Physics-informed neural networks (PINNs) are successful machine-learning methods for the solution and identification of partial differential equations (PDEs). We employ PINNs for solving the Reynolds-averaged Navier$\unicode{x2013}$Stokes (RANS) equations for incompressible turbulent flows without any specific model or assumption for turbulence, and by taking only the data on the domain boundaries. We first show the applicability of PINNs for solving the Navier$\unicode{x2013}$Stokes equations for laminar flows by solving the Falkner$\unicode{x2013}$Skan boundary layer. We then apply PINNs for the simulation of four turbulent-flow cases, i.e., zero-pressure-gradient boundary layer, adverse-pressure-gradient boundary layer, and turbulent flows over a NACA4412 airfoil and the periodic hill. Our results show the excellent applicability of PINNs for laminar flows with strong pressure gradients, where predictions with less than 1% error can be obtained. For turbulent flows, we also obtain very good accuracy on simulation results even for the Reynolds-stress components.

本文利用物理信息网络解决并识别局部微分方程组，应用此方法成功地解决了雷诺-平均纳维尔-斯托克斯方程，该方程适用于不可压的湍流流动，而且不需要特定的湍流模型或假设，并仅仅需要利用域边界数据来解决方程，研究结果表明该方法可以用于压力梯度强的层流，并且可以得到小于1％的误差，同时对于湍流流动的模拟结果也非常准确。

用于解决雷诺平均纳维尔-斯托克斯方程的受物理学指导的神经网络