# Physics-Informed Deep Learning For Traffic State Estimation: A Survey and the Outlook
## 2 Preliminaries and Related Work
### Traffic State Estimation
#### 2.2.1 Physics-Based Models  
The field of transportation has been buttressed by rich theories and models tracing back to as early as the 1930s [41]. A large amount of theories have since been successfully developed to explain real-world traffic phenomena, to prognose and diagnose anomalies for operation and management, and to make predictions for planning and management.  
These models make ample use of scientific knowledge and theories about transportation systems, ranging from closed-form solutions to numerical models and simulations. Transportation models have demonstrated their analytical and predictive power in the past few decades. For example, microscopic car-following models and macroscopic traffic flow models succeed in capturing transient traffic behavior, including shock waves and the stop-and-go phenomenon.  
The model-based approach relies on traffic flow models for single- or multi-lane and single- or multi-class traffic flow. Traffic models include first-order models such as Lighthill-Whitham-Richards (LWR) [42; 43], and second-order models such as Payne-Whitham (PW) [44; 45] and Aw-Rascle-Zhang (ARZ) [46; 47].  
Figure 3: Data types for TSE (adapted from [12], including fixed location sensors (blue hexagons), roadside camera, and collocation points (black crosses)).  
The first constitutive law that needs to be satisfied is a conservation law (CL) or transport equation, meaning that inflow equals outflow when there is no source or sink. Mathematically,  
\[(CL)\quad\rho_{t}+(\rho u)_{x}=0,\ (x,t)\in\mathcal{D}. \tag{6}\]  
A second equation that stipulates the relation between \(\rho,u\) can be a fundamental diagram (FD) (for the first-order traffic flow model) or a moment equation (for the second-order traffic flow model):  
\[(FD)\quad u=U(\rho),\ (\text{first-order}) \tag{7}\] \[u_{t}+uu_{x}=g(U(\rho)),\ (\text{second-order}).\]  
where \(U(\cdot)\) is a fundamental diagram, a mapping from traffic density to velocity, and \(g(U(\rho))\) is a nonlinear function of \(U(\rho)\). A fundamental diagram can also be a mapping from density to volume/flux, as exemplified in Figure 4, calibrated by a real-world traffic dataset. In the literature, several FD functions are proposed and interested readers can refer to [48] for a comprehensive survey.  
When a traffic flow model is selected as the underlying dynamical system, data assimilation is employed to find \"the most likely state\" and observations are used to correct the model prediction, including an extended Kalman filter (EKF) [15; 49; 50], unscented KF [51], ensemble KF [52], and particle filter [53].  
To quantify the uncertainty in TSE problems, the model-based approach usually makes a prior assumption about the distribution of traffic states by adding a Brownian motion on the top of deterministic traffic flow models, leading to Gaussian stochastic traffic flow models. There is the other school of literature that derives intrinsic stochastic traffic flow models with more complex probabilistic distributions [15; 16; 17; 18]. With a stochastic traffic flow model, a large population approximation or fluid limit is applied to extract the first and second moments of the stochastic processes to facilitate the application of filtering methods.