Physics Informed Neural Networks (PINNs) represent the intersection between physics-based modeling and deep learning, but successfully training PINNs in 3D for highly nonlinear PDEs on complex domains remains a challenging task. In this paper, PINNs are used to solve the 3D incompressible Navier-Stokes (NS) equations at high Reynolds numbers for complex geometries, using very sparsely distributed solution data in the domain. The effect of the amount of data provided and the PDE-based regularizers are investigated. Additionally, hybrid data-PINNs are used to create surrogate models to solve a realistic flow-thermal electronics design problem in near real-time, and it is found that the hybrid data-PINNs consistently outperform standard data-driven neural networks when tested on unseen query points. The findings of the paper show how PINNs can be effective when used in conjunction with sparse data for solving 3D nonlinear PDEs or for surrogate modeling of design spaces governed by them.

翻译：基于物理信息的神经网络（PINNs）代表了物理建模与深度学习之间的交叉领域，但在复杂三维域中成功训练PINNs以求解高度非线性偏微分方程仍是一项具有挑战性的任务。本文利用PINNs求解复杂几何形状下高雷诺数的三维不可压缩纳维-斯托克斯方程，且仅使用域中非常稀疏的分布解数据。研究了提供的数据量以及基于偏微分方程的正则化项的影响。此外，采用混合数据-PINNs构建代理模型，以近实时方式求解实际流动-热电子设计问题，结果发现，在未知查询点上测试时，混合数据-PINNs始终优于标准数据驱动的神经网络。本文的研究结果表明，PINNs与稀疏数据结合使用时，可有效求解三维非线性偏微分方程或为其控制的参数化设计空间进行代理建模。