## 3 Numerical Tests
### Couette Flow  
Couette flow describes the laminar flow in a two-dimensional infinite plate with non-slip boundary conditions. As shown in Figure 1, the length of the plate \(L\) is set to 1, the distance between plates \(d\) is set to 0.1 and the static pressure is set to 0.1 and 0.0 at the inlet and outlet, respectively. The corresponding boundary conditions can be written as:  
\[\left\{\begin{array}{l}\mathbf{u}(x,\,-0.05)=(0.0,\,0.0),\,0.0\leq x\leq 1.0 \\ \mathbf{u}(x,\,0.05)=(0.0,\,0.0),\,0.0\leq x\leq 1.0\\ p(0.0,\,y)=0.1,\,-0.05\leq y\leq 0.05\\ p(1.0,y)=0.0,\,-0.05\leq y\leq 0.05\end{array}\right. \tag{20}\]  
where \(x\) and \(y\) represent the streamwise and vertical components of the local coordinates, respectively. The analytic solutions of the flow can be expressed as:  
\[u{=(\frac{d^{2}}{4}-y^{2})\frac{\Delta p}{2\varphi L}} \tag{21}\]  
where \(\Delta p\) is the pressure difference between the inlet and outlet. In the following test, the viscosity \(v\) is varied to control the Reynolds number of the flow. The Reynolds number is defined with \(v\), \(d\) and the fluid velocity at the center line, \(u_{\max}\).  
To solve the above Couette flow using the PINN model, hard constraints are formulated for the specified boundary conditions, which can be written as:  
\[\left\{\begin{aligned} \hat{u}&=u_{\text{par}}+D_{u}( \mathbf{x})\overline{u}=0.0+a(\frac{d^{k}}{2^{k}}-|y|^{k})\overline{u}\\ \hat{v}&=v_{\text{par}}+D_{v}(\mathbf{x})\overline{v}=0. 0+a(\frac{d^{k}}{2^{k}}-|y|^{k})\overline{v}\\ \hat{p}&=p_{\text{par}}+D_{p}(\mathbf{x})\overline{p }=\left(\frac{x-x_{\text{in}}}{x_{\text{out}}-x_{\text{in}}}p_{\text{out}}+ \frac{x_{\text{out}}-x}{x_{\text{out}}-x_{\text{in}}}p_{\text{in}}\right)+(x- x_{\text{in}})(x_{out}-x)\overline{p}\end{aligned}\right. \tag{22}\]  
where \(x_{\text{in}}\) and \(x_{\text{out}}\) represent the streamwise coordinates of the inlet and outlet of the flow field, respectively; \(u_{\text{par}}\) and \(v_{\text{par}}\) are the particular solutions of the velocity components and are set to a constant of 0.0 to satisfy the non-slip boundary condition, while \(p_{\text{par}}\) is the particular solution of \(p\) and automatically satisfies the specified inlet and outlet pressure. For the \(D\) function of \(\hat{u}\), \(k\) and \(a\) are the parameters to be determined, where \(k\) denotes the degree of the function which determines the continuity property of the hard constraints while the scaling factor \(a\) adjusts the influence of the network output \(\overline{u}\) on the final result \(\hat{a}\). In addition, the \(D\) function of \(\hat{p}\) is referring to that in [PERSON]'s work [19], which is quite deterministic and not discussed. Overall, the key to formulating the hard constraints is the selection of the two parameters, i.e., \(k\) and \(a\), in the \(D\) functions of velocity.  
For each simulation case, 25,000 epochs are performed in the training of the PINN model to guarantee convergence. The loss function histories versus epochs at different degree values are given in Figure 2. It can be seen that the training converged after around 15,000 epochs.  
Figure 1: Schematic of Couette flow. Here, \(L\) is the plate length, \(d\) is the plate distance, \(u\) is the local streamwise velocity and \(u_{\text{max}}\) is the velocity at the center line.  
Figure 2: Training loss histories at different degree values of hard constraint in Couette flow problem.