Ci a ion: P iyada shini, J.;
P emala ha, M.; ˇ
Cep, R.; Jayasudha, M.;
Kali a, K. Analyzing Physics-Inspi ed
Me aheu is ic Algo i hms in Fea u e
Selec ion wi h K-Nea es -Neighbo .
Appl. Sci. 2023,13, 906. h ps://
doi.o g/10.3390/app13020906
Academic Edi o s: Juan A.
Gómez-Pulido and Michele Gi olami
Recei ed: 12 Sep embe 2022
Re ised: 30 Decembe 2022
Accep ed: 6 Janua y 2023
Published: 9 Janua y 2023
Copy igh : © 2023 by he au ho s.
Licensee MDPI, Basel, Swi ze land.
This a icle is an open access a icle
dis ibu ed unde he e ms and
condi ions o he C ea i e Commons
A ibu ion (CC BY) license (h ps://
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4.0/).
applied
sciences
A icle
Analyzing Physics-Inspi ed Me aheu is ic Algo i hms in
Fea u e Selec ion wi h K-Nea es -Neighbo
Jaya aju P iyada shini 1, Ma iappan P emala ha 1, Robe ˇ
Cep 2, Mu ugan Jayasudha 1and Kanak Kali a 3,*
1School o Compu e Science and Enginee ing, Vello e Ins i u e o Technology, Chennai 600027, India
2Depa men o Machining, Assembly and Enginee ing Me ology, Facul y o Mechanical Enginee ing,
VSB-Technical Uni e si y o Os a a, 17. Lis opadu 2172/15, 708 00 Os a a, Czech Republic
3Depa men o Mechanical Enginee ing, Vel Tech Ranga ajan D . Sagun hala R&D Ins i u e o Science
and Technology, A adi 600062, India
*Co espondence: [email p o ec ed]
Abs ac :
In ecen yea s, ea u e selec ion has eme ged as a majo challenge in machine lea ning. In
his pape , conside ing he p omising pe o mance o me aheu is ics on di e en ypes o applica ions,
six physics-inspi ed me apho algo i hms a e employed o his p oblem. To e alua e he capabili y o
dimensionali y educ ion in hese algo i hms, six di e se-na u ed da ase s a e used. The pe o mance
is compa ed in e ms o he a e age numbe o ea u es selec ed (AFS), accu acy, i ness, con e gence
capabili ies, and compu a ional cos . I is ound h ough expe imen s ha he accu acy and i ness o
he Equilib ium Op imize (EO) a e compa a i ely be e han he o he s. Finally, he a e age ank
om he pe spec i e o a e age i ness, a e age accu acy, and AFS shows ha EO ou pe o ms all
o he algo i hms.
Keywo ds: op imiza ion; non- adi ional algo i hms; ea u e educ ion; KNN; algo i hms
1. In oduc ion
Da a mining is he p ocess o inding meaning ul in o ma ion o ex ac ing knowledge
om la ge amoun s o da a. Da a mining has he challenging p oblem o dealing wi h huge
da a dimensions. When wo king wi h da a ha has a la ge numbe o dimensions, e en
he ad an ages o echnology can be a hassle [
1
]. The da a-mining p ocess may su e due
o a huge numbe o dimensions. I may also equi e a lo o compu ing ime and space.
T adi ional machine-lea ning (ML) me hods canno handle hese huge da ase s [
2
]. The
da ase is made up o se e al samples ha collec i ely gi e in o ma ion abou a speci ic
case o he p oblem. Each sample has a a ie y o a ibu es o ea u es. The da ase may
ha e se e al supe luous o duplica e a ibu es, in addi ion o i s huge dimensionali y. The
model may be complex, and he da ase may include a subs an ial amoun o noise. The bes
subse o he use ul ea u es ha will con ibu e o he ou pu is chosen ia a p e-p ocessing
echnique called ea u e selec ion (FS) [
2
]. FS can educe he aining ime as well as he huge
numbe o dimensions in he da a. Mo eo e , he model’s accu acy is enhanced in addi ion
o he simpli ica ion o he model and he bes u iliza ion o compu ing esou ces [3].
The wo main FS app oaches a e w appe me hods and il e me hods. The majo
d awback o he il e me hods is ha hey wo k independen ly o he ML classi ie s and
do no ake any inpu om hem [
4
]. Meanwhile, he w appe me hod uses he classi-
ie di ec ly and picks he ea u es using an op imiza ion algo i hm [
5
]. Op imiza ion
algo i hms p o ide he ad an age o choosing an op imal o nea ly op imal subse o
ea u es in a easonable amoun o ime as opposed o he con en ional exhaus i e sea ch.
An exhaus i e sea ch becomes imp ac ical, because i inds he solu ion by c ea ing all
easible ea u e subse s (2
m
di e en solu ions o
m
ea u es) [
6
]. In he li e a u e, op imiza-
ion algo i hms a e ca ego ized in o se e al g oups, such as e olu ion-based algo i hms,
Appl. Sci. 2023,13, 906. h ps://doi.o g/10.3390/app13020906 h ps://www.mdpi.com/jou nal/applsci
Appl. Sci. 2023,13, 906 2 o 19
swa m-based algo i hms, human beha io -inspi ed algo i hms, physics-inspi ed algo-
i hms, e c. [
7
]. Swa m-based algo i hms mimic he collec i e bu decen alized in elligence
o li ing c ea u es, such as bi ds [
8
], wol es [
9
], whales [
10
], bac e ia [
11
], e c. E olu iona y
algo i hms mimic he eme gence o he i es and heal hies indi iduals o e gene a ions.
A ew examples a e he Gene ic Algo i hm (GA) [
12
], Di e en ial E olu ion (DE) [
13
],
Biogeog aphy-Based Op imiza ion (BBO) [
14
], e c. Human beha io -inspi ed algo i hms
mimic he collec i e in elligen beha io o human beings in di e en eal-li e si ua ions,
such as poli ics [
15
], spo s [
16
], co po a ions [
7
], e c. Finally, physics-based algo i hms
a e inspi ed by he laws o na u e, such as he g a i a ional law [
17
], black holes [
18
],
galaxies [19], e c.
In ecen yea s, me apho -based algo i hms ha e ex ensi ely been used o sol e FS
p oblems om di e en domains. Examples include ea u e selec ion using Pa icle Swa m
Op imiza ion (PSO) o documen clus e ing [
20
], he use o a eal- alued G asshoppe Op i-
miza ion Algo i hm (GOA) o ea u e selec ion [
21
], hyb idiza ion o he Whale Op imiza ion
Algo i hm (WOA) and Simula ed Annealing (SA) o he ea u e selec ion p oblem [
22
],
ea u e selec ion o in usion de ec ion in wi eless mesh ne wo ks inco po a ing gene ic ope -
a o s in WOA [
23
], he inco po a ion o le y ligh and opposi ion-based lea ning in chao ic
Cuckoo Sea ch (CS) o ea u e selec ion [
24
], ea u e selec ion using Mo h Flame Op imiza ion
(MFO) [
25
], ea u e selec ion using he Fi e ly Algo i hm (FA) [
26
], he hyb idiza ion o SA
wi h Ha is Hawk Op imiza ion (HHO) o he ea u e selec ion p oblem [
27
] and ea u e
selec ion using bina y Teaching–Lea ning-Based Op imiza ion (TLBO).
Acco ding o he No-F ee-Lunch (NFL) heo em [
28
], no single op imiza ion algo i hm
is capable o sol ing e e y op imiza ion p oblem by ou pe o ming all o he op imiza ion
echniques. Because o his, one op imize can pe o m be e han he o he s on some
p oblems, bu no on all o hem. Hence, i is c ucial o compa e se e al op imiza ion
algo i hms on a a ie y o da ase s o ind he op imum solu ion o he ea u e selec ion
p oblem. Since he e a e hund eds o op imiza ion algo i hms in he li e a u e, in his s udy,
a ew well-known and highly ci ed physics-inspi ed algo i hms a e chosen o his pu pose.
The a ionale is o ca y ou a compa ison o he a ious me apho s d awn om physics
and e alua e hei e ec i eness. To e alua e he pe o mance o hese algo i hms, six small-
o-la ge-sized classi ica ion da ase s a e used. The accu acy, con e gence, and a e age
i ness o hese algo i hms a e compa ed. This pape has he ollowing con ibu ions:
•
The main no el y o ou pape lies in i s compa a i e analysis o six well-ci ed physics-
inspi ed me apho algo i hms o he p oblem o ea u e selec ion.
•
To he bes o ou knowledge, his is he i s ime hese physics-inspi ed algo i hms
ha e been compa ed o his speci ic p oblem, and ou indings p o ide aluable
insigh s in o hei pe o mance.
•
Ou s udy also has b oade implica ions o he ield o machine lea ning and da a min-
ing, as i helps o shed ligh on he e ec i eness o di e en op imiza ion algo i hms
o ea u e selec ion.
•
Ou wo k con ibu es o he g owing body o esea ch on me aheu is ics and hei
po en ial applica ions in machine lea ning and da a mining, and i highligh s he
po en ial alue o using physics-inspi ed op imiza ion algo i hms o ea u e selec ion.
•
Addi ionally, ou use o a iable-sized classi ica ion da ase s allows us o assess he
applicabili y o hese algo i hms on a wide ange o p oblems, making ou esul s
mo e gene alizable and applicable o p ac i ione s.
O e all, we belie e ha ou pape ep esen s a signi ican con ibu ion o he ield and
has he po en ial o impac he way p ac i ione s app oach he p oblem o ea u e selec ion.
The es o he pape is o ganized as ollows. The me hodology is discussed in Sec ion 2.
Sec ion 3, namely, he Resul s and Discussion, co e s he esul s and compa a i e analysis
o all six algo i hms, and he concluding ema ks a e gi en in Sec ion 4.
Appl. Sci. 2023,13, 906 3 o 19
2. Me hodology
2.1. W appe Me hod o Fea u e Selec ion
Fo ea u e selec ion, we employed a w appe me hod. To accomplish hei ask,
w appe echniques use a lea ning algo i hm ha applies a sea ch s a egy o explo e he
space o easible ea u e subse s, anking hem acco ding o he quali y o hei pe o mance
in a speci ic algo i hm. In mos cases, w appe app oaches ou pe o m il e me hods,
since he ea u e selec ion p ocess is ailo ed o he speci ic classi ica ion algo i hm being
employed. W appe me hods, on he o he hand, a e p ohibi i ely ime- and esou ce-
in ensi e o high-dimensional da a, since hey equi e e alua ing each ea u e se wi h he
classi ie algo i hm. Figu e 1depic s he way in which w appe me hods unc ion.
Appl. Sci. 2023, 12, x FOR PEER REVIEW 3 o 20
Sec ion 2. Sec ion 3, namely, he Resul s and Discussion, co e s he esul s and compa a-
i e analysis o all six algo i hms, and he concluding ema ks a e gi en in Sec ion 4.
2. Me hodology
2.1. W appe Me hod o Fea u e Selec ion
Fo ea u e selec ion, we employed a w appe me hod. To accomplish hei ask,
w appe echniques use a lea ning algo i hm ha applies a sea ch s a egy o explo e he
space o easible ea u e subse s, anking hem acco ding o he quali y o hei pe o -
mance in a speci ic algo i hm. In mos cases, w appe app oaches ou pe o m il e me h-
ods, since he ea u e selec ion p ocess is ailo ed o he speci ic classi ica ion algo i hm
being employed. W appe me hods, on he o he hand, a e p ohibi i ely ime- and e-
sou ce-in ensi e o high-dimensional da a, since hey equi e e alua ing each ea u e se
wi h he classi ie algo i hm. Figu e 1 depic s he way in which w appe me hods unc-
ion.
Figu e 1. W appe ea u e selec ion amewo k.
In his pape , K-Nea es Neighbo (k-NN) is used as he e alua o algo i hm. The k-
NN me hod uses a se o K neighbo s o de e mine how an objec should be ca ego ized.
A posi i e in ege alue o K is p e-decided be o e unning he algo i hm. To classi y a
eco d, he Euclidean dis ances be ween he unclassi ied eco d and he classi ied eco ds
a e de e mined and anked.
2.2. Fi ness Func ion
The e ec i eness o an op imize is e alua ed by i s i ness unc ion. The i ness unc-
ion in ea u e selec ion is dependen on he classi ica ion e o a e and he numbe o
ea u es used o classi ica ion. I is deemed o be a good solu ion i he selec ed ea u e
subse educes he classi ica ion e o a e and he numbe o ea u es chosen. The ollow-
ing i ness unc ion is used in his pape [29]:
↓ 𝐹𝑖𝑡𝑛𝑒𝑠𝑠 =𝜆𝛾𝑆(𝐷)+(1−𝜆)|𝑆|
|𝐹|
(1)
whe e 𝛾𝑆(𝐷) is he classi ica ion e o compu ed by he classi ie , |𝑆| is he educed
numbe o ea u es in he new subse , |𝐹| is he o al ea u es in he da ase , and 𝜆 𝜖 [0,1]
Figu e 1. W appe ea u e selec ion amewo k.
In his pape , K-Nea es Neighbo (k-NN) is used as he e alua o algo i hm. The
k-NN me hod uses a se o K neighbo s o de e mine how an objec should be ca ego ized.
A posi i e in ege alue o K is p e-decided be o e unning he algo i hm. To classi y a
eco d, he Euclidean dis ances be ween he unclassi ied eco d and he classi ied eco ds
a e de e mined and anked.
2.2. Fi ness Func ion
The e ec i eness o an op imize is e alua ed by i s i ness unc ion. The i ness
unc ion in ea u e selec ion is dependen on he classi ica ion e o a e and he numbe o
ea u es used o classi ica ion. I is deemed o be a good solu ion i he selec ed ea u e
subse educes he classi ica ion e o a e and he numbe o ea u es chosen. The ollowing
i ness unc ion is used in his pape [29]:
↓Fi ness =λγS(D) + (1−λ)|S|
|F|(1)
whe e
γS(D)
is he classi ica ion e o compu ed by he classi ie ,
|S|
is he educed numbe
o ea u es in he new subse ,
|F|
is he o al ea u es in he da ase , and
λ e [0, 1]
is a
ac o co esponding o he impo ance o he classi ica ion pe o mance and leng h o he
educed subse .
2.3. Physics-Inspi ed Me apho Algo i hms
In his pape , six well-ci ed physics-inspi ed me apho algo i hms a e employed o
sol e he p oblem o ea u e selec ion. In his sec ion, he unc ioning o hese algo i hms
and hei posi ion-upda ing mechanisms a e discussed.
Appl. Sci. 2023,13, 906 4 o 19
2.3.1. Simula ed Annealing
Simula ed Annealing is a undamen al na u e-inspi ed algo i hm ha was p oposed
in 1983 by Ki kpa ick e al. [
30
]. The sou ce o inspi a ion behind his algo i hm is he
annealing p ocess o me als. The p ocess o annealing, which s a s a a e y high empe -
a u e and p og essi ely cools down, is used o physically ha den me als. The algo i hm
in ol es h ee main pa ame e s, including he cooling a e (
c
), he inal empe a u e (
T
),
and he s a ing empe a u e (
T0
). The s a ing empe a u e is kep e y high ini ially, and
he cooling a e g adually educes un il i eaches he inal empe a u e. The p ocess is
mimicked by andomly gene a ing a candida e solu ion. The algo i hm uns i e a i ely, and
a new solu ion is gene a ed in he neighbo hood o he cu en solu ion in each i e a ion.
The i ness o he cu en and neighbo solu ions is compa ed. I he i ness o he new
solu ion is be e , hen he posi ion o he cu en solu ion is upda ed. Mo eo e , he bes
solu ion keeps he bes posi ion ound so a . The e mina ing condi ion o he epe i i e
p ocess is eaching he T . In each i e a ion, Tis upda ed as ollows:
T=T∗C, 0 <c<1 (2)
SA is a global op imiza ion algo i hm, because i can explo e as well as exploi he
sea ch space. The explo a ion is pe o med by upda ing he cu en solu ion wi h a wo se
neighbo ing solu ion in ea ly i e a ions based on he alue o
T
and he wo se alue o he
neighbo ing solu ion. The chance o accep ing he wo se neighbo is compu ed using he
ollowing equa ion:
exp−δ
T≤ (3)
whe e
exp
is he exponen ial unc ion,
δ
is equal o he i ness di e ence o cu en and
neighbo ing solu ions, and is andomly gene a ed in he ange [0, 1].
2.3.2. G a i a ional Sea ch Algo i hm
This algo i hm is inspi ed by New on’s law o g a i a ion and he second law o
mo ion [
17
]. I ea s each candida e solu ion in he sea ch space as an objec whose mass
is conside ed o be i s i ness. Hea ie objec s a e conside ed i e han ligh e objec s.
The objec s a e a ached o each o he wi h some g a i a ional o ce ha causes objec s o
explo e he sea ch space. The hea ies objec is conside ed he global bes solu ion. Since
he hea ie objec s a ac o he objec s wi h mo e o ce, he whole popula ion ul ima ely
con e ges owa d he hea ies objec , called he global bes solu ion. The algo i hm is
comp ised o a ew ma hema ical equa ions ha a e exp essed below.
Fo ce calcula ion: The o ce om an objec
j
on an objec
i
is calcula ed using he
ollowing equa ion:
Fd
ij( ) = G( )Mpi( )×Maj( )
Rij( ) + e(xd
j( )−xd
i( )) (4)
In he abo e equa ion,
G
is he g a i a ional cons an ha con ols he sea ch accu acy,
Mpi
is he passi e g a i a ional mass o solu ion
i
,
Maj
is he ac i e g a i a ional mass o
solu ion
j
, he dis ance be ween solu ion
i
and solu ion
j
is deno ed by
Rij
,
xd
is he posi ion
o a solu ion in d h dimension, and eis a small cons an .
The ul ima e o ce on a solu ion (mass) is calcula ed by aking he weigh ed sum o all
he o ces on ha solu ion om he kbes solu ions, which a e calcula ed as ollows:
Fd
i( ) = ∑
j∈kbes ,j6=i
andjFd
ij( )(5)
Appl. Sci. 2023,13, 906 5 o 19
Accele a ion calcula ion: Once he o al o ce on a solu ion in a pa icula dimension
d
is calcula ed, he accele a ion o he solu ion in ha dimension can be compu ed using he
ollowing equa ion:
ad
i( ) = Fd
i( )
Mii( )(6)
whe e Mii is he mass o ine ia o solu ion i.
Veloci y calcula ion: Based on he accele a ion, he eloci y o a solu ion can be
compu ed by adding he accele a ion o a ac ion o he p e ious eloci y o ha solu ion.
The equa ion o compu e eloci y is gi en below:
d
i= andi× d
i( ) + ad
i( )(7)
Posi ion upda ing: To upda e he posi ion o a solu ion, he upda ed eloci y is simply
added o he old posi ion o he solu ion, as o mula ed below:
xd
i( +1)=xd
i( ) + d
i( +1)(8)
G a i a ional cons an upda ing: To upda e G, he ollowing ela ion is used:
G( ) = G0exp−α
max (9)
whe e
G0
is he ini ial g a i a ional cons an , and
α
is a cons an .
and
max
ep esen he
cu en and inal i e a ion numbe s.
2.3.3. Sine Cosine Algo i hm
The Sine Cosine Algo i hm (SCA) [
31
] has a e y unique sou ce o inspi a ion. I
u ilizes wo sine and cosine unc ions o upda e he posi ion o solu ions when sea ching
he space o ind he global op imum. The posi ion-upda ing model o his algo i hm is
e y simple and is o mula ed below:
X +1
i=(X
i+ 1×sin( 2)× 3P
i−X
i, 4<0.5
X
i+ 1×cos( 2)× 3P
i−X
i, 4≥0.5 (10)
In he abo e equa ion,
Xi
deno es a solu ion in he
i h
dimension, and
Pi
deno es he
global bes solu ion, namely, he des ina ion solu ion in he pape . The abo e equa ion
in ol es a ew o he a iables ha a e de ined below.
1
is an adap i e pa ame e ha is linea ly educed wi h he cou se o i e a ions. I
s a s om a p e ixed alue and linea ly dec eases in each i e a ion. I is compu ed as
ollows:
1=α− α
T(11)
whe e αis cons an .
• 2is andomly gene a ed in he ange o 0 o 2π.
• 3is also a andom numbe ha is gene a ed in he ange o 0 o 2.
• 4
is also a andom numbe ha is gene a ed in he ange o 0 o 1, and based on i s
alue, i is decided whe he o use he sine unc ion o he cosine unc ion in upda ing
he posi ion o he cu en solu ion.
When mul iplied by
1
, he ange o alues p o ided by sin(
2
) and cos(
2
) shi s
om [
−
1, 1] o [
−
2, 2]. Due o a linea dec ease in he alues o he pa ame e
1
, he
ange begins a [
−
2, 2] and linea ly declines o [0, 0] du ing i e a ions. The posi ion-
upda ing equa ion o SCA c ea es wo egions a ound he des ina ion P: an inne egion
ha p omo es exploi a ion and an ou e egion o p omo e explo a ion. The p econdi ion
o sea ch he inne egion is
{−1<= 1Xcos( 2)<=1}
o
{−1<= 1Xsin( 2)<=1}
,

Appl. Sci. 2023,13, 906 6 o 19
and he p econdi ion o sea ch he ou e egion is
{ 1Xcos( 2)}
, o
{ 1Xsin( 2)}
gi es a
alue g ea e han 1 o lesse han −1.
2.3.4. A om Sea ch Op imiza ion
A om Sea ch Op imiza ion (ASO) [
32
], which is inspi ed by molecula dynamics,
has shown a emendous pe o mance on a a ie y o applica ions in he li e a u e. Each
a om is conside ed a candida e solu ion, and he mass is mapped wi h he i ness in he
op imiza ion algo i hm, whe e he highe he mass, he i e he solu ion. E e y a om
in he popula ion pulls o epels o he a oms in he sea ch space. The hea ie a oms
gene a e mo e o ce and pull ligh e objec s apidly, and he hea ie objec s a e pulled
slowly owa ds he o he s due o hei mass. The slowly mo ing a oms c ea e exploi a ion
in he algo i hm, because hey can sea ch mo e locally, whe eas he apidly mo ing a oms
allow he algo i hm o explo e he sea ch space because o longe and quicke jumps.
The algo i hm s a s wi h andom ini ializa ions. In e e y i e a ion, he a oms mo e and
accele a e, and he loca ion o he a om ha has pe o med he bes up o ha poin is also
likewise adjus ed. A omic accele a ion is also caused by wo o he ac o s: L-J po en ial
and cons ain o ces. The accele a ion helps o upda e he eloci y o he solu ions (a oms).
Finally, he eloci y is added o he p e ious posi ion o upda e he cu en posi ion o he
solu ion. The posi ion-upda ing mechanism o he algo i hm is discussed below.
The popula ion is gene a ed by andomly gene a ing posi ion and eloci y ec o s o
each a om in he popula ion.
Xi=hX1
i,X2
i, . . . , XD
ii(12)
Vi=hV1
i,V2
i, . . . , VD
ii(13)
The i ness o each solu ion in he popula ion is compu ed, and he global bes
Xbes
is
de e mined.
The mass o each a om is compu ed using he ollowing equa ion:
mi=Mi
∑N
j=1Mj
(14)
whe e
M
is compu ed om he i ness o he cu en solu ion, he bes solu ion, and he
wo s solu ion.
The alue o Kis compu ed, whe e Kdeno es he size o he subse o a oms:
K=N−(N−2)
T(15)
whe e Nis he size o he popula ion.
The in e ac ion o ce on an a om is calcula ed, which is accomplished using he
ollowing equa ion:
Fid=∑
j∈K
andjFijd(16)
whe e andjis a andom numbe in he ange o [0, 1].
The cons ain o ce is compu ed using he ollowing equa ion:
Gd
i=λXd
bes −Xd
i(17)
whe e λis he Lang angian mul iplie ha is compu ed as ollows:
λ=βe−20
T(18)
whe e βis he mul iplie weigh .
Appl. Sci. 2023,13, 906 7 o 19
Once he mass, cons ain o ces, and in e ac ion o ces a e compu ed, he accele a ion
is compu ed as ollows:
ad
i=Fd
i
md
i
+Gd
i
md
i
(19)
Once he accele a ion is compu ed, he eloci y o an a om can be compu ed as ollows:
Vd
i( +1)= 1Vd
i( ) + ad
i( )(20)
Using he upda ed eloci y, he posi ion o a solu ion is upda ed as ollows:
Xd
i( +1)=Xd
i( ) + Vd
i( +1)(21)
2.3.5. Hen y Gas Solubili y Op imiza ion
Hen y Gas Solubili y Op imiza ion (HGSO) is inspi ed by Hen y’s gas law [
33
], which
is s a ed below:
“A a cons an empe a u e, he amoun o a gi en gas ha dissol es in a gi en ype and
olume o liquid is di ec ly p opo ional o he pa ial p essu e o ha gas in equilib ium
wi h ha liquid”.
This law can be in e p e ed as he pa ial p essu e o a gas and he solubili y o ha
gas being di ec ly p opo ional. I one inc eases, hen he o he inc eases, oo. This ela ion
is exp essed h ough he ollowing equa ion:
Sg=H×Pg(22)
whe e he gas solubili y is deno ed by
Sg
, Hen y’s cons an is deno ed by
H
, and he
pa ial p essu e o he gas is ep esen ed by
Pg
. The p opo ionali y cons an
H
is highly
dependen on he empe a u e, as i a ies wi h he change in he empe a u e. In HGSO,
each gas pa icle is conside ed a candida e solu ion, whe eas all pa icles collec i ely make
up he popula ion. Ini ially, gas pa icles (popula ion) a e andomly gene a ed, and hen
gas pa icles upda e hei posi ions in he cou se o i e a ions by explo ing and exploi ing
he sea ch space. HGSO in ol es he ollowing s eps.
Popula ion ini ializa ion: A popula ion o
N
gas pa icles is andomly gene a ed using
he ollowing equa ion:
Xi( +1)=X{min}+ ×X{max}−X{min}(23)
whe e
Xi
deno es he ini ial posi ion o he
i h
solu ion,
X{min}
and
X{max}
a e he lowe
and uppe bounds o he p oblem unc ion unde conside a ion,
is a andomly gene a ed
eal numbe be ween 0 and 1, and is he i e a ion numbe .
The p ope ies o each sea ch agen in HGSO can be ini ia ed using he ollowing equa ion:
Hj( )=l1× and(0, 1),P{i,j}=l2× and(0, 1),Cj=l3× and(0, 1)(24)
whe e
Hj( )
ep esen s Hen y’s cons an o he
j h
clus e ,
P{i,j}
deno es he pa ial p essu e
o he
i h
pa icle in he
j h
clus e , and
Cj
indica es he ini ial cons an alue o he
j h
clus e .
Clus e ing: This s ep di ides he sea ch agen s in o
K
clus e s o map di e en ypes
o gases, whe e he same ypes o gases a e g ouped in o a clus e . The e o e, each clus e
has he same alue o Hen y’s cons an Hj.
Fi ness E alua ion: In his s ep, each sea ch agen in he
j h
clus e is e alua ed h ough
he objec i e unc ion o ind he bes solu ion
Xj,bes
in he
j h
clus e . Once all he clus e s
a e e alua ed, hen he gases a e anked o ind he global bes pa icle Xbes .
Appl. Sci. 2023,13, 906 8 o 19
Upda e Hen y’s coe icien : The pa ial p essu e o each gas pa icle changes in each
i e a ion. The e o e, he alue o Hen y’s coe icien
Hj
is upda ed using he ollowing equa ion:
H( +1)=exp−Cj×1
T−1
T0×Hj( ), ; T( ) = exp( −
{max}!) (25)
whe e
Hj
ep esen s he alue o Hen y’s cons an o he j
h
clus e ,
T
indica es he empe -
a u e,
T0
deno es a e e ence empe a u e equi alen o 298.15 K, and
{max}
ep esen s he
maximum i e a ions.
Upda e solubili y: In his s ep, he solubili y
S{i,j}
o he
i h
pa icle in he
j h
clus e is
upda ed using he ollowing equa ion:
S{i,j}( )=K×Hj( +1)×P{i,j}( )(26)
whe e Kis a cons an , and P{i,j}is he pa ial p essu e o gas iin clus e j.
Upda e posi ion: The p ope ies o pa icles compu ed in he p e ious s eps a e
u ilized o upda e he posi ion o he
i h
gas pa icle in he
j h
clus e acco ding o he
ollowing equa ion:
X{i,j}( +1)=X{i,j}( )+F× 1×γ×X{j,bes }( )−X{i,j}( )
+F× 2×a×S{i,j}( )×X{bes }( )−X{i,j}( )(27)
γ=β× expn acnF{bes }( )+eonF{i,j}( )+eoo, ; e=0.05 (28)
whe e he posi ion o he
i h
sea ch agen in he
j h
clus e is ep esen ed by
Xij
, he bes
agen in he
j h
clus e is deno ed by
Xj,bes
, and he global bes pa icle in he en i e
popula ion is ep esen ed by
Xbes
. Mo eo e ,
1
and
2
a e wo andom alues in he ange
[0, 1],
is he cu en i e a ion,
F
is a lag used o di e si ica ion pu poses and changes he
di ec ion o he solu ion,
γ
indica es he abili y o he
i h
pa icle in he
j h
clus e o in e ac
wi h o he agen s in i s clus e ,
a
ep esen s he impac o o he gases on he
i h
pa icle,
β
is
ixed as
β
= 1, F
i,j
is he i ness o he
i h
pa icle in he
j h
clus e , and
F{bes }
is he i ness o
he bes pa icle.
Escape om local op imum: To a oid s agna ion in local op ima, all he pa icles a e
e alua ed, and he wo s N
w
agen s a e selec ed and eini ialized using he ollowing equa ion:
Nw=N×( and(c2−c1)+c1), ; c1=0.1 ; and ;c2=0.2 (29)
whe e
N
is he popula ion size. Mo eo e ,
c1
and
c2
a e cons an s ha de ine he pe cen age
o wo s pa icles.
2.3.6. Equilib ium Op imize (EO)
Con ol olume mass balance models, which a e used o es ima e bo h dynamic and
equilib ium s a es, se e as an inspi a ion o he Equilib ium Op imize (EO), a ecen ly
p oposed physics-inspi ed algo i hm [
34
]. The pa icles a e conside ed o be he solu ions,
and hei posi ions map he concen a ion o he pa icles. This algo i hm cons uc s an
equilib ium pool o i e e e ence solu ions ( ou bes so- a pa icles and one a i hme ic
mean o hem) called equilib ium candida es. Each pa icle upda es i s posi ion wi h
e e ence o a andomly selec ed candida e om he pool. The algo i hm is aided by wo
ca e ully designed pa ame e s called he exponen ial e m (
F
) and he gene a ion a e
(
G
). Mo eo e , a concep o memo y sa ing is used, which allows a solu ion o upda e i s
concen a ion only i i imp o es as compa ed o i s p e ious concen a ion. The explo a ion,
exploi a ion, and he balance be ween hem a e con olled h ough hese pa ame e s: he
equilib ium pool and he gene a ion p obabili y.
Appl. Sci. 2023,13, 906 9 o 19
EO uses a mass-balance equa ion o desc ibe he conse a ion o mass wi hin a sys em.
The gene ic mass-balance equa ion is gi en as:
VdC
d =QC{eq}−QC +G(30)
whe e
VdC
d
ep esen s he a e o change o mass in a con ol olume,
Q
is he low a e,
he concen a ion a an equilib ium s a e is deno ed by
QC{eq}
, and
G
mimics he mass
gene a ion a e. He e,
dC
d
can also be sol ed in e ms o
Q
V
and
Q
V
deno ed by
λ
o he
u no e a e (i.e., λ=Q
V). The e o e, he abo e equa ion can be econs uc ed as:
dC
λC{eq}−λC+G
V
=d (31)
By aking he in eg a ion o he abo e equa ion, we ob ain:
C=C{eq}+C0−C{eq}F+G
λV(1−F)(32)
which is used as an upda ing ule o each pa icle, whe e Fis calcula ed as ollows:
F=exp[−λ( − 0)] (33)
whe e
0
and
C0
ep esen he ini ial s a ime and concen a ion. In his algo i hm, each
pa icle is a solu ion, and i s posi ion ep esen s i s concen a ion. The ma hema ical
o mula ion o EO is discussed in he ollowing s eps.
Ini ializa ion and unc ion e alua ion: The i s s ep is o ini ialize he pa icles’ con-
cen a ion acco ding o he ollowing equa ion:
X{ini }
{m}=X{min}+ andm(X{max}−X{min})(34)
whe e
X{ini }
{m}
ep esen s he ini ial concen a ion o he
m h
pa icle,
X{max}
shows he
maximum, and X{min}shows he minimum alues.
Equilib ium pool and candida es X
e
: In his algo i hm, ou equilib ium candida es
(good solu ions) a e de e mined o guide o he pa icles and p omo e explo a ion. Mo e-
o e , a pa icle cons uc ed by aking he a i hme ic mean o all hese candida es is also
used, which p omo es exploi a ion. These candida es a e hen assembled o o m an equilib-
ium pool. Each pa icle upda es i s posi ion wi h espec o a andomly selec ed candida e
om he pool.
Exponen ial e m (E): This e m is used in posi ion upda ing o balance explo a ion
and exploi a ion. The exponen ial e m is compu ed as ollows:
E=e{−{λ}( − 0)}(35)
whe e and 0a e compu ed by ollowing equa ions, espec i ely:
=1−I e
Max _i e {(a2I e
Max _i e )}
(36)
0=1
λln−a1sign({ }−0.5)h1−e{−{λ} }i+ (37)
In he abo e equa ion, a la ge alue o
a1
p omo es explo a ion, and a la ge alue
o
a2
p omo es exploi a ion. The
sign({ }−0.5)
con ols he di ec ion o explo a ion and
exploi a ion. Using he abo e equa ions, Eis compu ed as ollows:
E=−a1sign({ }−0.5)[e{−{λ} }−1]) (38)
Appl. Sci. 2023,13, 906 16 o 19
Table 6. A e age ank o all a e ages (Fi ness, accu acy, and AFS).
Da ase S a s SA GSA SCA ASO HGSO EO
DS1
A g. i ness ank
A g. accu acy ank
AFS ank
5
5
6
3
3
1
6
6
4
1
1
3
4
4
2
2
2
4
DS2
A g. i ness ank
A g. accu acy ank
AFS ank
6
6
4
5
5
5
3
3
1
2
2
6
4
4
3
1
1
2
DS3
A g. i ness ank
A g. accu acy ank
AFS ank
6
6
4
1
1
4
5
5
2
3
3
6
4
4
1
2
2
3
DS4
A g. i ness ank
A g. accu acy ank
AFS ank
6
5
6
4
4
4
3
3
1
5
5
5
2
2
2
1
1
3
DS5
A g. i ness ank
A g. accu acy ank
AFS ank
6
6
6
5
5
5
1
1
2
4
4
4
3
3
3
1
1
1
DS6
A g. i ness ank
A g. accu acy ank
AFS ank
6
6
6
3
3
5
4
4
2
1
1
4
5
5
1
2
2
3
A g. Rank 5.61 3.66 3.11 3.33 3.11 1.88
3.4. Compa ison wi h O he Me hods om he Li e a u e
In his sec ion, he op h ee physics-inspi ed me aheu is ic algo i hms a e compa ed
wi h he s a e o he a . Fo his compa ison, esul s om o he KNN-me aheu is ic
commina ions epo ed by Elminaam e al. [
37
] we e chosen. The me aheu is ics chosen
o compa ison d aw hei me apho inspi a ion om a ious sou ces. Fo example,
he G ey Wol Op imize (GWO) and Whale Op imiza ion Algo i hm (WOA) may be
classi ied as mammal inspi ed, whe eas mo h Flame Op imiza ion (MFO) and he Bu e ly
Op imiza ion Algo i hm (BFO) a e insec inspi ed. Simila ly, Ha is Hawk Op imiza ion
(HHO) and he Ma ine P eda o Algo i hm (MPA) a e inspi ed by p eying beha io seen
in na u e. Addi ionally, esul s based on popula ML algo i hms such as Nai e Bayes,
Logis ic Reg ession, Random Fo es , Suppo Vec o Machine (SVM), K-NN, Decision
T ee, and S ochas ic G adien Descen (SGD) a e also compa ed, along wi h hei p incipal
componen analysis (PCA)-enhanced e sions.
In e ms o classi ica ion accu acy (Table 7), in wo ou o he h ee da ase s compa ed,
EO ou pe o med all he o he me hods. In ac , o he b eas cance da ase and ionosphe e
da ase , EO was on a e age 12.75% and 7.12% be e , espec i ely, han he me aheu is ics
p esen ed in [
37
]. In he sona da ase , oo, EO and SCA we e wi hin 2.5% o he bes
solu ion epo ed in [
37
]. Addi ionally, when compa ed wi h he ML algo i hms, he EO
solu ion o he b eas cance da ase was on a e age 17.89% be e . An a e age supe io i y
o 5.68% was seen o EO when compa ed wi h he PCA-ML me hods epo ed in [38].
The a e age ea u es selec ed o he b eas cance , ionosphe e, and sona da ase s by he
a ious me aheu is ics a e epo ed in Table 8. I can be obse ed ha he ea u e educ ions by
he cu en physics-inspi ed me aheu is ics a e much highe . Fo he b eas cance , ionosphe e,
and sona da ase s, he a e age pe cen ea u e educ ion achie ed by he h ee physics-
inspi ed algo i hms was 85.11%, 88.24%, and 84.89%, espec i ely, and o he me aheu is ic
algo i hms om [37], i was only 67.62%, 64.71%, and 67.14%, espec i ely.
Thus, om he comp ehensi e compa isons shown so a , i is clea ha he cu -
en KNN hyb idized physics-inspi ed me aheu is ic algo i hms (especially EO, SCA, and
HGSO) a e supe io o hose epo ed in he li e a u e. Mo eo e , i is seen ha e en solu-
ions by hyb idized ML algo i hms ( o example, by dimensionali y educ ion echniques
such as PCA) we e in e io o cu en solu ions. This is wo h highligh ing, since he cu en
w appe me hods a e much simple in e ms o compu a ional complexi y as compa ed o
he PCA-hyb idized ML me hods.

Appl. Sci. 2023,13, 906 17 o 19
Table 7. Compa ison o classi ica ion accu acy wi h li e a u e esul s.
Me hod B eas Cance % Imp o emen &Ionosphe e % Imp o emen Sona % Imp o emen
EO 0.995 Bes Solu ion 0.986 Bes Solu ion 0.976 2.46%
SCA 0.986 0.91% 0.971 1.54% 0.976 2.46%
HGSO 0.986 0.91% 0.986 Bes Solu ion 0.951 5.15%
GWO [37] 0.970 2.58% 0.951 3.68% 0.970 3.09%
MFO [37] 0.605 64.46% 0.774 27.39% 0.547 82.82%
WOA [37] 0.973 2.26% 0.957 3.03% 0.976 2.46%
SSA [37] 0.982 1.32% 0.985 0.10% 1.000 Bes Solu ion
BOA [37] 0.903 10.19% 0.901 9.43% 0.881 13.51%
HHO [37] 0.929 7.10% 0.929 6.14% 0.833 20.05%
MPA [37] 0.982 1.32% 0.985 0.10% 0.976 2.46%
Nai e Bayes [38] 0.845 17.75% -- - -
Logis ic Reg ession [38] 0.879 13.20% -- - -
Random Fo es [38] 0.995 Bes Solu ion -- - -
SVM [38] 0.620 60.48% -- - -
K-NN [38] 0.900 10.56% -- - -
Decision T ee [38] 0.880 13.07% -- - -
SGD [38] 0.903 10.19% -- - -
PCA-Nai e Bayes [38] 0.975 2.05% -- - -
PCA-Logis ic Reg ession [38] 0.975 2.05% -- - -
PCA-Random Fo es [38] 0.962 3.43% -- - -
PCA-SVM [38] 0.942 5.63% -- - -
PCA-K-NN [38] 0.921 8.03% -- - -
PCA-Decision T ee [38] 0.905 9.94% -- - -
PCA-SGD [38] 0.916 8.62% -- - -
&% imp o emen achie ed by he bes solu ion wi h espec o he compa ed algo i hms.
Table 8. Compa ison o AFS wi h li e a u e esul s.
Me hod B eas Cance % Fea u e
Reduc ion &Ionosphe e % Fea u e
Reduc ion Sona % Fea u e
Reduc ion
EO 4.8 84% 5 85% 11.4 81%
SCA 4.8 84% 3.4 90% 8 87%
HGSO 3.8 87% 3.6 89% 7.8 87%
GWO [37] 7 77% 4 88% 11 82%
MFO [37] 6 80% 23 32% 31 48%
WOA [37] 8 73% 7 79% 26 57%
SSA [37] 11 63% 14 59% 16 73%
BOA [37] 12 60% 20 41% 26 57%
HHO [37] 12 60% 10 71% 20 67%
MPA [37] 12 60% 6 82% 8 87%
&
% Fea u e educ ion is calcula ed as 100% minus he a io o AFS by each algo i hm and maximum ea u es in
he co esponding da ase . A highe alue o % ea u e educ ion is desi ed.
4. Conclusions
In his pape , six well-ci ed physics-inspi ed me apho algo i hms we e employed o
ea u e selec ion. Fea u e selec ion is one o he majo challenges being aced in he ield o
da a mining and machine lea ning. The objec i e o his esea ch was o iden i y he mos
p omising physics-inspi ed algo i hms o he p oblem o ea u e selec ion. To accomplish
his, six small- o la ge-sized da ase s we e used. The pe o mance o EO was ound o be
supe io on mos o he da ase s, and he me ics ha we e used o he compa a i e analysis
we e aken om he li e a u e and included accu acy, i ness, he a e age numbe o ea u es
selec ed, and con e gence analysis. The cu en physics-inspi ed me apho algo i hms,
especially EO, SCA, and HGSO comp ehensi ely ou pe o med o he me aheu is ics, as
well as he ML-based solu ions seen in ecen li e a u e. Based on ou indings, we highly
ecommend using EO o he ea u e selec ion p oblem.
Au ho Con ibu ions:
Concep ualiza ion, R. ˇ
C. and K.K.; Da a cu a ion, J.P., M.P. and M.J.; Fo mal
analysis, J.P., M.P. and M.J.; In es iga ion, J.P., M.P. and M.J.; Me hodology, R. ˇ
C. and K.K.; So wa e,
R. ˇ
C. and K.K.; Valida ion, J.P., M.P. and M.J.; Visualiza ion, J.P., M.P. and M.J.; W i ing—o iginal d a ,
J.P., M.P. and M.J.; W i ing— e iew and edi ing, R. ˇ
C. and K.K. All au ho s ha e ead and ag eed o
he published e sion o he manusc ip .
Appl. Sci. 2023,13, 906 18 o 19
Funding: This esea ch ecei ed no ex e nal unding.
Ins i u ional Re iew Boa d S a emen : No applicable.
In o med Consen S a emen : No applicable.
Da a A ailabili y S a emen :
The da a p esen ed in his s udy a e a ailable h ough email upon
eques o he co esponding au ho .
Con lic s o In e es : The au ho s decla e no con lic o in e es .
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