17(2,0.1) This preprint is originally revised and extended from its previous
version (https://arxiv.org/abs/2101.06580v1), so there may exist some
overlaps. However, due to significant differences, both are kept separately on
ArXiv. It is recommended to read and make use of these two preprints
complementarily.  Please cite this paper as:
Shi, Rongye, et al.“A Physics-Informed Deep Learning Paradigm for Traffic
State and Fundamental Diagram Estimation.” IEEE Transactions on Intelligent
Transportation Systems (2021). DOI: 10.1109/TITS.2021.3106259

# A Physics-Informed Deep Learning Paradigm for Traffic State and Fundamental
Diagram Estimation

Rongye Shi, Zhaobin Mo, Kuang Huang, Xuan Di, and Qiang Du Manuscript received
March 19, 2020; revised November 9, 2020, January 23, 2021, June 5, 2021 and
August 17, 2021; accepted August 17, 2021. (Corresponding author: Xuan
Di.)Rongye Shi, Zhaobin Mo, and Xuan Di are with the Department of Civil
Engineering and Engineering Mechanics, Columbia University, New York, NY,
10027 USA (e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>sharon.di@columbia.edu).Kuang Huang and Qiang Du are with the Department of
Applied Physics and Applied Mathematics, Columbia University, New York, NY,
10027 USA (e-mail<EMAIL_ADDRESS>qd2125@columbia.edu).Xuan Di and Qiang
Du are also with the Data Science Institute, Columbia University, New York,
NY, 10027 USA.

###### Abstract

Traffic state estimation (TSE) bifurcates into two categories, model-driven
and data-driven (e.g., machine learning, ML), while each suffers from either
deficient physics or small data. To mitigate these limitations, recent studies
introduced a hybrid paradigm, physics-informed deep learning (PIDL), which
contains both model-driven and data-driven components. This paper contributes
an improved version, called physics-informed deep learning with a fundamental
diagram learner (PIDL+FDL), which integrates ML terms into the model-driven
component to learn a functional form of a fundamental diagram (FD), i.e., a
mapping from traffic density to flow or velocity. The proposed PIDL+FDL has
the advantages of performing the TSE learning, model parameter identification,
and FD estimation simultaneously. We demonstrate the use of PIDL+FDL to solve
popular first-order and second-order traffic flow models and reconstruct the
FD relation as well as model parameters that are outside the FD terms. We then
evaluate the PIDL+FDL-based TSE using the Next Generation SIMulation (NGSIM)
dataset. The experimental results show the superiority of the PIDL+FDL in
terms of improved estimation accuracy and data efficiency over advanced
baseline TSE methods, and additionally, the capacity to properly learn the
unknown underlying FD relation.

###### Index Terms:

Traffic state estimation, fundamental diagram learner, physics-informed deep
learning.

##  I Introduction

Ttraffic state estimation (TSE) refers to the data mining problem of
reconstructing traffic state variables, including but not limited to flow,
density, and speed, on road segments using partially observed data from
traffic sensors [1]. TSE approaches can be briefly divided into two main
categories: model-driven and data-driven [2]. A model-driven approach is based
on a priori knowledge of traffic dynamics, usually described by a physical
model, e.g., the Lighthill-Whitham-Richards (LWR) model [3, 4] and Aw-Rascle-
Zhang (ARZ) model [5, 6], to estimate the traffic state using partial
observation. It assumes the model to be representative of the real-world
traffic dynamics such that the unobserved values can be properly added using
the model with small data. The disadvantage is that existing models, which are
provided by different modelers, may only capture limited dynamics of the real-
world traffic, resulting in low-quality estimation in the case of
inappropriately-chosen models and poor model calibrations. Paradoxically,
sometimes, verifying or calibrating a model requires a large amount of
observed data, undermining the data efficiency of model-driven approaches.

A data-driven approach is to infer traffic states based on the dependence
learned from historical data using statistical or machine learning (ML)
methods. Approaches of this type do not use any explicit traffic models or
other theoretical assumptions, and can be treated as a “black box” with no
interpretable and deductive insights. The disadvantage is that in order to
maintain a good generalizable inference to long-term unobserved values,
massive and representative historical data are a prerequisite, leading to high
demands on data collection infrastructure and enormous installation-
maintenance costs.

To mitigate the limitations of the above-mentioned TSE approaches, hybrid TSE
methods are introduced, which integrate the traffic knowledge in the form of
traffic flow models to ML models for TSE. The hybrid methods based on the
learning paradigm of physics-informed deep learning (PIDL) are gaining
increasing attentions in recently years, and is the focus of this paper. PIDL
contains both a model-driven component (a physics-informed neural network for
regularization) and a data-driven component (a physics-uninformed neural
network for estimation), making possible the integration of the strengths of
both model-driven and data-driven approaches while overcoming the weaknesses
of either.

Despite that the addition of physics could guide the training of PIDL
efficiently, complicated mathematical formulas could instead make the PIDL
difficult to train. There are many theoretical attempts made to add
sophistication (usually in the form of complicated terms) to the FD relation
for an improved description of the dynamics. To balance the sophistication and
trainability of encoding physics, this paper explores a promising direction by
approximating the FD relation with an ML surrogate, instead of hard-encoding
an FD equation. Following this direction, we introduce an improved PIDL
paradigm, called physics-informed deep learning with a fundamental diagram
learner (PIDL+FDL), which integrates an ML surrogate (e.g., an NN) into the
model-driven component to represent the fundamental diagram (FD) and estimate
the FD relation. We focus on highway TSE with observed data from loop
detectors, using traffic density or velocity as traffic variables. Our
contributions are:

  * •

We propose the PIDL+FDL-based TSE method that possesses advantages to

  * -

Perform the TSE with improved estimation accuracy: A proper integration of ML
surrogates may avoid directly encoding the complicated terms in PIDL and trade
off between the sophistication of the model-driven aspect of PIDL and the
training flexibility, making the framework a better fit to the TSE problem;

  * -

Perform the FD estimation: The PIDL+FDL uses an ML surrogate to directly learn
the underlying FD relation without any FD output measurements, i.e., the ML
surrogate is purely trained under physical regularization from PIDL, making it
more likely to learn a suitable relation along with the TSE training. It can
also get around the calibration of parameters inside the FD equation;

  * -

Perform the model parameter identification: For a complete traffic model
reconstruction, in addition to the FD estimation, there may exist model
parameters outside the FD terms to be learned. The proposed PIDL+FDL conducts
the model parameter identification jointly.

  * •

We validate the PIDL+FDL performance with both numerical experiments and real-
world data: To demonstrate the strengths of the PIDL+FDL, we design the
PIDL+FDL architectures for the traffic dynamics governed by the Greenshields-
based LWR and ARZ models, respectively. Additionally, experiments using the
real-world data, the Next Generation SIMulation (NGSIM) dataset, are
conducted. The experimental results show the advantages of PIDL in terms of
estimation accuracy and data efficiency over baselines and the capacity to
properly estimate the FD relation and model parameters.

The rest of this paper is organized as follows. Section II briefs related work
on TSE and PIDL. Section III formalizes the PIDL+FDL framework for TSE.
Sections IV and V detail the designs and experiments of PIDL+FDL for
Greenshields-based LWR and Greenshields-based ARZ, respectively. Section VI
evaluates the PIDL+FDL on NGSIM data over baselines. Section VII concludes our
work.

##  II Related Work of Traffic State Estimation

Most model-driven estimation approaches are data assimilation (DA) based,
which find “the most likely state,” allowing observation to correct models’
prediction. Popular examples include the Kalman filter (KF) and its variants
[7, 8]. Other than KF-like methods, particle filter (PF) [9] with improved
nonlinear representation, adaptive smoothing filter (ASF) [10] for combining
multiple sensing data, were proposed to improve and extend different aspects
of the TSE process. In data-driven approaches, to handle complicated traffic
data, ML approaches were involved, including long short term memory (LSTM) and
deep embedded models [11, 12].

A paradigm integrating physics to ML has gained increasing interests recently.
Yuan et al [13] proposed a physics regularized Gaussian processfor macroscopic
traffic flow modeling and TSE. The hybrid methods using the PIDL framework
[14, 15] is becoming an active field. Huang et al [16] studied the use of PIDL
to encode the Greenshields-based LWR and validated it in the loop detector
scenarios using SUMO simulated data. Barreau et al [17, 18, 19] studied the
probe vehicle sensors and developed coupled micro-macro models for PIDL to
perform TSE. Shi et al [20] extended the PIDL-based TSE to the second-order
ARZ with observed data from both loop detectors and probe vehicles.

We want to highlight that, as to model reconstruction, which is another
feature of PIDL-based TSE, this paper only assumes a traffic flow conservation
equation and optionally, a momentum equation for the velocity field, without
specifying any mathematical relation between traffic quantities. While [17,
19] directly fit a velocity function using measured density and velocity from
probe vehicles before or during the PIDL training, we, in this paper, focus on
a more general case, where the output of the FD function is unobserved from
sensors, and the end-to-end FD relation is learned directly using ML
surrogates under the PIDL framework. In summary, this paper contributes to the
trend of developing hybrid methods for TSE and model reconstruction,
especially with both FD estimation and model parameter identification
involved.

##  III Mathematical Setting for PIDL+FDL

This section introduces the PIDL+FDL framework in the context of TSE at a high
level.

###  III-A PIDL for TSE

Consider a traffic flow dynamics of a road segment that is governed by a set
of non-linear equations (e.g., partial differential equations, PDEs):

$\mathcal{N}(\boldsymbol{M}(t,x),\boldsymbol{Q};\boldsymbol{\lambda})=\boldsymbol{0},x\in[0,L],t\in[0,T],$
(1)

where $L\in\mathbb{R}^{+}$, $T\in\mathbb{R}^{+}$. We use bold symbols to
denote vectors by default. The operator $\mathcal{N}$ contains the governing
non-linear equations of the traffic flow dynamics, while $\boldsymbol{M}(t,x)$
contains the traffic state variables, such as the traffic density $\rho(t,x)$
and velocity $u(t,x)$. $\boldsymbol{\lambda}$ contains the model parameters.
The model includes intermediate unobserved traffic variables $\boldsymbol{Q}$
that have some hidden relationship with $\boldsymbol{M}(t,x)$. Thus, the
dynamics can be represented by

$\mathcal{N}(\boldsymbol{M},\boldsymbol{Q}(\boldsymbol{M});\boldsymbol{\lambda})=\boldsymbol{0},$
(2)

and $\boldsymbol{M}$ stands for $\boldsymbol{M}(t,x)$. For general discussion,
the values of $\boldsymbol{Q}$ are not assumed to be directly observable, and
the relation is either unknown or deduced based on assumptions which may be
deficient. The TSE problem is to reconstruct the traffic states
$\boldsymbol{M}$ at each point $(t,x)$ in a continuous domain from partial
observation of $\boldsymbol{M}$. Accordingly, the continuous spatio-temporal
domain $D$ is a set of points: $D=\\{(t,x)|\forall t\in[0,T],x\in[0,L]\\}$. We
represent this continuous domain in a discrete manner using grid points $G\in
D$ that are evenly deployed throughout the domain. We define the set of grid
points as $G=\\{(t^{(r)},x^{(r)})|r=1,..,N_{g}\\}$. The total number of grid
points, $N_{g}$, controls the fine-grained level of $G$ as a representation of
the continuous domain.

PIDL approximates $\boldsymbol{M}(t,x)$ using a neural network with time $t$
and location $x$ as its inputs. This neural network is called physics-
uninformed neural network (PUNN) (or estimation network in our TSE study),
which is parameterized by $\theta$. We denote the approximation of
$\boldsymbol{M}(t,x)$ from PUNN as $\hat{\boldsymbol{M}}(t,x;\theta)$. When
$\mathcal{N}$, $\boldsymbol{Q}$ and $\boldsymbol{\lambda}$ are known, during
the learning phase of PUNN (i.e., to find the optimal $\theta$ for PUNN), the
following equiation defines the residual values of the approximation
$\hat{\boldsymbol{M}}(t,x;\theta)$:

$\boldsymbol{\hat{f}}(t,x;\theta):=\mathcal{N}(\hat{\boldsymbol{M}}(t,x;\theta),\boldsymbol{Q}(\hat{\boldsymbol{M}}(t,x;\theta));\boldsymbol{\lambda}),$
(3)

which is designed according to the traffic flow model in Eq. (2). The
calculation of residual $\boldsymbol{\hat{f}}(t,x;\theta)$ is done by a
physics-informed neural network (PINN). This network can compute
$\boldsymbol{\hat{f}}(t,x;\theta)$ directly using
$\hat{\boldsymbol{M}}(t,x;\theta)$, the output of PUNN, as its input. When
$\hat{\boldsymbol{M}}(t,x;\theta)$ is closer to the true value
$\boldsymbol{M}(t,x)$, the residual $\boldsymbol{\hat{f}}$ will be closer to
$\boldsymbol{0}$. PINN introduces no new parameters, and thus, shares the same
$\theta$ of PUNN.

In most cases, even the model is given, the model parameters
$\boldsymbol{\lambda}$ are unknown and can be made as learning variables in
PINN for model parameter identification. The residual $\hat{\boldsymbol{f}}$
is then redefined as

$\boldsymbol{\hat{f}}(t,x;\theta,\boldsymbol{\lambda}):=\mathcal{N}(\hat{\boldsymbol{M}}(t,x;\theta),\boldsymbol{Q}(\hat{\boldsymbol{M}}(t,x;\theta));\boldsymbol{\lambda}).$
(4)

This paper assumes unknown model parameters by default.

In PINN, $\boldsymbol{\hat{f}}(t,x;\theta,\boldsymbol{\lambda})$ is calculated
by automatic differentiation technique, which can be done by the function
tf.gradient in Tensorflow. The activation functions and the connecting
structure of neurons in PINN are designed to conduct the differential
operation in Eq. (4). We would like to emphasize that, the connecting weights
in PINN have fixed values which are determined by the traffic flow model and
$\boldsymbol{\lambda}$ are encoded as learning variables. Thus, the residual
$\boldsymbol{\hat{f}}$ is parameterized by both $\theta$ and
$\boldsymbol{\lambda}$.

The training data for PIDL consist of (1) observation points
$O=\\{(t^{(i)}_{o},x^{(i)}_{o})|i=1,...,N_{o}\\}$, (2) target values
$P=\\{\boldsymbol{M}^{(i)}|i=1,...,N_{o}\\}$ (i.e., the true traffic states at
the observation points), and (3) auxiliary points
$A=\\{(t^{(j)}_{a},x^{(j)}_{a})|j=1,...,N_{a}\\}$. $i$ and $j$ are the indexes
of observation points and auxiliary points, respectively. One target value is
associated with one observation point, and thus, $O$ and $P$ have the same
indexing system (indexed by $i$). This paper uses the term, observed data, to
denote $\\{O,P\\}$. Both $O$ and $A$ are subsets of the grid points $G$ (i.e.,
$O\in G$ and $A\in G$).

Observation points $O$ are limited to the time and locations that traffic
sensors can visit and record. In contrast, auxiliary points $A$ have neither
measurement requirements nor location limitations, and the number of $A$ is
controllable. $A$ are used for regularization purposes, and this is why they
are called “auxiliary”. To train a PUNN for TSE, the loss is used:

$\begin{split}\begin{gathered}Loss_{\theta,\boldsymbol{\lambda}}=\alpha\cdot
MSE_{o}+\beta\cdot MSE_{a}\\\
=\alpha\cdot\underbrace{\frac{1}{N_{o}}\sum\limits_{i=1}^{N_{o}}|\hat{\boldsymbol{M}}(t^{(i)}_{o},x^{(i)}_{o};\theta)-\boldsymbol{M}^{(i)}|^{2}}_{data\
discrepancy}\\\
+\beta\cdot\underbrace{\frac{1}{N_{a}}\sum\limits_{j=1}^{N_{a}}|{\boldsymbol{\hat{f}}}(t^{(j)}_{a},x^{(j)}_{a};\theta,\boldsymbol{\lambda})|^{2}}_{physical\
discrepancy}.\end{gathered}\end{split}$ (5)

where $\alpha$ and $\beta$ are hyperparameters for balancing the contribution
to the loss made by data discrepancy and physical discrepancy, respectively.
The data discrepancy is defined as the mean square error (MSE) between
approximation $\hat{\boldsymbol{M}}$ on $O$ and target values $P$. The
physical discrepancy is the MSE between residual values on $A$ and
$\boldsymbol{0}$, quantifying the extent to which the approximation deviates
from the traffic model.

Given the training data, we apply neural network training algorithms to solve
$(\theta^{*},\boldsymbol{\lambda}^{*})=\mathrm{argmin}_{\theta,\boldsymbol{\lambda}}\
Loss_{\theta,\boldsymbol{\lambda}}$. Then, the
$\boldsymbol{\lambda}^{*}$-parameterized traffic flow model of Eq. (4) is the
most likely physics that generates the observed data, and the
$\theta^{*}$-parameterized PUNN can then be used to approximate the traffic
states on $G$, which are consistent with the reconstructed traffic flow model
in Eq. (2).

###  III-B PIDL + FDL for TSE

As has been discussed previously, the PIDL-based TSE methods may perform
poorly when informed by a highly sophisticated traffic flow model. This is
because the models may contain complicated terms that are unfriendly to
differentiation-based learning (e.g., square root operators of learning
variables in the denominator, etc.), making the training and performance very
sensitive to the structural design of PINN. Many efforts such as variable
conversion, decomposition and factorization need to be made to have the PINN
trainable and the loss to converge. In our framework of Eq.(2), these
“unfriendly” terms can be contained as part of the hidden relation
$\boldsymbol{Q}$. To address the issues of PIDL-based TSE, we propose to use
an ML surrogate $\boldsymbol{\hat{Q}}$ to directly represent the
$\boldsymbol{Q}$ and learn the relation under the PIDL framework, instead of
hard-encoding a complicated term in PINN.

The advantages of properly introducing an ML surrogate of $\boldsymbol{Q}$ are
two-fold: (1) An ML term is usually differentiation-friendly, giving the PIDL
more flexibility to achieve an improved TSE accuracy. (2) No assumptions are
made to the hidden relation, and it is possible to learn a more suitable
$\boldsymbol{\hat{Q}}$ when trained under the physical regularization from
PINN.

This paper focuses on one kind of hidden relationships, the fundamental
diagram (FD), and the corresponding learning paradigm is called PIDL with an
FD Learner (PIDL+FDL). Specifically, the FD Learner, formalized as
$\boldsymbol{\hat{Q}}(\boldsymbol{\hat{M}};\omega)$, can be designed as a
neural network parameterized by $\omega$ to represent the unknown FD relation,
which takes the estimated traffic variables $\boldsymbol{\hat{M}}(t,x;\theta)$
as its input. The residual of Eq.(4) is redefined as the following

$\boldsymbol{\hat{f}}(t,x;\theta,\omega,\boldsymbol{\lambda}):=\mathcal{N}(\hat{\boldsymbol{M}}(t,x;\theta),\hat{\boldsymbol{Q}}(\hat{\boldsymbol{M}}(t,x;\theta);\omega);\boldsymbol{\lambda}).$
(6)

The loss function becomes

$\begin{split}\begin{gathered}Loss_{\theta,\omega,\boldsymbol{\lambda}}=\alpha\cdot
MSE_{o}+\beta\cdot MSE_{a}\\\
=\alpha\cdot\frac{1}{N_{o}}\sum\limits_{i=1}^{N_{o}}|\hat{\boldsymbol{M}}(t^{(i)}_{o},x^{(i)}_{o};\theta)-\boldsymbol{M}^{(i)}|^{2}\\\
+\beta\cdot\frac{1}{N_{a}}\sum\limits_{j=1}^{N_{a}}|{\boldsymbol{\hat{f}}}(t^{(j)}_{a},x^{(j)}_{a};\theta,\omega,\boldsymbol{\lambda})|^{2}.\end{gathered}\end{split}$
(7)

Using the training data, we apply neural network training algorithms to solve
$(\theta^{*},\omega^{*},\boldsymbol{\lambda}^{*})=\mathrm{argmin}_{\theta,\omega,\boldsymbol{\lambda}}\
Loss_{\theta,\omega,\boldsymbol{\lambda}}$. Then, in addition to TSE learning
and model parameter identification, the FD estimation is conducted
automatically. The $\omega^{*}$-parameterized $\boldsymbol{\hat{Q}}$ can be
used to represent the unknown hidden fundamental diagram relation. Note that
the values of $\boldsymbol{Q}$ are not assumed to be observable, and thus, are
not part of the data (i.e., to directly learn the $\boldsymbol{\hat{Q}}$ from
data cannot apply).

In some cases, the curve of the learned $\boldsymbol{\hat{Q}}$ may present
abnormal shapes on edge conditions. To mitigate this, one can encode prior
knowledge into the loss as an additional regularization term
$Reg(\boldsymbol{\hat{Q}})$ to reshape the FD. As an example, we can use
$\boldsymbol{Q}$ to represent the density-flow relation, i.e., the flux
function (one typical kind of FD), mapping the density $\rho$ to the flow
value, which is denoted as a scalar $Q(\rho)$. Existing theoretical works
usually assume $Q$ to be concave with respect to the traffic density $\rho$.
To impose the concavity property, we design the following regularization term:

$Reg(Q)=\int_{a}^{b}{\max(0,\frac{{\partial^{2}Q(\rho)}}{{\partial\rho^{2}}})d\rho},$
(8)

where the hyperparameters $a$ and $b$ determine the interval of $\rho$ on
which the reshaping takes effects without interfering the learning on other
regions. We propose this design because most abnormal shapes only occur on
edge region and it is not necessary to regularize over the whole traffic
density domain. We apply
$Loss_{\theta,\omega,\boldsymbol{\lambda}}=\alpha\cdot MSE_{o}+\beta\cdot
MSE_{a}+\xi\cdot Reg(\hat{Q})$ in the learning phase and properly reshape the
learned FD curves.

##  IV PIDL+FDL for Greenshields-Based LWR

The first numerical example aims to show the capability of our method to
estimate the traffic dynamics governed by the LWR model with a Greenshields
flux function.

Define flow rate $Q$ (a scalar) to be the number of vehicles passing a
specific position on the road per unit time, and traffic density $\rho$ to be
the average number of vehicles per unit length of the road. The traffic flux
$Q(\rho)$ describes $Q$ as a function of $\rho$, which is the FD relation of
interest in this numerical example. We treat $\rho$ as the basic traffic state
variable to estimate. Greenshields flux is a basic and popular choice of
$Q(\rho)$, which is defined as $Q(\rho)=\rho u_{max}(1-\rho/\rho_{max})$,
where $u_{max}$ and $\rho_{max}$ are maximum velocity and maximum (jam)
density, respectively. This flux function has a quadratic form with two
coefficients $u_{max}$ and $\rho_{max}$.

The LWR model [3, 4] describes the macroscopic traffic flow dynamics as
$\rho_{t}+(Q(\rho))_{x}=0$, which is derived from a conservation law of
vehicles. In order to reproduce more phenomena in observed traffic data, such
as delayed driving behaviors due to drivers’ reaction time, diffusively
corrected LWRs were introduced, by adding a diffusion term, containing a
second-order derivative $\rho_{xx}$. We focus on one version of the
diffusively corrected LWRs: $\rho_{t}+(Q(\rho))_{x}=\epsilon\rho_{xx}$, where
$\epsilon$ is the diffusion coefficient.

In this section, we study the Greenshields-based LWR traffic flow model of a
“ring road”:

$\begin{split}\begin{gathered}\rho_{t}+(Q(\rho))_{x}=\epsilon\rho_{xx},\
t\in[0,3],\ x\in[0,1],\\\ Q(\rho)=\rho\cdot
u_{max}\Bigl{(}1-\frac{\rho}{\rho_{max}}\Bigl{)}\ \ \ (FD\ relation),\\\
\rho(t,0)=\rho(t,1)\ \ \ (boundary\ condition\ 1),\\\
\rho_{x}(t,0)=\rho_{x}(t,1)\ \ \ (boundary\ condition\
2),\end{gathered}\end{split}$ (9)

where $\rho_{max}=1.0$, $u_{max}=1.0$, and $\epsilon=0.005$. $\rho_{max}$ and
$u_{max}$ are usually determined by physical restrictions of the road and
vehicles.

Given the bell-shaped initial $0.1+0.8e^{-(\frac{x-0.5}{0.2})^{2}}$,
$x\in[0,1]$, we apply the Godunov scheme to solve Eqs. (9) on 960 (time)
$\times$ 240 (space) grid points $G$ evenly deployed throughout the
$[0,3]\times[0,1]$ domain. In this case, the total number of grid points $G$
is $N_{g}=$960$\times$240\. The numerical solution is shown in Fig. 2 (see the
heat map background). From the figure, we can visualize the dynamics as
follows: At $t=0$, there is a peak density at the center of the road segment,
and this peak evolves to propagate along the direction of $x$, which is known
as the phenomenon of traffic shockwave. Since this is a ring road, the
shockwave reaching $x=1$ continues at $x=0$. This numerical solution of the
Greenshields-based LWR model is treated as the ground-truth traffic density.
We will apply a PIDL+FDL-based approach method to estimate the entire traffic
density field using observed data as well as to estimate the FD relation and
model parameters.

###  IV-A PIDL+FDL Architecture Design

The authors’ previous work [20] has shown the capacity of PIDL to perform both
TSE and model parameter identification when the closed traffic flow model is
given. Here we are only given the knowledge of conservation law and boundary
conditions, i.e., the FD relation is unknown and no direct observation of $Q$
is available.

We employ a neural network $\hat{Q}(\cdot;\omega)$ to estimate the traffic
flow from the traffic density and to represent the FD relation of interest.
Based on Eqs. (9), we define the residual value of PUNN’s traffic density
estimation $\hat{\rho}(t,x;\theta)$ as

$\hat{f}(t,x;\theta,\omega,\epsilon):=\hat{\rho}_{t}(t,x;\theta)+(\hat{Q}(\hat{\rho}(t,x;\theta);\omega))_{x}-\epsilon\hat{\rho}_{xx}(t,x;\theta).$
(10)

Note that the parameter $\boldsymbol{\lambda}$ contains the coefficient
$\epsilon$ only.

Given the definition of $\hat{f}$, the corresponding PIDL+FDL architecture is
shown in Fig. 1. This architecture consists of a PUNN for traffic density
estimation, followed by a PINN+FD Learner for calculating the residual Eq.
(10). The PUNN parameterized by $\theta$ is designed as a fully-connected
feedforward neural network with 8 hidden layers and 20 hidden nodes in each
hidden layer. Hyperbolic tangent function (tanh) is used as the activation
function for each hidden neuron in PUNN. In contrast, in PINN, connecting
weights are fixed and the activation function of each node is designed to
conduct specific nonlinear operation for calculating an intermediate (hidden)
value of $\hat{f}$. The flow value is calculated by a separate neural network
$\hat{Q}(\hat{\rho};\omega)$ with two hidden layers and 20 hidden nodes for
each. The model parameter $\epsilon$ is held by a variable node (blue
rectangular nodes).

To customize the training of PIDL+FDL to Eqs. (9), in addition to the training
data $O$, $P$ and $A$ defined in Section III-A, we need to introduce  boundary
auxiliary points
$B=\\{(t^{(k)}_{b},0)|k=1,...,N_{b}\\}\cup\\{(t^{(k)}_{b},1)|k=1,...,N_{b}\\}$,
for learning the two boundary conditions in Eqs. (9).

For experiments of state estimation with both parameter identification and FD
estimation, we design the following loss

$\begin{split}Loss_{\theta,\omega,\epsilon}&=\alpha\cdot MSE_{o}+\beta\cdot
MSE_{a}+\gamma\cdot B1+\eta\cdot B2\\\
&=\frac{\alpha}{N_{o}}\sum\limits_{i=1}^{N_{o}}|\hat{\rho}(t^{(i)}_{o},x^{(i)}_{o};\theta)-\rho^{(i)}|^{2}\\\
&+\frac{\beta}{N_{a}}\sum\limits_{j=1}^{N_{a}}|\hat{f}(t^{(j)}_{a},x^{(j)}_{a};\theta,\omega,\epsilon)|^{2}\\\
&+\frac{\gamma}{N_{b}}\sum\limits_{k=1}^{N_{b}}|\hat{\rho}(t^{(k)}_{b},0;\theta)-\hat{\rho}(t^{(k)}_{b},1;\theta)|^{2}\\\
&+\frac{\eta}{N_{b}}\sum\limits_{k=1}^{N_{b}}|\hat{\rho}_{x}(t^{(k)}_{b},0;\theta)-\hat{\rho}_{x}(t^{(k)}_{b},1;\theta)|^{2}.\end{split}$
(11)

$B1$, scaled by $\gamma$, is the MSE between estimations at the two boundaries
$x=0$ and $x=1$. $B2$, scaled by $\eta$, quantifies the difference of first
order derivatives at the two boundaries.

Figure 1: PIDL+FDL architecture for first-order traffic dynamics, consisting
of a PUNN for traffic density estimation and a PINN+FD Learner for calculating
the residual Eq. (10). The model parameter $\epsilon$ is held by a variable
node (blue rectangular nodes). All connecting weights are predefined and fixed
in PINN.

###  IV-B TSE+FDL using Observation from Loop Detectors

We justify the capacity of PIDL+FDL in Fig. 1 for estimating the traffic
density field using observation from loop detectors, i.e., only the traffic
density at certain locations where loop detectors are installed can be
observed. By default, loop detectors are evenly located along the road. To be
specific, the grid points at certain locations are used as the observation
points $O$, and their corresponding densities constitute the target values $P$
for training. There are $N_{a}=100,000$ auxiliary points in $A$ randomly
selected from grid points $G$. $N_{b}=650$ out of 960 grid time points (i.e.,
the time points on the temporal dimension of $G$) are randomly selected to
create boundary auxiliary points $B$. A sparse version of the deployment of
$O$, $A$ and $B$ in the spatio-temporal domain is shown in Fig. 2. Each
observation point is associated with a target value in $P$. Note $O$, $A$ and
$B$ are all subsets of $G$.

Figure 2: A sparse presentation of the deployment of observation points $O$ at
loop detectors, auxiliary points $A$ randomly selected from $G$, and boundary
auxiliary points $B$ deployed at the boundaries $x=0$ and $x=1$ for certain
time points. The heatmap is the numerical solution of Eqs. (9) using the
Godunov scheme. We treat the numerical solution as the ground truth.

We train the proposed PIDL+FDL on an NVIDIA Titan RTX GPU with 24 GB memory.
By default, we use the $\mathbbm{L}^{2}$ relative error on $G$ to quantify the
estimation error of the entire domain:

$Err(\hat{\rho},\rho)=\frac{\sqrt{\sum_{r=1}^{N_{g}}\bigl{|}\hat{\rho}(t^{(r)},x^{(r)};\theta)-\rho(t^{(r)},x^{(r)})\bigl{|}^{2}}}{\sqrt{\sum_{r=1}^{N_{g}}\bigl{|}\rho(t^{(r)},x^{(r)})\bigl{|}^{2}}}.$
(12)

The reason for choosing the $\mathbbm{L}^{2}$ relative error is to normalize
the estimation inaccuracy, mitigating the influence from the scale of true
density values. One remark is that there are some TSE methods (e.g., non
parametric ones) that do not perform any estimation on the observation points
and directly use the target values there. For these cases, the observation
points will be removed from $G$ before calculating Eq.(12).

We use the Xavier uniform initializer to initialize $\theta$ of PUNN and
$\omega$ of FD Learner (FDL). This neural network initialization method takes
the number of a layer’s incoming and outgoing network connections into account
when initializing the weights of that layer, which may lead to a good
convergence. The $\epsilon$ is initialized at 0. Then, we train the PUNN, FDL
and $\epsilon$ through the PIDL+FDL architecture using a popular stochastic
gradient descent algorithm, the Adam optimizer, for a rough training. A
follow-up fine-grained training is done by the L-BFGS optimizer [21] for
stabilizing the convergence, and the process terminates until the loss change
of two consecutive steps is no larger than $10^{-16}$. This training process
converges to a local optimum $\theta^{*}$, $\omega^{*}$ and $\epsilon^{*}$
that minimize the loss in Eq. (11).

We would like to clarify that in this paper, the training data are the
observed data from detectors, i.e., the traffic states on the points at
certain locations where loops are equipped.

Figure 3: Top: Estimation of the traffic density dynamics
$\hat{\rho}(t,x;\theta^{*})$ on grid points $G$ in the domain using the
trained PUNN. Bottom: Snapshots of estimated and true traffic density at
certain time points.

The results of applying the PIDL+FDL with 4 loops to the Greenshields-based
LWR dynamics is presented in Fig. 3, where PUNN is parameterized by the
optimal $\theta^{*}$. As shown in Fig. 3, the estimation
$\hat{\rho}(t,x;\theta^{*})$ is visually the same as the true dynamics
$\rho(t,x)$ in Fig. 2. By looking into the estimated and true traffic density
over $x$ at certain time points, there is a good agreement between two traffic
density curves. The $\mathbbm{L}^{2}$ estimation error $Err(\hat{\rho},\rho)$
is $1.287\times 10^{-2}$. Empirically, the difference cannot be visually
distinguished when the estimation error is smaller than $6\times 10^{-2}$.

TABLE I: Performance of PIDL+FDL on the LWR Dynamics

$loops$ | 2 | 3 | 4 | 5
---|---|---|---|---
$Err(\hat{\rho},\rho)$ | 0.6021 | 0.03327 | 0.01287 | 0.004646
$\epsilon^{*}$ | 3.12573 | 0.00495 | 0.00506 | 0.00509

  * •

$loops$ stands for the number of loop detectors. $\epsilon^{*}$ is the
estimated diffusion coefficient. Note the true $\epsilon=0.005$.

We change the number of loop detectors. For a fixed number of loop detectors,
we use grid search for hyperparameter tuning by default. Specifically, since
Adam optimizer is scale invariant, we fix the hyperparameter $\alpha$ to 100
and tune the other hyperparameters from [1, 10, 50, 100, 150, 200] with some
follow-up fine tuning. The minimal-achievable estimation errors of PIDL+FDL
over the numbers of loop detectors are presented in Table I. From the table,
we can see that the traffic density estimation errors improve as the number of
loop detectors increases. When more than two loop detectors are used, the
model parameters to be learned are able to converge to the true parameters
$\epsilon$. Specifically, with three loop detectors, in addition to a good
traffic density estimation error of 3.327$\times 10^{-2}$, the model parameter
converges to $\epsilon^{*}=0.00495$, which is very close to the true value
0.005. The results demonstrate that PIDL+FDL method can handle both TSE and
model parameter identification with three loop detectors for the traffic
dynamics of the Greenshields-based LWR.

In this experiment, the proposed method can reconstruct the exact Greenshields
FD when 5 loop detectors are used. The results are meaningful because neither
any assumptions on the FD relation are made nor the flow values are observed
directly. In addition to traffic density estimation and model parameter
identification, the PIDL+FDL is able to make full use of the conservation law
and boundary conditions to retrieve the density-flow relation automatically.
For the visualization of the results, please refer to Figure 4 in the
supplementary material in [22].

##  V PIDL+FDL for Greenshields-Based ARZ

The second numerical example aims to show the capacity of the proposed method
to handle the traffic dynamics governed by the Greenshields-based ARZ, a
second-order traffic flow model with both traffic density $\rho$ and velocity
$u$ as the traffic state variables.

An ARZ model involves both a conservation law of vehicles and a momentum
equation on velocity. Specifically, we study the following traffic flow
dynamics of a “ring road” in $t\in[0,3]$, $x\in[0,1]$:

$\begin{split}\begin{gathered}\rho_{t}+(\rho u)_{x}=0,\\\
(u+h(\rho))_{t}+u(u+h(\rho))_{x}=(U_{eq}(\rho)-u)/\tau,\\\
h(\rho)=U_{eq}(0)-U_{eq}(\rho)\ \ \ (traffic\ \ pressure),\\\
U_{eq}(\rho)=u_{max}(1-\rho/\rho_{max})\ \ \ (equilibrium\ \ speed),\\\
\rho(t,0)=\rho(t,1),u(t,0)=u(t,1)\ (boundary\
cond.),\end{gathered}\end{split}$ (13)

where we set the parameters irregularly as $\rho_{max}=1.13$, $u_{max}=1.02$,
and $\tau=0.02$. $U_{eq}$ is the equilibrium velocity, $h(\rho)$ defines the
traffic pressure and $\tau$ denotes the relaxation time scale. For more
explanations of this ARZ setting, we refer readers to our previous work in
[20].

Figure 4: a) is the bell-shaped initial $\rho$ and $u$ over $x\in[0,1]$; b)
and c) are numerical solutions for $\rho$ and $u$, respectively.

Given the bell-shaped initial of $\rho$ and $u$ as shown in Fig.4.a, we apply
the Lax–Friedrichs (LF) scheme to solve Eqs.(13) on grid $G$ with 960
(time)$\times$240 (space) points evenly deployed over the $[0,3]\times[0,1]$
domain. The LF numerical solutions of both $\rho$ and $u$ over the domain are
shown in Fig.4 as well. We treat this numerical solution as the ground-truth
to test our PIDL+FDL-based approach for the ARZ dynamics.

###  V-A PIDL+FDL Architecture Design for ARZ

We employ a neural network $\hat{U}_{eq}(\cdot;\omega)$ to estimate the
equilibrium velocity and $U_{eq}(\rho)$ is the target FD relation in this
numerical example. Based on Eqs. (13), we define the following residuals

$\begin{split}{\begin{array}[]{l}\hat{f}_{1}(t,x;\theta):=\hat{\rho}_{t}+(\hat{\rho}\hat{u})_{x},\\\
\hat{f}_{2}(t,x;\theta,\omega,\tau):=(\hat{u}+h(\hat{\rho}))_{t}+\hat{u}(\hat{u}+h(\hat{\rho}))_{x}\\\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
-(\hat{U}_{eq}(\hat{\rho};\omega)-\hat{u})/\tau,\end{array}}\end{split}$ (14)

where $\hat{\rho}$ and $\hat{u}$ are shorthands for $\hat{\rho}(t,x;\theta)$
and $\hat{u}(t,x;\theta)$, respectively outputted from a PUNN. By using the FD
surrogate $\hat{U}_{eq}$, the relaxation $\tau$ is the only model parameter to
be learned.

Given the definition of $(\hat{f}_{1}$, $\hat{f}_{2})$, the design of PINN+FDL
architecture is shown in Fig. 5. The structures of hidden layers of the PUNN
and FDL are the same with those of Section IV. For this experiment, we adjust
the learning loss as the following:

$\begin{split}\begin{gathered}Loss_{\theta,\omega,\tau}=MSE_{o}+MSE_{a}+B1\\\
=\frac{1}{N_{o}}\sum\limits_{i=1}^{N_{o}}\alpha_{1}|\hat{\rho}(t^{(i)}_{o},x^{(i)}_{o};\theta)-\rho^{(i)}|^{2}+\alpha_{2}|\hat{u}(t^{(i)}_{o},x^{(i)}_{o};\theta)-u^{(i)}|^{2}\\\
+\frac{1}{N_{a}}\sum\limits_{j=1}^{N_{a}}\beta_{1}|\hat{f_{1}}(t^{(j)}_{a},x^{(j)}_{a};\theta)|^{2}+\beta_{2}|\hat{f}_{2}(t^{(j)}_{a},x^{(j)}_{a};\theta,\omega,\tau)|^{2}\\\
+\frac{1}{N_{b}}\sum\limits_{k=1}^{N_{b}}(\gamma_{1}|\hat{\rho}(t^{(k)}_{b},0;\theta)-\hat{\rho}(t^{(k)}_{b},1;\theta)|^{2}\\\
+\gamma_{2}|\hat{u}(t^{(k)}_{b},0;\theta)-\hat{u}(t^{(k)}_{b},1;\theta)|^{2}).\end{gathered}\end{split}$
(15)

We solve
$(\theta^{*},\omega^{*},\tau^{*})=\mathrm{argmin}_{\theta,\omega,\tau}\
Loss_{\theta,\omega,\tau}$, and the results of traffic state estimation and
model parameter identification are presented in Table II. Empirically, the
difference between true and estimated values is visually indistinguishable
when the errors are smaller than 6.00 $\times 10^{-2}$ and 2.90 $\times
10^{-2}$ for density and velocity, respectively. Performances with accuracy
below these values are considered as “acceptable”. From Table II, we can
observe that the TSE performance of PIDL+FDL with more than three loops is
acceptable.

Figure 5: PIDL+FDL architecture for second-order traffic dynamics. The model
parameter $\tau$ is held by a variable node (blue rectangular node). TABLE II:
Performance of PIDL+FDL on the ARZ Dynamics

$loops$ | 2 | 3 | 4 | 5
---|---|---|---|---
$Err(\hat{\rho},\rho)$ | 0.5111 | 0.2249 | 0.04871 | 0.05243
$Err(\hat{u},u)$ | 0.1586 | 0.05914 | 0.01402 | 0.01389
$\tau^{*}$ | 0.02276 | 0.018994 | 0.019654 | 0.021619

  * •

$loops$ stands for the number of loop detectors. $\tau^{*}$ is the estimated
relaxation time. Note the true $\tau=0.02$.

The experiment also demonstrates that the FDL performance improves as the
number of loop detectors increases, and the proposed method with 4 loops and
above is able to correctly learn the Greenshields FD relation. For
visualization of the results, please refer to Figure 7 in the supplementary
material in [22].

##  VI PIDL+FDL-Based TSE on NGSIM Data

This section evaluates the PIDL+FDL-based TSE method using real-world traffic
data, the Next Generation SIMulation (NGSIM)
dataset111www.fhwa.dot.gov/publications/research/operations/07030/index.cfm,
and compares its performance to baselines.

###  VI-A NGSIM Dataset

NGSIM dataset is widely-used and contains real-world vehicle trajectories on
several road scenarios. We focus on a segment of the US Highway 101 (US 101),
monitored by a camera mounted on top of a high building on June 15, 2005. The
locations and actions of each vehicle in the monitored region for a total of
around 680 meters and 2,770 seconds were converted from camera videos.

We select the data from all the mainline lanes of the US 101 highway segment
to calculate the average traffic density for approximately every 30 meters
over a 1.5 seconds period. After preprocessing to remove the time when there
are non-monitored vehicles running on the road (at the beginning and end of
the video), there are 21 and 1770 valid cells on the spatial and temporal
dimensions, respectively. We treat the center of each cell as a grid point.
The spatio-temporal field of traffic density $\rho(t,x)$ and velocity $u(t,x)$
in the dataset can be visualized in Figure 8 in the supplementary in [22].

For TSE experiments in this section, loop detectors are used to provide
observed data with a recording frequency of 1.5 seconds. By default, they are
evenly installed on the highway segment. We assume that the loop detectors are
able to record the density and average velocity of cells on certain locations

###  VI-B TSE Methods for Real Data

We first introduce the PIDL+FDL-based methods for the real-world TSE problem,
and then, describe advanced baselines to compare with.

Figure 6:  The structure design of LWR-PIDL+FDL method for NGSIM data.

LWR-PIDL+FDL: This method is based on the PIDL+FDL encoded with the first-
order LWR, using the structure in Fig. 1 except for two modifications: 1) no
assumptions on diffusion effects are made and the traffic flow becomes
$\rho_{t}+(Q(\rho))_{x}=0$, which is commonly-used in literature [23]; and 2)
the estimation of velocity is calculated using an additional calculation node
$\hat{u}=\hat{Q}(\hat{\rho};\omega)/\hat{\rho}$. Specifically, the modified
structure of this TSE method is presented in Fig. 6. We select 80% of the grid
$G$ as the auxiliary points $A$. The loss in Eq. (11) (with the MSE on
velocity estimation added and the boundary conditions removed) is used for
training the PUNN and FDL using the observed data from loop detectors (i.e.,
both observation points $O$ and corresponding target state values $P$). After
tuning the hyperparameters with grid search, we present the minimal-achievable
estimation errors. The same for other baselines by default. Because the real
data could be noisy, leading to abnormal learned FD curves, the reshaping
regularization term in Eq. (8) is applied.

ARZ-PIDL+FDL: This method is based on the PIDL+FDL encoded with the second-
order ARZ, using the structure in Fig. 5. The Eq. (15) (with boundary
conditions removed) is applied as the loss function. Other experimental setups
are the same with those of the LWR-PIDL+FDL method.

Two-Dimensional Data Interpolation (Interp2): The two-dimensional linear
interpolation method is used as a baseline, which interpolates the traffic
states using the neighboring observed data in a linear manner.

Adaptive Smoothing (AS) Method: This method estimates the traffic state of a
cell using the sum of all the observed data weighted by some smoothing kernel
filters. We implement a generalized AS method proposed in [10] with parameters
suggested in [24].

Long Short Term Memory (LSTM) based Method: This baseline method employs the
LSTM architecture, which is customized from the LSTM-based TSE proposed by
[11]. This model can be applied to our problem by leveraging the spatial
dependency, i.e., to use the information of previous cells to estimate the
traffic density and velocity of the next cell along the spatial dimension.

Other baselines include the Pure Neural Network (NN) and the Extended Kalman
Filter (EKF) as well as the advanced PIDL-based TSE methods: LWR-based PIDL
(LWR-PIDL) and ARZ-based PIDL (ARZ-PIDL). For more descriptions regarding
these four baselines, we refer the readers to the authors’ previous work in
[20].

###  VI-C Results and Discussion

We apply PIDL+FDL-based, and baseline methods to TSE on the NGSIM dataset with
different numbers of loop detectors. The results are presented in Fig. 7.

Figure 7: TSE performance of PIDL+FDL-based methods and baselines.

From Fig. 7, we can observe that the PIDL-related methods generally perform
better than the model-driven and data-driven TSE baselines. The EKF/Interp2/AS
methods achieve better errors than the NN/LSTM methods when the number of loop
detectors are small, while the NN/LSTM methods’ performance catches up when
more loops are available.

The results are reasonable. The EKF is a model-driven approach, making
sufficient use of the traffic flow model to appropriately estimate unobserved
values when limited data are available. However, the model cannot fully
capture the complicated traffic dynamics in the real world, and as a result,
the EKF’s performance flattens out. The Interp2 is a non-parametric data-
driven method interpolating the unobserved fields using neighboring
observation in a linear manner. The AS method incorporates the characteristic
velocities of information propagation in free and congested traffic, by
skewing the principal axes of the smoothing kernel. The Interp2 and AS methods
have a relatively low complexity which can prevent over-fitting when the data
is small. However, they may not effectively handle subtle state changes in the
unobserved area due to the linearly-extrapolating nature or filtering nature,
respectively. The PIDL-based method’s errors are generally below the
baselines, because it can make efficient use of both the traffic flow model
and observed data. The ARZ-PIDL method is informed by a more advanced second-
order traffic model and its performance is superior to that of LWR-PIDL.

The PIDL+FDL-based methods can generally achieve the best estimation accuracy
and data efficiency over the above TSE baselines. The results demonstrate that
the proper integration of the NN-based FD surrogate to the PIDL can give the
learning framework more flexibility to achieve an improved TSE accuracy. One
interesting phenomenon is that the PIDL+FDL with the first-order LWR (LWR-
PIDL+FDL) can beat the one with a more sophisticated second-order ARZ model
(ARZ-PIDL+FDL). This observation supports our discussion that sophisticated
traffic models may not always lead to a better TSE performance, because the
model may contain complicated terms that makes the TSE performance sensitive
to the PINN structural design, and thus, the model becomes difficult to train.
Compared to the ARZ-PIDL+FDL, the LWR-PIDL+FDL can balance the trade-off
between the sophisticated level of PINN and the training flexibility more
properly, making it a better fit to the NGSIM scenario.

###  VI-D Discussions on Fundamental Diagram Estimation

The PIDL+FDL-based methods can further learn the hidden fundamental diagram
(FD) relation. We compare the FD curves learned via the PIDL+FDL-based methods
and PIDL-based methods in the density-flow space when small number of loops
are available. The results are presented in Fig. 8 where each dark blue dot is
a density-flow data point in the NGSIM dataset. Note, the flow values are not
part of the observed data during the training phase.

Figure 8: Comparison of the FD curves estimated by the PIDL+FDL-based TSE
methods and PIDL-based TSE methods. For consistent visualization, the learned
$\hat{U}_{eq}(\hat{\rho};\omega)$ is converted to
$\hat{\rho}\hat{U}_{eq}(\hat{\rho};\omega)$ to represent the estimated flow.

For the PIDL-based methods, the closed form of the flux function and velocity
function are given and the parameters in the PINN component are learned along
with the TSE training, and thus, the shape of the FD curves are predefined.
The corresponding FD curves with the learned model parameters are indexed as
“3” in Fig. 8. For consistent visualization, the learned
$\hat{U}_{eq}(\hat{\rho};\omega)$ is converted to
$\hat{\rho}\hat{U}_{eq}(\hat{\rho};\omega)$ to represent the estimated flow.
The LWR-PIDL method is encoded with the 3-parameter-based flux, and the ARZ-
PIDL is with the Greenshields function for the equilibrium velocity. The
former has a proper shape defined by the given mathematical formula, but due
to the complicated nature of the PINN for the 3-parameter flux and the noisy
quality of the data, the learned FD curves do not fit the density-flow points
to a satisfactory extent. The latter has a predefined quadratic shape and can
capture the density-flow characteristics to a limited level.

For the PIDL+FDL-based methods, using the standard training loss in Eq.(7),
the learned FD shapes (indexed as “1” in Fig. 8) fit the NGSIM density-flow
points well over the domain where the observed traffic state data are
distributed, i.e., around $\rho\in[0,0.5]$. However, the FD curves tend to
curl up in the large-density domain, where the data are sparse. To address
this abnormal shape, we apply the regularization term in Eq. (8) for reshaping
and impose the prior knowledge of concavity over a narrowed interval of
$\rho\in[0.6,0.7]$. To this end, we set the hyperparameters in Eq. (8) to
$a=0.6$ and $b=0.7$. The corresponding learned FD curves using the reshaping
regularization term are indexed as “2” in the figures, and they can properly
capture the density-flow characteristics to a satisfactory level. Because of
using the FD Learner, the LWR-PIDL+FDL contains no model parameters, and the
conservation law plus the $\hat{Q}(\rho;\omega^{*})$ constitutes the LWR model
reconstructed by the LWR-PIDL+FDL. The ARZ-PIDL+FDL contains one model
parameter, i.e., the relaxation time $\tau$. The learning with data from 3, 6
and 8 loops converges to $\tau^{*}=[23.36,25.99,27.66]$, which is reasonably
close to $\tau=[27.6,28.8,30.5]$ directly fitted from data. The conservation
law, the momentum of velocity with model parameter $\tau^{*}$, and the learned
$\hat{U}_{eq}(\rho;\omega^{*})$ constitute the ARZ model reconstructed by the
ARZ-PIDL+FDL.

The results demonstrate that the proposed PIDL+FDL-based TSE method (with the
regularization for reshaping) is able to efficiently conduct high-quality TSE,
model parameter identification and fundamental diagram estimation at the same
time with relatively small amounts of observed data.

##  VII Conclusion

We introduced the PIDL+FDL framework to the TSE problem on highways using loop
detector data and demonstrate the significant benefits of the integration of
an ML surrogate into the model-driven component in PIDL. This framework can be
used to handle traffic state estimation, model parameter identification, and
fundamental diagram estimation simultaneously. The experiments on real highway
data show that PIDL+FDL-based approaches can outperform baselines in terms of
estimation accuracy and data efficiency as well as the estimation of FD.

The limitations and potential future works of this paper are as follows: (1)
Similar to most deep learning methods, hyperparameter tuning is an issue of
PIDL-based TSE, and tuning such a large number of hyperparameters based on
approaches like cross-validation is too complicated for real-world
application, and the model basically has to be tuned for each scenario
(spatial resolution, temporal resolution, spatial grid size, temporal grid
size, observation error, etc), which limits its applicability to real-world
problems; (2) PIDL is known to have issues with noisy data, and how the high
noise and corruption in the real traffic data undermine the performance of
PIDL-based TSE needs further investigations; (3) It is worthy of considering
more ML surrogate components to represent other unobserved traffic quantities
in the traffic flow model, such as $h(\rho)$ and $U_{eq}(\rho-u)/\tau$, and
study to what extent the addition of surrogates affects the performance.

## Acknowledgment

This work is partially supported by NSF DMS-1937254, DMS-2012562 and
CCF-1704833. We would like to thank Professor Benjamin Seibold from Temple
University for inspiring us on FDL.

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