## 5 Experimental Setup and Discussion
model would be \(\lambda_{2}\cdot v_{x}+2\cdot\lambda_{1}\cdot v\cdot v_{x}+\lambda_{1}\cdot v_{t}=0\), where \(\lambda_{1}\) and \(\lambda_{2}\) are the parameters that define the nature of the relationship between traffic state variables, and these parameters are trained during the training of the neural network. The individual links are stitched together based on the connectivity matrix in Table 2. Each row (sub vector) in the matrix represents one link. The first, second, and third elements of a row denote link ID, start point, and end point, respectively. Node 1 or intersection 1 is the diverge point where traffic from link L1 diverges to links L2 and L3. Similarly, Node 2 is a merging point in the network where traffic from links L2 and L3 merges onto L1. Figure 4: (**a**) Speed contour plot showing cells where no vehicle detected; (**b**) speed contour where missing cells filled with the speed of neighboring cells. Figure 5: Speed, density, and flow plot of links in the network. To train the proposed model, 1200 data points are randomly selected from each link of the network. There are 10 layers and 30 neurons in each layer in PINNs architecture and the input training and residual data points are 1200 and 30,000, respectively. The input data show that the model is trained using only about 0.25% of the total data points in the network. Accuracy and performance are greatly influenced by the framework's width, depth, and learning rate. In line with the literature, 8 hidden layers are selected for the network and a sensitivity analysis is performed to determine its width and learning rate. Learning rate is crucial when seeking global minima. The studies showed that the computational cost of a small learning rate can increase even though it moves towards global minima gradually, whereas a large learning rate can skip the global minima altogether [53]. A sensitivity analysis is presented in Table 3, which compares the relative errors of the various links for the fixed tanh activation function. A model with 8 hidden layers and 30 neurons performs the best with a learning rate of 0.001. Keeping in view the optimal parameters, the PINNs architecture and the input data points are summarized in Table 4. There are 10 layers, including an input layer, 8 hidden layers, and 1 output layer. Keeping in view Figure 3 from the previous section, the conservation conditions _Q11 = Q12 + Q13_ and _Q21 = Q22 + Q23_ are integrated into the model for diverging and merging points, respectively. \(Q\) represents the flow \(V\) and the first and second digits in subscript refer to node and link, respectively. After training the model, the complete dataset of the network is passed to the model to estimate the traffic state. The predicted speeds for the links L1, L2, and L3 are respectively reported in Figure 6a-c. The predicted speed contours for all the links of the network are pretty similar to the actual speed contours counterparts in Figure 4b. The model has successfully captured the traffic dynamics of the network. The predicted data are further compared at different fixed locations along time and space dimensions. Figure 7 presents actual and predicted speed at three different times 125, 250, and 675 respectively given by \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Link** & **L1** & **L2** & **L3** \\ \hline No. of layers & 10 & 10 & 10 \\ Neurons & 30 & 30 & 30 \\ No. of Training points & 1200 & 1200 & 1200 \\ No. of Residual points & 30,000 & 30,000 & 30,000 \\ Total points in the link & 480,480 & 480,480 & 480,480 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: PINNs architecture and summary of data points in each link of the network.