## 3 Numerical Tests
### Stontic Flow/Anurysmal Flow
#### 3.3.2 Effect of Scaling Factor  
Similar to the previous tests, five hard constraints with different magnitudes of \(a\) (i.e., \(a\) = 1.0 \(\times\) 10\({}^{-4}\), 1.0 \(\times\) 10\({}^{-2}\), 1.0, 1.0 \(\times\) 10\({}^{2}\) and 1.0 \(\times\) 10\({}^{4}\)) and the same degree (i.e., \(k\)= 2.0) are formulated. The relative \(L_{2}\)-norm errors of the axial velocity predicted by PINN and FV-based CFD in the cases of \(A\) = 4.0 \(\times\) 10\({}^{-3}\), \(A\) = 7.0 \(\times\) 10\({}^{-3}\), \(A\) = \(-\)1.2 \(\times\) 10\({}^{-2}\) and \(A\) = \(-\)2.2 \(\times\) 10\({}^{-2}\) are listed in Table 6. Relative small errors are observed to be achieved when the scaling factor \(a\) varies between 1.0 \(\times\) 10\({}^{-2}\) and 1.0 \(\times\) 10\({}^{2}\), while the errors tend to become unacceptable when \(a\) becomes even smaller or larger, i.e., \(a\) = 1.0 \(\times\) 10\({}^{-4}\) and \(a\) = 1.0 \(\times\) 10\({}^{4}\). In addition, the relatively large errors with \(a\) = 1.0 \(\times\) 10\({}^{-4}\) and \(a\) = 1.0 \(\times\) 10\({}^{4}\) in Table 6 are still smaller than those with first-degree hard constraints in Table 5, further demonstrating the importance of degree in the PINN prediction accuracy. \begin{table}
\begin{tabular}{c c c} \hline
**Degree** & \(A\) & \(L_{2}\)**-Norm Error** \\ \hline \multirow{4}{*}{First} & 4.0 \(\times\) 10\({}^{-3}\) & 99.67\% \\  & 7.0 \(\times\) 10\({}^{-3}\) & 96.24\% \\  & \(-\)1.2 \(\times\) 10\({}^{-2}\) & 255.9\% \\  & \(-\)2.2 \(\times\) 10\({}^{-2}\) & 319.9\% \\ \hline \multirow{4}{*}{Second} & 4.0 \(\times\) 10\({}^{-3}\) & 6.850\% \\  & 7.0 \(\times\) 10\({}^{-3}\) & 9.120\% \\  & \(-\)1.2 \(\times\) 10\({}^{-2}\) & 9.960\% \\  & \(-\)2.2 \(\times\) 10\({}^{-2}\) & 7.810\% \\ \hline \multirow{4}{*}{Third} & 4.0 \(\times\) 10\({}^{-3}\) & 2.260\% \\  & 7.0 \(\times\) 10\({}^{-3}\) & 8.930\% \\ \cline{1-1}  & \(-\)1.2 \(\times\) 10\({}^{-2}\) & 4.930\% \\ \cline{1-1}  & \(-\)2.2 \(\times\) 10\({}^{-2}\) & 5.770\% \\ \hline \end{tabular}
\end{table} 
Table 5: The \(L_{2}\)-norm errors of axial velocity of stenotic/aneurysmal flow predicted by PINN with different degrees of hard constraints. Figure 7: Comparison of axial velocity distributions in stenotic/aneurysmal flow (\(A\) = 4.0 \(\times\) 10\({}^{-3}\)) between PINN-predicted results with different degrees of hard constraints and FV-based CFD results. The contours of axial velocity distributions are shown in Figures 9 and 10. For \(a\) = 1.0, although the \(L_{2}\)-norm error is slightly larger than those with \(a\) = 1.0 \(\times\)\(10^{-2}\) and \(a\)= 1.0 \(\times\)\(10^{2}\) (Table 6), the axial velocity distributions in the local stenotic/aneurysmal part are observed to best agree with the FV-based CFD result. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Scaling Factor** & \(A\) & \(L_{2}\)**-Norm Error** \\ \hline \multirow{3}{*}{\(1.0\times 10^{-4}\)} & \(4.0\times 10^{-3}\) & 99.64\% \\  & \(7.0\times 10^{-3}\) & 99.97\% \\  & \(-1.2\times 10^{-2}\) & 98.79\% \\  & \(-2.2\times 10^{-2}\) & 44.07\% \\ \hline \multirow{3}{*}{\(1.0\times 10^{-2}\)} & \(4.0\times 10^{-3}\) & 2.660\% \\  & \(7.0\times 10^{-3}\) & 12.63\% \\  & \(-1.2\times 10^{-2}\) & 7.980\% \\  & \(-2.2\times 10^{-2}\) & 8.610\% \\ \hline \multirow{3}{*}{\(1.0\times 10^{2}\)} & \(4.0\times 10^{-3}\) & 8.850\% \\  & \(7.0\times 10^{-3}\) & 8.012\% \\  & \(-1.2\times 10^{-2}\) & 9.960\% \\  & \(-2.2\times 10^{-2}\) & 9.710\% \\ \hline \multirow{3}{*}{\(1.0\times 10^{4}\)} & \(4.0\times 10^{-3}\) & 2.730\% \\  & \(7.0\times 10^{-3}\) & 99.90\% \\ \cline{1-1}  & \(-1.2\times 10^{-2}\) & 6.580\% \\ \cline{1-1}  & \(-2.2\times 10^{-2}\) & 48.96\% \\ \hline \multirow{3}{*}{\(1.0\times 10^{4}\)} & \(4.0\times 10^{-3}\) & 55.56\% \\  & \(7.0\times 10^{-3}\) & 94.67\% \\ \cline{1-1}  & \(-1.2\times 10^{-2}\) & 100.6\% \\ \cline{1-1}  & \(-2.2\times 10^{-2}\) & 98.75\% \\ \hline \end{tabular}
\end{table}