在本論文中，我們提出了一種修改的投影方法，利用物理信息神經網絡來求解不可壓縮的納維-斯托克斯方程。我們首先應用有限差分法結合投影方法來解決泰勒-格林渦旋，並將結果與解析解進行比較。我們的結果表明，該方法能夠以二階收斂速率準確預測泰勒-格林渦旋的流動和壓力場。隨後，我們使用結合投影方法的物理信息神經網絡來解決泰勒-格林渦旋。然而，我們的實驗結果表明，直接使用投影法會導致速度場的預測結果較差。為了解決這個問題，我們提出了一種修改的投影方法，同時求解流體函數和勢函數，並通過流體函數來更新速度場。我們的數值結果表明，這種方法能夠在方形、橢圓形和L形區域中準確預測泰勒-格林渦旋的流動和壓力場。

In this thesis, we propose a modified projection method for solving the incompressible Navier-Stokes equations using physics-informed neural networks (PINNs). We begin by applying the finite difference method combined with the projection method to solve the Taylor- Green vortex and compare the results with the analytical solution. Our results demonstrate that this approach accurately predicts the flow and pressure fields of the Taylor-Green vortex with a second-order convergence rate. We then use PINNs with the projection method to solve the Taylor-Green vortex. However, our experimental results indicate that, direct usage of the projection method leads to poor prediction results of the velocity field. To address this, we propose a modified projection method that simultaneously solves the stream function and potential function. Our numerical results show that this approach accurately predicts the flow and pressure fields of the Taylor-Green vortex in square, ellptical and L-shaped domains.

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