# A Scalable and Generalizable Pathloss Map Prediction

Ju-Hyung Lee, and Andreas F. Molisch The authors are with the Ming Hsieh
Department of Electrical and Computer Engineering, University of Southern
California, Los Angeles, USA; email {juhyung.lee<EMAIL_ADDRESS>The work
was financially supported by NSF projects 2133655 and 2008443. Part of this
work was presented at ICASSP 2023 [1] and Globecom 2023 [2].

###### Abstract

Large-scale channel prediction, i.e., estimation of the pathloss from
geographical/morphological/building maps, is an essential component of
wireless network planning. Ray tracing (RT)-based methods have been widely
used for many years, but they require significant computational effort that
may become prohibitive with the increased network densification and/or use of
higher frequencies in B5G/6G systems. In this paper, we propose a data-driven,
model-free pathloss map prediction (PMP) method, called PMNet. PMNet uses a
supervised learning approach: it is trained on a limited amount of RT (or
channel measurement) data and map data. Once trained, PMNet can predict
pathloss over location with high accuracy (an RMSE level of $10^{-2}$) in a
few milliseconds. We further extend PMNet by employing transfer learning (TL).
TL allows PMNet to learn a new network scenario quickly ($\times 5.6$ faster
training) and efficiently (using $\times 4.5$ less data) by transferring
knowledge from a pre-trained model, while retaining accuracy. Our results
demonstrate that PMNet is a scalable and generalizable ML-based PMP method,
showing its potential to be used in several network optimization applications.

###### Index Terms:

Pathloss map prediction, ray tracing, channel measurement, machine learning,
computer vision, transfer learning, network optimization, digital twin, 6G.

##  I Introduction

Digital twin (DT) technology is emerging as a key enabler for the artificial
intelligence (AI) and machine learning (ML)-driven design, simulation, and
optimization of 6G systems [3, 4]. A DT is a dynamic, digital replica of a
real-world network environment, providing real-time, accurate reflections of
physical network scenarios. However, implementing DT is challenging in 6G
networks, which are characterized by increased deployment density, complex
distributed architectures, and high-frequency operation in millimeter wave
(mmWave) and terahertz (THz) bands.

Individually and taken together, these developments necessitate dramatically
faster large-scale channel prediction methods.111The word ”channel prediction”
is often used for two different problems: (i) computation of the propagation
channel at a particular location based on maps of the environment, and (ii)
temporal prediction of the channel (often for a mobile device moving on a
trajectory), based on measurements in the immediate past. This paper only
considers the former case. Since traditional ray tracing (RT) tools are too
slow for the repeated runs required in such DT implementation processes, there
is a strong need for new, accurate, and fast methods for channel prediction
over a large-scale area (e.g., campus or city-map scale).

Several works have addressed this need by channel prediction using powerful ML
techniques. These works use ground-truth channel data (from RT simulations or
real channel measurements/soundings campaigns) to train neural networks (NNs).
This eventually provides an accurate and fast prediction of channel
information (e.g., received power, delay, angles, and so on) for a certain
area, a technique called ML-based site-specific radio propagation modeling.

Still, these ML-based approaches use supervised learning, meaning they are
trained to solve a specific network scenario with a certain labeled dataset.
In other words, the models may need to be rebuilt for a new network scenario,
e.g., different map scales, environmental aspects, and/or network
configuration - a process that can be time-consuming and expensive. This
creates a need for a method that can furthermore transfer knowledge of
propagation channels across different network scenarios and environments.

Figure 1: Overview of the pathloss map prediction (PMP) task and the cross-
scenario PMP. The input Map feature includes the transmitter (TX) location.

###  I-A Related Works

Due to the high cost and complexity of field measurements with channel
sounders, most cellular deployment planning has long replaced channel
measurements with _electromagnetic (EM) simulation-based_ approaches, such as
RT [5, 6] and ray launching [7] simulation.222Throughput this paper, we will
use the terms ”measurements”, ”ray tracing”, and ”ray launching”
interchangeably to refer to a suitable method for finding a site-specific
ground-truth for the pathloss. Over the past 30 years, the efficiency and
accuracy of RT have improved significantly [8], thanks to the prevalence of
GPUs (graphic processing units) that efficiently facilitate RT tasks.

However, due to the factors mentioned above (such as the need for more
detailed environmental consideration at higher frequencies and the need for
fast simulations with higher deployment density), RT simulations are too
computationally intensive for large-scale network deployment in 6G systems. As
a result, simplified _model-based_ approaches like the dominant path model
[9], or fine-tuning of generic pathloss models (e.g., 3GPP path gain model)
with limited measurement data [10, 11] have been proposed over the years.
However, these approaches have found only limited acceptance by network
operators due to their insufficient accuracy in predicting the propagation
characteristics of signals in complex environments.

In recent years, supervised ML has been applied to solve a variety of
challenging problems in wireless communication, including channel
measurement/prediction for 6G networks. Such an _ML-based_ approach can be
trained on a map of the environment (topology/morphology) and a relatively
small set of measurement data to learn how to provide a virtual replica (e.g.,
DT) of a large-scale network environment in real-time while accurately
modeling the behavior of channel characteristics.

On the one hand, models like WiNeRT [12] and NeRF2 [13] are specifically
developed to predict detailed channel information (e.g., power, delay, and
angle information) of each multi-path component (MPC) between TX and receiver
(RX) with the input of detailed information, including spatial configuration
and wireless configuration parameters. These models are particularly well-
suited for applications in small-scale indoor areas, where high-detailed
channel prediction is required (e.g., indoor sensing).

On the other hand, models like RadioUNet [14] and FadeNet [15] aim to predict
the path gain, received power, or coverage for TX-RX in a given area with the
input of a building map. These models are designed for large-scale channel
prediction, where fast operation is essential (e.g., network optimization).

In particular, several state-of-the-art works, such as Agile [16], PPNet [17],
and PMNet [1], are pushing the boundaries of predictive accuracy and
computational efficiency for large-scale channel prediction, as evidenced by
their performance in ML competitions such as the _RadioMap Challenge_ (see
details in [18]). This highlights the applicability and importance of large-
scale channel prediction in evolving wireless network optimization, which
aligns with our research direction.

###  I-B Contributions

This paper proposes a scalable and generalizable channel prediction approach
specifically designed for large-scale channel prediction, called PMP task. Our
contributions can be summarized as follows:

  * •

We design a PMP-oriented NN architecture, called PMNet, by leveraging
computer-vision techniques, generating highly accurate channel prediction
results for a given map in few milliseconds. PMNet achieves the best channel
prediction accuracy compared to two baselines: a model-based scheme (_3GPP-
UMi_ model [19]) and another ML-based scheme (_RadioUNet_ [14]) (see Table V
in Sec. IV) and also in different PMP datasets. PMNet achieved _1st_ -rank in
the ICASSP 2023 Radio Map Challenge [18].333In this competition, PMNet
demonstrated its high accuracy in the PMP task even on a different dataset
[20] with a different map scale and network configuration, and generated by a
different RT simulation tool, i.e., _WinProp_.

  * •

We build three sets of real-world channel measurement datasets using a RT
simulation tool, i.e., _Wireless Insite_ , for training and evaluation, which
reflects different network scenarios (e.g., different map scale, environment,
and network configuration) in two different light urban environments (the USC
and UCLA campuses) and a metropolitan area (the Boston area), see Table I in
Sec. III.

  * •

We propose a method of predicting pathloss in unseen network scenarios by
using transfer learning (TL) with a pre-trained model. We prepare three pre-
trained models for TL: VGG16 [21] and two pre-trained PMNet models trained
with 3GPP prediction results and RT simulation results, respectively, and
quantitatively and qualitatively evaluate their accuracy (see Table IX and
Fig. 8 in Sec. V).

  * •

We empirically demonstrate that our PMNet pre-trained model has generalization
capability for different network scenarios, adjusting to new network scenarios
$\times 5.6$ faster and using $\times 4.5$ less data than a baseline model
without TL, while still achieving high accuracy of an RMSE of $10^{-2}$ level
(see Fig. 6 and Table. VIII in Sec. V).

  * •

We release source code for the experiments to promote reproducible ML research
in wireless communication.444https://github.com/abman23/PMNet

###  I-C Paper Organization

The rest of the paper is organized as follows: Sec. II presents the background
on two important concepts: (1) _ray tracing simulation_ , which is used to
generate ground-truth channel information for training and evaluation; and (2)
_transfer learning_ , which enables us to transfer the knowledge learned from
a source task/dataset to a new task/dataset (e.g., unseen network scenario).
After introducing our dataset based on real geographical maps in Sec. III,
Sec. IV introduces the PMP task and our proposed NN architecture (PMNet) for
this channel prediction task. We also present the training and evaluation
process, as well as simulation results. Then, Sec. V presents our approach for
efficiently learning and predicting channels in unseen network environments by
transferring the pre-trained knowledge from other networks. We provide
extensive experimental results and quantitative and qualitative performance
analysis, followed by concluding remarks in Sec. VI.

Notation: Throughout this paper, we use the normal-face font to denote scalars
and the boldface font to denote vectors. We use $P(\cdot)$ and
$P(\cdot|\cdot)$ to represent a marginal probability distribution and
conditional distribution, respectively. We also use $\|\cdot\|$ to denote the
$L^{2}$-norm, which is an Euclidean norm. $\mathcal{N}(\mu,\sigma)$ denotes
the normal distribution with mean $\mu$ and standard deviation $\sigma$.

##  II Background

###  II-A Pathloss

The link gain between a TX at location ${\bf\it q}_{\mathrm{TX}}$ and an RX at
location ${\bf\it q}_{\mathrm{RX}}$ at time $t$ and frequency $f$ can be
expressed as follows:

$|h(t,f,{\bf\it q}_{\mathrm{TX}},{\bf\it
q}_{\mathrm{RX}})|^{2}=\frac{P_{\mathrm{RX}}(t,f,{\bf\it
q}_{\mathrm{RX}})}{P_{\mathrm{TX}}(t,f,{\bf\it q}_{\mathrm{TX}})}$ (1)

where $P_{\mathrm{RX}}$ and $P_{\mathrm{TX}}$ are received and transmitted
power, respectively. This link gain includes the effects of antenna gains at
TX and RX; when isotropic antennas are used, it becomes identical to the
channel gain. It exhibits variations in time and/or location due to small-
scale fading, shadowing, and large-scale distance changes. Averaging over
small-scale fading removes (under certain circumstances, see [22, Ch. 7]) the
dependence on frequency and time, providing the path gain (PG) that can be
written as a function of only the large-scale distance changes:

$\mathrm{PG}({\bf\it q}_{\mathrm{TX}},{\bf\it
q}_{\mathrm{RX}})=\dfrac{1}{T_{\mathrm{S}}}\dfrac{1}{B_{\mathrm{S}}}\int\limits_{T_{\mathrm{S}}}\int\limits_{B_{\mathrm{S}}}|h(t,f,{\bf\it
q}_{\mathrm{TX}},{\bf\it q}_{\mathrm{RX}})|^{2}\,df\,dt.$ (2)

Here, $T_{\mathrm{S}}$ and $B_{\mathrm{S}}$ denote the stationary-time and
-bandwidth, respectively. The path gain can be represented as the sum of the
powers of the $N$ MPCs, as discussed further in Sec. III-A. For later
reference, we note that the pathloss is the inverse of the path gain (or the
sign-flipped value when expressed in dB).

###  II-B Ray Tracing (RT) Simulation

RT is an approximate method for modeling the propagation of electromagnetic
waves in wireless communication scenarios. It works by tracing the paths of
individual rays as they propagate through the environment, whose features are
represented in a geographical database. The rays are reflected, deflected, and
scattered by the objects in the environment, with the various interaction
processes computed according to high-frequency approximations, namely (most
commonly) Snell’s laws for specular reflection and transmission, uniform
theory of diffraction (UTD) for diffraction, and Kirchhoff scattering theory
for diffuse scattering [22, Ch. 4].555RT can be implemented via image-theory-
based RT, or as ray launching. We will henceforth use the expression RT for
both those methods. In this paper, we employ a commercial RT tool, _Wireless
Insite_ from Remcom [7] for all RT simulations, both because of its user-
friendliness and the fact that its accuracy has been compared against a number
of channel sounder measurements [6, 23, 24].

RT can be used to predict channel information, such as received signal
strength, delay, and angles, in a variety of wireless environments, both
indoor and outdoor. The accuracy of RT simulations depends on various factors,
such as the complexity of the environment, the accuracy of the geographical
database, and the carrier frequency. The channel information obtained from the
RT can be utilized, inter alia, for various network optimization tasks,
including base station (BS) deployment planning, BS parameter optimization, as
well as beam management and localization.

###  II-C Transfer Learning (TL)

TL is a ML technique that involves reusing a pre-trained model on a new task.
This is particularly useful when there is limited data available for the new
task or when the new task is similar to a task that has already been learned.
For example, a model pre-trained on image classification can be used for
object detection or semantic segmentation.

One of the most popular pre-trained models is VGG16 [21], which is trained on
more than a million images from the ImageNet database for image
classification. VGG16 has been reused to improve the performance of a wide
variety of tasks, such as semantic segmentation [25] and object detection
[26].

However, it is important to note that the effectiveness of TL depends on the
_similarity_ between the pre-trained task and the target task. The
transferability of deep feature representations decreases as the discrepancy
between the pre-trained task and the target task increases [27]. In other
words, the further apart the task is, the less transferable the knowledge. One
example is catastrophic forgetting, which is a phenomenon that can occur when
fine-tuning a pre-trained model on a new task, resulting in a loss of
previously acquired knowledge [28].

Research has shown that well-generalized models, particularly those with
excellent pre-training performance [29], have the potential to require minimal
fine-tuning or even none at all (e.g., zero-shot learning) for new tasks [30].
These suggest the importance of selecting a pre-trained model suitable for the
target task.

##  III Dataset

((a)) USC campus (Map)

((b)) USC campus (Geometry in _Wireless Insite_)

((c)) UCLA campus (Map)

((d)) Boston (Map)

Figure 2: Map of USC, UCLA, and Boston used in RT simulation. Fig. 2(a) is
imported and converted to Fig. 2(b). The ground-truth pathloss map over the
USC campus is then obtained using _Wireless Insite_ RT simulation and pre-
processing (e.g., interpolation, gray conversion, and data augmentation).

In this section, we discuss the dataset preparation process for our pathloss
map datasets, reflecting real-world network scenarios in USC, UCLA, and Boston
areas.

We obtained the ground-truth channel measurement data using the commercial RT
tool _Wireless Insite_ [7], which takes into account the geographical and
morphological features of the propagation environment. We then pre-processed
the data (e.g., interpolation and data augmentation) to prepare the ground-
truth pathloss map.

###  III-A Channel Measurement

####  III-A1 RT simulation

As discussed in Sec. II-A, RT emulates the behavior of each MPC between TX and
RX, following physical principles including the free-space power loss and
interaction with different interacting objects (IOs). This allows us to
compute for each MPC the information of complex amplitude $a$, directions of
departure $\Omega$ and arrival $\Psi$, and delay $\tau$. The contribution of
$m$-th MPC can be expressed as [31]:

$\displaystyle
h_{m}(t,\tau,\Omega,\Psi)=a_{m}\delta(\tau-\tau_{m})\delta(\Omega-\Omega_{m})\delta(\Psi-\Psi_{m}),$
(3)

where the dependence of $\Omega,\Psi,\tau,a$ on $t$ is not written explicitly
on the r.h.s. The sum of contributions from all MPCs is given by

$\displaystyle h(t,\tau,\Omega,\Psi)=\sum_{m=1}^{N}h_{m}(t,\tau,\Omega,\Psi).$
(4)

Since $\Omega,\Psi,\tau,|a|$ are constant over a stationarity-time and
bandwidth, while $arg(a)$ varies over many periods of $2\pi)$, and assuming
isotropic antennas at TX and RX (so that $\Omega,\Psi$ do not matter), the
path gain averaged over the small-scale fading can be computed from (2) as

$\displaystyle\mathrm{PG}=\sum_{m=1}^{N}|h_{m}(\tau,\Omega,\Psi)|^{2}=\sum_{m=1}^{N}|a_{m}|^{2}.$
(5)

Note that our pathloss map uses the information of path gain (in [dB]) while
other information on angles and delay is not needed (though this information
can be used for further applications, e.g., beamforming algorithms).

Thus, $P_{\mathrm{RX}}$ (in [dBm]) can be expressed as a function of
$P_{\mathrm{TX}}$ (in [dBm]) as follows:

$\displaystyle P_{\mathrm{RX}}=P_{\mathrm{TX}}+\mathrm{PG}.$ (6)

Note that we set $P_{\mathrm{TX}}=0$ [dBm] in our RT dataset to simplify the
analysis, which makes $P_{\mathrm{RX}}$ in [dBm] equal to $\mathrm{PG}$ in
[dB].

To generate a ground-truth (labeled) dataset that simulates real-world network
scenarios, we conduct _Wireless Insite_ RT simulations on the geographical and
morphological maps of the University of Southern California (USC) campus, the
University of California, Los Angeles (UCLA) campus, and the Boston area. Both
campus areas are in Los Angeles, CA, and exhibit a (light) urban build-up,
with most buildings being five stories or less (with a few high-rises
interspersed), gaps between buildings along the street canyons, and some open
squares. The Boston area is in downtown of Boston, MA. It is a metropolitan
area with multiple high-rises; its streets are not arranged along a
rectangular grid. Each dataset has different network configurations and
environmental characteristics (e.g., map scale, and geographical features,
such as vegetation). See Fig. 2 and Table I for more details.666 It is worth
noting that the simulations are performed at the sub-6 GHz band, which is the
most widely used cellular band. Similar simulations can be performed in other
frequency bands, such as the mmWave and THz bands, with minor adjustments to
the parameters. However, at those high frequency bands, geographical data
bases with higher resolution might be required for comparable accuracy.

We stress that the goal of our work is the correct prediction of ”ground-
truth” pathloss by ML techniques. The pathloss obtained from the RT
simulations might deviate from measured values due to inaccuracies of the
database or inherent approximations of RTs. However, such deviations are
irrelevant to the assessment of our ML methods, since they only impact what is
used as ”ground-truth” and not the prediction process itself. In other words,
if the ground-truth is more accurate (similar to measurement results), our
prediction inherently becomes more accurate as well.

TABLE I: Parameters of USC, UCLA, and Boston datasets.

Parameter | Dataset
---|---
USC | UCLA | Boston
Map scale |  $880\times 880$ [m2] |  $760\times 760$ [m2] |  $553\times 553$ [m2]
Cropped map scale (per pixel) |  $221\times 221$ [m2] ($0.86\times 0.86$ [m2]) |  $225\times 225$ [m2] ($0.88\times 0.88$ [m2]) |  $187\times 187$ [m2] ($0.73\times 0.73$ [m2])
Terrain | ✓ | ✓ | ✓
Buildings | ✓ | ✓ | ✓
Foliage | ✗ | ✗ | ✓
Carrier frequency |  $2.5$ [GHz] |  $3.0$ [GHz] |  $3.0$ [GHz]
Transmit power |  $0$ [dBm] |  $0$ [dBm] |  $0$ [dBm]
TX antenna type777Isotropic and half-wave dipole antennas provide almost identical radiation patterns within a certain angular extent. MPC induced outside of the angular extent does not contribute significantly to the link. | Isotropic (vertical) | Half-wave dipole (vertical) | Half-wave dipole (vertical)
Total # of data/scene | $4754$ | $3776$ | $3143$

####  III-A2 3GPP model

The 3GPP 38.901 channel model [19] (henceforth simply called the ”3GPP model”
for conciseness) is a widely used model for wireless system standardization
that claims validity for frequencies spanning from $0.5$ to $100$ [GHz]. It
utilizes TDL or clustered delay line (CDL) to model the double-directional
impulse response in (4). This model defines clusters as collections of paths
that share the same delay but have slightly different angles. Cluster delays
can either be deterministic or random, with cluster power decreasing as the
delay increases.

For the purposes of this paper, we only consider the 3GPP modeling of the
pathloss, which follows the classical $\alpha-\beta$ model

$\mathrm{PL}_{\alpha-\beta}(d)=10\alpha\log_{10}(d)+\beta+S,$ (7)

where $S\sim\mathcal{N}(0,\sigma_{S})$ is a lognormally distributed random
variable (with variance $\sigma_{S}$) representing the shadow fading, and
$\alpha$, $\beta$, and $\sigma$ are parameters of the model that are based on
measurement campaigns and that are different in different environments.
Important for our later discussions, those parameters are also different
depending on whether an unobstructed optical line of sight (LoS) exists
between TX and RX or not.

Specifically, for urban environments, the following describes the path gain:

$\displaystyle\mathrm{PG}_{\mathrm{UMi-LoS}}=\begin{cases}\mathrm{PL}_{1},\
(10[\mathrm{m}]\leq d_{\mathrm{2D}}\leq d_{\mathrm{BP}})\\\ \mathrm{PL}_{2},\
(d_{\mathrm{BP}}\leq d_{\mathrm{2D}}\leq 5[\mathrm{km}])\end{cases}$ (8)
$\displaystyle\ \mathrm{PG}_{\mathrm{UMi-NLoS}}=\max($
$\displaystyle\mathrm{PG}_{\mathrm{UMi-LoS}},\mathrm{PL}_{3}),$ (9)
$\displaystyle\ \ \ \ \ \ (10[\mathrm{m}]\leq d_{\mathrm{2D}}\leq
5[\mathrm{km}])$

where the two-dimensional $xy$-distance is $d_{\mathrm{2D}}$ and the three-
dimensional $xyz$-distance is $d_{\mathrm{3D}}$,

$\displaystyle\mathrm{PL}_{1}$ $\displaystyle=$ $\displaystyle
32.4+21\log_{10}(d_{\mathrm{3D}})+20\log_{10}(f_{c}),$ (10)
$\displaystyle\mathrm{PL}_{2}$ $\displaystyle=$ $\displaystyle
32.4+40\log_{10}(d_{\mathrm{3D}})+20\log_{10}(f_{c})$
$\displaystyle-9.5\log_{10}((d_{\mathrm{BP}})^{2}+(h_{\mathrm{BS}}-h_{\mathrm{UT}})^{2}),$
$\displaystyle\mathrm{PL}_{3}$ $\displaystyle=$ $\displaystyle
22.4+35.3\log_{10}(d_{\mathrm{3D}})+21.3\log_{10}(f_{c})-$ $\displaystyle
0.6(h_{\mathrm{UT}}-1.5).$

Here, the breakpoint distance is $d_{\mathrm{BP}}=2\pi
h_{\mathrm{BS}}h_{\mathrm{UT}}f_{c}/c$ where $f_{c}$ is the center frequency
in [Hz] and $c=3.0\times 10^{8}$[m/s] is the speed of light. The antenna
heights at the TX (e.g., base station), $h_{\mathrm{BS}}$, and the RX (e.g.,
user terminal), $h_{\mathrm{UT}}$, are set to $1.5$ [m] and $10$ [m],
respectively. Note that the model differs for LoS and non-LoS (NLoS)
situations.

This model is employed as one of our baselines for the prediction (see Sec.
IV-E). While the 3GPP model also models shadowing, it incorporates it as
stochastic variations that cannot be related to particular map features; we
therefore omit them for the purposes of this paper.

###  III-B Pre-processing

The raw numeric data from the RT simulation is pre-processed using _gray
conversion_ and _interpolation_ methods to generate the ground-truth pathloss
map, _data augmentation_ methods to create an increased amount of labeled
data, and _sampling_ methods to divide them into training and testing sets.

Gray conversion. To generate the pathloss map, we begin by converting the
received power $P_{\mathrm{RX}}$ (in [dBm]) (or the path gain $\mathrm{PG}$ in
[dB]) into grayscale between $1$ and $255$ using _Min-Max_ normalization, with
the minimum value of $-254$ [dBm] and the maximum value of $0$ [dBm]. While
the upper value is higher than physically reasonable, this pair of values was
chosen for convenience to have a 1 [dBm] per gray value step mapping. A
smaller (or larger) step size does not have a significant impact on the
prediction performance.

The gray value $0$ is filled at pixels of building area, which is not our
region-of-interest (RoI), while, for our RoI, each pixel is filled with gray
values between $1$ and $255$, which corresponds to $P_{\mathrm{RX}}$. Then,
the pathloss map is generated after scaling the considered map scale into a
$256\times 256$ gray image. Note that the image size ($256\times 256$) has
nothing to do with the grayscale ($0-255$).

Interpolation. Since the RT simulations are carried out over a discrete set of
RX locations, and it is computationally challenging to gather the channel
information for every available RX location, there is missing channel
information in a few pixel locations. To fill the missing part of the pathloss
map, we utilize _bilinear interpolation_ , which approximates the missing
value with a weighted sum of the gray values of the adjacent locations.

Data augmentation. Typically, a larger dataset leads to improved performance
of NN training. In other words, the larger the data set, the better the
outcome. We thus use two augmentation methods - cropping and rotation - to
increase the size of our data set.

The entire map data is cropped into images of about a quarter of the size,
taking TX as an anchor point. This augments the size of the dataset by a
factor of $96$. The image is first cropped as a $64\times 64$ size image and
then upsampled to a $256\times 256$ size image. Note that some cropped images,
not including any TX, are skipped since the TX location will be used as our
second input feature. After cropping, the image sets are rotated by
$90^{\circ}$, $180^{\circ}$, and $270^{\circ}$, thus increasing the size of
the dataset by a further factor of $4$.

Sampling. In the training and testing of PMNet on the pathloss map dataset, we
employ an exclusive division scheme. Specifically, $90\%$ and $10\%$ of images
are randomly split into the training and validation set, while the images from
the same geographical map belong exclusively to either the training or the
validation set. This approach is taken to enhance the generalization
performance of PMNet.

##  IV Pathloss Map Prediction

Figure 3: Overview of the PMP task and the PMNet architecture

###  IV-A Task (1): Pathloss Map Prediction

We now formulate the prediction task in ML nomenclature. A domain (i.e.,
wireless channel prediction) is composed of a feature space $\mathcal{X}$,
where $x\in{\mathcal{X}}$. Given the domain, a PMP task is defined as
$\mathcal{T}=\\{\mathcal{Y},P(y|\mathbf{x})\\}$, which is composed of a label
space $\mathcal{Y}$, where $y\in{\mathcal{Y}}$. Given the task, a dataset is
defined as $\mathcal{D}=\\{\mathcal{X},\mathcal{Y}\\}$, which is a collection
of $|\mathcal{D}|=\mathcal{N}$ channel measurement data that belong to a
domain with a task $\mathcal{T}$.

For the PMP task, $\mathcal{X}$ consists of (1) a building map (including
terrain, building, and/or foliage) and (2) a TX location and $\mathcal{Y}$ is
a Pathloss map. The goal of the PMP task $\mathcal{T}$ is to find a predictive
function $f(\cdot)$, which accurately predicts $\mathcal{Y}$ for a given
$\mathcal{X}$.  It is worth noting that integrating RoI (denoted as
$\mathcal{A}^{\star}$) segmentation with path gain prediction simplifies the
PMP task and eliminates the need for separate pre- or post-processing steps
for the RoI segmentation for each map. Additionally, this integration helps NN
better understand the different IOs in a given building map.

In a nutshell, the PMP task is to predict the pathloss/path gain (and received
power $P_{\mathrm{RX}}$ using simple normalization) at RX locations ${\bf\it
q}_{\mathrm{RX}}$ given TX location ${\bf\it q}_{\mathrm{TX}}$ in RoI
$\mathcal{A}^{\star}$. This channel prediction task exploits _site-specific_
geographical information, focusing on the large-scale effects in the channel.

We employ a supervised ML method for the PMP task. We train the model on a
dataset of RT channel measurements for an area of $\mathcal{A}$, such as the
USC dataset in Table I; see Fig. 3 for an overview of the ML-based PMP
approach.

###  IV-B Network Architecture

In this subsection, we present the design process of our proposed PMP-oriented
NN architecture, referred to as _PMNet_. Our design principles are summarized
as follows: (1) several state-of-the-art techniques in the field of image
processing are carefully selected and tested, (2) some essential techniques
are selected following the concept of ablation study, and (3) the NN with
selected techniques is optimized with extensive trials.

PMNet
---
Encoder | Decoder
# | Type | Output Size | # | Type | Output Size
Input | Image | $2\times 256\times 256$ | Output | Image | $1\times 256\times 256$
1($\downarrow$) | Conv2d, MaxPool2d | $64\times 65\times 65$ |  1($\uparrow$) | Conv2d | $(128+2)\times 256\times 256$
2 | ResLayer | $256\times 65\times 65$ | 2 | Conv2d | $(256+64)\times 65\times 65$
3($\downarrow$) | ResLayer | $512\times 33\times 33$ | 3 | Conv2d | $(256+256)\times 65\times 65$
4($\downarrow$) | ResLayer | $512\times 17\times 17$ |  4($\uparrow$) | ConvTranspose2d | $(256+256)\times 65\times 65$
5 | ResLayer | $1024\times 17\times 17$ |  5($\uparrow$) | ConvTranspose2d | $(512+512)\times 33\times 33$
6 | Conv2d, AdaptiveAvgPool2d | $512\times 17\times 17$ | 6 | Conv2d | $(512+512)\times 17\times 17$

TABLE II: PMNet architectures and parameters. $\downarrow$ and $\uparrow$
represent the downsampling and upsampling layers, respectively.

####  IV-B1 Design choices

In the PMP task, the NN is required to perform image segmentation to identify
the RoI and predict received power within the RoI, while accounting for
complex wireless propagation physics. To accomplish this, our proposed PMNet
is designed based on such methods, _Encoder-Decoder_ and _Atrous convolution_.

Encoder-Decoder. Encoder-Decoder networks are a widely applied architecture
for many computer vision tasks, e.g., object detection [32], human pose
estimation [33], and semantic segmentation [34, 35, 36]. The encoder-decoder
architecture allows to learn a lower-dimensional representation from a higher-
dimensional dataset and utilize the learned representation for various tasks.
However, as the encoder shrinks the input feature maps, it may lose essential
information, leading to a _bottleneck problem_. Several architectures,
including U-Net [37], address the bottleneck problem by adding _skip
connections_ between the encoder and the decoder parts. Skip connections allow
the decoder to access feature maps from the encoder, which helps to propagate
context information to higher-resolution layers.

Atrous convolution.  _Receptive field_ of a convolutional layer is the region
of the input feature map that contributes to the output feature map at a given
location. The size of the receptive field is determined by the resolution of
the input feature map and the field-of-view (FoV) of the filter. There is a
logarithmic relationship between the localization accuracy of a model and the
size of its receptive field. This means the receptive field size should be
sufficient if the given dataset and task are observed with wide FoV. A
standard convolutional filter detects a particular feature by sliding over the
input feature map, resulting in the output feature map seeing only the
adjacent part of the input feature map. In terms of computational complexity,
having a wide receptive field with the standard convolutional filter is
expensive. Thus, broadly speaking, the receptive field of the standard
convolution filter is somewhat narrow, seeing only little context.

Atrous convolution, also known as dilated convolution, is a technique that
addresses this limitation [38]. It allows capturing a larger context with a
wider FoV by modifying the standard convolution operation. For the two-
dimensional case, atrous convolution is applied over the input feature map $f$
to produce the output feature map $g$ at location $\\{i,j\\}$ using the
convolution filter $w$. This operation can be expressed as follows:

$g_{\\{i,j\\}}=\sum_{m=1}^{k}\sum_{n=1}^{k}f_{\\{i+rm,j+rn\\}}w_{\\{m,n\\}}.$
(11)

Here, $k$ represents the kernel size, and $r$ is the atrous rate, which
determines the stride level. Notably, the atrous rate $r$ allows to adaptively
control the FoV of the filter. For example, an atrous rate of $r=2$ doubles
the FoV of the filter, while an atrous rate of $r=3$ triples it. The standard
convolution can be seen as a special case of (11) where $r=1$.

In the context of the PMP task, the encoder-decoder symmetric architecture of
PMNet facilitates efficient context propagation from the encoder to the
decoder, while atrous convolution enables it to handle scale variations and
capture broader context in map data, setting it apart from other UNet-based
networks [14, 15, 16, 17]. The combination of these two features enables PMNet
to efficiently and accurately predict pathloss maps, while also accounting for
complex wireless propagation physics.

####  IV-B2 Design parameters

PMNet architectures are composed of a stack of _ResLayers_ , each containing
multiple residual blocks [39]. These ResLayers can be configured with varying
numbers of blocks, atrous rates, multi-grids, and output strides. These
elements are summarized as follows:

  * •

_Number of blocks_ : The number of residual blocks in a ResLayer controls the
complexity and depth of the network. Increasing the number of blocks may
improve the accuracy of the model, but it also increases the computational
cost.

  * •

_Atrous rates_ : Atrous rates control the spacing between the convolutions in
a ResLayer. Larger atrous rates allow the network to capture more larger
spatial contexts in the PMP task.

  * •

_Multi-grids_ : Multi-grids allow the network to capture multi-scale
information from different levels of the CNN architecture.

  * •

_Output stride_ : The output stride of a ResLayer controls the ratio between
the resolution of the input image and the output image’s resolution. A higher
output stride results in a lower-resolution output image. This can be useful
to strike a balance between accuracy and speed.

Note that the impact of output stride in the PMP task is shown in Table IV in
Sec. IV-E (e.g., the case of $\frac{H}{8}\times\frac{W}{8}$). With these
design choices and parameters, PMNet effectively predicts pathloss maps even
for different channel measurement datasets (e.g., _RadioMapSeer_ [20]). For an
architectural overview, please refer to Fig. 3 and Table II. For more details,
please see our source code repository.

###  IV-C Training

Table III lists the hyper-parameters that are used for the training of PMNet.
We implement the PMNet using PyTorch and use an NVIDIA GeForce RTX 3080 Ti
GPU. For more stable training, we normalize the input values into $[0,1]$ via
scaling. During the training, we evaluate the PMNet by mean squared error
(MSE) on the validation set at the end of every epoch. For testing, we use the
parameters of PMNet with the best MSE score on the validation set.
Consequently, the pathloss map for a given map can be generated within a few
milliseconds after training.

Model | PMNet
---|---
Dataset (USC) |
Map | USC campus
Split for training (test) set |  $90\%$ ($10\%$) of dataset
Hyper-parameter |
Learning rate (LR) | $10^{-3}\sim 5\times 10^{-4}$
LR gamma, step size |  $0.5$, $10$
Batch size | $16\sim 32$
Optimizer | Adam
# of of epochs | $50$
TABLE III: Training configuration and hyper-parameters for PMNet training.

Case | Data Aug. ($\times 4$) | Feature Size | RMSE$\downarrow$ | RoI Segmentation Err.$\downarrow$ | Channel Prediction Err.$\downarrow$
---|---|---|---|---|---
w/o Data-Aug. | ✗ | $\frac{H}{16}\times\frac{W}{16}$ | 0.01637 | 0.00263 | 0.01860
w/ Data-Aug. | ✓ | $\frac{H}{16}\times\frac{W}{16}$ | 0.01259 | 0.00025 | 0.01403
$\frac{H}{8}\times\frac{W}{8}$ | ✓ | $\frac{H}{8}\times\frac{W}{8}$ | 0.01057 | 0.00096 | 0.01175

TABLE IV: Ablation study for PMNet training optimization. Lower values
indicate better performance.

###  IV-D Evaluation

Root mean square error (RMSE). RMSE is a widely used loss function in
regression analysis and is used as the primary evaluation metric for this
task. It measures the overall difference between the prediction $\hat{{\bf\it
y}}$ and ground-truth ${\bf\it y}$ and quantifies the overall accuracy of the
model. The formula for RMSE is:

$\mathrm{RMSE}(\hat{{\bf\it y}},{\bf\it
y})=\sqrt{\frac{1}{N}\sum_{n=1}^{N}(\hat{y}_{n}-y_{n})^{2}},$ (12)

where $\hat{y}_{n}\in\hat{{\bf\it y}}$ and $y_{n}\in{\bf\it y}$ denote
predicted and ground-truth gray value (corresponding $P_{\mathrm{RX}}$) at the
$n$-th pixel, respectively, and $N$ is the number of pixels in a pathloss map,
i.e., $256\times 256$. The RMSE averaged over all samples is the primary
evaluation metric for the PMP task.

RoI segmentation error. The RoI segmentation error, calculated using the
intersection over union (IoU) metric, quantifies the accuracy of RoI and non-
RoI area segmentation for all pixels in the ground-truth (${\\{i,j\\}}$) and
prediction ($\\{\hat{i},\hat{j}\\}$) - that is calculated as follows:

$\displaystyle\mathrm{RoI\leavevmode\nobreak\ Segmentation\leavevmode\nobreak\
Err.}=\frac{\sum_{i}\sum_{j}\mathrm{Err^{B}}_{\\{i,j\\}}}{\sum_{i}\sum_{j}\mathrm{Bld}_{\\{i,j\\}}}.$
(13)

Here, $\mathrm{Err^{B}}_{\\{i,j\\}}$ and $\mathrm{Bld}_{\\{i,j\\}}$ are
defined as:

$\displaystyle\mathrm{Err^{B}}_{\\{i,j\\}}$ $\displaystyle=\begin{cases}1,\
\\{i,j\\}\in\mathcal{B}\leavevmode\nobreak\ \mathrm{and}\leavevmode\nobreak\
\\{\hat{i},\hat{j}\\}\in\mathcal{A}^{\star}\\\ 1,\
\\{i,j\\}\in\mathcal{A}^{\star}\leavevmode\nobreak\
\mathrm{and}\leavevmode\nobreak\ \\{\hat{i},\hat{j}\\}\in\mathcal{B}\\\ 0,\
\mathrm{otherwise}\end{cases}$ (14) $\displaystyle\mathrm{Bld}_{\\{i,j\\}}$
$\displaystyle=\begin{cases}1,\ \\{i,j\\}\in\mathcal{B}\\\ 0.\
\mathrm{otherwise}\end{cases}$ (15)

Within a given map, the non-RoI area, denoted as black (gray value $0$), is
represented by $\mathcal{B}$, while the RoI area, denoted as non-black
(grayscale $1-255$), is represented by $\mathcal{A}^{\star}$. $\mathcal{B}$
and $\mathcal{A}^{\star}$ are complementary set within $\mathcal{A}$.
$\mathcal{B}$ can include buildings, foliage, and/or small objects.

Channel prediction error. Channel prediction error directly evaluates path
gain accuracy for pixels within the RoI area, evaluating power in [dBm] (or
path gain in [dB]) unlike RMSE, which quantifies differences based on gray
values.

To calculate channel prediction error, gray values within the RoI area of both
the predicted and ground-truth pathloss maps are converted into corresponding
received power values. The RMSE formula is then applied to these power values:

$\mathrm{RMSE}(\hat{{\bf\it p}},{\bf\it
p})=\sqrt{\frac{1}{N}\sum_{n=1}^{N}(\hat{p}_{n}-p_{n})^{2}},$ (16)

where $\hat{p}_{n}\in\hat{{\bf\it p}}$ and $p_{n}\in{\bf\it p}$ represent the
predicted and ground-truth $P_{\mathrm{RX}}$ at the $n$-th pixel,
respectively. Channel Prediction Error is then computed by averaging
$\mathrm{RMSE}(\hat{{\bf\it p}},{\bf\it p})$ across all given samples.

###  IV-E Simulation Result

####  IV-E1 Training optimization

Table IV presents an ablation study to identify the factors that significantly
contribute to PMNet’s performance in the PMP task, such as _data augmentation_
and _feature map size_.888Our extensive experiments tested other factors, such
as different sampling methods, training loss functions, and additional input
features (e.g., TX distance heatmap), but these factors did not show a
meaningful improvement to justify the additional complexity.

Impact of data augmentation. For the data augmentation, we do horizontal,
vertical and diagonal flips. In other words, including the original images, we
use the $\times 4$ number of images for training. Note that data augmentation
has several advantages in general: first, it enhances the diversity of the
training data by generating additional examples that capture various
variations of the original data. Second, it reduces overfitting by exposing
the model to a wider range of input patterns. Finally, data augmentation helps
to make the model more robust to noise and variability in the input data. As
shown in Table IV, it improves the performance of PMNet by $15.7$% in terms of
RMSE.

Impact of feature map size. We analyze the performances of PMNet according to
the size of the feature map, which is the output of the encoder. Table IV
compares the results with the feature map sizes $\frac{H}{8}\times\frac{W}{8}$
and $\frac{H}{16}\times\frac{W}{16}$, where $H$ and $W$ are the height and
width of an input image, respectively. To adjust the feature map size, we
modify the strides of the convolution layers in the encoder. We employ the
feature map size of $\frac{H}{8}\times\frac{W}{8}$ as the default option,
because PMNet yields better performance with the feature map size of
$\frac{H}{8}\times\frac{W}{8}$ than that of $\frac{H}{16}\times\frac{W}{16}$.

((a)) Prediction (3GPP)

((b)) Prediction (RadioUNet)

((c)) Prediction (PMNet)

((d)) Ground-truth (_Wireless Insite_)

((e)) Prediction (3GPP)

((f)) Prediction (RadioUNet)

((g)) Prediction (PMNet)

((h)) Ground-truth (_Wireless Insite_)

Figure 4: Comparison of the predicted pathloss map of 3GPP, RadioUNet, and
PMNet. [inner color=white, fill color= color0, outer color=color0]TX in
ground-truth represents the TX location. The scenes are randomly selected, not
cherry-picked.

Scheme | ML-based | RMSE$\downarrow$ | RoI Segmentation Err.$\downarrow$ | Channel Prediction Err.$\downarrow$
---|---|---|---|---
3GPP (with map info.) [19] | ✗ | 15.9451 | - | 17.5973
RadioUNet [14] | ✓ | 0.02634 | 0.00840 | 0.01249
PMNet | ✓ | 0.01057 | 0.00096 | 0.01175

TABLE V: Comparison study for PMP schemes (3GPP, RadioUNet, and PMNet). Lower
values indicate better performance, and the lowest errors are highlighted.

####  IV-E2 Accuracy

We compare the ML-based PMP with our proposed PMNet model to two other methods
for the PMP task: a model-based approach, _3GPP_ , and an ML-based approach,
_RadioUNet_. All three methods produce a single-channel $256\times 256$ image
of the pathloss map as the output, given the input of a two-channel $256\times
256$ image containing the geographical map and the TX location. Here are the
details of these baseline methods:

  1. 1.

3GPP (with map info.) As discussed in Sec. III-A2, the 3GPP model determines
the pathloss at a particular location based on the Euclidean distance and
whether the link between the TX and RX is in LoS or NLoS. To ensure a fair
comparison with other baselines, we utilize map information to determine the
LoS or NLoS condition of specific pixels to the TX.999 The original 3GPP
pathloss model uses a probabilistic model to determine LoS/NLoS condition at a
particular distance. However, to ensure a fair comparison, we use here the
deterministic LoS/NLoS condition determined from the map information in
calculating the pathloss gain.  Note that it does not require any NN training
as it is a model-based approach.

  2. 2.

RadioUNet [14] is an ML-based PMP method that extends the UNet architecture by
employing two UNets. Each UNet comprises $8$ encoder layers with convolution,
ReLU, and Maxpool layers, followed by $8$ decoder layers with transposed
convolution and ReLU layers. The encoders and decoders are concatenated, as in
the original UNet architecture. Here, RadioUNet employs curriculum training to
enhance training: in the first stage, the first UNet is trained for a specific
number of epochs, with the second UNet frozen. In the second stage, the second
UNet is trained using the two-channel input features and the output of the
first UNet, effectively making it a three-channel input network.

  3. 3.

PMNet (Proposed) is our proposed ML-based PMP method. This network employs
several parallel atrous convolutions with different rates and the encoder-
decoder network. The encoder consists of $6$ ResNet-based layers. Each ResNet
layer comprises several bottleneck layers consisting of convolution, batch
normalization, max pooling, and ReLU. The decoder consists of $6$ layers
consisting of convolution, adaptive average pooling, ReLU, transposed
convolution, and ReLU. Skip connections are used between encoders and
decoders.

Qualitative analysis. Fig. 4 shows the prediction results of the baselines.
Recall that each pixel in the RoI corresponds to the predicted received power
$P_{\mathrm{RX}}$ (or the path gain $\mathrm{PG}$). Note that some pixel
values in the ground-truth data appear noisy due to interpolation during the
gray conversion process after RT simulation.

3GPP exhibits a substantial deviation from ground-truth obtained through RT
simulation, highlighting the differences between how RT simulation and 3GPP
model calculate a pathloss. Specifically, for RX locations with LoS conditions
close to the TX, the results obtained using the 3GPP model approximately match
the ground-truth data obtained from _Wireless Insite_. However, for RX
locations farther from the TX or under non-LoS conditions, the 3GPP model
exhibits significant discrepancy from the ground-truth data. It is worth
noting that the 3GPP pathloss model does not provide results for near-field
within a link distance of 10 meters; so, we arbitrarily set the power in the
near-field area to gray value $255$, which does not introduce significant
errors. The 3GPP pathloss model is a simplified model that does not account
for the complex wireless propagation physics of reflection, diffraction, and
scattering (highlighted in [inner color=white, fill color= color0, outer
color=color0]A). Instead, it relies solely on two models for LoS and NLoS
locations, respectively, and only considers link distance and carrier
frequency. This simplified approach inevitably leads to significant
inaccuracies in the pathloss prediction.

RadioUNet demonstrates impressive RoI segmentation results, while its channel
prediction outputs appear somewhat blurry. It is worth noting that RadioUNet
conducts curriculum-based training with 50 epochs each in the first and second
stages, utilizing the same training/validation set as PMNet, which is trained
with a total of 50 epochs.

PMNet, on the other hand, achieves notable results for both RoI segmentation
and channel prediction. As highlighted in [inner color=white, fill color=
color0, outer color=color0]B, PMNet effectively captures the intricate
wireless propagation physics of reflection, diffraction, and scattering. This
can be attributed to PMNet’s ability to incorporate a broader contextual
understanding of the environment, enabling it to capture the representation of
wireless propagation physics in the surrounding environment.

Case | Model | Train Data | Eval. Data | RMSE$\downarrow$ | RoI Segmentation Err.$\downarrow$ | Channel Prediction Err.$\downarrow$
---|---|---|---|---|---|---
Vanilla | PMNet | USC | USC | 0.01057 | 0.00096 | 0.01175
Cross-scenario (UCLA) | PMNet | USC | UCLA | 0.19146 | 0.03925 | 0.21700
Cross-scenario (Boston) | PMNet | USC | Boston | 0.25842 | 0.04602 | 0.32436

TABLE VI: Numerical results of PMNet on an unseen network scenario. PMNet was
trained on the USC dataset and evaluated on the UCLA and Boston dataset.

Quantitative analysis. Table V compares our proposed PMNet model to the model-
based 3GPP method and the ML-based RadioUNet method in terms of three accuracy
metrics for the PMP task: RMSE, RoI segmentation error, and channel prediction
error. Note that the ground-truth dataset is made by RT simulation; therefore,
the error shows the difference between a scheme and the measurement by RT
simulation.

The model-based 3GPP method has inferior results compared to ML-based methods,
which can be explained by the oversimplifications inherent in this model, as
discussed above. While our proposed PMNet model achieves the best score on all
three metrics, another ML-based PMP method, RadioUNet, also achieves high
accuracy ($RMSE\leavevmode\nobreak\ \leq 0.03$). This result highlights the
capability of ML-based PMP approaches to learn a representation of the
wireless propagation physics implicit in the ground-truth RT measurement data.

##  V Transferable Pathloss Map Prediction

###  V-A Challenge: PMP for Unseen Network Scenario

((a)) Ground-truth (_Wireless Insite_)

((b)) Prediction (PMNet)

Figure 5: Prediction results of PMNet on an unseen network scenario (i.e.,
cross-scenario evaluation). The model is trained on the USC dataset and
evaluated on the Boston dataset.

As demonstrated in the previous section, PMNet exhibits high accuracy of the
PMP task for a given dataset. However, minimizing re-training efforts for new
network scenarios remains a challenge. To evaluate PMNet’s generalizability
across different scenarios, we conducted a cross-scenario evaluation, testing
PMNet trained on USC data on the Boston dataset.

As shown in Fig. 5 and Table VI, the PMNet achieves the RoI segmentation error
on the order of $10^{-2}$ and the channel prediction error on the order of
$10^{-1}$ in a new scenario. Such deterioration is due to differences in
network configuration and environmental characteristics between the two
scenarios (e.g., different map scales and geographical features). This
highlights the need for further development to improve PMNet’s performance
across different network scenarios, a task we refer to as _cross-scenario
PMP_.

###  V-B Task (2): Cross-scenario PMP

To enable better performance, we now allow cross-scenario PMP to improve the
model trained on a different network scenario through training with a reduced-
size training in the new scenario. This will allow the network to adapt to the
new scenario with less time and resource effort, while maintaining high
accuracy. To address this challenge, we leverage _transfer learning_ (TL).

Approach: Transfer learning. TL is an ML technique that allows knowledge
transfer from one task or dataset to another. In the context of cross-scenario
PMP, we can transfer the knowledge from the source scenario, which learns a
predictive function $f_{S}(\cdot)$ from a source dataset $\mathcal{D}_{S}$
(e.g., USC), to the target scenario, which learns a predictive function
$f_{T}(\cdot)$ from a target dataset $\mathcal{D}_{T}$ (e.g., UCLA and
Boston).

There are two main ways to use TL for the cross-scenario PMP.

  * •

_Feature extraction_ : We can train a feature extractor on a source scenario
and then use that feature extractor to extract features from data from a
target scenario. Once we have extracted the features, we can train a simple
model (e.g., a linear regressor) to predict the pathloss map for the target
scenario.

  * •

_Fine-tuning_ : We can fine-tune a pre-trained model on the target scenario.
This can be done by unfreezing some or all of the layers of the pre-trained
model and training the model on data from the target scenario.

The choice between those two methods depends on a number of factors, including
the size and complexity of the pre-trained model, the availability of training
data for the target dataset, and the computational resources available.

In this work, we focus on the fine-tuning TL approach with all of the layers
of the pre-trained model unfrozen.101010While we have performed sample
experiments with unfreezing certain layers, such as the encoder-frozen and
decoder-unfrozen, performance did not improve significantly. A more
comprehensive investigation of this topic is, however, beyond the scope of
this paper. This approach is simple yet effective, achieving higher accuracy
on various cross-scenario PMP tasks with less training data and shorter
training time, as elaborated in the following subsection.

We prepare and use the following pre-trained models in our experiments:

  1. (i)

VGG16ImgNet is the pre-trained CNN model trained on the ImageNet dataset,
which contains 140k images belonging to 22k categories. It is a powerful image
classification model that has been used to achieve state-of-the-art results on
a variety of image classification benchmarks.

  2. (ii)

PMNet3gpp is the pre-trained PMNet model trained on the 3GPP pathloss map
dataset. The 3GPP pathloss map dataset is prepared with the 3GPP pathloss
model in [19] (see 3GPP in Sec. IV-E2, Fig. 4, and Table V).

  3. (iii)

PMNetusc is the pre-trained PMNet model trained on the USC RT dataset. It is
similar to PMNet3gpp but is trained on a different dataset. This is our main
pre-trained model.

Each pre-trained model is available on our GitHub page.

Model |
---|---
Backbone | PMNet, VGG16
Pre-trained model |  PMNetusc, PMNet3gpp, VGG16ImgNet
Dataset (UCLA, Boston) |
Map | UCLA campus, Boston
Split for training (test) set |  $10\%\sim 90\%$ ($10\%$) of dataset
Hyper-parameter |
LR | $10^{-3}\sim 5\times 10^{-4}$
LR gamma, step size |  $0.5$, $10$
Batch size | $16$
Optimizer | Adam
# of of epochs | $50$

TABLE VII: Training configuration and hyper-parameters in cross-scenario PMP.
Figure 6: Comparison of the training efficiency of PMNet models with and
without TL. PMNet models are trained for 50 epochs and evaluated on the Boston
dataset.

###  V-C Simulation Results

As demonstrated in the cross-scenario evaluation results (in Fig. 5 and Table
VI), there is a need for further development to make PMNet adapt to different
network scenarios. To this end, our approach is fine-tuning a pre-trained
model with down-sized data for the new scenario. Here, the main questions in
performing cross-scenario PMP are: (1) How quickly and with how minimal data
PMNet can effectively adapt to new scenarios; and (2) Which pre-trained model
should be utilized for optimal performance in cross-scenario PMP.

####  V-C1 Efficiency

For cross-scenario PMP, rapidly adapting PMNet models to new network scenarios
using limited data is essential due to the time-consuming and expensive nature
of channel measurement using RT simulation or channel sounding. This is
particularly critical for applications like beam management and localization
using ML-based PMP, which demand quick adjustments for new scenarios.

Impact of TL. TL can significantly improve the training speed of PMNet models
for cross-scenario PMP. As shown in Fig. 6 and Table VIII, the TL case with
the PMNetusc pre-trained model achieves a given level of accuracy much faster
even with much less amount of training data. In particular, PMNetusc achieves
the same level of accuracy (RMSE $\leq 0.1$ and RMSE $\approx 0.03$) $\times
5.6$ and $\times 4.1$ faster, respectively, as the Vanilla case (highlighted
in [inner color=white, fill color= color0, outer color=color0]C), where we
define as ”Vanilla” the training from scratch in a particular environment.

Case | # of Required Step (Training Speed)
---|---
RMSE $\leq 0.1$ |  RMSE $\approx 0.03$
Vanilla ($90\%$ Data) |  5841  ($\mathbf{\times 1.0}$) |  6195  ($\mathbf{\times 1.0}$)
PMNetusc ($20\%$ Data) |  1040  ($\mathbf{\times 5.6}$) |  1520  ($\mathbf{\times 4.1}$)

TABLE VIII: Impact of TL on training speed ($=\frac{1}{\mathrm{steps}}$).
PMNet models with or without PMNetusc pre-trained model are trained and
evaluated on the Boston dataset.

Furthermore, the TL can also significantly save the required amount of data
for cross-scenario PMP. As shown in Fig. 7, the TL (PMNetusc) trained with
about $20\%$ of the Boston dataset achieves equivalent results to the Vanilla
case trained with about $90\%$ of the dataset.

It is worth noting that limited training data can easily induce overfitting,
as observed in the Vanilla case with $20\%$ Data (highlighted in [inner
color=white, fill color= color0, outer color=color0]D). For the same amount of
new scenario data, the TL case (PMNetusc ($20\%$)) does not experience the
overfitting issue. This suggests that TL also enhances training stability
(less overfitting issue with limited data) in cross-scenario PMP.

Our findings demonstrate that the pre-trained PMNetusc model efficiently
accelerates the training process by leveraging its knowledge of PMP tasks,
including the physics of wireless channel propagation and RoI segmentation,
and this model can be readily adapted to new scenarios with minimal data and
training steps.

Consequently, we confirm that fine-tuning with a _suitable_ pre-trained model
is an effective cross-PMP task method. Another key question is which pre-
trained model is suitable and which is not, which is discussed further in the
following.

((a)) RMSE

((b)) Channel prediction error

Figure 7: Impact of TL on training data requirements. PMNet models with or
without PMNetusc pre-trained model are trained with 50 epochs and evaluated on
the Boston dataset.

Case |  Pre-training | Model | RMSE$\downarrow$ | RoI Segmentation Err.$\downarrow$ | Channel Prediction Err.$\downarrow$
---|---|---|---|---|---
Vanilla |  ✗ | PMNet | 0.03415 | 0.02935 | 0.03844
TL (ImageNet) |  ✓ (ImageNet) | VGG16 | 0.04528 | 0.01814 |  0.05108 [inner color=white, fill color= color0, outer color=color0]E
TL (3GPP) |  ✓ (3GPP) | PMNet | 0.02809 | 0.00655 | 0.03238
TL (USC) |  ✓ (USC) | PMNet | 0.02792 | 0.01666 | 0.03145

((a)) UCLA

Case |  Pre-training | Model | RMSE$\downarrow$ | RoI Segmentation Err.$\downarrow$ | Channel Prediction Err.$\downarrow$
---|---|---|---|---|---
Vanilla |  ✗ | PMNet | 0.01736 | 0.02417 | 0.02125
TL (ImageNet) |  ✓ (ImageNet) | VGG16 | 0.01999 | 0.02040 | 0.02512
TL (3GPP) |  ✓ (3GPP) | PMNet | 0.01762 | 0.04030 | 0.02187
TL (USC) |  ✓ (USC) | PMNet | 0.00987 | 0.03530 | 0.01225

((b)) Boston

TABLE IX: Comparison of pre-trained models (VGG16ImgNet, PMNet3gpp, and
PMNetusc) in terms of accuracy. Models are evaluated on the UCLA and Boston
datasets, using $90\%$ of the data for training and $10\%$ of the data for
validation. $50$ epochs are used for training. Lower values indicate better
performance, and the lowest errors are highlighted.

((a)) Vanilla (UCLA)

((b)) VGG16ImgNet (UCLA)

((c)) PMNet3gpp (UCLA)

((d)) PMNetusc (UCLA)

((e)) Ground-truth (UCLA)

((f)) Vanilla (Boston)

((g)) VGG16ImgNet (Boston)

((h)) PMNet3gpp (Boston)

((i)) PMNetusc (Boston)

((j)) Ground-truth (Boston)

Figure 8: Comparison of the prediction results of pre-trained models
(VGG16ImgNet, PMNet3gpp, and PMNetusc). All models are trained using 50
epochs.

####  V-C2 Accuracy

As discussed in Sec. V, the source and target scenario (task or domain) should
be sufficiently similar for effective TL to occur. For instance, to
successfully apply TL to the target task of predicting wireless communication
channels, the NN should extract relevant features of wireless propagation
physics from the source task.

“Suitable” pre-trained model. Table IX compares the performance of the PMNet
model with and without TL. The baseline model, referred to as Vanilla, is
trained without any TL (without any pre-trained model). Additionally, we
compare the performance of TL using a pre-trained model trained on an
unrelated source scenario (i.e., VGG16 trained on ImageNet) with TL using a
pre-trained model trained on a related source scenario (i.e., PMNet trained on
USC or 3GPP datasets).

As shown in Table IX, both PMNet models trained on PMNetusc and PMNet3gpp
outperform the Vanilla case on all performance metrics, suggesting that using
a pre-trained model trained on a related source task can significantly improve
accuracy.

Interestingly, while the VGG16 model trained on ImageNet (VGG16ImgNet)
outperforms the Vanilla for RoI segmentation, it fails to do so for channel
prediction (highlighted in [inner color=white, fill color= color0, outer
color=color0]E). This discrepancy stems from the VGG16 pre-trained model,
which has an inherent understanding of segmentation and image representation
from its source task; however, does not have any knowledge of the physics of
wireless propagation.

Fig. 8 visually confirms the findings from Table IX. All models achieve high
accuracy for RoI segmentation, while only the TL case using a pre-trained
model trained on a related source scenario (e.g., PMNet3gpp and PMNetusc)
achieves high accuracy for channel prediction, capturing subtle details of the
wireless propagation physics. This suggests that our PMNet pre-trained model
is generalizable to different scenarios with its inherent knowledge of channel
propagation representation and that TL can further improve accuracy.

These results empirically demonstrate that pre-trained model’s source dataset
(task or domain) should be similar to the target dataset (task or domain) to
transfer useful information during TL. Specifically, for cross-scenario PMP,
it is important to use a pre-trained model that has been trained extensively
on data related to wireless propagation physics.

Therefore, we conclude that the suggested TL approach, fine-tuning with a
stable and closely related pre-trained model (such as PMNetusc), is a simple
yet effective way to address the cross-scenario PMP task, which is important
for practical applications.

##  VI Conclusion

This work introduces an ML-based large-scale channel prediction framework,
PMNet, which can create highly accurate pathloss predictions for a given map
in a few milliseconds. Utilizing an RT channel measurement dataset of real-
world scenarios (e.g., USC, UCLA, and Boston area), PMNet is verified for its
accuracy and training efficiency. In particular, TL with our PMNet pre-trained
model, which has generalization capability for different network scenarios,
enables the PMNet to adapt itself quickly and efficiently to a new network
scenario, while achieving an RMSE of $10^{-2}$ level.

The high accuracy and low runtime of the PMNet framework make it suitable for
deployment planning in dense networks as well as online optimization of
network parameters.

Still, it remains an open question whether the knowledge of wireless
propagation physics in our PMNet pre-trained model can be transferred to other
downstream tasks beyond the PMP task; this question will be the topic of our
future research.

## References

  * [1] J.-H. Lee, A. F. Molisch, and et al., “PMNet: Large-scale channel prediction system for ICASSP 2023 first pathloss radio map prediction challenge,” in _Proc. IEEE International Conf. on Acoustics, Speech and Signal Processing (ICASSP)_ , 2023.
  * [2] ——, “PMNet: Robust pathloss map prediction via supervised learning,” in _Proc. IEEE Global Commun. Conf. (GLOBECOM)_ , 2023.
  * [3] X. Lin, L. Kundu, C. Dick, E. Obiodu, T. Mostak, and M. Flaxman, “6G digital twin networks: From theory to practice,” _IEEE Commun. Magazine_ , pp. 1–7, 2023.
  * [4] A. Alkhateeb, S. Jiang, and G. Charan, “Real-time digital twins: Vision and research directions for 6g and beyond,” _IEEE Commun. Magazine_ , pp. 1–7, 2023.
  * [5] R. Valenzuela, “A ray tracing approach to predicting indoor wireless transmission,” in _Proc. IEEE Vehicular Technology Conf._ IEEE, 1993, pp. 214–218.
  * [6] S.-C. Kim and et al., “Radio propagation measurements and prediction using three-dimensional ray tracing in urban environments at 908 MHz and 1.9 GHz,” _IEEE Trans. on Veh. Technol._ , vol. 48, no. 3, 1999.
  * [7] REMCOM, “Wireless Insite,” https://www.remcom.com/wireless-insite-em-propagation-software.
  * [8] V. Degli-Esposti, F. Fuschini, E. M. Vitucci, and et al., “Measurement and modelling of scattering from buildings,” _IEEE Trans. on Antennas Propag._ , vol. 55, no. 1, pp. 143–153, 2007.
  * [9] R. Wahl, G. Wölfle, P. Wildbolz, and F. L, “Dominant path prediction model for urban scenarios,” in _IST Mobile and Wireless Commun._ , 2005.
  * [10] M. Lee and D. Han, “Voronoi tessellation based interpolation method for Wi-Fi radio map construction,” _IEEE Commun. Lett._ , vol. 16, no. 3, pp. 404–407, 2012.
  * [11] D. Romero and S.-J. Kim, “Radio map estimation: A data-driven approach to spectrum cartography,” _arXiv:2202.03269 [eess.SP]_ , 2022.
  * [12] T. Orekondy, P. Kumar, S. Kadambi, H. Ye, J. Soriaga, and A. Behboodi, “WiNeRT: Towards neural ray tracing for wireless channel modelling and differentiable simulations,” in _Proc. of International Conf. on Learning Representations (ICLR)_ , 2023.
  * [13] X. Zhao, Z. An, Q. Pan, and L. Yang, “NeRF2: Neural radio-frequency radiance fields,” in _Proc. of International Conf. on Mobile Computing and Networking (MobiCom)_ , 2023.
  * [14] R. Levie, G. Caire, and et al., “RadioUNet: Fast radio map estimation with convolutional neural networks,” _IEEE Trans. on Wireless Commun._ , vol. 20, pp. 4001–4015, 02 2021.
  * [15] V. V. Ratnam, H. Chen, and et al., “Fadenet: Deep learning-based mm-wave large-scale channel fading prediction and its applications,” _IEEE Access_ , vol. 9, pp. 3278–3290, 2021.
  * [16] E. Krijestorac, H. Sallouha, S. Sarkar, and D. Cabric, “Agile radio map prediction using deep learning,” in _Proc. IEEE International Conf. on Acoustics, Speech and Signal Processing (ICASSP)_ , 2023, pp. 1–2.
  * [17] K. Qiu, S. Bakirtzis, H. Song, I. Wassell, and J. Zhang, “Deep learning-based path loss prediction for outdoor wireless communication systems,” in _Proc. IEEE International Conf. on Acoustics, Speech and Signal Processing (ICASSP)_ , 2023, pp. 1–2.
  * [18] Çağkan Yapar, F. Jaensch, R. Levie, G. Kutyniok, and G. Caire, “The first pathloss radio map prediction challenge,” _arXiv preprint arXiv:2310.07658_ , 2023.
  * [19] 3GPP, “Study on channel model for frequencies from 0.5 to 100 GHz,” _TR 38.901 (V15.0.0)_ , Jul. 2018.
  * [20] Ç. Yapar, R. Levie, G. Kutyniok, and G. Caire, “Dataset of pathloss and ToA radio maps with localization application,” _arXiv:2212.11777 [cs.LG]_ , 2022. [Online]. Available: https://arxiv.org/abs/2212.11777
  * [21] K. Simonyan and A. Zisserman, “Very deep convolutional networks for large-scale image recognition,” _arXiv: 1409.1556 [cs.CV]_ , 2014.
  * [22] A. F. Molisch, _Wireless Communications: From Fundamentals to Beyond 5G_ , 3rd ed. Wiley-IEEE Press, 2023.
  * [23] F. Fuschini, H. El-Sallabi, V. Degli-Esposti, L. Vuokko, D. Guiducci, and P. Vainikainen, “Analysis of multipath propagation in urban environment through multidimensional measurements and advanced ray tracing simulation,” _IEEE Trans. on Antennas and Propag._ , vol. 56, no. 3, pp. 848–857, 2008\.
  * [24] T. Rautiainen, G. Wolfle, and R. Hoppe, “Verifying path loss and delay spread predictions of a 3D ray tracing propagation model in urban environment,” in _Proc. IEEE Vehicular Technology Conf._ , vol. 4, 2002, pp. 2470–2474 vol.4.
  * [25] J. Long, E. Shelhamer, and T. Darrell, “Fully convolutional networks for semantic segmentation,” in _Proc. IEEE Conf. on computer vision and pattern recognition_ , 2015, pp. 3431–3440.
  * [26] S. Ren, K. He, R. Girshick, and J. Sun, “Faster R-CNN: Towards real-time object detection with region proposal networks,” in _Proc. Advances in Neural Information Processing Systems (NIPS)_ , C. Cortes, N. Lawrence, D. Lee, M. Sugiyama, and R. Garnett, Eds., vol. 28, 2015.
  * [27] J. Yosinski, J. Clune, Y. Bengio, and H. Lipson, “How transferable are features in deep neural networks?” _arXiv:1411.1792 [cs.LG]_ , 2014.
  * [28] J. Kirkpatrick, R. Pascanu, and et al., “Overcoming catastrophic forgetting in neural networks,” _Proc. National Academy of Sciences_ , vol. 114, no. 13, pp. 3521–3526, Mar. 2017.
  * [29] S. Kornblith, J. Shlens, and Q. V. Le, “Do better imagenet models transfer better?” _arXiv:1805.08974 [cs.CV]_ , 2019.
  * [30] M. Huh, P. Agrawal, and A. A. Efros, “What makes imagenet good for transfer learning?” _arXiv:1608.08614 [cs.CV]_ , 2016.
  * [31] M. Steinbauer, A. F. Molisch, and E. Bonek, “The double-directional radio channel,” _IEEE Antennas and Propag. Magazine_ , vol. 43, no. 4, pp. 51–63, 2001.
  * [32] R. Araki, T. Onishi, and et al., “MT-DSSD: Deconvolutional single shot detector using multi task learning for object detection, segmentation, and grasping detection,” in _Proc. IEEE International Conf. on Robotics and Automation (ICRA)_ , 2020, pp. 10 487–10 493.
  * [33] A. Newell, K. Yang, and J. Deng, “Stacked hourglass networks for human pose estimation,” in _Computer Vision - ECCV 2016_. Springer, 2016.
  * [34] L.-C. Chen, Y. Zhu, G. Papandreou, F. Schroff, and H. Adam, “Encoder-decoder with atrous separable convolution for semantic image segmentation,” in _Computer Vision – ECCV 2018_. Springer, 2018.
  * [35] G. Lin, A. Milan, C. Shen, and I. Reid, “Refinenet: Multi-path refinement networks for high-resolution semantic segmentation,” in _Proc. IEEE Conf. on Computer Vision and Pattern Recognition (CVPR)_ , 2017, pp. 5168–5177.
  * [36] C. Peng, X. Zhang, G. Yu, G. Luo, and J. Sun, “Large kernel matters – improve semantic segmentation by global convolutional network,” in _Proc. IEEE Conf. on Computer Vision and Pattern Recognition (CVPR)_ , July 2017.
  * [37] O. Ronneberger, P. Fischer, and T. Brox, “U-Net: Convolutional networks for biomedical image segmentation,” in _Medical Image Computing and Computer-Assisted Intervention_. Springer, 2015.
  * [38] F. Yu and V. Koltun, “Multi-scale context aggregation by dilated convolutions,” in _Proc. International Conf. on Learning Representations (ICLR)_ , San Juan, Puerto Rico, 2016.
  * [39] K. He, X. Zhang, S. Ren, and J. Sun, “Deep residual learning for image recognition,” _Proc. IEEE Conf. on Computer Vision and Pattern Recognition (CVPR)_ , pp. 770–778, 2016.