## 2 Methodology  
### PINN Model  
The N-S equations to be solved by PINN can be mathematically expressed as:  
\[\
abla\cdot\mathbf{u}(\mathbf{x},t) =0\;\mathbf{x}\in\Omega\;t\in[0,T] \tag{1}\] \[\frac{\partial\mathbf{u}(\mathbf{x},t)}{\partial t}+(\mathbf{u}( \mathbf{x},t)\cdot\
abla)\mathbf{u}(\mathbf{x},t)+\frac{1}{\rho}\
abla p( \mathbf{x},t)-\
u\
abla^{2}\mathbf{u}(\mathbf{x},t) =0\;\mathbf{x}\in\Omega\;t\in[0,T]\] (2) \[\mathcal{B}(\mathbf{u}(\mathbf{x},t),\;p(\mathbf{x},t))=g( \mathbf{x},t)\;\mathbf{x}\in\partial\Omega,\;t\in[0,T]\] (3) \[\mathcal{L}(\mathbf{u}(\mathbf{x},0),\;p(\mathbf{x},t))=d( \mathbf{x})\;\mathbf{x}\in\Omega \tag{4}\]  
where both velocity \(\mathbf{u}(\mathbf{x},t)\) and pressure \(p(\mathbf{x},t)\) are functions of space coordinate \(\mathbf{x}\) and time coordinate \(t\); \(\rho\) and \(\
u\) denote the flow density and viscosity, respectively. For all the flows investigated in this study, \(\rho\) is set to be 1; \(\mathcal{B}\) and \(\mathcal{L}\) represent the boundary conditions (BC) and initial conditions (IC), respectively; \(g(\mathbf{x},t)\) and \(d(\mathbf{x})\) are known continuous functions.  
Similar to most ANNs, the PINN is a multi-layer perceptron that builds an explicit relationship between input and output. As for the above N-S equations with the predicted velocity and pressure of an \(n\)-layer PINN denoted as \(\overline{\mathbf{u}}\) and \(\overline{p}\), respectively, we have:  
\[[\overline{\mathbf{u}}(\mathbf{x},t,\mathbf{W},\mathbf{b}),\overline{p}( \mathbf{x},t,\mathbf{W},\mathbf{b})]=f_{n}\dots\dots f_{3}(f_{2}(f_{1}(\mathbf{ x},t))) \tag{5}\]  
where \(f_{i}\) represents the operation function corresponding to the \(i\)-th layer and is usually expressed as:  
\[\mathbf{q}_{i}=f_{i}\big{(}\mathbf{q}_{i-1}\big{)}=\phi_{i}\big{(}\mathbf{W}_ {i}\mathbf{q}_{i-1}+\mathbf{b}_{i}\big{)} \tag{6}\]  
where \(\mathbf{q}_{i}\) is the output of the \(i\)-th hidden layer, \(\phi_{i}\) is the activation function and \(\mathbf{W}_{i}\) and \(\mathbf{b}_{i}\) represent the related weights and thresholds, respectively.  
The primary idea of the solving method for N-S equations based on the PINN is to find a set of appropriate model parameters to minimize the residual of the equation with the related BC and IC enforced with the constraints, i.e.,  
\[\mathbf{W}^{*},\mathbf{b}^{*}\in\underset{\mathbf{W},\mathbf{b}}{\mathrm{ argmin}}, \tag{7}\]\[\xi = \left\|\frac{\partial\overline{u}}{\partial t}+(\overline{u}\cdot \
abla)\overline{u}+\frac{1}{\rho}\
abla\overline{p}-\
u\
abla^{2}\overline{u} \right\|+\left\|\
abla\cdot\overline{u}\right\| \tag{8}\] \[\text{s.t.}\bigg{\{} \begin{array}{l}\mathcal{B}(\overline{u}(\mathbf{x},t,\mathbf{W}, \mathbf{b}),\ \overline{p}(\mathbf{x},t,\mathbf{W},\mathbf{b}))=g(\mathbf{x},t)\ \mathbf{x}\in\partial\Omega,\ t\in[0,T]\\ \mathcal{L}(\overline{u}(\mathbf{x},t,\mathbf{W},\mathbf{b}),\ \overline{p}(\mathbf{x},t, \mathbf{W},\mathbf{b}))=d(\mathbf{x})\ \mathbf{x}\in\Omega\end{array} \tag{9}\]  
where \(\xi\) is the so-called \"physical-based loss function\". It reflects the degree to which the knowledge learned by the neural network fits with the given differential equations and definite solution conditions.