# Physics-Informed Deep Learning For Traffic State Estimation: A Survey and the Outlook
## 1 Introduction
### Scope of This Survey  
There are a variety of structure designs for PIDL. Here, we primarily focus on the PIDL framework that is represented by a _hybrid computational graph_ (HCG) consisting of two graphs: a physics-informed computational graph (PICG) and a physics-uninformed neural network (PUNN). Despite numerous benefits of PIDL, there are still many open questions that need to be answered, including the construction of the HCG, the choice of the physics and the architecture of the PICG, the architecture of the PUNN, the loss function, and the training algorithms. Among them, how to encode the physics into (deep) neural networks (DNNs) remains under-explored and varies case by case and across domains. In this paper, we will primarily establish a systematic design pipeline for hybrid computational graphs that would facilitate the integration of physics and DL. To this end, we will review the state-of-the-art of the PIDL architecture in the traffic state estimation (TSE) problem. This survey paper can be used as a guideline for researchers at large when they consider using PIDL for problems at hand.  
We have seen a growing number of studies that apply PIDL to physical and biological systems [4; 5]. However, its feasibility for social systems, in particular, human decision-making processes, such as driving behaviors, remains largely unexploited. Humans' decision making involves complex cognitive processes compounded by perception errors, noisy observations, and output randomness. Moreover, human driving behaviors exhibit highly unstable and nonlinear patterns, leading to stop-and-go traffic waves and traffic congestion [6; 7; 8]. Accordingly, neither model-driven nor data-driven approaches alone suffice to predict such behaviors with high accuracy and robustness. Therefore, we strongly believe that a hybrid method, which leverages the advantages of both model-driven and data-driven approaches, is promising [9; 10].  
There is a vast amount of literature on TSE [11; 12] and on PIDL [4; 5]. To distinguish this work from other survey papers in TSE, here, we primarily focus on data-driven approaches, PIDL in particular. Since PIDL for TSE is less studied than physics-based models, and the existing literature is focused on single roads, we will primarily examine TSE along links, and leave TSE on a road network which includes both link and node models for future research. To distinguish this work from other PIDL surveys, we primarily focus on the modular design of the hybrid PIDL paradigm and show how to customize various designs for accurate and robust identification of traffic dynamic patterns. Here, the _modular design_ refers to the architecture design of each component in the graph and how these components are wired, in other words, how physics laws are injected into DNNs. The generic architecture of a PIDL consists of two computational graphs: one is a DNN  
Figure 1: Comparison of pure physics-based, data-driven, and hybrid paradigms (adapted from [2]).  
(i.e., the data-driven component) for predicting the unknown solution, while the other is a computational graph (i.e., the physics-driven component, also called \"physics-informed neural networks (PINNs)\" in [1]), in which physics is represented, for evaluating whether the prediction aligns with the given physics. The physics encoded computational graph can be treated as a regularization term of the other deep neural network to prevent overfitting, i.e., high-variance. In summary, the hybrid of both components overcomes high-bias and high-variance induced by each individual one, rendering it possible to leverage the advantage of both the physics-based and data-driven methods in terms of model accuracy and data efficiency.