Fluid mechanics is a fundamental field in engineering and science. Solving the Navier-Stokes equation (NSE) is critical for understanding the behavior of fluids. However, the NSE is a complex partial differential equation that is difficult to solve, and classical numerical methods can be computationally expensive. In this paper, we present an innovative approach for solving the NSE using Physics Informed Neural Networks (PINN) and several novel techniques that improve their performance. The first model is based on an assumption that involves approximating the velocity component by employing the derivative of a stream function. This assumption serves to simplify the system and guarantees that the velocity adheres to the divergence-free equation. We also developed a second more flexible model that approximates the solution without any assumptions. The proposed models can effectively solve two-dimensional NSE. Moreover, we successfully applied the second model to solve the three-dimensional NSE. The results show that the models can efficiently and accurately solve the NSE in three dimensions. These approaches offer several advantages, including high trainability, flexibility, and efficiency.

翻译：流体力学是工程与科学中的基础领域。求解纳维-斯托克斯方程（NSE）对于理解流体行为至关重要。然而，NSE是一种难以求解的复杂偏微分方程，经典数值方法计算成本高昂。本文提出了一种创新方法，利用物理信息神经网络（PINN）及多种提升其性能的新技术来求解NSE。第一个模型基于一个假设，即通过采用流函数的导数来近似速度分量。这一假设简化了系统，并确保速度满足无散度方程。我们还开发了第二个更灵活的模型，无需任何假设即可近似求解。所提出的模型能够有效求解二维NSE。此外，我们成功应用第二个模型求解了三维NSE。结果表明，这些模型能够高效、准确地求解三维NSE。这些方法具有可训练性高、灵活性强和效率高等优势。