# Physics-Informed Deep Learning For Traffic State Estimation: A Survey and the Outlook
## 3 PIDL for Deterministic Tse
### Numerical Data Validation for Three-Parameter-Based LWR  
In this example, we show the ability of PIDL for the traffic dynamics governed by the three-parameter-based LWR traffic flow model on a ring road. Mathematically,  
\[\rho_{t}+(Q(\rho))_{x}=\epsilon\rho_{xx},\;x\in[0,1],\;t\in[0,3], \tag{9}\]  
where \(\epsilon=0.005\). The initial and boundary conditions are \(\rho(x,0)=0.1+0.8e^{-25(x-0.5)^{2}}\) and \(\rho(0,t)=\rho(1,t)\). In this model, a three-parameter flux function [21] is employed: \(Q(\rho)=\rho U(\rho)=\sigma(a+(b-a)\frac{\rho}{\rho_{max}}-\sqrt{1+y^{2}})\), where \(a=\sqrt{1+(\delta p)^{2}}\), \(b=\sqrt{1+(\delta(1-p))^{2}}\) and \(y=\delta(\frac{p}{\rho_{max}}-p)\). In the model, \(\delta\), \(p\), and \(\sigma\) are the three free parameters after which the function is named. The parameters \(\sigma\) and \(p\) control the maximum flow rate and critical density (where the flow is maximized), respectively. \(\delta\) controls the roundness level of \(Q(\rho)\). In addition to the above-mentioned three parameters, we also have \(\rho_{max}\) and the diffusion coefficient \(\epsilon\) as part of the model parameters. In this numerical example, we set \(\delta=5\), \(p=2\), \(\sigma=1\), \(\rho_{max}=1\), and \(\epsilon=0.005\). Given the bell-shaped initial density condition, we apply the Godunov scheme to solve Equation (9) on 240 (space) \(\times\) 960 (time) grid points evenly deployed throughout the \([0,1]\times[0,3]\) domain. The PIDL architecture that encodes the LWR model is shown in Figure 6. This architecture consists of a PUNN for traffic density estimation, followed by a PICG for calculating the residual \(r_{c}:=\rho_{t}+(Q(\rho))_{x}-\epsilon\rho_{xx}\) on collocation points. The estimated traffic density \(\rho\) \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{2}{c}{**Physics**} & \multicolumn{1}{c}{**Data**} & \multicolumn{1}{c}{**Descriptions**} & **Ref.** \\ \hline \multirow{4}{*}{LWR} & Synthetic & Integrated the Greenshields-based LWR to & \multirow{4}{*}{[25]} \\  & & (Lax–Hopf & PIDL and validated it using loop detectors & as well as randomly placed sensors. \\ \cline{3-4}  & & & & Presented the use of PIDL to solve \\ \multirow{4}{*}{LWR} & \multirow{2}{*}{LWR} & Numerical, & Greenshields-based and three-parameter-based & \multirow{4}{*}{[27]} \\  & & NGSIM & LWR models, and demonstrated its advantages & \\ \cline{3-4}  & & & using a real-world dataset. & \\ \cline{3-4}  & & & Studied the general partial-state reconstruction & \\ \multirow{4}{*}{LWR} & \multirow{2}{*}{Numerical} & \multirow{2}{*}{problem for traffic flow estimation, and used } & \multirow{2}{*}{[78]} \\  & & & PIDL encoded with LWR to counterbalance & \\ \cline{3-4}  & & & the small number of probe vehicle data. & \\ \cline{3-4}  & & & & Integrated a coupled micro–macro model, \\ \multirow{4}{*}{LWR} & \multirow{2}{*}{simulation} & \multirow{2}{*}{combinning the follow-the-leader (FL1) model and LWR model, to PIDL for TSE, which can use the velocity information from probe vehicles.} & \\ \hline \multirow{4}{*}{LWR} & \multirow{4}{*}{LWR} & \multirow{4}{*}{Numerical, } & Applied the PIDL-based TSE to the second- & \multirow{4}{*}{[28]} \\  & & & order ARZ with observations from both loop & \\ \cline{3-4}  & & & detectors and probe vehicles, and estimated both & \\ \cline{3-4}  & & & \(\rho\) and \(u\) in parallel. & \\ \cline{3-4}  & & & NGSIM &
\begin{tabular}{l} Proposed the idea of integrating ML surrogate \\ (e.g., an NN) into the physics-based component \\ in the PICG to represent the complicated FD \\ relation. Improved estimation accuracy achieved \\ and unknown FD relation learned. \\ \end{tabular} & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: State-of-the-art PIDL-TSE. is calculated by the PUNN \(f_{\theta}(x,t)\), which is an NN and maps a spatiotemporal point \((x,t)\) directly to \(\rho\), i.e., \(\rho=f_{\theta}(x,t)\).