Physics-informed neural networks (PINNs) have emerged as a major research focus. However, today's PINNs encounter several limitations. Firstly, during the construction of the loss function using automatic differentiation, PINNs often neglect information from neighboring points, which hinders their ability to enforce physical constraints and diminishes their accuracy. Furthermore, issues such as instability and poor convergence persist during PINN training, limiting their applicability to complex fluid dynamics problems. To address these challenges, a fast physics-informed neural network framework that integrates a simplified finite volume method (FVM) and residual correction loss term has been proposed, referred to as Fast Finite Volume PINN (FFV-PINN). FFV-PINN utilizes a simplified FVM discretization for the convection term, with an accompanying improvement in the dispersion and dissipation behavior. Unlike traditional FVM, the FFV-PINN outputs can be simply and directly harnessed to approximate values on control surfaces, thereby simplifying the discretization process. Moreover, a residual correction loss term is introduced to significantly accelerates convergence and improves training efficiency. To validate the performance, we solve a series of challenging problems -- including flow in the two-dimensional steady and unsteady lid-driven cavity, three-dimensional steady lid-driven cavity, backward-facing step flows, and natural convection at high Reynolds number and Rayleigh number. Notably, the FFV-PINN can achieve data-free solutions for the lid-driven cavity flow at Re = 10000 and natural convection at Ra = 1e8 for the first time in PINN literature, even while requiring only 680s and 231s. It further highlights the effectiveness of FFV-PINN in improving both speed and accuracy, marking another step forward in the progression of PINNs as competitive neural PDE solvers.

翻译：物理信息神经网络（PINNs）已成为主要研究焦点。然而，当前PINNs存在若干局限性。首先，在使用自动微分构建损失函数时，PINNs常忽略邻域点信息，这削弱了其施加物理约束的能力并降低了精度。此外，PINN训练过程中持续存在不稳定与收敛性差的问题，限制了其在复杂流体动力学问题中的应用。为应对这些挑战，本文提出了一种融合简化有限体积方法（FVM）与残差校正损失项的快速物理信息神经网络框架，称为快速有限体积PINN（FFV-PINN）。FFV-PINN采用简化FVM离散对流项，并伴随色散与耗散特性的改进。与传统FVM不同，FFV-PINN可直接利用网络输出来近似控制面上的数值，从而简化离散过程。此外，引入残差校正损失项可显著加速收敛并提升训练效率。为验证性能，我们求解了一系列具有挑战性的问题，包括二维稳态与非稳态顶盖驱动方腔流、三维稳态顶盖驱动方腔流、后向台阶流以及高雷诺数和高瑞利数下的自然对流。值得注意的是，FFV-PINN首次在PINN文献中实现了无数据求解Re=10000的顶盖驱动方腔流与Ra=1e8的自然对流，且仅需680秒与231秒。这进一步凸显了FFV-PINN在提升速度与精度方面的有效性，标志着PINN作为竞争性神经偏微分方程求解器迈出了新的一步。