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.

中文翻译：

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增强物理知识神经网络来求解纳维-斯托克斯方程

流体力学是工程和科学的一个关键领域。了解流体的行为需要求解纳维-斯托克斯方程 (NSE)。然而，NSE 是一个复杂的偏微分方程，求解起来可能具有挑战性，并且经典数值方法的计算成本可能很高。在本文中，我们建议通过修改残差损失函数并结合新的计算深度学习技术来增强物理信息神经网络（PINN）。我们提出了两个求解 NSE 的增强模型。第一个模型涉及开发经典的 PINN 来求解 NSE，基于速度分量的流函数方法。我们在该模型中添加了压力训练损失函数，并集成了新的计算训练技术。此外，我们提出了第二种更灵活的模型，它直接近似 NSE 的解，而不做任何假设。该模型在保持高精度的同时显着减少了训练时间。此外，我们还成功应用该模型来求解三维 NSE。结果证明了我们方法的有效性，具有多种优势，包括高可训练性、灵活性和效率。