# Physics-Informed Deep Learning For Traffic State Estimation: A Survey and the Outlook
## 3 PIDL for Deterministic Tse
### Real-World Data Validation  
It would be interesting to see the performance of state-of-the-art methods based on either physics or data-driven approaches in order to better quantify the added value of the proposed class of approaches. We will use a widely used real-world open dataset, the next generation simulation (NGSIM) dataset, detailed in Table 3. Figure 8 plots the traffic density heatmap using data collected from the US 101 highway. The performances of two baseline models and four PIDL variants for deterministic TSE are presented in Figure 9. As shown on the y-axis, the REs of the traffic density and velocity are used for evaluation. The comparison is made under representative combinations of the probe vehicle ratios (see x-axis) and numbers of loop detectors (see the titles of the sub-figures). We implement an EKF and a pure NN model as the representative pure data-driven and physics-driven baseline approaches, respectively. The EKF makes use of  
Figure 7: Estimated traffic density \(\rho\) (**left**) and traffic velocity \(u\) (**right**) of the PIDL when the number of loop detectors is 3, where the horizontal black lines in the heatmap represent the sensor positions. In each half, the prediction heatmap and snapshots at certain time points are presented. Note that the PUNN does not predict \(u\) directly, and instead, it is calculated by \(Q(\rho)/\rho\) in the post-processing. the three parameter-based LWR as the core physics when conducting the estimation. The NN only contains the PUNN component in Figure 6, and uses the first term in Equation (10) as the training loss. Among the PIDL variants, the PIDL-LWR and PIDL-ARZ are the PIDL models that encode the three parameter-based LWR and Greenshields-based ARZ, respectively, into the PICG. PIDL-LWR-FDL and PIDL-ARZ-FDL are the variant models of PIDL-LWR and PIDL-ARZ, replacing the FD components in the PICG with an embedded neural network (i.e., the FD learner). Note, the FD leaner technique is the one introduced by [31]. **Performance of baseline models**: From the experimental results, it is observed that the performance of all of the models improves as the data amounts increase. The EKF method performs better than the NN method, especially when the number of observations is small. The results are reasonable because EKF is a physics-driven approach, making sufficient use of the traffic flow model to appropriately estimate unobserved values when limited data are available. However, the model cannot fully capture the complicated traffic \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Site** & **Location** & **Date** & **Length (m)** & **Sampling Rate (s)** & **Lane \#** \\ \hline US 101  & LA, CA & 6/15/2005 & 640 & 0.1 & 5 \\ \hline \hline \end{tabular} Note: Vehicular information includes the position, velocity, acceleration, occupied lane, and vehicle class. Time periods include 7:50-8:05 a.m., 8.05-8:20 a.m., and 8:20-8:35 a.m. and the data is collected on 15 June 2005. The data remains accessible to date. We average traffic states across all lanes; 1 www.fhwa.dot.gov/publications/rese arch/operations/07030/index.cfm; accessed on 22 May 2023.  
\end{table} 
Table 3: Real-world data description. Figure 8: Average traffic density and speed on US 101 highway. Heatmap for the traffic density (**left**) and velocity (**right**). Figure 9: Results of the deterministic PIDL models for the NGSIM dataset. “#loop” stands for the number of loop detectors. (**a**) RE of the traffic density; (**b**) RE of the traffic velocity. dynamics in the real world and the performance improves slowly as the data amount increases. The NN is able to catch up with the EKF when the amount of data is relatively large (see the case with loop = 2 and ratio = 3.0%). However, its data efficiency is low and large amounts of data are needed for accurate TSE. **Comparison between PIDL-based models**: The PIDL-based approaches generally outperform the baseline models.