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Re ining sa elli e ajec o ies wi h celes ial body ea u es using neu al
ne wo ks
José Calde ón a,c,∗, Daniel Ayala a,c, Ra ael Ayala b, Luis Valencia-Cab e a c,
Inma He nández a,c, Da id Ruiz a,c
aDEAL, ETSII, Uni e si y o Se ille, A . Reina Me cedes, s/n, 41012, Se ille, Spain
bMolecula C yo-Elec on Mic oscopy Uni , OIST, 904-0411, Okinawa, Japan
cSCORE Lab, I3US, Uni e si y o Se ille, A . Reina Me cedes, s/n, 41012, Se ille, Spain
A R T I C L E I N F O
Da ase link: SCD-ML
Keywo ds:
Sa elli es
O bi p opaga ion
Neu al ne wo ks
Fea u es enginee ing
A B S T R A C T
Sa elli e o bi p opaga ion in ol es p edic ing a sa elli e’s u u e posi ion and eloci y based on ini ial
condi ions. T adi ional physical models, such as SGDP4, simpli y he o ces ha ac on he sa elli e o achie e
high compu a ional e iciency a he cos o educed p edic ion accu acy, especially o e longe ime in e als
whe e e o accumula es. Mo e sophis ica ed models like HPOP o e imp o ed accu acy a he cos o high
p edic ion imes, ende ing hem unusable o eal ime long- e m p edic ions. Recen ad ancemen s ha e
in oduced machine lea ning echniques o e ine hese p edic ions and educe e o s. Howe e , hey o en lack
an analysis o model design choices, such as inpu ea u e selec ion and a chi ec u al con igu a ions. Exis ing
models do no inco po a e ea u es ela ed o he s a e o celes ial bodies, such as he posi ions o he Moon
o Sun, which can in luence he sa elli e’s ajec o y. This pape p oposes a no el model ha in eg a es such
ea u es a bo h he ini ial ime and h oughou he p edic ion in e al, le e aging hei po en ial impac on he
o bi o he sa elli e. The model is based on a neu al ne wo k a chi ec u e employing GRU laye s o encoding
sequen ial da a abou he celes ial condi ions. Ou esul s demons a e ha he inclusion o hese sequen ial
ea u es signi ican ly educes p edic ion e o s. Addi ionally, we ha e e alua ed a a ie y o design choices such
as independen sub-models o speci ic spa ial coo dina es and ime in e als, u he enhancing pe o mance.
These inno a ions lead o subs an ial imp o emen s in bo h sho - and long- e m o bi p edic ions, p o iding
a mo e obus and accu a e al e na i e o sa elli e o bi p opaga ion.
1. In oduc ion
On Oc obe 4 h, 1957, he So ie Union launched he i s a i icial
sa elli e, Spu nik-1. Since hen, almos 20000 objec s ha e been sen
in o space, o which mo e han 13000 a e s ill in o bi , acco ding o
he UN (Uni ed Na ions O ice o Ou e Space A ai s, 2024). This
massi e olume o objec s makes i c ucial o ack hem o ensu e
he e iciency, sa e y, and success o space missions and he p ope
unc ioning o sa elli es (Ayala, Ayala, Vidal, & Ruiz, 2023; Le i &
Ma shall, 2011; Re aa , Badawy, Ash y, & Adel, 2018).
O bi p opaga ion consis s in he p edic ion o he posi ion and
eloci y o a sa elli e a some u u e da e and ime (known as epoch
in he con ex o as onomy), gi en some ini ial condi ions (Flo es,
Bu hani, & Fan ino, 2021). The ajec o y o a sa elli e is in luenced
by a ious o ces, mainly g a i a ional and cen i ugal, as well as o he
pe u ba ions such as he in luence o he Sun, he Moon, and o he
∗Co esponding au ho a : SCORE Lab, I3US, Uni e si y o Se ille, A . Reina Me cedes, s/n, 41012, Se ille, Spain.
E-mail add esses: [email p o ec ed] (J. Calde ón), [email p o ec ed] (D. Ayala), [email p o ec ed] (R. Ayala), [email p o ec ed] (L. Valencia-Cab e a),
[email p o ec ed] (I. He nández), [email p o ec ed] (D. Ruiz).
celes ial bodies (Shou, 2014). To make hese p edic ions, wo main
ypes o models a e used: analy ical models and high-p ecision models.
Analy ical models, also known as simpli ied pe u ba ion models,
include models such as SGP4 (Simpli ied Gene al Pe u ba ions) and
SDP4 (Simpli ied Deep-space Pe u ba ions) (Hoods & Roeh ich, 1988),
which we collec i ely e e o as SGDP4. These models o e an ap-
p oxima e way o model he sa elli e o bi , allowing o e y as bu
less p ecise long- e m p edic ions. They a e use ul o applica ions in
which speed is c ucial and accu acy can be sac i iced in a ou o
compu a ional e iciency.
High-p ecision models, such as HPOP (High P ecision O bi P op-
aga o ), use di e en ial equa ions o model all he o ces ac ing on
he sa elli e. These models a e ex emely p ecise, bu equi e high
compu ing powe , making hem expensi e and slow o apply, especially
o long- e m p edic ions, as can be seen in Table 1.
h ps://doi.o g/10.1016/j.eswa.2025.127453
Recei ed 23 Oc obe 2024; Recei ed in e ised o m 31 Janua y 2025; Accep ed 25 Ma ch 2025
Expe Sys ems Wi h Applica ions 281 (2025) 127453
A ailable online 7 Ap il 2025
0957-4174/© 2025 The Au ho s. Published by Else ie L d. This is an open access a icle unde he CC BY-NC-ND license ( h p://c ea i ecommons.o g/licenses/by-
nc-nd/4.0/ ).
J. Calde ón e al.
Table 1
Compu a ional ime o di e en p edic ion imes wi h HPOP and SGDP4 p opaga o s.
P edic ion Compu a ional Compu a ional
ime ime ime
(hou s) HPOP SGDP4
0,5 8,69 s 602.54 μs
1 13,57 s 617.03 μs
2 23,55 s 605.36 μs
5 34,93 s 637.44 μs
10 57,45 s 615.19 μs
24 4,09 min 608.23 μs
48 7,99 min 627.64 μs
These models conside in de ail g a i a ional pe u ba ions, sola
adia ion p essu e, a mosphe ic d ag, among o he ac o s, p o iding a
mo e accu a e p edic ion o he sa elli e o bi (Ba e, Muelle , & Whi e,
1971; Mon enb uck, Gill, & Lu ze, 2002; Vallado, 2001). In u n, hey
equi e he inpu o de ailed in o ma ion abou he sa elli e’s p ope ies,
such as i s mass, a ea, and o he s.
In his con ex , Machine Lea ning eme ges as a powe ul ool o
add ess his challenge: assis in p o iding a solu ion d ama ically as e
han HPOP, and wi h a signi ican imp o emen in accu acy wi h
espec o SGDP4. Ad ances in Machine Lea ning algo i hms and he
a ailabili y o la ge olumes o da a ha e he po en ial o he de elop-
men o models ha signi ican ly imp o e some adi ional o ecas ing
me hods (Lam e al., 2023; Xie, Yao, Li, Wang, Zheng, & Chen, 2024).
Addi ionally, Machine Lea ning models can con inuously imp o e as
mo e da a becomes a ailable, ensu ing ha o bi p opaga ion me hods
s ay cu en and e ec i e.
To imp o e he accu acy and e iciency o o bi p opaga o s, some
au ho s ha e de eloped hyb id models, which combine a dynamic
model wi h a machine lea ning me hod. Fo sequen ial sa elli e posi-
ion da a, app oaches such as linea eg ession and Long Sho -Te m
Memo y (LSTM) neu al ne wo ks (Ren e al., 2019), o combina ions
o au oencode s and andom o es s (Liu, Ta low, Akba , Donnellan,
& Senkow, 2021), ha e been p oposed. In cases whe e da a lacks
a sequen ial na u e, al e na i e me hods such as dense neu al ne -
wo ks (San-Juana, Pé ezb, Ve ga ac, San Ma ınd, Lópeze, Wi ig ,
& Izzog, 2018), suppo ec o machines (Peng & Bai, 2017) o a
combina ion o PCA and XGBoos (Zhai, Huyan, Hu, Jiang, & Li, 2022).
Despi e hese ad ancemen s, hey only es speci ic con igu a ions,
missing he chance o s udy he in luence o ce ain a chi ec u es
o da a o e he ou come o he models. Addi ionally, hese wo ks
end o ocus solely on sa elli e-speci ic da a, excluding he in eg a ion
o ex e nal con ex ual in o ma ion, such as en i onmen al ac o s o
ex e nal o ces, which could u he e ine p edic ions.
Among he di e en Machine Lea ning echniques, neu al ne wo ks
s and ou o hei abili y o model complex non-linea ela ionships in
da a, making hem pa icula ly sui able o p edic ion and classi ica ion
asks. Neu al ne wo ks ha e been shown o ou pe o m o he Machine
Lea ning algo i hms in e ms o accu acy and e iciency in ce ain
con ex s (Ayala, Bo ego, He nández, & Ruiz, 2020; Ayala, He nández,
Ruiz, & To o, 2019; LeCun, Bengio, & Hin on, 2015; Schmidhube ,
2015).
Fo his eason, his wo k ocuses on explo ing how neu al ne wo ks
can be applied in conjunc ion wi h simpli ied pe u ba ion models,
de eloping a hyb id model wi h SGDP4 o op imize o bi al p edic ion
as illus a ed in Fig. 1, educing e o s in p edic ed ajec o ies while
keeping a low p edic ion ime. Th oughou his wo k, we p esen
a ious models designed o add ess his speci ic ask, e alua ing he
pe o mance o each one compa ed o he e e ence SGDP4 model and
analysing i s impac on accu acy. Ou mos signi ican con ibu ion is
he s udy o he impac o including explici ea u es ela ed o he s a e
o he celes ial bodies ha may in luence he sa elli e ajec o y. Ou
wo king hypo hesis is ha in eg a ing hese no el ea u es is c ucial o
he de elopmen o mo e e ec i e p edic ion models.
Fig. 1. Use o a model o e ine o bi p opaga ion. The ou pu o he model co esponds
o he g een a ow.
To achie e his goal, we c ea ed a da ase wi h da a om Kosmos
2514, a sa elli e om he GLONASS cons ella ion. Then, we de eloped
and es ed some neu al ne wo k models wi h di e en inpu da a, ind-
ing g ea e o educ ion in some o hem. Ou expe imen s show ha
he inclusion o ea u es ela ed o celes ial bodies esul s in signi ican
imp o emen s, and is key when i comes o ob aining accep able esul s
in sho - e m p edic ions, in which he baseline p edic ion p o ided by
he SGDP4 model is mo e accu a e and he e o e ha de o imp o e o
a leas no o de e io a e.
2. Rela ed wo k
Se e al au ho s ha e al eady e alua ed he ou comes o applying
machine lea ning echniques o e ine ajec o y p edic ion. These s ud-
ies highligh he po en ial o machine lea ning algo i hms o enhance
p ecision, showing imp o emen s o e only using adi ional me hods.
Ren e al. (2019) p oposed a combina ion o an LSTM neu al ne -
wo k and linea eg ession. They use LSTM laye s o p edic o bi al
inclina ion, o bi al eccen ici y and a e age displacemen , aining one
model o p edic each a iable. Besides, linea eg ession is used o
es ima e he ascending node, pe igee angula dis ance and nea poin
angle. A e p edic ing he six elemen s o he TLE o bi , he posi ion o
he spacec a can be ob ained by a simple ans o ma ion ha allows
calcula ing he e o . The inpu s o he six models a e he same: he
six o bi al elemen s, he imes amp o which hese elemen s co espond
and he ime in e al o he p edic ion. In hese LSTM ne wo ks, each
o he six o bi al pa ame e s is p o ided as a i e-elemen sequence. To
ain hese models, da a om he IRIDIUM 118 sa elli e1 was used. The
da a was di ided in o aining and es ing se s, wi h 80% o aining
and 20% o es ing. The au ho s conside he p edic ed posi ion e o
o be wi hin an accep able ange and i s a ia ion o s ay ela i ely
s able agains he p og ess o ime.
Peng and Bai (2017) explo ed he iabili y o employing a Suppo
Vec o Machine (SVM) eg ession model o o ecas he e o in an
assumed dynamical model o he ENVISAT sa elli e.2 Ini ially, hey
model h ee s a ions o gene a e disc e e measu emen s, when he
a ge sa elli e is isible o hem, acco ding o a ‘‘ u h’’ dynamical
model. Then hey apply leas squa es es ima ion o ge he s a e o he
sa elli e. A e ob aining es ima ions o all he acks, he p edic ion
p ocess is s aigh o wa d, he SVM model is asked wi h p edic ing he
e o in his o ecas o e ine he model es ima es. The inpu s o he
SVM model a e he du a ion o he p edic ion; posi ion and eloci y
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Expe Sys ems Wi h Applica ions 281 (2025) 127453
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a he cu en epoch; es ima ed d ag coe icien a he cu en epoch;
maximal measu ed ele a ion in he cu en ack and he co esponding
ange and azimu h; and p edic ed posi ion and eloci y. The ou pu o
he SVM is a six-componen ec o de ailing he e o in posi ion and
eloci y ac oss each axis. The conclusion d awn om hei s udy was
ha he SVM model, once ained, had limi ed applicabili y o p e-
dic ions ex ending oo a in o he u u e. The e o e, hey ad ised ha
o bi p edic ions should be made wi hin a ela i ely sho ime ame o
ensu e accu acy.
Peng and Bai (2018) la e in oduced a neu al ne wo k app oach,
employing dense neu al ne wo ks ha used iden ical inpu s om he
same sa elli e as in he p e iously desc ibed SVM model. The ou pu s o
he neu al ne wo k mi o ed hose o he SVM, wi h a no able a ia ion:
he app oach in ol ed aining a di e en model o each componen ,
esul ing in a o al o six unique models. They s udied also he abili y o
hese models o gene alize o u u e epochs and o di e en bu nea by
Residen Space Objec s (RSOs) in subsequen epochs. In hei analysis
o his second ask, he au ho s analysed he ange o lea ning a iables,
excluding any a iables whose anges exhibi ed signi ican dispa i ies
be ween he aining and es ing da ase s. Thei indings sugges ed ha
he neu al ne wo k models demons a e s ong gene aliza ion capabil-
i ies o u u e epochs. Fu he mo e, i was concluded ha hese neu al
ne wo ks could be gene alized o a ela i ely b oad spec um o nea by
RSOs no included in he aining da ase , showcasing hei e sa ili y
and po en ial o p edic i e accu acy in dynamic space en i onmen s.
Liu e al. (2021) de eloped a hyb id model, combining SGP4 wi h
au oencode s and andom o es s o educe he p opaga ion e o . Fi s ,
hey used he encode o ob ain a ep esen a ion o he dis ance e o
om SGP4. Then, a andom o es model was applied o p edic he
embedding ec o o he nex ime s ep and inally, a decode ob ained
he dis ance e o . This was done o each posi ional coo dina e (𝑥, 𝑦,
𝑧) esul ing in h ee di e en models. The inpu o he model was a
30-days ime se ies o he SGDP4 dis ance e o . The da a hey used
come om h ee objec s in low ea h o bi (LEO): a esea ch Cube-
Sa (QuakeSa by S an o d Uni e si y3), a sa elli e payload (COSMOS
20984), and a sa elli e deb is (PEGASUS DEB deb is5). These da a we e
collec ed om he Space-T ack API (SAIC, 2024). Wi h his app oach,
hey ob ained an a e age 20%–30% imp o emen on 30-days o bi
p edic ion.
Zhai e al. (2022) p oposed a combina ion o PCA and XGBoos
model o imp o e he o bi p edic ion accu acy. The inpu s o he
model a e p edic ion du a ion, p edic ed mo emen , posi ion a he
ini ial epoch, eloci y a he epoch, d ag coe icien , p edic ed posi ion
and p edic ed eloci y. The da a used o aining and e alua ion we e
sou ced om sa elli e simula ion en i onmen s. Fi s , hey ained an
XGBoos model o choose he mos app op ia e combina ion o ea-
u es. Then, based on ha pa ame e s, he PCA–XGBoos model was
ained,in which PCA was used o educe he dimensionali y o he
da a. The a ge a iable was he ue e o p edic ion, which consis s
o six elemen s, h ee axis o posi ion and ano he h ee o eloci y,
such ha a o al o six models a e ained. They concluded ha his
p oposal imp o es sa elli e p edic ion in hei simula ion en i omen
and claimed ha i s capabili y o gene aliza ion is good o all six
componen s.
Despi e he p e ious app oaches, he applica ion o machine lea n-
ing echniques o sa elli e p opaga ion has se e al unexplo ed a eas
ha p esen oppo uni ies o signi ican ad ancemen s. The impac o
some decisions ega ding he de elopmen o models is unclea . Fo
example, when Liu e al. use a model o each posi ional coo dina e,
i is unknown o wha deg ee he sepa a ion o a model in o sepa a e
coo dina es con ibu es owa ds imp o ing esul s. O he aspec s su e
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om a simila lack o insigh s, such as he consequences o e alua ing
a e inemen model ac oss di e en ime windows (sho , medium, and
long e m) o e en aining models ha ocus on a single ime window.
Bu pe haps he mos ema kable gap is he absence o models ha
inco po a e ea u es ela ed o he s a e o celes ial bodies. Fea u es
used by exis ing models a e based on he desc ip ion o he s a e o he
sa elli e a he ime o in e ence, namely i s posi ion and eloci y, and
ea u es ha a e de i ed om hese and he applica ion o he SGDP4
model such as o bi al elemen s. Howe e , he accu a e p opaga ion
o a sa elli e o bi elies on he posi ion and speed o all su ounding
celes ial bodies, which a e no included in exis ing models. These a e
also mo e challenging o p ope ly exploi , since hei posi ion changes
du ing p opaga ion ime, and he e o e he way in which hey impac
he sa elli e posi ion.
Rela ed o sa elli e ayec o y p edic ions, we encoun e ime se ies
p ocessing. The li e a u e commonly add esses he classi ica ion o
mul i a ia e ime se ies (Xiao e al., 2023). In hese cases, models lea n
disc imina i e ep esen a ions o assign a label o an en i e sequence
o empo al da a. Howe e , ou objec i e di e ges om his app oach.
Ra he han classi ying ime se ies in o disc e e labels, ou goal is o
p edic con inuous alues a a speci ic u u e ime s ep.
Ano he dominan pa adigm in he ield o machine lea ning o
ime se ies is sequence p edic ion, as seen in a ic o ecas ing mod-
els (Wu, Pan, Long, Jiang, & Zhang, 2019) o me eo ology (Pa hak
e al., 2022). In hese cases, he goal is o in e he u u e alues o he
en i e ime se ies based on p e ious obse a ions. A p ominen example
o such models is he use o T ans o me s (Vaswani, 2017), which ha e
p o en highly e ec i e in cap u ing long- e m dependencies wi hin
empo al sequences. Howe e , in ou case, we do no aim o p edic
he en i e u u e e olu ion o he sequence bu a he only he s a e o
he sa elli e a a speci ic ins an .
In summa y, while ou s udy sha es undamen al aspec s wi h exis -
ing esea ch on ime se ies, such as le e aging his o ical da a o in e
u u e alues, ou objec i e is di e en . We nei he seek o classi y
empo al sequences in o ixed ca ego ies no o model hei en i e
u u e ajec o y. Ins ead, ou ocus is on p ecise s a e es ima ion a
a pa icula u u e ins an .
3. Ou p oposal
We p opose a hyb id model ha combines he SGDP4 p opaga o
wi h a neu al ne wo k o imp o e sa elli e p opaga ion p edic ions
while le e aging no el ea u es. The model ope a es in wo s eps:
1. Ini ial P edic ion: Le 𝐩SGDP4(𝑡) deno e he posi ion p edic ion
o he sa elli e a ime 𝑡 gene a ed by he SGDP4 p opaga o .
2. E o Co ec ion: The ini ial p edic ion 𝐩SGDP4(𝑡) is used as
inpu o a neu al ne wo k , which o ecas s he e o ec o
𝐞(𝑡) made by he SGDP4 p opaga o . The co ec ed posi ion
p edic ion 𝐩co ec ed(𝑡) is hen gi en by:
𝐩co ec ed(𝑡) = 𝐩SGDP4(𝑡) + 𝐞(𝑡)(1)
By in eg a ing he adi ional SGDP4 p opaga o wi h he ad anced
capabili ies o a neu al ne wo k, ou hyb id model aims o enhance he
accu acy o sa elli e posi ion p edic ions. This app oach le e ages he
s eng hs o bo h me hods: he es ablished eliabili y o SGDP4 and he
adap i e lea ning po en ial o neu al ne wo ks. Neu al ne wo ks allow
us o eed ou model a a ie y o ea u es 𝐟 o s udy hei e ec on he
ou come, and assess which a ian s o e he bes esul s.
In he ollowing sec ions, we desc ibe he aims and scopes o ou
wo k in o de o ou line i s limi a ions, he speci ic goals o ou ex-
pe imen s, he neu al ne wo k a chi ec u es we implemen ed, and he
design o he da ase s we used in ou e alua ion.
3.1. Aims and scopes
Ou esea ch is ocused on he e alua ion o neu al ne wo k models
when applied o he p edic ion o sa elli e ajec o ies, in combina ion
Expe Sys ems Wi h Applica ions 281 (2025) 127453
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J. Calde ón e al.
Fig. 2. A chi ec u e o he Densely Connec ed Ne wo k.
wi h he adi ional p opaga o SGDP4. Ou aim is o compa e he
impac on pe o mance o di e en s a egies o assess hei p edic i e
po en ial. Pa icula ly, we aim o s udy how he addi ion o ea u es
ela ed o he s a e o celes ial bodies impac s pe o mance, since we
conside ha he ad ance o he applica ion o machine lea ning in his
ield g ea ly elies on he de elopmen o no el ways o model ele an
in o ma ion as ea u es.
Ou goal is no o eplace adi ional models and hei physical
simula ions. Ins ead, we aim o e ine and build upon he ounda ion
p o ided by adi ional models, le e aging hei obus ness and accu-
acy o in oduce p og essi e imp o emen s and insigh s in o which
s a egies a e mos e ec i e.
We ocus on he c ea ion o models o a single sa elli e, as opposed
o c ea ing a model ha , a e being ained, can be used o in e he
ajec o y o any sa elli e. The la e would equi e modelling de ailed
in o ma ion abou he physical cha ac e is ics o each sa elli e, which
is gene ally no a ailable, in o de o p ope ly make p edic ions abou
di e en sa elli es (e.g. he sec ion o he sa elli e used o measu e
he o ce o adia ion used by he HPOP model). Ou expe imen s
co espond o a use case in which he use is able o ga he da a
abou he p ecise ajec o y o a sa elli e in o de o ain a model
and enable mo e p ecise p edic ion o i s ajec o y in he u u e. By
ocusing models on a single sa elli e, he model is implici ly adjus ed
o he speci ic condi ions o ha sa elli e, elimina ing he need o know
exac ly all he pa ame e s ha in luence i s ajec o y, hus simpli ying
he model de elopmen p ocess.
We do no in end o p opose a de ini i e and closed solu ion o
ajec o y p opaga ion, bu a he o alida e he e ec i eness o ce -
ain echniques in a con olled con ex . The i s ea u es se used in
ou expe imen a ion co esponds o he ea u es used by he echniques
in he S a e o he A , se ing as a baseline ep esen ing he in o -
ma ion ypically used in exis ing p oposals, namely he posi ion and
eloci y o he sa elli e. The conclusions de i ed om ou esul s a e
o independen in e es o any echnique. Likewise, we do no seek
an exhaus i e e alua ion o complex neu al ne wo k a chi ec u es o
ex ensi e combina ions o hype pa ame e s. Ins ead, we ocus on using
simple and e ec i e a chi ec u es ha a e su icien o exploi he
in o ma ion o in e es .
3.2. Goals
Ou objec i e is o assess he e ec o conside ing di e en a chi-
ec u es wi h di e en inpu da a on neu al ne wo k pe o mance o
sa elli e p opaga ion. In pa icula , ou expe imen s aim o assess he
impac o he ollowing a iables:
1. Base Fea u es: The base ea u es used o he p edic ion o he
ajec o y, including he posi ion 𝐩(𝑡) and eloci y 𝐯(𝑡) o he
sa elli e, as well as he SGDP4 p edic ion 𝐩SGDP4(𝑡).
2. Celes ial Bodies a Ini ial Epoch: The inclusion o ea u es e-
la ed o celes ial bodies a he ini ial epoch 𝑡0, and he inclusion
o celes ial bodies beyond he mos in luen ial ones.
3. Sequence o Celes ial Bodies: The inclusion o ea u es ela ed
o celes ial bodies in a sequence o posi ions om he ini ial
epoch 𝑡0 o he in e ence epoch 𝑡. Speci ically, we eed he
models a sequence o ixed leng h 10, co esponding o e enly
dis ibu ed ime poin s be ween 𝑡0 and 𝑡.
4. Sepa a e Models o Coo dina es: The c ea ion o a sepa a e
model o each p edic ed coo dina e, as opposed o a model ha
p edic s all coo dina es simul aneously, eplacing wi h 𝑥,
𝑦, and 𝑧 o he 𝑥, 𝑦, and 𝑧 coo dina es, espec i ely.
5. Time F ame Speci ic Models: The c ea ion o a model o single
ime ames, as opposed o a gene ic model applicable o any
ime ame, eplacing wi h 0.5ℎ, 1ℎ, 2ℎ and so on.
3.3. Ne wo k a chi ec u e
Nex , we desc ibe he neu al ne wo k a chi ec u es used in ou
models. In pa icula , we implemen ed h ee di e en ne wo ks: a
simple one o p ocessing s a ic ea u es, a second one wi h GRU (Ga ed
Recu en Uni ) laye s (Chung, Gulceh e, Cho, & Bengio, 2014) and a
hi d a chi ec u e combining a T ans o me encode wi h GRU laye s
o p ocess he sequences o ea u es ela ed o celes ial bodies.
The i s model is a densely connec ed ne wo k (DCN), also known
as a ully connec ed ne wo k, which p ima ily uses dense laye s. These
laye s connec e e y inpu ea u e o e e y ou pu ea u e wi h a unique
weigh o each connec ion. They a e pa icula ly use ul o asks in
which he ea u es do no ha e an inhe en o de o spa ial ela ion-
ships be ween hem, as is he case wi h mos o he ea u es we use,
such as he p edic ion p o ided by he SGDP4 model, o he sa elli e
posi ion and eloci y a he ini ial epoch. This model, depic ed in Fig.
2, con ains 7 in e media y dense laye s o sizes 128, 128, 256, 256,
128, 128, 64. Each laye is ollowed by a ReLU ac i a ion laye , and
a skip connec ion is added a e e e y laye wi h he same ou pu size
as he p e ious one. Finally, he ou pu laye has size 3 co esponding
o he h ee p edic ed coo dina es, o size 1 o he cases in which an
independen model is ained o each coo dina e.
The use o esidual skip connec ions was essen ial, specially in
models ha ecei ed sequen ial da a. This change in he a chi ec u e
led o big imp o emen s in he p edic ion pe o mance, educing he
e o in 80% app oxima ely.
The second model adds GRU laye s. This model uses as i s main
componen one o said laye s, which a e designed o consume se-
quences o da a and p oduce an ou pu o each elemen in he se-
quence. The ou pu co esponding o he las elemen o he sequence
will be in luenced by he en i e sequence, making GRU laye s ideal
o p ocess sequen ial da a (Cho e al., 2014). GRU ne wo ks ha e
been applied o all kinds o p oblems in which he inpu con ains
some kind o sequence, such as elec oca diog am classi ica ion (Lynn,
Pan, & Kim, 2019) o sho - e m powe load o ecas ing (Zheng e al.,
2018). Compa ed o o he ecu en neu al ne wo ks, GRU ne wo ks
a e simple and as e o ain hanks o hei inne simplici y, ha ing
po en ial o as e con e gence in cases whe e he inpu sequence is
no e y long. In ou model, he inpu sequence has a ixed leng h o
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Fig. 3. A chi ec u e o he GRU ne wo k.
10, co esponding o he 10 e enly dis ibu ed ime poin s be ween
he ini ial and in e ence epochs. In o mal es s con i med ha he
pe o mance o GRU laye s was sligh ly be e han ha o popula
LSTM laye s.
As shown in Fig. 3, ou GRU ne wo k handles wo ypes o inpu s:
sequen ial and non-sequen ial da a. Fo he sequen ial da a, we de ine
an inpu laye wi h a shape ha ma ches he sequence leng h (10) and
he numbe o sequen ial ea u es. This inpu is p ocessed by a GRU
laye wi h 64 uni s using he anh ac i a ion unc ion. Fo he non-
sequen ial da a, we de ine ano he inpu laye wi h a shape ma ching
he numbe o non-sequen ial ea u es. This inpu is p ocessed by a
dense laye wi h 64 uni s. The ou pu s om he GRU laye and he
dense laye a e conca ena ed o o m a combined inpu . This combined
inpu is hen passed h ough a se ies o laye s equi alen o laye s om
he i s model, s a ing a laye s o size 258.
The hi d model inco po a es a T ans o me encode , a mecha-
nism known o i s success in handling sequen ial da a ac oss a ious
domains. T ans o me s ely on sel -a en ion mechanisms o cap u e
ela ionships be ween all elemen s in a sequence, ega dless o hei
dis ance, o e coming he limi a ions o ecu ence-based models like
GRU o LSTM, which s uggle wi h long- ange dependencies. T ans-
o me encode s, speci ically, ocus on encoding inpu sequences in o
a ich, con ex -awa e ep esen a ion by a ending o e e y elemen in
he sequence and weighing hei ele ance (Vaswani, 2017).
Ou T ans o me -Based model, shown in Fig. 4, handles wo ypes
o inpu s: sequen ial and non-sequen ial da a. Fo he sequen ial da a,
we de ine an encode o p ocess sequences o leng h 10. Fi s , we
conca ena e posi ional encoding, which a y acco ding o he posi ion
in he sequence, and celes ial body encoding, which a e ainable and
a y acco ding o he celes ial body. A e ha , we inco po a e mul i-
head a en ion o p ocessing he sequence wi h he encodings. This
a en ion mechanism uses 3 heads wi h an embedding dimension o 16,
helping he ne wo k cap u e he spa ial and con ex ual ela ionships in
he da a.
The ou pu o he a en ion mechanism is ollowed by a skip connec-
ion wi h he same ou pu size as i s inpu and a no maliza ion laye .
A e ha comes wo dense laye s wi h 13 and 27 neu ons and ReLU
ac i a ions, espec i ely. The ou pu o his second laye is ollowed
by a skip connec ion wi h he inpu o he i s dense laye and a
no maliza ion laye .
A e he encode , he a chi ec u e aligns wi h ha o a GRU ne -
wo k. Howe e , he numbe o uni s in he GRU laye has been in-
c eased o 256.
3.4. Compu a ional complexi y
3.4.1. Densely connec ed ne wo k
Le us deno e by 𝐿 he o al numbe o dense laye s and by 𝑛𝑖 he
numbe o uni s in laye 𝑖, wi h 𝑛0 being he numbe o inpu ea u es.
The co e ope a ion in each laye is he ma ix mul iplica ion be ween
he inpu o size 𝑛𝑖−1 and he laye ’s ainable weigh s o size 𝑛𝑖−1 ×𝑛𝑖.
Hence, he o wa d pass o laye 𝑖 cos s (𝑛𝑖𝑛𝑖−1). Summing ac oss all
dense laye s yields:
(𝐿
∑
𝑖=1
𝑛𝑖𝑛𝑖−1).
Residual o skip connec ions p ima ily add elemen -wise ope a ions,
which emain (𝑛𝑖) pe skip link and do no al e he o al complexi y.
The memo y oo p in is de e mined by he ainable pa ame e s
inside hese dense laye s, which o al ∑𝐿
𝑖=1(𝑛𝑖𝑛𝑖−1 +𝑛𝑖) including bi-
ases. The e o e, he dense a chi ec u e is well-sui ed when he inpu
dimensionali y is mode a e, as is he case.
3.4.2. GRU-based model
In his a chi ec u e, le 𝑇 be he sequence leng h, 𝑑 he dimension o
each ime s ep’s inpu , and ℎ he numbe o hidden uni s pe GRU laye .
A GRU laye p ocesses each ime s ep sequen ially, mixing he cu en
inpu wi h he p e ious hidden s a e. Each s ep equi es mul iple ma ix
mul iplica ions o size 𝑑×ℎ and ℎ×ℎ, esul ing in a pe -s ep complexi y
o ((𝑑+ℎ)ℎ). O e 𝑇 ime s eps, his yields:
(𝑇(𝑑+ℎ)ℎ).
The ollowing dense laye s add an addi ional cos o (∑𝐿
𝑖=1 𝑛𝑖𝑛𝑖−1).
Thus, he compu a ional complexi y emains p opo ional o 𝑇, 𝑑,
and ℎ, and can be exp essed as:
(𝑇(𝑑+ℎ)ℎ+
𝐿
∑
𝑖=1
𝑛𝑖𝑛𝑖−1).
In e ms o pa ame e coun , a single GRU laye has on he o de o
3ℎ(ℎ+𝑑) ainable pa ame e s (including biases), because each ga e in
he GRU cell has i s own se o weigh s.
O e all, o sho sequences o smalle hidden dimensions, conca e-
na ing he GRU ou pu wi h non-sequen ial dense componen s adds
mode a e o e head. In ou implemen a ion, 𝑑= 3, 𝑇= 10, and ℎ= 64,
which should esul in easonable aining imes.
3.4.3. T ans o me -based model
In his a chi ec u e, a classical T ans o me encode is applied be-
o e he GRU laye . Le 𝐸 be he dimension o each ime s ep’s inpu
a e he addi ion o posi ional and celes ial body embeddings, 𝐻 he
numbe o a en ion heads, and 𝑑 he desi ed size o he que y, key,
and alue p ojec ions.
The addi ion o embeddings has a complexi y o (𝑇 𝐸).
The T ans o me encode consis s o mul i-head sel -a en ion mech-
anisms and eed- o wa d neu al ne wo ks. The sel -a en ion mecha-
nism in ol es p ojec ing he inpu ea u es in o que y, key, and alue
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Fig. 4. A chi ec u e o he T ans o me -Based Ne wo k.
ec o s, each wi h a complexi y o (𝑇 𝐸𝑑). The sel -a en ion mecha-
nism i sel has a complexi y o (𝑇2𝑑) pe head, and wi h 𝐻 heads,
his becomes (𝐻𝑇 2𝑑). The eed- o wa d ne wo k adds a complexi y
o (𝑇 𝑑2).
The o al complexi y o he T ans o me encode laye is:
(𝑇 𝐸𝑑 +𝐻𝑇 2𝑑+𝑇 𝑑2).
A e he T ans o me encode , he GRU laye p ocesses he se-
quence wi h a complexi y o (𝑇(𝑑+ℎ)ℎ). The ollowing dense laye s
add an addi ional cos o (∑𝐿
𝑖=1 𝑛𝑖𝑛𝑖−1).
The e o e, he o e all compu a ional complexi y o he T ans o me -
based model is:
(𝑇(𝐸𝑑 +𝐻𝑇 𝑑 +𝑑2+ (𝑑+ℎ)ℎ) +
𝐿
∑
𝑖=1
𝑛𝑖𝑛𝑖−1)
The T ans o me encode has on he o de o 𝐻𝐸𝑑 pa ame e s
o he a en ion mechanism and 2𝑑2 pa ame e s o he eed- o wa d
ne wo k pe laye . The GRU laye has 3ℎ(ℎ+𝑑) ainable pa ame e s,
including biases.
In ou implemen a ion, 𝑑= 16, 𝑇= 10, ℎ= 256, 𝐸= 3 + 16 + 8 = 27,
and 𝐻= 3. These alues we e chosen acco ding o in o mal es s in
which we ound ha hey p o ided he bes pe o mance.
3.5. Da ase design
Fo he c ea ion o he da ase we used da a om he RINEX
eposi o y o he Ins i u e o S a is ics and Ca og aphy o Andalusia
(IECA) (Ins i u o de Es adís ica y Ca og a ía de Andalucía, 2024),
speci ically om he Có doba s a ion (CRDB). The da a consis ed o
RINEX (Recei e Independen Exchange Fo ma ) iles om GPS and
GLONASS sa elli es co e ing he pe iod om 2019 o 2022. RINEX
iles a e a s anda d da a o ma used o s o ing and exchanging GNSS
(Global Na iga ion Sa elli e Sys em) da a. De eloped o acili a e he
exchange o GNSS da a be ween di e en ecei e manu ac u e s and
p ocessing so wa e, RINEX iles p o ide posi ions and eloci ies o
sa elli es.
Fo he ea men o he da a and he applica ion o o bi al p opaga-
o s we used as eRisk (Ayala e al., 2023), an R package o compu a-
ion o sa elli e posi ions. This package p o ides ools o p ecise o bi al
mechanics calcula ions, ensu ing accu a e de e mina ion o sa elli e
ajec o ies.
The Kosmos 2514 sa elli e om he GLONASS cons ella ion p o-
ided he mos ex ensi e da a (numbe o messages de ailing he posi-
ion and eloci y a a gi en epoch). Using his in o ma ion, we c ea ed
an auxilia y da ase , which con ained posi ions, eloci ies and da es o
each a ailable ins an .
We selec ed a lis o p edic ion ime windows o ou s udy, includ-
ing 30 min, 1 h, 2 h, 5 h, 10 h, 24 h, 48 h, 120 h, 240 h, and 720 h.
These in e als ep esen ed he a ious imes a which we aimed o
make p edic ions and ain and es ou model, co e ing a wide ange
o magni udes anging om hal an hou o 30 days.
To c ea e he inal da ase , we s a ed by selec ing andom da es
om he auxilia y da ase . Fo each chosen da e and each p edic ion
ime, we gene a ed a new ow in he da ase . These new ows include:
he selec ed ini ial epoch, he ue posi ion and eloci y o he sa elli e
a ha epoch, he ue posi ion and eloci y a he p edic ion epoch,
and he posi ion and eloci y p edic ed by he SGDP4 me hod o ha
p edic ion epoch.
Fo ou SGDP4 p edic ion, we i s needed o con e he posi ions
and eloci ies om RINEX iles in o he TLE (Two-Line Elemen Se )
o ma Vallado and Ce ola (2012), which is equi ed by he SGDP4
model. The TLE o ma is a s anda dized o ma used o ep esen
o bi al elemen s o sa elli es.
In addi ion o he o bi al elemen s, we also had o p o ide he 𝐵∗
d ag coe icien . The 𝐵∗ coe icien is a pa ame e ha encapsula es he
e ec s o a mosphe ic d ag on he sa elli e o bi . Howe e , calcula ing
he 𝐵∗ d ag coe icien equi es speci ic da a abou he shape o he
sa elli e and i s mass ha is no a ailable. Gi en his limi a ion and he
ac ha he 𝐵∗ alue is ypically e y small, we decided o se i o 0
o all o ou examples. This simpli ica ion allowed us o p oceed wi h
he p edic ions using he a ailable da a.
Du ing his p ocess, he e we e some cases in which he e we e no
sa elli e da a a ailable o ce ain p edic ion epochs. To add ess his
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Fig. 5. Da ase samples collec ion. Black do s ep esen epochs a which posi ional
in o ma ion is a ailable. A ows ep esen samples wi h di e en desi ed p edic ion
imes. I he e is no da a o a desi ed p edic ion ime, he nea es poin can be used
ins ead.
Table 2
Numbe o samples in he da ase o each ime band.
P edic ion ime Samples
0.5 h 3929
1 h 3557
2 h 3231
5 h 2399
10 h 3405
24 h 3890
48 h 3918
120 h 3914
240 h 3917
720 h 3839
issue, we implemen ed a h eshold o selec a ime close o he in ended
p edic ion ime. Fo ins ance, i da a we e no a ailable o a p edic ion
ime o 5 h, we allowed he selec ion o da a om a nea by poin , such
as 5.5 h. This app oach ensu ed ha he p edic ion epoch was nea he
in ended ime windows. This is exempli ied in Fig. 5.
A e c ea ing he da ase , we inco po a ed da a om a ious celes-
ial bodies wi hin ou sola sys em. This da ase includes he posi ions
and eloci ies o he Sun, he Moon, Me cu y, Venus, Ma s, Jupi e ,
Sa u n, U anus, Nep une, and Plu o, all e e enced ela i e o Ea h. Ad-
di ionally, we ga he ed in o ma ion on luna lib a ion, which desc ibes
he oscilla ing mo ion o he Moon as obse ed om Ea h.
These celes ial bodies we e chosen by i ue o being he majo
g a i a- ional o ces ha should in luence he ajec o y o a sa elli e.
No e ha o he celes ial bodies, such as as e oids o come s, we e no
included in ou da ase . This decision was made due o he negligible
e ec ha he g a i a ional in luence exe ed by such objec s has in
he ajec o y o a i icial sa elli es in he imescales conside ed in his
s udy.
We we e able o acqui e hese da a h ough he JPL DE440 model
(Pa k, Folkne , Williams, & Boggs, 2021), de eloped by NASA. The JPL
DE440 is a highly ad anced model ha enables quick and accu a e
calcula ions o he posi ions and eloci ies o celes ial bodies, as well
as o he ele an as onomical da a.
We added celes ial da a o ou da ase in wo di e en e sions. In
he i s e sion, we inco po a ed he da a a he ini ial epoch o each
ow. In he second e sion, we included he celes ial da a in sequences
o leng h 10, e enly dis ibu ed be ween he ini ial and p edic ion
epochs. This cap u es he p og ession o celes ial condi ions o e he
en i e p edic ion pe iod, du ing which he plane s exe g a i a ional
o ce on he sa elli e.
The inal da ase has a o al o 36000 samples, dis ibu ed as shown
in Table 2.
4. Expe imen al esul s
Fo aining all he models in ou expe imen s, he hold-ou ech-
nique was used o da a spli ing. The da a was di ided equi ably,
wi h 50% assigned o he aining se and he emaining 50% o he
es se . The Adam op imize was selec ed o dynamically adjus he
lea ning a es and he Mean Squa ed E o (MSE) loss unc ion was
used o e alua e he model du ing aining. Addi ionally, a callback
was implemen ed o au oma ically sa e he bes model based on he
Mean Absolu e E o (MAE) me ic, ensu ing he selec ion o he mos
accu a e model du ing he aining p ocess.
Ma hema ically, he MSE loss unc ion is de ined as:
MSE =1
𝑛
𝑛
∑
𝑖=1
(𝐲𝑖−
𝐲𝑖)2(2)
whe e 𝐲𝑖 is he ue alue and
𝐲𝑖 is he p edic ed alue.
The MAE me ic is de ined as:
MAE =1
𝑛
𝑛
∑
𝑖=1 |𝐲𝑖−
𝐲𝑖|(3)
To ob ain a eliable measu e o model pe o mance, each model
unde wen 10 aining sessions, each wi h a di e en andom seed,
anging om seed 667 o 676. The mean e o was calcula ed ac oss
hese 10 sessions, aking in o accoun he SGDP4 co ec ion in all
models.
All ou expe imen s we e conduc ed on a compu e equipped wi h
an In el Co e i9-9900K CPU, 64 GB o DDR4 RAM and an N idia
RTX 3080-Ti GPU. Thei code is a ailable on h ps://gi hub.com/DEAL-
US/sa elli e-p opaga ion- e inemen
When we indica e ha a model ecei es inpu s like posi ion, dis-
placemen , o eloci y, we e e o he h ee coo dina es o componen s
o he co esponding a iable (𝑋, 𝑌, and 𝑍). This app oach is simila ly
applied o o he magni udes, such as luna lib a ion. Da es a e p o ided
in UNIX o ma o consis ency and ease o p ocessing.
All models in he expe imen ecei e he ollowing common inpu
ea u es: he ini ial epoch 𝑡0 , he posi ion 𝐩(𝑡0) o he sa elli e a ha
ins an , and he epoch 𝑡 on which he p edic ion is o be made. Thus,
hese inpu s a e e e ed o as he base ea u es.
We compa e he di e en models agains he SGDP4 baseline, since
ou pu pose is o imp o e he esul s ob ained by said model. The ull
esul s o ou expe imen s can be ound a Appendix, in which he
a e age o each model we de eloped is included.
Nex , we p esen he esul s o ou expe imen s in a p og essi e
manne , s a ing wi h he simples models and g adually in oducing
mo e complex models wi h addi ional ea u es. This allows us o iden-
i y he impac o di e en ea u es and addi ions, se ing as an abla ion
s udy o unde s and he impo ance o each componen .
The expe imen a ion began wi h an expe imen on he DCN (Densely
Connec ed Ne wo k) models, e alua ing he impac o inco po a ing
eloci y 𝐯(𝑡) as pa o he inpu da a and whe he i was mo e e ec i e
o use he p edic ion o he SGDP4 model as an absolu e posi ion
𝐩SGDP4(𝑡) o as he ela i e mo emen 𝛥𝐩SGDP4(𝑡) = 𝐩SGDP4(𝑡) − 𝐩(𝑡0).
The aining pa ame e s con igu a ion was 100 epochs and a ba ch size
o 128.
We also es ed he pe o mance o h ee exis ing echniques in he
S a e o he A , pa icula ly hose by Peng and Bai (2017), Peng and
Bai (2018), and Zhai e al. (2022).
Fig. 6 shows he esul s o his expe imen . I is om he 48-hou
p edic ions onwa ds ha some models s a o ou pe o m SGDP4, wi h
he imp o emen being e y clea in he 720-hou (30-day) ime win-
dow. This ep esen s a signi ican ad ance o long- e m p edic ions,
which should be easie o he model, since a ha poin p edic ions by
he SGDP4 model de ia e la gely. A smalle ime windows, i is ha d
o he model o gene alize, esul ing in a dis o ed p edic ion wi h a
much highe e o .
The inclusion o he SGDP4 p edic ion as an absolu e posi ion
𝐩SGDP4(𝑡) esul s in mino imp o emen a some ime windows (2 h, and
24 h onwa ds), while he inclusion o he ini ial eloci y 𝐯(𝑡0) seems
o be i ele an . Howe e , in subsequen expe imen s using addi ional
ea u es, he use o he p edic ed mo emen 𝛥𝐩SGDP4(𝑡) esul ed in a
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J. Calde ón e al.
Fig. 6. Spa ial e o in base models and s a e-o - he-a p oposals. ‘‘mo ’’ deno es
he inclusion as a ea u e o he SGDP4 p edic ed posi ion as a di e en ial o
posi ion (𝛥𝐩SGDP4(𝑡)). ‘‘pos’’ deno es he inclusion o he o me as an absolu e posi ion
(𝐩SGDP4(𝑡)). ‘‘ el’’ deno es he inclusion o he eloci y o he sa elli e a he ini ial
epoch (𝐯(𝑡0)). ‘‘s m’’ deno es he echnique by Peng and Bai (2017). ‘‘ann’’ deno es he
echnique by Peng and Bai (2018).‘‘pca + xgboos ’’ deno es he echnique by Zhai e al.
(2022).
Fig. 7. Spa ial e o in celes ial da a models. ‘‘smj’’ deno es he inclusion as ea u es
o he posi ion o he Sun, Moon, and Jupi e a he ini ial epoch. ‘‘s mmj’’ deno es
he inclusion o he posi ion o he Sun, Venus, Moon, Ma s, and Jupy e . ‘‘plane s’’
deno es he inclusion o he posi ion o all Sola Sys em plane s (including Plu o). ‘‘lib’’
deno es he inclusion o he luna lib a ion.
sligh ly be e a e age. The e o e, om now on we epo on he esul s
ob ained wi h he ‘‘mo ’’ ea u es.
Ou o he echniques in he li e a u e, Peng and Bai (2017) ob ains
he bes esul s, bu s ill achie es wo se pe o mance han ou models.
Nex , we pe o m he same expe imen s, bu adding as ea u es
di e en combina ions o he posi ion o he bodies in he sola sys em
and he luna lib a ion, main aining he aining con igu a ion.
Fig. 7 shows he esul s o his expe imen . The inclusion o hese
ea u es leads o a signi ican imp o emen o he esul s. Now, we can
obse e some imp o emen s a ing a he 24 h ime windows, as well
as be e o e all esul s. Howe e , sho - e m p edic ions a e s ill wo se
han he baseline.
Fig. 8. Spa ial e o in sequen ial celes ial da a models. ‘‘g u(...)’’ deno es he inclusion
o a se o ea u es as a sequence ha is ed o GRU laye s o he ne wo k.
Fig. 9. Spa ial e o in sepa a e coo dina e models. ‘‘(XYZ)’’ deno es he use o h ee
independen models o he p edic ion o he h ee a ge coo dina es.
I is no ewo hy ha he inclusion o ea u es abou addi ional
celes ial bodies does no esul in be e esul s. On he con a y, he
simples se o ea u es (‘‘smj + mo ’’) yields sligh ly be e esul s a
sho e ime windows.
Subsequen ly, Residual GRU was implemen ed, o include he same
ea u es while expanding hose ela ed o celes ial bodies o a sequence
o en poin s. The numbe o epochs was adjus ed o 300 in o de o
accoun o he inc eased complexi y o he ne wo k.
Fig. 8 shows he esul s o his expe imen . While he ea u es we e
he same, hei inclusion as an expanded sequence esul s in signi ican
imp o emen s. In he sho - e m, said imp o emen allows he dis o -
ion in oduced by he neu al ne wo k o be ela i ely minimal, while
in he long- e m he e is a subs an ial imp o emen . The SGDP4 model
is now imp o ed a he 10 h ime window.
In e es ingly enough, he use o ea u es in ol ing addi ional celes-
ial models (‘‘s mmj’’) now esul s in a sligh ly be e median.
In he nex expe imen , we es a a ian o he models in which
coo dina e sepa a ion was pe o med by di iding he p edic ions in o
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J. Calde ón e al.
Fig. 10. Example o e olu ion o spa ial e o o a single s a ing posi ion.
h ee dis inc componen s, esul ing in he c ea ion o h ee models
ins ead o one: 𝑥, 𝑦, and 𝑧.
This p ocess was ca ied ou o he models ha showed he bes
pe o mance, which we e all o he Residual GRU a chi ec u e. A i s
glance, he models may pe o m simila ly. Howe e , he loss om one
coo dina e migh nega i ely impac he o he s. This can lead o wo se
o e all pe o mance, as e o s in p edic ing one coo dina e can a ec
he p edic ions o he o he s.
The e o e, we es i aining a sepa a e model o each coo dina e,
despi e being mo e cos ly, imp o es he p edic ion esul s. By ha ing
indi idual models, each one can ocus on he speci ic cha ac e is ics
and pa e ns o i s coo dina e wi hou in e e ence om o he s.
Fig. 9 shows he esul s o his expe imen . The imp o emen in
p edic ions is p esen a e e y p edic ion ime. This indica es ha
sepa a ing coo dina es has a posi i e impac on bo h sho - e m and
long- e m p edic ion accu acy. No e ha his addi ion leads o he i s
models ha consis en ly do no wo sen he p edic ion o he SGDP4
model a sho - e m, bu achie e an equi alen o be e median a
e e y ime window.
Fig. 10 shows an example in which he sa elli e o an o bi is
p opaga ed om a andom single poin , displaying he spa ial e o
in ime o he SGDP4 model and he co ec ion wi h he bes neu al
ne wo k e inemen . The model is, howe e , s ill no o much use o
p edic ions below he 10 h ime window. I is ou hypo hesis ha he
gene al aining leads he model o ocus on he longe p edic ion imes
ha cause a highe loss. The e o e, we explo e aining models o
each ime window ha may lead o be e pe o mance compa ed o
a gene al app oach.
Based on he success o he inco po a ion o a GRU laye we im-
plemen ed a T ans o me -Based a chi ec u e, inco po a ing an encode
and main aining he same ea u es as in he las model. The numbe o
epochs was adjus ed o 200.
Fig. 11 shows he esul s o his expe imen . Al hough he ea u es
emained unchanged, he inco po a ion o he encode signi ican ly
inc eased bo h he aining and he execu ion ime. Howe e , his
modi ica ion did no enhance he p edic i e pe o mance compa ed o
he p e ious GRU model.
The a chi ec u e showing he bes accu acy was g u(s mmj) +
mo + el (XYZ). Based on his a chi ec u e, 10 models we e ained
co esponding o each ime window. In his app oach, o a speci ic
p edic ion ime, he nea es specialized model is used, such as applying
he 10-hou window model o an 8-hou p edic ion.
Fig. 12 shows he esul s o his expe imen . The di e ence o
pe o mance is d ama ic a he sho - e m, whe e i has achie ed be e
esul s ac oss all ime windows. A lowe imp o emen can be obse ed
a he 2 h ime window, which may be caused by some cyclical
as onomic ac o .
In he medium and long e m, he new model also ou pe o ms
he SGDP4 baseline. Howe e , i is impo an o no e ha i does no
su pass i s equi alen gene al e sion, p obably because o he p esence
o mo e aining da a when no es ic ing he model o a single ime
window.
Rega ding compu a ional e iciency, he aining o he bes model,
g u(s mmj) + mo + el (XYZ), ook app oxima ely 27 min, and he
specialized models ook a ound 2.3 min each. Conside ing ha he
model ac ually equi es he aining o h ee sepa a e models, his is a
easonable ime ha would allow he use o ain wi h a bigge da ase
o o explo e model hype pa ame e s i mo e complexi y is needed o
he model. I he use is in e es ed in u he educing he aining ime,
apa om adi ional gene ic echniques, a lowe numbe o poin s o
he sequence o celes ial bodies could be used.
5. Conclusions and u u e wo k
We ha e p esen ed a ho ough s udy on he applica ion o neu al
ne wo ks o sa elli e p opaga ion, speci ically o e ine he p edic ions
p o ided by he SGDP4 model. While exis ing li e a u e con ains a
a ie y o machine lea ning echniques applied o his domain, ou
esea ch dis inguishes i sel h ough a numbe o no el con ibu ions
ha allow he educ ion o he e o in he p opaga ion by an o de
o magni ude while. An exhaus i e e alua ion o di e en a ian s has
enabled us o p ope ly s udy he con ibu ion o se e al design choices,
such as he c ea ion o dis inc models o each coo dina e.
Fig. 11. Spa ial e o in T ans o me -Based models.
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