## 1 Introduction  
The traffic conditions on road segments in a network are usually described by macroscopic traffic state variables, such as flow rate, vehicle speed, and vehicle density as traffic streams. Transport planners identify congestion levels and traffic demands, as well as bottlenecks on roadways, through these indicators [1]. However, these important measurements are not available for all locations or times due to physical and budget constraints of the measurements; even if the availability issue is not a concern, undesired noises in the measurements have been making ITS traffic management operations difficult [2]. Combining factors such as the cost of sensor installation, the accuracy of vehicle detection techniques, and restrictions on data storage and transmission can often lead to partial observations of traffic state variables [3]. To deliver effective traffic management, it is vital to estimate traffic state variables at locations without sensor data. Traffic state estimation (TSE) refers to the prediction of traffic state variables such as flow, density, and speed of road segments using partially observed traffic data [4]. Approaches to TSE can be roughly categorized into two groups based on the a priori knowledge they depend on: model-driven and data-driven. In model-driven approaches, traffic flow state predictions are based on prior knowledge of the flow, usually embodied in physics-based models such as the Lighthill-Whitham-Richards (LWR) model or the Aw-Rascle-Zhang (ARZ) model [5, 6]. Physical processes play a key role in how these models work, such as the way the state variable changes across space and time. These models are much less limited by data availability and can make predictions without the training data. A model-based approach assumes that the physics-based models are representative of traffic dynamics and can be utilized to determine unobserved values based on partially observed data. Despite the independence from data requirements, the physics-basedmodels suffer from a few challenges that may lead to low-quality estimates. Some of these challenges include: (i) they can only capture a limited set of traffic dynamics in real-world situations; (ii) they are derived under ideal conditions, entail large effort for parameter calibrations; and (iii) they are difficult to apply with noisy/fluctuating traffic data. The other group of TSE methods is the data-driven models based on machine learning. Data-driven models employ statistical relationships in data and are model-free so that formulating specific underlying physics is not necessary. Data-driven methods have been intensively studied recently due to the increasing availability of data, as they do not require specific theoretical assumptions and have a remarkably low computing cost during the testing phase. However, the data-driven nature of those machine learning (ML) models leaves them vulnerable to the following situations: (i) insufficient training data to learn the system's complexity, (ii) training samples with misleading information, and (iii) test samples that are not representative of the training samples. Dramatic performance drops alongside large and/or biased estimations are not rare in these scenarios, which unfortunately are very common in the real world. In addition, the machine learning models are difficult to interpret because of their 'black-box' nature. Very recent studies on the hybrid approaches integrate physics knowledge from traffic flow models and machine learning models, in order to mitigate the limitations of the above-mentioned approaches. For instance, [PERSON] et al. (2021) presented the physics regularized Gaussian process (PRGP) method for TSE by integrating macroscopic traffic flow models with Gaussian process (GP) [7]. [PERSON] et al. were the first ones to propose implementations of fully connected neural networks that incorporate residuals from partial differential equations (PDEs) into their loss function as regularizers, which restrains the feasible solution space [8].