Quantifying Uncertainty In Traffic State Estimation Using Generative Adversarial Networks

Traffic states, represented by traffic velocity or density, can contain uncertainty from various sources, including model errors, measurement noise, inherent stochasticity in driving behavior, and random initial or boundary conditions. Imposed with these stochastic terms, traffic states are essentially spatiotemporal (ST) random fields. In this paper, we are primarily focused on uncertainty arising from driving behaviors and their resultant effect on traffic states. We aim to infer the distribution of traffic state random fields using sparse data collected from fixed-location sensors, i.e., loop detectors.

Traffic state estimation (TSE), a classical traffic problem, has long been studied aiming to estimate traffic states using various sensor data [1]. The methods that predict traffic states and characterize the robustness of the prediction can be categorized into model-based and data-driven methods. The model-based methods assume that observed data are generated by underlying models and filtering methods can be applied to these models to propagate uncertainty. The data-driven methods learn distributions of uncertainty directly from data without using a prior assumption of underlying physics. The model based methods suffer from limitations such as the non-Gaussian likelihoods and high-dimensional posterior distributions [2], while the data-driven approaches require a huge amount of data to learn the posterior distributions from high-dimensional random inputs. Thus, an integration of model based and data-driven methods has become a rapidly growing research arena, namely, physics-informed deep learning (PIDL) [3, 4, 5, 6].

Recent years have seen a growing trend of learning random fields using machine learning models, including but not limited to: Gaussian Processes [7], Bayesian neural networks [8], and most recently, generative models [2]. Among the generative models, generative adversarial networks (GAN) have demonstrated its robustness in various applications, from image or video generation [9] to uncertainty quantification (UQ) [10]. For UQ problems, physics-informed GANs (PhysGAN) is becoming increasingly popular to quantify uncertainty in stochastic differential equations [11, 2, 12, 13]. Little research, though, has been documented when randomness inherently arises from human behaviors or inherent dynamics. In other words, the existing literature on PhysGAN integrates a deterministic physics model into GANs because it is assumed that the equations that generate data are deterministic and the prediction contains stochasticity due to randomness from initial or boundary conditions. In this paper, in contrast, we will assume that the models that generate data are themselves stochastic due to randomness in parameters, and thus, stochastic physics models need to be integrated to capture the randomness from these sources.

The contributions of this paper include:
(1) Physics-informed GANs are used to quantify uncertainty in TSE. Because traffic, constituted by human drivers, is inherently stochastic due to randomness in driving behaviors, stochastic traffic flow models are incorporated into GANs to help regularize the training process;
(2) Parameters in physics are estimated jointly with traffic state prediction;
(3) PhysGAN-TSE is applied to the real-world dataset, Next Generation SIMulation (NGSIM), to demonstrate its predictive accuracy compared to other baselines.

The rest of the paper is organized as follows: Section II reviews the state-of-the-art of PIDL on TSE. Section III introduces the preliminaries of GAN based UQ. Section IV introduces how we integrate PIDL and GAN into the framework of PhysGAN for TSE problems. Section V demonstrates how PhysGAN is applied to NGSIM to characterize uncertainty from the real-world data. Section VI concludes our work and projects future research directions in this promising arena.

For studies that use model-based UQ for TSE, a prior assumption is usually made about the distribution of randomness in inputs or traffic states. The existing practice includes two ways: one to add a Brownian motion on the top of the deterministic traffic flow models, leading to Gaussian stochastic traffic flow models [14]; and the other to derive intrinsic stochastic traffic flow models with more complex probabilistic distributions [15]. The former is more amenable to applying standard filtering methods while the latter is more challenging to deal with and requires a large population approximation to extract mean and variance before any filtering can be applied.

If the noise in the output traffic states is non-Gaussian and follows complex distributions, the filtering methods could fail. GANs have been used to learn complex distributions of stochasticity residing in random fields from ST data [11]. Key issues with GANs include mode collapse and training instability [16]. One solution is to regularize the training of GANs by incorporating physics into the structure of generators (and potentially discriminators), where unlabeled data are generated from generators that are guided by physics-based models, so that the search space of parameters is narrowed down.

PhysGAN have been applied to quantify real-world data uncertainty in various domains, which includes flood prediction [17], blood alcohol concentration prediction [18], and porous media flow modeling [19]. PhysGANs have not been used to quantify uncertainty in traffic state estimation. Traffic, different from physical systems, can exhibit highly nonlinear randomness arising from inherent human driving behavior. Thus, it poses new challenges when PhysGANs are applied.

Let us start with a partial differential equation (PDE). Define location $x\in[0,L]$ and time $t\in[0,T]$ and $L,T\in\mathbb{R}^{+}$. Then the ST domain of interest is a continuous set of points: $D=\{(x,t)|x\in[0,L],t\in[0,T]\}$. A PDE defined over the ST domain is written as:

where, $\mathcal{N}_{x}(\cdot)$ is the nonlinear differential operator, $\mathcal{B}_{x}(\cdot)$ is the boundary condition operator, $\mathbf{u}(x,t)$ is the exact solution of the PDE, and $\lambda$ is the physics parameter vector.

Now we will approximate the PDE solution, $\mathbf{u}(x,t)$, by a (deep) neural network (DNN) parametrized by $\theta$, i.e. $\hat{\mathbf{u}}(x,t)=f_{\theta}(x,t)$, which is called “physics uninformed neural network (PUNN).” If this PUNN is exactly equivalent to the PDE solution, then we have

Otherwise we define a residual function:

If PUNN is well trained, the residual needs to be as close to zero as possible. Since NNs are normally trained with discrete data points, below we will define the potentially accessible training data in the context of TSE.

Assume the dataset consists of (1) (labeled) observation data $O=\{(x_{o}^{(i)},t_{o}^{(i)};\mathbf{u}_{o}^{(i)})|i=1,...,N_{o}\}$, and (2) (unlabeled) collocation points $C=\{(x_{c}^{(j)},t_{c}^{(j)})|j=1,...,N_{c}\}$. $i$ and $j$ are the indexes of observation points and collocation points, respectively. The numbers of observed data and collocation states are denoted as $N_{o},N_{c}$, respectively. Observation data $O$ are limited to the time and locations where traffic sensors are placed. In contrast, collocation points $C$ have neither measurement requirements nor location limitations, and thus are controllable. The target values associated with collocation points are used to regularize the loss function. Accordingly, we define a loss function as the weighted average of data discrepancy and physics discrepancy. Denote the weight of the data discrepancy in the loss function as $\alpha$. The loss function is:

We will introduce stochasticity into the PDE solution. Below we will discuss how the probabilistic physics-informed neural networks [2] can be used for UQ. When considering the stochasticity of a PDE, we assume the uncertainty mainly comes from two sources, the intrinsic uncertainty of the physics parameter $\lambda$ and the measurement uncertainty of the variable $\mathbf{u}$. In connection with this, we treat the physics parameter as a random variable, i.e. $\lambda\sim p_{\lambda}(\lambda)$, and assume the variable $\mathbf{u}$ follows a conditional distribution of the spatial-temporal coordinates, i.e. $\mathbf{u}\sim p_{\mathbf{u}}(\mathbf{u}|x,t)$.

After formulating the uncertainty of the physics $\lambda$ and the variable $\mathbf{u}$, a UQ problem further assumes that there exists a mapping between the variable $\mathbf{u}$ and a latent random variabe $\mathbf{z}\sim p_{\mathbf{z}}(\mathbf{z})$, i.e. $G:(x,t,\mathbf{z})\rightarrow\mathbf{u}$, such that the PDE holds in a stochastic manner:

where $\mathbf{z}$ encodes possible randomness sources that lead to stochasticity in the observable output $\mathbf{u}$.

With Eq. 5 generating noised output $\mathbf{u}$ that are constrained by a PDE, the UQ problem based on adversarial inference is to match the conditional distribution of the generated data, denoted as $\hat{\mathbf{u}}=G_{\theta}(x,t,\mathbf{z})\sim p_{\theta}(\hat{\mathbf{u}}|x,t)$ parametrized by $\theta$, and the conditional distribution of the observed output, which is $p_{\mathbf{u}}(\mathbf{u}|x,t)$. One widely used metric to match two distributions is the reverse Kullback-Leibler (KL) divergence.

Now we will formulate the UQ problem in the context of conditional GANs. The generator is a neural network surrogate model, denoted as $G_{\theta}$, that approximates the ground-truth mapping $G$ mentioned before. The objective of the generator $G_{\theta}$ is to fool an adversarially trained discriminator $D_{\phi}$. Then the adversarial UQ problem becomes a min-max game:

where $\theta$ and $\phi$ are the parameters of the generator and the discriminator, respectively. $q(x,t,\mathbf{u})=p_{\mathbf{u}}(\mathbf{u}|x,t)q(x,t)$ is the joint distribution of the spatial-temporal coordinates and the output $\mathbf{u}$, and $q(x,t)$ is the marginal distribution of the spatial-temporal coordinates. $p_{\mathbf{z}}(\mathbf{z})$ is a standard normally distributed random variable.

With the physics introduced into the generator, the loss functions of the generator $G_{\theta}$ and the discriminator $D_{\phi}$ can be decoupled and defined below, respectively:

where $\hat{\mathbf{u}}_{o}^{(i)}=G_{\theta}(x_{o}^{(i)},t_{o}^{(i)},\mathbf{z}^{(i)})$ and $\hat{\mathbf{u}}_{c}^{(j)}=G_{\theta}(x_{c}^{(j)},t_{c}^{(j)},\mathbf{z}^{(j)})$ are predictions of the generator on observation and collocation points, respectively. Note that ${\cal L}_{D}$ only depends on the observation points, ${\cal L}_{phy}$ only depends on the collocation points, and ${\cal L}_{G}$ depends on both the observation and the collocation points.

In this section, we will introduce how PhysGANs are introduced to solve the UQ problem with latent variables.

To validate the performance and effectiveness of our proposed model and algorithm, in this section, we will apply LWR-PhysGAN and ARZ-PhysGAN to the real-world dataset.

This paper proposes PhysGANs to quantify the uncertainty in the TSE problem. The effectiveness of the proposed model is verified by conducting experiments on the real-world NGSIM dataset, where the ARZ-PhysGAN has the best performance, followed by the LWR-PhysGAN. In summary, we show that the GAN model can achieve an enhanced prediction accuracy in terms of the RE and KL divergence if physics information is used to guide the training. We also show that the ARZ-PhysGAN outperforms the LWR-PhysGAN because the ARZ model can better capture the real-world traffic dynamics.

This work can be further improved in two directions. First, although adding physics can help stabilize the training process, GAN is still hard to train due to the min-max game. One possible solution is to use the normalizing flow [24] as the generator, which can explicitly compute the data likelihood and thus the discriminator is not needed. Second, the proposed model needs to be re-trained if applied to other roads or to the same road but within a new time slot, which limits the real-world application of the proposed model. Thus, the model generalizability for unseen scenarios is worth investigating.