We leverage Physics-Informed Neural Networks (PINNs) to learn solution functions of parametric Navier-Stokes Equations (NSE). Our proposed approach results in a feasible optimization problem setup that bypasses PINNs' limitations in converging to solutions of highly nonlinear parametric-PDEs like NSE. We consider the parameter(s) of interest as inputs of PINNs along with spatio-temporal coordinates, and train PINNs on generated numerical solutions of parametric-PDES for instances of the parameters. We perform experiments on the classical 2D flow past cylinder problem aiming to learn velocities and pressure functions over a range of Reynolds numbers as parameter of interest. Provision of training data from generated numerical simulations allows for interpolation of the solution functions for a range of parameters. Therefore, we compare PINNs with unconstrained conventional Neural Networks (NN) on this problem setup to investigate the effectiveness of considering the PDEs regularization in the loss function. We show that our proposed approach results in optimizing PINN models that learn the solution functions while making sure that flow predictions are in line with conservational laws of mass and momentum. Our results show that PINN results in accurate prediction of gradients compared to NN model, this is clearly visible in predicted vorticity fields given that none of these models were trained on vorticity labels.

翻译：我们利用物理信息神经网络（PINNs）来学习参数化纳维-斯托克斯方程（NSE）的解函数。所提出的方法构建了一个可行的优化问题框架，从而避免了PINNs在收敛于如NSE等高非线性参数化偏微分方程解时的局限性。我们将感兴趣的参数与时空坐标一同作为PINNs的输入，并基于参数化偏微分方程实例的数值解对PINNs进行训练。我们在经典的二维圆柱绕流问题上进行实验，旨在学习以雷诺数范围为感兴趣参数的流速和压力函数。通过提供数值模拟生成的训练数据，能够对参数范围内的解函数进行插值。因此，我们在此问题框架下比较了加入无约束传统神经网络（NN）的PINNs，以探究在损失函数中考虑偏微分方程正则化的有效性。结果表明，所提出的方法在优化PINN模型学习解函数的同时，确保了流动预测符合质量和动量守恒定律。我们的结果还表明，与NN模型相比，PINN能够更准确地预测梯度，这一点在预测涡量场中尤为明显——尽管这两种模型均未使用涡量标签进行训练。