Physics-in o med Machine Lea ning-based
Me hodology o Pla ed Th ough Holes Li e ime
Es ima ion in P in ed Ci cui Boa ds
Ma co Spe i1, Chinmay Nawghane1, Ba Vande elde1, Nicolas Lammens2, Ma hias Ve beke3,4
1imec, Leu en, Belgium
2Siemens Indus y So wa e n , S a egy & Inno a ion, Leu en, Belgium
3Depa men o Compu e Science, KU Leu en, Belgium
4Flande s Make@KU Leu en, Belgium
[email p o ec ed]
Abs ac —The inc easing demand o eliable elec onics un-
de sco es he need o p edic i e ools o es ima e componen
li e imes and mi iga e key ailu e isks and associa ed cos s.
This s udy ocuses on Pla ed Th ough Holes (PTHs) in P in ed
Ci cui Boa ds, which a e c i ical o sys em eliabili y bu p one
o ailu es unde s anda d he mal cycling due o s ain om
coe icien o he mal expansion misma ches. A physics-in o med
machine lea ning-based me hodology is p oposed, in eg a ing
da a om Fini e Elemen Me hod simula ions and expe imen al
da a om deg ada ion es s. Two machine lea ning models a e
combined o es ima e he Remaining Use ul Li e o PTHs: a
eed o wa d neu al ne wo k (FFNN) able o p edic he numbe
o cycles o ailu e o a gi en s uc u e and ained on a Design
o Expe imen da ase wi h geome ic and ma e ial pa ame e s,
and a Long Sho -Te m Memo y (LSTM) ne wo k o p edic
he empo al deg ada ion end measu ed by eal senso s on
he boa d. The combina ion o hese wo models allows he
implemen a ion o a Physics-in o med Neu al Ne wo k whe e
he physics lea ned based on he FFNN is used as a physical
cons ain in he cos unc ion o he LSTM o guide he
p edic ion o he deg ada ion.
Keywo ds—P in ed ci cui boa ds, pla ed h ough holes, da a-
d i en app oach, deg ada ion, physics-in o med neu al ne wo k
I. INTRODUCTION
The inc easing complexi y and minia u iza ion o elec-
onic sys ems ha e ampli ied he impo ance o ensu ing
hei eliabili y. Failu es in c i ical componen s can lead o
signi ican economic losses and ope a ional dis up ions, espe-
cially in sec o s such as ae ospace, au omo i e, and elecom-
munica ions. Among he key eliabili y challenges in hese
sys ems a e P in ed Ci cui Boa ds (PCBs), which se e as
he backbone o elec onic in e connec ions. Wi hin PCBs,
Pla ed Th ough Holes (PTHs) play a i al ole in es ablishing
elec ical connec ions be ween laye s, bu hei suscep ibili y
o he mal and mechanical s esses makes hem a common
ailu e poin . PTH ailu es a e o en induced by cyclic he mal
loading, which a ises due o he misma ch in he Coe icien
o The mal Expansion (CTE) be ween he coppe pla ing and
he su ounding composi e ma e ial. This misma ch gene a es
localized s ains leading o c ack ini ia ion and p opaga ion
h ough he coppe laye . Such ailu es comp omise he elec-
ical in eg i y o he sys em and a e a majo conce n o
PCB manu ac u e s and end-use s. Expe imen al s udies and
Fini e Elemen Me hod (FEM) simula ions ha e ex ensi ely
in es iga ed hese phenomena, highligh ing he impo ance o
accu a e li e ime p edic ion models o PTHs [1]–[5]. T adi-
ional p e en i e main enance elies on ixed schedules o
inspec ions and componen eplacemen s. While his app oach
minimizes he isk o unexpec ed ailu es, i o en esul s
in unnecessa y in e en ions and inc eased cos s. P edic i e
main enance (PdM), on he o he hand, le e ages eal- ime
condi ion moni o ing and ad anced analy ical models o es i-
ma e he Remaining Use ul Li e ime (RUL) o componen s,
enabling imely in e en ions ha op imize esou ce u iliza-
ion and sys em eliabili y [6], [7]. By employing p edic i e
main enance, sys ems can educe down ime, a oid ca as ophic
ailu es, and enhance ope a ional e iciency. Fo example,
ime se ies p edic ion algo i hms and neu al ne wo k-based
me hods ha e been used o p edic deg ada ion ajec o ies
and p eemp i ely add ess ailu es, showcasing he ad an ages
o PdM o e adi ional s a egies [7]. Two p ima y me hod-
ologies domina e he ield o RUL es ima ion: physics-based
models and da a-d i en models. Physics-based models ely
on undamen al p inciples o desc ibe ma e ial beha io unde
cyclic loading [4], [5]. These models p o ide in e p e abili y
and eliabili y bu o en equi e ex ensi e calib a ion and
compu a ional esou ces. Con e sely, da a-d i en app oaches
le e age senso da a and machine lea ning (ML) algo i hms
o p edic ailu es based on his o ical and eal- ime da a.
Machine lea ning has become a co ne s one o da a-d i en
RUL es ima ion, wi h algo i hms like Long Sho -Te m Mem-
o y (LSTM) ne wo ks demons a ing ema kable success in
cap u ing empo al dependencies in senso da a. LSTMs a e
pa icula ly e ec i e in modeling deg ada ion pa e ns and
p edic ing RUL, as hey can p ocess sequen ial da a while
mi iga ing issues o anishing g adien s common in adi ional
ecu en neu al ne wo ks [8], [9].
Howe e , challenges such as da a sca ci y, o e i ing, and
he black-box na u e o ML models emain signi ican hu dles.
Fig. 1. Wo k low o he p oposed model o li e ime p edic ion o PTHs. The diag am illus a es he da a low and he in eg a ion o he Machine Lea ning
echniques used. S a ing om a sample subjec ed o he mal load, he model ollows wo dis inc b anches. (a) Uppe b anch: a Feed o wa d Neu al Ne wo k
(FFNN) p edic s he Numbe o cycles o ailu e based on da a om FEM simula ions and on he knowledge o he inpu ea u es (geome y, ma e ial
p ope ies, he mal load). (b) Lowe b anch: an LSTM is ained on an expe imen al deg ada ion da ase o p edic he esis ance e olu ion o e ime. The
PINN amewo k in eg a es bo h p edic ions by combining da a-d i en loss and physics-based loss.
In eg a ing domain knowledge, such as using PINNs, has
p o en e ec i e in enhancing bo h he accu acy and in e -
p e abili y o p edic ions [7], [10]. While da a-d i en models
excel in handling la ge da ase s and cap u ing complex in e -
ac ions, hey may lack physical in e p e abili y and obus ness
when ex apola ing beyond he aining da a [6], [11]. Hyb id
models, such as Physics-In o med Neu al Ne wo ks (PINNs),
aim o b idge his gap by embedding physical laws in o he
cos unc ions o da a-d i en models, ensu ing p edic ions
emain consis en wi h known physics [11].
This s udy builds on hese ad ancemen s by p oposing a no el
physics-in o med machine-lea ning me hodology o es ima e
he RUL o PTHs subjec ed o he mal cycling. The app oach
in eg a es FEM simula ions o model s ain dis ibu ions and
expe imen al da a om PCB es boa ds. ML models a e
ained on hese da ase s o p edic s ain ampli ude and
deg ada ion ends, o e ing a scalable and e icien solu ion
o p edic i e main enance in elec onic sys ems. The no el y
o his app oach lies in he in eg a ion o da a om FEM sim-
ula ions and expe imen al measu emen s, allowing he models
o le e age bo h syn he ic and eal-wo ld deg ada ion pa e ns.
This hyb id me hodology enhances p edic i e accu acy by
ensu ing ha he lea ned ea u es inco po a e bo h physics-
based cons ain s and obse ed deg ada ion beha io .
II. RELATED WORK
The applica ion o da a-d i en me hodologies and physics-
in o med app oaches in eliabili y assessmen has been widely
explo ed in he li e a u e. Howe e , hei combina ion, pa -
icula ly in he con ex o elec onics p ognos ics, emains
a ela i ely unexplo ed ield. In his sec ion, we p o ide
an o e iew o exis ing app oaches, i s discussing da a-
d i en me hods o eliabili y assessmen and hen e iewing
physics-in o med me hodologies o p ognos ics. Finally, we
highligh how ou p oposed me hod uniquely in eg a es eal-
wo ld senso da a wi h physics-based simula ions wi hin a
PINN, imp o ing obus ness and p edic ion accu acy.
A. Da a-D i en app oaches
In ecen yea s, he applica ion o ML and Deep Lea ning
(DL) in elec onics p ognos ics has gained signi ican a en-
ion. Va ious s udies ha e explo ed da a-d i en me hodologies
o es ima e he RUL o elec onic componen s and sys ems,
le e aging ML echniques o enhance p edic i e accu acy.
These app oaches o en complemen o eplace adi ional
physics-based models, o e ing scalable and adap i e solu ions
o eliabili y assessmen . Bha e al. p o ide a comp ehensi e
e iew on he applica ion o machine lea ning algo i hms
in p ognos ics and heal h moni o ing (PHM) o elec onic
sys ems, highligh ing key challenges and ad ancemen s in
da a-d i en eliabili y assessmen [12]. Fe ando-Villalba and
Vande elde de eloped an A i icial Neu al Ne wo k (ANN)
eg esso o p edic inelas ic s ain in PBGA solde join s
unde he mal cycling, demons a ing ha ML models can
signi ican ly educe compu a ional cos s compa ed o a-
di ional FEM simula ions while main aining an accep able
p edic ion accu acy [13]. Yao e al. p oposed a physics-based
nes ed ANN app oach o p edic he eliabili y o Fan-Ou
Wa e -Le el Packages (FOWLP), in eg a ing ma e ial p ope y
es ima ion and mechanical esponse p edic ion o imp o e
accu acy and compu a ional e iciency [14]. Chou and Chiang
de eloped an ANN eg ession model o assess he long-
e m eliabili y o wa e -le el packages (WLPs), le e aging
FEM simula ions combined wi h empi ical mechanics heo ies
o p edic solde join a igue li e e icien ly [15]. Alghassi
e al. de eloped a compu a ionally e icien and embeddable
p ognos ic app oach o powe elec onics, le e aging a da a-
d i en me hod o p edic he RUL o insula ed-ga e bipola
ansis o s (IGBTs) based on accele a ed aging expe imen s
and s ochas ic deg ada ion modeling [16].
B. Physics-In o med app oaches
Beyond da a-d i en s a egies, physics-based models p o-
ide an al e na i e means o p edic ing sys em deg ada ion
by le e aging undamen al ma e ial p ope ies and ailu e
mechanisms. These app oaches, such as ini e elemen simula-
ions and empi ical a igue models, o e high in e p e abili y
and gene alizabili y bu o en equi e ex ensi e calib a ion
and compu a ional esou ces. Zheng e al. p oposed a LSTM
ne wo k o RUL es ima ion, demons a ing i s e ec i eness
in cap u ing sequen ial dependencies in senso da a and ou -
pe o ming adi ional eg ession-based me hods and Con o-
lu ional Neu al Ne wo ks (CNNs) in p ognos ics applica ions
[17]. Meszme e al. explo ed he use o deep neu al ne wo ks
o s ess p ognos ics in encapsula ed elec onic packages,
demons a ing ha LSTM and GRU a chi ec u es e ec i ely
p edic s ess dis ibu ion and deg ada ion pa e ns based on
ime-se ies da a om in-si u condi ion moni o ing du ing
he mal shock es s [18]. Zhang e al. p oposed a LSTM
Recu en Neu al Ne wo k (RNN) o p edic ing he RUL
o li hium-ion ba e ies, demons a ing i s abili y o cap u e
long- e m dependencies in ba e y deg ada ion da a and im-
p o e p edic ion accu acy compa ed o adi ional machine
lea ning me hods [19]. Habibollahi Naja Abadi and Moda es
in oduced a guided neu al ne wo k amewo k o p edic ing
sys em deg ada ion, in eg a ing a physics disco e y neu al
ne wo k wi h a p edic i e model o enhance he accu acy and
obus ness o li e ime es ima ion in enginee ing sys ems [11].
Wen e al. in oduced a PINN amewo k o PHM o li hium-
ion ba e ies, in eg a ing empi ical and physics-based models
o imp o e RUL es ima ion accu acy and model gene alizabil-
i y [20]. Li e al. in oduced a PINN app oach o compac
de ice modeling, inco po a ing undamen al de ice physics
in o neu al ne wo k a chi ec u es o ensu e physically consis-
en and compu a ionally e icien p edic ions [21]. Habibollahi
Naja Abadi and Moda es p oposed a deep lea ning ame-
wo k o disco e ing he unde lying physics o deg ada ion
in da a-d i en p ognos ics, in eg a ing a p edic i e model wi h
a physics disco e y ne wo k o imp o e in e p e abili y and
gene aliza ion in RUL es ima ion [22].
The me hodology p oposed in his pape b idges he gap
be ween hese wo pa adigms by in eg a ing da a om eal-
wo ld senso measu emen s and ini e elemen simula ions
wi hin a PINN. Unlike p e ious s udies ha ely exclusi ely
on ei he expe imen al deg ada ion da a o physics-based sim-
ula ions, ou app oach le e ages bo h, ensu ing a mo e obus
and gene alizable p edic ion model. By embedding physics
cons ain s di ec ly in o he lea ning p ocess, we enhance he
model’s abili y o ex apola e beyond he aining da ase while
main aining consis ency wi h es ablished ailu e mechanisms.
This hyb id me hodology signi ican ly imp o es p edic i e
accu acy in PCB eliabili y assessmen , o e ing a scalable and
in e p e able solu ion o PTH li e ime es ima ion.
III. BACKGROUND
Pla ed Th ough Holes a e c i ical in e connec s uc u es in
PCBs, ensu ing elec ical con inui y be ween di e en laye s
and di e en componen s o he ci cui . Howe e , hei e-
liabili y is challenged by he momechanical s esses induced
by CTE misma ches be ween he coppe pla ing and he
su ounding composi e ma e ial ( ypically FR4). These s ains
accumula e o e epea ed he mal cycles, leading o ma e ial
a igue and e en ual ailu e [1].
A. Failu e Mechanisms in PTHs
PTH ailu es p edominan ly o igina e om low-cycle a-
igue (LCF), whe e cyclic plas ic de o ma ion induces c ack
ini ia ion and p opaga ion wi hin he elec opla ed coppe
walls [2]. The p ima y causes o hese ailu es include:
•CTE misma ch-induced s ain: Due o he mal cycling
(e.g., be ween -50°C and 125°C in au omo i e appli-
ca ions [23]), he coppe expands and con ac s a a
di e en a e compa ed o he PCB lamina e. This esul s
in localized plas ic de o ma ion, pa icula ly a s ess
concen a ion poin s (e.g., knee egions o he PTH) [3].
•Mic os uc u al de ec s: Elec opla ed coppe in PTHs
exhibi s g ain bounda y weaknesses and esidual s esses
om he deposi ion p ocess. These ac o s accele a e
a igue ailu e by p omo ing c ack nuclea ion and oid
o ma ion [5]. This e ec is no conside ed in he model.
To p edic he RUL o PTHs, classical a igue models a e
employed, p ima ily he Co in-Manson equa ion [2] and he
Pa is law [24], [25] o c ack p opaga ion.
The a igue li e N o a gi en s uc u e can be es ima ed
using he Co in-Manson ela ionship:
∆ε=σ′
E(2N )b+ε′(2N )c(1)
whe e ∆εis he s ain ampli ude de ined as he di e ence
o he s ain accumula ed a he ex eme poin s o he applied
he mal load. I is de ined as he sum o an elas ic and a plas ic
Fig. 2. Axisymme ic model o he PTH o he FEM simula ion.
e m. In Eq. 1, band ca e espec i ely he elas ic and plas ic
exponen s, σ′/E is called he cyclic esis ance coe icien , and
ε′is he a igue duc ili y coe icien .
C acks p opaga e ollowing Pa is’ law (Eq. 2), which de-
sc ibes he c ack g ow h a e da
dN as a unc ion o he s ess
in ensi y ac o ange ∆K:
da
dN =A(∆K)m(2)
whe e Aand ma e empi ical cons an s dependen on
he ma e ial mic os uc u e [26]. This phase is c ucial o
unde s anding he ime o inal ailu e cycle once a c ack
has ini ia ed. Fini e Elemen Analysis (FEA) and expe imen-
al he mal cycling es s alida e hese a igue models by
co ela ing s ain dis ibu ions wi h obse ed c ack ini ia ion
and p opaga ion loca ions [2]. S udies con i m ha educing
PTH aspec a io, inc easing coppe pla ing hickness, and
op imizing PCB ma e ial selec ion can signi ican ly enhance
a igue li e [3].
B. Machine Lea ning echniques o a igue li e p edic ion
Machine Lea ning echniques ha e gained inc easing ele-
ance in a igue li e p edic ion by cap u ing complex s ess-
s ain ela ionships and es ima ing he a igue li e N o
c i ical elec onic componen s. This pa ag aph p o ides an
o e iew o he Feed o wa d Neu al Ne wo ks, Long Sho -
Te m Memo y ne wo ks, and Physics-In o med Neu al Ne -
wo ks, which a e le e aged in he me hodological app oach
p oposed in his pape .
•A Feed o wa d Neu al Ne wo k (FFNN) is one o he
simples a chi ec u es in deep lea ning, whe e in o ma ion
lows only in one di ec ion, om he inpu laye o he
ou pu laye , h ough one o mo e hidden laye s. Each
neu on p ocesses a weigh ed sum o he inpu s, ollowed
by a non-linea ac i a ion unc ion (σin Eq. 3). The
ma hema ical o mula ion is:
h(l)=σW(l)h(l−1) +b(l)(3)
Fig. 3. Applied he mal load
whe e h(l)and h(l−1) a e espec i ely he ou pu and he
inpu o he laye l,W(l)is he weigh ma ix, and b(l)
is he bias ec o [27], [28]. The aim o his ne wo k,
depic ed in he uppe b anch in Fig. 1, is he li e ime
p edic ion o a gi en unseen inpu pa e n, conce ning
he geome y, ma e ial p ope ies, and he mal loads.
•Unlike FFNNs, which do no cap u e empo al depen-
dencies, Long Sho -Te m Memo y (LSTM) ne wo ks,
which uni cell is depic ed a e a ype o Recu en Neu al
Ne wo k (RNN) speci ically designed o model sequen ial
da a. This makes LSTMs pa icula ly e ec i e o a igue
c ack g ow h p edic ion, whe e pas loading condi ions
in luence u u e damage accumula ion [8], [9].
•While con en ional ML models like FFNN and LSTMs
ely pu ely on da a, Physics-In o med Neu al Ne wo ks
(PINN) inco po a e physical knowledge (e.g. pa ial di -
e en ial equa ions) in o he lea ning p ocess, ensu ing
ha p edic ions emain consis en wi h he physics o he
p oblem ( he deg ada ion p ocess due o s ain accumula-
ion in his s udy). The loss unc ion o he PINN in Eq.
4 is composed by wo e ms:
L=Lda a +λLphysics (4)
whe e Lda a ep esen s he adi ional da a loss (e.g., Mean
Squa ed E o ), Lphysics en o ces physical consis ency, and
λis a weigh ing ac o ha balances he wo e ms [21],
[22].
IV. METHODOLOGY
The aim o his wo k is o de elop a me hodology o
p edic he emaining use ul li e ime wi h a mo e obus model,
h ough he combina ion o wo di e en ype o da a desc ibed
in he ollowing pa ag aphs.
A. Da a om FEM simula ions
Fini e Elemen Me hod (FEM) is widely used o model
he beha iou o s uc u es made by di e en ma e ials and
Fig. 4. Axial s ain dis ibu ion a T = 125°C in he coppe laye
subjec ed o he mal loads o es ima e he s ess and s ain
dis ibu ion. Due o he o a ional symme y o he s uc u e,
an axisymme ic model has been implemen ed o educe he
compu a ional ime compa ed o a 3D one, as shown in Fig.
2, ollowing he app oach p oposed in [29]. The ma e ials
in ol ed a e: Coppe , modeled as elas ic-plas ic iso opic;
FR4, he composi e ma e ial o he boa d, ea ed as elas ic-
plas ic o ho opic ma e ial, and so, i s ma e ial p ope ies a e
ep esen ed by a enso . Since he model is 2D, he mesh
is made by 4-noded elemen s and i has been made ine a
he in e ace be ween he wo ma e ials whe e mo e accu acy
is needed in he calcula ion o s ess and s ain. Due o he
symme y along he X di ec ion, only hal o he s uc u e
was e ained o u he inc ease he simula ion speed. The
he mal p o ile in Fig. 3 is applied o he s uc u e and he
s ain dis ibu ion is e alua ed a i s ex eme poin s: -50°C and
125°C. Fig. 4 shows he s ain dis ibu ion in he Coppe laye
a T = 125°C and he olume wi h he maximum alues, he
damage olume, is he one in he middle. The same happens
a he lowe empe a u e. So, he a e age s ain is ex ac ed
om hese elemen s o he mesh and he s ain ampli ude is
e alua ed wi h he o mula:
∆ε=εa g(T= 125◦C)−εa g(T=−50◦C)(5)
so ha Eq. 1, he Co in-Manson model, can be used o
e alua e he Numbe o Cycles o Failu e.
To p epa e a aining da ase o he ML-based app oach,
a Design o Expe imen s has been buil by selec ing some
ea u es desc ibed in Table 1 and by using he La in hype cube
sampling me hod [30], [31]. The esul ing da ase has 10.000
samples wi h di e en geome y, ma e ial p ope ies and he -
mal loads, and consequen ly also di e en ∆εcompu ed as in
Eq. 5.
TABLE I
DESIGN OF EXPERIMENTS. FEATURES AND RANGES
Fea u e Min Max
dVIA (mm) 0.25 0.6
PCB (mm) 1.0 2.5
Cu (µm) 25 40
EFR4,z (GPa) 4 10
CTEFR4,z (ppm/◦C) 50 70
YCu (MPa) 150 250
Tmin (◦C) -55 0
Tmax (◦C) 85 150
The s ain ampli ude has been con e ed in o Numbe o
Cycles o Failu e h ough he Co in-Manson model p e iously
op imized and he e o e now he eg ession ask no longe only
in ol es he p edic ion o a eal numbe , bu o a dis ibu ion,
and he e o e o mean alue and s anda d de ia ion.
B. Da a om expe imen al es
Expe imen al da a we e collec ed om PCB boa ds con ain-
ing 12 a ays o PTHs (10×10 a ays, 100 PTHs pe a ay).
Each a ay has a dis inc geome ic con igu a ion (inne land
con igu a ion and diame e ). The schema ic o he boa d is
depic ed in Fig. 5a and 5b. Du ing es ing, elec ical esis ance
o each a ay is moni o ed unde he mal cycling. Resis ance
inc eases indica e ailu e in one o mo e PTHs in he a ay.
These da a (i.e. he esis ance measu emen s o he a ays
wi h Con igu a ion A a e epo ed in Fig. 5c) a e used o ain
an LSTM model ha p edic s esis ance e olu ion o e cycles,
allowing he es ima ion o N . To ob ain he N om hese
cu es, a pa ame ic model has been c ea ed o he equi alen
esis ance o a single a ay wi h he ollowing assump ions:
•All he PTHs in an a ay a e independen . Valida ion o
his hypo hesis has been ob ained h ough FEM simula-
ion o a 3D a ay.
•The zoom in Fig. 5c shows ha , apa om noise, he
inc ease in esis ance alue occu s ia s eps. Fo he
model i was assumed ha each single esis ance, when
ailu e occu s a a ce ain N , inc eases as a s epwise
unc ion: The s epwise unc ion:
RPTH,i(N) = (Rini , N < N
∞, N ≥N
(6)
whe e N ep esen s he cycles.
The index iin Eq. 6 uns om 1 o 100 and ep esen s he
single PTH wi hin an a ay.
The equi alen esis ance o an a ay can be exp essed as:
Req(N) = Rs+ 1
P100
i=1 1
RPTH,i(N)!(7)
whe e Rsis a se ies esis ance connec ed o he a ay. The
co esponding model is shown in Fig. 6a. Req(N)depends
Fig. 5. (a) Schema ic o he boa d used o expe imen al es s: measu emen s a e made on 12 a ays o 100 PTHs connec ed in pa allel and each a ay is
cha ac e ized by a geome ic con igu a ion (b) Schema ic o he con igu a ion o he non- unc ional coppe pads (o inne lands) wi hin he hickness o he
PCB. (c) Resis ance measu emen s o he PTH a ays wi h con igu a ion A
on 100 alues o N ha mus be op imized o i he cu e
o he model wi h he expe imen al ones. The esul o his
i ing (Fig. 6b) p o ides a dis ibu ion o he N o a gi en
a ay (and so, o a gi en geome ical con igu a ion) in e ms
o mean alues and s anda d de ia ion. Eq. 2 has been used o
compu e he numbe o cyles om he c ack ini ia ion o he
end o he c ack p opaga ion as explained in [32], [33] and
hese dis ibu ions o he N a e used o une he empi ical
pa ame e s o he Co in-Manson model (Eq. 1) o compu e
he N in e ms o mean alue and s anda d de ia ion s a ing
om he p edic ed ∆ε.
Some p ep ocessing is needed be o e he aining p ocess o
imp o e he p edic ion pe o mances: (a) The Mo ing A e age
il e ing me hod [34] wi h 5%window size was applied o
smoo h he cu es and emo e noise and spikes om he
measu emen s and (b) A scaling as desc ibed in Eq. 8 and
9 was pe o med o no malize he cu es so ha hey all s a
a 100%, wi h hei alues dec easing as he numbe o applied
cycles inc eases.
Rno m,k(N) = Rk(N)−Rk(N= 0)
Rmax,k −Rk(N= 0) (8)
SoHk(N) = 100 ·[1 −Rno m,k(N)] (9)
whe e he index k uns om 1 o 12 and ep esen s he
a ay. Rk(N)is he ime e olu ion o he esis ance o a ay
k,Rk(N= 0) and Rmax,k a e he ini ial and maximum alues
o he esis ance o a ay k.
The ans o ma ion allows he esis ance alues o be con-
e ed in o he o e all S a e o Heal h (SoH) [19], [20] o he
en i e a ay.
C. Li e ime p edic ion
Th ough his analysis he wo di e en ypes o da a a e
consis en o build he PINN including he physical knowledge
in he i s FFNN in o he loss unc ion o he LSTM and
o ha e a guided p edic ion o he esis ance inc ease as a
unc ion o he applied cycles.
I is impo an o ema k he ac ha he ou pu o he LSTM
is he alue o he esis ance a he nex cycle, and so i is
able o p edic he en i e esis ance cu e Rk,p ed(N). Thanks
o he i ing s a egy desc ibed o ob ain he dis ibu ion o
he numbe o cycles o ailu e, i is possible o associa e a
dis ibu ion N ,p ed o he p edic ed cu e. These wo ou pu s
a e shown in he lowe b anch in Fig. 1.
Lphysics =∥N ,physics −N ,p ed∥(10)
Lda a =
NEXP
X
N=1
[Rk(N)−Rk,p ed(N)]2(11)
I he geome ic and mechanical pa ame e s o he s uc u e
unde measu emen , as lis ed in Table I, a e known, he FFNN
can be used o p edic he N dis ibu ion. This dis ibu ion
ac s as he physical cons ain (N ,physics), as i is de i ed
Fig. 6. (a) Model o he equi alen esis ance o a single a ay on he boa d,
desc ibed by Eq. 6 and 7. (b) Fi ing be ween he model and expe imen al
cu e
om a neu al ne wo k ained on a da ase gene a ed h ough
FEM simula ions, ensu ing ha i inhe en ly inco po a es he
physics o he p oblem. Thus, du ing each epoch, he PINN
aining p ocess is ca ied ou by compu ing he loss unc ion,
using Eq. 4 as he sum o : (a) a da a-based e m, which is
he MSE be ween he eal and p edic ed esis ance cu es
(Eq. 11) and (b) a physics-based e m in Eq. 10, which
e alua es he di e ence be ween he p edic ed and physical
N dis ibu ions.
Eq. 10 desc ibes he physical cons ain . The dis ance be-
ween wo dis ibu ions can be measu ed as MSE o , wi h a
mo e igo ous app oach, h ough he Kullback-Leible di e -
gence c i e ia [35]. This me hod has been implemen ed in he
amewo k desc ibed in his pape .
V. EXPERIMENTS AND RESULTS
In his sec ion he applica ion and esul s o he h ee models
desc ibed in Fig. 1, he FFNN, he LSTM and he PINN, will
be shown and discussed. The aim is o demons a e ha he
in eg a ion o he p edic ions made by he FFNN ained on he
FEM da a imp o es he p edic ion o he end o he elec ical
esis ance along he applied he mal cycles.
•The FFNN has been ained wi h 80%o he da ase
and es ed wi h he emaining 20%. The a chi ec u e is
composed by 2 hidden laye s wi h 64 neu ons and ReLU
ac i a ion unc ion. The a chi ec u al selec ion has been
done by selec ing he one wi h he smalles e o on he
alida ion se , andomly ex ac ing 20%samples om he
aining se . The wo ou pu s ep esen he mean alue
and he s anda d de ia ion o N . A so plus ac i a ion is
added o he s anda d de ia ion ou pu neu on o impose
i o be posi i e, while a linea one is used o he mean
alue. The p edic ion capabili y o his FFNN can be
obse ed in he sca e plo in Fig. 7.
•To ain he LSTM he da ase was di ided in o h ee
subse s o aining, alida ion, and es ing. Speci ically, 8
ime se ies (app oxima ely 70%o he o al 12 esis ance
measu emen s) we e used o aining, while 2 ime se ies
we e alloca ed o alida ion, and he emaining 2 se ies
we e assigned o he es se . The spli was pe o med
andomly o ensu e a balanced dis ibu ion o da a ac oss
he h ee phases.
Fo each ime se ies, a sliding ime window con aining
N esis ance alues was de ined. The LSTM model was
ained o p edic he nex esis ance alue based on he
i s N-1 alues wi hin he window. The pa ame e N
di ec ly impac s bo h he aining ime and he model’s
compu a ional complexi y: la ge window sizes allow o
cap u ing long- e m empo al dependencies mo e e ec-
i ely bu also inc ease compu a ional cos and he isk o
o e i ing. A window size o N=5 yielded good esul s,
e en in e ms o p edic ions on he es se .
Since LSTM is a deep lea ning app oach, i is c ucial
o minimize o e i ing and o imp o e he gene aliza ion
pe o mance o he p edic ed ou pu .
Va ious egula iza ion echniques we e es ed, and he
Fig. 7. P edic ed s Real numbe o cycles o ailu e o he samples in he
es se o he FFNN eg esso
Fig. 8. Time se ies p edic ion. Compa ison be ween LSTM and PINN app oaches o he esis ance cu es in he es se , co esponding o he second and
eigh h a ays on he boa d: (a) Con igu a ion A, d = 300 µm and (b) Con igu a ion C, d = 300 µm
mos e ec i e one was ound o be D opou a 30%, as i
esul ed in he model ha minimized he alida ion loss.
The esul s o he a ay 2, which belong o he es se ,
a e summa ized in Table II.
TABLE II
LSTM RESULTS, ARRAY 2
Regula iza ion P edic ion E o
NO 31.141
30% d opou 17.953
MSE was used o compu e hese me ics. Wi hou egula -
iza ion, he ne wo k o e i s by memo izing pa e ns and
noise, leading o poo gene aliza ion. D opou mi iga es
his by andomly deac i a ing neu ons, o cing he model
o lea n mo e obus ep esen a ions. While his dec eases
accu acy on he aining se , i imp o es pe o mance on
unseen da a, as shown by he lowe es e o .
TABLE III
PREDICTION ERROR COMPARISON, TEST SET
A ay LSTM PINN
2 17.953 6.562
8 19.924 8.054
•Once he wo a chi ec u es ha e been ained and op i-
mized o make p edic ions on hei espec i e da a ypes,
he PINN can be cons uc ed as explained in Sec ion IV
and illus a ed in Fig. 1.
The λpa ame e in Eq. 4 has been se o 0.1.
This app oach ensu es ha he model emains consis en
wi h he unde lying physical p inciples while le e aging
da a-d i en lea ning o enhanced p edic i e accu acy.
The p edic ion e o o he end o he esis ance co -
esponding o a ays in he es se , compu ed wi h Eq.
11, be ween he p edic ed cu e wi h PINN app oach
and he eal cu e a e epo ed in Table III, esul ing
in an imp o emen o he p edic ion pe o mances o
app oxima ely 63% o a ay 2 and 59% o a ay 8.
Fig. 8a and 8b p esen a compa ison be ween he wo
app oaches, demons a ing how inco po a ing he physics o
he p oblem leads o imp o ed p edic ion pe o mance, pa ic-
ula ly in he mos ele an sec ion o ailu e p edic ion, whe e
he esis ance inc eases. The plo s show he esul s o A ay
2 and 8 o he boa d.
In eg a ing he physics o he p oblem in o he loss unc-
ion o be minimized signi ican ly enhances he accu acy o
esis ance end p edic ion. P e iously, he LSTM model elied
solely on da a, leading o po en ial o e i ing and limi ed
gene aliza ion when applied o unseen con igu a ions.
By inco po a ing physics-based cons ain s h ough he
PINN amewo k, he model is guided owa ds physically
consis en p edic ions, educing e o and imp o ing obus -
ness. The compa a i e analysis, summa ized in Table III,
demons a es ha ac oss all es ed a ays, he PINN app oach
consis en ly achie es lowe p edic ion e o s han he pu ely
da a-d i en LSTM model.
To ob ain mo e eliable and obus esul s, he model was
ained and alida ed using di e en con igu a ions o he
h ee se s ( aining, alida ion, and es ), consis en ly yielding
simila esul s o hose epo ed.
VI. CONCLUSION
In his wo k, we demons a ed ha implemen ing a Physics-
In o med Neu al Ne wo k (PINN) o p edic ing he deg a-
da ion o e ime o Pla ed Th ough Holes (PTHs) signi -
ican ly imp o es pe o mance compa ed o a pu ely da a-
d i en app oach. Gi en ha PTHs a e c i ical componen s
p esen in all PCBs, and ha hey can also be used as cana y
senso in elec onic sys ems, unde s anding hei deg ada ion
beha iou is essen ial. A key aspec o ou app oach is ha ,
despi e inco po a ing physics-based cons ain s, i emains
almos en i ely da a-d i en. Unlike adi ional PINNs, we do
no include di e en ial equa ions in he loss unc ion. The only
physical cons ain applied is he bounda y condi ion om he
FEM simula ions. As a esul , we success ully ained a neu al
ne wo k ha accu a ely p edic s deg ada ion s a es using a e y
limi ed deg ada ion da ase (only 12 signals). Mo eo e , con-
s uc ing he physics-based da ase did no equi e designing
and execu ing a long- e m expe imen al campaign, which is
o en a majo bo leneck in his ield. Ins ead, ou Design o
Expe imen s (DoE) app oach enabled us o ob ain s ain ampli-
ude da a o 10.000 samples ac oss di e en con igu a ions in
jus ou days. This e ec i ely add esses a well-known issue in
he li e a u e ( he lack o deg ada ion da a) while signi ican ly
educing ime and esou ce cons ain s. These esul s highligh
he po en ial o combining physics-in o med and da a-d i en
me hodologies o enhance he p edic i e capabili ies o elia-
bili y models o elec onic componen s, pa ing he way o
mo e e icien and scalable p ognos ic solu ions in PCB heal h
moni o ing.
ACKNOWLEDGMENT
This esea ch was conduc ed wi hin he MIRELAI p ojec ,
Funded by he Eu opean Union (G an Ag eemen No.
101072491). Views and opinions exp essed a e howe e hose
o he au ho (s) only and do no necessa ily e lec hose o he
Eu opean Union o he Eu opean Resea ch Execu i e Agency.
Nei he he Eu opean Union no he g an ing au ho i y can be
held esponsible o hem.
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