Using Physics-Informed neural networks for solving Navier-Stokes equations in fluid dynamic complex scenarios

Navier-Stokes equations used to model fluid dynamic processes are fundamental to address several real-world problems related to energy production, aerospace applications, automotive design, industrial process, etc. However, since in most cases they do not admit any analytical solution, numerical simulations are required in industrial contexts to assess fluid dynamic behaviors in specific setups. Computational Fluid Dynamics (CFD) methods, like those using finite volume or element approaches, are exploited to find Navier-Stokes solutions and carry out simulations. However, such methods require expensive hardware resources, relevant computational times, and manual efforts for the definition of dense meshes on which equations are evaluated iteratively for each time step of the simulation. Physics-Informed Neural Networks (PINNs), which are deep neural networks where physical laws are directly embedded into the training process, offer a promising approach for solving Navier-Stokes equations, thus alleviating hardware and time requirements. PINNs bypass some CFD limitations by using neural networks to produce solutions based on governing equations, thus reducing the need for large datasets, dense meshing, and iterative estimation over time. This paper evaluates the application of PINNs in near real-world scenarios, while considering various geometries. The study focuses on the achieved accuracy, by comparing PINN estimates with CFD solutions obtained via OpenFOAM, and the required training times; this includes evaluating different neural network architectures, activation functions, and numbers of sampling points. Additionally, several training strategies such as fine-tuning, multi-resolution learning, and parametrized training are proposed to enhance efficiency and obtain speed up. Results demonstrate that PINNs can achieve comparable accuracy to CFD methods (with a velocity magnitude mean absolute error inferior to ${10}^{-2}$) and significantly reduce computational costs. Our findings demonstrated that with appropriate training techniques PINNs can be effectively used in industrial applications requiring rapid and accurate fluid dynamic simulations, thus paving the way for their broader adoption in practical engineering problems.