# Physics-Informed Deep Learning For Traffic State Estimation: A Survey and the Outlook
## 4 PIDL for UQ
### Real-World Data Validation  
We apply PIDL-UQ models to the NGSIM dataset. We use four model types, i.e., PI-GAN, PI-GAN-FDL, PI-Flow, and TrafficFlowGAN for demonstration. Each model can be informed by either the LWR or ARZ equations, resulting in eight model variants \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Model** & **Descriptions** & **Pros** & **Ref.** \\ \hline \multirow{4}{*}{\begin{tabular}{c} **P-Flow** \\ **P-Flow** \\ \end{tabular} } & \begin{tabular}{c} The normalizing flow model is \\ used as the generator for explicit \\ computation of the likelihood; the \\ physics loss \(L_{c}\) is the same as \\ PI-GAN. \\ \end{tabular} & Simple & \\ \hline \multirow{4}{*}{\begin{tabular}{c} **P-Flow** \\ **P-Flow** \\ \end{tabular} } & \begin{tabular}{c} The normalizing flow model is \\ used as the generator for explicit \\ computation of the likelihood; a \\ convolutional neural network is \\ used as the discriminator to ensure \\ high sample quality physics loss \(L_{c}\) \\ is the same as PI-GAN. \\ \end{tabular} & Combines the & \\ \cline{1-1}  & Traffic Flow & \
\begin{tabular}{c} Convolutional neural network is \\ used as the discriminator to ensure \\ high sample quality physics loss \(L_{c}\) \\ is the same as PI-GAN. \\ \end{tabular} & PhysGAN and \\ \cline{1-1}  &
\begin{tabular}{c} **P-Flow** \\ \end{tabular} & PhysFlow & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: _Cont._  
Figure 12: Estimated traffic density \(\rho\) **(left)** and traffic velocity **(right)** of the PI-GAN when the number of loop detectors is equal to 3, where the horizontal black lines in the heatmap represent the positions of the loop detectors. In each half, the prediction heatmap and snapshots at certain time points are presented. in total. EKF and GAN are used as baselines. The EKF uses the three parameter-based LWR as the physics. The results are shown in Figure 13. As shown on the y-axis, the upper panels are the REs of the traffic density and velocity, and the lower panels are the Kullback-Leibler divergence (KL) of the traffic density and velocity. The comparison is made under representative combinations of probe vehicle ratios (see x-axis) and numbers of loop detectors (see the titles of sub-figures). We interpret the results from three perspectives:  
**Effect of loop data. ** When the probe vehicle ratio is fixed to 0.5%, the performance of all of the models is significantly improved as the loop number increases from 0 to 2, which is because the loop data provides more information. This improvement is not significant when the probe vehicle ratio is 3%, as the probe data alone has provided sufficient information. Figure 13: Results of the PIDL-UQ models for the NGSIM dataset. **(a)** RE of the traffic density; **(b)** RE of the traffic velocity; **(c)** KL of the traffic density; **(d)** KL of the traffic velocity; **(e)** summary of RE and KL of the traffic density and velocity of all data sizes. **Effects of using FDL. ** When the loop number is 2 and the probe vehicle ratio is 0.5%, PI-GAN-FDL achieves significantly lower REs and KLs compared to PI-GAN and PI-Flow, while this advantage becomes less significant when the data is sparse. This is because the ML surrogate requires more data to train. Furthermore, PI-GAN-FDL achieves lower KLs than PI-GAN in general, indicating that PI-GAN-FDL can better capture uncertainty. **Comparison between the ARZ-based model and the LWR-based model. ** The ARZ-based model outperforms the LWR-based model in general, which shows that the second-order physics is more suitable for the real-world scenario. **Comparison between PIDL-UQ models. ** As the data amount increases, the performance improves and the performance difference across models becomes small. Among all of the PIDL-based models, TrafficFlowGAN generally achieves the least error in terms of both RE and KL, because it combines the advantages of both PhysGAN and PhysFlow.