# Physics-Informed Deep Learning For Traffic State Estimation: A Survey and the Outlook
## 2 Preliminaries and Related Work  
### Pidl  
Definition 1: Generic framework for physics-informed deep learning: Define location \(x\in[0,L]\) and time \(t\in[0,T]\) and \(L,T\in\mathbb{R}^{+}\). Then, the spatiotemporal (ST) domain of interest is a continuous set of points: \(\mathcal{D}=\{(x,t)|x\in[0,L],t\in[0,T]\}\). Denote the state as \(\mathbf{s}\) and its observed quantity as \(\mathbf{s}\). Denote the (labeled) observation \(\mathcal{O},\mathcal{B},\mathcal{I}\) and the (unlabeled) collocation points \(\mathcal{C}\) below:  
\[\left\{ \begin{aligned} &\mathcal{O}=\{(\mathbf{x}^{(i)},t^{(i)}; \mathbf{s}^{(i)})\}_{i=1}^{N_{\mathcal{L}}}:\text{within-domain observation},\\ &\mathcal{B}=\{t^{(i)};\mathbf{s}^{(i)}\}_{i=1}^{N_{\mathcal{L} }}:\text{boundary observation},\\ &\mathcal{I}=\{\mathbf{x}^{(i)};\mathbf{s}^{(i)}\}_{i=1}^{N_{ \mathcal{L}}}:\text{initial observation},\\ &\mathcal{C}=\{(\mathbf{x}^{(i)},t^{(i)})\}_{i=1}^{N_{\mathcal{ L}}}:\text{ collocation points},\end{aligned} \right. \tag{1}\]  
where \(i\) and \(j\) are the indices of the observation and collocation points, respectively. \(i_{b},i_{0}\) are the indices of the boundary and initial data, respectively. The number of observed data, boundary and initial conditions, and collocation states are denoted as \(N_{o},N_{b},N_{0},N_{c}\), respectively. The subscripts \(b,0\) represent the boundary and initial condition indices, respectively. We design a hybrid computational graph (HCG) consisting of two computational graphs: (1) a PLINN, denoted as \(f_{\theta}(\mathbf{x},t)\), to approximate mapping \(\mathbf{s}^{(i)}\), and (2) a PICG, denoted as \(f_{\Lambda}(\mathbf{x},t)\), for computing traffic states of \(\mathbf{s}^{(j)}\) from collocation points. In summary, a general PIDL model, denoted as \(f_{\theta,\lambda}(\mathbf{x},t)\), is to train an optimal parameter set \(\theta^{*}\) for PUINN and an optimal parameter set \(\lambda^{*}\) for the physics. The PLINN parameterized by the solution \(\theta^{*}\) can then be used to predict a traffic state \(\mathbf{\hat{s}}_{new}\) on a new set of observed ST points \(\mathcal{O}_{new}\subseteq\mathcal{D}\), and \(\lambda^{*}\) is the most likely model parameters that describe the intrinsic physics of the observed data._  
One important application of PIDL is to solve generic partial differential equations (PDE). We will briefly introduce the underlying rationale. Define a PDE over the ST domain as:  
\[\mathbf{s}_{t}(x,t)+\mathcal{N}_{x}[\mathbf{s}(x,t)]=0,(x,t)\in \mathcal{D}, \tag{2}\] \[\mathcal{B}[\mathbf{s}(x,t)]=0,\ (x,t)\in\partial\mathcal{D},\] \[\mathcal{I}[\mathbf{s}(x,0)]=0,\]  
where \(\mathcal{N}_{x}(\cdot)\) is the nonlinear differential operator, \(\mathcal{B},\mathcal{I}\) are the boundary and initial condition operators, respectively, \(\partial\mathcal{D}=\{(0,t)|t\in[0,T]\}\cup\{(L,t)|t\in[0,T]\}\) is the set of ST points on the boundary of the domain \(\mathcal{D}\), and \(\mathbf{s}(x,t)\) is the exact solution of the PDE. Now, we will approximate the PDE solution, \(\mathbf{s}(x,t)\), by a DNN parametrized by \(\theta\), \(f_{\theta}(x,t)\), which is PUNN. If this PUNN is exactly equivalent to the PDE solution, then we have  
\[f_{\theta}(x,t)+\mathcal{N}_{x}[f_{\theta}(x,t)]=0,(x,t)\in D. \tag{3}\]  
Otherwise, we define a residual function \(r_{c}(x,t)=[f_{\theta}(x,t)]_{t}+\mathcal{N}_{x}[f_{\theta}(x,t)]\). If PUNN is well trained, the residual needs to be as close to zero as possible. Figure 2 describes the schematic of using PUNN to approximate PDE solutions. The physics is usually encoded as _governing equations, physical constrains_, or _regularity terms_ in loss functions when training PUNNs [13]. When \"soft regularization\" was implemented by [14], the loss in PIDL was a weighted sum of two distance measures: one for the distance between the observed action and predicted one (also known as _data discrepancy_) and the other for the distance between the computed action from the physics and the predicted one (also known as _physics discrepancy_).