Utility-Driven Adaptive Model Selection for Digital Twinning

U ili y-D i en Adap i e Model Selec ion o Digi al Twinning
Kons an inos Vlachasa,∗, An onios Kama io isa, Eleni Cha zia
aDepa men o Ci il, En i onmen al, and Geoma ic Enginee ing, ETH Zü ich, Zü ich, Swi ze land
Abs ac
Digi al wins ha e become cen al o mode n enginee ing by p o iding in e connec ed i ual ep esen a ions ha
mi o he beha io o physical asse s. Thei e ec i eness, howe e , depends c i ically on he ideli y and e iciency
o he unde lying compu a ional models used o decision suppo . This wo k in oduces an adap i e Bayesian ame-
wo k o quan i ying and op imizing he ideli y, e iciency, and o e all u ili y o such i ual ep esen a ions when
app oxima ing sys em esponses in decision-o ien ed con ex s. By in eg a ing educed-o de models (ROMs) wi hin a
Bayesian in e ence and decision- heo e ic se ing, he amewo k iden i ies, a any poin in ime, he lowes -cos model
capable o deli e ing he equi ed p edic i e accu acy ac oss all quan i ies o in e es , while igo ously accoun ing o
he unce ain y associa ed wi h each model esolu ion. The app oach maximizes an expec ed-u ili y unc ion ha
balances wo compe ing a ibu es: (i) p ecision, e lec ing he e ec o o e - o unde -es ima ing a ge quan i ies on
decision quali y, and (ii) compu a ional e iciency, ensu ing easibili y o eal- ime in e ence. Le e aging con inu-
ously assimila ed moni o ing da a, he amewo k pe o ms his model selec ion ecu si ely, enabling he i ual win
o e ol e in andem wi h he physical sys em. The esul ing me hodology suppo s au oma ed decisions on whe he o
e ain he cu en model, swi ch o an al e na i e esolu ion, o igge e aining— hus es ablishing a pa hway owa d
adap i e, us wo hy digi al wins o enginee ing decision suppo .
Keywo ds: Digi al Twinning, Reduced O de Models (ROMs), Bayesian Model Upda ing, Bayesian In e ence,
unce ain y quan i ica ion
1. In oduc ion
In as uc u e asse s a e inhe en ly complex, ea u ing la ge-scale models, nonlinea dynamics, and conside able2
unce ain y. As nex -gene a ion sys ems come o he o e on , he e is a clea shi owa d digi al and hyb id win
ep esen a ions ha p o ide i ual coun e pa s o physical asse s [1, 2]. The p ocess o c ea ing hese ep esen a-4
ions—commonly e e ed o as i ualiza ion o winning—aims o eplica e and in e ac wi h he beha io o eal
sys ems [3, 4]. To ensu e such i ual asse s ope a e e ec i ely, hey mus ely on compu a ional models ha a e bo h6
accu a e and e icien [5]. Wi hou his balance, eal- ime applica ions cen al o Ope a ions and Managemen (O&M),
S uc u al Heal h Moni o ing (SHM), and decision suppo become in easible [6, 7].8
In he exis ing li e a u e, nume ous amewo ks aim o de elop low-dimensional models o nume ical su oga es
ha enable apid compu a ion while e aining he essen ial physics o high- ideli y sys ems [8, 9, 10]. B oadly, hese10
app oaches all in o wo ca ego ies: da a-d i en me hods and physics-awa e me hodologies. Da a-d i en models a e
pa icula ly e ec i e o sys ems wi h complex o chao ic dynamics [11, 12]. Howe e , hey o en ely on ad hoc12
aining p ocedu es ha cap u e only limi ed inpu –ou pu ela ionships and end o deg ade in pe o mance unde
en i onmen al o pa ame ic a iabili y [13]. Mo eo e , such me hods ypically s uggle o ex apola e ac oss ime,14
inpu condi ions, o spa ial disc e iza ions o he a ge sys em [14]. As a esul , many da a-d i en su oga es emain
p oblem-speci ic, mesh-dependen , and unable o gene alize beyond he a iable used o aining [15, 16]. These16
limi a ions es ic hei p ac icali y o de eloping ac ionable digi al models sui able o in eg a ion in o highe -le el
O&M, SHM, o decision-making amewo ks.18
∗Co esponding au ho
Email add ess: [email p o ec ed] (Kons an inos Vlachas )
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In con as , physics-based and physics-awa e app oaches p ese e he in e nal s uc u e o he dynamical sys em
and embed physical knowledge h ough explici cons ain s o induc i e biases wi hin he i ual model [17, 18].20
This imp o es in e p e abili y, s abili y, and accu acy in ex apola ion and gene aliza ion asks, he eby enhancing he
applicabili y and eliabili y o he esul ing digi al ep esen a ions o highe -le el enginee ing wo k lows [19]. The e -22
ec i eness o such s a egies has been demons a ed in se e al s udies, including [20] and i s ex ension in [21], whe e
educed-o de models (ROMs) se e as o wa d simula o s in in e se p oblem se ings o aul de ec ion and inpu 24
es ima ion in eal ime. Simila ly, [22] employs s ochas ic ROMs o damage de ec ion in a mock-up ai c a wing,
while [23] showcases he in eg a ion o ROMs wi h he Unscen ed Kalman Fil e o sys em iden i ica ion and unce -26
ain y quan i ica ion unde en i onmen al and pa ame ic a iabili y. Beyond model-based in e ence, ela ed ad ances
using physics-in o med su oga es ha e also eme ged [24, 25, 26, 27, 28]. In his wo k, we ocus on model-based28
me hodologies, which p o ide a s uc u ed, physics-g ounded amewo k o suppo enginee ing decisions ac oss he
ope a ion and main enance (O&M) li e-cycle o in as uc u e asse s [29, 30]. The co e o such amewo ks lies in30
nume ical models ha condense complex sys ems in o compu a ionally ac able o ms while e aining hei essen ial
dynamic beha io , a iabili y, and unce ain y.32
Howe e , he use ulness o i ual models ul ima ely depends on balancing compu a ional cos agains model
ideli y, pa icula ly when hei ou pu s suppo high-s akes enginee ing decisions. De eloping digi al ep esen a ions34
ha main ain his balance is inhe en ly challenging, as achie ing consis en e iciency wi hou comp omising accu acy
becomes inc easingly di icul o e ol ing and da a- ich sys ems [31]. In his wo k, we add ess his challenge h ough36
an adap i e, da a-in o med amewo k ha le e ages con inuous moni o ing in o ma ion o dynamically selec he
mos sui able model using Bayesian decision heo y. By e alua ing a hie a chy o models wi h di e en le els o 38
e inemen and ideli y, he p oposed app oach allows he i ual ep esen a ion o adap and e ol e in andem wi h
he physical sys em.40
Wi hin his con ex , a numbe o hyb id me hodologies ha e been p oposed o de elop ep esen a ions ha adap
o changes in he sys em’s dynamics. Fo ins ance, se e al s udies [32, 33, 34, 35] employ physics-based indica o s42
o igge model e aining o e inemen s eps, o en suppo ed by da a-d i en en ichmen o he p ojec ion basis
o p oblems such as ac u e dynamics. O he app oaches, such as [36], in oduce basis en ichmen h ough de-44
composi ion, spli ing selec ed educed-basis ec o s in o disjoin componen s and employing ec o -space sie ing o
e inemen ee echniques o adap he educed-o de space. Ex ensions o hese ideas add ess online model upda es,46
ei he h ough basis comp ession [37] o addi i e low- ank upda es o educed spaces [38]. Finally, ac i e lea ning
s a egies [39, 40] and p obabilis ic schemes based on Gaussian P ocesses and Bayesian ne wo ks [41, 42, 43, 44, 45]48
ha e been p oposed o assimila e senso da a and build adap i e, da a-d i en su oga es capable o e ining hei p e-
dic ions as new in o ma ion becomes a ailable.50
Howe e , exis ing app oaches only pa ially add ess he challenge o c ea ing dynamically e ol ing digi al asse s,
as model e aining o e inemen is ypically igge ed by esponse-based e o indica o s. Such schemes neglec he52
o e all u ili y o he i ual ep esen a ion and canno explici ly assess he Value o In o ma ion (VoI)— ha is, he
bene i gained om knowledge ha imp o es decision-making [46, 47]. In con as , his wo k in oduces a Bayesian54
decision- heo e ic amewo k ha quan i ies he expec ed ideli y and u ili y o a ailable i ual models when used
o gene aliza ion o ex apola ion. The p oposed app oach ackles a key challenge in he i ualiza ion p ocess: he56
ecu si e selec ion o he lowes -cos model capable o achie ing he equi ed p edic i e accu acy o a gi en engi-
nee ing con ex [4]. In doing so, i es ablishes a pa hway owa d i - o -pu pose digi al ep esen a ions ha maximize58
decision alue and eliabili y h oughou he asse ’s li e-cycle [48].
In his con ex , selec ing he op imal model esolu ion o p edic ing he esponse o enginee ed sys ems om60
a pool o candida e models ep esen s a pe sis en and c i ical challenge [49, 50, 51]. These candida e models—o
i ual ep esen a ions—can ange om simpli ied analy ical o mula ions o high- ideli y ini e elemen models, each62
o e ing a di e en ade-o be ween accu acy, compu a ional cos , and unce ain y [23]. E ec i e model selec ion
enables eliable and e icien decision-making o sys em design and ope a ion by exploi ing a ailable moni o ing da a64
o con inuously educe unce ain y [52]. Th ough Bayesian model upda ing (BMU) and he ecu si e assimila ion
o new in o ma ion [53, 54], one can es ima e he pos e io unce ain y associa ed wi h each candida e model o 66
esolu ion, he eby suppo ing da a-in o med and anspa en decisions. Consequen ly, he model selec ion p oblem
can be na u ally o mula ed as a ask o decision-making unde unce ain y [55]. The ul ima e goal is o iden i y he68
i - o -pu pose model ha o e s he bes balance be ween accu acy in eco e ing quan i ies o in e es (QoIs) and
compu a ional e iciency. To achie e his, he p oposed amewo k app oaches he model selec ion ask h ough he70
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lens o Bayesian decision heo y, enabling a ional and adap i e choices among compe ing i ual ep esen a ions [56].
The p oposed amewo k begins by de i ing hyb id model ep esen a ions o a ying ideli y le els. Le e ag-72
ing he ex apola ion and gene aliza ion capabili ies o Reduced-O de Models (ROMs) [57], hese a e o mula ed
alongside a Full-O de Model (FOM) ollowing [58, 59], wi h a gene ic pa ame iza ion ha accoun s o en i on-74
men al a iabili y, ope a ional changes, and e ol ing phenomena such as damage o de e io a ion. The ROMs a e
hen embedded in a sequen ial Bayesian in e ence amewo k ha quan i ies he pos e io unce ain y o each model76
esolu ion and guides he selec ion o he mos sui able ep esen a ion [60]. By ecu si ely assimila ing moni o ing
da a, he amewo k adap i ely acks he e ol ing s a e o he physical asse and selec s he model ha maximizes78
expec ed u ili y, balancing p edic ion accu acy and compu a ional e iciency o eal- ime applica ions. As illus a ed
in Figu e 1, he me hod au onomously decides whe he o e ain he cu en model, swi ch o ano he , o igge 80
e aining— hus enabling adap i e, unce ain y-awa e digi al wins ailo ed o he decision con ex .
Con inuous
moni o ing Bayesian In e ence
(o any o he equi alen amewo k)
Mop |d=a gmax
{M1,M2,...,Mn}
Ep|d,Mi[U]
P e ained pa ame ic
esolu ions Mi(p)
Hype
ROM ROM FOM
U ili y unc ion
U( ideli y,e iciency,p|Mi)
Op imal model
a any gi en ime
ˆ
p→Model Upda ing
Response QoIs
Expec ed U ili y
In e media e es ima es
Highes e iciency
Lowes ideli y
Lowes e iciency
Highes ideli y
Figu e 1: G aphical abs ac o he p oposed adap i e digi al- win amewo k. Reduced-o de and ull-o de models
o a ying ideli y a e in eg a ed wi hin a Bayesian decision- heo e ic scheme ha ecu si ely assimila es moni o ing
da a, upda es pa ame e s, and selec s he model ha maximizes expec ed u ili y, ensu ing an adap i e and e icien
ep esen a ion o he physical asse .
This pape is o ganized as ollows: In sec ion 2, he educed-o de modeling me hodology employed o de i e he82
p e ained model esolu ions in Figu e 1 is p esen ed in sho . Sec ion 3 ou lines a Bayesian decision- heo e ic ap-
p oach o model selec ion. In sec ion 4, we desc ibe he nume ical case s udies used o alida ion, and he nume ical84
esul s ha highligh he e iciency and e ec i eness o he p oposed amewo k. Las ly, sec ion 5 concludes he pape
by summa izing ou con ibu ions, o e ing insigh s, and sugges ing di ec ions o u u e esea ch.86
2. Pa ame ic Reduced O de Models
The i s componen o he p oposed amewo k in ol es de eloping pa ame ic educed-o de models (ROMs)88
h ough physics-based educ ion, yielding low-dimensional su oga es o he ull-o de model (FOM). These ROMs,
illus a ed in Figu e 1, balance compu a ional speed wi h p edic i e accu acy unde a ying condi ions, making hem90
well-sui ed o downs eam asks in S uc u al Heal h Moni o ing (SHM) and P ognos ics and Heal h Managemen
(PHM) [61]. The ollowing subsec ions ou line he p oblem o mula ion ia he go e ning nonlinea equa ions o 92
mo ion and desc ibe he employed ROM me hodology.
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2.1. P oblem S a emen 94
We assume a ailabili y o a Full O de Model (FOM), which co esponds o a Fini e Elemen (FE) model ha
is sui able o nonlinea s uc u al dynamics simula ions. The co esponding dynamical sys em is dependen on he96
inpu ec o p=[p1, ..., pk]T∈Ω⊂Rk, which cap u es all sys em- and exci a ion- ele an pa ame e s. Thus, he
esponse is desc ibed by he ollowing se o go e ning equa ions:98
M¨
u( )+g(u( ),˙
u( ),p)=F( ,p),(1)
whe e u( )∈Rndeno es he esponse in e ms o displacemen s, M∈Rn×n he mass ma ix, and F( ,p)∈Rn he
induced exci a ion. Fo simplici y, he pa ame ic dependence o he mass ma ix is omi ed. Nonlinea beha io 100
is cap u ed h ough he es o ing o ce e m g!(u( ),˙
u( ),p)∈Rn, which may ep esen e ec s such as plas ici y,
hys e esis, o in e ace nonlinea i ies, depending on he sys em esponse and pa ame e ec o p. The ini e elemen 102
(FE) model se ing as he ull-o de model (FOM) is a disc e ized o m o Equa ion (1). I s ull-o de dimension n
de ines he size o he coo dina e space—and hus he o al numbe o deg ees o eedom—go e ning he model’s104
compu a ional cos , as nume ical ope a ions scale wi h n.
2.2. P ojec ion-based model o de educ ion106
The educ ion app oach employed he ein begins wi h a Gale kin p ojec ion scheme, wi h mo e gene al ex en-
sions such as he Pe o –Gale kin me hod discussed in [62, 63]. As ou lined in sec ion 1, a physics-based educ-108
ion is adop ed o enhance in e p e abili y and applicabili y wi hin highe -le el S uc u al Heal h Moni o ing (SHM)
amewo ks. This app oach assumes ha he sys em’s dynamics—i.e., he solu ion o Equa ion (1)— eside in a low-110
dimensional subspace o ank ≪n, whe e nis he ull-o de dimension. Acco dingly, he sys em esponse can be
exp essed as:112
u(p)≈V(p)q(2)
whe e V∈RN× ep esen s he p ojec ion basis ha exp esses he a o emen ioned subspace o he Reduced O de
Model (ROM) and q∈R is he low-o de coo dina e ec o . Via subs i u ion o uin o Equa ion (1) and a e 114
mul iplying he go e ning se wi h uT, hus pe o ming a Gale kin p ojec ion, he ollowing can be de i ed:
˜
M¨
q( )+˜
g(¨
q,˙
q,p)=˜
F(p, )(3)
whe e ˜
M=VTMV,˜
g=VTgand ˜
F=VTF. Se e al echniques exis o cons uc ing he p ojec ion basis V[8]. In116
his wo k, he P ope O hogonal Decomposi ion (POD) is employed, using a se o FOM simula ions o e a aining
pa ame e space o assemble he solu ion snapsho s as:118
ˆ
S=hˆ
Up1ˆ
Up2. . . ˆ
UpNs i (4)
whe e ˆ
Upi∈RN×N con ains he displacemen ime his o y o a gi en pa ame ic ealiza ion pi, hence o h e med
as a snapsho , and, as a esul , ˆ
S∈RN×(N ×Ns)is e med he snapsho ma ix. The a iable N ep esen s he numbe 120
o simula ion ( ime) s eps and Nsis he o al numbe o snapsho s. Via Singula Value Decomposi ion (SVD) o ˆ
S he
ROM p ojec ion basis can be assembled as ollows122
ˆ
S=ΛΣZT(5)
and a e unca ing Λ:
V=hΛ1Λ2. . . Λ i(6)
whe e Λiis he icolumn o ma ix Λ, loosely de ined as a P ope O hogonal Decomposi ion (POD) mode. Since he124
low-o de dimension o Equa ion (2) is de ined as , he abo e unca ion is applied o ob ain he i s o hono mal
componen s o V. To de ine , a sui able e o measu e is employed based on he singula alue decay [64, 65].126
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2.3. VpROM: A condi ional Va ia ional Au oencode (cVAE)-boos ed ROM
The dynamic beha io o he sys em, go e ned by he equa ions in Equa ion (1), is highly dependen on he128
pa ame e ec o p. As a esul , a emp ing a p ojec ion-based educ ion wi h a single basis, as desc ibed in Equa-
ion (2), migh necessi a e a la ge numbe o modes, esul ing in a p ohibi i ely la ge dimension and a ROM ha 130
p o es ine icien o imp ac ical. To handle pa ame ic a iabili y and enable esponse in e ence ac oss di e en op-
e a ing condi ions, a common s a egy is o cons uc a se o local p ojec ion bases Vi, each de i ed om FOM132
snapsho s ˆ
U(pi) co esponding o a speci ic pa ame e ealiza ion pi. Subsequen in e pola ion [66] o clus e ing [65]
me hods can hen app oxima e sys em esponses a unseen pa ame e alues [67]. Building upon p io wo k by he134
au ho s [68, 58, 59], he p oposed amewo k employs a condi ional Va ia ional Au oencode (cVAE) as a nonlinea
gene a i e model o lea n a smoo h, gene alizable mapping be ween pa ame e s and basis ep esen a ions. The e-136
sul ing VpROM no only accele a es esponse p edic ion bu also quan i ies con idence in i s es ima es, enhancing
i s alue o SHM applica ions. Fu he de ails on he amewo k’s componen s can be ound in [69] and ela ed138
e e ences.
To add ess pa ame ic a iabili y a he ROM le el, he wo-s age p ocess in oduced in [59] is used. Fi s , a140
collec ion o local bases is ob ained as desc ibed and he ollowing sys em is sol ed in he leas -squa es sense:
Vipi=Vglobal ∗Xipi(7)
whe e Vi∈Rn× is an ins ance o he collec ion o local bases, Vglobal ∈Rnט cap u es he dynamics ac oss he en i e142
domain and Xi∈R˜ × is a coe icien ma ix. ˜ signi ies he o al numbe o unca ed odes e ained on Vglobal and
can be compu ed simila ly o . Second, in e pola ion is pe o med on he coe icien ma ices Xi. These comp ise a144
educed size (˜ ≪ ), hus emo ing any dependency on he FOM dimension no e ing addi ional e iciency. A e
in e pola ing he coe icien ma ices Xi, he co esponding local ROM basis Vcan be ob ained o any alida ion146
pa ame ic sample. In [59] he local bases Va e p ojec ed i s o he angen space o he p ope G assmannian
mani old, whe e he in e pola ion ope a ions pe o med will ou pu a local basis ha e ains key p ope ies like o -148
hogonali y [70, 71]. Al e na i ely, a e-o hogonaliza ion s ep can be pe o med a e in e pola ing he coe icien
ma ices and eco e ing he p ojec ion basis.150
This wo k builds upon he amewo k in oduced in [58], which employs a condi ional Va ia ional Au oencode
(cVAE) as a gene a i e model capable o ecu si ely in e ing he local basis om ea u es ex ac ed om moni o ing152
da a. The cVAE ac s as a nonlinea gene a o and app oxima o o he coe icien ma ices X, and consequen ly o he
local p ojec ion subspaces Vand associa ed ROMs. Pa ame ic a iabili y is inco po a ed di ec ly a he ROM le el154
by condi ioning he cVAE on known sys em pa ame e s o on ea u es de i ed om measu ed esponses. In p e ious
o mula ions [69, 58], he VpROM was designed in a gene alized manne , assuming no p io knowledge o he ue156
pa ame e ec o pdu ing aining o deploymen ; ins ead, he cVAE in e s he local basis om obse ed esponse
ea u es. When he pa ame e s pa e known, hey can di ec ly se e as condi ioning inpu s. Finally, he p obabilis ic158
na u e o he cVAE allows he VpROM o quan i y p edic ion unce ain y: he la en space de ines a lea ned p obabili y
dis ibu ion whose sampling enables p opaga ion o unce ain y h ough he ROM, yielding con idence bounds and160
e o es ima es on he p edic ed esponses.
The o e all amewo k is illus a ed in Figu e 2. The condi ioning ea u es Wa e conca ena ed wi h he inpu X162
and he la en a iables Z. Fo mally, he encode lea ns he condi ional dis ibu ion qθ(Z|X,W), while he decode
econs uc s he da a ia pϕ(X|Z,W). Fo simplici y, we se W=p, since pa ame e in e ence is handled sepa a ely164
wi hin he Bayesian in e ence amewo k shown in Figu e 1. Acco dingly, he sys em a iabili y in Equa ion (1) is
ep esen ed by p, and he mapping om p o he local bases Vin Equa ion (3) is lea ned h ough he ela ionship166
be ween pand he educed coe icien ma ices Xin Equa ion (7).
Once ained, he condi ional VAE se es as a gene a i e model capable o sampling he educed basis coe icien s168
Xide ined in Equa ion (7). Gi en an in e ed pa ame e ealiza ion ˆ
p om Bayesian in e ence, samples d awn om
he la en dis ibu ion a e decoded o gene a e he co esponding X alues. In his wo k, a diagonal Gaussian p io is170
adop ed o he la en space, as depic ed in Figu e 3.
2.4. Unce ain y Bounds ia he Unscen ed T ans o m172
Once he decode is ained, p edic ions a e ob ained by sampling he in e ed a ia ional dis ibu ion o he la-
en space and decoding hese samples. A ealiza ion o he andom a iable ϵis i s d awn and ans o med in o Z174
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X
W
+Encode
qθ(Z|X,W)
µθ
σθ
ε
⊙
+Z
W
N(0,1)
+Decode
pϕ(ˆ
X|Z,W)ˆ
X
Figu e 2: The a chi ec u e o he employed cVAE. The condi ioning ea u es Wa e injec ed ia conca ena ion wi h
he inpu ec o Xand he la en space Z.
µθ
σθ
ε
⊙
+Z
ˆ
p
Ob ained ia Bayesian In e ence
+
N(0,1)
Decode
pϕ(ˆ
X|Z,p)ˆ
Xˆ
V
ˆ
V=Vglobal ∗ˆ
X
Figu e 3: A chi ec u e o he cVAE in basis gene a ion mode: The p io dis ibu ion ϵis sampled, and he la en
ec o s a e aken a e conca ena ing wi h he (in e ed) ec o ˆ
p. Dimensionali y se es only illus a i e pu poses.
using he de e minis ic mean µθand s anda d de ia ion σθin e ed by he encode . The la en ec o Zis hen con-
ca ena ed wi h he in e ed pa ame e s ˆ
pand passed h ough he decode o gene a e samples om pϕ(X|Z,p)—each176
ep esen ing a possible ealiza ion o he educed coe icien s X. By epea ing his p ocess, he mean and a iance o
he es ima ed coe icien s and p ojec ion bases can be e alua ed. Subsequen pa allel ROM simula ions using hese178
sampled bases, as de ined in Equa ion (3), enable unce ain y p opaga ion om he la en space o he ou pu esponse.
The esul ing ensemble s a is ics yield nume ical e o bounds ha quan i y p edic ion unce ain y, as demons a ed180
in [68, 58].
The sampling p ocedu e desc ibed abo e can be compu a ionally expensi e, as hund eds o samples may be e-182
qui ed o cap u e he sys em’s esponse unce ain y [58]. To imp o e e iciency, we adop he Unscen ed T ans o m
(UT) [72, 73], which exploi s he Gaussian na u e o he la en dis ibu ions o gene a e a de e minis ic se o sigma184
poin s ins ead o nume ous andom samples. These sigma poin s a e p opaga ed h ough he decode , and he esul -
ing ou pu s a e combined using p ede ined weigh s ha p ese e he mean and co a iance o he o iginal Gaussian186
dis ibu ion, yielding accu a e and e icien unce ain y es ima es.
2.5. Hype - educ ion188
Hype - educ ion p o ides a second-le el app oxima ion ha alle ia es he compu a ional cos o upda ing and e-
cons uc ing he nonlinea e m ˜
gin Equa ion (3) [74]. He e, we employ he Ene gy Conse ing Mesh Sampling and190
Weigh ing (ECSW) me hod [75], a well-es ablished physics-based echnique. Comp ehensi e discussions o ECSW
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and ela ed app oaches can be ound in [76, 77, 78]. In b ie , ECSW selec s a subse o ini e elemen s om he192
FE disc e iza ion—used he e as he FOM—by sol ing a nonlinea op imiza ion p oblem ha ensu es he weigh ed
p ojec ions o he nonlinea e ms accu a ely app oxima e he o al in e nal wo k. Once he op imal subse and co e-194
sponding weigh s a e de e mined, a Hype -ROM is ob ained, as depic ed in Figu e 1, o e ing a subs an ial educ ion
in compu a ional cos . Among he a ailable model esolu ions, he Hype -ROM achie es he highes e iciency, he196
FOM he highes ideli y, and s anda d ROMs occupy he in e media e ange.
3. A Bayesian decision- heo e ic app oach o model selec ion198
The amewo k ou lined in he p e ious sec ion can be employed o gene a e di e en model esolu ions o assis
wi h decision-making and SHM asks. Speci ically, we ha e highligh ed h ee esolu ions in Figu e 1, each one200
co esponding o a ade-o be ween compu a ional e iciency and ideli y: a) The ull-o de FE model o he sys em,
e med FOM, which o e s he highes possible p ecision, b) he ROM ha s ike a balance be ween p ecision and202
e iciency, and c) he Hype ROM, which co esponds o he de eloped ROM equipped wi h hype - educ ion, and
po en ially o e s he highes e iciency. Fu he mo e, he p oposed amewo k allows o he de elopmen o i ual204
models condi ioned on sys em dependencies, hus cap u ing pa ame ic a iabili y.
As physical in as uc u e asse s e ol e o e ime, a ecu si e challenge eme ges: de e mining which pa ame ic206
ins ance and model esolu ion p o ide he bes balance be ween compu a ional cos and p edic i e accu acy. This
model selec ion ask, condi ioned on incoming esponse da a, can be na u ally o mula ed as a decision-making unde 208
unce ain y (DMUU) p oblem, whe e he op imal choice is he model ha maximizes he expec ed u ili y [79, 80].
In decision heo y, u ili y se es as a o mal measu e o e alua e he ela i e desi abili y o al e na i e ac ions. I 210
is de ined h ough an objec i e unc ion ha maps he a ibu es o a decision o a nume ical alue ep esen ing i s
o e all bene i . In he p esen con ex , wo p ima y a ibu es go e n he u ili y o a model:212
1. Model ideli y, in e ms o eco e ing a ge Quan i ies o In e es (QoIs), and
2. Model cos , in e ms o compu a ional complexi y and un ime.214
These a ibu es a e cen al o he p oblem, as i ual models used in decision-making and SHM applica ions
inhe en ly balance p ecision agains compu a ional cos . They a e also compe ing by na u e: highe p ecision ypi-216
cally en ails highe cos . Bayesian decision heo y [56, 79] o e s a igo ous amewo k o sol ing decision-making
unde unce ain y (DMUU) p oblems, whe e in o ma ion becomes a ailable ecu si ely and p og essi ely educes218
unce ain y in decision-making.
Le θdeno e he andom ec o o unce ain model pa ame e s. Fo simplici y, we assume hese coincide wi h220
he modeling pa ame e s o he FOM and ROMs in Eqs. 1, 3, and Figu e 3, i.e., θ=ˆ
p. This assump ion is no
es ic i e— he amewo k emains alid p o ided θ⊆p. Hence, he model pa ame iza ion in oduced in sec ion 2222
is kep in en ionally gene al, as he pa ame e ec o di ec ly go e ns model selec ion. In u n, a p io p obabilis ic
model πp (ˆ
p) needs o be assigned. The incoming eco ded signals, collec i ely deno ed as moni o ing da a d, can224
be le e aged o pe o m in e ence and Bayesian model upda ing (BMU), which ou pu s he pos e io dis ibu ion o
ˆ
pgi en d, deno ed by πpos(ˆ
p|d). The op imal model o be selec ed Mop condi ional on he moni o ing da a dcan be226
hen ob ained as:
Mop |d=a gmax
{M1,M2,...,Mn}
Eˆ
p|d,Mi[Ui],(8)
whe e Ui=U( ideli y|Mi,e iciency|Mi,ˆ
p|Mi)
whe e Uis he conside ed u ili y unc ion ha ela es he unce ain pa ame e ec o ˆ
pand models M, hus mapping228
possible ou comes o hei u ili y as illus a ed in Figu e 4. The expec a ion Ein Equa ion (8) is e alua ed wi h espec
o he pos e io dis ibu ion πpos(ˆ
p|y,Mi). The possible decisions a e summa ized in he se {M1,M2,...,Mk}in he230
case o kcompe ing models, whe e Mi ep esen s he decision o employ he i- h model o in e ing he pa ame e
and eco e ing he dynamic beha io o he enginee ed sys em o in e es .232
As discussed, wo compe ing a ibu es de ine he u ili y unc ion Uin his wo k: model ideli y and compu a ional
e iciency. Illus a i e examples o he esul ing u ili y b anches a e shown in Figu e 4. In Figu e 4a he conside ed234
u ili y cu e wi h espec o he abili y o he model o eco e he quan i y o in e es is depic ed, also accoun ing o
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−20 −10 0 10 20
Measu ed QoI - P edic ion (%)
0.0
0.2
0.4
0.6
0.8
1.0
U ili y U
Unde es ima ion
O e es ima ion
(a) U ili y U as a unc ion o he model ideli y.
0.1 1 10
E alua ion ime (secs)
0.0
0.2
0.4
0.6
0.8
1.0
U ili y Ue
(b) U ili y Ueas a unc ion he model un ime.
Figu e 4: Example u ili y unc ions o he conside ed compe ing a ibu es, namely i) model ideli y, which accoun s
o o e /unde -es ima ing a ge quan i ies, and (ii) compu a ional un ime o ensu e eal- ime in e ence when possible.
o e - and unde -es ima ion e ec s. Unde es ima ion o quan i ies o in e es is he e penalized mo e as i migh lead236
o unexpec ed ailu es and ca as ophic e en s, which a e conside ed mo e c i ical in asse managemen compa ed o
po en ial addi ional cos s caused by o e es ima ion. Since he Bayesian in e ence amewo k is also employed o pa-238
ame e es ima ion, he u ili y wi h espec o he model’s p ecision is e alua ed using eadings om edundan senso
measu emen s ha a e no employed on he in e ence o ˆ
p. In Figu e 4b he implemen ed ela ionship be ween he240
e alua ion ime o he model and i s u ili y is epo ed. The o e all u ili y can be compu ed as a weigh ed combina ion
o he compe ing a ibu es, allowing he decision-make o weigh he conside ed a ibu es acco ding o he decision242
a hand. The co esponding ma hema ical o mula ion eads:
Ui( ideli y|Mi,e iciency|Mi,ˆ
p|Mi)=γ∗Ui( ideli y|Mi,ˆ
p|Mi)+(1−γ)∗Ui(e iciency|Mi,ˆ
p|Mi) (9)
Ui( ideli y|Mi,ˆ
p|Mi)=U (ˆ
p|Mi)=












1−w1∗y −ˆy
y 2
∗(w1∗ϵmax)−1,y −ˆy >0
1−w2∗y −ˆy
y 2
∗(w2∗ϵmax)−1,y −ˆy ≤0
(10)
Ui(e iciency|Mi,ˆ
p|Mi)=Ue(ˆ
p|Mi)=1−log2 Mi
log2( max)(11)
whe e y deno es he obse ed alues o he QoI and ˆy he model-app oxima ed ones using he in e ed ˆ
pand he se-244
lec ed model esolu ion Mi. The a iable Mideno es he e alua ion ime o model esolu ion Mi, max is he maximum
accep able e alua ion ime ha co esponds o ze o u ili y and ϵmax he maximum p edic ion e o ha is penalized246
wi h ze o u ili y. The weigh s w1,w2a e chosen p ope ly o accoun o o e - and unde -es ima ion e ec s (w1>w2)
and o no malize he compu ed disc epancy so ha he maximum possible u ili y is equal o 1, as illus a ed in Fig-248
u e 4, whe e w1=3,w2=1, max =10s, ϵmax =20%. The ac o γweighs he impo ance o ideli y in he ask a
hand, also e lec ing a balancing ac wi h he co esponding e iciency.250
Fo he Bayesian model upda ing (BMU) ask, he imp o ed T ansi ional Ma ko Chain Mon e Ca lo (iTMCMC)
me hod is employed [49, 81], adap ed om he open-sou ce implemen a ion a ailable a h ps://gi hub.com/252
ERA-So wa e/O e iew. An ini ial ensemble o ns=1000 samples is d awn om he p io dis ibu ion πp (ˆ
p).
The upda ing s ep is hen pe o med condi ionally on he obse ed da a y o ob ain he pos e io dis ibu ion o he254
unce ain pa ame e s, πpos(ˆ
p|y). The disc epancy be ween he measu ed accele a ion signals yand he model-p edic ed
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esponses ˆ
yis modeled p obabilis ically as:256
nRMS E(ˆ
p,Mi)=
u
u
Pn
j=1yj−ˆ
yj(ˆ
p,Mi)2
Pn
j=1y2
j
∈Rn ∼ N0,Σ=diag(c2||y||2)(12)
whe e N(0,Σ) deno es he mul i a ia e no mal (MVN) dis ibu ion wi h a ze o-mean ec o and co a iance ma ix Σ.
A diagonal co a iance ma ix is assumed, wi h he a iance o each componen assumed p opo ional o he L2-no m258
o y. The ac o ccan be ega ded as a coe icien o a ia ion, and i s chosen alue e lec s he o al p edic ion e o .
I is he e assumed ha c=0.10. The likelihood unc ion can be hen w i en as:260
L(p;y)∼ Nη;0,Σ,(13)
whe e N(·;0,Σ) deno es he alue o he MVN densi y unc ion a a speci ied loca ion. Thus, he expec ed ideli y
u ili y U o employ model Mican be app oxima ed using nspos e io samples om πpos(ˆ
p|y) as:262
Eˆ
p|y,Mi[U (ˆ
p|Mi)] ≈1
ns
ns
X
j=1
U Mi,ˆ
pj,(14)
The equi alen is implemen ed o Ue.
4. Nume ical case s udies264
The p oposed amewo k is alida ed in wo s uc u al dynamics case s udies, in ol ing hys e e ic, geome ic,
and ma e ial nonlinea i ies. In bo h cases, he app oach pe o ms ecu si e model selec ion o iden i y he op imal266
esolu ion o s uc u al asse s whose s a e, pa ame e s, and heal h condi ion e ol e o e ime. The conside ed model
esolu ions a e summa ized in Table 1.268
Table 1: Re e ence able o he conside ed model esolu ions.
Re e ence name Desc ip ion
FOM The ull-o de FE model
VpROM ROM acco ding o he o mula ion in subsec ion 2.3.
H–VpROM VpROM addi ionally equipped wi h hype - educ ion
Pe u bed FOM The ull-o de FE model wi h a ine mesh and pe u bed ma e ial p ope ies.
Used o gene a ing esponse signals wi h a 8% noise le el o es ing.
Gi en he limi ed a ailabili y o expe imen al o ield moni o ing da a, simula ed measu emen s a e used o
alida ion. These a e gene a ed om an independen nume ical ini e elemen (FE) model, e e ed o as he pe u bed270
FOM, as summa ized in Table 1. To clea ly dis inguish he moni o ing da a used o es ing om he FOM employed
in cons uc ing he ROMs, he ollowing measu es a e adop ed:272
•The pe u bed FOM employs a ine ini e elemen mesh han he in-house FOM used o ain he ROMs.
•The Young’s modulus Eis andomly pe u bed, wi h elemen -wise alues d awn om a no mal dis ibu ion274
wi h mean E e and s anda d de ia ion 5,GPa.
•Measu emen noise co esponding o 8% o he oo -mean-squa e (RMS) alue o each signal is added.276
The pe u bed FOM is employed o gene a e he es ing da a in o de o a oid he in e se c ime [82, 83], which
a ises when he same o nea ly iden ical model is used bo h o gene a e and o in e sys em esponses in an in e se278
p oblem [84].
The pe o mance o he p oposed amewo k in pa ame e and esponse in e ence (see Figu e 1) is assessed by280
ep oducing he ime-his o y esponse o selec ed quan i ies o in e es o hei spa ial dis ibu ion a speci ic simula-
ion snapsho s. Pa ame e in e ence is based on ib a ion measu emen s om accele ome e s assumed o be ins alled282
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4.2. Simpli ied uselage panel
The second nume ical example conce ns a simpli ied uselage panel, a ep esen a i e load-ca ying componen in418
bo h ma i ime and ae ospace s uc u es, and he e o e o b oad p ac ical ele ance. This example is loosely inspi ed
by [93], whe e dynamic subs uc u ing was applied o analyze c acking in hin-walled ae onau ic s uc u es. He e,420
we adop and ex end he implemen a ion p oposed in [20] o model assembly, modi ying i o accoun o la ge
de o ma ions and he esul ing geome ic nonlinea i ies.422
Figu e 11: Simpli ied uselage panel geome y, loading, and senso loca ions. The blue senso s a e used o in e ence
while he ed edundan senso is employed o pe o mance and u ili y e alua ion.
The uselage panel geome y, bounda y condi ions, loading, and senso layou a e shown in Figu e 11. The panel is
cylind ical wi h a adius o 2.5 m and a uni o m hickness o 3 mm. An aluminum ma e ial is assumed, wi h a a iable424
Young’s modulus Epin oduced o ampli y la ge de o ma ions and hus cap u e geome ic nonlinea i ies. Unlike
p e ious s udies employing ee– ee o ully ixed condi ions, he p esen se up applies ixed bounda y condi ions426
only along he le edge, while he igh edge is connec ed h ough sp ings o a iable s i ness kb o ep esen bol ed
join loosening. The s i ness pa ame e anges om 0 ( ee end) o 1 ( ully ixed). The uselage is disc e ized428
using 4,393 MITC4 elemen s, esul ing in 47,334 deg ees o eedom. A dynamic p essu e load is applied uni o mly,
ollowing a whi e-noise-like signal wi h pa ame ic magni ude pin he equency ange [1, 250] Hz o e a du a ion430
o 2 s. Time in eg a ion is pe o med using he implici Newma k scheme wi h a s ep size o ∆ =2×10−4s.
Table 5: Sys em p ope ies and ange o pa ame ic ai s.
Pa ame e : Ep(×25 GPa)kb(×1e16)p(kN/m2) Poisson’s a io Densi y (kg/m3)
Range: [1.0,2.0] [0.0,1.0] [0.1,1.0] ν=0.3ρ=2700
The sys em p ope ies and he pa ame ic dependencies o he model a e summa ized in Table 5. The pa ame e 432
ec o o be in e ed is p=[Ep,kb,p]. The ma e ial pa ame e Epcan simula e po en ial damage on he uselage and
in luence he geome ically nonlinea beha io di ec ly. Simila ly, pa ame e kbcon ols he igidi y o he bounda y434
and can model ixed- ee o ixed- ixed condi ions and bol loosening o equi alen damage e en s in a simpli ied
manne . Las ly, he ampli ude o he exci a ion pin luences he de o ma ion ampli ude, and hus, he nonlinea 436
beha io .
Fo aining he ROMs lis ed in Table 1, =32 basis modes a e employed along wi h 500 aining samples o p438
gene a ed ia La in Hype cube Sampling (LHS). Tes ing is pe o med using 200 pa ame ic samples. The moni o ing
se up, summa ized in Table 6, includes eigh accele a ion senso s placed a dis inc nodes (blue do s in Figu e 11),440
each eco ding accele a ion along he exci a ion di ec ion. These measu emen s a e used o he pa ame e in e ence
ask, as desc ibed in Figu e 1. A edundan senso (senso 9, shown in ed in Figu e 11) p o ides independen ib a ion442
da a o igge adap i e model selec ion and assess model ideli y. The u ili y unc ions ollow he same o mula ion
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as in Equa ions 9–11 and Figu e 4, excep ha in he e iciency u ili y, he maximum penalized e alua ion ime (ze o444
u ili y) is se o 100 ins ead o 10 seconds.
Table 6: De ails o he moni o ing se up o he uselage panel.
Scena io Panel exhibi s geome ic nonlinea i y and pa ame ic ma e ial and bounda y
ai s while expe iencing 2s exci a ion windows o a ying ampli ude.
Objec i e Selec model o be used o SHM- ela ed downs eam asks
Moni o ing Da a Nodal accele a ions
Eigh nodes moni o ed (blue do s in Fig. 11)
QoIs o U ili y compu a ion Fideli y: Max. accele a ion a edundan senso 9 (No malized RMSE)
E iciency: A g. model e alua ion ime (s)
U ili y unc ions Equa ions 9-11, max =100s,w1=3,w2=1, ϵmax =25%
Moni o ing window 0.5 seconds olling windows
As summa ized in Table 6, he es ing scena ios consis o epea ed 2-second windows wi h e ol ing pa ame ic446
ai s. The exci a ion ampli ude p a ies eely wi hin each sequence, while he ma e ial modulus Epand bounda y
s i ness kbdec ease p og essi ely o emula e s uc u al de e io a ion. Fo ins ance, Epmay dec ease om 1.65 o448
1.42, 1.12, and so o h. Pe o mance alida ion includes 50 such combina ions, wi h pa ame e ealiza ions d awn ia
LHS. A ep esen a i e example o he Bayesian pa ame e in e ence esul s o he exci a ion ampli ude and bounda y450
s i ness is shown in Figu e 12; simila ends a e obse ed o he ma e ial pa ame e Ep. The epo ed esul s
co espond o he bes -pe o ming 2-second window among he olling sequences. Despi e mino disc epancies, he452
p oposed amewo k success ully econs uc s he e ol ing pa ame ic con igu a ion om he sensing da a, enabling
adap i e model upda es in eal ime.454
0.2 0.4 0.6 0.8 1.0
0
2
4
6
8
10
12
14
Densi y
T ue Value
P io
H-VpROM
VpROM
FOM
(a) In e ence o exci a ion ampli ude coe icien kb.
0.2 0.4 0.6 0.8 1.0
0
2
4
6
8
10
12
Densi y
T ue Value
P io
H-VpROM
VpROM
FOM
(b) In e ence o bounda y s i ness coe icien p.
Figu e 12: Bayesian pa ame e in e ence o ob aining he pos e io dis ibu ion o he unce ain model pa ame e s.
A ep esen a i e a e age pe o mance example is shown.
As shown in Figu e 12, he FOM exhibi s he highes p ecision in cap u ing he unde lying pa ame e s, as ex-
pec ed. This end is con i med in Figu e 13a, whe e he FOM achie es he g ea es ideli y u ili y U . Howe e , his456
accu acy comes a a subs an ial compu a ional cos , as e lec ed by he e y low e iciency u ili y Uein Figu e 13b.
Con e sely, he H-VpROM p o ides hype -accele a ed e alua ions, yielding high Ue alues while main aining accep -458
able accu acy. The VpROM demons a es a simila end, a aining a di e en bu s ill a o able accu acy–e iciency
ade-o . These esul s highligh he impo ance o he u ili y-based o mula ion, which quan i a i ely balances com-460
pe ing a ibu es and enables da a-in o med model selec ion and alida ion.
ollowing Equa ion (9), he compe ing u ili y a ibu es can be balanced h ough an app op ia e choice o he462
weigh ing coe icien γ. This pa ame e can be de ined by he decision make , allowing ask-speci ic p io i iza ion
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23456
0.4
0.6
0.8
1
Time (s)
U ili y U
H-VpROM VpROM FOM
(a) U ili y U w model’s p ecision.
23456
0
0.5
1
Time (s)
U ili y Ue
H-VpROM VpROM FOM
(b) U ili y Uew model’s e iciency.
Figu e 13: E alua ions o he ideli y and e iciency u ili y unc ions o he conside ed model esolu ions o a ep e-
sen a i e example es ing scena io. Red do ed lines indica e pa ame e changes.
be ween ideli y and e iciency. Two illus a i e examples o he esul ing o al u ili y o di e en γ alues a e p e-464
sen ed in Figu e 14. As shown in Figu e 13, he H-VpROM consis en ly achie es he highes u ili y unde he equal
weigh ing scheme, owing o i s supe io compu a ional pe o mance, and is hus iden i ied as he op imal model o 466
he gi en ask (see Figu e 14a). When p ecision is p io i ized (γ=0.75), howe e , he VpROM becomes mo e a-
o able, as shown in Figu e 14b. These esul s emphasize he alue o u ili y-based model selec ion, which enables468
con ex -awa e adap a ion by quan i ying ade-o s be ween compe ing objec i es.
23456
0.4
0.6
0.8
1
Time (s)
U ili y U
H-VpROM VpROM FOM
(a) Equally impo an u ili y a ibu es (γ=0.50).
23456
0.6
0.7
0.8
0.9
Time (s)
U ili y U
H-VpROM VpROM FOM
(b) P ecision p io i ized o e e iciency (γ=0.75).
Figu e 14: To al u ili y e alua ions o di e en weigh ed combina ions o he conside ed a ibu es in Figu e 13.Red
do ed lines indica e pa ame e changes.
Following he model upda ing and selec ion p ocess, and based on he pos e io dis ibu ions in Figu e 12, he470
H-VpROM is employed o esponse p edic ion. Owing o i s physics-awa e o mula ion, he model can eco e
quan i ies o in e es ac oss all physical ields. Samples d awn om he pos e io dis ibu ions a e p opaga ed h ough472
he H-VpROM o quan i y con idence bounds on he p edic ed esponses. By u he combining hese samples wi h
d aws om he model’s p obabilis ic la en space ia he Unscen ed T ans o m (see subsec ion 2.4), he esul ing474
bounds also accoun o unce ain y in he educed ep esen a ion i sel . The co esponding H-VpROM p edic ions a e
shown in Figu es 15–16 o he sys em’s displacemen and accele a ion esponses. The epo ed a e age and maximum476
e o es ima es a e compu ed ac oss all es ing con igu a ions, using he highes -u ili y H-VpROM iden i ied in he
18
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ini ial olling windows.478
0 0.5 1 1.5 2
−1
0
1
2
Time (s)
Displacemen (mm)
Pe . FOM
H-VpROM
Con . bounds
(a) A e age app oxima ion.
0 0.5 1 1.5 2
−1
0
1
2
Time (s)
Displacemen (mm)
Pe . FOM
H-VpROM
Con . bounds
(b) Maximum e o app oxima ion.
Figu e 15: P edic ed displacemen esponse a he edundan node wi h associa ed con idence bounds. Repo ed
alues include he a e age and maximum app oxima ion e o s ac oss all es ing samples.
As shown in Figu e 15a, he amewo k e ec i ely balances compu a ional e iciency and accu acy, deli e ing
highly consis en esponse es ima es on a e age. Mo eo e , he p obabilis ic o mula ion and embedded Bayesian480
componen s enable he amewo k o p o ide con idence bounds ha enclose he ue esponse, e en unde he maxi-
mum e o scena io depic ed in Figu e 15b. Simila ends a e obse ed o he accele a ion esponse in Figu e 16. As482
discussed in he p e ious case s udy, he decision-make can de ine a p ecision u ili y h eshold U o au oma ically
igge model e aining in eal ime, a oiding delays associa ed wi h ull FOM e alua ions. This app oach mi iga es484
local minima in Figu e 13 and imp o es high-e o cases such as ha in Figu e 15b. Al e na i ely, by adjus ing
he weigh ing scheme be ween compe ing a ibu es, he amewo k can p io i ize e iciency and selec he VpROM486
ins ead, as illus a ed in Figu e 14b.
0.5 1 1.5 2
−50
0
50
Time (s)
Accele a ion (m/s2)
Pe . FOM
H-VpROM
Con . bounds
(a) A e age app oxima ion.
0 0.5 1 1.5 2
−100
−50
0
50
100
Time (s)
Accele a ion (m/s2)
Pe . FOM
H-VpROM
Con . bounds
(b) Maximum e o app oxima ion
Figu e 16: F amewo k’s accele a ion esponse app oxima ion a he edundan node and co esponding e o bounds.
The a e age and maximum e o pe o mance ac oss es ing samples is epo ed.
19
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5. Discussion488
The de eloped amewo k add esses a cen al challenge in eal- ime Ope a ions and Managemen (O&M), S uc-
u al Heal h Moni o ing (SHM), and decision suppo — he adap i e selec ion o he lowes -cos model capable o 490
deli e ing p edic ions o he equi ed ideli y unde e ol ing condi ions [4]. A i s co e, he p oposed amewo k
in eg a es a gene a i e educed-o de modeling (ROM) scheme, condi ioned on pa ame ic inpu s, wi h a Bayesian in-492
e ence laye ha ecu si ely es ima es he unde lying sys em pa ame e s om sensing da a. This combina ion enables
con inuous model adap a ion, quan i ies p edic ion con idence, and suppo s decision-making h ough a u ili y-based494
o mula ion ha balances wo compe ing a ibu es: (i) model ideli y and (ii) compu a ional e iciency. By maximiz-
ing he expec ed u ili y, he amewo k iden i ies he i - o -pu pose model a each ime s ep, de e mining whe he o496
accep , swi ch, o e ain a candida e ep esen a ion.
Two nume ical case s udies—a hys e e ic shea ame wi h une en damage and a geome ically nonlinea uselage498
panel—demons a e he amewo k’s abili y o selec op imal model esolu ions and adap o changing sys em condi-
ions. The app oach eliably iden i ies when educed-o de models lose alidi y and e aining is equi ed, pa icula ly500
du ing ex eme e en s beyond he aining domain. Mo eo e , he p obabilis ic o mula ion p o ides unce ain y quan-
i ica ion, gene a ing esponse con idence bounds ha cap u e he ue beha io e en unde low-p ecision condi ions.502
Se e al limi a ions ou line a enues o u u e esea ch. Fi s , he cu en iTMCMC-based Bayesian upda ing
emains compu a ionally demanding o high-dimensional p oblems; u u e wo k will explo e mo e e icien in e ence504
s a egies o eal- ime deploymen . Second, he amewo k assumes ha all ele an physics a e ep esen ed wi hin
he a ailable models. A ully adap i e sys em should also handle un o eseen o unmodeled phenomena, whe e e en506
high- ideli y models may yield low u ili y—a challenge poin ing owa d sel -e ol ing digi al ep esen a ions. Thi d,
while his s udy uses pe u bed simula ion da a o a oid in e se c ime, expe imen al alida ion will be essen ial o508
assess obus ness in eal-wo ld condi ions. Finally, he pa ame ic o mula ions employed may no ully cap u e e ec s
such as exci a ion equency con en o mode in e ac ions; hese aspec s will be explo ed alongside he ex ension510
owa d sel -e ol ing model s uc u es.
Da a A ailabili y512
The models and da a suppo ing his s udy’s indings a e a ailable om he au ho KV upon eques .
Decla a ion o Compe ing In e es 514
The au ho s decla e ha hey ha e no known compe ing inancial in e es s o pe sonal ela ionships ha could
ha e appea ed o in luence he wo k epo ed in his pape .516
Acknowledgemen s
The au ho KV, EC g a e ully acknowledges he unding om he Eu opean Commission unde he Ho izon Eu-518
ope unding gua an ee, o he p ojec s ‘INBLANC - INdus ialisa ion o Building Li ecycle da a Accumula ion,
Nume acy and Capi alisa ion’ (g an ag eemen No: 101147225) and ‘TURING - T us wo hy Uni ied Robus In el-520
ligen Gene a i e Sys ems’ (g an ag eemen No: 101215032).
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