Though PINNs (physics-informed neural networks) are now deemed as a
complement to traditional CFD (computational fluid dynamics) solvers rather
than a replacement, their ability to solve the Navier-Stokes equations without
given data is still of great interest. This report presents our
not-so-successful experiments of solving the Navier-Stokes equations with PINN
as a replacement for traditional solvers. We aim to, with our experiments,
prepare readers for the challenges they may face if they are interested in
data-free PINN. In this work, we used two standard flow problems: 2D
Taylor-Green vortex at Re=100 and 2D cylinder flow at Re=200. The PINN method
solved the 2D Taylor-Green vortex problem with acceptable results, and we used
this flow as an accuracy and performance benchmark. About 32 hours of training
were required for the PINN method's accuracy to match the accuracy of a 16x16
finite-difference simulation, which took less than 20 seconds. The 2D cylinder
flow, on the other hand, did not produce a physical solution. The PINN method
behaved like a steady-flow solver and did not capture the vortex shedding
phenomenon. By sharing our experience, we would like to emphasize that the PINN
method is still a work-in-progress, especially in terms of solving flow
problems without any given data. More work is needed to make PINN feasible for
real-world problems in such applications.

本文介绍了用 PINN（受物理启发的神经网络）作为传统求解器替代品解决 Navier-Stokes 方程的实验，并使用 2D 的 Taylor-Green 涡旋和圆柱流问题作为实验数据。结果表明，尽管 PINN 方法在 Taylor-Green 问题上表现良好，但在圆柱流问题上未能产生物理解。这表明 PINN 方法在无已知数据的情况下求解流动问题仍有待进一步探究。

物理信息神经网络在流体仿真中的经验报告：陷阱与挫折

Experience report of physics-informed neural networks in fluid simulations: pitfalls and frustration

Physics-informed neural networks allow models to be trained by physical laws
described by general nonlinear partial differential equations. However,
traditional architectures struggle to solve more challenging time-dependent
problems due to their architectural nature. In this work, we present a novel
physics-informed framework for solving time-dependent partial differential
equations. Using only the governing differential equations and problem initial
and boundary conditions, we generate a latent representation of the problem's
spatio-temporal dynamics. Our model utilizes discrete cosine transforms to
encode spatial frequencies and recurrent neural networks to process the time
evolution. This efficiently and flexibly produces a compressed representation
which is used for additional conditioning of physics-informed models. We show
experimental results on the Taylor-Green vortex solution to the Navier-Stokes
equations. Our proposed model achieves state-of-the-art performance on the
Taylor-Green vortex relative to other physics-informed baseline models.

本文提出了一种新颖的基于物理信息的神经网络框架，用于解决时间依赖偏微分方程，利用离散余弦变换对空间频率进行编码，再利用循环神经网络处理时间演化，从而实现对问题的时空动态的潜在表达，提高了物理相关模型的效率和灵活性，并在 Navier-Stokes 方程的 Taylor-Green 涡旋解上实现了最先进的性能。