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REVIEW ARTICLE

A review on the design and optimization of antennas using

machine learning algorithms and techniques

Hilal M. El Misilmani | Tarek Naous | Salwa K. Al Khatib

Department of Electrical and Computer

Engineering, Beirut Arab University,

Debbieh, Lebanon

Correspondence

Hilal M. El Misilmani, Department of

Electrical and Computer Engineering,

Faculty of Engineering, Beirut Arab

University, P.O. Box 11-5020 Beirut, Riad

El Solh, 1107 2809, Debbieh, Lebanon.

Email: hilal.elmisilmani@ieee.org

Abstract

This paper presents a focused and comprehensive literature survey on the use

of machine learning (ML) in antenna design and optimization. An overview of

the conventional computational electromagnetics and numerical methods used

to gain physical insight into the design of the antennas is first presented. The

major aspects of ML are then presented, with a study of its different learning

categories and frameworks. An overview and mathematical briefing of regres-

sion models built with ML algorithms is then illustrated, with a focus on those

applied in antenna synthesis and analysis. An in-depth overview on the differ-

ent research papers discussing the design and optimization of antennas using

ML is then reported, covering the different techniques and algorithms applied

to generate antenna parameters based on desired radiation characteristics and

other antenna specifications. Various investigated antennas are sorted based

on antenna type and configuration to assist the readers who wish to work with

a specific type of antennas using ML.

KEYWORDS

antenna design, computational electromagnetics, machine learning, neural networks, regression

models

1|INTRODUCTION

Over the past few decades, the art of machine learning

(ML) has taken the world by storm with its pervasive

applications in automating mundane tasks and offering

disruptive insights across all walks of science and engi-

neering. Though arguably still in its infancy, ML has all

but revolutionized the technology industry. ML practi-

tioners have managed to alter the foundations of count-

less industries and fields of study, including lately the

design and optimization of antennas. In the light of the

Big Data era the world is experiencing, ML has gar-

nered a lot of attention in this field. ML shows great

promise in the field of antenna design and antenna

behavior prediction, whereby the significant accelera-

tion of this process can be achieved while maintaining

high accuracy.

Known for their complex shapes, antennas typically

do not have closed-form solutions. Computational Elec-

tromagnetics (CEM)

1-3

are applied to model the interac-

tion of electromagnetic fields with antennas using

Maxwell's equations. Approximate solutions are usually

used to gain physical insight into the design of the

antenna. With the advancements in numerical methods,

integral equations were used to solve linear antennas.

Later on, with the advancements in computers, it became

possible to solve Maxwell's equations using integral and

differential equation solvers. Method of moments

(MoM)

4

was then introduced to also solve the integral

equations. For a more complicated antenna structure,

additional unknowns are added to the equations. Differ-

ential equation solvers were then developed with a sim-

pler implementation even though they contain a larger

number of unknowns. Memory and CPU usages are

Received: 22 January 2020 Revised: 18 May 2020 Accepted: 23 June 2020

DOI: 10.1002/mmce.22356

Int J RF Microw Comput Aided Eng. 2020;e22356. wileyonlinelibrary.com/journal/mmce © 2020 Wiley Periodicals LLC 1of28

https://doi.org/10.1002/mmce.22356

among the main drawbacks of the integral and differen-

tial equation solvers since they scale with the size of the

antenna. Fast integral equation solvers were then devel-

oped, for which the integral equations are solved using

iterative methods, with reduced memory requirements.

The most widely known CEM methods in antenna

design can be classified into numerical methods and high

frequency methods. Three numerical analysis methods

that are commonly used in antenna simulations and test-

ing are namely: finite difference time domain (FDTD),

5-7

finite element method (FEM),

8,9

and MoM.

10,11

Using

physical optics approximation method, the radiation field

of high frequency reflector antennas can be also

obtained. Typically, most of the work involving antenna

simulations require solving partial differential equations,

with defined boundary conditions, using computers.

High frequency methods include current based Physical

optics (PO)

12

and field based Geometric optics (GO).

13

Other methods are also found, such as generalized multi-

pole technique (GTM), multiple multipole program

(MMP), conjugate gradient method (CGM), and trans-

mission line matrix method (TLM).

14

The most widely used commercial CEM software for

antenna design and simulations are ADS, HFSS, CST, and

IE3D. These software tools also lack several important fea-

tures. For instance, 3D structures cannot be modeled using

ADS, structures with finite details cannot be simulated using

IE3D, and the execution time of HFSS and CST is high and

increases as the size of the antenna structure is enlarged.

Due to their inherent nonlinearities, ML has been

considered thoroughly as a complimentary method to

CEM in designing and optimizing various types of anten-

nas

15-18

for several advantages, as will be discussed fur-

ther in this paper. ML is a large area within artificial

intelligence (AI), as shown in Figure 1, that focuses on

getting useful information out of data, thus explaining

why ML has been frequently associated with statistics

and data science. Indeed, the data-driven approach of ML

has allowed us to design systems like never before, taking

the world steps closer to building truly autonomous sys-

tems that can match, compete, and sometimes out-

perform human capabilities and intuition. However, the

success of ML approaches relies heavily on the quality,

quantity, and availability of data, which can be challeng-

ing to obtain in certain cases. From an antenna design

perspective, this data need to be acquired, if not already

available, since no standardized dataset for antennas,

such as the ones available for computer vision, are yet

available. This can be achieved by simulating the desired

antenna on a wide range of values using CEM simulation

software. Based on the obtained results, a dataset can be

created and divided into a train, cross-validation, and test

sets, for the purpose of training a ML model and validat-

ing whether this model succeeds in generalizing on new

inputs. At this point, it is up to the designer's clairvoy-

ance and expertise to know how to diagnose the model to

improve performance. Some common steps to follow in

this regard would be to plot the learning curves and to

FIGURE 1 Relationship between artificial intelligence, machine learning, and deep learning

2of28 EL MISILMANI ET AL.

check the values for the bias and variance. Typically, a

large part in optimizing a model's performance depends

on the intuition of the designer, specifically when using

neural networks, where the best possible architecture

and hyper-parameters need to be found out for optimal

performance.

This paper presents and investigates the use of ML in

antenna design and optimization and provides a compre-

hensive survey of all the antennas designs found in the

literature that have employed different ML techniques. It

serves as a guide to researchers in the antenna commu-

nity with minimal ML expertise seeking to employ this

technology in their work. The different antenna design

papers investigated are sorted according to the type and

category of the antenna, which makes it simpler for

readers interested in beginning research on antenna

design and optimization using ML.

The rest of this paper is organized as illustrated in

Figure 2 and as follows: a detailed overview of the CEM

methods is presented in Section 2. Section 3 presents an

overview on ML covering the different categories of

learning, in addition to ML frameworks and applications.

Section 4 investigates the regression models built with

ML algorithm and used for antenna design. Section 5 pre-

sents the in-depth overview on the different works in the

literature discussing the design and optimization of

antenna parameters using ML. Section 6 presents another

aspect of the literature, where ML was used to enhance

different types of optimization algorithms in designing

antennas. Concluding remarks, challenges, and future

directions, follow in Section 7. A list of most of the acro-

nyms used in paper is also presented in Table 1.

2|CEM OVERVIEW

Using central-difference approximations, FDTD is based

on discretizing the time-dependent Maxwell's equations

to the space and time partial derivatives.

19,20

It basically

contains a grid of points containing the computational

domain with boundary conditions. Field equations are

used to find physical quantities using post processing.

21

In FEM, linear equations are formed by meshing compu-

tational domain problems using weighted residual

method.

22,23

As for MoM, the computational area is split

into various segments. Each segment is then meshed and

evaluated using basis functions.

24-26

The current of each

segment and the strength of each moment are studied

using Green's functions.

Nevertheless, all of these methods suffer from several

drawbacks that affect their results. For FDTD, the accu-

racy of computation is affected by the reflection from the

boundary. Truncation techniques can be used to reduce

these reflections; however, the truncation also affects the

FIGURE 2 Diagrammatic view of the organization of this survey

EL MISILMANI ET AL.3of28

accuracy of the computations.

27

For high order absorbing

boundary conditions, the time and memory resources

needed get higher as the computational domain is larger.

Many methods were developed to remedy some of these

drawbacks. For instance, perfectly matched layer (PML)

can be used to decrease the reflections by absorption of

EM. However, this comes at the cost of increasing the

required CPU time and computational domain.

28

Stair-

case approximation was also proposed for discretization,

but also increases the reflection and affects the accuracy

of computation.

29

Another approach is to use 3D FDTD

which employs two different time step increments;

however, strong electric fields largely affects the stability

of this method.

30

Other methods are also found, such as

semi implicit schemes (SIS),

31

sub-cell algorithm,

32,33

FDTD-alternating direction implicit method,

34

one-

dimensional-finite-difference time-domain method,

35

domain decomposition-Laguerre-FDTD Method,

36

and

Runge-Kutta Higher Order FDTD.

37

Knowing that each

one of these methods has its own advantages, such as the

enhanced accuracy, and the reduced CPU time, most of

these methods are still considered as time consuming.

Also, they have difficulties in modeling thin wires, fre-

quency dependent materials, and have dispersion

errors.

38

As for FEM, which is widely used in modeling wave-

guides, Yagi-Uda antennas, horn antennas, and vehicular

antennas, it also suffers from certain drawbacks. For

instance, as a result of its unstructured mesh, large radia-

tion problems are difficult to be modeled using FEM, as

they require excessive computation that could result in

computational errors. Several methods have been pro-

posed with FEM to remedy some of its drawbacks. For

instance, Direct FE solver has been proposed for better

accuracy with 3D structures, but also suffers from CPU

time and memory storage requirements.

39

Dual prime,

40

which can be used with 3D structure problems, Vivaldi

arrays, and other array problems, has a faster conver-

gence time but also suffers from a trade-off between accu-

racy and computational cost.

41

Element Tearing and

interconnecting full-dual-primal have been also proposed

for the analysis of 3D large-scale problems, but also suffer

from memory and CPU time requirements.

42

Despite its

parallelization difficulty, finite element-boundary

integral-multilevel fast multipole (FE-BI-MLFMA) algo-

rithm method has been also proposed and used in bio-

medical and space applications, in addition to antenna

arrays, as a result of its efficiency and accuracy.

43

Other

methods are also found, such as non-conforming FETI,

and domain decomposition based preconditioner (FE-BI-

MLFMA) algorithm, but also suffer from memory

requirements, and have difficulty working with lossless

3D objects with high permittivity and permeability.

44,45

Additionally, FEM has also difficulties modeling thin

wires.

As for MoM, errors can occur as a result of the choice

of the testing and basis functions.

46

Typically, many

issues are associated with MoM, such as low-frequency

breakdown and singularity.

47

In addition, MoM is not

efficient to inhomogeneous and composite structures.

47

Although some solutions are found for several drawbacks

of MoM, such as the use of pre-conditioners to solve the

low-frequency breakdown,

47

recovery to solve the charge

cancelation problem,

48

and the multi-resolution

approach to improve the spectrum of MoM,

49

the

TABLE 1 List of acronyms

Acronym Definition

ANN Artificial neural network

BR Bayesian regularization

BRANN Bayesian regularized artificial neural network

CEM Computational electromagnetics

DE Differential evolution

FDTD Finite difference time domain

FEM Finite element method

FFBP Feed forward backpropagation

GA Genetic algorithm

GD Gradient descent

GPR Gaussian process regression

K-NN K-Nearest neighbors

LASSO Least absolute shrinkage and selection operator

LBE Learning-by-example

LM Levenberg–Marquardt

LR Linear regression

ML Machine learning

MLP Multi-layer perceptron

MoM Method of moments

MoM-LP Method of moments based on local periodicity

MSE Mean squared error

PIFA Planar inverted-F antenna

PSO Particle swarm optimization

RBF Radial basis function

RPROP Resilient backpropagation

SDG Stochastic gradient descent

SIW Substrate integrated waveguide

SOM Self-organizing map

SVM Support vector machines

SVR Support vector regression

VSWR Voltage standing wave ratio

4of28 EL MISILMANI ET AL.

computational cost, CPU memory and timing required

can be further enhanced. MoM is also considered as com-

putationally expensive since it requires dense systems of

equations to solve the integral equations.

3|MACHINE LEARNING

OVERVIEW

In conjunction with the standard CEM methods, artifi-

cial neural networks (ANNs) can be used to minimize

the energy function obtained by FEM.

50-52

Due to their

stability, ANNs have also been used as a solution to

MoM in Reference 53. Taking advantage of today's

advances in distributed computing, ANNs can be used

to efficiently solve large and complex EM problems, as

well as integral equations, due to their parallel and

distributed processing capabilities.

54

Tospeedupthe

solution of EM problems, ANNs were also used with

FDTD and proved to increase the computational

speed. For instance, they were used in Reference 55 to

provide a global modeling approach for Microwave

and Millimeter-Wave Circuits design, in a much faster

approach than the traditional FDTD.

Generally speaking, ANN models possess advanta-

geous characteristics that are beneficial in solving EM

problems. They are characterized by their ability to

approximate nonlinear input-output mappings which

optimizes the relation between the input data and the

required output, their adaptivity to changes in the envi-

ronment, their uniformity of analysis and design, and

neurobiological analogy.

56

One of ML's major advantages in this field is the

reduction of the large computational times found in the

presented CEM techniques, especially when several

parameters are to be optimized, or when a large structure

is to be designed. The formulations of several antenna

geometries, especially those with innovative structuring,

complex geometries, or nonlinear loads, are still difficult

to be treated analytically with known antenna theories,

especially that some of them still suffer from low accu-

racy.

57

ML can be applied to model and predict scattering

problems and analyze and optimize antennas in real-

time.

58,59

ANNs can be easily realized using several avail-

able frameworks, implemented on high-performance

computers, and can efficiently model electromagnetic

structure in much less time with very low computational

resources, and negligible degrees of errors.

60,61

In the

antenna design sense, where closed-form solutions are

hard to be found, ML can be the perfect solution to elimi-

nate the time consumed in trial-and-error simulations

when optimizing geometrical parameters to achieved

some specific design requirements such as the desired

radiation characteristics, especially if some of these char-

acteristics are to be modified in real time.

Although the idea behind ML dates back to the

1950s,

62

recent times have witnessed an unanticipated

surge of interest in ML algorithms. This interest has been

stimulated by the large availability of data in the digital

age the world has been witnessing, the access to high per-

formance computing, and the better mathematical formu-

lation and comprehension of learning techniques. Having

revolutionized many aspects in research and industry,

multiple breakthroughs in ML have occurred such as deep

reinforcement learning

63

and generative adversarial net-

works (GANs).

64

Although some ML algorithms, specifi-

cally deep neural networks (DNNs), are perceived as

“Black Box”tools, they work very well in practice and

have outperformed some well-disciplined approaches.

3.1 |Categories of learning

ML can be generally divided to three key categories:

supervised learning, unsupervised learning, and rein-

forcement learning, shown in Figure 3.

3.1.1 |Supervised learning

It is a learning task in which a model generalizes on a set

of labeled input-output pairs to consequently make pre-

dictions on unseen input. There is a distinction between

training and testing data in supervised learning, where

training samples are associated with labels or targets

which the test samples are missing. Supervised learning

can be divided into parts:

•Regression: It is a supervised learning problem in

which data are used to predict real-valued labels of

unseen data. Regression algorithms include linear

regression (LR),

65-67

kernel ridge regression,

68

support

vector regression (SVR),

69-71

and least absolute shrink-

age and selection operator (LASSO).

72

•Classification: In classification, the goal is to label data

from a finite set of classes. Binary classifications refer

to classification based on a set of two classes, and

multi-class classification refers to classification based

on a set of three or more classes.

3.1.2 |Unsupervised learning

After receiving an unlabeled dataset, an unsupervised

learning model then predicts certain labels for new data.

Unlike the case in supervised learning, there is no

EL MISILMANI ET AL.5of28

distinction between train and test data in unsupervised

learning.

73

Two learning problems are recognized in

unsupervised learning:

•Clustering: Often used for large datasets, clustering is a

learning problem that aims to identify regions or

groups within these datasets.

•Dimensionality reduction: Also known as manifold

learning, it is the process of reducing the dimensions

in which data are represented while maintaining some

principal features of the initial representation.

3.1.3 |Reinforcement learning

It is a learning paradigm in which the learner, also

referred to as the agent, actively interacts with the learning

environment to achieve a common goal. Used in control

theory, optimization, and cognitive sciences, this paradigm

depends on the notion of rewards given to the agent in

amounts proportional to the achievements of the agent,

which he aims to maximize. A model that is widely

adopted in this field is Markov decision processes (MDPs)

which represents the environment and the interactions

with it. Since the transition and reward probabilities do

not rely on the entire history of the model and only on its

current state, the model is considered Markovian.

74

3.2 |Machine learning frameworks

Numerous open-source frameworks are available to

apply machine and deep learning concepts for solving

real world problems. These platforms that are based

on optimized codes written in Python, R, Java, or any

other programming language, offer flexible and fast

usage of several algorithms, thus making them essen-

tial and critical tools in research and development.

These libraries include but are not limited to:

Tensorflow,

75

Scikit-Learn,

76

ApacheSpark,

77

CAFFE,

78

Microsoft CNTK,

79

LIBSVM,

80

and many

others. In addition to these, off-the-shelf tools such as

the WEKA software

81

are available for people with

domain-expertise but minimal ML experience where

they would only have the task of acquiring data and

tuning the hyperparameters.

3.3 |Applications

There is an abundance of areas where ML can have an

impact ranging from molecular dynamics for predicting

atomic behavior,

82

to serving as an analysis tool in

bioinformatics,

83

or building reliable financial predic-

tors.

84

In the realm of electrical and computer engineer-

ing, a plethora of works presented in the literature can

be found where ML has contributed to the enhancement

of previous systems, or in finding new approximate solu-

tions to recurring problems. These techniques have also

been widely employed in communication technology,

whetheritbeatthephysicallayerortheupperlayers,

among which we mention: deep learning based detec-

tion and decoding,

85

antenna selection in MIMO,

86

wireless and cellular networks,

87

cognitive radios,

88

wireless sensor networks,

89

cybersecurity,

90

and

others.

91

FIGURE 3 The three main categories of ML. ML, machine learning

6of28 EL MISILMANI ET AL.

4|BUILDING REGRESSION

MODELS WITH LEARNING

ALGORITHMS

Regression algorithms are the essential tools needed

when applying ML in the design of antennas. By using

these algorithms and dataset of a considerable size, a

model representing the mapping function of the non-

linear relationship between the antenna's geometrical

parameters and characteristics can be derived. The most

widely used ML algorithms for antenna design are

ANNs

92-95

and SVR.

96,97

Other regression methods that

are less widely used are LR, LASSO, Gaussian process

regression (GPR),

98,99

and Kriging Regression.

100,101

This

section provides a mathematical briefing of these ML

algorithms that are applied in antenna design.

4.1 |Linear regression

Considered to be one of the simplest regression algo-

rithms, LR is a statistical tool used to trace a linear rela-

tionship between some variables and their respective

numeric target values. For an unknown stochastic envi-

ronment, we consider a set of labeled examples

xi,yi

ðÞ

fg

N

i=1 for the goal of building a model

102

:

fw,b

fg

xðÞ=wx +bð1Þ

where Nis the size of the set, x

i

is a D-dimensional vector

of example i=1,…,N,y

i

∈R is the numeric target value,

w

i

is a D-dimensional vector of unknown, but fixed

parameters, and band Dare real numbers.

102

For the

model to produce the most accurate prediction of y, the

optimal values of wand bneed to be reached. To that

end, we consider the following cost function to be

minimized:

lw,bðÞ=1

NXN

i=1 fw,bxi

ðÞ−yi

2ð2Þ

This squared error loss function represents the aver-

age loss, or empirical risk, obtained after applying the

model to the training data. It accounts for the average

penalties for misclassification of examples i=1,…,N.

Gradient descent optimization algorithm (GD),

103

is used

to minimize the cost function. GD is used in LR to itera-

tively find the minimum of the function by gradually tak-

ing steps toward the negative of the gradient. The first

step of GD is calculating the partial derivative of every

parameter in the cost function as follows:

∂l

∂w=1

NXN

i=1−2xiyi−wxi+bðÞðÞ

ð3Þ

∂l

∂b=1

NXN

i=1−2yi−wxi+bðÞðÞ

ð4Þ

where the partial derivatives were calculated using the

chain rule. The parameters w

0

and b

0

are initialized by

zero. It is worth noting that the correct initialization of

parameters is integral in the success of the optimization

algorithm. After initializing the parameters, training data

(x

i

,y

i

) are iterated through, where in each iteration the

parameters are updated as follows:

wi=α−2xiyi−wi−1xi+bi−1

ðÞðÞ

Nð5Þ

bi=α−2yi−wi−1xi+bi−1

ðÞðÞ

Nð6Þ

where αdenotes the learning rate, and w

i

and b

i

denote the

respective values of wand bafter using the training example

(x

i

,y

i

). The algorithm stops iterating when the values of the

parameters remain relatively constant upon the end of an

epoch, where an epoch is a pass over all training examples.

4.2 |Least absolute shrinkage and

selection operator

LASSO algorithm, also known as Sparse Linear Regres-

sion, integrates L1 regularization and mean-squared error

with a linear model.

104

L1 regularization is known to

result in a sparse solution, where sparsity refers to having

parameters with an optimal value of zero. Thus, this algo-

rithm can be used for feature selection. The LASSO esti-

mate is defined by

105

:

X

N

i=1

wxi+b−yi

ðÞ

2+λw

kk

1ð7Þ

where λis the regularization parameter, and kwk

1

is the

L1 norm obtained by Pd

i=1 wi

jj

.

4.3 |Artificial neural networks

A neurobiological analogy of the brain, an ANN is a ML

technique that derives its computing power from the

massive interconnections between its “neurons,”which

EL MISILMANI ET AL.7of28

are the computing cells, and from its ability to generalize

based on experiential knowledge. ANNs are known to be

great function approximators

106

and are widely used for

regression problems. In general, an ANN consists of an

input layer of nodes that is not counted since no compu-

tations occur at this layer, an output layer of computation

nodes, and zero or more hidden layers whose computa-

tion nodes are referred to as hidden nodes. An example is

shown in Figure 4 where a deep neural network with two

hidden layers is sketched. The architecture of the net-

work when it comes to the number of layers and the

number of nodes at each layer depends on the algorithm

used in the learning process and the desired output of the

network.

107

Consider the following nested function that repre-

sents an ANN:

y=fNN xðÞ ð8Þ

The internal functions of layer indices lof the nested

function have the following form:

flzðÞ=glWlz+bl

ðÞ ð9Þ

where g

l

represents an activation function. Activation

functions are fixed non-linear functions used as tools to

compute the output of a computation node, which is then

fed as input to the subsequent nodes.

108

We present three

types of commonly used activation functions:

Logistic function: Also known as the sigmoid func-

tion, the logistic function is defined as follows

108

:

gxðÞ=1

1+e−xð10Þ

As shown in Figure 5 the logistic function saturates

and becomes less sensitive to input at high or low values

of x, while they exhibit sensitivity for values of

xnear zero.

Hyperbolic tangent function: Also known as tanh

function, shown in Figure 6. It is characterized as

108

:

tanh xðÞ=ex−e−x

ex+e−xð11Þ

ReLU Function: Typically used in all hidden layers,

the ReLU function is a rectified linear unit function

shown in Figure 7 and defined as follows

108

:

relu xðÞ=

0ifx<0

xotherwise

8

<

:

ð12Þ

To non-linearly estimate the gradient of the cost func-

tion of an ANN, which is the cross-entropy loss, we con-

sider a popular, widely used training algorithm called

backpropagation.

109

Based on GD, Backpropagation is a computational

iterative procedure that aims to find a local minimum of

the cost function. It consists of forward and backward

passes. During the forward passes, the outputs of the acti-

vation functions are computed and stored to be used in

the following pass.

During backward passes, partial derivatives of the

cost function are calculated using the chain rule starting

from the final layer to eventually update the parameters.

The error is said to be “back propagated”from layer to

layer. It is worth noting that the non-convex nature of

the cost function in this case implies that a local rather

than a global optimum is reached.

110

In Backpropagation,

the change Δw

ji

(k) in the weight of a connection between

two neurons iand jis given by the following:

FIGURE 4 Schematic of a deep neural network with two

hidden layers FIGURE 5 Sigmoid function

8of28 EL MISILMANI ET AL.

Δwji kðÞ=αδ jxi+μΔwji k−1ðÞ ð13Þ

where the input is x

i

,αis the learning rate, δ

j

determines

whether the neuron jis a hidden neuron or an output

neuron, and μis the momentum coefficient.

4.4 |Support vector regression

Support vector machines (SVM), a widely popular mod-

ern ML algorithm used for classification has inspired

another algorithm used for regression: SVR.

111

Similar to

its classification counterpart, the idea behind SVR is to

separate the data points into two sets: points which fit

within a predefined tube of width ϵ> 0 and which are

not penalized, and points which fall outside this bound-

ary and are thus penalized as shown in Figure 8.

For a set of linear hypothesis functions

111

:

H=x↦w:Φxi

ðÞ+b:w∈RN,b∈R

ð14Þ

where Φis the feature mapping corresponding to a posi-

tive definite symmetric kernel function K, and w.Φ(x

i

)is

the dot product of the feature mapping Φ(x

i

) and w.

Training this model involves reaching optimal values of

wand bby minimizing the corresponding cost function.

The cost function to be minimized is as follows

111

:

1

2wjj

2+CXm

i=1 yi−w:Φxi

ðÞ+bðÞjj

εð15Þ

where |.|

ϵ

denotes the ϵ-insensitive loss as shown in

Figure 8.

It is worth noting that the choice of the parameter ε

plays a role in determining the sparsity and accuracy of

the model, where assigning large values to εresults in

sparser solutions.

111

Gaussian kernels, otherwise known as radial basis

function (RBF), is a kernel Kdefined over R

N

as:

8x,x0∈RN,Kx,x0

ðÞ= exp −x−x0

jj

2

2σ2

!

ð16Þ

for any constant σ> 0. These kernels are the most com-

monly used kernels in this and other applications.

111

4.5 |Gaussian process regression

In GPR, the objective function is considered as a sam-

ple of a Gaussian stochastic process. By using the

available data samples, the distribution of the func-

tion value for new samples can be predicted. Consid-

ering a set of labeled examples {(x

i

,y

i

)}, new

predictions at a certain input x

*

can be obtained by

the following

112

:

FIGURE 6 Tanh function

FIGURE 7 ReLU function

FIGURE 8 SVR epsilon-bounded data. SVR, support vector

regression

EL MISILMANI ET AL.9of28

^

yx

*

=μ+rTR−1y−IμðÞ ð17Þ

where Iis a n×1 vector of ones, μis the mean of the pre-

dictive distribution, R

i,j

= Corr(x

i

,x

j

) is a correlation

function with i,j=1,2,…,n, and r= [Corr(x

*

,x

1

), Corr

(x

*

,x

2

), …, Corr(x

*

,x

n

)].

4.6 |Kriging regression

A less widely used regression method in the design of

antennas is the Kriging Regression algorithm. In this

method, the relationship between auxiliary variables and

a target is modeled using known values of auxiliary vari-

ables. This algorithm can be defined as follows

113

:

^

zs

0

ðÞ=Xp

k=0

^

βkqks0

ðÞ ð18Þ

where ^

βkare the regression coefficients, pis the number

of auxiliary variables q, and ^

zs0

ðÞis the predicted value of

a target variable given an input s

0

.

5|TRAINING MACHINE

LEARNING MODELS WITH

OPTIMIZATION ALGORITHMS

Optimization algorithms are an important aspect in ML,

since they allow to find the optimal weight and bias

parameters of the ML model. Specifically, these algo-

rithms are not ML algorithms but are used in the training

process of a ML model to minimize the cost function and

find the optimal values for the parameters. The optimiza-

tion algorithm used has a direct impact on the perfor-

mance of the ML model that results after training, and

the choice of this optimizer is based purely on the type

and amount of data available and on the designer's intui-

tion. The most commonly used optimizers in antenna

design can be listed as follows:

5.1 |Gradient descent

GD algorithm, also known as batch GD, is known to be

slow since it updates the parameters once after calculat-

ing the gradient of the whole dataset. Another drawback

of GD is its vulnerability to being stuck in local minima

before converging to the global minimum in a non-

convex surface. In the era of Deep Learning, where we

may have millions of data samples, vanilla GD would not

do. Hence, several optimization algorithms are available

to use inside the architecture of a ML algorithm. Alterna-

tives include Stochastic Gradient Descent (SGD),

114

where the parameters would be updated for each training

example, and mini-batch GD,

115

where the gradient of a

small amount of data samples are computed before per-

forming updates.

5.2 |Adaptive moment estimation

A more recent, computationally efficient, and faster algo-

rithm is the adaptive moment estimation (ADAM) algo-

rithm, where the learning rates are computed for each

parameter.

116

This algorithm is especially useful in the

case of optimization problems with relatively huge

amounts of data or with big numbers of parameters.

5.3 |Levenberg-Marquardt algorithm

Used for nonlinear least-squares estimation problems,

the Levenberg-Marquardt (LM) algorithm is a batch-form

trust region optimization algorithm that is widely used in

a variety of disciplines to find the local minimum of a

function.

117

The LM algorithm is a mix between Gauss-

Newton iterations and GD, making it faster in conver-

gence than vanilla GD. It is most efficient for usage in

cases of small or medium sized patterns and offers a solu-

tion for nonlinear least squares minimization.

118

5.4 |Bayesian regularization

Bayesian regularization (BR) is mostly used to train

ANNs instead of error backpropagation, with the main

advantage of bypassing the need for lengthy cross-vali-

dation.

119

Bayesian regularized artificial neural net-

works (BRANNs) are known to be difficult to

over-train and over-fit making them an attractive

choice for usage.

5.5 |Evolutionary algorithms

Evolutionary algorithms are a category of algorithms that

are inspired by the biological behavior and evolutionary

process of living creatures.

120

This class of algorithms,

that contains genetic algorithms (GA), differential evolu-

tion (DE), particle swarm optimization (PSO), and others,

is usually used in global optimization, and has been

extensively used in electromagnetic optimization

121-123

and can also be used to train ML models in the case of

antenna design.

10 of 28 EL MISILMANI ET AL.

6|PREDICTING ANTENNA

PARAMETERS WITH MACHINE

LEARNING MODELS

A large body of literature exists where ML has been used

to design and optimize antennas. Most of these works

have employed the usage of ANNs to find direct relation-

ships between different antenna parameters, such as

between the geometrical properties of the antenna and

the antenna characteristics. As the complexity of an

antenna's structure increases, the number of geometrical

parameters increase, and it becomes hard to derive rela-

tionships between these parameters and values for the

resonant frequency and other radiation characteristics.

The usual approach for optimizing a design is simulating

the antenna to finally reach the desired values, a process

described as computationally heavy and time demanding.

Instead, ML can accelerate the design process by provid-

ing a mapping between whatever the desired inputs and

outputs may be. In general, the following procedure can

be adopted:

1. Numeric values corresponding to the desired inputs

with their respective outputs are obtained by simula-

tions and are stored in a database

2. Once this dataset is created, it is split into training,

cross-validation, and test-sets, where the percentage of

each depends on the amount of data samples

3. A ML algorithm is chosen to learn from this data. The

choice of the algorithm relies on the complexity of the

problem, the amount of data at hand, and the mathe-

matical formulation of the algorithm

4. After training and testing the model, it can be used to

predict output values for the desired inputs

Although this process demands going into simula-

tions to create a dataset for training, once a model is

obtained, predictions can be made for any desired inputs

at very high speeds, and within very low error margins

compared to simulated results. Several metrics have been

in the literature to quantify this error, among which are:

The Output Error: obtained by calculating the differ-

ence between the output obtained by simulations and the

output predicted by the ML model. The unit of this error

depends the parameter being predicted and could be in

dB, Hz, mm, or any other unit. It is expressed by:

ei=yd−yið19Þ

where e

i

is the output error, y

d

is the desired output, and

y

i

is the output predicted by the ML model.

The mean squared error (MSE) expressed by:

MSE =1

NXN

i=1 ei

ðÞ

2ð20Þ

where Nis the size of the training samples.

The error percentage is obtained by the following:

Error%=jyd−yi

yd

j×100 ð21Þ

In this section, we investigate the different papers

found in the literature on the design and optimization of

antennas using ML procedures. These papers are sorted

according to the type and configuration of antennas,

starting with the typical rectangular and circular patch

antennas, fractal shape antennas, elliptical shape anten-

nas, monopole and dipole antennas, planar inverted-F

antenna (PIFA), substrate integrated waveguide (SIW),

special patch design, reflectarray antennas, in addition to

some other types of antennas.

6.1 |Microstrip antennas

6.1.1 |Rectangular patch

The simplest form of antenna design using ML is the

design of rectangular patch antennas. In References 124

and 125, multi-layer perceptron (MLP) neural networks

have been used for the synthesis and analysis of rectan-

gular microstrip antennas. During the synthesis phase,

the height and permittivity of the substrate, denoted by

Hand ϵ

r

in Figure 8, in addition to the resonance fre-

quency of the antenna, are used to generate the length

and width of the rectangular patch, denoted by Land

Win Figure 9. During the analysis phase, the width and

length, in addition to the height and effective permittivity

of the substrate are used to generate the resonance fre-

quency. The obtained neural network results were com-

pared to those available in the literature where an MSE

FIGURE 9 Substrate configuration

EL MISILMANI ET AL.11 of 28

of 10

−5

was obtained, showing good agreement. In Refer-

ence 125, RBF networks were used in the proposed

approach, where results showed that the RBF network

gave the best results with an error percentage of 0.91%

compared with the MLP approach that reached 3.47%.

Other works have employed SVRs in the design of

rectangular microstrip antennas. The optimization of the

resonant frequency f

r

, operation bandwidth (BW), and

input impedance R

in

of a rectangular microstrip patch

antenna using SVR was presented in References 126 and

97. The results obtained from the proposed approach

were compared to those obtained from an ANN-based

approach. It was determined that SVR computed the

above design parameters with higher accuracy than the

ANN approach. SVR error percentages reached 1.21% for

f

r

, 2.15% for BW, and 0.2% for R

in

while the ANN

approach achieved accuracy percentages of 1.67% for

f

r

,1.19% for BW, and 1.13% for R

in

.

Similarly, a rectangular patch antenna was designed

using SVR with a Gaussian Kernel in Reference 127. The

training and test sets were obtained by FDTD simulations

where accurate values of the antenna's performance

parameters such as the resonant frequency, gain, and

voltage standing wave ratio (VSWR) were obtained with

the corresponding values for the width and length of the

rectangular patch. This data was then used to train the

SVM, where the geometrical properties of the antenna

are predicted based on desired values for the performance

parameters that are given as the input.

In Reference 128, the resonance magnitude of a rect-

angular patch antenna with a two-section feed was

predicted using SVR. The patch antenna, which has

dimensions of 50.7 ×39.4 mm

2

and an operating fre-

quency of 1.8 GHz, has two feeds of about 20 mm in

length. A total number of 23 samples has been obtained

through varying the widths of the feeds, out of which

21 were used for training and 2 for testing. During train-

ing, the two width values were taken as input parame-

ters, and the resonance frequency as the output. Different

kernel configurations including linear, polynomial of

order 3, sigmoid and radial kernels were tested, and the

radial kernel was used. It was shown that the average

predicted error between the simulations results and the

predicted value was around 3 dB on average.

In Reference 129, the slot-position and slot-size of

rectangular microstrip antenna were predicted using a

SVR model and ANN model. Two asymmetrical and two

symmetrical slots were inserted on the radiating and

grounding surface respectively after which the models

were used to predict the slot-size and slot-position. Ana-

lytical results showed that SVR was more accurate and

time efficient than the ANN, where the SVR was ~10%

more accurate and had a speedup rate of 416 times.

Using ANN, the radiation characteristics of a slotted

rectangular patch antenna, including its resonance fre-

quency, gain, and directivity, have been used to generate

the required slot-size and substrate air-gap dimensions in

Reference 130. Multiple optimization algorithms have

been tested to train the ANN with the LM algorithm

proving to be the most efficient by providing the most

accurate results in the shortest training time and least

number of iterations. A prototype antenna was also fabri-

cated to validate the accuracy of the obtained model,

where the measured results showed great agreement with

the simulated and predicted ones where a low percentage

error of 0.208% was achieved.

More recently in Reference 131, rectangular patch

antenna was designed using a new ANN architecture.

ANNs, based on feed forward backpropagation (FFBP)

algorithm, resilient backpropagation (RPROP) algorithm,

LM algorithm, and RBF, were trained and tested using

MATLAB. The input parameters of the models were the

dielectric constant, width and length of the patch, and

the substrate thickness, with the output being the reso-

nance frequency of the antenna. After comparing the per-

formance error of the four algorithms used, it was

concluded that the RBF-based network produced the

most accurate results with a value of 3.49886 ×10

−14

for

the error.

In Reference 132, PSO was used to train an ANN in

the design of rectangular patch antennas. The ANN used

the sigmoid function as an activation function, with input

units representing the resonance frequency, the height,

and permittivity of the substrate material, and output

units representing the dimensions of the patch. It was

shown that training required less than 5 minutes of com-

puter work. In addition, an RBF ANN was also used to

produce the value of inset feed distance d, shown in

Figure 10, corresponding to the suitable normalized input

resistance. The results produced by the proposed

approach and those obtained from conventional simula-

tions were determined to be in good agreement with an

MSE value of 0.104.

In Reference 133, a GA was used to train an ANN

instead of backpropagation for the purpose of optimizing

a rectangular microstrip antenna. The ANN was able to

predict the resonant frequency of the antenna, having the

substrate dielectric constant ε

r

, the width Wand length

Lof the patch, and the shorting post position as inputs.

Although the results obtained showed good.

agreement with the experimental ones with an aver-

age error of 0.013545 GHz for the resonant frequency, it

was concluded that despite optimizing the parameters

accurately, using GA to train the ANN was not very time

efficient and could have been achieved in less time by

employing backpropagation instead.

12 of 28 EL MISILMANI ET AL.

6.1.2 |Circular patch

Another well-known and simple type of microstrip

antennas is the circular patch antenna, shown in

Figure 11. The design of a circular patch antenna with

thin and thick substrates, using ANN, was presented in

Reference 134. The ANN took as input the radius of the

patch, the height and permittivity of the substrate, to gen-

erate the resonance frequency using MLP and RBF net-

works. The effectiveness of five learning algorithms in

the training of MLPs was investigated, the delta-bar-delta

(DBD), the extended delta-bar-delta (EDBD), the quick-

propagation (QP), the directed random search (DRS),

and the GA. After comparing train, test, and total errors

of the mentioned algorithms, it was deduced that EBDB

attained the best results with a test error of 2 MHz com-

pared with 13, 142 and 271 MHz in error for the DBD,

DRS, and GA approaches respectively. As for the RBF-

based network, its learning strategy was used for train-

ing. Additionally, a neural network trained by EDBD

and backpropagation was used to compute the charac-

teristic impedance and the effective permittivity of

asymmetric coplanar waveguide (ACPW) backed with a

conductor.

In Reference 135, ANNs were used in the design and

determination of feed position for a circular microstrip

antenna. The first network, an MLP neural model with

two hidden layers was used to predict the radius a, effec-

tive radius, and directivity of the patch. The inputs were

thickness of the substrate h, relative dielectric constant of

the substrate, and resonant frequency, and the optimiza-

tion algorithm used was LM algorithm. The trained net-

work was tested on 45 various samples, and the reported

MSE was 9.70 ×10

−4

, 9.80 ×10

−4

, and 7.76 ×10

−4

for

the respective inputs. The second network, an RBF neu-

ral model with one hidden layer, was used to predict the

input impedance. The input was a representation of the

various radial distances from the center of the patch. The

network, which was trained with 200 input-output pairs,

had an MSE of 2.69 ×10

−4

upon testing.

An MLP ANN was used in Reference 136, to model,

simulate, and optimize multilayer circular microstrip

antennas. Chosen from 11 tested learning algorithms,

LM algorithm was used to train the network. The reso-

nance frequency was calculated for any arbitrary values

of the patch radius, dielectric constant of different layers

and their thickness. The results showed good agreement

with reference results, where the average error percent-

age of the resonant frequency was 0.35%, 0.065%, 0.43%,

and 0.066% for circular microstrip antenna with and

without cover, spaced dielectric antenna, and microstrip

antenna with two superstrates respectively.

In Reference 137, the resonance frequency of a circu-

lar patch antenna was also modeled using a conjugate

gradient model of an ANN. A closed form expression of

the resonance frequency of the antenna, based on the cir-

cular patch radius, height, and permittivity of the sub-

strate, was used to generate data for ANN modeling and

testing. Forward modeling and reverse modeling were

used to either predict the resonance frequency of the

antenna or the circular patch radius. A comparison

between the simulated results and the results predicted

by the ANN showed 0.10721% error for the resonant fre-

quency and 0.1956% error for the patch radius.

FIGURE 10 Rectangular patch antenna with line feed

FIGURE 11 Rectangular patch antenna with insert feed

EL MISILMANI ET AL.13 of 28

The optimization of various design parameters of

a circular microstrip patch antenna using ANN

trained with LM algorithm was presented in Refer-

ence 138. A FFBP neural network was used to esti-

mate the following seven parameters: return loss

(RL), VSWR, resonance frequency, BW, gain, direc-

tivity, and antenna efficiency. The input parameters

were the patch radius, in addition to the height and

permittivity of the substrate. Results from testing the

model were in good agreement with simulated results

and achieved an MSE of 9.96 ×10

−7

.

6.1.3 |Fractal patch

Fractal patch antennas are another type of microstrip

antennas that have their design procedure dominated by

ANNs. In Reference 139, the resonance frequency, RL,

and the gain of a coaxial fed elliptical fractal patch

antenna were calculated using an ANN with bac-

kpropagation algorithm. IE3D software was used to gen-

erate the dataset for different values of feed positions and

for different iterations of the antenna fractal shape. These

values were used for training a model to find the position

of the feed point of the coaxial feed for optimized imped-

ance matching of the antenna.

In another work, the optimization of a square fractal

antenna using ANN was presented in Reference 140.

With the aim to reduce the size of the broadband reso-

nance antenna, selected iterated structures resulting from

the ANN were simulated in HFSS to obtain optimal reso-

nance characteristics. Also, the design of quasi-fractal

patch antennas using ANNs has been presented in Refer-

ence 141. Several values of the antenna parameters that

allow operation at a specific resonance frequency have

been obtained by simulations. The dataset was then used

to train the network, which resulted in a prediction

model that provides a mapping between parameters and

frequency of operation.

6.1.4 |Elliptical patch

ANNs were used in two works for the design of elliptical

patch antennas. In Reference 142, ANNs using RBF were

utilized in the design of elliptical microstrip patch

antenna. The resonance frequency for even mode, sub-

strate height and permittivity, and the eccentricity of

elliptical patch were used as input parameters to compute

the resonance frequency for odd mode and the semi-

major axis. When comparing the obtained results with

the results of conventional simulations, the error percent-

age reached as low as 0.006% and 0.043%.

In a different approach, the design and modeling of

an elliptical microstrip patch antenna using ANN was

presented in.

143

For the purpose of computing the RL

and the gain of the antenna, a FFBP neural network was

trained in MATLAB using the three major axes of the

connected ellipses as input parameters. A dataset was

obtained from CST simulations. The obtained results

were compared to those of the simulated and measured

results of a fabricated antenna, and a good agreement

was revealed with error values as low as 0.0202 dB for

the Gain and 0.2014 dB for the RL.

6.1.5 |Monopole and dipole antennas

The design of a circular monopole antenna was facili-

tated using an ANN in Reference 144. The feed-gap, a

design parameter required for the antenna to operate

within a specific frequency band, was calculated using an

ANN trained with dataset obtained from IE3D simula-

tions. The model was later tested with five input-output

pairs, and the resultant error percentage was determined

to be within 0.4 and 4.6%.

The LASSO technique, a sparse LR method, was used

in Reference 145, to design a reference dual band double

T-shaped monopole antenna. Five design parameters that

cooperatively represent the shape and structural geome-

try of the antenna were repeatedly obtained from HFSS

simulations. After the model was fitted using the LASSO

method, which is characterized by variable selection and

regularization, optimum predicted design parameters

were reached. The resulting model was able to predict

values 495 616 design points after being trained on a

dataset that consists of only 450 training samples. Even

given the relatively small size of the dataset, the model

was able to analyze an exponentially higher number of

data points in a very brief amount of time without the

need to perform further electromagnetic simulations.

This work was later developed in Reference 146, where

two more ML techniques, namely ANNs and the k-

nearest neighbor (k-NN) algorithm, were used to opti-

mize the same antenna. It was shown that by using

ANNs or LASSO, better predictions can be achieved than

the k-NN approach that had an error percentage

of 2.90%.

The first took the microstrip line impedance Zas

input, and the line width as output. The second took Z

and the substrate dielectric constant as input, and the

width and height of the substrate as output. The third

took Z as input, the substrate dielectric constant, width,

and height, as output. The synthesis ANN was further

tested on a printed dipole antenna with integrated balun,

where the results of the proposed neuro-computational

14 of 28 EL MISILMANI ET AL.

model were compared to those of a developed FDTD

analysis tool. The input parameters to the model were

three voltage standing-wave ratio numbers and two fre-

quencies, and the output was two geometric parameters.

6.1.6 |Planar inverted-F antenna

The optimization of design parameters of a PIFA with

magneto dielectric nano-composite substrate using ANNs

trained with the BR algorithm was presented in Reference

147. The model was trained on two databases obtained

from CST simulations. Taking as inputs the particle radius

and volume fraction of the nano-magnetic material, differ-

ent antennas parameters, such as gain, BW, radiation effi-

ciency, and resonance frequency can be generated using

neural networks with error percentages close to zero.

Working with the same antenna, the algorithm used in

Reference 147 has been further optimized in

148

for the

same input and output parameters. In addition, a reverse

technique has been also addressed using ML, for which the

corresponding design space of possible material parameters

can be generated based on given antenna parameters.

6.1.7 |Substrate integrated waveguide

ANNs were used to predict the geometrical parameters of a

SIW patch antenna in Reference 149, taking as inputs the

desired resonance frequency and the RL. Feed-forward MLP

and backpropagation were used for training the ANN in

MATLAB, using dataset obtained from HFSS simulations.

In Reference 150, the design of a broadband millimeter-

wave SIW cavity-backed slot (CBS) antenna using ML was

presented. A ML assisted optimization method (MLOM)

wasusedasthereferencemethodcomparedtoaproposed

ML assisted method with additional feature (MLOMAF),

which utilizes the population-based metaheuristic optimiza-

tion method. HFSS was used in the design and analysis pro-

cess of the antenna structures. Following database

initialization, the initial training set was sampled using the

Latin hypercube sampling (LHS) and the resultant data was

exploited in building a Gaussian process (GP) surrogate

model. It was shown that this algorithm was able to reach

the stopping criterion 12 iterations before the MLOM did,

and that the proposed antenna exhibited notable features

related to BW and ease of fabrication.

6.1.8 |Special patch designs

Other types of special patch designs have been also

designed in the literature using different ML techniques.

The analysis and design of a frequency re-configurable

planar antenna using a MLP ANN and a self-organizing

map (SOM) neural network respectively was presented in

Reference 151. In the analysis phase, the operational fre-

quency bands of the antenna at different reconfigured

conditions were located by the trained MLP network. In

the design phase, switches to be turned on for a specific

desired frequency response were identified using a SOM

neural network, trained using Kohonen learning algo-

rithm. Frequency responses from the output of the MLP

network were used to feed the input layer, whereas the

output of the networks was four clusters of frequency

responses, used to approximate the position of the

switches to be turned on.

In Reference 152, the design of a three-layer annular

microstrip ring antenna with pre-specified operational

features was facilitated using ANNs, where structural

design parameters were computed. A dataset of reflec-

tion coefficient vs frequency was used to train an MLP

ANN to generate the geometrical properties of the patch

as well as the physical properties of the substrate. Upon

testing, the root MSE of the model was determined to

be 1%.

In Reference 153, the design of a two-slot rectangular

microstrip patch antenna was facilitated using MLP and

RBF, based on an ANN trained with different learning

algorithms. The MLP-based networks were trained using

five learning algorithms: LM algorithm, scale conjugate

gradient backpropagation, Fletcher Powell CG bac-

kpropagation, gradient decent with momentum, and

adaptive gradient decent. It was determined that the LM

algorithm resulted in the least MSE compared to the

other MLP-based algorithms, but the RBF-based network

resulted in a lower error percentage of 0.09%.

A spiral microstrip antenna has been designed in Ref-

erence 154 using ANN. Antenna parameters were

mapped to characteristics such as the resonance fre-

quency, RL, and VSWR. After training the network on

data samples obtained by the simulator, it was shown

that accurate prediction results may be obtained which

allows bypassing the computational burdens of conven-

tional simulation methods.

In Reference 155, the design of an aperture-coupled

microstrip antenna using an ANN with a hybrid network

architecture was presented. The RBF and the bac-

kpropagation algorithm were combined to develop the

hybrid network. Using the hybrid ANN, the antenna

parameters including the dimensions of the ground

plane, the aperture, and the radiating element, in addi-

tion to the dimensions of the feed and its position, were

determined based on different resonance frequencies.

The obtained ANN results were compared with those

using backpropagation and RBF models, which showed

EL MISILMANI ET AL.15 of 28

performance superiority after showing an error percent-

age of 0.27%.

The design of single-feed circularly polarized square

microstrip antenna (CPSMA) with truncated corners, has

been facilitated in Reference 156 using ANN synthesis

model. A total of 5000 data samples were generated by

calculating the resonance frequency, in addition to the Q-

factor of the antenna by analytical formulations, out of

which 3500 were used for training the model. The LM

algorithm was used for training, which resulted in a

faster and simpler structure. During the synthesis pro-

cess, for a desired operating frequency and a given sub-

strate, the size of the truncated corners can be obtained

for CP operation. An ANN with three hidden layers was

found to give the highest accuracy. The average relative

error and the maximal relative error have been calculated

to test the accuracy of the model. The proposed model

was compared with simulation results. It was shown that

the antenna with the calculated parameters achieved a

circular polarization with less than 2 dB of axial ratio,

with some discrepancy in the frequency of operation of

less than 5%. Eight different CPMAs were also fabricated

and tested, for which the measured results showed an

axial ratio of less than 1 dB, with a discrepancy between

the measured and synthesis values of 3.6% for the physi-

cal dimensions, and 2.3% for the frequency of operation,

with an average relative error of less than 1%.

In Reference 157, the optimization of design parame-

ters of a tulip-shaped microstrip patch antenna using

ANN was presented. Taking as input the resonance fre-

quency band and the RL of the lower and higher reso-

nance frequencies, the ANN was used to generate the

patch dimensions. Backpropagation was used to train the

ANN in MATLAB using a dataset obtained from HFSS

simulations.

The design of a pentagonal-shaped flexible antenna,

shown in Figure 12, for ultra-wide band (UWB) wireless

applications such as WLAN, 5G, and WiMAX applica-

tions using ANN was presented in Reference 158. The

aim was to determine the two frequencies representing

the structure BW from the radius.

of the pentagonal shape. This approach was used due

to the complexity behind finding the non-linear relation-

ship between these parameters and representing it in an

equation, and due to some time and cost concerns. For

this, an ANN based on LM algorithm was trained and

tested using a dataset obtained from Ansoft simulator.

The error resulting from learning, validation, and testing

was 6%.

In Reference 159, the resonance frequency of rectan-

gular patch antennas printed on isotropic or uniaxially

anisotropic substrate, with or without air gap, was

modeled using ANN. Spectral dyadic Green's function

was used in conjunction with a developed single neural

network. To reduce the computational complexities,

required time, and amount of data needed to maintain

the accuracy of the ANN model, a single matrix was used

to present the effective parameters. Two types of anten-

nas were tested with two ANN models: a circular patch

antenna where the radius of the patch has been deter-

mined based on the resonance frequency, substrate thick-

ness, and permittivity, in addition to a modified PIFA

antenna with a chip resistance where the feed position

was determined based on the input impedance.

In Reference 160, an ANN has been used in the analy-

sis and synthesis of Short-Circuited Ring-Patch Antennas

(SCRP). The importance of the ANN in this work was to

solve the drawbacks of the analytical calculations that do

not accurately model the effect of the thickness and per-

mittivity of the substrate on the resonance frequency of

the antenna in TM

10

mode. For the training process, the

internal and external radius, in addition to the substrate

permittivity and thickness, were varied to obtain

275 training samples from simulations. During the train-

ing phase that required less than 1-minute, least squares

cost function with a backpropagation method have been

used. In the analysis case, the resonance frequency was

obtained from the ANN given the other patch dimensions

for 30 test cases. The comparison between the frequency

obtained by the trained ANN and that obtained by simu-

lations showed a percentage error of less than 0.3%. In

the synthesis case, the importance of using ANN for this

type of antennas was seen in estimating the value needed

for the external radius from other parameters and for a

desired resonance frequency. Usually, the external radius

FIGURE 12 Circular patch antenna

16 of 28 EL MISILMANI ET AL.

is estimated via analytical formulations and adjusted by

trial and error using the simulation software. In this case,

the comparison between the external radius obtained by

the trained ANN and the simulations achieved an error

of less than 3.2%. A SCRP has been also fabricated based

on the trained ANN, for which the measured results

showed closer results to those estimated by the ANN than

the simulated ones.

Some works have explored the design of multiple

patch types at the same time. In Reference 161, a combi-

nation of ANN and adaptive-network-based fuzzy infer-

ence system (ANFIS) was used to calculate the resonance

frequencies of rectangular, circular, and triangular micro-

strip antennas. The MLP ANN, trained with the BR algo-

rithm, was utilized to compute the resonance frequencies

of the antennas. As for the ANFIS, it was trained using

the hybrid learning algorithm, which is a combination of

least square method and backpropagation. The inputs to

this hybrid model were the geometrical parameters of the

patch and the dielectric constant of the substrate,

whereas the calculated output was the resonant frequen-

cies. The MLP ANN was used in computing the resonant

frequencies, and the ANFIS was used in compensating

for the inaccuracies in the ANN results. Finally, the

results were compared to those of the single neural

models, conventional methods, and approaches based on

GA and Tabu search algorithm (TSA). It was determined

that the proposed hybrid method provided results of

higher accuracy.

6.2 |Reflectarray antennas

Several works focusing on the accelerated design and

analysis of very large reflectarrays, as the one shown in

Figure 13, using ML, have been presented in the litera-

ture. Among these, many have employed ANNs as the

main design and analysis tool. In Reference 162, an ANN

was utilized in the optimization of microstrip patches

unit cell parameters of broadband reflectarray antennas

with Malta cross unit cell configuration, as shown in

Figure 14. To this end, the ANN was used to accurately

characterize the non-linear relationship between the

phase behavior of patch radiator and its geometric

parameters. The proposed network used MLP. The hyper-

bolic tangent was chosen as the activation function for

the two-layered network. The model was trained using

the error backpropagation algorithm. It was shown that,

when compared to direct evaluation, the ANN approach

had results of similar accuracy that were attained with an

enhanced speed.

This work was later expanded in Reference 163 where

the results of a modified ANN were compared with those

of a full-wave method of moments based on local period-

icity (MoM-LP) approach. The sigmoid function was used

here as the activation function. The reflection coefficient

corresponding to any re-radiating element of a reflec-

tarray approximated by the two approaches were in

agreement, further showing that the ANN method can

maintain the desired level of accuracy while significantly

reducing the computational cost.

In Reference 164, an ANN was used to characterize

the elements of a rectangular planar surface reflectarray

composed of 70 ×74 elements, for satellite applications

covering the Eutelsat footprint. Each ANN took as input

the angle of incidence and patch dimensions, in addition

to the resonance frequency as a constant parameter.

Feed-forward MLP topology was used, along with

FIGURE 13 Pentagonal-shaped CPW antenna. CPW,

coplanar waveguide

FIGURE 14 Reflectarray with unit cells

EL MISILMANI ET AL.17 of 28

backpropagation training algorithm, to optimize the

reflectarray patch dimensions by training the model on a

dataset obtained from MoM-LP-based computations. The

obtained ANN results of the gain pattern and phase dis-

tribution of the reflectarray were compared with those

obtained from MoM-LP computations and showed good

agreement while having a speed up factor of 2 ×10

2

.

An MLP-ANN has been used in Reference 165 to opti-

mize and speed up the design and analysis of

reflectarrays. The reflection phase characteristics of the

unit cell element was first trained and tested using CST.

A dataset of 990 samples was used for training, while

660 data samples were used for testing. In the analysis

model, the edge length of the patch, the ratio of the cavity

to the edge length, the substrate thickness and resonance

frequency, were taken as input in the analysis model.

The LM algorithm was used for training. The MSE was

calculated to be 3.5992 ×10

−4

for training and

4.0192 ×10

−4

for testing. The reconstructed phase varia-

tions and the target ones were of high similarity, validat-

ing the efficiency of the proposed model. A reflectarray

with the optimized Minkowski elements was then tested

to validate the overall optimized performance of the

antenna (Figure 15).

The design of reflectarrays composed of second-order

phoenix cells was presented in Reference 166. Fast char-

acterization of these cells was made possible by using

ANNs, allowing to obtain a spherical mapping that com-

plies with the results obtained by full-wave simulations

with the local periodicity assumption.

A reflectarray antenna using modified Malta-Cross

cells was designed using an ANN in Reference 167.

Starting by a dataset obtained through full-wave simula-

tions, the ANN was trained using error backpropagation,

resulting in a model that allows the computation of

reflection coefficients from any input value for the geo-

metrical and re-radiating field parameters of the reflec-

tarray in both cases of horizontal and vertical

polarizations. The obtained ANN model allows high

accuracy predictions for lower memory usage with less

computation time and load.

A contour-shaped reflectarray antenna was analyzed

in Reference 168. A trained ANN is used to predict the

complex reflection coefficient's amplitude and phase by

taking six geometrical parameters, the incident angle in

terms of azimuth and elevation, and the frequency as

inputs. The results were compared to full-wave electro-

magnetic computations and showed great agreement

while having a speed up factor of 700.

Other techniques have been also employed in the

design of reflectarray antennas. Using an advanced

learning-by-example (LBE) method, namely the Kriging

method, the design of high-performance reflectarrays

was presented in Reference 169. The problem of

predicting the scattering matrix of complex reflectarray

elements was addressed using this LBE algorithm that, if

trained on a set of known input-output relationships, can

accurately predict the output of new input-output pairs.

The Kriging method is not only proficient in dealing with

deterministic noiseless processes but can also facilitate

vectorized outputs. A set of preliminary numerical results

were used to validate the accuracy and time-efficiency of

the proposed model. It was confirmed that this method,

while maintaining a prediction error below 5%, allowed

for a 99.9% time saving percentage when compared to

standard full-wave approaches.

The prediction of the electromagnetic response of

complex-shaped reflectarray elements was presented in

Reference 170, where the authors presented an innova-

tive LBE method based on Ordinary Kriging to obtain

reliable predictions. Full-Wave simulations were used in

order to generate a training set composed of the elevation

angle, azimuth, operating frequency, and the degrees-of-

freedom (DoF) for each array element, along with the

corresponding field distribution. The relationship

between such parameters is highly non-linear which

encourages the usage of ML techniques to find an accu-

rate input/output mapping of these parameters without

having to go through simulations. The authors compared

the performance of their approach to the performance of

SVR and Augmented RBF neural networks used for the

same problem. In addition, several unit-cell shapes have

been considered such as the several cross-slot, ring-slot,

and square/rectangular Phoenix shapes. Results showed

FIGURE 15 Malta cross patch

18 of 28 EL MISILMANI ET AL.

that the customized LBE approach achieved lower error

rates for the same number of training examples com-

pared with SVR and Augmented RBF neural networks.

A framework that employs a ML technique in its

architecture was proposed in Reference 171, where the

algorithm focuses mainly on improving the antenna per-

formance. A surrogate model was obtained using the

SVMs. SVM was used in References 172-174 to design

shaped-beam reflectarrays and for modeling dual-

polarized reflectarray unit cells in Reference 175. Using

SVM, the computational burden resulting from the use of

Full-Wave Local-Periodicity for the design and analysis

was reduced. This has been tested and proven where an

acceleration factor of 880 was achieved compared to sim-

ulations based on the MoM-LP, with error percentages as

low as 0.43%.

6.3 |Other antenna types

The indirect use of an ANN for predicting the input

impedance of broadband antennas through a parametric

frequency model was proposed in Reference 176. While

the antenna geometry parameters and frequency are rou-

tinely used as inputs of the ANN, the resistance was ini-

tially parametrized by a Gaussian model and the ANN

was later used to reach an approximate non-linear rela-

tionship between the antenna geometry and the model

parameters. This novel method was used to obtain a

smaller network size with a smaller number of hidden

units for an ultimately faster training time. For testing, a

loop-based broadband antenna with three tuning arms

was used, where the results were compared to those of a

direct approach. It was found that the proposed model

was considerably more time efficient as it required

10 times less the amount of electromagnetic computa-

tions when training the ANN.

The design of a loop antenna was facilitated using

competitive learning ANN in Reference 177. The aim was

to determine the physical dimensions for frequencies in

the range of 200 to 300 MHz by calculating the best com-

bination of conductor thickness and loop radius using a

SOM. 11 sets of efficiency values corresponding to fre-

quencies related to frequencies for 11 pairs of loop radius

aand wire radius bwere used to train the SOM. The

SOM was later used to produce the desired set of (a, b)

that has the required radiation efficiency, which was veri-

fied by comparison to theoretical results. The design was

shown to respond well to input parameter changes

of 50%.

An MLP-ANN was used to model and predict the

radar cross section of a non-linearly loaded antenna in

Reference 57. After training the MLP-ANN with

backpropagation, the slope information of the resulting

model was used to optimize the antenna. Theoretical for-

mulations have been proposed for this aim, verified by

numerical simulation results. A nonlinear loaded dipole

antenna was used for simplicity as an example. The har-

monic balance technique

178

was used to calculate

101 data samples for training, and 100 data samples for

testing. Comparing the predicted values by extension of

MLP-ANN with those calculated from the harmonic bal-

ance technique, it was concluded that the proposed

method is accurate and can obtain the required results in

less time.

A multi-grade ANN model was proposed in Reference

179 for the design of finite periodic arrays. To take into

consideration the mutual coupling and the array environ-

ment, this approach introduced an innovative approach

where two sub ANNs were used. The first-grade ANN is

called the element-ANN that can provide the non-linear

relationship between the geometrical parameters and

electromagnetic behavior, represented by a certain trans-

fer function (TF) coefficients, of the array element with-

out considering mutual coupling. The output of this

element-ANN is then fed as the input to the second-grade

ANN called the array-ANN. The array-ANN is then capa-

ble of producing outputs of the electromagnetic behavior

of the whole array, with mutual coupling considered.

This approach allows to obtain the mapping between the

geometrical parameters of the element and the electro-

magnetic response of the whole array, while separating

array and element information. Several arrays types were

used to verify the effectiveness of the proposed approach

including a linear phased array, a six-element printed

dipole array, and a U-slot microstrip array. Results

showed training and testing errors smaller than previous

approaches that do not use the multi-grade ANN

approach.

In Reference 180, ANNs were used to optimize the

parameters of a pyramidal horn antenna. The ANN used

RBF as the activation function in its layers and was

trained on data obtained by a full wave simulator. Taking

as inputs the desired frequency of operation and gain, the

ANN generated the required antenna dimensions such as

the height and width of the flared end, the height and

width of the waveguide, and the length of the horn

antenna. Results showed that the trained model can give

very accurate results compared to those obtained by a

simulator with an error percentage as low as 1.3%.

In Reference 181, an ANN was used for analysis and

synthesis simulations of profiled corrugated circular

horns, and then compared with the conventional mode

matching- combined field integral equation (CFIE) tech-

nique. During analysis, the ANN takes as inputs the aper-

ture radius, the horn length, the corrugation height, the

EL MISILMANI ET AL.19 of 28

metal-void ratio, in addition to the number of corruga-

tions per wavelength. The output of the ANN was the

RL, in addition to the co- and cross-polar patterns limited

to 0to 40range, with 2step. To accelerate the analysis,

several ANNs are used. The input space is formed of

10 hypercubes, with each single hypercube mapped to a

subspace of the output space. This approach has been

also presented in Reference 182. The ANN was then

trained in the synthesis procedure to approximate the

function that can relate the main beam width and the

maximum level of the cross-polar level, to the corrugated

horn geometrical parameters. The RL was not taken into

consideration as an input to the ANN during analysis,

since with this type of horn, low levels of RL can be easily

obtained. In addition, some of the geometrical parame-

ters were assumed to be constant, and not varied during

the synthesis process. An example of an optimized pro-

filed corrugated horn using the proposed ANN has been

also fabricated and measured. The results have been com-

pared with the traditional electromagnetic analysis. The

results showed less accuracy when compared with those

obtained from a careful optimization process. Neverthe-

less, the cost and the time needed to design the antenna

as per the required parameters were highly reduced.

ANNs were also used in the design of a W-Band slot-

ted waveguide array antenna in Reference 183. The

model was trained, cross-validated, and tested on dataset

obtained by HFSS simulations. The seven design parame-

ters that were used as input were the lengths and orienta-

tion angles of the coupling slots, in addition to the length

of the radiating slots. The antenna was later fabricated

using Stereolithography 3D printing techniques, and the

measured and simulated results were compared, which

were in good agreement with slight errors.

In Reference 184 a novel multibranch ANN modeling

technique was proposed as a solution to the non-

uniqueness problem in the design of antenna arrays. The

nonuniqueness issue can be defined as the case where

the desired output can be mapped to several inputs,

resulting in conflicting output values for similar inputs in

a dataset, and leading to a large training error and poor

ML model accuracy. This work presented a novel tech-

nique based on calculus to provide a solution for the non-

uniqueness problem, where the training data is separated

into several groups after obtaining the data boundary

locations and monotonicity of this data. These groups

can be used separately to train different ANN branches,

forming one multibranch ANN model that can predict

the antenna's geometrical or physical parameters based

on the desired input EM characteristics.

This technique was tested on Short Dipole Planar

Array where the obtained model had a train and test set

errors of 0.25% and 0.28% respectively, considered to be a

significant improvement on the non-multibranch con-

ventional approach that has a 20.38% training set error

and a 20.40% test set error. Further testing was made on

sparse linear dipole array resulting in 0.37% and 0.68%

training and test set errors respectively.

As a summary to the in-depth investigation of the dif-

ferent antenna design papers using ML presented in the

literature and studied in this work, Table 2 lists the differ-

ent papers investigated sorted by ML and as per the

antenna type and configuration.

7|MACHINE LEARNING-

ASSISTED ANTENNA

OPTIMIZATION

Another line of researcher focuses on embedding ML

models inside an optimization algorithm that is used to

reach optimal parameters and performance of an

antenna. By integrating a ML model within the opti-

mizer, the design and optimization process would speed

up since less simulations would be required. This

section presents the work that has been done in this

regard, along with the various results obtained. A Sum-

mary of these antennas along with the used algorithms

can be found in Table 3.

Interpolation combined with GA used for the design

of an UWB ring monopole antenna was presented in Ref-

erences 186 and 187 where fitness function behaviors

such as the BW, the RL, and the central frequency divi-

sion (CFD) were estimated. After optimizing those

parameters, comparison was held between a simulated

antenna and a real prototype manufactured from the

obtained values.

Different numbers of datasets were used in training

the model and it was determined that the perception on

the behavior of the objectives (BW, RL, and CFD)

increases as the size of the dataset increases.

The design of stacked patch antennas using ANNs

was presented in References 188-191, where a trained

ANN embedded in PSO was used to obtain multi-band

characteristics. After having decided upon the geometri-

cal parameters of the antenna by the PSO, a function

mapping “black-box”was built by the ANN, and the fre-

quencies and associated bandwidths were related to the

dimensional antenna parameters. The obtained ANN

results were then compared with measured results of a

fabricated antennas, where good agreement has been rev-

ealed with an error of order 10

−5

.

In Reference 192, DE and the Kriging algorithm were

used in the design optimization of an E-shaped antenna.

Six antenna variables were optimized, which were feed

position, the slot position, the length and width of the

20 of 28 EL MISILMANI ET AL.

patch, and the slot width and length. Good prediction

accuracy was exhibited by the model after reaching opti-

mal solutions by the model. It was concluded that the

proposed approach reduced the number of necessary sim-

ulations significantly.

In Reference 193, a new algorithm named (SADEA)

based on surrogate model assisted (SMA-DE) and GPR

was proposed. This method was found efficient in the

design of antennas, where it has been tried on three types

of antennas that are namely: an inter-chip antenna, a

four-element linear array, and a 2-D array. It was shown

that SADEA can speed up the design and optimization

procedure by more than four times compared with DE.

Slots antennas were optimized in References 194 and

195 by using Space Mapping as an optimization engine.

Computational costs were reduced by implementing

Bayesian SVR (BSVR)

196

as the coarse response surface

model instead of relying on electromagnetic simulations.

The parameters of a CPW-fed Slot Dipole Antenna and a

CPW-fed T-shaped Slot Antenna were optimized using

this procedure which resulted in satisfactory designs.

8|DISCUSSION AND

CONCLUSION

The aforementioned discussion has highlighted the

importance and usefulness of using ML techniques in the

design and analysis of many antennas. However, many

challenges arise when adopting this approach instead of

relying on computational electromagnetics. The first

challenge relates to the lack of standardized datasets for

antenna structures that can be used directly to train a

certain model and obtain results. Instead, data need to be

generated by simulations beforehand to create a database

of selected input and output variables. This can be a

tedious and time-consuming task since the initial goal of

using ML in the context of antennas is obtaining an

TABLE 2 Investigated antennas designed using ML

ML

algorithm Antenna type References

ANN Rectangular patch [124,125,130-133]

Circular patch [134-138]

Fractal patch [139-141]

Elliptical patch [142,143]

Monopole antenna [144]

Dipole antenna [95,185]

PIFA [147,148]

SIW [149]

Special patch structures [151-161]

Reflectarrays [162-168]

Broadband antenna [176]

Loop antenna [177]

Non-linearly loaded

dipole antenna

[57]

Antenna arrays [179]

Corrugated circular horn

antenna

[181]

Pyramidal horn antenna [180]

Slotted waveguide

antenna array

[183]

SVR/SVM Rectangular patch [97,126-129]

Reflectarrays [171-175]

GPR SIW [150]

Kriging

regression

Reflectarrays [169,170]

LASSO Monopole antenna [145,146]

Abbreviations: ANN, artificial neural networks; GPR, Gaussian pro-

cess regression; LASSO, least absolute shrinkage and selection oper-

ator; ML, machine learning; PIFA, planar inverted-F antenna; SIW,

substrate integrated waveguide; SVR/SVM, support vector regres-

sion/support vector machines.

TABLE 3 Investigated antennas

designed using ML assisted

optimization

ML algorithm Optimization algorithm Antenna type References

Interpolation GA Ring monopole antenna [186,187]

ANN PSO Stacked patch antenna [188-191]

Kriging DE E-shaped antenna [192]

GPR SMA-DE Inter-chip antenna [193]

Four-element array

2-D array

BSVR Space mapping Slots antenna [194,195]

Abbreviations: ANN, artificial neural network; BSVR, Bayesian SVR; DE, differential evolution;

GA, genetic algorithm; GPR, Gaussian process regression; ML, machine learning; PSO, particle

swarm optimization; SMA-DE, surrogate model assisted differential evolution.

EL MISILMANI ET AL.21 of 28

accelerated design and characterization process while

maintaining high accuracy. Having to go through simula-

tions to obtain a dataset also translates into a heavier

computational load.

Another aspect to be considered is selecting the best

model hyperparameters that can lead to the optimal

results. It can be clearly deduced that ANNs have domi-

nated this research area by being the most popular choice

of ML technique with many frameworks and software

packages available for their quick and efficient employ-

ment, and by showing resilience in providing highly accu-

rate results compared to conventional CEM approaches.

The importance of ANNs in antenna design becomes more

recognizable as the complexity of the antenna structure

increases. Therefore, it is necessary to investigate what

type of training and optimization method, network archi-

tecture, regularization techniques, choice of activation

functions, and similar factors that affect the model's per-

formance, would be most suitable for each antenna type.

While ML stands out as an attractive antenna design

and analysis tool that can perform predictions with high

accuracy in a shorter period of time compared to simula-

tion approaches, having to generate the training data

would seem unattractive and demanding. For this reason,

a good approach to address this issue is the development

of an antenna design software based purely on ML

models to replace simulators. Such a tool would of course

be limited in terms of designer flexibility and would have

to be targeted on specific antenna types and structures

but may be extended to cover a large number of anten-

nas. Having a fast, accurate, and optimized design tool

would allow quick characterization of the selected

antenna type, where the user would only need to input

the design requirement to obtain the geometrical predic-

tions. However, this software falls short in cases where a

special structure is desired, which forces the designer of

going through simulations.

This paper provided a comprehensive survey on the

usage of ML in antenna design and analysis. ML is

expected to reduce the computational burdens imposed

by simulators and accelerate the design process. The dif-

ferent research papers presented in the literature that

have employed ML algorithms in their design have been

investigated. An overview on a variety of ML concepts

has also been presented, thus enabling readers that are

interested in antenna research but have minimal ML

expertise with the basic and fundamental understandings

needed to use these effective tools in their projects.

ORCID

Hilal M. El Misilmani https://orcid.org/0000-0003-

1370-8799

Tarek Naous https://orcid.org/0000-0003-0049-9318

Salwa K. Al Khatib https://orcid.org/0000-0002-9588-

8473

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