# Physics-Informed Deep Learning For Traffic State Estimation: A Survey and the Outlook
## 4 PIDL for UQ  
It is a widely studied modeling and computational challenge to quantify how uncertainty propagates within dynamical systems that could result in cascading errors, unreliable predictions, and worst of all, non-optimal operation and management strategies. It is thus  
Figure 10: Ratios of the contributions made by the physics-based component and the data-driven component to the optimal performance of PIDL. crucial to characterize uncertainties in traffic state estimators and, consequently, in traffic management that relies on TSE predictors. UQ for TSE using PIDL is still in its nascent stage. **Definition 4**. : _Uncertainty quantification (UQ) aims to assess the robustness of the developed model and bound the prediction errors of dynamical systems by estimating the probability density of quantities with input features and boundary conditions [79]. Stochastic effects and uncertainties potentially arise from various sources, including variability and errors in measurements, dynamic actions of various entities, model biases, and discretization and algorithmic errors. In summary, there are two types of uncertainty in the context of TSE [79; 80]:_  
1. _Aleatoric uncertainty (or data uncertainty): an endogenous property of data and thus irreducible, coming from measurement noise, incomplete data, or a mismatch between training and test data._
2. _Epistemic uncertainty (or knowledge uncertainty, systematic uncertainty, model discrepancy): a property of a model arising from inadequate knowledge of the traffic states. For example, traffic normally constitutes multi-class vehicles (e.g., passenger cars, motorcycles, and commercial vehicles). A single model can lead to insufficiency in capturing diversely manifested behaviors._  
UQ is \"as old as the disciplines of probability and statistics\" [79]. In recent years, its explosive growth in large-scale applications has been bolstered by the advance of big data and new computational models and architectures. The conventional UQ techniques include but are not limited to: sensitivity analysis and robust optimization [81]; probabilistic ensemble methods and Monte- arlo methods with multi-level variants [82; 83; 84]; stochastic spectral methods [79]; and methods based on the Frobenius-Perron and Koopman operators [85; 86] for dynamic systems. Epistemic uncertainty arising from model discrepancies, often bias, can be compensated for by improved domain knowledge, which has received increasing attention, especially with the advance of PIDL. In the computational architecture of PIDL, the data component can be treated as a compensation term for inaccurate or biased physics supplied by the physics-based component. Thus, it is natural to generalize PIDL to UQ, where the physics-based component provides partial domain knowledge when stochasticity is propagated within highly nonlinear models, and the data-driven component learns extra randomness arising from both data and model errors. **Definition 5**. : _UQ for traffic state estimation (UQ-TSE) aims to capture the randomness of traffic states \(\hat{\mathbf{s}}=\{\hat{\rho},\hat{a}\}\) by probabilistic models. It is assumed that \(\hat{\mathbf{s}}\) follows the observational distribution, i.e., \(\hat{\mathbf{s}}\sim p_{\text{data}}(\hat{\mathbf{s}}|x,t)\). The goal of the UQ-TSE problem is to train a probabilistic model \(G_{\theta}\) parameterized by \(\theta\) such that the distribution of the prediction \(\mathbf{s}\sim p_{\theta}(\mathbf{s}|x,t)\) resembles the distribution of the observation \(\hat{\mathbf{s}}\sim p_{\text{data}}(\hat{\mathbf{s}}|x,t)\). One widely used metric to quantify the discrepancy between \(p_{\text{data}}\) and \(p_{\theta}\) is the reverse Kullback-Leibler (KL) divergence._  
Since the majority of the literature on UQ-PIDL employs deep generative models, including generative adversarial networks (GANs) [87], normalizing flow [88], and variational autoencoder (VAE) [89], here we will focus on how to leverage deep generative models for UQ problems.