Enhancing physics informed neural networks for solving Navier-Stokes equations

Fluid mechanics is a critical field in both engineering and science. Understanding the behavior of fluids requires solving the Navier-Stokes equation (NSE). However, the NSE is a complex partial differential equation that can be challenging to solve, and classical numerical methods can be computationally expensive. In this paper, we propose enhancing physics-informed neural networks (PINNs) by modifying the residual loss functions and incorporating new computational deep learning techniques. We present two enhanced models for solving the NSE. The first model involves developing the classical PINN for solving the NSE, based on a stream function approach to the velocity components. We have added the pressure training loss function to this model and integrated the new computational training techniques. Furthermore, we propose a second, more flexible model that directly approximates the solution of the NSE without making any assumptions. This model significantly reduces the training duration while maintaining high accuracy. Moreover, we have successfully applied this model to solve the three-dimensional NSE. The results demonstrate the effectiveness of our approaches, offering several advantages, including high trainability, flexibility, and efficiency. We propose two enhanced approaches of physics informed neural networks (PINN) for solving the challenging Navier-Stokes equation (NSE). The first approach improves the model by approximating the velocity components and integrating a pressure-based loss function. The second approach directly approximates the NSE solution without assumptions, significantly reducing training duration while maintaining high accuracy. We successfully apply this approach to solve the three-dimensional NSE, demonstrating the advantages of our models in terms of trainability, flexibility, and efficiency.image