Belangrijkste concepten
The core message of this paper is to design a helicity-conservative physics-informed neural network (PINN) model for solving the incompressible Navier-Stokes equations, which can exactly preserve the fluid helicity without any discretization error.
Samenvatting
The paper presents a helicity-conservative physics-informed neural network (PINN) model for solving the incompressible Navier-Stokes equations. The key highlights are:
The authors design a PINN model that can preserve the fluid helicity, which is an important conserved quantity in the Navier-Stokes system. This is the first attempt to develop a helicity-preserving neural network model for the Navier-Stokes equations.
Unlike standard finite element methods that are based on the weak formulation of the PDEs, the PINN model is based on the strong form of the PDEs. This makes it easier to enforce conservation properties, such as helicity conservation, without introducing auxiliary variables.
The authors provide theoretical justifications for the helicity conservation property of the proposed PINN model. They show that the PINN model can exactly preserve the fluid helicity, unlike the traditional finite element method which only approximates the helicity up to discretization errors.
Numerical experiments are conducted to demonstrate the error analysis and the helicity conservation property of the PINN model. The results show that the PINN model can accurately preserve the fluid helicity, outperforming the traditional finite element method.
The authors also discuss an alternative PINN-based approach, called the ωNN network, which directly generates the vorticity field. However, they show that this approach does not guarantee exact helicity conservation due to the difficulty in representing the divergence-free condition of the vorticity field.
Overall, the paper presents a novel and effective PINN-based approach for solving the incompressible Navier-Stokes equations while exactly preserving the important fluid helicity.