Application of Physics-Informed Neural Network on the Example of Modeling Hydrodynamic Processes that Allow an Analytical Solution

We consider an actual approach to develop a physically based neural network for solving model problems for the Kovazhny flow, for the geophysical Beltrami flow, and for the flow in a section of the river by the shallow water theory. Physics-informed neural networks (PINN) allow to significantly reduce the computation time compared to conventional computations. There is a different analytical solution for each model flow. The architecture of the DeepXDE software library, its composition by modules, and fragments of program code in the Python programming language are discussed.  The PINN model is tested on a test sample. The prediction is evaluated using the MSE metric. The fully connected neural network can contain 4, 7, 10 hidden layers with the number of neurons 50, 50, 100 respectively.  The influence of hyperparameters of the neural network on the magnitude of the prediction error is discussed. The calculations performed on a server with Nvidia GeForce RTX 3070 card can significantly reduce the training time for PINN.

1. Raissi M., Perdikaris P., Karniadakis G.E. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, Journal of Computational Physics 378 (2019) 686-707.

2. Raissi M., Yazdani A., Karniadakis G.E. Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations, Science 367 (2020) 1026-1030.

3. Лойцянский Л.Г. Механика жидкости и газа. Учебник для вузов. - 7-е изд., испр. - М.: Дрофа, 2003. - 840 с.

4. Петров А.Г. Аналитическая гидродинамика. М.: Физматлит. 2010. - 520 с.

5. Cuomo, Salvatore & Schiano Di Cola, Vincenzo & Giampaolo, Fabio & Rozza, Gianluigi & Raissi, Maziar & Piccialli, Francesco. (2022). Scientific Machine Learning Through Physics–Informed Neural Networks: Where we are and What’s Next. Journal of Scientific Computing. 92. 10.1007/s10915-022-01939-z.

6. Николенко С., Кадурин А., Архангельская Е. Глубокое обучение. Погружение в мир нейронных сетей. 2020. – 480 с.

7. Lu Lu, et al. DeepXDE: A Deep Learning Library for Solving Differential Equations. SIAM REVIEW. 2021. Vol. 63, No. 1, pp. 208–228.

8. https://github.com/lululxvi/deepxde

9. Kovasznay L.I.G. Laminar flow behind a two-dimensional grid. Math. Proc. Cambridge. 1948. V. 44. Is. 1. P. 58–62.

10. Бубенчиков А.М., Попонин В.С., Мельникова В.Н. Разработка универсальной техники реализации неоднородных граничных условий Дирихле и Неймана в методе спектральных элементов, Вестн. Томск. гос. ун-та. Матем. и мех., 2008, номер 2(3), 87–98.

11. Dong S., Karniadakis G.E., Cryssostomidis C. A robust and accurate outflow boundary condition for incompressible flow simulations on severely-truncated unbounded domains. J. Comput. Phys. 2014. V. 261. P. 83–105.

12. Jin X., Cai S., Li H., Karniadakis G.E. NSFnets (Navier-Stokes flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations. Journal of Computational Physics, Volume 426, 2021, 109951.

13. Беликов В.В., Алексюк А.И. Модели мелкой воды в задачах речной гидродинамики. Российская академия наук, Отделение наук о Земле, Институт водных проблем РАН. – Москва: Российская академия наук, 2020. – 346 с. – ISBN 978-5-907036-22-2.

14. Зиновьев А.Т., Кошелев К.Б., Дьяченко А.В., Коломейцев А.А. Экстремальный дождевой паводок 2014 года в бассейне Верхней Оби: причины, прогноз и натурные наблюдения // Водное хозяйство России: проблемы, технологии, управление. – 2015. – № 6. – С. 93-104.

15. Cedillo S. et al. Physics‑Informed Neural Network water surface predictability for 1D steady‑state open channel cases with different flow types and complex bed profile shapes. Adv. Model. and Simul. in Eng. Sci. (2022) 9:10.

16. Rosofsky S.G. et al. Applications of physics informed neural operators. Mach. Learn.: Sci. Technol. 4 (2023) 025022.

17. Feng D., Tan Z., He Q.Z. Physics-informed neural networks of the Saint-Venant equations for downscaling a large-scale river model. Water Resources Research, (2023). 59, e2022WR033168.