Compu a ional and Applied Ma hema ics (2025) 44:308
h ps://doi.o g/10.1007/s40314-025-03273-0
Residual-based da a-d i en a ia ional mul iscale educed
o de models o pa ame e -dependen p oblems
Bi gul Koc1·Samuele Rubino2·Tomás Chacón Rebollo2·T aian Iliescu3
Recei ed: 5 Janua y 2025 / Re ised: 22 Ap il 2025 / Accep ed: 16 May 2025
© The Au ho (s) 2025
Abs ac
In his pape , we p opose a no el esidual-based da a-d i en closu e s a egy o educed o de
models (ROMs) o unde - esol ed, con ec ion-domina ed p oblems. The new ROM closu e
model is cons uc ed in a a ia ional mul iscale (VMS) amewo k by using he a ailable ull
o de model da a and a model o m ansa z ha depends on he ROM esidual. We emphasize
ha his closu e modeling s a egy is undamen ally di e en om he cu en da a-d i en
ROM closu es, which gene ally depend on he ROM coe icien s. We in es iga e he new
esidual-based da a-d i en VMS ROM closu e s a egy in he nume ical simula ion o h ee
es p oblems: (i) a one-dimensional pa ame e -dependen ad ec ion-di usion p oblem; (ii)
a wo-dimensional ime-dependen ad ec ion-di usion- eac ion p oblem wi h a small di u-
sion coe icien (ε=1e−4); and (iii) a wo-dimensional low pas a cylinde a Reynolds num-
be Re =1000. Ou nume ical in es iga ion shows ha he new esidual-based da a-d i en
VMS-ROM is mo e accu a e han he s anda d coe icien -based da a-d i en VMS-ROM.
Keywo ds Reduced o de models ·Va ia ional mul iscale ·Da a-d i en modeling ·Residual
Ma hema ics Subjec Classi ica ion 65M60 ·76-04
1 In oduc ion
Reduced o de models (ROMs) ha e been ins umen al in signi ican ly educing he com-
pu a ional cos o ull o de models (FOMs) (e.g., ini e elemen o ini e olume me hods)
BBi gul Koc
[email p o ec ed]
Samuele Rubino
[email p o ec ed]
Tomás Chacón Rebollo
[email p o ec ed]
T aian Iliescu
[email p o ec ed]
1Depa amen o EDAN, Uni e sidad de Se illa, Se illa, Spain
2Depa amen o EDAN & IMUS, Uni e sidad de Se illa, Se illa, Spain
3Depa men o Ma hema ics, Vi ginia Tech, Blacksbu g, USA
0123456789().: V,- ol 123
308 Page 2 o 28 B. Koc e al.
in applica ions ha equi e epea ed model uns, such as, design and con ol, unce ain y
quan i ica ion, in e se p oblems, and da a assimila ion. Howe e , in con ec ion-domina ed
p oblems (e.g., u bulen lows), s anda d ROMs gene ally yield inaccu a e esul s. Indeed,
o ensu e a low compu a ional cos , ela i ely low-dimensional ROMs a e gene ally used in
p ac ice. Con ec ion-domina ed p oblems, howe e , usually equi e a la ge numbe o ROM
basis unc ions in o de o accu a ely ep esen he unde lying dynamics. Thus, s anda d, low-
dimensional ROMs usually yield spu ious nume ical oscilla ions ha signi ican ly deg ade
he solu ion accu acy. To alle ia e his inaccu a e beha io , he s anda d ROMs a e gene ally
equipped wi h (i) nume ical s abiliza ions o di e en ypes (e.g., p ojec ion-based (Azaïez
e al. 2021; Chacón Rebollo e al. 2022; No o and Rubino 2021), subspace o a ion (Balajew-
icz e al. 2016), a ia ional mul iscale (Be gmann e al. 2009; Iliescu and Wang 2013,2014;
Reyes and Codina 2020), s eamline-upwind Pe o -Gale kin (Pa ish e al. 2020), il e -based
(Gi oglio e al. 2021), and physically cons ain s (Sande se 2020)); o (ii) ROM closu es,
which a e e ms ha a e added o he s anda d ROM o model he e ec o he un esol ed
ROM scales. The e a e se e al ypes o ROM closu e s a egies, which a e e iewed in Ahmed
e al. (2021).
The a ia ional mul iscale (VMS) ROM closu es a e a popula class o ROM closu e
s a egies. The VMS-ROM closu es le e age he VMS amewo k (Hughes e al. 1998),
which has been ex ensi ely used a he FOM le el (see Ahmed e al. (2017) o a e iew).
Speci ically, he physical space is i s decomposed in he ROM space (i.e., he space used o
cons uc he ROM) and he space o un esol ed scales. Then, he VMS-ROM equa ions a e
ob ained by p ojec ing he unde lying equa ions on o he ROM space. We emphasize ha he
VMS-ROM includes bo h s anda d ROM ope a o s ( ha depend on he ROM space), as well
as a ROM closu e e m, i.e., a ROM ope a o ha depends on he space o un esol ed scales.
To ob ain a p ac ical, sel -con ained ROM, his closu e e m in he VMS-ROM needs o be
modeled by using exclusi ely he ROM space. This, in a nu shell, is he celeb a ed ROM
closu e p oblem.
Se e al success ul VMS-ROM closu e s a egies ha e been p oposed o e he yea s. A
e iew o hese app oaches is pe o med in Sec ion IV.A.5 o Ahmed e al. (2021). Nex , we
ou line se e al VMS-ROM closu e models. In Be gmann e al. (2009), a esidual-based VMS
model was p oposed as a ROM s abiliza ion s a egy. In Reyes and Codina (2020), a wo-
scale VMS-ROM equipped wi h ime-dependen o hogonal sub-g id scales was de eloped.
In Iliescu and Wang (2013), he au ho s p oposed a VMS-ROM closu e ha includes an
a i icial iscosi y added only o he small esol ed scales o he g adien . The nume ical
es s in Iliescu and Wang (2013) showed he inc eased nume ical s abili y and accu acy o
he VMS-ROM o e he s anda d G-ROM and illus a ed he heo e ical con e gence a es. In
pa icula , a p oblem displaying shock-like phenomena was conside ed (a 2D a eling wa e)
a a mode a e Pécle numbe (ν=10−4). In Iliescu and Wang (2014), he VMS-ROM was
ex ended and s udied o he incomp essible Na ie -S okes equa ions. Recen VMS-ROM
de elopmen s can be ound in, e.g., E oglu e al. (2017); Pa ish and Du aisamy (2017); Roop
(2013); S abile e al. (2019); Tello e al. (2019).
A pa adigm shi in he de elopmen o VMS-ROM closu es occu ed in Mou e al. (2021)
(see also Xie e al. (2018) o ele an wo k), whe e he classical, physical modeling used o
de elop VMS-ROM closu es was eplaced wi h da a-d i en modeling. Ins ead o using a-
di ional a gumen s (e.g., eddy iscosi y), he no el da a-d i en VMS-ROM (d2-VMS-ROM)
p oposed in Mou e al. (2021) was cons uc ed by le e aging a ailable da a. Speci ically,
he d2-VMS-ROM was buil by i s pos ula ing a model o m o he closu e e m (i.e., a
linea o quad a ic model), and hen sol ing a leas squa es p oblem o de e mine he model
pa ame e s ha yielded he closes i be ween he model o m and he a ailable da a. The
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o iginal leas squa es o mula ion in he d2-VMS-ROM was eplaced wi h a machine lea ning
s a egy in Ahmed e al. (2023) (see Xie e al. (2020) o ele an wo k). The d2-VMS-ROM
has been ex ended in di e en di ec ions (e.g., adding physical cons ain s (Mohebujjaman
e al. 2019), p o iding ma hema ical suppo (Koc e al. 2022), and de eloping a s ochas ic
amewo k o e icien da a assimila ion (Mou e al. 2023)), and has been success ully used
in challenging nume ical simula ions (e.g., om he quasigeos ophic equa ions (Mou e al.
2020) o he u bulen channel low (Mou 2021)).
Despi e i s signi ican achie emen s, he d2-VMS-ROM has been exclusi ely used in
i s coe icien -based o m. Speci ically, he da a-d i en closu e e m in he d2-VMS-ROM
depends exclusi ely on he ROM coe icien s. In his pape , we p opose a undamen ally
di e en d2-VMS-ROM s a egy, in which he ROM closu e e m is a unc ion o he ROM
esidual. The main ad an age o he new esidual-based ROM closu e e m is ha i is
consis en : As he ROM dimension inc eases, he esidual dec eases, and hus he ROM
closu e e m dec eases as well. This beha io is consis en wi h he physical ole o he ROM
closu e e m: As he ROM dimension inc eases, mo e physical scales a e esol ed by he
d2-VMS-ROM, and hus he ole o he ROM closu e e m dec eases. We emphasize ha ,
in con as o he new esidual-based d2-VMS-ROM, he cu en coe icien -based d2-VMS-
ROM a e no consis en (i.e., as he ROM dimension inc eases, he coe icien -based ROM
closu es do no necessa ily dec ease). In his pape , we pe o m a nume ical in es iga ion o
he new esidual-based d2-VMS-ROM and show ha i is mo e accu a e and e icien han
he classical coe icien -based d2-VMS-ROM.
The ou line o he pape is as ollows: In Sec . 2, we b ie ly ou line he s anda d Gale kin
ROM o he incomp esible Na ie -S okes equa ions. In Sec . 3, we desc ibe he small-
scale o la ge-scale decomposi ion ha unde pins he VMS-ROM amewo k. In Sec . 4,we
in oduce wo s a egies o he cons uc ion o he no el esidual-based d2-VMS-ROM, and
ou line he s anda d coe icien -based d2-VMS-ROM.
In Sec . 5, we pe o m a nume ical in es iga ion o he new esidual-based d2-VMS-
ROMs in he simula ion o h ee es p oblems: (i) a one-dimensional pa ame e -dependen
ad ec ion-di usion equa ion; (ii) a wo-dimensional ime-dependen ad ec ion-di usion-
eac ion wi h small iscosi y; and (iii) a wo-dimensional low pas a ci cula cylinde a
Reynolds numbe Re =1000. To assess he pe o mance o he new esidual-based d2-VMS-
ROMs, we compa e hem wi h he s anda d Gale kin ROM, he classical coe icien -based
d2-VMS-ROM, and an ideal VMS-ROM, in which he closu e e m is compu ed om he
FOM da a. As benchma k o ou nume ical in es iga ion, we use he FOM da a. In Sec . 6,
we conclude he pape wi h a sho summa y and u u e esea ch di ec ions.
2 Gale kin educed o de model (G-ROM)
This sec ion p o ides a b ie o e iew o he s anda d Gale kin ROM (G-ROM) s a egy,
which is one o he mos common ypes o ROMs o luid lows (Hes ha en e al. 2016; Noack
e al. 2011; Qua e oni e al. 2015). As a ma hema ical model, we use he incomp essible
Na ie -S okes equa ions (NSE) (1)–(2):
∂u
∂ −Re−1u+u·∇u+∇p= ,(1)
∇·u=0,(2)
whe e uis he eloci y, p he p essu e, he o ce, and Re he Reynolds numbe . Fo cla i y
o p esen a ion, we use homogeneous Di ichle bounda y condi ions and ze o o ce, i.e.,
=0.
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In Algo i hm 1, we ou line he cons uc ion o G-ROM, which is ca ied ou using he
eloci y ield. In he ROM amewo k, we employ a di e gence- ee eloci y basis (u ilizing
Sco -Vogelius elemen s in he ini e elemen se ing). Consequen ly, he p essu e e m is
omi ed in he G-ROM. Fo ROMs ha include he p essu e app oxima ion, see, e.g., Balla in
e al. (2015); Be gmann e al. (2009); Chacón Rebollo e al. (2022); Hes ha en e al. (2016);
Noack e al. (2005); No o and Rubino (2021); Qua e oni e al. (2015); Reyes and Codina
(2020); S abile and Rozza (2018).
Algo i hm 1 Gale kin ROM (G-ROM)
1: Use a ailable FOM da a o cons uc dominan modes by using he p ope o hogonal decomposi ion (POD),
{ϕ1,...,ϕL},Ld(whe e dis he dimension o he inpu da ase ), which co espond o he la ges
ela i e kine ic ene gy con en and ep esen he dominan spa ial s uc u es o he gi en es p oblem;
2: Cons uc a ROM eloci y app oxima ion:
uL=
L
j=1
(aL)jϕj,(3)
as a linea combina ion o ROM basis unc ions ϕjwi h ROM coe icien s (aL)j;
3: Replace uin he gi en es p oblem wi h he ROM solu ion uLgi enin(3);
4: Use he Gale kin p ojec ion, which p ojec s he sys em ob ained in s ep 3 on o he ROM eloci y space
XLspanned by {ϕ1,...,ϕL}.
By using Algo i hm 1 o he NSE equa ions (1)–(2), we ob ain he ollowing G-ROM:
daL
d =ALL aL+aLBLLL aL,(4)
whe e ALL is an L×Lma ix wi h en ies Aij := −Re−1(∇ϕi,∇ϕj)and BLLL is an
L×L×L enso wi h en ies Bijk := −(ϕi,ϕj·∇ϕk).TheG-ROM(4)isanL-dimensional
sys em o ODEs ha can be used o ime in e als and/o pa ame e s di e en om hose
used in he aining egime (i.e., in he cons uc ion o he G-ROM).
3 Va ia ional mul iscale educed o de model (VMS-ROM)
In his sec ion, we cons uc he VMS-ROM amewo k, which will be used in he nex
sec ions o build he d2-VMS-ROMs.
Fi s , we no e ha when all he a ailable ROM modes a e used o c ea e a ROM solu ion,
he ROM app oxima ion becomes
ud=
d
j=1
(ad)jϕj.(5)
In his case, udis he mos accu a e ROM app oxima ion o he FOM solu ion wi h he gi en
da a in he POD sense (i.e., om he ene ge ic poin o iew (Volkwein 2013)).
Fo lamina lows, using a ew (Ld) ROM basis unc ions is enough o cap u e he
main dynamics o he gi en p oblem, i.e., we a e in he esol ed egime.In his egime,a
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low-dimensional ROM solu ion uL, wi h small Ld, yields an accu a e app oxima ion o
he FOM solu ion.
Howe e , o u bulen lows, he low-dimensional ROM solu ion (3) wi h small Ld
is no an accu a e app oxima ion o he FOM solu ion, i.e., we a e in he unde - esol ed
egime. To inc ease he accu acy o he L-dimensional ROM solu ion (3), we gene ally ha e
wo op ions: (i) inc ease he G-ROM dimension, L, o (ii) add nume ical s abiliza ion o a
low-dimensional closu e e m o he G-ROM. In his pape , we aim o inc ease he nume ical
accu acy wi hou signi ican ly inc easing he compu a ional cos . Thus, we choose he second
op ion. Nex , we explain wha ROM closu e modeling is (see Ahmed e al. (2021) o a e iew)
and how i is pe o med in a VMS se ing.
The o hogonali y o he ROM basis unc ions (which is in insic o he POD ame-
wo k) allows us o decompose he ROM space as ollows: Xd=XL⊕XS,whe e
Xd:= span{ϕ1, ..., ϕd},XL:= span{ϕ1, ..., ϕL},andXS:= span{ϕL+1, ..., ϕd}.Byusing
his decomposi ion, we de ine he la ge-scale and sub-scale solu ions o he mos accu a e
(in he POD sense) ROM solu ion, ud, as ollows:
uL:=
L
j=1
(aL)jϕj,(6a)
uS:=
d
j=L+1
(aS)jϕj.(6b)
Nex , we no e ha he mos accu a e ROM app oxima ion in (5), ud, sol es he ollowing
d-dimensional weak o m o NSE equa ions (1)–(2)
D ( d,ud)+a( d,ud)+b( d,ud,ud)=0∀ d∈Xd,(7)
whe e he bilinea o ms a e D ( d,ud):= ( d,∂
ud)and a( d,ud):= Re−1(∇ d,∇ud),
and he nonlinea o m is b( d,ud,ud):= ( d,ud·∇ud). Fu he mo e, (·,·) ep esen s
he inne p oduc in L2(). By using he VMS me hod and choosing L=ϕL∈XLand
S=ϕS∈XS( d= L+ S), we can decompose (7) in o wo p oblems as ollows:
D (ϕL,uL+uS)+a(ϕL,uL+uS)+b(ϕL,uL+uS,uL+uS)=0(8a)
D (ϕS,uL+uS)+a(ϕS,uL+uS)+b(ϕS,uL+uS,uL+uS)=0.(8b)
The ma ix- ec o o ms o (8a)–(8b) a e as ollows:
daL
d =ALL aL+ALS aS+a
LBLLLaL+a
LBLLSaS+a
SBLSLaL+a
SBLSSaS,
(9a)
daS
d =ASS aS+ASL aL+a
LBSLLaL+a
LBSLSaS+a
SBSSLaL+a
SBSSSaS.
(9b)
The VMS-ROM idea can be explained by using he ma ix o mula ion in (9a)–(9b). Fi s ,
we no e ha his ma ix o m was ob ained by using he a ia ional o mula ion in (8a)–(8b).
Second, we no e ha he ma ix o mula ion in (9a)–(9b) is a mul iscale o mula ion since aL
and aSco espond o he la ge and small scales in he sys em, espec i ely. Thus, he ma ix
o mula ion in (9a)–(9b) is uly a a ia ional mul iscale ROM o mula ion.
The VMS-ROM a ionale is ha , since Ld,aLcan be compu ed e icien ly. In con as ,
since LS, we should y o a oid he expensi e compu a ion o aS. Howe e , he challenge
is ha he equa ions o aLand aSin (9a)–(9b) a e coupled.
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308 Page 6 o 28 B. Koc e al.
Fig. 1 Diag am o he algeb aic o m o he coupled sys em (9a)–(9b)
The VMS-ROM s a egy cen e s on wo simple ideas, which a e illus a ed in he schema ic
in Fig. 1: The i s idea is ha VMS-ROMs educe he la ge sys em o equa ions (9a)–(9b) o
a low-dimensional sys em o equa ions o aL. The second idea is ha , o ob ain an accu a e
app oxima ion o aL, he e ec o aSneeds o be modeled (i.e., he closu e p oblem needs
o be add essed). In he nex sec ion, we p esen wo new da a-d i en s a egies o modeling
he e ec o aS(Sec . 4.1–4.2).
4 Da a-d i en a ia ional mul iscale ROM (d2-VMS-ROM)
In his sec ion, we explain how we build he low-dimensional closu e e m in d2-VMS-
ROM (10), which aims o inc ease he nume ical accu acy by modeling he e ec o he
sub-scales:
daL
d =ALL aL+aLBLLL aL+(Closu e-Te m).(10)
In he ollowing sec ions, we p esen wo undamen ally di e en ypes o Closu e-Te m.
In Sec . 4.1–4.2, we p opose no el esidual-based d2-VMS-ROMs in which he sub-scale
in o ma ion in he closu e e m is modeled by using he la ge-scale ROM esidual, ResS(aL).
Spec ically, in Sec . 4.1 we p opose R1-ROM which is associa ed wi h a single Closu e-Te m,
and in Sec . 4.2 we p opose R2-ROM which le e ages wo Closu e-Te ms o be e con ol
sub-scale e ec s. In Sec . 4.3, we desc ibe he classical coe icien -based d2-VMS-ROM,
which has ecen ly been p oposed (Mou e al. 2021). In his model, he sub-scale in o ma ion
in he Closu e-Te m is modeled by using he la ge-scale ROM coe icien ec o , aL.Ou
goal in he nume ical in es iga ion in Sec . 5is o show ha he new esidual-based d2-
VMS-ROMs (R1-ROM and R2-ROM) a e mo e accu a e and e icien han he s anda d
coe icien -based d2-VMS-ROM.
To build he d2-VMS-ROM (10), we i s no e ha , o ensu e ha i is an e icien , L-
dimensional model, i s closu e e m should be modeled using only he la ge-scale ROM
coe icien , aL. The nex s ep in he d2-VMS-ROM cons uc ion is o pos ula e a model
o m o he Closu e-Te m, deno ed Ansa z(aL). To ind he Ansa z-Ope a o s ha yield he
mos accu a e esul s, we use da a-d i en modeling. Speci ically, in he o line phase, we
sol e a leas squa es p oblem ha minimizes he di e ence be ween Ansa z(aL)and he ue
sub-scale e m, deno ed Sub-Scale-Te m(aL,aS), e alua ed wi h he FOM sub-scale and
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Residual-based da a-d i en a ia ional mul iscale… Page 7 o 28 308
la ge-scale coe icien ec o s:
min
Ansa z-Ope a o s
M
k=1
Sub-Scale-Te m(ak
L,ak
S)−Ansa z(ak
L)
2
L2,(11)
whe e M ep esen s he numbe o snapsho s. Then, in he online phase, he d2-VMS-ROM
wi h he Ansa z compu ed om (11) is used o ime in e als and/o pa ame e s di e en
om hose used in he aining s age.
We no e ha al e na i e da a-d i en s a egies o ROM closu e ha e been used in, e.g.,
Maulik e al. (2020); P akash and Zhang (2024), and e iewed in Sande se e al. (2024).
4.1 Residual-based d2-VMS-ROM wi h one ansa z (R1-ROM)
In his sec ion, we in oduce he i s esidual-based d2-VMS-ROM, R1-ROM.
In Algo i hm 2, we ou line he cons uc ion o he R1-Closu e-Te m.
Algo i hm 2 Residual-Based Closu e Te m wi h One Ansa z (R1-Closu e-Te m)
1: In oduce he Sub-Scale-Te m, which needs o be modeled, and he o m o Ansa z, which appea s in he
Closu e-Te m o R1-ROM (14):
Sub-Scale-Te m =aS(12)
≈Ansa z =
AResS(aL)+ResS(aL)
BResS(aL),
whe e he la ge-scale ROM esidual, de i ed om (9b), has he o m ResS(aL):= ASL aL+a
LBSLL aL.
2: Sol e he leas squa es p oblem (11) o ob ain he S-dimensional ( he dimension o he sub-scale)
Ansa z-Ope a o s
Aand
B, using he Sub-Scale-Te m and Ansa z de ined in (12).
3: Subs i u e all aS e ms in he la ge-scale equa ion (9a) wi h Ansa z om (12) o de i e he R1-Closu e-
Te m (13):
R1-Closu e-Te m ≈ALS Ansa z +a
LBLLS Ansa z (13)
+Ansa zBLSLaL+Ansa zBLSS Ansa z.
Replacing he Closu e-Te m in he d2-VMS-ROM (10) wi h he R1-Closu e-Te m (13)
om Algo i hm 2, we de i e he ollowing R1-ROM o mula ion:
daL
d =ALL aL+a
LBLLL aL+(R1-Closu e-Te m).(14)
We highligh ha he la ge-scale ROM esidual, ResS(aL),in(12) does no include a
ime-de i a i e e m due o he o hogonali y o he POD modes in he L2inne p oduc .
The idea behind he R1-ROM s a egy is o a oid sol ing he expensi e, high-dimensional
equa ion (9b). Ins ead, we only le e age he in o ma ion in (9b) (i.e., he ac ha aSdepends
on he esidual ResS(aL)) o model he sub-scales in (9a).
We emphasize ha he closu e e m in he esidual-based R1-ROM (14) is consis en since
i depends on he la ge-scale ROM esidual, ResS(aL). In con as , he closu e e m o he
s anda d coe icien -based C-ROM (which is p esen ed in Sec . 4.3) is no consis en since i
depends on he la ge-scale ajec o y, aL.
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308 Page 8 o 28 B. Koc e al.
4.2 Residual-based d2-VMS-ROM wi h wo ansa zes (R2-ROM)
In his sec ion, we in oduce he second esidual-based d2-VMS-ROM, R2-ROM. The main
di e ences be ween he R1-ROM and R2-ROM a e he ollowing: (i) he R2-ROM uses wo
ansa zes whe eas he R1-ROM uses only one ansa z, which is a common ansa z, and (ii)
because he R2-ROM uses mo e ansa zes, i has mo e in o ma ion ela ed o he sub-scales.
In Algo i hm 3, we ou line he cons uc ion o R2-ROM.
Algo i hm 3 Residual-Based Closu e Te m wi h Two Ansa zes (R2-Closu e-Te m)
1: In oduce a i s sub-scale e m Sub-Scale-Te m1, which needs o be modeled, and he o m o Ansa z1.
The Sub-Scale-Te m1 and i s co esponding Ansa z1 a e modeled simila ly o R1-ROM (see (12)) o
app oxima e he sub-scale ROM coe icien , i.e., aSin (9a):
Sub-Scale-Te m1 =aS(15)
≈Ansa z1 =
A1ResS(aL)+ResS(aL)
B1ResS(aL).
2: Sol e he leas squa es p oblem (11) o compu e he i s se o Ansa z ope a o s,
A1and
B1, which a e o
dimension S( he sub-scale dimension), u ilizing he Sub-Scale-Te m1 and Ansa z1 as de ined in (15);
3: Replace all ins ances o aSin he la ge-scale equa ion (9a) wi h Ansa z1 om (15);
4: In oduce a second sub-scale e m (Sub-Scale-Te m2) and he co esponding Ansa z2 o enhance nume ical
accu acy:
Sub-Scale-Te m2 := ResL(Ansa z1)=ALS (Ansa z1)+(Ansa z1)BLSS (Ansa z1)(16)
≈Ansa z2 =
A2ResL(aS)+(ResL(aS))
B2ResL(aS);
5: Sol e he leas squa es p oblem (11) o ob ain he second se o Ansa z ope a o s,
A2and
B2, o dimension
L( he la ge-scale dimension), based on Sub-Scale-Te m2 and Ansa z2, as de ined in (16);
6: Replace he e m ResL(aS)in he la ge-scale equa ion (9a) wi h Ansa z2 om (16), and subs i u e all
emaining ins ances o aSin he la ge-scale equa ion wi h Ansa z1 om (15), he eby de i ing he R2-
closu e e m:
R2-Closu e-Te m ≈
A2ResL(Ansa z1)+(ResL(Ansa z1))
B2ResL(Ansa z1)(17)
+a
LBLLS Ansa z1 +Ansa z1BLSL aL.
Replacing he Closu e-Te m in he d2-VMS-ROM (10) wi h he R2-Closu e-Te m (17)
om Algo i hm 3, we de i e he ollowing R2-ROM o mula ion:
daL
d =ALL aL+a
LBLLL aL+(R2-Closu e-Te m).(18)
We emphasize ha bo h he la ge-scale ROM esidual, ResS(aL)in (15), and he sub-scale
ROM esidual, ResL(aS),in(16) do no con ain a ime-de i a i e e m. This is due o he
o hogonali y o he POD modes in he L2inne p oduc . Fu he mo e, he inclusion o a
second ansa z allows us o e ine he app oxima ion o ResL(Ansa z1)using sub-scale FOM
da a.
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The expec a ion is ha he esidual-based R2-ROM (18) could yield mo e accu a e esul s
han he esidual-based R1-ROM (14) since he R2-Closu e-Te m (17) g adually models
he sub-scale and has mo e sub-scale in o ma ion han he R1-Closu e-Te m (13). Howe e ,
in nume ical simula ions, since he R2-Closu e-Te m (17) is mo e complex, i could yield
inaccu a e esul s.
4.3 Coe icien -based d2-VMS-ROM (C-ROM)
In his sec ion, we ou line he coe icien -based C-ROM s a egy (Mou e al. 2021).
In Algo i hm 4, we ou line he cons uc ion o C-Closu e-Te m.
Algo i hm 4 Coe icien -Based Closu e Te m wi h One Ansa z (C-Closu e-Te m)
1: In oduce he sub-scale e m Sub-Scale-Te m, which needs o be modeled, and he o m o Ansa z, which
appea s in he Closu e-Te m o C-ROM (21):
Sub-Scale-Te m =ALS aS+a
LBLLS aS+a
SBLSL aL+a
SBLSS aS(19)
≈Ansa z =∗
AaL+a
L
Ba
L,
whe e Sub-Scale-Te m is de i ed om (9a).
2: Sol e he leas squa es p oblem (11) o compu e he Ansa z-Ope a o s
Aand
B, which a e o dimension
L( he la ge-scale dimension), u ilizing he Sub-Scale-Te m and Ansa z de ined in (19).
3: Subs i u e Sub-Scale-Te m in he la ge-scale equa ion (9a) wi h Ansa z om (19) o de i e he C-Closu e-
Te m (20):
C-Closu e-Te m ≈
AaL+a
L
Ba
L.(20)
Replacing he Closu e-Te m in he d2-VMS-ROM (10) wi h he C-Closu e-Te m (20)
om Algo i hm 4, we de i e he ollowing C-ROM o mula ion:
daL
d =ALL aL+a
LBLLL aL+(C-Closu e-Te m).(21)
4.4 Ideal a ia ional mul iscale ROM (I-ROM)
In his sec ion, we ou line he ideal ROM (I-ROM), which is used as a benchma k model o
discuss he e ec o Closu e-Te m in (10). We emphasize ha I-ROM is a pu ely heo e ical
model, used o assess he accu acy o ROM closu e models. Speci ically, he closu e e m in
I-ROM is compu ed di ec ly om bo h he la ge-scale and sub-scale FOM da a:
I-Closu e-Te m =ALS aS+a
LBLLS aS+a
SBLSL aL+a
SBLSS aS.(22)
Thus, he I-ROM closu e e m canno be used in p ac ical se ings, whe e FOM da a is
no a ailable. Replacing he Closu e-Te m in he d2-VMS-ROM (10) wi h he I-Closu e-
Te m (22), we ob ain he I-ROM:
aL
d =ALL aL+a
LBLLL aL+(I-Closu e-Te m).(23)
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Fig. 7 ADR equa ion; L2-POD decay o eigen alues
Table 4 ADR equa ion; a e age
ela i e L2-p ojec ion e o (30)
o G-ROM, I-ROM, C-ROM,
R1-ROM, and R2-ROM o
a ious L alues (numbe o
POD modes)
LG-ROM I-ROM C-ROM R1-ROM R2-ROM
2 2.14e+00 3.87e-01 1.75e+00 1.60e+00 1.81e+00
4 2.19e+00 4.01e-01 2.00e+00 1.73e+00 1.10e+00
6 2.13e+00 4.59e-01 2.02e+00 1.68e+00 5.25e-01
8 2.06e+00 6.02e-01 1.99e+00 1.66e+00 5.38e-01
10 1.99e+00 9.25e-01 1.95e+00 1.76e+00 5.17e-01
In Fig. 8, we plo he ela i e ROM e o in ime o L=4,6, and 10. We obse e ha
he ela i e R2-ROM e o eco e s he I-ROM as Linc eases. Fu he mo e, o L=10, he
ela i e R2-ROM e o is lowe han he ela i e I-ROM e o .
Finally, in Fig. 9, we plo he FOM and all ROM solu ions a he inal ime, T=1, o
L=8. We obse e ha he C-ROM does no diminish he oscilla o y bump, whe eas he R1-
ROM ies o dec ease he magni ude o he bump. The R2-ROM dec eases he oscilla ions
he mos , bu i also p oduces ano he oscilla o y bump.
5.3 Two-dimensional low pas a cylinde
In his sec ion, we conside a 2D channel low pas a ci cula cylinde a Reynolds numbe
Re =1000. As c i e ia in ou nume ical in es iga ion, we use he a e age L2ROM e o s,
kine ic ene gy, o ex shedding equency, and Pa e o plo s.
Compu a ional Se ing
As a ma hema ical model, we use he NSE (1)–(2). The compu a ional domain is a 2.2×0.41
ec angula channel wi h a adius =0.05 cylinde , cen e ed a (0.2,0.2);seeFig.10.
We p esc ibe no-slip bounda y condi ions on he walls and cylinde , and he ollowing
in low and ou low p o iles (John 2004; Mohebujjaman e al. 2019,2017; Rebholz and Xiao
2017):
u1(0,y, )=u1(2.2,y, )=6
0.412y(0.41 −y), (31)
u2(0,y, )=u2(2.2,y, )=0,(32)
whe e u=u1,u2. The e is no o cing and he low s a s om es .
Snapsho Gene a ion
Fo he spa ial disc e iza ion, we use he poin wise di e gence- ee, LBB s able (P2,Pdisc
1)
Sco -Vogelius FE pai on a ba ycen e e ined egula iangula mesh o he ba ycen e (John
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Fig. 8 ADR equa ion; ela i e ROM e o o e he ex apola ion es ing ime in e al o a ious L alues
e al. 2016). The mesh p o ides 103K(102962) eloci y and 76K(76725) p essu e deg ees
o eedom. We u ilize he commonly used linea ized BDF2 empo al disc e iza ion and a
ime s ep size =0.002 o bo h he FOM and he ROM ime disc e iza ions. In he i s
ime s ep, we use a backwa d Eule scheme so ha we ha e wo ini ial ime-s ep solu ions,
as equi ed o he BDF2 scheme.
ROM Cons uc ion and Tes ing
The FOM simula ion achie es he s a is ically s eady s a e a e =13 in he nume ical
in es iga ion. To build he ROM basis unc ions and ope a o s, we decided o use 3 ime
uni s o FOM da a. Thus, we collec FOM snapsho s om =13 o =16 and label hem
as aining FOM da a. In Fig. 11, we plo he decay o eigen alues o he ROM basis.
To ain he Ansa z Ope a o s in (11), which a e used o cons uc he Closu e-Te m
in (10), we use he aining FOM da a. Due o he pe iodic beha io o he low and in o de
o dec ease he compu a ional cos o cons uc ing he Ansa z Ope a o s, we use a hal -pe iod
o aining FOM da a, i.e., 68 FOM snapsho s, om =13 o =13.134. Then we es all
ROMs in he ex apola ion es ing ime in e al, =16 o =23.
In Sec . 5.3.1–5.3.3, we compa e he quali y o all ROMs based on h ee c i e ia: L2-no m
e o , kine ic ene gy e o , and o ex shedding equency, espec i ely.
In Table 5, we lis he a e age L2FOM consis ency e o (26), compu ed using a educed
basis o d=22 POD modes. This choice o d, which is subs an ially lowe han he ull ank
o he sys em, is used consis en ly in Sec . 5.3.1 and 5.3.2. F om he esul s in Table 5,we
obse e ha bo h R1-ROM and R2-ROM ( o bo h ansa zes) ha e lowe FOM consis ency
e o s han C-ROM. Fu he mo e, he FOM consis ency e o s o R1-ROM and R2-ROM
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Fig. 9 ADR equa ion; ROM solu ions a he inal ime, T=1, o L=8
Fig. 10 Geome y o he low pas a ci cula cylinde nume ical expe imen
Fig. 11 Flow pas a cylinde ; L2-POD decay o eigen alues
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Table 5 Flow pas a cylinde ; d=22; a e age L2FOM consis ency e o (26)o C-ROM,R1-ROM,and
R2-ROM o a ious L alues (numbe o POD modes).
LC-FOM-Consis ency R1-FOM-Consis ency R2-FOM-Consis ency
Ansa z1 Ansa z2
2 2.28e-01 2.59e-02 2.05e-02 7.37e-02
5 1.49e+00 1.72e-02 2.62e-02 3.61e-03
8 4.01e-01 1.22e-04 6.60e-05 3.03e-05
11 1.35e+00 1.27e-08 1.54e-08 3.63e-09
14 2.46e-01 5.39e-03 4.24e-03 2.25e-05
17 7.29e-01 6.19e-03 6.19e-03 1.03e-04
20 1.43e-01 2.34e-03 2.52e-03 8.35e-04
22 0 0 0 0
Table 6 Flow pas a cylinde ; d=14; a e age L2FOM consis ency e o (26)o C-ROM,R1-ROM,and
R2-ROM o a ious L alues (numbe o POD modes).
LC-FOM-Consis ency R1-FOM-Consis ency R2-FOM-Consis ency
Ansa z1 Ansa z2
2 2.27e-01 2.29e-02 2.29e-02 7.19e-03
3 2.59e+00 6.39e-04 3.58e-03 1.41e-03
5 1.49e+00 1.63e-02 1.63e-02 1.82e-03
7 7.63e-02 4.28e-02 4.28e-02 6.37e-03
9 2.10e+00 1.50e-02 1.50e-02 4.84e-04
11 1.33e+00 1.77e-02 1.14e-02 6.61e-03
13 1.15e+00 7.90e-03 7.90e-03 2.28e-03
14 0 0 0 0
each he lowes alue a L=11, and hen inc ease again. Since he leas squa es p oblem (11)
is sensi i e based on using he size o he da a, o in es iga e his beha io in Table 5,inTable6,
we also lis he a e age L2FOM consis ency e o (26), calcula ed using a educed basis wi h
d=14 POD modes. In Table 6, we obse e a simila beha io as ha obse ed in Table 5
bu less signi ican inc easing, om L=5.
Fo he low pas a cylinde low, due o he complex nonlinea in e ac ions in he Na ie -
S okes equa ions, he consis ency e o has a mo e complex beha io han in he linea ADR
case. Speci ically, he consis ency e o dec eases as he dimension Lo he educed space
inc eases, up o some alue L=Lop (depending on he o al numbe o modes aken in o
accoun o build he ROM, d), and hen i app oxima ely s agna es: Lop =11 when d=22,
and Lop =3whend=14.
5.3.1 L2-no m e o
In his sec ion, we compa e he nume ical accu acy o he d2-VMS-ROMs, I-ROM, and
G-ROM de ined in Sec . 4by using he a e age ela i e L2ROM p ojec ion e o (30).
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308 Page 20 o 28 B. Koc e al.
Table 7 Flow pas a cylinde ;
a e age ela i e L2-p ojec ion
e o (30) o G-ROM, I-ROM,
C-ROM, R1-ROM, and R2-ROM
o a ious L alues (numbe o
POD modes).
LG-ROM I-ROM C-ROM R1-ROM R2-ROM
2 1.94e+00 1.33e-02 6.96e-01 6.07e-01 3.05e-01
3 1.54e+00 4.80e-03 9.20e-01 1.03e-01 1.13e-01
4 1.19e+00 4.61e-03 3.26e-01 1.95e-01 9.55e-02
5 1.19e+00 2.15e-03 9.51e-01 5.09e-01 4.03e-01
6 5.76e-01 3.09e-02 2.41e-01 7.10e-02 3.09e-02
7 4.91e-01 2.00e-02 4.56e-01 8.60e-02 3.51e-01
8 2.58e-01 4.23e-03 1.53e-01 4.12e-02 6.40e-02
Fig. 12 Flow pas a cylinde ; Pa e o plo o a e age ela i e L2e o o d2-VMS-ROMs, i.e., R1-ROM,
R2-ROM, and C-ROM
In Table 7, we lis he a e age ela i e L2ROM e o s (27) o di e en ROM dimensions,
L. We obse e ha bo h R1-ROM and R2-ROM yield mo e accu a e esul s han C-ROM and
G-ROM. O e all, he R2-ROM is mo e accu a e han he R1-ROM, excep o L =3,7,8.
In Fig. 12, we p esen a Pa e o plo o he d2-VMS-ROMs, i.e., R1-ROM and R2-ROM,
and C-ROM, a e aging he L2e o and o line ansa z cos o e he low ROM dimensions,
i.e., L =2,3,4,5, and he high-ROM dimensions, i.e., L =6,7,8. Fo he low-ROM
dimension, we obse e ha he R2-ROM yields he mos accu a e model, al hough i is he
mos expensi e model. Fo he high-ROM dimension L =6,7,8, he R1-ROM is he mos
accu a e and leas expensi e model.
5.3.2 Kine ic ene gy e o
In his sec ion, we compa e he nume ical accu acy o he d2-VMS-ROMs, I-ROM, and
G-ROM de ined in Sec . 4by using he kine ic ene gy (KE) c i e ion:
Ekin := 1
2u2
L2=1
2
|u|2d. (33)
The aim o ou nume ical in es iga ion is o obse e how he kine ic ene gy o ROMs e ol es
in he ex apola ion ime in e al, e.g. whe he i dissipa es, blows up, o s agna es.
EKE =M
k=1|EFOM
kin ( k)−EROM
kin ( k)|
M
k=1|EFOM
kin ( k)|.(34)
In Table 8, we lis he a e age ela i e kine ic ene gy e o (34) o he ROMs. We obse e
ha R2-ROM gene ally yields a lowe kine ic ene gy e o han R1-ROM, and a much lowe
kine ic ene gy e o han C-ROM and G-ROM o low ROM dimensions. Fo high ROM
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Table 8 Flow pas a cylinde ;
a e age ela i e kine ic ene gy
e o (34) o G-ROM, I-ROM,
C-ROM, R1-ROM, and R2-ROM
o a ious L alues (numbe o
POD modes)
LG-ROM I-ROM C-ROM R1-ROM R2-ROM
2 4.99e-01 5.91e-03 1.40e-02 1.16e-01 4.34e-03
3 1.62e-01 1.35e-03 1.25e-01 4.57e-03 2.48e-02
4 1.08e-01 2.14e-03 2.22e-02 5.79e-02 4.52e-03
5 8.35e-02 9.35e-04 2.40e-01 1.32e-01 9.41e-02
6 2.14e-01 1.94e-03 1.17e-01 2.74e-02 2.73e-03
7 2.06e-01 2.51e-03 2.08e-01 1.44e-02 1.14e-01
8 7.53e-02 1.73e-03 3.47e-02 8.57e-03 1.31e-02
Fig. 13 Flow pas a cylinde ; Pa e o plo o a e age ela i e KE e o o d2-VMS-ROMs, i.e., R1-ROM,
R2-ROM, and C-ROM
dimensions, R1-ROM yields a lowe kine ic ene gy e o han R2-ROM, and a much lowe
kine ic ene gy e o han C-ROM and G-ROM.
In Fig. 13, we p esen he Pa e o plo o he d2-VMS-ROMs, i.e., R1-ROM, R2-ROM,
and C-ROM, a e aging he a e age ela i e kine ic ene gy and o line ansa z cos o e he
low ROM dimensions, i.e., L =2,3,4,5, and he high ROM dimensions, i.e, L =6,7,8.
Fo he low L alues, we obse e ha R2-ROM yields he mos accu a e esul s, al hough
i s compu a ional cos is he highes . On he o he hand, o he high L alues, he R1-ROM
is he mos accu a e and i s compu a ional cos is he lowes . The conclusion o he kine ic
ene gy Pa e o plo in Fig. 13 is consis en wi h he conclusion o he a e age L2Pa e o plo
in Fig. 12.
In Fig. 14, we plo he kine ic ene gy (33) o he FOM p ojec ion, G-ROM, I-ROM,
C-ROM, R1-ROM, and R2-ROM o he ROM dimension alues L =2,4,6,8 o e he
ex apola ion ime in e al. These plo s suppo he esul s o Table 8.
5.3.3 Vo ex shedding equency ma ching be ween FOM and d2-VMS-ROMs
In his sec ion, we compu e he a e age o ex shedding equency so d2-VMS-ROMs
and FOM based on he o ex shedding pe iod Ts, which is de ined as ollows:
Ts=1
Ns
Ns
k=1
( s(k+1)− s(k)), (35)
whe e s(k)deno es he ime ins ances co esponding o successi e peaks in he kine ic ene gy
wi hin he ex apola ion es ing ime in e al [18,23]. These peaks a e used o es ima e he
dominan o ex shedding cycle o he espec i e models.
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Fig. 14 Flow pas a cylinde ; kine ic ene gy o FOM p ojec ion, G-ROM, I-ROM, C-ROM, R1-ROM, and
R2-ROM o L =2,4,6,8
In Table 9, we lis he o ex shedding equency, i.e., s=1/Ts, o he FOM and d2-
VMS-ROMs, i.e., C-ROM, R1-ROM, and R2-ROM, o a ious L alues, i.e., L=2,4,6,8.
O e all, he esul s in Table 9show ha R2-ROM yields he mos accu a e p edic ions o he
S ouhal numbe , especially o low L alues.
In Table 10, we lis he ela i e o ex shedding equency e o s o d2-VMS-ROMs:
Es=| FOM
s− ROM
s|
FOM
s
.(36)
Based on he ela i e e o s in o ex shedding equency lis ed in Table 10,R2-ROM
demons a es supe io accu acy in cap u ing he o ex shedding pe iod o he FOM among
he d2-VMS-ROM a ian s.
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Table 9 Flow pas a cylinde ;
o ex shedding equency, i.e.,
s o FOM, C-ROM, R1-ROM,
and R2-ROM o a ious L
alues (numbe o POD modes)
LFOM C-ROM R1-ROM R2-ROM
2 3.77e+00 3.74e+00 3.83e+00 3.79e+00
4 3.77e+00 3.79e+00 3.73e+00 3.78e+00
6 3.77e+00 3.82e+00 3.78e+00 3.77e+00
8 3.77e+00 3.78e+00 3.77e+00 3.77e+00
Table 10 Flow pas a cylinde ;
ela i e o ex shedding
equency e o (36) o C-ROM,
R1-ROM, and R2-ROM o
a ious L alues (numbe o
POD modes)
LC-ROM R1-ROM R2-ROM
2 9.14e-03 1.40e-02 3.79e-03
4 4.21e-03 1.04e-02 4.19e-04
6 1.88e-02 8.39e-04 4.19e-04
8 1.26e-03 0 0
Fig. 15 Flow pas a cylinde ; o ex shedding pe iods o d2-VMS-ROMs, i.e., R1-ROM, R2-ROM, and C-
ROM o L=2,6
In Fig. 15, we p esen he sequence o kine ic ene gy peaks o he FOM and he d2-VMS-
ROMs o L=2andL=6, hus isualizing he o ex shedding pe iods exhibi ed by each
model.
Fo L=6, in Fig. 15, we obse e ha bo h R1-ROM and R2-ROM mo e accu a ely
eco e he o ex shedding pe iods o he FOM han C-ROM. Fu he mo e, he ampli udes
o he kine ic ene gy peaks in C-ROM a e no iceably highe han hose in R1-ROM and
R2-ROM. These obse a ions a e consis en wi h he lowe ROM e o s (see Table 7)and
kine ic ene gy e o s (see Table 8) exhibi ed by R1-ROM and R2-ROM.
Fo L=2, R2-ROM demons a es he bes ma ch o FOM in e ms o bo h o ex
shedding equency and ampli ude o kine ic ene gy peaks. Al hough he o ex shedding
equencies o C-ROM and R1-ROM a e simila , R1-ROM signi ican ly o e es ima es he
peak ampli udes compa ed o FOM.
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Fig. 16 Flow pas a cylinde ; he eloci y o he I-ROM and mos accu a e d2-VMS-ROM, i.e., R2-ROM, o
L=2,4,6a T=23
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Finally, in Fig. 16, we plo he eloci y ield o I-ROM and he mos accu a e d2-VMS-
ROMs, i.e., R2-ROM, o L=2,4,6 a he inal ime, T=23. We obse e ha R2-ROM
eco e s he main ea u es o he low wi h good accu acy.
6 Conclusions and ou look
In his pape , we p oposed a no el esidual-based da a-d i en ROM closu e model o unde -
esol ed, con ec ion-domina ed p oblems. The new da a-d i en ROM closu e model was
cons uc ed by le e aging he VMS amewo k and he a ailable da a. To build he new da a-
d i en VMS-ROM (d2-VMS-ROM), we i s pos ula ed a closu e model o m ansa z ha
depends on he ROM esidual, and hen we sol ed a leas squa es p oblem o ind he ansa z
pa ame e s ha yield he closes i be ween he ansa z and he FOM da a. We also conside ed
wo ypes o esidual-based ansa zes, which yielded wo ypes o d2-VMS-ROMs, deno ed
R1-ROM and R2-ROM. The main no el y o he p oposed esidual-based d2-VMS-ROMs
is ha hey depend on he ROM esidual ins ead o he ROM coe icien s (which is he
s anda d app oach in cu en da a-d i en ROM closu es). To assess he no el esidual-based
d2-VMS-ROMs, we compa ed hem wi h he s anda d coe icien -based d2-VMS-ROM,
deno ed C-ROM (Mou e al. 2021; Xie e al. 2018). Finally, o compa ison pu poses, we
also in es iga ed a s anda d G-ROM in which nei he s abiliza ion no closu e was used. We
in es iga ed he new esidual-based d2-VMS ROMs, i.e., R1-ROM and R2-ROM, as well as
he s anda d C-ROM and G-ROM, in he nume ical simula ion o h ee es p oblems: (i) a
one-dimensional pa ame e -dependen ad ec ion-di usion p oblem (Sec ion 5.1); (ii) a wo-
dimensional ime-dependen ad ec ion-di usion- eac ion p oblem wi h a small di usion
coe icien ε=1e−4 (Sec ion 5.2); and (iii) a wo-dimensional low pas a cylinde a
Reynolds numbe Re =1000 (Sec ion 5.3).
Ou nume ical in es iga ion yielded he ollowing conclusions: The no el esidual-based
d2-VMS-ROMs, R1-ROM and R2-ROM, we e signi ican ly mo e accu a e han he s an-
da d coe icien -based d2-VMS-ROM, C-ROM. Fu he mo e, R2-ROM gene ally yielded
sligh ly mo e accu a e esul s han R1-ROM, bu he e we e cases in which R1-ROM was
mo e accu a e. Finally, all he d2-VMS-ROMs (i.e., R1-ROM, R2-ROM, and C-ROM) we e
signi ican ly mo e accu a e han he s anda d G-ROM. Since R1-ROM has a simple o mula-
ion han R2-ROM and is easie o cons uc , and since he R1-ROM and R2-ROM accu acies
a e simila , R1-ROM appea s p e e able o R2-ROM in p ac ice.
The e a e se e al esea ch di ec ions ha can be pu sued nex . P obably he mos impo -
an is he ex ension o he new esidual-based d2-VMS-ROM amewo k o mo e complex
con ec ion-domina ed p oblems, such as unde - esol ed u bulen lows. Ano he impo an
esea ch di ec ion is p o iding ma hema ical suppo o he new esidual-based d2-VMS-
ROM. Fo example, we plan o p o e he esidual-based d2-VMS-ROM’s e i iabili y, i.e.,
o show ha when he closu e model e o dec eases, he ROM e o dec eases a he same
a e. The i s s ep in his di ec ion has been aken in Koc e al. (2022),whe ewep o ed he
e i iabili y o he s anda d coe icien -based d2-VMS-ROM, C-ROM.
Acknowledgemen s The i s au ho is pa ially suppo ed by P ojec PID2021-123153OB-C21 unded
by MCIN/AEI/10.13039/501100011033/FEDER, UE and Juan de la Cie a 2022 wi h p ojec numbe
2023/1061. The second and hi d au ho s a e unded by P ojec PID2021-123153OB-C21 unded by
MCIN/AEI/10.13039/501100011033/FEDER, UE. The ou h au ho is unded by ARIA MSCA-RISE EU
G an 872442 and Na ional Science Founda ion g an DMS-2012253.
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