Д.Sukhinov A.I., Protsenko S.V., Protsenko E.A., Panasenko N.D.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://www.cmit-journal.ru/jour/article/view/209

Введение. Экстремальные штормы со скоростью ветра более 30–35 м/с представляют серьёзную угрозу для судоходства и прибрежной инфраструктуры Азовского моря. Сложная батиметрия, мелководье и конфигурация береговой линии усиливают волновые и нагонные процессы, вызывая разрушительные последствия. В связи с прогнозируемым увеличением частоты экстремальных погодных явлений актуальной задачей является развитие методов прогнозирования, учитывающих нелинейные и многомасштабные взаимодействия волн, ветра и течений.Материалы и методы. Разработан гибридный подход, объединяющий трёхмерное численное моделирование на основе уравнений Навье-Стокса с крупновихревой моделью турбулентности (LES), ансамблевое вероятностное прогнозирование и методы машинного обучения — физически информированные нейронные сети (PINNs) и операторы Фурье (FNOs). Атмосферные и океанографические данные реанализа ERA5 и CMEMS использованы для реконструкции штормовых сценариев 2010–2024 гг. Взаимодействие волн с судами описано в шести степенях свободы. Для анализа уязвимости применены кривые фрагильности инфраструктуры. Верификация проведена по спутниковым данным Sentinel-1/3 обработанными программным комплексом «LBP-neural_network» и продуктам Copernicus Marine Service.Результаты исследования. Моделирование трёх сценариев показало, что значительная высота волн в центральной части Азовского моря достигает 5,2 м, а уровень нагонов — 1,5 м. Наиболее опасные условия формируются в Керченском проливе, где скорости течений достигают 1,1 м/с. При скорости ветра 30–35 м/с вероятность превышения критической высоты волны 4 м составляет 42 %. Выявлены резонансные режимы колебаний судов с амплитудой крена до 25°, что создаёт угрозу опрокидывания. Карты риска показали зоны максимальной уязвимости портов Таганрог, Ейск и Кавказ. Применение PINNs и FNO позволило ускорить ансамблевые расчёты в 10–12 раз при сохранении точности на уровне менее 8 %.Обсуждение. Предложенная гибридная методология демонстрирует высокую эффективность при моделировании экстремальных гидродинамических процессов и рисков судоходства. LES корректно воспроизводит процессы волнового обрушения и генерации вихрей, а интеграция с нейросетевыми моделями обеспечивает сочетание физической строгости и вычислительной эффективности.Заключение. Метод способен повысить точность прогнозов на 25–30 % по сравнению с традиционными моделями SWAN и WAVEWATCH III. Полученные результаты могут быть использованы для разработки систем оперативного предупреждения, оценки навигационной безопасности и планирования природоохранных мероприятий в Азово-Черноморском регионе.

Introduction. Extreme storms with wind speeds exceeding 30–35 m/s pose a significant threat to navigation and coastal infrastructure in the Azov Sea. The complex bathymetry, shallow water, and coastal geometry amplify wave and surge effects, causing severe destruction. The increasing frequency of extreme weather events requires next-generation forecasting systems capable of capturing nonlinear multiscale interactions between wind, waves, and currents.Materials and Methods. A hybrid approach was developed, combining three-dimensional numerical hydrodynamic modelling based on the Navier-Stokes equations with Large-Eddy Simulation (LES) turbulence closure, ensemble probabilistic forecasting, and machine learning methods — including Physics-Informed Neural Networks (PINNs) and Fourier Neural Operators (FNOs). Atmospheric and oceanographic data from ERA5 and CMEMS reanalyses were used to reconstruct storm scenarios for 2010–2024. Ship-wave interactions were modeled in six degrees of freedom, while coastal infrastructure fragility was evaluated using probabilistic vulnerability curves. Validation was performed using Sentinel-1/3 satellite data processed by the “LBP-neural_network” software package and Copernicus Marine Service products.Results. Three representative storm scenarios were simulated. The significant wave height in the central Azov Sea reached up to 5.2 m, with surge amplitudes up to 1.5 m. The most hazardous conditions occurred in the Kerch Strait, where current velocities reached 1.1 m/s. Under wind speeds of 30–35 m/s, the probability of exceeding the critical 4 m wave height was 42%. Resonant ship motions with roll amplitudes up to 25° were detected, indicating a high capsizing risk. Risk maps identified the most vulnerable zones near Taganrog, Yeysk, and Port Kavkaz. The integration of PINNs and FNOs accelerated ensemble simulations by a factor of 10–12 while maintaining prediction errors below 8%.Discussion. The proposed hybrid methodology proved highly effective for modelling extreme hydrodynamic processes and navigation risks. The LES framework accurately reproduced wave breaking and vortex generation processes, while coupling with neural network surrogates combined physical consistency with computational efficiency.Conclusion. The approach improved forecast accuracy by 25–30% compared with conventional spectral models (SWAN, WAVEWATCH III). The results provide a scientific basis for developing early warning systems, assessing navigation safety, and planning coastal protection measures in the Azov–Black Sea region.

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