11institutetext: Center for Artificial Intelligence (C4AI) – University of Sao
Paulo, Brazil
11email: {marlon.mathias,wesleyalmeida,marcel.barros,jfialho,lfreitasp2001,
<EMAIL_ADDRESS>
Av. Prof. Lúcio Martins Rodrigues, 370, Butantã, São Paulo, CEP 05508-020
https://c4ai.inova.usp.br/22institutetext: Instituto de Estudos Avançados –
University of São Paulo, Brazil 33institutetext: Escola Politécnica –
University of Sao Paulo, Brazil 44institutetext: Instituto Oceanográfico –
University of Sao Paulo, Brazil

# Augmenting a Physics-Informed Neural Network for the 2D Burgers equation by
addition of solution data points

Marlon S. Mathias 1122 0000-0002-2415-5723 Wesley P. de Almeida 1133
0000-0002-3253-0191 Marcel R. de Barros 1133 0000-0002-0759-2497 Jefferson F.
Coelho 1133 0000-0002-5421-0789 Lucas P. de Freitas 1133 0000-0002-3440-1633
Felipe M. Moreno 1133 0000-0001-9611-1116 Caio F. D. Netto 1133
0000-0002-6627-4931 Fabio G. Cozman 1133 0000-0003-4077-4935 Anna H. R. Costa
1133 0000-0001-7309-4528 Eduardo A. Tannuri 1133 0000-0001-7040-413X Edson S.
Gomi 1133 0000-0003-1267-9519 Marcelo Dottori 1144 0000-0003-2382-4136

###### Abstract

We implement a Physics-Informed Neural Network (PINN) for solving the two-
dimensional Burgers equations. This type of model can be trained with no
previous knowledge of the solution; instead, it relies on evaluating the
governing equations of the system in points of the physical domain. It is also
possible to use points with a known solution during training. In this paper,
we compare PINNs trained with different amounts of governing equation
evaluation points and known solution points. Comparing models that were
trained purely with known solution points to those that have also used the
governing equations, we observe an improvement in the overall observance of
the underlying physics in the latter. We also investigate how changing the
number of each type of point affects the resulting models differently.
Finally, we argue that the addition of the governing equations during training
may provide a way to improve the overall performance of the model without
relying on additional data, which is especially important for situations where
the number of known solution points is limited.

###### Keywords:

Physics-Informed Neural Networks Burgers Equation.

##  1 Introduction

The rapid development of Machine Learning (ML) models has led to significant
progress in various areas and allowed a new approach to problems involving
time-series forecasts. By training these models with large datasets, the
models can recognize complex patterns and address the prediction of physical
systems. However, a purely data-driven approach may also not be entirely
desirable, as such models require substantial amounts of data to train, which
might not always be available, and they may lead to non-physical solutions
when presented to previously unseen scenarios [4].

Physics-Informed Machine Learning models present a compromise between purely
data-driven and purely physics-based approaches in an attempt to combine their
advantages and minimize their shortcomings. The literature presents several
ways of combining both models [5, 9, 12]. These approaches include, but are
not limited to: using ML to estimate the error in the physics-based models
[14]; using ML to increase the resolution of known flow fields [8, 2];
substituting the governing equations by trained neural networks [13]; adding
physical constraints to the ML model [1, 10].

In this work, we implement a Physics-Informed Neural Network (PINN) that uses
the governing equations of the physical system in its fitness evaluation. [7]
calls this approach a neural-FEM, comparing it to the Finite Elements Method
(FEM) of solving partial differential equations. In this analogy, the neural
network works as one large and complex finite element, which spans the whole
domain and solves the equation within its solution space, similarly to the
classical FEM, in which several elements, each with a relatively simple
solution space, are spread along with the domain in a mesh.

In this paper, we will find solutions for the Burgers equation in a two-
dimensional space. Computational Fluid Dynamics researchers widely use this
equation as a toy problem intended to test novel ideas that may be used to
improve fluid dynamics solvers.

One advantage of PINN models over regular physics-based models is that the
neural network may be trained using the governing equations and a set of
points where the solution is known beforehand, comprising a hybrid solution
between physics-driven and data-driven models. This paper compares neural
networks that were trained solely by the governing equations, data points, or
an array of combinations between both scenarios.

In real-world uses of machine learning models, there are situations where a
limited amount of data is available for training. By adding physical knowledge
to the model, it might be possible to reduce the amount of data needed without
impacting its quality. Furthermore, it may aid the model in making accurate
predictions even for previously unseen situations.

The remainder of this paper is structured as follows: In Section 2, we present
the governing equations of our physical system and its boundary conditions and
describe the PINN implementation and training procedure. Section 3 shows the
results obtained after training was complete. Finally, Section 4 finishes with
some conclusions and suggestions for future works.

##  2 Methods

We begin this section by describing the governing equations of the physical
system and its initial and boundary conditions. Then, we proceed to define the
neural network and the loss function that is minimized during training.

###  2.1 Governing Equations

The Burgers equation with two spatial dimensions is given by the system:

$\frac{\partial U}{\partial t}+U\frac{\partial U}{\partial x}+V\frac{\partial
U}{\partial y}=\nu\left(\frac{\partial^{2}U}{\partial
x^{2}}+\frac{\partial^{2}U}{\partial y^{2}}\right),$ (1)

$\frac{\partial V}{\partial t}+U\frac{\partial V}{\partial x}+V\frac{\partial
V}{\partial y}=\nu\left(\frac{\partial^{2}V}{\partial
x^{2}}+\frac{\partial^{2}V}{\partial y^{2}}\right),$ (2)

where $x$ and $y$ are the spatial coordinates, and $U$ and $V$ are the
velocities in each direction, respectively. $t$ is the solution time, and
$\nu$ is the viscosity. If $\nu$ is set to zero, the solution would eventually
lead to extremely high gradients, akin to shock waves in compressible fluids,
by setting a positive value to $\nu$, the right-hand side of both equations
allows some dissipation, which leads to a smoother solution field.

The solution will be evaluated in the domain $0\leq t\leq 1$, $0\leq x\leq 1$,
$0\leq y\leq 1$, with Dirichlet boundary conditions setting both velocities to
zero at the domain boundaries. The initial condition is given by:

$U(t=0,x,y)=\sin(2\pi x)\sin(2\pi y),$ (3)

$V(t=0,x,y)=\sin(\pi x)\sin(\pi y).$ (4)

Figure 1 shows $U$ and $V$ at this condition.

Figure 1: Initial condition for $U$ and $V$.

###  2.2 Physics-Informed Neural Network

The neural network in our implementation is a Multilayer Perceptron (MLP) and
receives coordinates $t$, $x$, and $y$ and outputs the variables $U$ and $V$
for any point of the domain. The MLP network is fully connected and has 4
hidden layers of 20 neurons each, with a hyperbolic tangent activation
function. Larger amounts of layers and neurons were tested with little impact
on the results. Residual connections are added every two layers, which allows
for better computations of the gradients and aids the convergence during
training [3]. There are two outputs: $\tilde{U}$ and $\tilde{V}$, one for each
variable in the problem. The tilde denotes that these variables are the direct
output of the MLP and may not observe the boundary conditions of the problem.

After the MLP, there is a Boundary-encoded output layer, as described by [11].
This layer makes sure that the Dirichlet boundary conditions are always
observed. The layer works by combining the output of the MLP to a known
function that perfectly observes the Dirichlet boundary condition in the
following manner:

$U=d(t,x,y)\tilde{U}+\left(1-d(t,x,y)\right)U_{p},$ (5)

where $U_{p}$ is the particular solution for the velocity in the $x$ direction
and $d$ is a distance function, which is continuous and differentiable and is
equal to zero, where the value of $U$ is set by either a Dirichlet boundary
condition or by the initial condition and non-zero everywhere else. The
concept behind this definition is that the value of $U$ is fixed to $U_{p}$ at
the boundary and initial conditions but remains under the control of the
neural network everywhere else in the domain. This operation is analogous to
$V$. We have defined the following equations for $d$:

$d=16x(1-x)y(1-y)\tanh(\alpha t).$ (6)

The hyperbolic tangent function is used so that the value of $d$ is constant
and close to one when away from the initial condition. $\alpha$ is set to
26.4, which causes $d$ to reach 0.99 at $t=0.1$. The constant 16 is used, so
$d$ becomes close to 1 at the center of the domain. In case other types of
boundary conditions, such as Neumann, were present, they would have to be
implemented via an additional loss function during training, as they cannot be
set by using this method, similarly to the implementation of all boundary
conditions by [9].

We have set the particular solution of this case as $U$ and $V$ being constant
in time and equal to the initial condition values. Therefore:

$U_{p}(t,x,y)=\sin(2\pi x)\sin(2\pi y).$ (7)

$V_{p}(t,x,y)=\sin(\pi x)\sin(\pi y).$ (8)

In summary, Figure 2 shows the final configuration of the network.

Figure 2: Summary of the PINN network.

###  2.3 Loss function

In this work, there are two distinct loss functions whose values are added to
obtain the overall loss. First, we will define the loss function relative to
the observance of the governing equations, and its value is defined as the sum
of the mean absolute value of the residual of each governing equation.

Therefore, given that $U_{i}$ and $V_{i}$ are the PINN’s output at coordinate
($t_{i}$,$x_{i}$,$y_{i}$), as given by equation 5 and its analogous for $V$,
the residuals relative to the governing equations are computed as:

$R_{f,1,i}=\left|\frac{\partial U_{i}}{\partial t}+U_{i}\frac{\partial
U_{i}}{\partial x}+V_{i}\frac{\partial U_{i}}{\partial
y}-\nu\left(\frac{\partial^{2}U_{i}}{\partial
x^{2}}+\frac{\partial^{2}U_{i}}{\partial y^{2}}\right)\right|$ (9)

and

$R_{f,2,i}=\left|\frac{\partial V_{i}}{\partial t}+U_{i}\frac{\partial
V_{i}}{\partial x}+V_{i}\frac{\partial V_{i}}{\partial
y}-\nu\left(\frac{\partial^{2}V_{i}}{\partial
x^{2}}+\frac{\partial^{2}V_{i}}{\partial y^{2}}\right)\right|.$ (10)

The second type of loss is relative to the data-driven part of the training
and is given by the mean absolute distance between the predicted values and
known solution values. Naturally, this loss depends on previous knowledge of
the solution at given domain points.

The loss function of the model is given by:

$\mathcal{L}=\frac{1}{n_{f}}\sum_{i=1}^{n_{f}}\left(R_{f,1,i}+R_{f,2,i}\right)+\frac{1}{2n_{s}}\sum_{i=1}^{n_{s}}\left(\left|U_{i}-U^{gt}_{i}\right|+\left|V_{i}-V^{gt}_{i}\right|\right).$
(11)

Where $U^{gt}_{i}$ and $V^{gt}_{i}$ are the ground truth values of $U$ and $V$
at point $i$. $n_{s}$ is the number of points with known solutions. The
governing equations are evaluated at $n_{f}$ random points uniformly
distributed along with the domain. One characteristic of PINNs is that there
is an infinite pool of points for the loss to be evaluated. Consequently,
there is no limit for the value of $n_{f}$, while $n_{s}$ is limited to the
set of points where the solution is known beforehand.

###  2.4 Implementation

Our code is written in Python and uses the PyTorch module for machine learning
tasks. The governing equations’ partial derivatives are obtained using
PyTorch’s autograd feature. The optimization is performed with the Adam
algorithm. All tests were executed on a Nvidia® GeForce™ RTX3080 graphics
card, with 10 GB of available memory. The code is available on
GitHub111https://github.com/C4AI/PINN_for_Burgers.

###  2.5 Generation of reference data

The data-driven part of the models requires previous knowledge of the solution
at some points in the domain. Therefore, we have used numerical
differentiation and integration techniques to approximate a solution for the
2D Burgers equation. For this, a mesh of 401 nodes in each direction was
uniformly distributed in $x$ and $y$. A sixth-order compact finite differences
scheme [6] was used to approximate the spatial derivatives of the governing
equations, while a fourth-order Runge-Kutta scheme was used to integrate the
solution through time, with a time step of $10^{-5}$. For training, $n_{s}$
points are randomly chosen from this solution. This model was implemented in
Matlab™and will also be available on Github at the time of publication.

##  3 Results

In this section, we compare the solutions obtained by models trained with an
array of values for the number of function evaluation points ($n_{f}$) and the
number of solution sample points ($n_{s}$). We have chosen five values for
each variable: 0, 100, 1000, 10000, and 100000. All combinations of values
were run with the obvious exception of $n_{f}=n_{s}=0$. When $n_{f}=0$, no
knowledge of the governing equation is used, and training is performed solely
from the reference data. Similarly, when $n_{s}=0$, no reference data is used,
and the training is purely physics-based.

These cases will also be compared against our reference data, which was
generated by more conventional techniques, as described in Section 2.5. For
all cases, the value to $\nu$ is set to $0.01/\pi$.

###  3.1 Sample Results

Figure 3 shows contours of both velocity components as modeled by a PINN
trained with $10^{5}$ samples of each type. As the field evolves in time,
large gradients form in the middle of the $x$ domain for the $U$ component and
at the end of the $y$ domain for the $V$ component. This behavior is expected
for the Burgers equation and is representative of shock waves in compressible
gasses. The dissipative part, corresponding to the right-hand side of
equations 1 and 2, causes these large gradients to dissipate as time passes,
as can be seen in the images to the right-hand side of the figure. If $\nu$
were set to zero, we would be solving the inviscid Burgers equation, which
leads to larger and larger gradients, which are notoriously hard to solve with
the techniques presented in this paper.

Figure 3: Contours of $U$ (top) and $V$ (bottom) for five equally spaced
instants from $t=0$ to $t=1$ (left to right), as modeled by a PINN trained
with $n_{f}=n_{s}=10^{5}$.

###  3.2 Comparison to Reference Data

The numerical solution described in Section 2.5 was evaluated in a mesh of 401
by 401 equally spaced nodes, with a time step of $10^{-5}$. Figure 4 shows the
velocity contours for the reference data.

Figure 4: Contours of $U$ (top) and $V$ (bottom) for five equally spaced
instants from $t=0$ to $t=1$ (left to right), as modeled by the finite
differences numerical solver.

The solution by the PINN and by the finite differences solver – Figs. 3 and 4,
respectively – are visually similar for most of the domain. Nonetheless, both
solutions have a clear difference near the $Y=1$ boundary, especially for the
$V$ velocity. This variable presents large gradients in this region,
physically equivalent to a shock wave in a compressible gas. Interestingly, a
similar region of large gradients for $U$ near $X=0$ has reached a much better
agreement between the PINN and the reference data. Figure 5 overlaps contours
of both cases at $t=0.25$. This result leads us to believe that there might be
some influence on the implementation of the boundary conditions; perhaps the
function chosen for $d(t,x,y)$ in Equation 6 is too smooth near the boundaries
and makes it difficult for the MLP to model large gradients in this region.
Further investigation on the optimal choice of $d(t,x,y)$ is needed.

Figure 5: Contours of $U$ (left) and $V$ (right) at $t=0.25$ for the reference
data, in blue, and for the PINN with $n_{f}=n_{s}=10000$, in red.

###  3.3 Governing Equations Residuals

The PINN was trained for all combinations of $n_{f}$ and $n_{s}$ in the range
0, 100, 1000, 10000 and 100000, with the natural exception of $n_{f}=n_{s}=0$.
After training, the governing equation residuals were measured on a uniformly
spaced grid of points across the domain in its three dimensions, one temporal
and two spatial. Figure 6 shows the root mean square of the residuals for each
combination. It is possible to reduce the residual and thus improve the model
by increasing either $n_{f}$ or $n_{s}$. One significant result we can observe
in this figure is when we increase $n_{f}$ from 0 to 100 at the lower values
of $n_{s}$, such as 100 and 1000. This result illustrates that adding physical
knowledge ($n_{f}>0$) to cases where few data points are known may be a key to
increasing the overall model’s accuracy without additional data.

Figure 6: Root mean square of the governing equations’ residuals for different
combinations of function evaluation points ($n_{f}$) and solution sample
points ($n_{s}$).

For further understanding the effect of adding physical knowledge to a data-
based approach, we have plotted the value of $U$ along the $x$ axis (with
$y=0.25$ and $t=0.5$) for various values of $n_{f}$, and $n_{s}$ fixed at
1000. This plot can be seen on the left-hand side of Figure 7. Interestingly,
the case with no physical knowledge ($n_{f}=0$) follows the reference data
more closely than some cases with physical knowledge, despite the higher
residual. Nonetheless, the higher residual can be easily explained by noting
that the values of $U$ for the $n_{f}=0$ case are not as smooth as those with
$n_{f}>0$. Therefore, despite coming close to the absolute value of the
reference data, the derivatives are considerably off, which directly
influences the residuals of the governing equations. For $n_{f}=100$, the
curve is much smoother than for the $n_{f}=0$ case, and, despite being
considerably further from the reference data in absolute values, the residuals
are lower. For $n_{f}=1000$ or higher, the curves are much closer to the
reference data both in terms of absolute values and in terms of its
derivative. The right-hand side of Figure 7 shows a similar plot, but of $V$
with respect to $y$ at $x=0.25$ and $t=0.5$. This time, only the
$n_{f}=100000$ case can approach the reference data in terms of absolute
values, but the overall conclusion remains.

Figure 7: (Left) Plot of $U$ with respect to $x$ at $y=0.25$ and $t=0.5$ for
$n_{s}=1000$ and various values of $n_{f}$. (Right) Plot of $V$ with respect
to $y$ at $x=0.25$ and $t=0.5$ for $n_{s}=1000$ and various values of $n_{f}$.

For completeness, a similar behaviour can be seen when the same plots are made
for both $n_{s}=0$ and varying $n_{f}$ and for $n_{f}=0$ and varying $n_{s}$,
shown in Figures 8 and 9, respectively. For the cases with no sample data
($n_{s}=0$), as the number of function evaluation points increases, the values
of $U$ and $V$ move away from null values and approximate the reference
solution. On the other hand, when the model has no physical reference
($n_{f}=0$), the predictions quickly approximate the reference data, even for
low values of $n_{s}$; however, only models trained with larger values of
$n_{s}$ can reach a smooth prediction, which more closely observes the
governing equations.

Figure 8: (Left) Plot of $U$ with respect to $x$ at $y=0.25$ and $t=0.5$ for
$n_{s}=0$ and various values of $n_{f}$. (Right) Plot of $V$ with respect to
$y$ at $x=0.25$ and $t=0.5$ for $n_{s}=0$ and various values of $n_{f}$.
Figure 9: (Left) Plot of $U$ with respect to $x$ at $y=0.25$ and $t=0.5$ for
$n_{f}=0$ and various values of $n_{s}$. (Right) Plot of $V$ with respect to
$y$ at $x=0.25$ and $t=0.5$ for $n_{f}=0$ and various values of $n_{s}$.

The errors of each model concerning the reference data were estimated by
computing the model at a grid of uniformly spaced points and comparing it to
the reference. Figure 10 summarizes the root mean square of the errors of each
case. It is possible to note that the physics-less case often produces lower
errors when compared to the physics-informed model, especially at the larger
values of $n_{s}$. Nonetheless, by further increasing $n_{f}$, the errors
decrease considerably. This phenomenon can be explained by looking back at
Figure 7, where we have argued that, despite reaching values that are closer
to the reference data in an absolute sense, the solution produced by the
physics-less model is not smooth, which incurs a larger non-observance of the
governing equations.

Figure 10: Root mean square of the error for different combinations of
function evaluation points ($n_{f}$) and solution sample points ($n_{s}$).

###  3.4 Computational Cost

Each case’s memory footprint was measured using PyTorch’s native method
`cuda.max_memory_allocated`, which returns the largest amount of GPU memory
allocated during execution. Naturally, increasing either $n_{f}$ or $n_{s}$
has caused more memory to be allocated. Both values presented a roughly linear
relation to the allocation size. Nonetheless, the number of function
evaluations had a much larger impact than the number of sample points. This
result is caused by the much larger complexity in computing the residuals of
the governing equations as opposed to simply comparing the model’s output to
the reference data. The case with $n_{f}=0$ and $n_{s}=100000$ has allocated
just over 100 MB of memory, while the $n_{f}=100000$ and $n_{s}=0$ has needed
over 2.5 GB. Figure 11 shows the values for each combination of $n_{f}$ and

Figure 11: Maximum allocated GPU memory,in megabytes, for different
combinations of function evaluation points ($n_{f}$) and solution sample
points ($n_{s}$).

##  4 Discussion

In this paper, we implemented a Physics-Informed Neural Network to solve the
2D Burgers equation. During training, this model can use both function
evaluation points, where the residual of the governing equations is sought to
be minimized, and solution sample points, where the solution’s value is known
and training seeks to minimize the difference between the model’s output and
the reference data. The network was trained with different amounts of function
evaluation points ($n_{f}$) and solution sample points ($n_{s}$), and the
resulting models were compared.

By looking at the residual of the governing equations, we improved the models
by increasing either $n_{f}$ or $n_{s}$. Nonetheless, we have observed that
this improvement happens in different manners for each variable. Increasing
$n_{s}$ causes the model to approximate the reference values quickly. However,
only larger sample sizes can reach smooth curves, which more closely follow
the governing equations. On the other hand, by setting $n_{s}$ to zero and
relying only on function evaluation points, the solution obtained is usually
smooth, but it can only approach the actual solution for larger batch sizes
($n_{f}\geq 10000$ in our case).

These results can be beneficial for scenarios where the availability of
reference data is limited. Adding function evaluation points allows one to
improve the model without requiring additional data. Nonetheless, care must be
taken so that enough function evaluation points are used. Otherwise, training
may lead to solutions that do not follow the reference data, despite having
lower residuals in the governing equations.

For future works, we suggest investigating how the balance between the
physics-based loss and the data-based loss influences the resulting model.
Perhaps different loss functions would be able to take advantage of the
lowered governing equation residuals offered by adding function evaluation
points without the downside of causing the model to drift away from the
reference data, as we have observed at low values of $n_{f}$.

## Acknowledgements

This work was carried out at the Center for Artificial Intelligence (C4AI-
USP), with support by the São Paulo Research Foundation (FAPESP) under grant
number 2019/07665-4, and by the IBM Corporation. This work is also supported
in part by FAPESP under grant number 2020/16746-5, the Brazilian National
Council for Scientific and Technological Development (CNPq) under grant
numbers 310085/2020-9, 310127/2020-3, 312180/2018-7, Coordination for the
Improvement of Higher Education Personnel (CAPES, Finance Code 001), and by
Itaú Unibanco S.A. through the Programa de Bolsas Itaú (PBI) program of the
Centro de Ciência de Dados (C2D) of Escola Politécnica of USP.

## Preprint disclaimer

This preprint has not undergone peer review or any post-submission
improvements or corrections. The Version of Record of this contribution is
published in the Lecture Notes in Computer Science book series (LNAI,volume
13654), and is available online at
https://doi.org/10.1007/978-3-031-21689-3_28

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# A Deep Reinforcement Learning based Approach for NOMA-based Random Access
Network with Truncated Channel Inversion Power Control

Ziru Chen1, Ran Zhang2, Lin X. Cai1, Yu Cheng1, and Yong Liu3  1Department of
Electrical and Computer Engineering, Illinois Institute of Technology,
Chicago, USA  2 College of Engineering and Computing, Miami University,
Oxford, OH, USA  3 Department of Physics and Telecommunication Engineering,
South China Normal University, Guangzhou, China<EMAIL_ADDRESS><EMAIL_ADDRESS>{lincai<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

As a main use case of 5G and Beyond wireless network, the ever-increasing
machine type communications (MTC) devices pose critical challenges over MTC
network in recent years. It is imperative to support massive MTC devices with
limited resources. To this end, Non-orthogonal multiple access (NOMA) based
random access network has been deemed as a prospective candidate for MTC
network. In this paper, we propose a deep reinforcement learning (RL) based
approach for NOMA-based random access network with truncated channel inversion
power control. Specifically, each MTC device randomly selects a pre-defined
power level with a certain probability for data transmission. Devices are
using channel inversion power control yet subject to the upper bound of the
transmission power. Due to the stochastic feature of the channel fading and
the limited transmission power, devices with different achievable power levels
have been categorized as different types of devices. In order to achieve high
throughput with considering the fairness between all devices, two objective
functions are formulated. One is to maximize the minimum long-term expected
throughput of all MTC devices, the other is to maximize the geometric mean of
the long-term expected throughput for all MTC devices. A Policy based deep
reinforcement learning approach is further applied to tune the transmission
probabilities of each device to solve the formulated optimization problems.
Extensive simulations are conducted to show the merits of our proposed
approach.

###### Index Terms:

NOMA, random access, truncated channel inversion power control, Deep
reinforcement learning

##  I Introduction

The numbers of machine-type communication (MTC) devices and the corresponding
mobile data volume have grown rapidly with the development of smart metering,
smart traffic surveillance, environmental monitoring, smart grid and other
Internet of Things (IoT) applications. More than $14.7$ billion machine type
devices are anticipated to be connected to the internet by the year of 2023
[1]. Due to the fact that the channel conditions are too costly to measure and
update for low power machines, random access (RA) has attracted great
attention for MTC networks [2].

Non-orthogonal multiple access (NOMA), as a promising technology in 5G and
Beyond, allows more than one device sharing the same time-frequency resource
block, which improves the spectral efficiency considerably [3]. Recently, the
notion of NOMA is applied to slotted ALOHA system in order to achieve higher
throughput for MTC network [4, 5]. Specifically, the transmission power of
devices need to be tuned according to the channel state information (CSI) to
guarantee that the received signal strength equals to one of some predefined
values at the receiver. By empowering each device to decide its transmission
power with different probabilities, NOMA significantly improves the network
throughput performance by resolving collisions via successive interference
cancellation (SIC) technique. However, due to the power limitation, IoT
devices may have various constraints in a realistic IoT network. For instance,
some IoT devices may not be able to transmit in all power levels. In order to
analyze this scenario, the author in [6] developed an analytical model, in
which two type of devices has been considered and two power levels are
available at the receiver side. Based on the analytical model, two algorithms
have been proposed to find the optimal transmission probabilities to attain
the maximum throughput and max-min fairness respectively. However, the
analytical model requires the knowledge of the number of devices for each
type, and the proposed algorithms cannot be applied to the case with more than
two power levels are available in the system due to the complexity of
formulation.

Machine learning (ML) plays an very important role in human life these days.
Reinforcement learning (RL), as a promising solution to handle many ML
problems, has been applied extensively in NOMA based MTC networks in recent
years. In order to minimize random access channel collision, a Q-learning
algorithm has been implemented for each MTC device to dynamically select RA
slots and transmit power for its transmission [7]. The authors in [8] extended
the work in [7] by further considering the short-packet communication and
imperfect successive interference cancellation. In [9], the case with
unsaturated traffic has been further considered. However, the proposed
Q-learning methods in [7, 8, 9] cannot handle the situation when the number of
supported devices changes dynamically due to the limitation of the Q table.
Besides, the fairness between devices cannot be guaranteed. In order to tackle
those problems, NOMA based slotted ALOHA scheme may better suit the MTC
network. The author in [10] applied reinforcement learning method to an
adaptive NOMA based p-persistent slotted ALOHA protocol. However, the reversed
power control has not been considered which may cause the performance
degradation.

In this work, we study the network throughput performance of the NOMA-based
slotted ALOHA in MTC network. To capture the realistic power constraints of
IoT devices and stochastic wireless fading channels, truncated channel
inversion power control is considered. We first analyze the power level design
strategy in MTC network. With given power level design, devices in the network
are categorized into different types, with some types of devices being capable
to utilize all power levels, while other types of devices only using lower
power levels. In order to guarantee the fairness between different types of
devices, two optimization problems have been formulated, namely, to maximize
the minimum long-term expected throughput and to maximize the geometric mean
of the long-term expected throughput for devices in the network. To solve the
optimization problems effectively, a stateless deep RL learning approach has
been proposed. To the best of our knowledge, our approach is the first to
optimize network performance of the NOMA based slotted ALOHA in MTC network
with truncated channel inversion power control, in which more than two types
of devices exist in the network. Extensive simulations have been conducted to
validate the performance of our proposed approach.

The remainder of the paper is organized as follows. Section II describes the
system model. The power level design analysis is presented in Section III.
Optimization problems are formulated and a deep RL approach has been proposed
to solve the optimization problem in Section IV. The performance of our
proposed approach has been validated and analyzed in Section V, followed by
concluding remarks and future work in Section VI.

##  II System Model

Figure 1: System model of a NOMA based uplink MTC network.

As depicted in Fig. 1, a single cell uplink MTC network system has been
considered, in which an access point (AP) is located at the center of a
circular coverage area with a radio of $R$ meters, and multiple IoT devices
are randomly distributed in its coverage area. The number of IoT devices may
change dynamically to capture the mobility feature of the MTC network.

During the uplink transmission, IoT devices send data to the AP by using NOMA
based $p$-persistent slotted ALOHA protocol. Specifically, time is slotted.
Power-domain NOMA is exploited to allow the AP to receive data from multiple
IoT devices in each time slot. During each transmission, IoT devices can
adjust its transmission power to ensure the received signal strength at the AP
side belongs to a fixed predefined set $\mathcal{V}$ where
$\mathcal{V}=\\{V_{1},V_{2},\dots,V_{M}\\}$, and $V_{1}<V_{2}<\dots<V_{M}$.
Due to the fact that IoT devices have limited transmission power, truncated
channel inversion power control has been considered in this scenario. This is,
IoT devices in the coverage area may not be able to adjust their transmission
power to achieve all power levels. Therefore, $M$ type of devices could exist
in the system. i.e., The first type of devices can only utilize the power
level $V_{1}$ as their received signal strength at the AP side. The second
type of devices can use all power levels less than or equal to $V_{2}$ and
etc. Thus, the $M$-th type of devices can exploit all $M$ power levels. Let
$\mathcal{N}=\\{\mathcal{N}_{1},\mathcal{N}_{2},\dots,\mathcal{N}_{M}\\}$ be
the set of IoT devices in the system, in which $\mathcal{N}_{n}$ denotes the
set of $n$-th type of IoT devices and $1\leq n\leq M$.

Similar to $p$-persistent slotted ALOHA protocol, a transmission probabilities
matrix $\mathbb{P}$ has been introduced to guide the uplink transmission of
IoT devices. The transmission probabilities matrix is updated by AP and will
broadcast to all devices after every $T$ time slots. If we use $m$ to indicate
the index of the power level, the probability of a $n$-th type of device to
transmit by using $m$-th power level is denoted as $\tau_{n,m}$. Since the
transmission probability for each type of devices is less than or equal to 1,
we have $\sum_{m=1}^{M}\tau_{n,m}\leq 1,\forall n\leq M$. And
$\tau_{n,m}=0,\forall n<m$ due to the fact that $n$-th type of devices cannot
transmit with $m$-th power level when $n<m$. Thus, the transmission
probabilities matrix $\mathbb{P}$ is a triangular matrix which can be written
as

$\mathbb{P}=\begin{bmatrix}\tau_{1,1}&0&\cdots&0\\\
\tau_{2,1}&\tau_{2,2}&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
\tau_{M,1}&\tau_{M,2}&\cdots&\tau_{M,M}\\\ \end{bmatrix}.$

Multiple devices may transmit simultaneously during each time slot in a random
access network. If we use $\mathcal{N}^{\prime}_{m}$ to denote the set of
devices transmitting by using $V_{m}$ as received power, the received signal
at the AP side can be written as

$\displaystyle
y=\sum_{V_{m}\in\mathcal{V}}\sum_{l\in\mathcal{N}^{\prime}_{m}}\sqrt{V_{m}}x_{l}+z,$
(1)

where $x_{l}$ denotes the transmitted signals of the device $l$, and $z$ is
the background noise.

The AP can decode signals sequentially by applying SIC technique based on the
descending order of the signal strength. Specifically, the AP starts the
decoding from the signal with the highest receiving power under the
interference from all other signals which is transmitting concurrently.
Without loss of generality, we assume that the AP can decode the signal
successfully only when the SINR of the decoding signal larger than or equal to
a threshold $\Gamma$. Once the signal has been successfully decoded, it will
be canceled by the AP. Thus, the rest signals would not be interfered by it.
Throughput of the signal that has been decoded successfully is given by

$\displaystyle\mathbf{Th}=\log_{2}(1+\mathbf{SINR}).$

In contrast, if the signal has not been decoded successfully, it cannot be
canceled and will interfere the decoding of the following signals. The
corresponding throughput of the signal is thus $0$. It is worth noting that
since the AP decode signals on the descending order of the signal strength,
once a signal with high power level cannot be decoded successfully, all
following signals cannot be decoded successfully as well. It is possible that
there are more than one devices transmitting by using the same power level. In
this case, the AP decodes their signals sequentially in random order.

If all signals before the $i$-th decoding signal with the power level $V_{m}$
have been decoded and canceled successfully, the interference comes from all
following signals that haven’t been decoded. In this case, the SINR of the
signal can be written as

$\displaystyle\mathbf{SINR}_{m}^{i}=\frac{V_{m}}{V_{m}(|\mathcal{N}^{\prime}_{m}|-i)+\sum_{x<m}V_{x}|\mathcal{N}^{\prime}_{x}|+\delta^{2}},$
(2)

in which $|\cdot|$ denotes the number of devices in the corresponding device
set and $i\leq|\mathcal{N}^{\prime}_{m}|$, $\delta^{2}$ is the normalized
background noise.

##  III  Power Level Analysis

In this section, we analyze the constraints of power levels after inversion
power control. To take the advantage of NOMA, the predefined power level set
$\mathcal{V}$ needs to satisfy that

$\displaystyle\frac{V_{m}}{\sum_{x<m}V_{x}+\delta^{2}}\geq\Gamma,\forall
1<m\leq M$ (3)

and $V_{1}/\delta^{2}\geq\Gamma$.

######  Proposition 1.

If we use $V_{max}$ to indicate the maximum achievable power level of the
system, the maximum number of power levels we can have is given by

$\displaystyle
M=\lfloor\frac{\log(\frac{V_{max}}{\delta^{2}\Gamma})}{\log(1+\Gamma)}+1\rfloor,$
(4)

in which $\lfloor x\rfloor$ denotes the highest integer smaller than or equal
to $x$.

###### Proof.

To find the maximum number of power levels, the gap between all power levels
should be as small as possible. Under this circumstance, equation (3) can be
rewritten as follow,

$\displaystyle V_{m}$ $\displaystyle=\Gamma(\sum_{x<m}V_{x}+\delta^{2})$
$\displaystyle=\Gamma V_{m-1}+\Gamma(\sum_{x<m-1}V_{x}+\delta^{2})$
$\displaystyle=\Gamma V_{m-1}+V_{m-1}.$ (5)

Thus, we have $V_{m}=V_{max}/(1+\Gamma)^{(M-m)}$. To ensure
$V_{1}\geq\delta^{2}\Gamma$,

$\displaystyle\frac{V_{max}}{(1+\Gamma)^{(M-1)}}$
$\displaystyle\geq\delta^{2}\Gamma$
$\displaystyle\frac{V_{max}}{\delta^{2}\Gamma}$
$\displaystyle\geq(1+\Gamma)^{(M-1)}$
$\displaystyle\frac{\log(\frac{V_{max}}{\delta^{2}\Gamma})}{\log(1+\Gamma)}+1$
$\displaystyle\geq M.$ (6)

∎

It is also possible to design power levels in other way as long as equation
(3) is satisfied, and the deep RL method we proposed can also solve the
problem. However, in the following paper, we will focus on the case that
$V_{m}=V_{max}/(1+\Gamma)^{(M-m)}$ which helps us to utilize more power levels
for the MTC network.

##  IV Proposed RL Method

###  IV-A Optimization Problems

Our goal is to maximize the network performance by tuning the transmission
probabilities matrix $\mathbb{P}$. The most commonly used performance matrix
for IoT network is the total expected throughput. However, total expected
throughput and fairness are generally conflicting performance metrics in
heterogeneous IoT networks. Specifically, devices belong to the same type can
achieve long term fairness in a random access network, yet devices belong to
different types use different transmission probabilities and achieve different
expected throughput. The NOMA transmission may favor the type of devices which
could help to achieve the highest total expected throughput and stop the
transmission of other type of devices to avoid channel congestion. With
considering the fairness, two optimization problems has been considered. The
first objective function we considered in this paper is to achieve the max-min
fairness of devices in the network. In this case, we formulate the decision
problem of tuning $\mathbb{P}$ as a optimization problem

$\displaystyle(\mathbf{P1})\qquad$
$\displaystyle\underset{\mathbb{P}}{\text{maximize}}$
$\displaystyle\min\left(\mathbf{\overline{Th}}_{l}\right),\qquad\forall
l\in\mathcal{N},$ (7)

in which $\mathbf{\overline{Th}}_{l}$ is the expected throughput of device $l$
over $T$ time slots.

Geometric mean of the expected throughput for all devices, on the other hand,
is a performance matrix that also considered the fairness[10]. It is zero if
any device in the network do not have chance to transmit. With the increasing
of the geometric mean, we ensure that no device is starved without any chance
to transmit, and the expected throughput of most devices are increasing. The
corresponding optimization problem can be written as

$\displaystyle(\mathbf{P2})\qquad$
$\displaystyle\underset{\mathbb{P}}{\text{maximize}}$
$\displaystyle\left(\prod_{l\in\mathcal{N}}\mathbf{\overline{Th}}_{l}\right)^{1/|\mathcal{N}|},$
(8)

where $|\mathcal{N}|$ is the number of devices in the IoT network.

The objective functions in $\mathbf{P1}$ and $\mathbf{P2}$ are mathematically
intractable. Therefore, we propose a data-driven approach where a policy-based
deep RL agent is applied at the AP to learn the transmission probabilities
matrix $\mathbb{P}$ automatically.

###  IV-B Deep RL basic

The goal of a RL approach is to find an optimal strategy, i.e., a sequence of
actions that maximizes the long-term expected accumulated discounted reward.
Policy based RL methods, specifically, are well-known in addressing tasks with
continuous action space. There are several policy based RL algorithms which
has been developed recently, i.e., REINFORCE, trust region policy optimization
(TRPO), deep deterministic policy gradient (DDPG), and proximal policy
optimization (PPO). Among these algorithms, PPO draws great attention due to
the fact that it is efficient, easy to be implemented and tuned [11].

In policy based RL algorithms, the most commonly used estimator is given by

$\displaystyle
L(\pi_{\theta_{a}})=E_{\tau}[\pi_{t}(a_{t}|s_{t};\theta_{a})A_{t}(s_{t},a_{t})],$
(9)

where $s_{t}$ and $a_{t}$ represent state and action at time $t$ respectively,
$\pi_{t}(a_{t}|s_{t};\theta_{a})$ represents the policy at time $t$ and
$\theta_{a}$ is the parameter of actor neural network which is used to
generate policy $\pi_{t}(a_{t}|s_{t};\theta_{a})$. With given parameter
$\theta_{a}$, the action can be generated by using Gaussian distribution,

$\displaystyle\pi_{t}(a_{t}|s_{t};\theta_{a})=\frac{1}{\sqrt{2\pi\sigma(s_{t},\theta_{\sigma})^{2}}}\exp-\frac{(a_{t}-\mu(s_{t},\theta_{u})^{2})}{2\sigma(s,\theta_{\sigma})^{2}},$
(10)

in which $\mu(s_{t},\theta_{u})$ and $\sigma(s,\theta_{\sigma})$ are generated
parameters from actor neural network. $A_{t}(s_{t},a_{t})=G_{t}-B_{t}$ is the
advantage value where $G_{t}$ is the discounted future reward after time $t$
and $B_{t}$ is baseline.

It is worth noting that in PPO, data generated in previous episode can also be
used to update current policy. In order to reuse the historical data, a
clipping function is used to avoid large changes between current updated
policy and the old policy. The clip function is given by,

$\displaystyle
clip(y,1-\epsilon,1+\epsilon)\\!\\!=\\!\\!\begin{cases}y,&\mbox{if
$1-\epsilon<y<1+\epsilon$},\\\ 1-\epsilon,&\mbox{if $y\leq 1-\epsilon$},\\\
1+\epsilon,&\mbox{if $y\geq 1+\epsilon$}.\end{cases}$ (11)

The changes between current updated policy and old policy can be written as

$\displaystyle
r_{t}=\frac{\pi_{t}(a_{t}|s_{t};\theta_{new})}{\pi_{t}(a_{t}|s_{t};\theta_{old})}.$
(12)

Thus, the new estimator can be modified as

$\displaystyle
L(\pi_{\theta_{a}})=E_{\tau}[min(r_{t}A_{t},clip(r_{t},1-\epsilon,1+\epsilon)A_{t})].$
(13)

It can be found that if $A_{t}(s,a)$ is negative, the estimator is bounded by
$(1-\epsilon)A_{t}$. On the other hand, when $A_{t}(s,a)$ is positive, the
estimator is at most $(1+\epsilon)A_{t}$.

As an actor-critic algorithm, PPO learns the baseline by using a critic neural
network. The loss function of the critic network is given by

$\displaystyle Loss(s_{t},\theta_{v})=G_{t}-V(s_{t};\theta_{v}),$ (14)

where $V(s_{t};\theta_{v})$ is the value generated by the critic neural
network and $\theta_{v}$ is the parameter of the critic network. During each
episode, the actor neural network optimize the estimator $L(\pi_{\theta_{a}})$
in (13) with respect to $\theta_{a}$ and minimize the loss function
$Loss(s_{t},\theta_{v})$ with respect to $\theta_{v}$.

###  IV-C Overview of Our Approach

A stateless deep RL approach has been taken to solve our optimization problems
$\mathbf{P1}$ and $\mathbf{P2}$, in which RL agent (PPO network) at the AP
generates a transmission probabilities matrix $\mathbb{P}$ based on given
power level set $\mathcal{V}$; once $\mathbb{P}$ has been generated, AP
broadcasts it to all devices in the network. AP knows how many devices exist
in the network but do not know the type of each device. Devices then start to
upload packets to the AP with corresponding transmission probabilities in
$\mathbb{P}$. Without loss of generality, devices are aware of their own type,
so the transmission probabilities can be decided by a device once $\mathbb{P}$
is received. After $T$ time slot, the AP calculates the reward, updates the
actor critic network and generates new transmission probabilities matrix. The
stateless deep RL problems can be formulated as Markov Decision Process
(MDPs), consisting of two key elements:

  * •

actions: the transmission probabilities matrix $\mathbb{P}$.

  * •

reward: the reward collected during $T$ time slots.(e.g., Minimum expected
throughput, Geometric mean of the expected throughput).

####  IV-C1 Actions

Recall that for the transmission probabilities matrix, we need to guarantee
$\sum_{m=1}^{n}\tau_{n,m}\leq 1,\forall n\leq M$. One way to generate
reasonable $\mathbb{P}$ is to introduce the Beta distribution. Beta
distribution defines on the interval $[0,1]$, in which two positive parameters
$\alpha$ and $\beta$ control the shape of the distribution. For $n$-th type of
devices, in order to generate $\tau_{n,m},\forall 1\leq m\leq n$, we first
generate continuous number $a_{n}$ and $b_{n}$ from the agent, then calculate
$\alpha_{n}=\exp(a_{n})$, $\beta_{n}=\exp(b_{n})$ to ensure $\alpha_{n}$ and
$\beta_{n}$ are larger than or equal to 0. With given $\alpha_{n}$ and
$\beta_{n}$, the cumulative distribution function (CDF) of the beta
distribution can be calculated easily. Let $F_{n}(x)$ be the value of the CDF
at $x$, we have $F_{n}(0)=0$ and $F_{n}(1)=1$. The probability that $n$-th
type of devices transmit by using $m$-th power level $\tau_{n,m}$ can be
calculated as

$\displaystyle\tau_{n,m}=F_{n}\left(\frac{m}{n+1}\right)-F_{n}\left(\frac{m-1}{n+1}\right),\forall
1\leq m\leq n.$ (15)

Thus, the probability that the $n$-th type of devices do not transmit is
$1-F_{n}(\frac{n}{n+1})$.

####  IV-C2 Reward

In order to find the $\mathbb{P}$ that achieves the max-min fairness, the
reward is defined as the following:

$\displaystyle R=\min\left(\mathbf{\overline{Th}}_{l}\right),\qquad\forall
l\in\mathcal{N}.$ (16)

If the goal is to find the $\mathbb{P}$ which maximizes the geometric mean of
the expected throughput for all devices, the reward is given by

$\displaystyle
R=\left(\prod_{l\in\mathcal{N}}\mathbf{\overline{Th}}_{l}\right)^{1/|\mathcal{N}|}.$
(17)

We summarize our proposed approach in $\mathbf{Algoithm1}$.

Goal: $\mathbb{P}$;

Initialize: $T$, $\mathcal{V}$, $\Gamma$;

while  _True_ do

1\. Generates $\\{a_{1},b_{1},a_{2},b_{2},\dots,a_{M},b_{M}\\}$ through actor
neural network;

2\. Calculate
$\\{\alpha_{1},\beta_{1},\alpha_{2},\beta_{2},\dots,\alpha_{M},\beta_{M}\\}$;

3\. Generate $\mathbb{P}$ by using (15);

4\. Broadcast $\mathbb{P}$ to all IoT devices in the network;

5\. Wait $T$ time slots and calculate $\mathbf{\overline{Th}}_{l},\forall
l\in\mathcal{N}$;

6\. Calculate reward $R$ by using (16) or (17);

7\. Update weights of actor neural network and critic neural network by
optimizing (13) and minimizing (14) respectively;

end while

Algorithm 1 Agent Learning Process

##  V Performance Evaluation

(a) Total expected throughput as the reward function.

(b) Geometric mean of the expected throughput as the reward function.

(c) Minimum expected throughput as the reward function.

Figure 2: Comparison between different reward functions with respect to the
system performance.

(a) Total expected throughput as the reward function.

(b) Geometric mean of the expected throughput as the reward function.

(c) Minimum expected throughput as the reward function.

Figure 3: Comparison between different reward functions with respect to the
performance of different type of devices.

The performance of a NOMA-based random access network with truncated channel
inversion, which adopts the proposed deep RL approach has been evaluated in
this section. The IoT network parameters we used in the simulation are $M=5$,
$T=3000$, $V_{max}=16$,
$\\{|\mathcal{N}_{1}|,|\mathcal{N}_{2}|,\dots,|\mathcal{N}_{5}|\\}=\\{5,5,8,10,12\\}$,
$\Gamma=1$, $\delta^{2}=1$ if not otherwise specified. The power level set can
be calculated as mentioned in Lemma. 1, in which $\mathcal{V}=[16,8,4,2,1]$.
The parameters for the deep RL approach are shown in follow: learning rate of
the actor and critic network are $1e^{-4}$, and the clipping parameter
$\epsilon$ is $0.3$.

In Fig. 2, the network performance such as the arithmetic mean of the expected
throughput, geometric mean of the expected throughput, and the minimum
expected throughput, are shown for the cases of (a) using total expected
throughput as the reward function, (b) using geometric mean of the expected
throughput as the reward function, and (c) using the minimum expected
throughput as the reward function respectively. It can be intuitively seen
that our approach has fast convergence speed and is capable to find the
optimal solution. The corresponding average throughput for each type of
devices are shown in Fig. 3.

For the case (a), after convergence, the geometric mean of the expected
throughput and the minimum expected throughput are all zeros as shown in Fig.
2a. It can also be observed in Fig. 3a that only type 5 devices who can
utilize all power levels are transmitting. This is due to the fact that the
transmission of other type of devices will only increase the congestion
probability of the channel, the best strategy to maximize the total expected
throughput is to maximize the performance of the type 5 devices while other
type of devices stop their transmission. When it comes to the case (b), the
network performance and the average throughput for each type of devices are
shown in Fig. 2b and Fig. 3b respectively. As can be observed that after
convergence, all type of devices are capable to transmit while the total
expected throughput is still relatively high. However, the average throughput
of devices with more power levels to utilize is higher than the average
throughput of devices with less power levels options to transmit. In other
word, the network is not totally fair for all devices. In Fig. 2c, we plot the
network performance of case (c). As shown in this figure, after convergence,
the gap between arithmetic mean of the expected throughput, geometric mean of
the expected throughput and the minimum expected throughput are small. This is
because after maximize the minimum expected throughput, the average throughput
of each type of devices are generally equal as shown in Fig. 3c. Otherwise, if
the average throughput of a type of devices is smaller than the average
throughput of other type of devices, the transmission probabilities of other
type of devices should be decremented to promote the average throughput of the
type of devices with minimum expected throughput, which helps to increase the
minimum expected throughput of the network. It is worth noting that even
thought the case (c) achieves fairness and improves the minimum throughput of
the system, the total expected throughput is lower than that of case (b),
which indicates that in order to achieve max-min fairness, the average
throughput of devices with more power levels to select has been severely
influenced.

To evaluate the effect of the power level set $\mathcal{V}$ on the system
performance, we depict the reward of the proposed RL approach of case (b) and
case (c) in Fig. 4 and Fig. 5 respectively. The results shown that better
network performance could be achieved with more power levels available in the
system.

Figure 4: Reward of the RL approach with different $\mathcal{V}$ when case (b)
is considered. Figure 5: Reward of the RL approach with different
$\mathcal{V}$ when case (c) is considered.

##  VI Conclusion

In this paper, we introduced a novel deep RL approach for NOMA-based slotted
ALOHA network with truncated channel inversion power control. The proposed
approach enables the AP to tune the transmission probabilities matrix which
guides the uplink transmission of IoT devices to improve the network
performance. Instead of optimizing the total expected throughput which ignored
the fairness in this heterogeneous scenario, two optimization problems have
been formulated to maximize the geometric mean of the expected throughput and
the minimum expected throughput for all devices. Extensive simulations shown
that our approach helps us to find the optimal solutions of our optimization
problems. In our future work, we will incorporate intelligent reflecting
surface (IRS) in the NOMA-based random access network.

##  VII Acknowledgement

This work is supported in part by NSF Career award ECCS1554576 and NSF grants
CNS-1320736.

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# Diffeological Symplectic Frobenius Reciprocity

Gabriele BARBIERI† & Mauro SPERA††111Corresponding author
† Dipartimento di Matematica e Applicazioni
Università di Milano-Bicocca, Via Cozzi 55, 20125 Milano, Italy
and
Dipartimento di Matematica “Felice Casorati”,
Università di Pavia,
Via Ferrata 5, 27100 Pavia, Italy
e-mail<EMAIL_ADDRESS>
void
†† Dipartimento di Matematica e Fisica,
Università Cattolica del Sacro Cuore,
Via della Garzetta 48, 25133 Brescia, Italy
e-mail<EMAIL_ADDRESS>
###### Abstract

In this note we prove that the symplectic Frobenius Reciprocity established in
the paper [9] as a set bijection is indeed a diffeological diffeomorphism, as
conjectured by its authors. The same holds in the prequantum space context.

Keywords: Symplectic reduction; Frobenius reciprocity; diffeology.

MSC 2020: 53D20, 81R05, 20C35, 58A40.

##  1 Introduction

The celebrated group theoretical Frobenius Reciprocity (FR for short) has been
widely extended via modern representation theory, see e.g. [8]. Subsequently,
V. Guillemin and S. Sternberg reformulated the basic ingredients of the
representation theoretic approach in symplectic terms (see the foundational
[4]) and proved it for coadjoint orbits of compact Lie groups in [5].
Recently, T. Ratiu and F. Ziegler ([9]) succeeded in establishing a symplectic
FR as a set-theoretical bijection between manifolds and conjectured that this
bijection is indeed a diffeological diffeomorphism. The present note aims at
proving that this is indeed true. Diffeology, pioneered by J.M. Souriau [11,
12] and then mainly developed by P. Iglesias-Zemmour (see e.g. [6]) proved to
be quite effective in analyzing singular spaces such as orbit spaces of group
actions - which are also alternatively studied by noncommutative geometry via
operator theoretic tools (see e.g. [3]) - by virtue of its flexibility and
intuitive content mildly departing from that of classical differential
geometry.

The outline of the paper is the following. In the first section we collect
basic tools from diffeology theory in order to ease readability. Then, in
Section 3, we recall the symplectic framework of [9] together with their
”induction in stages” theorem. Subsequently, in Section 4, we provide the
proof of our diffeological FR and this will be done by building on basic
constructions in [13]. After discussing the prequantum space version of FR in
[9] (Section 5), its diffeological version will be proved in Section 6. The
final section points out possible future research directions.

##  2 Tools in diffeology

In this preliminary section we are going to gather together basic
diffeological notions and results, referring to [6] for a thorough treatment.

######  Definition 2.1.

Let $X$ be a non empty set. A diffeology on $X$ is a set $\mathcal{D}$ of
parametrizations (§1.4 of [6]) of $X$, such that the following axioms are
satisfied:

  1. 1.

The set $\mathcal{D}$ contains the constant parametrizations $P:U\to
X,P(r)=x,\forall r\in U$, where $x\in X$,
$U\subseteq{\mathbb{R}}^{p},p\in{\mathbb{N}}$.

  2. 2.

Let $P:U\to X,U\subseteq{\mathbb{R}}^{p},p\in{\mathbb{N}}$ be a
parametrization. If for any $r\in U$ there exists an open neighborhood $V$ of
$r$ such that $P|_{V}\in\mathcal{D}$, then $P\in\mathcal{D}$.

  3. 3.

For any $P:U\to X$ in $\mathcal{D}$, for any
$V\subseteq{\mathbb{R}}^{p},p\in{\mathbb{N}}$ and for any
$F\in\mathcal{C}^{\infty}(V,U)$, $P\circ F\in\mathcal{D}$.

The pair $(X,\mathcal{D})$ is called diffeological space and the elements of
$\mathcal{D}$ are called plots of the space $X$.

######  Definition 2.2.

Let $X,X^{\prime}$ be diffeological spaces. A map $f:X\to X^{\prime}$ is said
to be smooth if, for any plot $P$ of $X$, $f\circ P$ is a plot of
$X^{\prime}$. A map $f:X\to X^{\prime}$ is a diffeomorphism if it is bijective
and both $f$ and $f^{-1}$ are smooth.

######  Definition 2.3.

Let $(X,\mathcal{D})$ and $(X^{\prime},\mathcal{D}^{\prime})$ be two
diffeological spaces. The diffeology $\mathcal{D}$ is said to be finer than
$\mathcal{D}^{\prime}$ if $\mathcal{D}\subset\mathcal{D}^{\prime}$. In this
case, $\mathcal{D}^{\prime}$ is said to be coarser than $\mathcal{D}$.

######  Definition 2.4.

Let $f:X\to X^{\prime}$ be a map, where $X$ is a set and
$(X^{\prime},\mathcal{D}^{\prime})$ is a diffeological space. The pullback
diffeology on $X$ is the coarsest diffeology such that $f$ is smooth and it is
denoted by $f^{*}(\mathcal{D}^{\prime})$. In particular, $P\in
f^{*}(\mathcal{D}^{\prime})$ if and only if $f\circ P\in\mathcal{D}^{\prime}$.

######  Definition 2.5.

Let $f:X\to X^{\prime}$ be a map between diffeological spaces. $f$ is said to
be an induction if it is injective and
$f^{*}(\mathcal{D}^{\prime})=\mathcal{D}$, i.e. for any plot $P$ of $X$,
$f\circ P$ is a plot of $X^{\prime}$.

######  Definition 2.6.

Let $X$ be a diffeological space and let $A\subset X$. The subset diffeology
is the set of the plots $P$ of $X$ with values in $A$.

######  Remark 2.1.

Let $X$ be a diffeological space and $A\subset X$, endowed with the subset
diffeology. The inclusion $j:A\xhookrightarrow{}X$ is an induction (§1.36 of
[6]).

######  Definition 2.7.

Let $f:X\to X^{\prime}$ be a map between a diffeological space
$(X,\mathcal{D})$ and a set $X^{\prime}$. There exists a finest diffeology on
$X^{\prime}$ such that $f$ is smooth; it is called pushforward diffeology and
it is denoted by $f_{*}(\mathcal{D})$. A parametrization $P^{\prime}:U\to
X^{\prime}$ is a plot of $f_{*}(\mathcal{D})$ if and only if for any $r\in U$
there exists an open neighbourhood $V$ of $r$ such that $P^{\prime}|_{V}$ is
constant or there exists a plot $Q:V\to X$ of $X$ such that
$P^{\prime}|_{V}=f\circ Q$.

######  Definition 2.8.

Let $f:X\to X^{\prime}$ be a map between diffeological spaces. The function
$f$ is said to be a subduction if it is surjective and
$f_{*}(\mathcal{D})=\mathcal{D}^{\prime}$.

######  Definition 2.9.

Let $(X,\mathcal{D})$ be a diffeological space and let $\mathcal{R}$ be an
equivalence relation on $X$. Let $\pi:X\to X/\mathcal{R}$ be the projection
onto the quotient. The quotient diffeology on $X/\mathcal{R}$ is defined as
$\pi_{*}(\mathcal{D})$.

######  Remark 2.2.

The projection onto the quotient, endowed with the quotient diffeology, is a
subduction.

######  Definition 2.10.

Let $\\{(X_{i},\mathcal{D}_{i})\\}_{i\in I}$ be a collection of diffeological
spaces indexed by some $I$. The product diffeology on $\Pi_{i\in{I}}X_{i}$ is
the coarsest diffeology $\mathcal{D}$ such that the projections
$\pi_{i}:\Pi_{j\in I}X_{j}\to X_{i}$ are smooth. The pair $(\Pi_{i\in
I}X_{i},\mathcal{D})$ is called diffeological product.

######  Remark 2.3.

Under the assumptions of the previous definition, the projections $\pi_{i}$
are subductions (§1.56 of [6]).

######  Definition 2.11.

Let $X,X^{\prime}$ be two diffeological spaces and let
$\mathcal{C}^{\infty}(X,X^{\prime})$ be the set of smooth maps between $X$ and
$X^{\prime}$, $f\in\mathcal{C}^{\infty}(X,X^{\prime}),x\in X$ . Let
$ev(f,x)=f(x)$ be the evaluation map. A functional diffeology is any
diffeology defined on $\mathcal{C}^{\infty}(X,X^{\prime})$ such that $ev$ is
smooth. The standard functional diffeology is the coarsest functional
diffeology defined on $\mathcal{C}^{\infty}(X,X^{\prime})$.

######  Definition 2.12.

Let $V$ be a vector space over a field $\mathbb{K}$. A diffeology on $V$ is a
vector space diffeology if the sum of two vectors and the multiplication of a
vector by a scalar are smooth.

######  Proposition/Definition 2.4.

Let $V$ be a vector space over a field $\mathbb{K}$. There exists a finest
vector space diffeology on $V$ and it is generated by a family of
parametrizations $P$ such that

$P:r\mapsto\sum_{i\in I}\lambda_{i}(r)v^{i},$

where $I$ is finite, $v^{i}\in V$ and the $\lambda_{i}$ are smooth
parametrizations of $\mathbb{K}$, defined on the domain of $P$. Equivalently,
the plots of this diffeology are parametrizations $P:U\to V$ such that for any
$r_{0}\in U$ there exists an open neighborhood $W\subseteq U$ of $r_{0}$, a
family of smooth parametrizations $\lambda_{i}:W\to\mathbb{K}$ and a family of
vectors $v^{i}\in V$ indexed by a finite set of indices $I$ such that:

$P|_{W}:r\mapsto\sum_{i\in I}\lambda_{i}(r)v^{i}.$

This diffeology is called the fine diffeology.

###### Proof.

See §3.7 of [6]. ∎

######  Definition 2.13.

An $n$-dimensional manifold $M$ can be regarded as a diffeological space whose
diffeology is generated by local charts (§4.1 of [6]).

######  Definition 2.14.

Let $X$ be a diffeological space. A differential $k$-form on $X$ is a map
$\alpha$ that associates to each plot $P$ of $X$ a differential $k$-form
$\alpha(P)$ defined on the domain of $P$, such that, for any smooth
parametrization $F$ in the domain of $P$, the following holds:

$\alpha(P\circ F)=F^{*}(\alpha(P)).$

######  Proposition/Definition 2.5.

Let $X$ and $X^{\prime}$ be two diffeological spaces, $f:X\to X^{\prime}$ a
smooth map and $\alpha^{\prime}$ a $k$-form on $X^{\prime}$. Then there exists
a differential $k$-form on $X$, denoted by $f^{*}(\alpha^{\prime})$, such that

$f^{*}(\alpha^{\prime})(P)=\alpha^{\prime}(f\circ P),$

for any $P$ plot of $X$. The $k$-form $f^{*}(\alpha^{\prime})$ is called
pullback of $\alpha^{\prime}$ by $f$.

###### Proof.

See §6.32 of [6]. ∎

######  Proposition 2.6.

Let $X$ and $X^{\prime}$ be two diffeological spaces. Let $\pi:X\to
X^{\prime}$ be a subduction and let $\alpha$ be a differential $k$-form on
$X$. Then $\alpha$ is the pullback of a $k$-form $\beta$ defined on
$X^{\prime}$ if and only if, for any $P,Q$ plots of $X$ such that $\pi\circ
P=\pi\circ Q$, $\alpha(P)=\alpha(Q)$.

###### Proof.

See §6.38 of [6]. ∎

######  Definition 2.15.

A diffeological group $G$ is a group endowed with a compatible diffeology such
that multiplication and inversion are smooth maps.

######  Definition 2.16.

Let $X$ be a diffeological space. Being a subset of
$\mathcal{C}^{\infty}(X,X)$, $\mathrm{Diff}(X)$ inherits a functional
diffeology. It can also be endowed with the structure of diffeological group:
it is the coarsest group diffeology such that the evaluation map is smooth. A
parametrization $P$ of $\mathrm{Diff}(X)$ is a plot of the standard diffeology
of group of diffeomorphisms if and only if $r\mapsto P(r)$ and $r\mapsto
P^{-1}(r)$ are plots for the functional diffeology defined on
$\mathcal{C}^{\infty}(X,X)$ .

######  Definition 2.17.

Let $X$ be a diffeological space and $G$ a diffeological group. A smooth
action of $G$ on $X$ is any smooth homomorphism $\rho:G\to\mathrm{Diff}(X)$,
where $\mathrm{Diff}(X)$ is endowed with the functional diffeology of group of
diffeomorphisms.

##  3 Symplectic induction in stages

In this Section we shall act within the framework of [9], using the same
notation therein. All manifolds and groups involved will be finite
dimensional.

Let $H\subset G$ be a closed subgroup of a Lie group $G$ and let us consider a
Hamiltonian $H$-space $(Y,\omega_{Y},\Psi)$. Let $\overline{\omega}\in T^{*}G$
such that $\overline{\omega}(\delta p):=\langle p,\delta q\rangle$, where
$\pi:T^{*}G\to G$ is the canonical projection and $\pi(p)=q,\delta p\in
T_{p}(T^{*}G)$, $\delta q:=\pi_{*}(\delta p)\in T_{\pi(p)}G$. The manifold
$N:=T^{*}G\times Y$ can be endowed with the symplectic form
$\omega:=d\overline{\omega}+\omega_{Y}$ and with a $G\times H$-action:

$(g,h)(p,y):=(gph^{-1},h(y)),$

where $h(y)$ is the action of $h\in H$ on $y\in Y$. By [9], the equivariant
moment map associated to this action is
$\phi\times\psi:N\to\mathfrak{g}^{*}\times\mathfrak{h}^{*}$, where

$\phi(p,y)=pq^{-1}$ $\psi(p,y)=\Psi(y)-q^{-1}p_{|\mathfrak{h}}.$

######  Definition 3.1.

The induced manifold is the reduction of $N$ at $0\in\mathfrak{h}^{*}$:

$\mathrm{Ind}_{H}^{G}Y:=N\sslash H=\psi^{-1}(0)/H.$

######  Remark 3.1.

Since the action of $H\subset G$ on $T^{*}G$ is free and proper, it is free
and proper on $N$ as well (see Appendix for a proof). This implies that the
induced manifold is actually a manifold.

######  Theorem 3.2.

([9]) Let $H\subset K\subset G$, where $H$ and $K$ are closed subgroups of the
Lie group $G$. Then:

$\mathrm{Ind}_{K}^{G}\mathrm{Ind}_{H}^{K}Y=\mathrm{Ind}_{H}^{G}Y.$

######  Remark 3.3.

The equality of the previous theorem means that there exists a diffeomorphism
between the manifolds $\mathrm{Ind}_{K}^{G}\mathrm{Ind}_{H}^{K}Y$ and
$\mathrm{Ind}_{H}^{G}Y.$ If we regard manifolds as diffeological spaces
endowed with the manifold diffeology, the above diffeomorphism can be viewed
as a diffeomorphism between diffeological spaces.

##  4 Symplectic Frobenius Reciprocity

In this section we are going to prove our result. First, we recall the
symplectic reformulation of the space of intertwiners in classical
representation theory (see e.g. [8]) given in §6 of [4].

######  Definition 4.1.

$\mathrm{Hom}_{G}(X_{1},X_{2}):=(X_{1}^{-}\times X_{2})\sslash G,$

where $X_{1}^{-}$ is the Hamiltonian $G$-space $X_{1}$ endowed with opposite
$2$-form and moment map.

${M}$${N}$${\overline{\psi}^{-1}(0)}$${(\overline{\phi}\times\overline{\psi})^{-1}(0)}$${{\psi}^{-1}(0)}$${M\sslash
H}$${\overline{\Phi}^{-1}_{M\sslash H}(0)}$${(M\sslash H)\sslash K}$${N\sslash
H}$$\scriptstyle{r}$$\scriptstyle{j_{1}}$$\scriptstyle{j}$$\scriptstyle{s}$$\scriptstyle{j_{3}}$$\scriptstyle{\pi_{1}}$$\scriptstyle{\pi}$$\scriptstyle{j_{2}}$$\scriptstyle{\pi_{2}}$$\scriptstyle{\pi_{3}}$$\scriptstyle{t}$
Figure 1: Commutative diagram of [9], used in the proof of Theorems 2.1 and
3.4 of [9] and Theorem 4.1.

We are in a position to state the following:

######  Theorem 4.1 (Symplectic Frobenius Reciprocity).

Let $H$ and $G$ be Lie groups with $H\subset G$. Let $X$ be a Hamiltonian
$G$-space and $Y$ a Hamiltonian $H$-space. Then

$\mathrm{Hom}_{G}(X,\mathrm{Ind}_{H}^{G}Y)=\mathrm{Hom}_{H}(\mathrm{Res}_{H}^{G}X,Y),$

where the equality denotes a diffeological diffeomorphism.

###### Proof.

The first part of the proof coincides with the one of Theorem 3.4 of [9]; we
briefly resume the notation and some intermediate results that will be used in
the diffeological part of the proof.

Following the usage in [9], let us rename $G$ as $K$ for convenience and set
$N:=X^{-}\times Y$, $M:=X^{-}\times T^{*}K\times Y$; these definitions allow
us to perform a proof similar to the one of the symplectic induction in
stages. An $H$-action is defined on N:

$h(x,y)=(h(x),h(y)),$

and the corresponding moment map is $\psi:N\to\mathfrak{h}^{*}$,

$\psi(x,y)=\Psi(y)-\Phi(x)|_{\mathfrak{h}};$ (1)

a $K\times H$-action is defined on $M$:

$(k,h)(x,\overline{p},y)=(k(x),k\overline{p}h^{-1},h(y)),$

with moment map
$\overline{\phi}\times\overline{\psi}:M\to\mathfrak{k}^{*}\times\mathfrak{h}^{*},$

$\overline{\phi}(x,\overline{p},y)=\overline{p}\,\overline{q}^{-1}-\Phi(x)|_{\mathfrak{h}}$
(2)
$\overline{\psi}(x,\overline{p},y)=\Psi(y)-\overline{q}^{-1}\overline{p}|_{\mathfrak{h}}.$
(3)

Let us denote by $j,j_{1},j_{2},j_{3}$ the canonical inclusions and by
$\pi,\pi_{1},\pi_{2},\pi_{3}$ the canonical projections involved in the
construction of the reductions. Let $\overline{\Phi}_{M\sslash H}:M\sslash
H\to\mathfrak{k}^{*}$ be the moment map for the action of $K$ on the quotient
$M\sslash H$. Then, by Definition 4.1:

$N\sslash H=(X^{-}\times Y)\sslash
H=\mathrm{Hom}_{H}(\mathrm{Res}_{H}^{K}X,Y);$ $(M\sslash H)\sslash
K=(X^{-}\times(T^{*}K\times Y)\sslash H)\sslash K=$
$=(X^{-}\times\mathrm{Ind}_{H}^{K}Y)\sslash
K=\mathrm{Hom}_{K}(X,\mathrm{Ind}_{H}^{K}Y).$

Before giving the diffeological proof, let us recall the definitions and the
properties of the maps $r$, $s$ and $t$ of the diagram, which are proved in
Theorem 3.4 of [9]. Set $r:M\to N$ and
$r(x,\overline{p},y)=(\overline{q}^{-1}(x),y),\overline{p}\in
T^{*}_{\overline{q}}K;$ let
$s:(\overline{\phi}\times\overline{\psi})^{-1}(0)\to\psi^{-1}(0)$, defined as
the map $r\circ j\circ j_{1}$, with range restricted to $\psi^{-1}(0)$. Under
this restriction it becomes surjective. Moreover, $s$ is equivariant with
respect to the $K\times H$ action on
$(\overline{\phi}\times\overline{\psi})^{-1}(0)$ and to the $H$-action on
$\psi^{-1}(0)$; furthermore the fibers of $s$ are $K$-orbits. These facts
imply that $s$ descends to a bijection $t:(M\sslash H)\sslash K\to N\sslash H$
such that the diagram is commutative, i.e. $t\circ\pi_{2}\circ\pi=\pi_{3}\circ
s$. These constructions are portrayed in Fig. 1.

Now we can prove that $t$ is a diffeological diffeomorphism, that is, both $t$
and $t^{-1}$ are smooth (in a diffeological sense). The scheme of the proof is
similar to the one of Lemma 3.17 and Proposition 3.43 of [13]. Let us prove
that $t:(M\sslash H)\sslash K\to N\sslash H$ is smooth: let $p:U\to(M\sslash
H)\sslash K$ be a plot of $(M\sslash H)\sslash K$, with
$U\subseteq{\mathbb{R}}^{p},p\in{\mathbb{N}}$. Then, by definition of quotient
diffeology, for any $u\in U$ there exists $V\subseteq U$ and
$q:V\to\overline{\Phi}^{-1}_{M\sslash H}(0)$, a plot of
$\overline{\Phi}^{-1}_{M\sslash H}(0)$, such that $u\in V$ and
$p|_{V}=\pi_{2}\circ q$. For the same reason, upon further restrictions on
$V$, there exists $\rho:V\to(\overline{\phi}\times\overline{\psi})^{-1}(0)$, a
plot of $(\overline{\phi}\times\overline{\psi})^{-1}(0)$, such that
$q=\pi\circ\rho$. Since the diagram is commutative we get:

$t\circ p|_{V}=t\circ\pi_{2}\circ\pi\circ\rho=\pi_{3}\circ s\circ\rho,$

hence $t$ is smooth if $s$ is smooth. Now, $s$ is smooth if, for any plot
$\sigma$ of $(\overline{\phi}\times\overline{\psi})^{-1}(0)$, $s\circ\sigma$
is a plot of $\psi^{-1}(0)$. Since $j$ is an inclusion, it is an injection
with respect to the subset diffeology and then $j\circ\sigma$ is a plot of
$\overline{\psi}^{-1}(0)$; moreover, $j_{1}\circ j\circ\sigma$ is a plot of
$M$ and by commutativity of the diagram

$r\circ j_{1}\circ j\circ\sigma=j_{3}\circ s\circ\sigma.$

By Proposition 1.31 of [6], $s$ is smooth if $r$ is smooth. By Lemma 4.3
below, $r$ is smooth, then $s$ is smooth and $t$ is smooth.

Now let us prove that $t^{-1}:N\sslash H\to(M\sslash H)\sslash K$ is smooth.
Let $p:U\to N\sslash H$ be a plot of $N\sslash H$, with
$U\subseteq{\mathbb{R}}^{p},\ p\in{\mathbb{N}}$. By definition of quotient
diffeology, for any $u\in U$ there exists $V\subseteq U$ and
$q:V\to\psi^{-1}(0)$, a plot of $\psi^{-1}(0)$, such that $u\in V$ and
$p|_{V}=\pi_{3}\circ q$. By Lemma 4.4 below, $s$ is a subduction and by
proposition 1.48 of [6] there exists, upon further restrictions of $V$, a plot
$\rho:V\to(\overline{\phi}\times\overline{\psi})^{-1}(0)$ of
$(\overline{\phi}\times\overline{\psi})^{-1}(0)$ such that $q=s\circ\rho$.
Commutativity of the diagram then yields

$t^{-1}\circ p|_{V}=t^{-1}\circ\pi_{3}\circ q=t^{-1}\circ\pi_{3}\circ
s\circ\rho=\pi_{2}\circ\pi\circ\rho.$

whence $t^{-1}$ is smooth, being the projections smooth. Summarizing, the idea
of the proof is to use the diffeological properties of the projections and the
inclusions to climb up the diagram so that proving that $t$ and $t^{-1}$ are
smooth is tantamount to prove that $r$ is smooth and $s$ is a subduction,
respectively.

Finally, we prove that $t$ preserves the diffeological forms on the quotients.
Indeed, $M$ and $N$ can be endowed, as in Theorem 2.1 of [9], with a
symplectic form, which is also a diffeological form. Passing to quotients,
note that $(M\sslash H)\sslash K$ and $N\sslash H$ are not necessary manifolds
under our hypotheses, which is the reason why we cannot assert that $t$ is a
diffeomorphism between manifolds. However, considering the subsets of a
diffeological space and the quotients of a diffeological space, the symplectic
reduction can be endowed with a diffeological space structure and a reduced
diffeological form. Indeed, let $(M,\omega,\phi)$ be a Hamiltonian $G$-space.
Even if $0\in\mathfrak{g}^{*}$ is not a regular value with respect to $\phi$,
$\phi^{-1}(0)\subset M$ can be endowed with the subset diffeology. Moreover,
we can consider $\phi^{-1}(0)/G=:M\sslash G$ as a diffeological space, endowed
with the quotient diffeology. Let $j:\phi^{-1}(0)\hookrightarrow M$ be the
inclusion and $\pi:\phi^{-1}(0)\to\phi^{-1}(0)/G$ the projection onto the
quotient. Since $\pi$ is a subduction, there exists a diffeological form on
the quotient $\omega_{M\sslash G}$ such that
$j^{*}\omega=\pi^{*}\omega_{M\sslash G}$ if and only if, for any $P,Q$ plots
of $\phi^{-1}(0)$ such that $\pi\circ P=\pi\circ Q$, one has
$j^{*}\omega(P)=j^{*}\omega(Q)$. From the condition on the projection of the
plots it follows that $P$ and $Q$ differ only by the action of $G$; being
$\omega$ $G$-invariant, we indeed have $j^{*}\omega(P)=j^{*}\omega(Q)$, hence
there exists $\omega_{M\sslash G}$ on $M\sslash G$ such that
$j^{*}\omega=\pi^{*}\omega_{M\sslash G}$. Thus there exists a diffeological
form on the reduction, defined in the same way as the reduced symplectic form.
Therefore, we can proceed as in the proof of Theorem 2.1 of [9]: we have
$s^{*}j_{3}^{*}\omega_{N}=j^{*}j_{1}^{*}\omega_{M}$ and by commutativity of
the diagram $\pi^{*}\pi_{2}^{*}t^{*}\omega_{N\sslash
H}=\pi^{*}\pi_{2}^{*}\omega_{(M\sslash H)\sslash K}.$ Since $\pi_{2}\circ\pi$
is surjective, its pullback is injective and we have $t^{*}\omega_{N\sslash
H}=\omega_{(M\sslash H)\sslash K}$, as diffeological forms. ∎

######  Remark 4.2.

In the induction in stages situation $M$ and $N$ were defined as $T^{*}G\times
T^{*}K\times Y$ and $T^{*}G\times Y$, respectively. The very presence of
$T^{*}G$ and $T^{*}K$ made the group actions of $H$ and $K$ free and proper
(as in Remark 3.1), thus paving the way to quotient manifold structures. In
the proof of the previous Theorem, the definitions of $M$ and $N$ do not
involve the cotangent spaces $T^{*}G$ and $T^{*}K$, thus the action is not
necessarily free and proper and the quotients are no longer manifolds.

The above result strengthens Theorem 3.4 of [9], in which the equality was
set-theoretical.

We are now going to prove the missing lemmas.

######  Lemma 4.3.

The map $r:M\to N$ is smooth.

###### Proof.

Let us consider $M$ and $N$ as in the proof of the previous theorem and the
corresponding $K\times H$ and $H$-action, respectively. The map $r:M\to N$
maps $(x,\overline{p},y)\mapsto(\overline{q}^{-1}(x),y)$, where
$\overline{p}\in T^{*}_{\overline{q}}K$. Let us consider a plot $P$ of $M$,
endowed with the product diffeology; then $P$ can be defined as follows:
$P:U\to M$, $U\subset{\mathbb{R}}^{p},p\in{\mathbb{N}}$,
$P(u)=(P^{X}(u),P^{T}(u),P^{Y}(u))$, where $P^{X},P^{T},P^{Y}$ are plots of
$X,T^{*}K,Y$, respectively. In particular, since $T^{*}K=\\{(k,\alpha(k))|k\in
K,\alpha\in\Omega^{1}(K)\\}$, it can also be endowed with the product
diffeology; furthermore, the first factor can be equipped with a Lie group
diffeology (§7.1 of [6]), which is both a group and a manifold diffeology, and
the second one with a functional diffeology. These considerations allow us to
write the plot of $T^{*}K$ as $P^{T}(u)=(P^{K}(u),P^{\Omega}(u)(P^{K}(u)))$,
where $P^{K}$ and $P^{\Omega}$ are plots of $K$ and $\Omega^{1}(K)$,
respectively. The map $r$ is smooth if $r\circ P$ is a plot of $N$:

$r\circ
P(u)=r((P^{X}(u),P^{T}(u),P^{Y}(u)))=(P^{K}(u)^{-1}(P^{X}(u)),P^{Y}(u)).$

This is ascertained as follows: first, notice that $r$ maps identically the
third component of $P$ to the second one of the image, so we only have to
focus on the mapping of the first two components of $P$ to the first one of
the image. The latter is smooth because, being the action of $K$ smooth on $X$
in the usual sense, the action of $K$, viewed as a diffeological group acting
on $X$, is also smooth. Hence $r$ is smooth. ∎

######  Lemma 4.4.

The map $s:(\overline{\phi}\times\overline{\psi})^{-1}(0)\to\psi^{-1}(0)$ is a
subduction.

###### Proof.

We already know from the proof of Theorem 3.4 of [9] that $s$ is surjective
onto $\psi^{-1}(0)$ and from the proof of Theorem 4.1, combined with Lemma
4.3, $s$ is also smooth; then, proving that $s$ is a subduction is tantamount
to prove the existence, for any plot $P:U\to\psi^{-1}(0)$ of $\psi^{-1}(0)$,
of a plot $Q:V\to(\overline{\phi}\times\overline{\psi})^{-1}(0)$ of
$(\overline{\phi}\times\overline{\psi})^{-1}(0)$ such that $P|_{V}=s\circ Q$,
where $V\subseteq U$. Let
$P:U\to\psi^{-1}(0),U\subseteq{\mathbb{R}}^{p},p\in{\mathbb{N}}$ be a plot of

$\psi^{-1}(0)=\\{(x,y)\in X^{-}\times Y|\Psi(y)=\Phi(x)|_{\mathfrak{h}}\\},$

which can be endowed with the product diffeology and each component with the
manifold diffeology. Suppose that
$Q:V\to(\overline{\phi}\times\overline{\psi})^{-1}(0),V\subseteq U$ is a plot
of $(\overline{\phi}\times\overline{\psi})^{-1}(0)$ such that $s\circ
Q=P|_{V}$. Since $(\overline{\phi}\times\overline{\psi})^{-1}(0)\subset M$,
$Q(v)=(Q^{X}(v),Q^{T}(v),Q^{Y}(v))$, where $Q^{X},Q^{T},Q^{Y}$ are plots of
$X,T^{*}K,Y$ and, as before,

$Q^{T}(v)=(Q^{K}(v),Q^{\Omega}(v)(Q^{K}(v)))=:(\overline{q}(v),\overline{p}(v)),$

where $\overline{p}(v)\in T^{*}_{q(v)}K$. Let $r\in V$, then

$s\circ
Q(r)=P(r)\iff(\overline{q}(r)^{-1}(Q^{X}(r)),Q^{Y}(r))=(P^{X}(r),P^{Y}(r))$
$\iff\overline{q}(r)^{-1}(Q^{X}(r))=P^{X}(r)\mbox{ and }Q^{Y}(r)=P^{Y}(r)$
$\iff Q^{X}(r)=\overline{q}(r)(P^{X}(r))\mbox{ and }Q^{Y}(r)=P^{Y}(r).$

Now we can evaluate equation (1) in $P$, and equations (2) and (3) in $Q$.
Since the latter are plots of $\psi^{-1}(0)$ and
$(\overline{\phi}\times\overline{\psi})^{-1}(0)$, respectively, we find
successively:

$\Psi(P^{Y}(r))=\Phi(P^{X}(r))|_{\mathfrak{h}}$ (4)
$\overline{p}(r)\overline{q}(r)^{-1}=\Phi(Q^{X}(r))=\Phi(\overline{q}(r)(P^{X}(r))=\overline{q}(r)(\Phi(P^{X}(r)))$
(5)
$\Psi(P^{Y}(r))=\Psi(Q^{Y}(r))=\overline{q}(r)^{-1}\overline{p}|_{\mathfrak{h}}\implies\Phi(P^{X}(r))|_{\mathfrak{h}}=\overline{q}(r)^{-1}\overline{p}|_{\mathfrak{h}}.$
(6)

Then it follows from (5) that

$\Phi(P^{X}(r))=\overline{q}(r)^{-1}\overline{p}(r)\overline{q}(r)^{-1},$

and substituting in (6) we get

$\overline{q}(r)^{-1}\overline{p}(r)|_{\mathfrak{h}}\overline{q}(r)^{-1}=\overline{q}(r)^{-1}\overline{p}(r)|_{\mathfrak{h}}.$

Then
$\overline{p}(r)|_{\mathfrak{h}}\overline{q}(r)^{-1}=\overline{p}(r)|_{\mathfrak{h}}$,
leading to $\overline{q}(r)^{-1}=1_{K}$. This holds for all $r\in V$, which
implies $\overline{p}(r)=\Phi(P^{X}(r))$ and $P^{X}(r)=Q^{X}(r)$. Hence

$Q(r)=(P^{X}(r),(1_{K},\Phi(P^{X}(r))),P^{Y}(r)),$

and $V$ can be taken equal to $U$. Now we have to prove that the above
determined $Q$ is indeed a plot of
$(\overline{\phi}\times\overline{\psi})^{-1}(0)$. If we consider the product
diffeology on $(\overline{\phi}\times\overline{\psi})^{-1}(0)$, the first and
the third component of $Q$ are, by definition, plots of $X$ and $Y$,
respectively; hence we are left to prove that $\Phi(P^{X}(r))$ is a plot of
$\mathfrak{k}^{*}$ with respect to a functional diffeology. If we consider
$\mathfrak{k}$ and ${\mathbb{R}}$ endowed with their fine vector space
diffeology, by Proposition 3.12 of [6], $\Phi(P^{X}(r))$ is a plot of
$\mathfrak{k}^{*}$. Indeed, if we take $F$ an $n$-dimensional vector subspace
of $\mathfrak{k}$, $\Phi(P^{X}(r))|_{F}$ writes

$\Phi(P^{X}(r))|_{F}=\sum_{i=1}^{n}a_{i}(P^{X}(r))v_{i}^{*},$

where $(v_{1},...,v_{n})$ is any basis of $F$,
$a_{i}\in\mathcal{C}^{\infty}(X)$ and $P^{X}$ is a plot of the manifold
diffeology of $X$. In particular, since $a_{i}\in\mathcal{C}^{\infty}(X)$ and
$P^{X}$ is a plot of $X$, the coefficients are smooth parametrizations of
${\mathbb{R}}$, hence $\Phi(P^{X}(r))$ is a plot of $\mathfrak{k}^{*}$. In
this case, “smooth” refers both to the usual and to the diffeological case,
since the finest diffeology on ${\mathbb{R}}$ coincides with the smooth one by
Exercise 66 of [6]. This concludes the proof of the lemma and that of Theorem
4.1. ∎

##  5 Prequantum $G$-spaces

In this section we recall the prequantum $G$-space framework of [9], again in
view of extending the corresponding FR to the diffeological context.

######  Definition 5.1.

[10]). A prequantum manifold $\tilde{X}$ is a manifold endowed with a
differential 1-form $\overline{\omega}$ such that, setting
$\sigma:=d\overline{\omega}$, the following hold:

$\mathrm{dim}(\mathrm{ker}\sigma)=1$
$\mathrm{dim}(\mathrm{ker}\overline{\omega}\cap\mathrm{ker}\sigma)=0.$

Moreover, there exists a $U(1)$-action on $\tilde{X}$ such that:

$z(\tilde{x})=\tilde{x}\iff z=1,\forall z\in U(1),\tilde{x}\in\tilde{X},$

and a vector field $i$ on $\tilde{X}$, called Reeb vector field, such that

$\sigma(i(\tilde{x}))=0,\,\overline{\omega}(i(\tilde{x}))=1,\forall\tilde{x}\in\tilde{X}.$

The definitions of moment map, dual space, product and induction can be
rephrased in the prequantum framework.

######  Proposition/Definition 5.1.

Let $(\tilde{X},\overline{\omega})$ be a prequantum manifold. Then
$(\tilde{X},d\overline{\omega})$ is a presymplectic manifold whose null leaves
are the orbits of $U(1)$ and $d\overline{\omega}$ restricts to a symplectic
form $\underline{\omega}$ on the quotient $X=\tilde{X}/U(1)$. If a Lie group
$G$ acts on $\tilde{X}$ and preserves $\overline{\omega}$, then it commutes
with $U(1)$ and the equivariant moment map

$\Phi:\tilde{X}\to\mathfrak{g}^{*}$
$\langle\Phi(\tilde{x}),Z\rangle=\overline{\omega}(Z(\tilde{x})),\tilde{x}\in\tilde{X},Z\in\mathfrak{g}$

descends to an equivariant moment map $\underline{\Phi}:X\to\mathfrak{g}^{*}$.
Thus $X$ becomes a Hamiltonian $G$-space prequantized by the prequantum
$G$-space $(\tilde{X},\overline{\omega})$ (5.2 of [9]).

######  Definition 5.2.

We define $\tilde{X}^{-}$ as the $G$-space that coincides with $\tilde{X}$,
but it is endowed with opposite 1-form $-\overline{\omega}$ and, consequently,
with opposite Reeb field and $U(1)$-action; it prequantizes
$(X^{-},-\underline{\omega},-\underline{\Phi})$ (5.3 of [9]).

######  Definition 5.3.

Let $\tilde{X}_{1}$ and $\tilde{X}_{2}$ be prequantum $G$-spaces. Then
$\tilde{X}_{1}\times\tilde{X}_{2}$, endowed with the diagonal $G$-action, is a
$U(1)\times U(1)$-space. Moreover, the orbits of the action of
$\Delta:=\\{(z^{-1},z)|z\in U(1)\\}$ are the characteristic leaves of the sum
of the 1-forms of the prequantum spaces:
$\overline{\omega}_{1}+\overline{\omega}_{2}$. Passing to the quotient we
obtain
$\tilde{X}_{1}\boxtimes\tilde{X}_{2}:=(\tilde{X}_{1}\times\tilde{X}_{2})/\Delta$
and this space prequantizes $X_{1}\times X_{2}$. Notice that, by the previous
definition, the $\Delta$-action on $\tilde{X}_{1}^{-}\times\tilde{X}_{2}$ is
$z(\tilde{x}_{1},\tilde{x}_{2})=(z(\tilde{x}_{1}),z(\tilde{x}_{2}))$ (5.4 of
[9]).

######  Proposition/Definition 5.2.

Let $G$ be a Lie group that acts freely and properly on $\tilde{X}$ and let
$L:=\Phi^{-1}(0)$. Being $\Phi$ a moment map, the tangent space to the orbit
through $\tilde{x}$ verifies
$\mathfrak{g}(\tilde{x})\subset\mathrm{Ker}(\overline{\omega}|_{L})\cap\mathrm{Ker}(d\overline{\omega}|_{L})$.
Due to the $G$-invariance of $\overline{\omega}|_{L}$, it descends to a
contact 1-form on the quotient $\tilde{X}\sslash G:=\Phi^{-1}(0)/G$. The
latter prequantizes the symplectic reduction $X\sslash
G=\underline{\Phi}^{-1}(0)/G$ (5.5 of [9]).

In view of the above definitions it is possible to construct an induced
prequantum $G$-space, as in section 6 of [9]. Let $H$ be a closed subgroup of
the Lie group $G$ and let $(\tilde{Y},\overline{\omega}_{\tilde{Y}})$ be a
prequantum $H$-space with moment map denoted by $\Psi$. Let
$\tilde{N}:=T^{*}G\times\tilde{Y}$ be the prequantum $(G\times H)$-space
endowed with the 1-form
$\overline{\omega}_{T^{*}G}+\overline{\omega}_{\tilde{Y}}$ and action
$(g,h)(p,\tilde{y}):=(gph^{-1},h(\tilde{y}))$. This action admits an
equivariant moment map
$\phi\times\psi:\tilde{N}\to\mathfrak{g}\times\mathfrak{h}^{*}$, where

$\phi(p,\tilde{y})=pq^{-1}$
$\psi(p,\tilde{y})=\Psi(\tilde{y})-q^{-1}p_{|\mathfrak{h}}.$

Proceeding as in Remark 3.1, it is possible to define the induced prequantum
manifold

$\mathrm{Ind}_{H}^{G}\tilde{Y}:=\tilde{N}\sslash H=\psi^{-1}(0)/H.$

The induced prequantum $G$-space $\mathrm{Ind}_{H}^{G}\tilde{Y}$ prequantizes
the symplectically induced manifold $\mathrm{Ind}_{H}^{G}{Y}$.

In the prequantum framework the induction in stages holds:

######  Theorem 5.3.

([9]) Let $H\subset K\subset G$, where $H$ and $K$ are closed subgroups of the
Lie group $G$. Then:

$\mathrm{Ind}_{K}^{G}\mathrm{Ind}_{H}^{K}\tilde{Y}=\mathrm{Ind}_{H}^{G}\tilde{Y}.$

##  6 Prequantum Frobenius Reciprocity

######  Definition 6.1.

The intertwiner space between two prequantum $G$-spaces $\tilde{X}_{1}$ and
$\tilde{X}_{2}$ is
$\mathrm{Hom}_{G}(\tilde{X}_{1},\tilde{X}_{2}):=(\tilde{X}_{1}\boxtimes\tilde{X}_{2})\sslash
G$ (8.1 of [9]).

We may now state and prove the following:

######  Theorem 6.1 (Prequantum Frobenius Reciprocity).

Let $\tilde{X}$ be a prequantum $G$-space and $\tilde{Y}$ a prequantum
$H$-space, then

$\mathrm{Hom}_{G}({\tilde{X}},\mathrm{Ind}_{H}^{G}{\tilde{Y}})=\mathrm{Hom}_{H}(\mathrm{Res}_{H}^{G}{\tilde{X}},{\tilde{Y}}),$

where the equality denotes a diffeological diffeomorphism.

${\tilde{\tilde{M}}:=\tilde{X}^{-}\times
T^{*}G\times\tilde{Y}}$${\tilde{\tilde{N}}:=\tilde{X}^{-}\times\tilde{Y}}$${\tilde{M}:=\tilde{X}^{-}\boxtimes
T^{*}G\times\tilde{Y}}$${\tilde{\tilde{N}}:=\tilde{X}^{-}\times\tilde{Y}}$${M:=X^{-}\times
T^{*}G\times Y}$${N:=X^{-}\times
Y}$$\scriptstyle{\tilde{\tilde{r}}}$$\scriptstyle{\rm{mod}\,\Delta}$$\scriptstyle{\rm{mod}\,\Delta}$$\scriptstyle{\tilde{r}}$$\scriptstyle{r}$$\scriptstyle{\rm{mod}\,(U(1)\times
U(1))/\Delta}$$\scriptstyle{\rm{mod}\,(U(1)\times U(1))/\Delta}$ Figure 2:
Commutative diagram 8.1 of [9].

###### Proof.

The first part of the proof coincides with the one of Theorem 8.2 of [9].
Referring to the commutative diagram in Fig.2, we can define
$\tilde{\tilde{r}}(\tilde{x},p,\tilde{y})=(q^{-1}(\tilde{x}),\tilde{y})$,
where $p\in T^{*}_{q}G$. The map $\tilde{\tilde{r}}$ descends to a map
$\tilde{r}$ and to a map $r$, that coincides with the one of Theorem 4.1;
indeed, at each level of the diagram in Fig.2 we have a copy of the one in
Fig.1, where we find the tilded versions of the maps $t$ and $s$. By
definition 6.1, proving the Theorem is tantamount to show that
$\tilde{t}:(\tilde{M}\sslash H)\sslash G\to\tilde{N}\sslash H$ is a
diffeological diffeomorphism. By Theorem 8.2 of [9], $\tilde{t}$ is a
bijection. It is also (diffeologically) smooth together with its inverse,
since Theorem 4.1 and Lemmas 4.3 and 4.4 also hold in the prequantum case:
their proofs can be performed as before, starting from diffeological spaces
$\tilde{X}$ and $\tilde{Y}$ in the tilded version of the diagram in Fig.1.
Thus $\tilde{t}$ is a diffeological diffeomorphism.

∎

##  7 Conclusions and outlook

In this paper, a proof of the Frobenius Reciprocity theorem in the symplectic
framework – together with a prequantum $G$-space version – enhancing the
Ratiu-Ziegler set-theoretical one has been given, resorting to well
established techniques in diffeology theory [11, 12, 6, 13]. This is an
encouraging signal towards further applications of the theory in various
directions, including infinite dimensional contexts like loop space geometry
and geometrical hydrodynamics (see e.g. [2, 1, 7]).

## Appendix

Here, for the sake of completeness, we provide a proof of the statement made
in Remark 3.1 (known in the literature in different guises): the free and
proper action of $H\subset G$ on $T^{*}G$ is free and proper on $N$ as well.

###### Proof.

The action of $H$ on $N$ is defined by $h(p,y)=(ph^{-1},h(y))$. Let $h\in
H,(p,y)\in N$ such that $h(p,y)=(p,y)$, then $(ph^{-1},h(y))=(p,y)$ and since
the right action of $H$ on $T^{*}G$ is free and $ph^{-1}=p$, we have
$h=1_{H}$, hence the $H$-action is free on $N$. Furthermore, this implies that
$\psi$ is a submersion and $\psi^{-1}(0)$ is a submanifold. Now let us prove
that the $H$-action on $N$ is proper as well. This will imply that $N\sslash
H$ is a manifold. Let

$f:H\times N\to N\times N,(h,(p,y))\mapsto(h(p,y),(p,y)),$

and let $U\times V$ be a compact subset of $N\times N$; we want to prove that

$f^{-1}(U\times V)=\\{(h,(p,y))\in H\times V|h(p,y)\in U\\}$

is compact. Being $U$ and $V$ compact subsets of $N=T^{*}G\times Y$, their
projections on $T^{*}G$ and $Y$, say $U_{1}$, $U_{2}$, $V_{1}$, $V_{2}$,
respectively, are compact. Let us also consider the $H$-action on $T^{*}G$ and
the map

$\phi:H\times T^{*}G\to T^{*}G\times T^{*}G,(h,p)\mapsto(ph^{-1},p).$

Since the $H$-action on $T^{*}G$ is proper, $\phi^{-1}(U_{1}\times
V_{1}):=\tilde{H}\times V_{1}$ is a compact subset of $H\times V_{1}$ and, in
particular, $\tilde{H}$ is compact. Finally, if $h(p,y)\in U_{1}\times U_{2}$,
where $(p,y)\in V$, then $(ph^{-1},p)=\phi(h,(p,y))\in U_{1}\times V_{1}$,
thus $h\in\tilde{H}$ and

$f^{-1}(U\times V)=\\{(h,(p,y))\in\tilde{H}\times V|h(p,y)\in U\\}.$

Being $U$ closed, $f^{-1}(U\times V)$ is a closed subset of the compact set
$\tilde{H}\times V$, then it is compact and the $H$-action is proper on $N$. ∎

Acknowledgements. The first named author is currently benefiting from a three
year Ph.D grant (issued by “Consorzio Milano Bicocca-Pavia”; project:
Geometric quantization and ramifications). The second named author’s research
is supported by UCSC D1-funds. Both authors thank A.M. Miti for discussions.
Their research is carried out within INDAM-GNSAGA’s framework.

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