$\displaystyle=\left(\;0,\;\partial_{r}\phi^{1},\;\partial_{\theta}\phi^{1}-\frac{1}{\sqrt{g_{rr}}}\phi^{2},\;-\frac{\sin\theta}{\sqrt{g_{rr}}}\phi^{3}\right)\>,\hskip
14.22636ptF_{\mu}^{\;2}=\left(\;0,\;\partial_{r}\phi^{2},\;\partial_{\theta}\phi^{2}+\frac{1}{\sqrt{g_{rr}}}\phi^{1},-\cos\theta\phi^{3}\right)\>,$
$\displaystyle F_{\mu}^{\;3}$
$\displaystyle=\left(\;0,\;\partial_{r}\phi^{3},\;\partial_{\theta}\phi^{3},\;\frac{\sin\theta}{\sqrt{g_{rr}}}\phi^{1}+\cos\theta\phi^{2}\right)\>$
(121)

The field-strength $1$-form and the connection coefficients computed in
section 5 may now be plugged in into (107). The resulting set of equations of
motions can be compiled into three partial differential equations:

$\displaystyle\Box\phi^{1}-\frac{2}{r^{2}g_{rr}}\phi^{1}-\frac{2\cot\theta}{r^{2}\sqrt{g_{rr}}}\phi^{2}-\frac{2\partial_{\theta}\phi^{2}}{r^{2}\sqrt{g_{rr}}}\;$
$\displaystyle=\;0\>,$ (122a)
$\displaystyle\Box\phi^{2}-\left(\frac{\cot^{2}\theta}{r^{2}}+\frac{1}{r^{2}g_{rr}}\right)\phi^{2}+\frac{2\partial_{\theta}\phi^{1}}{r^{2}\sqrt{g_{rr}}}\;$
$\displaystyle=\;0\>,$ (122b)
$\displaystyle\Box\phi^{3}-\left(\frac{\cot^{2}\theta}{r^{2}}+\frac{1}{r^{2}g_{rr}}\right)\phi^{3}\;$
$\displaystyle=\;0\>,$ (122c)

where $\Box=\nabla^{\mu}\nabla_{\mu}$ is the d’Alembert operator. These
equations can be interpreted as describing different components of the
$0$-form field that have different relative amplitudes depending on the
location in spacetime. The equations (122a) and (122b) are coupled in a non-
trivial way. In the limit of large radii $r\rightarrow\infty$ the system of
equations reduces to three independent wave equations.

With knowledge of the field-strength $1$-form (121), we can compute the
components of the energy momentum tensor via Equation (118):

$\displaystyle T_{tt}$
$\displaystyle=-\frac{\alpha}{2}g_{tt}\left[\frac{1}{g_{rr}}\left((\partial_{r}\phi^{1})^{2}+(\partial_{r}\phi^{2})^{2}+(\partial_{r}\phi^{3})^{2}\right)+\frac{1}{r^{2}}\left((\partial_{\theta}\phi^{1})^{2}+(\partial_{\theta}\phi^{2})^{2}+(\partial_{\theta}\phi^{3})^{2}\right)\right.$
$\displaystyle\;+\left.\frac{1}{r^{2}g_{rr}}\left(2(\phi^{1})^{2}+(\phi^{2})^{2}+(\phi^{3})^{2}\right)+\frac{\cot^{2}\theta}{r^{2}}((\phi^{2})^{2}+(\phi^{3})^{2})+\frac{2\cot\theta}{r^{2}\sqrt{g_{rr}}}\phi^{1}\phi^{2}\right.$
$\displaystyle\;-\left.\frac{2}{r^{2}\sqrt{g_{rr}}}\left((\partial_{\theta}\phi^{1})\phi^{2}-(\partial_{\theta}\phi^{2})\phi^{1}\right)\right]\>+\frac{\beta}{4}g_{tt}\left[\frac{(\partial_{r}g_{rr})^{2}}{r^{2}g_{rr}^{4}}+\frac{1}{r^{4}}\left(1-\frac{1}{g_{rr}}\right)^{2}\right]$
(123a) $\displaystyle T_{rr}$
$\displaystyle=\frac{\alpha}{2}\left[\left(\partial_{r}\phi^{1}\right)^{2}+(\partial_{r}\phi^{2})^{2}+(\partial_{r}\phi^{3})^{2}\right]-\frac{\alpha}{2}g_{rr}\left[\frac{1}{r^{2}}\left((\partial_{\theta}\phi^{1})^{2}+(\partial_{\theta}\phi^{2})^{2}+(\partial_{\theta}\phi^{3})^{2}\right)\right.$
$\displaystyle\;+\left.\frac{1}{r^{2}g_{rr}}\left(2(\phi^{1})^{2}+(\phi^{2})^{2}+(\phi^{3})^{2}\right)+\frac{\cot^{2}\theta}{r^{2}}((\phi^{2})^{2}+(\phi^{3})^{2})+\frac{2\cot\theta}{r^{2}\sqrt{g_{rr}}}\phi^{1}\phi^{2}\right.$
$\displaystyle\;-\left.\frac{2}{r^{2}\sqrt{g_{rr}}}\left((\partial_{\theta}\phi^{1})\phi^{2}-(\partial_{\theta}\phi^{2})\phi^{1}\right)\right]+\frac{\beta}{4}g_{rr}\left[-\frac{(\partial_{r}g_{rr})^{2}}{r^{2}g_{rr}^{4}}+\frac{1}{r^{4}}\left(1-\frac{1}{g_{rr}}\right)^{2}\right]\>.$
(123b)

The equations of motion for the metric components are derived from the
Einstein equations

$G_{\mu\nu}\;=\;8\pi T_{\mu\nu}\>.$ (124)

On the left hand side of equation (124), only functions of the radial
coordinate appear when choosing the radially symmetric metric ansatz (119).
However, the energy momentum tensor components explicitly depend the radial
distance $r$ and the polar angle $\theta$. This indicates a poloidal
deformation of spacetime since the energy-momentum distribution encoded in
$T_{\mu\nu}$ is anisotropic. For more general considerations, the components
of the $0$-form field $\phi^{a}$ and metric components will have to depend on
the polar angle $\theta$. For the scope of this work, the simple ansatz chosen
in equations (120) and (119) is acceptable, since it was only meant as a proof
of concept to illustrate the application of our formalism for a concrete
physical case.
For numerical applications, one could substitute the simple ansatz (119) with
an axially symmetric ansatz for a metric that depends on both $r$ and
$\theta$. Another option would be to take an average of $T_{\mu\nu}$ over the
polar angle $\theta$. The tensor $T_{\mu\nu}$ on the r.h.s of equation (124)
would then have to be replaced by its poloidal average
$\left<T_{\mu\nu}\right>_{\theta\in[0;\pi]}$. The equations (124) then reduce
to two ordinary differential equations in the radial distance $r$. A future
numerical study is necessary to assess the quality of the averaging
prescription $\left<T_{\mu\nu}\right>_{\theta\in[0;\pi]}$ compared to using a
fully axially symmetric ansatz for the field and the metric.

###  6.2 Numerical Implications

To solve the $k$-form equations numerically, it is important to know the type
of equations and initial conditions. We discuss these points for the case of
the $0$-form field introduced in the previous subsection.

#### Analyzing the Equations of Motion

We start by analyzing the type of the partial differential equations
(122a)-(122c). All three of the equations are linear second order PDEs. There
exists a unique classification for linear second order PDEs with two
variables. Let $f(r,\theta)$ be a smooth and (at least) twice differentiable
function of two variables $r$ and $\theta$. Then, every second order linear
PDEs can be written as

$A\partial_{r}^{2}f(r,\theta)+C\partial_{\theta}^{2}f(r,\theta)+2B\partial_{r}\partial_{\theta}f(r,\theta)+D\partial_{r}f(r,\theta)+E\partial_{\theta}f(r,\theta)+F=0\>,$
(125)

with smooth functions $A,B,C,D,E,F$ that depend on $r$ and $\theta$. All
second order linear partial differential equations can now be categorized into
hyperbolic, elliptic and parabolic. A second order linear PDE is called
hyperbolic if $B^{2}-AC>0$ for all $r$ and $\theta$. It is called elliptic if
$B^{2}-AC<0$ and parabolic if $B^{2}-AC=0$ for all $r$ and $\theta$. In our
case, for the equations (122a)- (122a), one obtains

$B^{2}-AC=-\frac{1}{r^{2}g_{rr}}\>.$ (126)

Since $g_{rr}>0$ for all radial distances $r\in(0,\infty)$ it follows that
$B^{2}-AC<0$ for all $r$ and $\theta$. Thus, equations (122a)-(122c) are a
system of elliptic partial differential equations. In a numerical analysis,
standard solution methods for elliptic partial differential equations can be
applied to find an explicit solution. One of the inputs needed for this
methods are appropriate boundary conditions.

#### Boundary Conditions

For the ansatz chosen in this work, there are four boundaries at which
appropriate boundary conditions must be applied. These four boundaries are
located at $(r,\theta)=(0,\theta)$, $(r,\theta)=(R,\theta)$,
$(r,\theta)=(r,0)$ and $(r,\theta)=(r,\pi)$, where $R$ denotes the largest
radial position of a two dimensional polar grid. For the boundary located at
$(r,\theta)=(R,\theta)$, a consistent boundary condition is given by

$\phi^{a}(R,\theta)=\phi^{a}(R)\>.$ (127)

Here $\phi^{a}(R)$ is independent of the angle $\theta$ and must be chosen
depending on the physical scenario one wants to study. One possible choice is
to let the field vanish at large distances $R$.
This boundary condition is valid for sufficiently large $R$ since for
$r\rightarrow\infty$ the equations (122a)-(122c) reduce to decoupled free wave
equations. For the coordinate poles $\theta=\\{0,\pi\\}$, typical pole fixing
boundary conditions can be applied, for example reflective or absorption
boundary conditions.
For the remaining boundary condition at the center of the grid at
$(r,\theta)=(0,\theta)$ it is instructive to investigate the proportionality
behaviour of the field for small radii. Evidently, equations (122a)-(122c) all
contain terms proportional to $1/r^{n},n\in\mathbb{N}$ which are singular at
$r=0$. Therefore, one can not simply choose a set of constant values
$\phi^{a}(0,\theta)=\phi_{0}^{a}(\theta)=const$. Instead, one needs to expand
the field components $\phi^{a}$ in a polynomial series in the radial distance
$r$

$\phi^{a}(r,\theta)=\sum_{n=0}^{\infty}\phi^{(n),a}(\theta)\,r^{n}\>.$ (128)

By inserting the boundary condition (128) into equations (122a)-(122c) and
imposing regularity at $r=0$ one can obtain a set of constraints for the
radius-independent functions $\phi^{(n),a}(\theta)$. We subsequently obtain
the following boundary condition

$\phi^{a}(r,\theta)=\phi^{(3),a}(\theta)\>r^{3}+\mathcal{O}(r^{4})\>,\hskip
42.67912ptr\rightarrow 0\>.$ (129)

The yet unknown functions $\phi^{(3),a}(\theta)$ can be obtained by inserting
the boundary condition into equations (122a)-(122c). The resulting equations
are three second order ordinary differential equations in $\theta$:

$\displaystyle\partial_{\theta}^{2}\,\phi^{(3),1}(\theta)+\cot\theta\,\partial_{\theta}\,\phi^{(3),1}(\theta)+10\;\phi^{(3),1}(\theta)-2\cot\theta\,\partial_{\theta}\,\phi^{(3),2}(\theta)\;=\;0\;$
(130a)
$\displaystyle\partial_{\theta}^{2}\,\phi^{(3),2}(\theta)+\cot\theta\,\partial_{\theta}\,\phi^{(3),2}(\theta)-(\cot^{2}\theta-11)\,\phi^{(3),2}(\theta)+2\partial_{\theta}\,\phi^{(3),1}(\theta)\;=\;0\>,$
(130b)
$\displaystyle\partial_{\theta}^{2}\,\phi^{(3),3}(\theta)+\cot\theta\,\partial_{\theta}\,\phi^{(3),3}(\theta)+10\;\phi^{(3),1}(\theta)-(\cot^{2}\theta-11)\,\phi^{(3),3}(\theta)\;=\;0\>.$
(130c)

Here, the asymptotic flatness conditions $g_{tt}(0)=g_{rr}(0)=1$ and
$\partial_{r}g_{tt}(0)=\partial_{r}g_{rr}(0)=0$ were assumed. This is only
valid for the spherically symmetric metric ansatz. Coupling to matter can now
be realized by either solving the equation fully self-consistently (with the
metric depending on $r$ and $\theta$), or by using the averaging procedure for
the energy-momentum tensor, as explained in the last paragraph of section 6.1.
We note that the polar grid has a singularity at $r=0$. In practice, one has
to remove this point from the domain to ensure regularity at all points of the
grid. Numerically the set of equations (130) can be treated as a set of
constraint equations which must be satisfied for sufficiently small
$r=\epsilon$. These equations constitute consistent boundary conditions that
have to be fulfilled at the boundary $(r,\theta)=(\epsilon,\theta)$. With
this, we conclude our discussion on boundary conditions.

###  6.3 Higher Order $k$-Form Fields

In the following, we briefly address the steps to study $1$-form and $2$-form
fields in a similar fashion as laid out for the $0$-form field in section 6.1.
In this work, we refrain from studying $3$-form fields. However we note that
the process to extend the description to $3$-form fields is in principle
straightforward. Following similar steps as for the $0$-form field, one can
derive the field strengths and the field equations for the $1$-form field from
the most general form of equation (69). If we let
$A\in\Omega^{1}(M;\mathfrak{g})$ be a Lie algebra valued $1$-form field and
$B\in\Omega^{1}_{con}(M;\mathfrak{g})$ be the connection computed in chapter
5, then the corresponding $2$-form field strength
$F\in\Omega^{2}(M;\mathfrak{g})$ is given by:

$F=dA+[B,A]\;=\frac{1}{2}F_{\mu\nu}^{a}\left(dx^{\mu}\wedge
dx^{\nu}\right)\;\mathbin{\mathop{\otimes}\limits}T_{a}\>.$ (131)

where the $F_{\mu\nu}^{a}$ denote the components of the Lie algebra valued
$2$-form $F$.
Similarly, for the $2$-form field $A\in\Omega^{2}(M;\mathfrak{g})$, one can
derive the following expression for the $3$-form field-strength
$F\in\Omega^{3}(M;\mathfrak{g})$:

$F=dA+[B,A]\;=\frac{1}{3!}F_{\mu\nu\rho}^{a}\left(dx^{\mu}\wedge
dx^{\nu}\wedge dx^{\rho}\right)\;\mathbin{\mathop{\otimes}\limits}T_{a}\>.$
(132)

The components of the field strengths defined in (131) and (132) can also be
expressed in terms of the field components $A_{\mu}^{a}$, $A_{\mu\nu}^{a}$ and
the connection coefficients components $B_{\mu}^{a}$ as

$\displaystyle F_{\mu\nu}^{a}$
$\displaystyle=2\nabla_{[\mu}A_{\nu]}^{a}+2if^{a}_{\;bc}\;B_{[\mu}^{b}\;A_{\nu]}^{c}\>,$
(133a) $\displaystyle F_{\mu\nu\rho}^{a}$
$\displaystyle=3\nabla_{[\mu}A_{\nu\rho]}^{a}+3if^{a}_{\;bc}\;B_{[\mu}^{b}\;A_{\nu\rho]}^{c}\>.$
(133b)

The squared brackets denote the usual total antisymmetrization with respect to
the Lorentz indices (see paragraph below Def. 2). To derive the full equations
of motion for the $1$-form field, we follow the same steps as in the previous
section, where the derivation of the $0$-form equations of motion was laid out
in detail. In coordinate based notation, the resulting field equations have
the same structure as for the $0$-form field. They can be written for the
$1$-form field using the components of the $2$-form field strength defined in
equation (133a)

$\nabla_{\mu}F^{\mu\nu a}+i\,f^{a}_{\;bc}B_{\mu}^{b}F^{\mu\nu c}=0\>.$ (134)

With similar steps we obtain the field equations for the $2$-form field
$A\in\Omega^{2}(M;\mathfrak{g})$ using the 3-form field strength defined in
equation (133b)

$\nabla_{\mu}F^{\mu\nu\rho a}+i\,f^{a}_{\;bc}B_{\mu}^{b}F^{\mu\nu\rho c}=0\>.$
(135)

The corresponding energy momentum tensor for the 1-form field
$A\in\Omega^{1}(M,\mathfrak{g})$ is given in terms of the corresponding field
strength

$T_{\mu\nu}\;=\alpha\sum_{a=1}^{\textrm{dim}(\mathfrak{g})}\left(\;F_{\mu}^{\;\alpha
a}F_{\nu\alpha
a}-\frac{1}{4}g_{\mu\nu}F_{\alpha\beta}^{\;\;a}F^{\alpha\beta}_{a}\right)+\beta\sum_{a=1}^{\textrm{dim}(\mathfrak{g})}\left(\;H_{\mu}^{\;\alpha
a}H_{\nu\alpha
a}-\frac{1}{4}g_{\mu\nu}H_{\alpha\beta}^{\;\;a}H^{\alpha\beta}_{a}\right)\>.$
(136)

In an identical fashion one can derive the following expression for the energy
momentum tensor for the $2$-form field $A\in\Omega^{2}(M;\mathfrak{g})$

$T_{\mu\nu}\;=\alpha\sum_{a=1}^{\textrm{dim}(\mathfrak{g})}\left(\;\frac{1}{2}F_{\mu\rho\kappa}^{a}F_{\nu
a}^{\;\rho\kappa}-\frac{1}{12}g_{\mu\nu}F_{\rho\kappa\lambda}^{a}F^{\rho\kappa\lambda}_{a}\right)+\beta\sum_{a=1}^{\textrm{dim}(\mathfrak{g})}\left(\;H_{\mu}^{\;\alpha
a}H_{\nu\alpha
a}-\frac{1}{4}g_{\mu\nu}H_{\alpha\beta}^{\;\;a}H^{\alpha\beta}_{a}\right)\>\>.$
(137)

Both energy momentum tensors are hence completely determined by the
corresponding field strengths. From this point on, to perform any specific
calculation, one must choose a concrete ansatz for the $1$-form/$2$-form field
$A$. Future work might use approaches based upon studies of $SU(2)$ Yang-Mills
stars, such as [76, 77], as a potential ansatz for the $k$-form fields.

##  7 Conclusion and Outlook

In this work, we developed a rigorous way to incorporate differential $k$-form
fields $A$ with a local gauge symmetry $G$ into curved spacetimes. The
symmetry of the fields was modelled using a compact, semisimple Lie group. In
section 3, we constructed a generalized action for Lie algebra valued
differential $k$-form fields in $n$ dimensions. For this, we introduced a
covariant derivative with corresponding connection $B$, which respects both
the inner group structure of the fields and the spacetime curvature. We showed
that the action is self-consistent and that it is invariant under local
symmetry transformations in $G$ and $SO(n-1,1)$. From this action, we derived
equations of motion for the field $A$ and the connection $B$ and discussed
their degrees of freedom.
In section 5, we explored possible choices for the connection $B$. We showed
that it can be related to the spin-connection of spacetime. As an example, we
explicitly studied the case of a spherically symmetric spacetime with a local
symmetry group $G=SU(2)$. In this case, we were able to fully express the
connection coefficients of $B$ using the spacetime metric and group structure
coefficients. We also found that in some cases, the dimension of the local
group space and of spacetime can be different form each other. This makes is
necessary to complete the basis using ”extra dimensions” and rectangular
vielbein basis vectors. Calculations of the vielbeins for the group $SU(3)$
are shown in Appendix A.
In section 6, we rewrote the equations of motion for the $k$-form field and
the connection in index notation and used this to derive differential
equations for a $0$-form field (i.e. a scalar function) with internal $SU(2)$
symmetry in spherical symmetry. We also discussed possible applications and
numerical solution strategies.

Our work can be applied in a variety of scenarios. One is boson stars, which
are a class of self-gravitating bosonic field solutions of the Einstein
equations. In most recent studies (see citations in the introduction), the
bosonic field is usually taken to be a classical scalar (or vector) field with
a global $U(1)$ symmetry. They are often motivated through light dark matter
particles or theories of modified gravity. An extension to internal symmetries
of the fields could thus allow to probe different types of dark matter or
extensions of gravity. Gravitational condensates of pions – and other
particles with internal symmetry groups – could also be modelled using our
approach because in a classical picture they can be viewed as scalar particles
with an internal $SU(2)$ symmetry.
Another field of interest is string theory, which features various
differential $k$-form fields with different degrees $k$ and internal symmetry
structures. In certain superstring theories, differential $k$-form fields
appear either as low energy effective modes or as excitations of $D$-branes.
In particular $D$-branes would be interesting to study with the formalism in
this work. The group index of the field would then correspond to the numbering
of multiple branes in specific orientations.

Our formalism also has some limitations. It is valid for $k$-form fields $A$
with internal symmetries. These symmetries are described by semisimple compact
Lie groups. It also features a connection $B$, which is independent of the
field $A$. The description of the fields in terms of Lie algebra valued
differential forms allows us to study them with various internal symmetries.
These groups cover a large amount of symmetry groups widely used in physics.
However, it restricts us to model physical bosonic fields with spin $0$ and
spin $1$. Also, the allowed Lie groups do not permit to study all possible
groups that might be interesting for physical applications.

Our work can be further be expanded on in two main ways: modifying the
symmetry groups and modifying the fields.
Regarding the groups, one could consider more complicated internal symmetry
groups such as product groups. Relaxing the assumption of semisimplicity and
compactness of the Lie groups might also be worthwhile to study – although
relaxing semisimplicity leads to a degenerate Cartan–Killing form.
Another option is to add additional types of symmetries such as global
symmetries. This is interesting in the context of boson stars where the
bosonic fields are usually modeled as complex fields with a global $U(1)$
symmetry (see [9] for an extensive review). Using internal symmetries for
boson stars might be a way to circumvent Derrick’s theorem [78, 79, 80], which
forbids scalar boson stars without conserved Noether currents.
Regarding the fields, another avenue of study would be higher order tensor
fields with internal symmetries. This would open the way to model arbitrary
integer spin fields. These would not be described using differential forms.
The action would thus need to be modified.
Another interesting option might be to incorporate fermionic fields into the
formalism. This amounts to adding half-integer differential $k$-form fields.
They are described by the $k+\frac{1}{2}$ representations of the double cover
of the Lorentz group $SO(n-1,1)$.
Modifications to the connection $B$ are also interesting. Promoting $B$ to a
fully dynamical field might allow different physical interpretations.
Considering different choices for the connection as a non-dynamical object
could also be interesting.

But even without further modifications, there are already a large number of
cases that can be studied using our formalism. It would be interesting to
formulate solutions to the equations of motion for various $k$-form fields and
internal symmetry groups.
In this work, we made a choice for the vielbeins where we matched specific
dimensions of the local symmetry space and the global spacetime.
Alternatively, one could also consider to expand the rectangular vielbeins in
a different manner – e.g. matching the $\mathfrak{su}(2)$ algebra space to the
spacetime components ($0,1,2$) instead of the spatial spacetime components
($1,2,3$).
Regarding concrete computations of solutions to the equations of motion, one
might use a more complete ansatz for the fields and the metric such that both
depend on the coordinates $(r,\theta)$.
Other extensions include the use of different potential forms in the action,
adding coupling terms to matter or modifying the underlying theory of gravity.
We have summarized a number of other extensions in more detail in chapter 3.5.
The large amount of options aside, the next steps should be to study the
solutions evaluated in this work numerically. This entails solving
differential equations to obtain the values of the field $A$ and the
spacetime. We leave the detailed analysis for a future work.

## Acknowledgements

We thank Leon Menger for his invaluable insights, discussions and proof-
reading the manuscript. We also thank Laura Sagunski for providing support
during the initial stages of the project.

## Appendix

##  Appendix A Vielbeins for SU(3)

In the following section, we briefly outline the basic computational steps to
calculate vielbein components for the case when the dimension of the symmetry
group $G$ is larger than the dimension $n$ of the spacetime manifold $M$. For
the sake of this example, we choose $G=SU(3)$. Let further $g$ be the
spacetime metric. We also assume that $g$ it is static and spherically
symmetric. Let $K$ be the Cartan–Killing metric of the Lie algebra
$\mathfrak{su}(3)$ with components $K_{ab}=-N\delta_{ab}$, where $N=3$. The
vielbeins $e$ may then be determined using the relation

$g_{\mu\nu}=e_{\mu}^{a}K_{ab}e_{\nu}^{b}\>.$ (138)

From equation (138) it is apparent that the vielbeins are not quadratic. The
reason is that the dimension of the algebra $dim(\mathfrak{g})=8$ is larger
than the dimension of the spacetime manifold $n=4$. In section 5.2, we
discussed the procedure to calculate vielbeins for cases where
$dim(\mathfrak{g})<n$. This was done by completing the basis of vielbeins
using additional ”extra dimensions” of the group. When $dim(\mathfrak{g})>n$,
we may employ the same procedure by adding additional spacetime dimensions
instead. There are many possible choices on how to choose the manifold that
incorporates additional dimensions. In this example we use 6-dimensional
spherical coordinates $\\{r,\theta_{j}\\}_{\;j\in[1;6]}$ with the
corresponding metric line element

$ds^{2}=g_{tt}(r)dt^{2}+g_{rr}(r)dr^{2}+r^{2}\left(d\theta_{1}^{2}+\sum_{m=2}^{6}\left(\prod_{j=1}^{m-1}\sin^{2}\theta_{j}\right)\;(d\theta_{m})^{2}\right)\>.$
(139)

Using the metric (139), it is straightforward to compute the vielbein
components (because both $g$ and $K$ are symmetric in the chosen basis). We
list all resulting non-zero vielbein components $e_{\mu}^{a}$ for $SU(3)$
below:

$\displaystyle e_{t}^{1}$ $\displaystyle=i\;\sqrt{\frac{g_{tt}}{3}}\hskip
42.67912pte_{r}^{2}=i\;\sqrt{\frac{g_{rr}}{3}}\hskip
42.67912pte_{\theta_{1}}^{3}=i\;\frac{r}{\sqrt{3}}\hskip
42.67912pte_{\theta_{2}}^{4}=i\;\frac{r\sin\theta_{1}}{\sqrt{3}}$
$\displaystyle e_{\theta_{3}}^{5}$
$\displaystyle=i\;\frac{r\sin\theta_{1}\sin\theta_{2}}{\sqrt{3}}\hskip
39.83368pte_{\theta_{4}}^{6}=i\;\frac{r\sin\theta_{1}\sin\theta_{2}\sin\theta_{3}}{\sqrt{3}}\hskip
39.83368pte_{\theta_{5}}^{7}=i\;\frac{r\sin\theta_{1}\sin\theta_{2}\sin\theta_{3}\sin\theta_{4}}{\sqrt{3}}$
$\displaystyle e_{\theta_{6}}^{8}$
$\displaystyle=i\;\frac{r\sin\theta_{1}\sin\theta_{2}\sin\theta_{3}\sin\theta_{4}\sin\theta_{5}}{\sqrt{3}}\>.$
(140)

Using these vielbeins one can compute the connection coefficients
$B_{\mu}^{a}$ in an identical manner as was shown for $SU(2)$ in section 5.2.
The connection coefficients can be computed using the structure coefficients
$f^{a}_{\;bc}$ of $\mathfrak{su}(3)$ and equation (80).
Due to the artificial extension of the spacetime dimension from $dim(M_{4})=4$
to $dim(M_{8})=8$, additional components will be introduced to all tensorial
and non-tensorial objects. This includes the connection, spacetime metric and
also the equations of motion. But the the additional dimensions were added as
a mere mathematical tool to make the vielbeins well-defined. These extra
dimensions are thus unphysical and must be removed before evaluating any
physically relevant expressions. This is formally achieved by taking the
following intersection:

$M_{4}=\bigcap_{\theta_{j}=0\;\forall\;j\in[3;6]}\;M_{8}\>.$ (141)

This effectively reduces all final expressions to the physical four-
dimensional spacetime and sets any contributions from the ”extra dimensions”
to zero.

##  Appendix B Derivation of the Equations of Motion

Equation (66) provides a formal definition of a variation for an action that
contains Lie algebra valued differential $k$-form fields
$A\in\Omega^{k}(M;\mathfrak{g})$. It is defined by the limit

$\delta S_{KFT}=\lim_{\epsilon\rightarrow 0}\frac{S_{KFT}[A+\epsilon\delta
A]-S_{KFT}[A]}{\epsilon}\>.$ (142)

Before performing the variation, we rewrite the KFT part of the action (55)
using the $k$-form field $A$. By inserting the field strength, defined in
equation (56), the action can be expressed solely in terms of the field $A$
and the connection $B$. For the remainder of this derivation, summation over
repeated group indices is assumed. We also adopt the same notation, $d$
instead of $d_{\omega}$, for the covariant derivative relative to the
spacetime connection $\omega\in\Omega_{con}(M;\mathfrak{so}(n-1,1))$ as in
section 4:

$\begin{split}S_{KFT}&=\int_{M}\left(dA^{a}\wedge\star_{g}dA^{b}\mathbin{\mathop{\otimes}\limits}Tr(T_{a}T_{b})+2(B^{b}\wedge
A^{c})\wedge\star_{g}dA^{a}\mathbin{\mathop{\otimes}\limits}Tr([T_{b},T_{c}]T_{a})\right.\\\
&+\left.(B^{a}\wedge A^{b})\wedge\star_{g}(B^{c}\wedge
A^{d})\mathbin{\mathop{\otimes}\limits}Tr([T_{a},T_{b}][T_{c},T_{d}])\right)\>.\end{split}$
(143)

We now turn to the calculation of the variation of the action (143). First, we
write out the terms from the variation defined in equation (142) explicitly

$\begin{split}&S_{KFT}[A^{a}+\epsilon\delta
A^{a}]=\int_{M}Tr\left[d(A^{a}+\epsilon\delta
A^{a})\wedge\star_{g}d(A^{b}+\epsilon\delta
A^{b})\mathbin{\mathop{\otimes}\limits}T_{a}T_{b}\right.\\\
&+\left.2d(A^{a}+\epsilon\delta
A^{a})\wedge\star_{g}(B^{b}\wedge(A^{c}+\epsilon\delta
A^{c}))\mathbin{\mathop{\otimes}\limits}T_{a}[T_{b},T_{c}]\right.\\\
&\left.+(B^{b}\wedge(A^{a}+\epsilon\delta
A^{a}))\wedge\star_{g}(B^{d}\wedge(A^{c}+\epsilon\delta
A^{c}))\mathbin{\mathop{\otimes}\limits}[T_{b},T_{a}][T_{d},T_{c}]\right]\>.\end{split}$
(144)

All terms of the form $B^{b}\wedge(A^{c}+\epsilon\delta A^{c})$ can be
rewritten as

$B^{b}\wedge(A^{c}+\epsilon\delta A^{c})=B^{b}\wedge A^{c}+\epsilon
B^{b}\wedge\delta A^{c}\>.$ (145)

We further use that the scalar product (see equation (21)) is symmetric.

$\begin{split}&S_{KFT}[A^{a}+\epsilon\delta
A^{a}]=S_{KFT}[A]+\epsilon\int_{M}Tr_{\mathfrak{g}}\left[2dA^{a}\wedge\star_{g}d(\delta
A^{b})\mathbin{\mathop{\otimes}\limits}T_{a}T_{b}\right.\\\
&+\left.2\left(d\delta A^{a}\wedge\star_{g}(B^{b}\wedge
A^{c})+dA^{a}\wedge\star_{g}(B^{b}\wedge\delta
A^{c})\right)\mathbin{\mathop{\otimes}\limits}T_{a}[T_{b},T_{c}]\right.\\\
&+\left.\left((B^{b}\wedge\delta A^{a})\wedge\star_{g}(B^{d}\wedge
A^{c})+(B^{d}\wedge\delta A^{c})\wedge\star_{g}(B^{b}\wedge
A^{a})\right)\mathbin{\mathop{\otimes}\limits}[T_{b},T_{a}][T_{d},T_{c}]\right]\\\
&+\epsilon^{2}\int_{M}Tr_{\mathfrak{g}}\left[d(\delta
A^{a})\wedge\star_{g}d(\delta
A^{b})\mathbin{\mathop{\otimes}\limits}T_{a}T_{b}+2B^{b}\wedge\delta
A^{c}\wedge\star_{g}d(\delta
A^{a})\mathbin{\mathop{\otimes}\limits}[T_{b},T_{c}]T_{a}\right.\\\
&+\left.B^{a}\wedge\delta A^{b}\wedge\star_{g}B^{c}\wedge\delta
A^{d}\mathbin{\mathop{\otimes}\limits}[T_{a},T_{b}][T_{c},T_{d}]\right]\>.\end{split}$
(146)

Now, we again use the symmetry of the scalar product as stated in Def. 1 as
well as Stokes’ theorem for differential $k$-forms. After applying these
relations, the following terms remain

$\begin{split}&S_{KFT}[A^{a}+\epsilon\delta
A^{a}]=S_{KFT}[A^{a}]+2\epsilon\int_{M}Tr_{\mathfrak{g}}\left[d(\delta
A^{b})\wedge\star_{g}dA^{a}\mathbin{\mathop{\otimes}\limits}T_{a}T_{b}\right.\\\
&+\left.\left(d(\delta A^{a})\wedge\star_{g}(B^{b}\wedge
A^{c})+(B^{d}\wedge\delta
A^{c})\wedge\star_{g}dA^{a}\right)\mathbin{\mathop{\otimes}\limits}T_{a}[T_{b},T_{c}]\right.\\\
&+\left.(B^{b}\wedge\delta A^{a})\wedge\star_{g}(B^{d}\wedge
A^{c})\mathbin{\mathop{\otimes}\limits}[T_{b},T_{a}][T_{d},T_{c}]\right]\\\
&+O(\epsilon^{2})\>.\end{split}$ (147)

We again make use of Stokes theorem(see Def.(23)) and also make use of the
anti-symmetry of the wedge-product to shift the terms $\delta A$ to the left.
We obtain

$\begin{split}&S_{KFT}[A^{a}+\epsilon\delta
A^{a}]=S_{KFT}[A^{a}]+2\epsilon\int_{M}Tr_{\mathfrak{g}}\left[(-1)^{k+1}\delta
A^{b}\wedge(d\star_{g}dA^{a})\mathbin{\mathop{\otimes}\limits}T_{a}T_{b}\right.\\\
&+\left.\left((-1)^{k+1}\delta A^{a}\wedge(d\star_{g}(B^{b}\wedge
A^{c}))+(-1)^{k}\delta A^{c}\wedge
B^{b}\wedge\star_{g}dA^{a}\right)\mathbin{\mathop{\otimes}\limits}T_{a}[T_{b},T_{c}]\right.\\\
&+\left.(-1)^{k}\delta A^{a}\wedge B^{b}\wedge\star_{g}(B^{d}\wedge
A^{c})\mathbin{\mathop{\otimes}\limits}[T_{b},T_{a}][T_{d},T_{c}]\right]\\\
&+O(\epsilon^{2})\>.\end{split}$ (148)

Now, we can factorize the $dA^{a}$-term and rename a few indices, if necessary

$\begin{split}&=S_{KFT}[A^{a}]+2\epsilon(-1)^{k}\int_{M}Tr_{\mathfrak{g}}\left[\delta
A^{b}\wedge\left\\{-(d\star_{g}dA^{a})\mathbin{\mathop{\otimes}\limits}T_{a}T_{b}\right.\right.\\\
&-\left.\left.d\star_{g}(B^{a}\wedge
A^{c})\mathbin{\mathop{\otimes}\limits}T_{b}[T_{a},T_{c}]+(B^{c}\wedge\star_{g}dA^{a})\mathbin{\mathop{\otimes}\limits}T_{b}[T_{c},T_{a}]\right.\right.\\\
&+\left.\left.(B^{a}\wedge\star_{g}(B^{d}\wedge
A^{c}))\mathbin{\mathop{\otimes}\limits}[T_{a},T_{b}][T_{d},T_{c}]\right\\}\right]\\\
&+O(\epsilon^{2})\>.\end{split}$ (149)

If we impose that the variation must vanish for arbitrary values of $\delta
A^{a}$, it follows that the term in the curly brackets must be zero. Now, we
can make use of the trace identity $Tr([A,B][C,D])=Tr([[C,D],A]B)$ for
matrices $A,B,C,D\in\mathbb{C}^{n\times n}$ to rewrite the term in the curly
brackets as

$\begin{split}0&=Tr_{\mathfrak{g}}\left[\left\\{d\star_{g}dA^{b}T_{b}+d\star_{g}(B^{b}\wedge
A^{c})[T_{b},T_{c}]+B^{b}\wedge(\star_{g}dA^{c})[T_{b},T_{c}]+B^{b}\wedge\star_{g}(B^{d}\wedge
A^{c})[T_{b},[T_{d},T_{c}]]\right\\}T_{a}\right]\\\
&=Tr_{\mathfrak{g}}\left[\left\\{d\star_{g}dA+d\star_{g}[B,A]+[B,\star_{g}dA]+[B,\star_{g}[B,A]]\right\\}T_{a}\right]\>.\end{split}$
(150)

To shorten notation, we dropped all tensor products. Now we can identify all
terms corresponding to the field strength $F$. This results in the final
equations of motion (one for each open index $a$):

$\displaystyle
Tr_{\mathfrak{g}}\left((d\star_{g}F+[B,\star_{g}F])T_{a}\right)=0\>.$ (151)

The internal structure of the equation can be made more explicit by writing
out the matrix indices of the field strength $F$ and the generators $T_{a}$:

$\displaystyle(d\star_{g}F_{ij}+[B,\star_{g}F]_{ij})(T_{a})_{ji}=0\>,$ (152)

where
$(F_{\mu_{1}...\mu_{k+1}})_{ij}\hat{=}(F_{\mu_{1}...\mu_{k+1}})^{a}\mathbin{\mathop{\otimes}\limits}(T_{a})_{ij}$.

##  Appendix C Transformation Properties of the Fields and Equations of Motion

#### Invariance Under Local $G$ Group Transformations

The field strength $F\in\Omega^{k+1}(M;\mathfrak{g})$ for a general $k$-form
$A\in\Omega^{k}(M;\mathfrak{g})$ and a general connection
$B\in\Omega_{\text{con}}(M;\mathfrak{g})$ can be defined as

$F=dA+[B,A]\>.$ (153)

The following computations are equally valid when $d=d_{\omega}$ is the
spacetime covariant derivative.
The field $A$ and the connection $B$ transform under Lie group transformations
with group element $U\in G$ as

$A\rightarrow A^{{}^{\prime}}=\text{adj}(U)=UAU^{-1},\hskip
56.9055ptB\rightarrow
B^{{}^{\prime}}=\text{adj}(U)-(dU)U^{-1}=UBU^{-1}-(dU)U^{-1}\>.$ (154)

The transformed field strength can be calculated as

$\begin{split}F^{{}^{\prime}}&=d(A^{{}^{\prime}})+[B^{{}^{\prime}},A^{{}^{\prime}}]\\\
&=d(UAU^{-1})+[UBU^{-1}-(dU)U^{-1},UAU^{-1}]\\\
&=U(dA)U^{-1}+(dU)AU^{-1}+UA(dU^{-1})+[UBU^{-1},UAU^{-1}]-[(dU)U^{-1},UAU^{-1}]\\\
&=U(dA)U^{-1}+(dU)AU^{-1}+UA(dU^{-1})+(UBU^{-1})\wedge(UAU^{-1})-(UAU^{-1})\wedge(UBU^{-1})\\\
&-(dU)AU^{-1}+UAU^{-1}(dU)U^{-1}\\\
&=U(dA)U^{-1}+(dU)AU^{-1}+UA(dU^{-1})+U(B\wedge A)U^{-1}-U(A\wedge
B)U^{-1}-(dU)AU^{-1}-UA(dU^{-1})\\\ &=U(dA)U^{-1}+U(B\wedge A)U^{-1}-U(A\wedge
B)U^{-1}\\\ &=U(dA+[B,A])U^{-1}\\\
&=UFU^{-1}\in\Omega^{k+1}(M;\mathfrak{g})\>,\end{split}$ (155)

where we used the relation $(dU)U^{-1}=-UdU^{-1}$.

#### Invariance Under Local $SO(n-1,1)$ Transformations

The total action (55) is invariant under local $SO(n-1,1)$ transformations if
the individual terms are. The Palatini–Cartan term is known to be invariant,
see [64]. We now show the invariance of the remaining terms individually.
Lorentz scalars are invariant under local $SO(n-1,1)$ transformations if they
are constructed from a contraction of tensors. It is therefore sufficient to
show that the field strength $F$ and the curvature form $H$ behave like
tensors under local $SO(n-1,1)$ transformations.
The field $A$ and the connection $B$ transform like a $k$-form and like a
$1$-form (vector) under local $SO(n-1,1)$ transformations, respectively. Note
that the connection $B$ is only a connection with respect to $G$ and not
$SO(n-1,1)$. Therefore, contractions and products of $A$ and $B$ will also be
tonsorial with respect to $SO(n-1,1)$. The only non-trivial parts remaining in
the definitions of $F$ and $H$ are the covariant derivatives $d_{\omega}A$ and
$d_{\omega}B$, with respect to the spacetime connection
$\omega\in\Omega_{con}(M,\mathfrak{so}(n-1,1))$. These expressions will be
covariant because covariant derivatives of tensors are also tensors.

##  Appendix D Additional Potential Forms

We define a potential form $V$ as an $n$-form, which is invariant under
transformations in $G$ and $SO(n-1,1)$. There are many – potentially infinite
– different ways to construct valid potential forms. This depends on e.g. the
spacetime dimension $n$ and the degree of the $k$-forms, which are used to
construct $V$. In this section we give a brief overview of valid choices and
we give instructions how to construct certain classes of potential forms.

#### Potential Forms Involving Only the Fields $A$

Apart from the mass term mentioned in section 3.5, any term involving wedge
products of $A$ with itself is a valid potential form. This is the case if a
trace is taken over the whole expression and if the resulting form is an
$n$-form. For example, contracting a $1$-form $A\in\Omega^{1}(M;\mathfrak{g})$
$n$ times with itself will be a valid term. It will have the form

$\displaystyle V_{A}(A)=Tr_{\mathfrak{g}}\left[\underbrace{A\wedge\cdots\wedge
A}_{n\text{ times}}\right]\hskip 28.45274pt\text{if}\hskip
14.22636ptA\in\Omega^{1}(M;\mathfrak{g})\>.$ (156)

Another possibility is to combine terms including Hodge stars, because
$(\text{any term})\wedge\star(\text{any term})$ will always be an $n$-form,
provided that the given term is at most an $n$-form. Thus, terms of the
following form will always be possible:

$\displaystyle V_{A}(A)=Tr_{\mathfrak{g}}\left[A\wedge\star_{g}A\right]\hskip
28.45274pt\text{for all}\hskip 14.22636ptA\in\Omega^{k}(M;\mathfrak{g})\>.$
(157)

Incidentally, this is the mass term, which was mentioned in the main text.
Such terms can also be nested like this:

$\displaystyle
V_{A}(A)=Tr_{\mathfrak{g}}\left[\left(A\wedge\star_{g}A\right)\wedge\star_{g}\left(A\wedge\star_{g}A\right)\right]\hskip
28.45274pt\text{for all}\hskip 14.22636ptA\in\Omega^{k}(M;\mathfrak{g})\>.$
(158)

This construction can be iterated an arbitrary number of times, allowing for
an infinite amount of valid potential forms.
For certain spacetime dimensions $n$, terms that are quartic in the field $A$
can also be constructed like this:

$\displaystyle V_{A}(A)=Tr_{\mathfrak{g}}\left[\left(A\wedge
A\right)\wedge\star_{g}\left(A\wedge A\right)\right]\hskip
28.45274pt\text{for}\hskip 14.22636ptA\in\Omega^{k}(M;\mathfrak{g})\>,\>k\leq
n/2\>.$ (159)

Terms of this form with the wedge product of three or more fields are also
possible, depending on the degree of the field $A$. Interestingly, for
$0$-forms, an arbitrary amount of wedge products can be taken and crafted into
an n-form using a Hodge star:

$\displaystyle
V_{A}(A)=Tr_{\mathfrak{g}}\left[\left(\underbrace{A\wedge\cdots\wedge
A}_{\text{arbitrary
amount}}\right)\wedge\star_{g}\left(\underbrace{A\wedge\cdots\wedge
A}_{\text{arbitrary amount}}\right)\right]\hskip 28.45274pt\text{if}\hskip
14.22636ptA\in\Omega^{0}(M;\mathfrak{g})\>.$ (160)

The above examples should be enough to enable the interested reader to
construct further terms as needed.

#### Potential Forms Involving Only the Connection $B$

The connection $B$ does not transform like an algebra element under
transformations in $G$. This excludes potential forms which simply are wedge
products of $B$ with itself or with an algebra element. However, the
corresponding field strength $H=dB+\frac{1}{2}[B,B]$ does transform like an
algebra element and can easily be combined into valid terms for a potential
form $V_{B}(B)$. A difference to the $k$-form field is that $H$ will always be
a $2$-form, since $B$ is a connection. This limits the possible amount of non-
trivial terms (apart from the infinitely nesting configuration discussed in
(158)). In $n=4$ dimensions, the only valid non-trivial potential forms using
only the connection $B$ are

$\displaystyle V_{B}(B)$
$\displaystyle=Tr_{\mathfrak{g}}\left[H\wedge\star_{g}H\right]\>,$ (161a)
$\displaystyle V_{B}(B)$ $\displaystyle=Tr_{\mathfrak{g}}\left[H\wedge
H\right]\>.$ (161b)

The first term is the kinetic term for the connection $B$ and the second term
is a topological term. The kinetic term is already included in the full KFT
action. The topological term is optional, depending on whether one wants to
study topological terms.

#### Potential Forms Involving the Field $A$ and the Connection $B$

Combining the field $A$ and the connection $B$ is more involved due to the
different transformation properties of $A$ and $B$. For example, terms that
are simply wedge products between $A$ and $B$ will not work because they break
invariance under transformations in $G$. This leaves the possibility to
combine the field strengths $F$ and $H$ with the fields $A$. This is possible
since all these objects individually transform like an algebra element. Apart
from the terms discussed in equations (61) and (62), other possible terms
include:

$\displaystyle V_{AB}(A,B)$ $\displaystyle=Tr_{\mathfrak{g}}\left[(A\wedge
A)\wedge\star_{g}H\right]\hskip 28.45274pt\text{for}\hskip
14.22636ptA\in\Omega^{1}(M;\mathfrak{g})\>,\forall\,n\geq 2\>,$ (162a)
$\displaystyle V_{AB}(A,B)$ $\displaystyle=Tr_{\mathfrak{g}}\left[(A\wedge
A)\wedge\star_{g}F\right]\hskip 28.45274pt\text{for}\hskip
14.22636ptA\in\Omega^{1}(M;\mathfrak{g})\>,\forall\,n\geq 2\>,$ (162b)
$\displaystyle V_{AB}(A,B)$ $\displaystyle=Tr_{\mathfrak{g}}\left[(F\wedge
F)\wedge\star_{g}H\right]\hskip 28.45274pt\text{for}\hskip
14.22636ptA\in\Omega^{0}(M;\mathfrak{g})\>,\forall\,n\geq 2\>,$ (162c)
$\displaystyle V_{AB}(A,B)$ $\displaystyle=Tr_{\mathfrak{g}}\left[(F\wedge
F)\wedge\star_{g}(F\wedge F))\right]\hskip 28.45274pt\text{for}\hskip
14.22636ptA\in\Omega^{k}(M;\mathfrak{g})\>,k\leq n/2\>,$ (162d) $\displaystyle
V_{AB}(A,B)$ $\displaystyle=Tr_{\mathfrak{g}}\left[(F\wedge F\wedge F\wedge
F)\wedge\star_{g}(F\wedge F\wedge F\wedge F)\right]\hskip
28.45274pt\text{for}\hskip
14.22636ptA\in\Omega^{0}(M;\mathfrak{g})\>,\forall\,n\geq 4\>,$ (162e)
$\displaystyle V_{AB}(A,B)$
$\displaystyle=Tr_{\mathfrak{g}}\left[(H\wedge\star_{g}H)\underbrace{\wedge
A\wedge\cdots\wedge A}_{\text{arbitrary amount}}\right]\hskip
28.45274pt\text{for}\hskip
14.22636ptA\in\Omega^{0}(M;\mathfrak{g})\>,\forall\,n\geq 2\>.$ (162f)

This is only a small selection of the infinite amount of possible terms. In
practical applications, some terms might also be excluded for different
reasons. For example, one might exclude topological terms or terms above a
certain order in the field or the connection.

## References

  * [1] Manuel Tecchiolli  “On the Mathematics of Coframe Formalism and Einstein–Cartan Theory—A Brief Review”  In _Universe_ 5.10, 2019, pp. 206  DOI: 10.3390/universe5100206
  * [2] Frank Gronwald and Friedrich W. Hehl  “On the gauge aspects of gravity”  In _International School of Cosmology and Gravitation: 14th Course: Quantum Gravity_ , 1995, pp. 148–198  arXiv:gr-qc/9602013
  * [3] Friedrich W. Hehl, Paul Heyde, G. Kerlick and James M. Nester  “General relativity with spin and torsion: Foundations and prospects”  In _Rev. Mod. Phys._ 48 American Physical Society, 1976, pp. 393–416  DOI: 10.1103/RevModPhys.48.393
  * [4] Thomas A. DeGrand  “Lattice QCD at the end of 2003”  In _Int. J. Mod. Phys. A_ 19, 2004, pp. 1337–1394  DOI: 10.1142/S0217751X04017574
  * [5] Raúl A. Briceño, Zohreh Davoudi and Thomas C. Luu  “Nuclear Reactions from Lattice QCD”  In _J. Phys. G_ 42.2, 2015, pp. 023101  DOI: 10.1088/0954-3899/42/2/023101
  * [6] Charles W. Misner, K.. Thorne and J.. Wheeler  “Gravitation”  San Francisco: W. H. Freeman, 1973
  * [7] Alberto S. Cattaneo, Leon Menger and Michele Schiavina  “Gravity with torsion as deformed $BF$ theory”, 2023  arXiv:2310.01877 [math-ph]
  * [8] David J. Kaup  “Klein-Gordon Geon”  In _Phys. Rev._ 172, 1968, pp. 1331–1342  DOI: 10.1103/PhysRev.172.1331
  * [9] Steven L. Liebling and Carlos Palenzuela  “Dynamical boson stars”  In _Living Rev. Rel._ 26.1, 2023, pp. 1  DOI: 10.1007/s41114-023-00043-4
  * [10] Richard Brito, Vitor Cardoso, Carlos A.. Herdeiro and Eugen Radu  “Proca stars: Gravitating Bose–Einstein condensates of massive spin 1 particles”  In _Phys. Lett. B_ 752, 2016, pp. 291–295  DOI: 10.1016/j.physletb.2015.11.051
  * [11] Vitor Cardoso, Caio F.. Macedo, Kei-ichi Maeda and Hirotada Okawa  “ECO-spotting: looking for extremely compact objects with bosonic fields”  In _Class. Quant. Grav._ 39.3, 2022, pp. 034001  DOI: 10.1088/1361-6382/ac41e7
  * [12] Katsuki Aoki and Masato Minamitsuji  “Resolving the pathologies of self-interacting Proca fields: A case study of Proca stars”  In _Phys. Rev. D_ 106.8, 2022, pp. 084022  DOI: 10.1103/PhysRevD.106.084022
  * [13] Y. Brihaye, Th. Delplace and Y. Verbin  “Proca Q Balls and their Coupling to Gravity”  In _Phys. Rev. D_ 96.2, 2017, pp. 024057  DOI: 10.1103/PhysRevD.96.024057
  * [14] Claudio Lazarte and Miguel Alcubierre  “$\ell$-Proca stars”, 2024  arXiv:2401.16360 [gr-qc]
  * [15] Jae-Weon Lee  “Is dark matter a BEC or scalar field?”  In _J. Korean Phys. Soc._ 54, 2009, pp. 2622  DOI: 10.3938/jkps.54.2622
  * [16] R. Sharma, S. Karmakar and S. Mukherjee  “Boson star and dark matter”, 2008  arXiv:0812.3470 [gr-qc]
  * [17] L. Urena-Lopez and Argelia Bernal  “Bosonic gas as a Galactic Dark Matter Halo”  In _Phys. Rev. D_ 82, 2010, pp. 123535  DOI: 10.1103/PhysRevD.82.123535
  * [18] Tanja Rindler-Daller and Paul R. Shapiro  “Angular Momentum and Vortex Formation in Bose-Einstein-Condensed Cold Dark Matter Haloes”  In _Mon. Not. Roy. Astron. Soc._ 422, 2012, pp. 135–161  DOI: 10.1111/j.1365-2966.2012.20588.x
  * [19] David J.. Marsh and Ana-Roxana Pop  “Axion dark matter, solitons and the cusp–core problem”  In _Mon. Not. Roy. Astron. Soc._ 451.3, 2015, pp. 2479–2492  DOI: 10.1093/mnras/stv1050
  * [20] Riccardo Della Monica and Ivan Martino  “Narrowing the allowed mass range of ultralight bosons with the S2 star”  In _Astron. Astrophys._ 670, 2023, pp. L4  DOI: 10.1051/0004-6361/202245150
  * [21] F.. Guzman and J.. Rueda-Becerril  “Spherical boson stars as black hole mimickers”  In _Phys. Rev. D_ 80, 2009, pp. 084023  DOI: 10.1103/PhysRevD.80.084023
  * [22] F.. Vincent et al.  “Imaging a boson star at the Galactic center”  In _Class. Quant. Grav._ 33.10, 2016, pp. 105015  DOI: 10.1088/0264-9381/33/10/105015
  * [23] João Luís Rosa and Diego Rubiera-Garcia  “Shadows of boson and Proca stars with thin accretion disks”  In _Phys. Rev. D_ 106.8, 2022, pp. 084004  DOI: 10.1103/PhysRevD.106.084004
  * [24] Carlos A.. Herdeiro et al.  “The imitation game: Proca stars that can mimic the Schwarzschild shadow”  In _JCAP_ 04, 2021, pp. 051  DOI: 10.1088/1475-7516/2021/04/051
  * [25] João Luís Rosa, Paulo Garcia, Frédéric H. Vincent and Vitor Cardoso  “Observational signatures of hot spots orbiting horizonless objects”  In _Phys. Rev. D_ 106.4, 2022, pp. 044031  DOI: 10.1103/PhysRevD.106.044031
  * [26] Roldao Rocha  “Minimal geometric deformation of Yang-Mills-Dirac stellar configurations”  In _Phys. Rev. D_ 102.2, 2020, pp. 024011  DOI: 10.1103/PhysRevD.102.024011
  * [27] R. Bartnik and J. Mckinnon  “Particle - Like Solutions of the Einstein Yang-Mills Equations”  In _Phys. Rev. Lett._ 61, 1988, pp. 141–144  DOI: 10.1103/PhysRevLett.61.141
  * [28] Franz E. Schunck and Eckehard W. Mielke  “General relativistic boson stars”  In _Class. Quant. Grav._ 20, 2003, pp. R301–R356  DOI: 10.1088/0264-9381/20/20/201
  * [29] Yves Brihaye, Betti Hartmann and Eugen Radu  “Boson stars in SU(2) Yang-Mills-scalar field theories”  In _Phys. Lett. B_ 607, 2005, pp. 17–26  DOI: 10.1016/j.physletb.2004.12.020
  * [30] Vladimir Dzhunushaliev, Kairat Myrzakulov and Ratbay Myrzakulov  “Boson stars from a gauge condensate”  In _Mod. Phys. Lett. A_ 22, 2007, pp. 273–282  DOI: 10.1142/S0217732307022669
  * [31] Y. Brihaye and Y. Verbin  “Spherical Non-Abelian Solutions in Conformal Gravity”  In _Phys. Rev. D_ 81, 2010, pp. 044041  DOI: 10.1103/PhysRevD.81.044041
  * [32] Amarjit Soni and Yue Zhang  “Hidden SU(N) Glueball Dark Matter”  In _Phys. Rev. D_ 93.11, 2016, pp. 115025  DOI: 10.1103/PhysRevD.93.115025
  * [33] Jhan N. Martinez, Jose F. Rodriguez, Yeinzon Rodriguez and Gabriel Gomez  “Particle-like solutions in the generalized SU(2) Proca theory”  In _JCAP_ 04, 2023, pp. 032  DOI: 10.1088/1475-7516/2023/04/032
  * [34] Mudit Jain  “Soliton stars in Yang-Mills-Higgs theories”  In _Phys. Rev. D_ 106.8, 2022, pp. 085011  DOI: 10.1103/PhysRevD.106.085011
  * [35] Amarjit Soni and Yue Zhang  “Gravitational Waves From SU(N) Glueball Dark Matter”  In _Phys. Lett. B_ 771, 2017, pp. 379–384  DOI: 10.1016/j.physletb.2017.05.077
  * [36] A.. Henriques, Andrew R. Liddle and R.. Moorhouse  “Combined Boson - Fermion Stars: Configurations and Stability”  In _Nucl. Phys. B_ 337, 1990, pp. 737–761  DOI: 10.1016/0550-3213(90)90514-E
  * [37] Chris Kouvaris and Peter Tinyakov  “Can Neutron stars constrain Dark Matter?”  In _Phys. Rev. D_ 82, 2010, pp. 063531  DOI: 10.1103/PhysRevD.82.063531
  * [38] Gianfranco Bertone and Malcolm Fairbairn  “Compact Stars as Dark Matter Probes”  In _Phys. Rev. D_ 77, 2008, pp. 043515  DOI: 10.1103/PhysRevD.77.043515
  * [39] Kathryn M. Zurek  “Asymmetric Dark Matter: Theories, Signatures, and Constraints”  In _Phys. Rept._ 537, 2014, pp. 91–121  DOI: 10.1016/j.physrep.2013.12.001
  * [40] Chris Kouvaris  “The Dark Side of Neutron Stars”  In _Adv. High Energy Phys._ 2013, 2013, pp. 856196  DOI: 10.1155/2013/856196
  * [41] Robin Fynn Diedrichs et al.  “Tidal deformability of fermion-boson stars: Neutron stars admixed with ultralight dark matter”  In _Phys. Rev. D_ 108.6, 2023, pp. 064009  DOI: 10.1103/PhysRevD.108.064009
  * [42] Fabrizio Di Giovanni, Nicolas Sanchis-Gual, Pablo Cerdá-Durán and José Antonio Font  “Can fermion-boson stars reconcile multimessenger observations of compact stars?”  In _Phys. Rev. D_ 105.6, 2022, pp. 063005  DOI: 10.1103/PhysRevD.105.063005
  * [43] Cédric Jockel and Laura Sagunski  “Fermion Proca Stars: Vector Dark Matter Admixed Neutron Stars”  In _Particles_ 7.1, 2024, pp. 52–79  DOI: 10.3390/particles7010004
  * [44] Mikko Partanen and Jukka Tulkki  “Gravity based on internal symmetry of quantum fields”, 2023  arXiv:2310.01460 [gr-qc]
  * [45] Ashoke Sen  “Covariant action for type IIB supergravity”  In _Journal of High Energy Physics_ 2016.7 Springer ScienceBusiness Media LLC, 2016  DOI: 10.1007/jhep07(2016)017
  * [46] Jnanadeva Maharana  “S-duality and compactification of type IIB superstring action”  In _Physics Letters B_ 402.1-2 Elsevier BV, 1997, pp. 64–70  DOI: 10.1016/s0370-2693(97)00444-9
  * [47] Joseph Polchinski  “Dirichlet Branes and Ramond-Ramond Charges”  In _Physical Review Letters_ 75.26 American Physical Society (APS), 1995, pp. 4724–4727  DOI: 10.1103/physrevlett.75.4724
  * [48] Joseph Polchinski and Edward Witten  “Evidence for heterotic — Type I string duality”  In _Nuclear Physics B_ 460.3 Elsevier BV, 1996, pp. 525–540  DOI: 10.1016/0550-3213(95)00614-1
  * [49] Edward Witten  “String theory dynamics in various dimensions”  In _Nuclear Physics B_ 443.1-2 Elsevier BV, 1995, pp. 85–126  DOI: 10.1016/0550-3213(95)00158-o
  * [50] Edward Witten  “Bound states of strings and p-branes”  In _Nuclear Physics B_ 460.2 Elsevier BV, 1996, pp. 335–350  DOI: 10.1016/0550-3213(95)00610-9
  * [51] Edward Witten  “Five-brane effective action in M-theory”  In _Journal of Geometry and Physics_ 22.2 Elsevier BV, 1997, pp. 103–133  DOI: 10.1016/s0393-0440(97)80160-x
  * [52] Dimitri Polyakov  “RR-dilaton interaction in a type IIA superstring”  In _Nuclear Physics B_ 468.1-2 Elsevier BV, 1996, pp. 155–162  DOI: 10.1016/0550-3213(96)00100-9
  * [53] Juan M. Maldacena  “Black Holes in String Theory”, 1996  arXiv:hep-th/9607235 [hep-th]
  * [54] John H. Brodie and Amihay Hanany  “Type IIA superstrings, chiral symmetry, and N = 1 4D gauge theory dualities”  In _Nuclear Physics B_ 506.1-2 Elsevier BV, 1997, pp. 157–182  DOI: 10.1016/s0550-3213(97)00482-3
  * [55] Olaf Hohm, Seung Ki Kwak and Barton Zwiebach  “Double field theory of type II strings”  In _Journal of High Energy Physics_ 2011.9 Springer ScienceBusiness Media LLC, 2011  DOI: 10.1007/jhep09(2011)013
  * [56] John H Schwarz  “Multicharge superstrings”  In _Journal of Physics A: Mathematical and Theoretical_ 46.45 IOP Publishing, 2013, pp. 453001  DOI: 10.1088/1751-8113/46/45/453001
  * [57] M… Hamilton  “The field and Killing spinor equations of M-theory and type IIA/IIB supergravity in coordinate-free notation”, 2016  arXiv:1607.00327 [math.DG]
  * [58] Prof. Bernig  “Analysis auf Mannigfaltigkeiten (lecture notes, German)”, 2020  URL: https://www.uni-frankfurt.de/87238312/AmfkSS20.pdf
  * [59] J.M. Lee  “Manifolds and Differential Geometry”, Graduate studies in mathematics  American Mathematical Society, 2009  URL: https://books.google.de/books?id=QqHdHy9WsEoC
  * [60] L.W. Tu  “An Introduction to Manifolds”, Universitext (1979)  Springer, 2008  URL: https://books.google.de/books?id=y6pIngEACAAJ
  * [61] J. Jost  “Riemannian Geometry and Geometric Analysis”, Universitext  Springer International Publishing, 2017  URL: https://books.google.de/books?id=Dec5DwAAQBAJ
  * [62] MIT S.Helgason  “Introduction To Lie Groups”, 2001  URL: https://ocw.mit.edu/courses/18-755-introduction-to-lie-groups-fall-2004/
  * [63] Roberto Zucchini  “Operational total space theory of principal 2-bundles II: 2-connections and 1- and 2–gauge transformations”  In _J. Geom. Phys._ 156, 2020, pp. 103825  DOI: 10.1016/j.geomphys.2020.103825
  * [64] Merced Montesinos, Ricardo Escobedo, Jorge Romero and Mariano Celada  “Canonical analysis of $n$-dimensional Palatini action without second-class constraints”  In _Phys. Rev. D_ 101.2, 2020, pp. 024042  DOI: 10.1103/PhysRevD.101.024042
  * [65] Bruno J.Barros, Zahra Haghani, Harko Tiberiu and Francisco S..  “Static spherically symmetric three-form stars”, 2021  arXiv:2101.04445 [gr-qc]
  * [66] Tomi S. Koivisto and Nelson J. Nunes  “Inflation and dark energy from three-forms”  In _Phys. Rev. D_ 80 American Physical Society, 2009, pp. 103509  DOI: 10.1103/PhysRevD.80.103509
  * [67] Tomi S. Koivisto, David Fonseca Mota and Cyril Pitrou  “Inflation from N-Forms and its stability”  In _Journal of High Energy Physics_ 2009, 2009, pp. 092–092  URL: https://api.semanticscholar.org/CorpusID:18694402
  * [68] Antonio De Felice and Shinji Tsujikawa  “f(R) theories”  In _Living Rev. Rel._ 13, 2010, pp. 3  DOI: 10.12942/lrr-2010-3
  * [69] Francisco S.. Lobo  “The Dark side of gravity: Modified theories of gravity”, 2008  arXiv:0807.1640 [gr-qc]
  * [70] Shin’ichi Nojiri and Sergei D. Odintsov  “Introduction to modified gravity and gravitational alternative for dark energy”  In _eConf_ C0602061, 2006, pp. 06  DOI: 10.1142/S0219887807001928
  * [71] Shin’ichi Nojiri and Sergei D. Odintsov  “Unified cosmic history in modified gravity: from F(R) theory to Lorentz non-invariant models”  In _Phys. Rept._ 505, 2011, pp. 59–144  DOI: 10.1016/j.physrep.2011.04.001
  * [72] Shinji Tsujikawa  “Modified gravity models of dark energy”  In _Lect. Notes Phys._ 800, 2010, pp. 99–145  DOI: 10.1007/978-3-642-10598-2˙3
  * [73] Alejandro Corichi, Irais Rubalcava-García and Tatjana Vukašinac  “Actions, topological terms and boundaries in first-order gravity: A review”  In _Int. J. Mod. Phys. D_ 25.04, 2016, pp. 1630011  DOI: 10.1142/S0218271816300111
  * [74] V H Hamity and D E Barraco  “First order formalism of f(R) gravity”  In _General Relativity and Gravitation; (United States)_ 25:5, 1993  DOI: 10.1007/BF00756965
  * [75] G.. Volovik  “From elasticity tetrads to rectangular vielbein”  In _Annals Phys._ 447, 2022, pp. 168998  DOI: 10.1016/j.aop.2022.168998
  * [76] Mikhail S. Volkov and Dmitri V. Galt’sov  “Gravitating non-Abelian solitons and black holes with Yang–Mills fields”  In _Physics Reports_ 319.1, 1999, pp. 1–83  DOI: https://doi.org/10.1016/S0370-1573(99)00010-1
  * [77] Yuewen Chen and Shing-Tung Yau  “Stable regular solution of Einstein-Yang-Mills equation”, 2022  arXiv:2210.09861 [gr-qc]
  * [78] G.. Derrick  “Comments on nonlinear wave equations as models for elementary particles”  In _J. Math. Phys._ 5, 1964, pp. 1252–1254  DOI: 10.1063/1.1704233
  * [79] Alberto Diez-Tejedor and Alma X. Gonzalez-Morales  “No-go theorem for static scalar field dark matter halos with no Noether charges”  In _Phys. Rev. D_ 88, 2013, pp. 067302  DOI: 10.1103/PhysRevD.88.067302
  * [80] Sante Carloni and João Luís Rosa  “Derrick’s theorem in curved spacetime”  In _Phys. Rev. D_ 100.2, 2019, pp. 025014  DOI: 10.1103/PhysRevD.100.025014


# Physics-informed Machine Learning for Calibrating Macroscopic Traffic Flow
Models

Yu Tang, Li Jin and Kaan Ozbay This work was in part supported by US NSF Award
CMMI-1949710, USDOT Award # 69A3551747124 via the C2SMART Center, NYU Tandon
School of Engineering, SJTU UM Joint Institute, and J. Wu & J. Sun Endowment
Fund.Y. Tang is with C2SMART Center, Department of Civil & Urban engineering,
Tandon School of Engineering, New York University, USA. L. Jin is with the UM
Joint Institute and with the Department of Automation, Shanghai Jiao Tong
University, China. K. Ozbay is with C2SMART Center, Department of Civil &
Urban engineering, Tandon School of Engineering, New York University, 11201,
USA. (emails<EMAIL_ADDRESS><EMAIL_ADDRESS>kaan.ozbay@nyu.edu).

(Extended Abstract)

Key Words: Physics-informed machine learning, traffic flow models, parameter
identification.

##  1 Introduction

###  1.1 Motivation

Macroscopic traffic flow models have been shown to be capable of reproducing
congestion propagation and explaining complicated phenomena, such as capacity
drops [1] and stop-and-go waves [2]. They provide a solid foundation for the
performance analysis of traffic systems [3, 4] and control design for freeway
management [5, 6]. However, before macroscopic models are put into practice,
they should be carefully calibrated to accurately replicate real-life
complications briefly mentioned above.

Extensive studies have been devoted to the calibration of traffic flow models
[7, 8, 9, 10]. They mainly utilized optimization algorithms to determine model
parameters over a certain period (i.e., morning or evening peaks) with
multiple days’ data, but only a few of them, with the goal of testing for
parameter transferability [10], validated the calibrated models for the same
period but on different days. This transferability can be poor since road
traffic is prone to perturbations induced by demand variations, weather
conditions, driving behavior, and so on, as illustrated by the example in
Figure 1: our data analysis indicates that daily re-calibration is necessary
for quantifying parameter uncertainties.

Figure 1: Fundamental diagrams varying day by day: a) a stretch of Interstate
210 Eastbound (I-210 E) from Irwindale Ave. to Grand Ave. in California, b)-h)
seven days fundamental diagrams revealed by sensor 772872. Clearly, traffic
flow parameters, such as capacity, are subject to day-to-day changes. Obvious
capacity drops happened on 2019-03-19, 2019-03-20 and 2019-03-21.

Admittedly, one can apply optimization-based approaches to day-to-day
calibration, but this can be cumbersome in practice. First, these methods
bring about heavy computation costs when being applied to long-term modeling
up to several months or years. They typically involve non-convex and even non-
smooth problems, and it is hard to solve them using optimization, let alone to
repeat the calibration procedures for each day. The second problem arises from
data quality. Traditional traffic sensors, such as widely-used inductive
loops, are infamously unreliable. For instance, it is reported that only
around 64% of detectors work any given day in the California’s freeway system
[11]. Figure 2 illustrates practical observation rates of induction loops that
are closely related with how many convincing observations are collected.
Clearly, day-to-day optimization methods are vulnerable to the fluctuation in
data quality. However, limited research discussed how to fulfill parameter
identification on corrupted data in a robust way.

Figure 2: Observation rates of sensors along I-210 E from Irwindale Ave. to
Grand Ave. in California, which stand for percentages of observed data, e.g.,
zero percent indicates sensor breakdown without any measurements.

In response to the above challenges, we develop a novel physics-informed,
learning-based approach of identifying traffic flow parameters across days,
including capacity, free flow speed, jam density and congestion wave speed.
The proposed method belongs to the unsupervised machine learning category
since the actual values of parameters are unknown in advance. We inform our
machine learning model of physics knowledge about traffic flows so that it can
achieve the calibration even without ground truths of the parameters. Once
well trained, our method is expected to yield reasonable parameter values
given boundary conditions and traffic measurements with possibly missing
values. It provides efficient calibration over massive periods with robustness
to incomplete data.

Figure 3: Frameworks of autoencoder and our extensions for physics-informed,
learning-based calibration: a) deep autoencoder comprising two neural networks
(NNs), one acting an encoder and the other one acting as a decoder, b)
autoencoder-based parameter identification, c) denoising autoencoder-based
parameter identification.

###  1.2 Our contributions

We try to address two main questions:

  1. (i)

How can we train an encoder, given perfect freeway measurements, to yield
appropriate traffic parameters that are comparable to those calibrated by
traditional methods?

  2. (ii)

Can we also complete the task in (i), even given corrupted traffic data?

To resolve (i), we extend the deep autoencoder [12], a classical unsupervised
machine learning approach for dimension reduction, i.e., converting high-
dimensional data into latent variables, typically with lower dimensions, by
minimizing the error between encoder inputs and decoder outputs; see Figure
3a). Generally, the original low-dimensional representations do not have
significant physical meanings; they cannot be recognized as parameters of
traffic flow models. To address this problem, we also feed encoder outputs
into traffic flow models and inform our decoder of extra discrepancies between
the simulation result and its own output; see Figure 3b). Clearly, minimizing
the new error encourages the decoder to learn physical laws of traffic flows,
and the total error decreases only when the encoder yields appropriate
parameters of traffic flow models. Besides, we introduce the concept of
conditional generation. That is, the decoder relies not only on latent
variables (parameters), but also on boundary conditions, such as upstream
traffic volumes. This is a straightforward but indispensable extension since
traffic observations, mimicked by the decoder, are determined by these
conditions.

Note this paper considers the classical first-order cell transmission model
(CTM). Assuredly, high-order models like Payne-Whitham (PW) model [13, 14],
along with its discretized version METANET [15], and Aw-Rascle-Zhang (ARZ)
model [16, 17] can reproduce traffic phenomena with higher accuracy, but they
have more parameters that could complicate the training process. To the best
of our knowledge, this is the first attempt to estimate traffic flow
parameters using unsupervised training. It is thus worthwhile starting with a
simple model. We leave identification of high-order models as a future
research task. Specially, it will be interesting to investigate whether we can
accelerate it via calibrating first-order models since first- and high-order
models have common traffic flow parameters.

To answer (ii), we integrate denoising autoencoder [18], a simple but robust
variant of the original autoencoder, into the calibration approach. That is,
we use unspoiled sensor readings to generate new data with partially missing
values, which mimics the pattern of real unreliable data, and then apply this
synthesized data to training; see Figure 3c). It allows to deploy the
parameter identification on real data with missing values after training.
Ideally, if there are too many missing values, any method will learn nothing
and end up with poor calibration. Thus we also provide a sensitivity analysis
by controlling the missing rates to empirically reveal how our approach
degenerates.

###  1.3 Related work

Most of the previous work on calibrating macroscopic traffic flow models
employed optimization methods applied over specific days. One approach is to
estimate the fundamental diagram (FD), especially for first-order traffic flow
models [19, 20, 21]. It requires little computation, but may suffer from
accuracy losses. First, fitting the left part of FD, namely free-flow regime,
is usually easy, but the same task can be hard for the right part since
traffic data collected during congestion periods are sparse and scattered.
Besides, individual calibration of FDs along a freeway corridor fails to
capture flow interactions. More studies formulated the calibration problem as
mathematical programming that considered flow dynamics specified by first- or
high-order models [22, 7, 23, 9]. In this case, one needs to solve a non-
convex optimization problem with locally optimal solutions, and thus heuristic
optimization/search algorithms, such as simultaneous perturbation stochastic
approximation, simulated annealing, etc, can be used; please see [10] for a
comprehensive review.

Recently, some researchers discussed the idea of learning-based calibration in
the context of traffic state estimation (TSE). That is, they integrated
traffic flow models into machine learning methods to enhance TSE [24, 25, 26,
27, 28], to incorporate simultaneous parameter identification and state
estimation techniques. It should be pointed out that these studies are
different from what is proposed in this paper. First, they put emphasis on TSE
that used partial observations to infer full states. Thus, they divided
training and testing data by sensor locations to evaluate transferability over
space. By contrast, our method takes in full observations and returns traffic
parameters. We desire transferability over time periods and thereby separate
training and testing data by time. Second, although the current studies can
update traffic parameters, they still require relatively good initial values
that are normally obtained from classical calibration approaches [29] because
poor initial traffic parameters do not guarantee convergence [28]. This kind
of good initial parameter estimates, however, is not necessarily required by
our approach.

Our proposed method is inspired by deep autoencoder-based system
identification which has emerged in recent years. Depending on identification
goals, the latent variables, obtained from the autoencoder, can be recognized
as either states or parameters. If one wants to fit a dynamical model
approximating to the physical one, the encoder gives the states [30, 31, 32];
see the full framework in Figure 4a). If the target is exact parameters of
physical models, the encoder can yield calibration results as well; see Figure
4b). In that case the decoder is a physical model rather than a NN. Up to now,
this framework has only been applied to identification of linear time-
invariant (LTI) systems [33]. Though it has the same objective as our method,
it cannot be directly applied in our problem setting for the following
reasons. First, LTI systems have closed-form solutions, and it is convenient
to compute gradients with respect to parameters, which implies easy training.
However, when physical models, like traffic flow models, are too complicated
to have analytical solutions, gradient calculation becomes very hard by making
the training inefficient. Second, it is assumed that step signals are standard
inputs for all LTI systems and they are thus ignored in the identification
framework. In practice, however, we cannot always manipulate model inputs,
i.e., boundary conditions, and should include them in the learning-based
calibration.

Figure 4: Current frameworks of autoencoder-based system identification: a)
identifying approximate dynamics, b) identifying parameters.

##  2 Problem Statement

In this section, we generally state the calibration problem and then
explicitly present the considered freeway model. It consists of a dynamics
model, the CTM incorporating capacity drop, and an observation model.

###  2.1 Learning-based calibration problem

We consider a freeway corridor with $K$ mainline _cells_ , $K$ on-ramp
_buffers_ , and $K$ off-ramps, as shown in Figure 5. The $k$th buffer has a
state of queue length, denoted by $q_{k}(t)$, and the $k$th cell is
characterized by traffic density, denoted by $\rho_{k}(t)$. Note that the
first buffer is not an actual on-ramp; instead, it represents the upstream
freeway section and stores the upstream mainline traffic. The $k$th buffer is
subject to a time-varying _demand_ $\alpha_{k}(t)\in\mathbb{R}_{\geq 0}$. In
addition, we apply _mainline ratio_ $\eta_{k}(t)\in[0,1]$ to model off-ramp
flows. This ratio denotes the fraction of traffic from cell $k$ entering cell
$k+1$; the remaining traffic flow leaves the freeway at the $k$th off-ramp. We
also assume that the last cell $K$ discharges outflows at a speed of
$v_{K}(t)$, which can be measured as the downstream boundary condition.
Finally, we denote by $f_{k}(t)$ the flows from cell $k$ to the downstream
cell and by $r_{k}(t)$ the flow from buffer $k$ to cell $k$.

Figure 5: A freeway corridor with states $(\rho(t),q(t))$ and model inputs
$u(t):=(\alpha(t),\eta(t),v_{K}(t))$, where
$\rho(t):=[\rho_{1}(t),\rho_{2}(t),\cdots,\rho_{K}(t)]^{\mathrm{T}}$,
$q(t):=[q_{1}(t),q_{2}(t),\cdots,q_{K}(t)]^{\mathrm{T}}$,
$\alpha(t):=[\alpha_{1}(t),\alpha_{2}(t),\cdots,\alpha_{K}(t)]^{\mathrm{T}}$
and $\eta(t):=[\eta_{1}(t),\eta_{2}(t),\cdots,\eta_{K}(t)]^{\mathrm{T}}$.

Now we suppose that the following freeway model:

$\displaystyle\rho(t+1)=$ $\displaystyle F(\rho(t),q(t),u(t);\theta),$ (1a)
$\displaystyle q(t+1)=$ $\displaystyle G(\rho(t),q(t),u(t);\theta),$ (1b)
$\displaystyle y(t)=$ $\displaystyle H(\rho(t),q(t)),$ (1c)

where $F$ and $G$ in (1a)-(1b) denote the dynamics models parameterized by
$\theta$, and $H$ in (1c) is the observation model with the ideal senor output
$y(t)$, i.e., without missing values. We let $y^{\mathrm{obs}}(t)$ represent
real observations of $y(t)$, which can contain missing values.

Clearly, the traffic flow model assumes constant parameters. Though we admit
variation in traffic parameters, it is still acceptable to assume stationary
values over a certain period, i.e., morning or evening peaks [10]. This is the
basis of our method. Our calibration does not address abrupt parameter changes
due to incidents and other unexpected disruptions. This kind of real-time
parameter identification that may be solved by online machine learning
algorithms is a future research task. For convenience of notation, we let
$\theta_{p}$,
$\mathbf{M}_{p}:=(y^{\mathrm{obs}}(0),y^{\mathrm{obs}}(1),\cdots,y^{\mathrm{obs}}(T))_{p}$
and $\mathbf{B}_{p}:=(\rho(0),u(0),u(1),\cdots,u(T))_{p}$ denote the period-
specific parameter, real measurements and boundary conditions, respectively.

The calibration problem is formulated as follows. Suppose
$\mathcal{P}^{\mathrm{train}}$ is a set of periods, each of them with $T+1$
time steps. Given observations
$\\{\mathbf{M}_{p}\\}_{p\in\mathcal{P}^{\mathrm{train}}}$ and boundary
conditions $\\{\mathbf{B}_{p}\\}_{p\in\mathcal{P}^{\mathrm{train}}}$, we aim
at training a machine learning model so that it can calibrate the model
parameters not only on the training data set but also on the testing ones
$\mathcal{P}^{\mathrm{test}}$,
$\\{\mathbf{M}_{p}\\}_{p\in\mathcal{P}^{\mathrm{test}}}$ and
$\\{\mathbf{B}_{p}\\}_{p\in\mathcal{P}^{\mathrm{test}}}$. Since we do not have
true values of $\\{\theta_{p}\\}_{p\in\mathcal{P}^{\mathrm{train}}}$ and
$\\{\theta_{p}\\}_{p\in\mathcal{P}^{\mathrm{test}}}$, we evaluate the
calibrated results $\\{\hat{\theta}_{p}\\}_{p\in\mathcal{P}^{\mathrm{train}}}$
and $\\{\hat{\theta}_{p}\\}_{p\in\mathcal{P}^{\mathrm{test}}}$ by comparing
real measurements with those from the model (1c).

###  2.2 Freeway model

In the following, we first use the CTM to specify the dynamics functions $F$
and $G$, and the parameter $\theta$ in (1a)-(1b). The CTM is favored due to
its simplicity and wide use, and it is not a necessary requirement. Then we
consider induction loops for the observation function $H$ and the sensor
output $y(t)$ in (1c). Although Lagrangian sensors, such as floating cars,
with higher accuracy have been previously studied [34, 35], their observations
could be sparse given low market penetration rates. Besides, there is limited
public access to them. By contrast, we have rich sources of induction loop
data, which supports training and validation of learning-based models.

####  2.2.1 Dynamics model

The flow from buffer $k$ to cell $k$, $r_{k}$, is specified by

$r_{k}(t)=\min\\{\alpha_{k}(t)+q_{k}(t)/\delta_{t},U_{k},w_{k}(\rho^{\max}_{k}-\rho_{k}(t))\\},$
(2)

where $\delta_{t}$ denotes time step size, $U_{k}$ denotes capacity of buffer
$k$, $\rho_{k}^{\max}$ denotes jam density of cell $k$, and $w_{k}$ denotes
congestion wave speed of cell $k$. Then the flows between cells, $f_{k}$ for
$k=1,2,\cdots,K$, are given by

$\displaystyle
f_{k}(t)=\eta_{k}(t)\min\big{\\{}v_{k}\rho_{k}(t),Q_{k}(t),w_{k+1}(\rho^{\max}_{k+1}-\rho_{k+1}(t))-r_{k+1}(t)\big{\\}},~{}1\leq
k\leq K-1$ (3a) $\displaystyle f_{K}(t)=v_{K}(t)\rho_{K}(t)$ (3b)

where $\delta_{t}$ denotes time step size, $v_{k}$ denotes free-flow speed of
cell $k$, $Q_{k}(t)$ denotes capacity of cell $k$, $\rho_{k}^{\max}$ denotes
jam density of cell $k$, and $w_{k}$ denotes congestion wave speed of cell
$k$. The flow functions (3a)-(3b) indicate higher merging priority of on-ramp
flows and the first-in-first-out rule for off-ramp flows [36]. Besides, note
that we consider time-varying capacity $Q_{k}(t)$ which allows to model
capacity drop as follows:

$Q_{k}(t)=\begin{cases}Q^{\mathrm{nominal}}_{k},&\rho_{k}(t)\leq\rho_{k}^{\mathrm{critical}},\\\
Q^{\mathrm{drop}}_{k},&\rho_{k}(t)>\rho_{k}^{\mathrm{critical}},\end{cases}$
(4)

where $\rho_{k}^{\mathrm{critical}}:=Q_{k}^{\mathrm{nominal}}/v_{k}$ denotes
critical density of cell $k$; see more discussions and implementations of
capacity drop in [37].

Then, by the conservation law of flows, the traffic dynamics is given by

$\displaystyle q_{k}(t+1)=$ $\displaystyle
q_{k}(t)+\delta_{t}(\alpha_{k}(t)-r_{k}(t)),~{}1\leq k\leq K,~{}t=0,1,\cdots,$
(5a) $\displaystyle\rho_{1}(t+1)=$
$\displaystyle\rho_{1}(t)+\frac{\delta_{t}}{\ell_{1}}(r_{1}(t)-\frac{f_{1}(t)}{1-\eta_{1}(t)}),~{}t=0,1,\cdots,$
(5b) $\displaystyle\rho_{k}(t+1)=$
$\displaystyle\rho_{k}(t)+\frac{\delta_{t}}{\ell_{k}}(r_{k}(t)+f_{k-1}(t)-\frac{f_{k}(t)}{1-\eta_{k}(t)}),~{}2\leq
k\leq K,~{}t=0,1,\cdots,$ (5c)

where $\ell_{k}$ denotes the length of cell $k$.

Note that the model above assumes infinite-sized buffers. It helps to store
boundary inflows given insufficient cell space that is probably caused by the
selection of inappropriate parameters during the training process. We also
introduce $U_{k}$, $k=1,2,\cdots,K$, to prevent unrealistically large inflows
when there are long queues at buffers. All of these parameters can be
specified in advance. Then the parameters to be calibrated are presented
below:

$\theta=(\\{v_{k}\\}_{k=1}^{K-1},\\{Q_{k}^{\mathrm{nominal}}\\}_{k=1}^{K-1},\\{Q_{k}^{\mathrm{drop}}\\}_{k=1}^{K-1}\\{\rho_{k}^{\max}\\}_{k=1}^{K},\\{w_{k}\\}_{k=1}^{K}).$

####  2.2.2 Observation model

In practice, induction loops update measurements of flow rates and speed at a
certain frequency $\Delta_{t}$ that is larger than the time step size
$\delta_{t}$ of the traffic model. We suppose $\Delta_{t}=m\delta_{t}$ with a
multiple $m\in\mathbb{Z}_{>0}$. Then the sensor outputs are given by

$\displaystyle\bar{r}_{k}(t)=$
$\displaystyle\sum_{i=t-m+1}^{t}r_{k}(i)/m,~{}1\leq k\leq K,~{}t=m,2m,...,$
(6a) $\displaystyle\bar{f}_{k}(t)=$
$\displaystyle\sum_{i=t-m+1}^{t}f_{k}(i)/m,~{}1\leq k\leq K,~{}t=m,2m,...,$
(6b) $\displaystyle\bar{v}_{k}(t)=$
$\displaystyle\sum_{i=t-m+1}^{t}f_{k}(i)v_{k}(i)/\sum_{i=t-m+1}^{t}f_{k}(i),~{}1\leq
k\leq K,~{}t=m,2m,...,$ (6c)

where $v_{k}(t):=f_{k}(t)/\rho_{k}(t)$ denotes traffic speed of cell $k$ at
time $t$. Clearly, (6a)-(6c) yield

$y(t)=[\bar{r}_{1}(t),\bar{r}_{2}(t),\cdots,\bar{r}_{K}(t),\bar{f}_{1}(t),\bar{f}_{2}(t),\cdots,\bar{f}_{K}(t),\bar{v}_{1}(t),\bar{v}_{2}(t),\cdots,\bar{v}_{K}(t)]^{\mathrm{T}},t=m,2m,\cdots.$

##  3 Proposed Method

In this section, we first introduce the autoencoder-based parameter
identification that assumes complete traffic measurements and then present its
extension for corrupted data.

###  3.1 Autoencoder-based parameter identification

At first, the encoder outputs the estimated parameter $\hat{\theta}_{p}$ given
the measurement $\mathbf{M}_{p}$ and the boundary condition $\mathbf{B}_{p}$.
Then the parameter, along with the boundary condition, is passed into the
encoder and the CTM to obtain $\hat{\mathbf{M}}_{p}$ and
$\tilde{\mathbf{M}}_{p}$, respectively. The computations are presented below:

$\displaystyle\hat{\theta}_{p}=$ $\displaystyle
E(\mathbf{M}_{p},\mathbf{B}_{p};w_{E}),$ (7a)
$\displaystyle\hat{\mathbf{M}}_{p}=$ $\displaystyle
D(\hat{\theta}_{p},\mathbf{B}_{p};w_{D}),$ (7b)
$\displaystyle\tilde{\mathbf{M}}_{p}=$ $\displaystyle
C(\hat{\theta}_{p},\mathbf{B}_{p}),$ (7c)

where $E$, $D$, $C$ denote the encoder, the decoder and the CTM. Note that
both the encoder and the decoder are neural networks, parameterized by weights
$w_{E}$ and $w_{D}$ respectively. The estimated parameters $\hat{\theta}_{p}$
should fall within feasible ranges; otherwise it cannot be passed into the
CTM. Thus we exploit the sigmoid function for rescaling in the last layer of
the encoder as follows:

$\hat{\theta}_{p}=(\theta^{\max}-\theta^{\min})\odot\mathrm{sigmoid}(\hat{\theta}^{\mathrm{unscaled}}_{p})+\theta^{\min}$
(8)

where $\odot$ denotes element-wise product,
$\hat{\theta}_{p}^{\mathrm{unscaled}}$ is the output from the previous layer,
and $\theta^{\mathrm{min}}$ (resp. $\theta^{\mathrm{max}}$) is a lower bound
(resp. an upper bound) of traffic parameters.

As for training, we consider minimizing the following loss function:

$\min_{w_{E},w_{D}}L=\min_{w_{E},w_{D}}\sum_{p\in\mathcal{P}^{\mathrm{train}}}||\hat{\mathbf{M}}_{p}-\tilde{\mathbf{M}}_{p}||_{2}^{2}+\gamma\sum_{p\in\mathcal{P}^{\mathrm{train}}}||\mathbf{M}_{p}-\hat{\mathbf{M}}_{p}||_{2}^{2}$
(9)

where $\gamma$ is a loss weight. Clearly, the loss function includes two
parts. Minimizing the first part ensures that the decoder learns the physical
laws of the CTM, and then minimizing the second part induces the encoder to
yield appropriate traffic flow parameters.

###  3.2 Denoising autoencoders-based parameter identification

Considering the observation $\mathbf{M}_{p}$ could have missing values, we
define a binary matrix $\mathbf{I}_{p}$, with the same dimension as
$\mathbf{M}_{p}$, to indicate whether the corresponding data is missing. We
first sample from the set of complete observations and then from
$\\{\mathbf{I}_{p}\\}_{p\in\mathcal{P}^{\mathrm{train}}}$ to assemble a new
observation $\mathbf{M}_{p}^{*}$. Although it may have missing values, their
corresponding ground truths are known. We feed the artificial observation
$\mathbf{M}_{p}^{*}$ to the encoder. That is, instead of (7a), we use the
following

$\hat{\theta}_{p}=E(\mathbf{M}_{p}^{*},\mathbf{B}_{p};w_{E})$ (10)

but we still apply (9) to training.

##  4 Experimental Design

###  4.1 Data preparation

We consider the freeway segment, up to 6.2 kilometers, shown in Figure 1. It
has 6 on-ramps and 4 off-ramps. We divide it into 9 cells based on locations
of on-ramps, off-ramps and sensors. We also collected 5-min traffic flow and
speed data from 2017 to 2019 via the PeMS [38]. The preliminary data analysis
showed recurrent morning and evening peaks. Thus we separately calibrate for
these two periods for each day. We select the data of 2017 and 2018 as the
training dataset and the data of 2019 as the testing dataset.

###  4.2 Preliminary results

We have trained our machine learning model for the calibration given complete
sensor measurements. We set the structure of neural networks, as the encoder
and the decoder, based on the LeNet [39] with customized input and output
layer sizes. We also selected the loss weight $\gamma=0.8$. Figure 6 presents
the training loss.

Figure 6: Training loss of learning-based calibration given intact sensor
data.

We tested the trained model on March 26, 2019. For that, we first used the
encoder to yield the parameters required by the CTM and then applied the
traffic flow model to numerical simulation. Figure 7 and 8 presented the
comparison in terms traffic speed and density, respectively. These results
showed that our machine learning model can yield reasonable calibration so
that the traffic flow model can reproduce the congestion pattern.

Figure 7: Speed evaluation on March 26, 2019: a) ground truth, b) simulation
results. Figure 8: Traffic density evaluation on March 26, 2019: a) ground
truth, b) simulation results.

##  5 Summary and Future Work

In this paper, we propose a physics-informed, learning-based calibration
approach, inspired by autoencoders comprising of one encoder and one decoder,
that is expected to achieve performances comparable to those of optimization-
based methods while not requiring day-to-day optimization after training. We
consider calibrating the CTM, a widely-used traffic flow model. In our
approach, the encoder takes as input traffic measurements and boundary
conditions, and yields parameters required by CTM; the decoder recovers the
measurements from the encoder output and the boundary conditions. Specially,
we feed the decoder input to CTM and inform the autoencoder of a novel error
between the decoder output and the simulation results besides the conventional
error between the traffic measurements and the decoder output. This encourages
the encoder to produce reasonable parameters so that the new error is
minimized. We also introduce the denoising autoencoder into our calibration
method so that it can handles with corrupted data. It is expected to
demonstrate the effectiveness of the proposed method through a case study of
I-210 E in California.

We are currently continuing to train our machine learning model to test the
calibration with corrupted data. In the full version of the paper, we will
also compare our method with several benchmarks. We will first consider the
case of complete sensor measurements. We will then apply case-by-case
optimization on the testing data, one fitting the fundamental diagrams [21]
and the other one solving non-convex optimization [7]. Then, for the case of
corrupted data, we will test these two methods for different scenarios, one
only with raw data and the other one with imputed data.

## References

  * [1] M. M. Khoshyaran and J. P. Lebacque, “Capacity drop and traffic hysteresis as a consequence of bounded acceleration,” _IFAC-PapersOnLine_ , vol. 48, no. 1, pp. 766–771, 2015.
  * [2] J. A. Laval and L. Leclercq, “A mechanism to describe the formation and propagation of stop-and-go waves in congested freeway traffic,” _Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences_ , vol. 368, no. 1928, pp. 4519–4541, 2010.
  * [3] K. Huang, X. Di, Q. Du, and X. Chen, “Scalable traffic stability analysis in mixed-autonomy using continuum models,” _Transportation research part C: emerging technologies_ , vol. 111, pp. 616–630, 2020.
  * [4] X. Shi and X. Li, “Constructing a fundamental diagram for traffic flow with automated vehicles: Methodology and demonstration,” _Transportation Research Part B: Methodological_ , vol. 150, pp. 279–292, 2021.
  * [5] G. Gomes and R. Horowitz, “Optimal freeway ramp metering using the asymmetric cell transmission model,” _Transportation Research Part C: Emerging Technologies_ , vol. 14, no. 4, pp. 244–262, 2006.
  * [6] I. Papamichail, A. Kotsialos, I. Margonis, and M. Papageorgiou, “Coordinated ramp metering for freeway networks–a model-predictive hierarchical control approach,” _Transportation Research Part C: Emerging Technologies_ , vol. 18, no. 3, pp. 311–331, 2010.
  * [7] A. Spiliopoulou, M. Kontorinaki, M. Papageorgiou, and P. Kopelias, “Macroscopic traffic flow model validation at congested freeway off-ramp areas,” _Transportation Research Part C: Emerging Technologies_ , vol. 41, pp. 18–29, 2014.
  * [8] A. Spiliopoulou, I. Papamichail, M. Papageorgiou, Y. Tyrinopoulos, and J. Chrysoulakis, “Macroscopic traffic flow model calibration using different optimization algorithms,” _Operational Research_ , vol. 17, no. 1, pp. 145–164, 2017.
  * [9] S. Mohammadian, Z. Zheng, M. M. Haque, and A. Bhaskar, “Performance of continuum models for realworld traffic flows: Comprehensive benchmarking,” _Transportation Research Part B: Methodological_ , vol. 147, pp. 132–167, 2021.
  * [10] Y. Wang, X. Yu, J. Guo, I. Papamichail, M. Papageorgiou, L. Zhang, S. Hu, Y. Li, and J. Sun, “Macroscopic traffic flow modelling of large-scale freeway networks with field data verification: State-of-the-art review, benchmarking framework, and case studies using metanet,” _Transportation Research Part C: Emerging Technologies_ , vol. 145, p. 103904, 2022.
  * [11] Caltrans, “ _Caltrans Performance Measurement System_ ,” 2022, https://pems.dot.ca.gov/?logout=1.
  * [12] G. E. Hinton and R. R. Salakhutdinov, “Reducing the dimensionality of data with neural networks,” _science_ , vol. 313, no. 5786, pp. 504–507, 2006\.
  * [13] H. Payne, “Models of freeway traffic and control. mathematical models of public systems. simulation councils,” _Inc., Vista, CA, USA_ , 1971.
  * [14] G. B. Whitham, “Linear and nonlinear waves(book),” _New York, Wiley-Interscience, 1974. 651 p_ , 1974.
  * [15] A. Messner and M. Papageorgiou, “Metanet: A macroscopic simulation program for motorway networks,” _Traffic engineering & control_, vol. 31, no. 8-9, pp. 466–470, 1990.
  * [16] A. Aw and M. Rascle, “Resurrection of” second order” models of traffic flow,” _SIAM journal on applied mathematics_ , vol. 60, no. 3, pp. 916–938, 2000\.
  * [17] H. M. Zhang, “A non-equilibrium traffic model devoid of gas-like behavior,” _Transportation Research Part B: Methodological_ , vol. 36, no. 3, pp. 275–290, 2002.
  * [18] P. Vincent, H. Larochelle, Y. Bengio, and P.-A. Manzagol, “Extracting and composing robust features with denoising autoencoders,” in _Proceedings of the 25th international conference on Machine learning_ , 2008, pp. 1096–1103.
  * [19] L. Muñoz, X. Sun, D. Sun, G. Gomes, and R. Horowitz, “Methodological calibration of the cell transmission model,” in _Proceedings of the 2004 American Control Conference_ , vol. 1. IEEE, 2004, pp. 798–803.
  * [20] G. Dervisoglu, G. Gomes, J. Kwon, R. Horowitz, and P. Varaiya, “Automatic calibration of the fundamental diagram and empirical observations on capacity,” in _Transportation Research Board 88th Annual Meeting_ , vol. 15. Citeseer, 2009, pp. 31–59.
  * [21] R. Zhong, C. Chen, A. H. Chow, T. Pan, F. Yuan, and Z. He, “Automatic calibration of fundamental diagram for first-order macroscopic freeway traffic models,” _Journal of Advanced Transportation_ , vol. 50, no. 3, pp. 363–385, 2016.
  * [22] J.-B. Lee and K. Ozbay, “Calibration of a macroscopic traffic simulation model using enhanced simultaneous perturbation stochastic approximation methodology,” Tech. Rep., 2008.
  * [23] S. Mudigonda and K. Ozbay, “Robust calibration of macroscopic traffic simulation models using stochastic collocation,” _Transportation Research Part C: Emerging Technologies_ , vol. 59, pp. 358–374, 2015.
  * [24] J. Huang and S. Agarwal, “Physics informed deep learning for traffic state estimation,” in _2020 IEEE 23rd International Conference on Intelligent Transportation Systems (ITSC)_. IEEE, 2020, pp. 1–6.
  * [25] Y. Yuan, Q. Wang, and X. T. Yang, “Traffic flow modeling with gradual physics regularized learning,” _IEEE Transactions on Intelligent Transportation Systems_ , 2021.
  * [26] Y. Yuan, Z. Zhang, X. T. Yang, and S. Zhe, “Macroscopic traffic flow modeling with physics regularized gaussian process: A new insight into machine learning applications in transportation,” _Transportation Research Part B: Methodological_ , vol. 146, pp. 88–110, 2021.
  * [27] R. Shi, Z. Mo, K. Huang, X. Di, and Q. Du, “A physics-informed deep learning paradigm for traffic state and fundamental diagram estimation,” _IEEE Transactions on Intelligent Transportation Systems_ , 2021.
  * [28] R. Shi, Z. Mo, and X. Di, “Physics-informed deep learning for traffic state estimation: A hybrid paradigm informed by second-order traffic models,” in _Proceedings of the AAAI Conference on Artificial Intelligence_ , vol. 35, no. 1, 2021, pp. 540–547.
  * [29] Y. Yuan, Z. Zhang, and X. T. Yang, “Highway traffic state estimation using physics regularized gaussian process: Discretized formulation,” _arXiv preprint arXiv:2007.07762_ , 2020.
  * [30] D. Masti and A. Bemporad, “Learning nonlinear state–space models using autoencoders,” _Automatica_ , vol. 129, p. 109666, 2021.
  * [31] G. Beintema, R. Toth, and M. Schoukens, “Nonlinear state-space identification using deep encoder networks,” in _Learning for Dynamics and Control_. PMLR, 2021, pp. 241–250.
  * [32] D. Gedon, N. Wahlström, T. B. Schön, and L. Ljung, “Deep state space models for nonlinear system identification,” _IFAC-PapersOnLine_ , vol. 54, no. 7, pp. 481–486, 2021.
  * [33] T. Nagel and M. F. Huber, “Autoencoder-inspired identification of lti systems,” in _2021 European Control Conference (ECC)_. IEEE, 2021, pp. 2352–2357.
  * [34] D. B. Work, O.-P. Tossavainen, Q. Jacobson, and A. M. Bayen, “Lagrangian sensing: traffic estimation with mobile devices,” in _2009 American Control Conference_. IEEE, 2009, pp. 1536–1543.
  * [35] Y. Yuan, J. Van Lint, R. E. Wilson, F. van Wageningen-Kessels, and S. P. Hoogendoorn, “Real-time lagrangian traffic state estimator for freeways,” _IEEE Transactions on Intelligent Transportation Systems_ , vol. 13, no. 1, pp. 59–70, 2012.
  * [36] A. Ferrara, S. Sacone, and S. Siri, _Freeway traffic modelling and control_. Springer, 2018.
  * [37] M. Kontorinaki, A. Spiliopoulou, C. Roncoli, and M. Papageorgiou, “First-order traffic flow models incorporating capacity drop: Overview and real-data validation,” _Transportation Research Part B: Methodological_ , vol. 106, pp. 52–75, 2017.
  * [38] C. Chen, _Freeway performance measurement system (PeMS)_. University of California, Berkeley, 2002.
  * [39] Y. LeCun, L. Bottou, Y. Bengio, and P. Haffner, “Gradient-based learning applied to document recognition,” _Proceedings of the IEEE_ , vol. 86, no. 11, pp. 2278–2324, 1998.