Rosa, Ma iapia De; Pompameo, Lau a; Li inenko, Alexande ; Cuomo, Sal a o e
A icle — Published Ve sion
HOMO-PINN: Hype pa ame e Op imiza ion o a Mul i-
ou pu Physics-In o med Neu al Ne wo k
Ope a ions Resea ch Fo um
P o ided in Coope a ion wi h:
Sp inge Na u e
Sugges ed Ci a ion: Rosa, Ma iapia De; Pompameo, Lau a; Li inenko, Alexande ; Cuomo, Sal a o e
(2025) : HOMO-PINN: Hype pa ame e Op imiza ion o a Mul i-ou pu Physics-In o med Neu al
Ne wo k, Ope a ions Resea ch Fo um, ISSN 2662-2556, Sp inge In e na ional Publishing, Cham,
Vol. 6, Iss. 4,
h ps://doi.o g/10.1007/s43069-025-00561-7
This Ve sion is a ailable a :
h ps://hdl.handle.ne /10419/330385
S anda d-Nu zungsbedingungen:
Die Dokumen e au EconS o dü en zu eigenen wissenscha lichen
Zwecken und zum P i a geb auch gespeiche und kopie we den.
Sie dü en die Dokumen e nich ü ö en liche ode komme zielle
Zwecke e iel äl igen, ö en lich auss ellen, ö en lich zugänglich
machen, e eiben ode ande wei ig nu zen.
So e n die Ve asse die Dokumen e un e Open-Con en -Lizenzen
(insbesonde e CC-Lizenzen) zu Ve ügung ges ell haben soll en,
gel en abweichend on diesen Nu zungsbedingungen die in de do
genann en Lizenz gewäh en Nu zungs ech e.
Te ms o use:
Documen s in EconS o may be sa ed and copied o you pe sonal
and schola ly pu poses.
You a e no o copy documen s o public o comme cial pu poses, o
exhibi he documen s publicly, o make hem publicly a ailable on he
in e ne , o o dis ibu e o o he wise use he documen s in public.
I he documen s ha e been made a ailable unde an Open Con en
Licence (especially C ea i e Commons Licences), you may exe cise
u he usage igh s as speci ied in he indica ed licence.
h ps://c ea i ecommons.o g/licenses/by/4.0/
Ope a ions Resea ch Fo um (2025) 6:153
h ps://doi.o g/10.1007/s43069-025-00561-7
RESEARCH
HOMO-PINN: Hype pa ame e Op imiza ion o a
Mul i-ou pu Physics-In o med Neu al Ne wo k
Ma iapia De Rosa1·Lau a Pompameo3·Alexande Li inenko2·
Sal a o e Cuomo3
Recei ed: 12 June 2025 / Accep ed: 25 Sep embe 2025
© The Au ho (s) 2025
Abs ac
The good choice o hype pa ame e s is c ucial o he success ul applica ion o deep
lea ning (DL) ne wo ks in o de o ind accu a e solu ions o he bes pa ame e in
sol ing pa ial di e en ial equa ions (PDEs), ha a e sensi i e o e o s in coe -
icien es ima ion. Fo his pu pose, hype pa ame e op imiza ion o mul i-ou pu
physics-in o med neu al ne wo ks (HOMO-PINNs) is based on he op imal sea ch
o PINN hype pa ame e s o sol ing PDEs wi h unce ain coe icien s in he unce -
ain y quan i ica ion (UQ) ield. By es ing his no el me hodology on di e en PDEs,
he ela ionship be ween ac i a ion unc ions, he numbe o ou pu neu ons, and he
deg ee o coe icien unce ain y can be obse ed. The expe imen al esul s show ha
adding ou pu neu ons o he neu al ne wo k (NN), e en i a heo e ically inco ec
ac i a ion unc ion is chosen, keeps he p edic ed solu ion accu a e.
Keywo ds Physics-in o med neu al ne wo k ·Nume ical me hods ·PDE ·
Unce ain y quan i ica ion ·PINN ·Hype pa ame e op imiza ion
1 In oduc ion
Pa ial di e en ial equa ions (PDEs) a e undamen al o modelling a wide ange o
physical phenomena in a ious scien i ic and enginee ing disciplines. In o de o ind an
accu a e solu ion o di e en ial p oblems, physics-in o med neu al ne wo ks (PINNs)
we e de eloped by Raissi e al. [20], combining ma hema ical models and da a. These
ne wo ks a e capable o using physical equa ions as a unc ional o minimize, he eby
c ea ing a NN ha is sui able o sol ing p oblems in a manne ha is consis en
wi h he p inciples o physics. Consequen ly, PINNs ha e been employed o add ess
BSal a o e Cuomo
sal [email p o ec ed]
1Is i u e o Bios uc u es and Bioimaging (IBB) o he Na ional Resea ch Council (CNR), Via
Tommaso De Amicis, 95, 80145 Naples, I aly
2RWTH Aachen Uni e si y, Pon d iesch, 14-16, 52062 Aachen, Ge many
3Uni e si y o Naples Fede ico II, Via Cin ia, 26, 80126 Naples, I aly
0123456789().: V,- ol 123
153 Page 2 o 19 Ope a ions Resea ch Fo um (2025) 6:153
eal-wo ld issues in a mul i ude o disciplines, such as luid dynamics [3], lexible
mecha onics and so obo ics [17], elec onics [25], and ailway acking [6]. As is
well documen ed in he li e a u e, he coe icien s o a PDE can ha e a signi ican
impac on he solu ion [9] and i s nume ical app oxima ion [10,13]. Consequen ly,
he ex en o noise p esen in hese coe icien s necessi a es he es ima ion o he e o
in p edic ing a solu ion by a PINN. Fo example, Qian e al. [19] p o ide an e o
analysis o PINN o he wa e equa ion. While De Ryck e al. [7] p o e bounds on he
e o s caused by he app oxima ion o he incomp essible Na ie –S okes equa ions
wi h PINNs.
PINNs may help o sol e PDEs wi h unce ain coe icien s o andom noise. In
he unce ain y quan i ica ion (UQ) con en , he chosen PDE is sol ed mul iple imes
unde andom pe u ba ions o coe icien s o assess solu ion a iabili y. This app oach
allows o e alua ing he accu acy o he mean p edic ed solu ion agains an analy ical
e e ence. In he con ex o elas ic de o ma ion in he e ogeneous solids, Bha adwaja
e al. [2] demons a ed ha he mean and s anda d de ia ion o PINN solu ions align
well wi h Mon e Ca lo ini e elemen esul s. Simila ly, Li e al. [16] in oduced a
UQ-based me hod o iden i y colloca ion poin s whe e PINNs exhibi la ge p edic ion
e o s, p o iding a sys ema ic way o enhance solu ion eliabili y.
Yang and Fos e [24] p oposed mul i-ou pu PINNs (MO-PINNs), which a e capa-
ble o sol ing bo h o wa d and in e se p oblems go e ned by PDEs, e en in he
p esence o noisy da a. This app oach no only p edic s he solu ion, bu also p o ides
an associa ed unce ain y es ima e, allowing quan i ica ion o he con idence in he
esul s, pa icula ly sui ed o he UQ ask. Fu he mo e, Liu e al. [18] use MO-PINNs
o one- and wo-dimensional nonlinea ime dis ibu ed-o de models. In addi ion,
Hao e al. [11] u ilize MO-PINN o p edic he e olu ion o ime- a ying coe icien s
and p obabili y densi y unc ion o quan i y o in e es (QoI). While Chang e al. [4]
show how MO-PINNs educe he ela i e e o by an o de o magni ude in he con ex
o pa ame e iden i ica ion. Fo he pu pose o his pape , MO-PINNs a e used due o
hei unique abili y o ain a single NN on each o he coe icien -based PDEs, and
hen isualize all ou pu s simul aneously.
Consequen ly, he selec ion o he numbe o ou pu neu ons, in conjunc ion wi h he
o he NN hype pa ame e s, has a signi ican impac on he e iciency o PINNs. Fo
ins ance, Escapil-Inchauspé and Ruz [8] in oduce a hype -pa ame e op imiza ion
(HPO) algo i hm and es i on he Helmhol z equa ion. While Sha ma e al. [22]
pe o m an op imiza ion p ocess o hea conduc ion p oblems wi h a discon inuous
solu ion on many hype pa ame e s, including he numbe o hidden laye s, lea ning
a e, and ac i a ion unc ions. In pa icula , he ou pu ac i a ion unc ion, which is
ypically selec ed based on he ange o alues whe e he p oblem esides, plays a
c ucial ole in he NN aining phase. While he hype bolic angen ( anh) ac i a ion
unc ion is commonly used as he de aul op ion o PINNs, some s udies ha e indica ed
ha o he ac i a ion unc ions can be mo e e ec i e in speci ic scena ios (see [21]).
Fo ins ance, esea ch by Jag ap e al. [12] and Al Sa wan e al. [1] sugges s ha he
sigmoid and swish unc ions may o e ad an ages in ce ain cases. Howe e , hese
s udies do no p o ide a de ini i e conclusion on he op imal ac i a ion unc ion o
a ious PDE p oblems. Fu he in es iga ion and compa ison ac oss di e en ypes o
PDEs a e needed o es ablish a clea e p e e ence o ac i a ion unc ions in PINNs.
123
Ope a ions Resea ch Fo um (2025) 6:153 Page 3 o 19 153
In o de o de e mine he op imal choice o hype pa ame e s (e.g., he numbe o
neu ons and he ac i a ion unc ion o he ou pu laye ), i is necessa y o de elop a
obus me hodology ha can be e p edic he mos accu a e solu ion o he di e -
en ial p oblem, e en when he e o on one o mo e coe icien s ha appea in he
PDE inc eases (in he sense o being a ec ed by a noise o la ge a iance). This
s udy p esen s such me hodology, in oducing hype pa ame e op imiza ion o MO-
PINNs (HOMO-PINNs), which a e capable o combining an op imal sea ch s a egy
o hype pa ame e s wi h he UQ o solu ions o PDEs a ec ed by unce ain ies on
coe icien s.
The no el y o his pape is as ollows:
•in oduc ion o he HOMO-PINNs, ob ained by combining mul i-ou pu PINNs
wi h Hype pa ame e Op imiza ion, while p esen ing he quan i ica ion o unce -
ain ies in PDE;
• es ing o his me hodology o g oundwa e low and Poisson equa ions in o de
o showcase he links be ween hype pa ame e choices and di e en pe cen ages
o e o ;
•disco e ing ha a w ong ac i a ion unc ion equi es mo e ou pu neu ons in o de
o pe o m well when he coe icien e o is la ge.
The pape is s uc u ed as ollows: Sec ion 2p o ides he ma hema ical p esen a ion
o PINNs, ocusing on mul i-ou pu s, and ou lines he hype pa ame e op imiza ion
s a egy; Sec ion 3is de o ed o expe imen al esul s o he HOMO-PINN me hodol-
ogy on he wo es PDEs; Sec ion 4includes he concluding ema ks, summa izing
he indings ob ained h ough es ing.
2 Me hodology
This sec ion ocuses on he desc ip ion o he HOMO-PINN app oach (see Fig. 1),
om he ma hema ical aspec s o PINNs, h ough hei mul i-ou pu a ian , o he
sea ch o op imal hype pa ame e s.
2.1 Physics-In o med Neu al Ne wo ks
PINNs ep esen a signi ican ad ancemen in he ield o compu a ional modeling,
b idging he gap be ween adi ional da a-d i en app oaches and physics-based sim-
ula ions. By embedding physical laws (desc ibed by PDEs) di ec ly in o he neu al
ne wo k’s lea ning p ocess, PINNs enhance he model’s p edic i e capabili ies, espe-
cially in scena ios wi h limi ed o noisy da a.
The PINN a chi ec u e is based on a ully-connec ed eed- o wa d neu al ne wo k
(FC-FFNN), whe e each neu on in each laye is connec ed o e e y neu on in he
p e ious laye . The ma hema ical ope a ions, wi hin a single laye , can be desc ibed in
e ms o ma ix mul iplica ion ollowed by he applica ion o a (linea o non-linea )
ac i a ion unc ion. Le x∈Rd, o hel- h laye in he ne wo k, he ou pu is
h(l)=φ(l)W(l)h(l−1)+b(l),
123
153 Page 4 o 19 Ope a ions Resea ch Fo um (2025) 6:153
Fig. 1 Pipeline o he HOMO-PINN. Once he hype pa ame e s o be es ed ha e been selec ed, all he
possible combina ions o hem a e gene a ed as ec o s. Fo each o hese, a MO-PINN is ained, which
e u ns he p edic ed solu ions associa ed wi h di e en samples o he coe icien . The mean o hese solu-
ions is hen compa ed wi h he analy ical solu ion o iden i y he op imal combina ion o hype pa ame e s
h ough g aphical analysis
whe e φ(l)is he ac i a ion unc ion, W(l)∈Rnl×nl−1is he weigh ma ix, b(l)∈Rnl
is he bias ec o and nlis he numbe neu ons in he l- h laye . No e ha h(l−1)is he
ou pu o he p e ious laye and h(0)=x. Then, he FC-FFNN ou pu is gi en by
uθ(x)=h(L+1)◦h(L)◦···◦h(0)(x), (2.1.1)
whe e Lis he numbe o hidden laye s and θindica es he se o hype pa ame e s o
he NN.
A dis inc i e ea u e o PINNs is he loss unc ion exp essed by he disc epancy
be ween he NN ou pu Eq. 2.1.1 and he physical law. Le us conside a gene al PDE
o he o m
F(u(x), ∇u(x), ∇2u(x),...)=0,
I(u(x)) =0,B(u(x)) =0,(2.1.2)
whe e uis he solu ion o he PDE, xis he ec o o space- ime coo dina es, while
F,I, and Ba e di e en ial ope a o s e e ing o he PDE, i s ini ial condi ions (IC),
and bounda y condi ions (BC), espec i ely. The esiduals o Eq. 2.1.2 a e de ined as
Rθ
PDE(x)=F(uθ(x), ∇uθ(x), ∇2uθ(x),...)
Rθ
IC(x)=I(uθ(x)), Rθ
BC(x)=B(uθ(x))
Then, he loss unc ion can be de ined as ollows
L(θ) := λPDE
NPDE
NPDE
i=1
|Rθ
PDE(xi)|2+λIC
NIC
NIC
j=1
|Rθ
IC(xj)|2+λBC
NBC
NBC
k=1
|Rθ
BC(xk)|2,
(2.1.3)
whe e NPDE,NIC and NBC a e he numbe o colloca ion poin s used o e alua e he
PDE esidual, ini ial condi ion esidual, and bounda y condi ion esidual, espec i ely
123
Ope a ions Resea ch Fo um (2025) 6:153 Page 5 o 19 153
and λPDE,λ
IC and λBC hei weigh s. The e o e, he PDE is app oxima ed by inding
an op imal se θ∗o NN hype pa ame e s ha minimizes Eq. 2.1.3 as ollows
θ∗:= a gmin
θ
L(θ), (2.1.4)
using an op imiza ion algo i hm (e.g., ADAM [14]). A e iden i ying he op imal
hype pa ame e se Eq. 2.1.4, he solu ion uθ∗(x)is de i ed, which p o ides he bes
app oxima ion o he ac ual solu ion o he p oblem.
2.2 Mul i-ou pu PINNs (MO-PINNs)
To add ess he inhe en unce ain ies in PDE coe icien s a ising om ac o s such as
measu emen e o , ma e ial he e ogenei y, and model app oxima ions, MO-PINNs
ex end adi ional PINNs and enable he simul aneous p edic ion o mul iple possible
solu ions unde a ious unce ain y scena ios. MO-PINNs a e also use ul o app oxi-
ma ing s a is ical p ope ies such as he mean and a iance o he solu ion. The key idea
is ha ins ead o aining mul iple PINNs o di e en ealiza ions o unce ain pa am-
e e s, a single neu al ne wo k is designed wi h mul iple ou pu s, each ep esen ing a
di e en ealiza ion o he solu ion.
In his con ex , UQ is na u ally in eg a ed by conside ing a s ochas ic ep esen a ion
o he PDE coe icien s. Speci ically, in he amewo k in oduced in Subsec ion 2.1,
le us suppose ha he ope a o Fin Eq. 2.1.2 also depends on an unce ain coe icien
˜
K(x,ω), i.e.,
F(˜
K(x,ω),u(x), ∇u(x), ∇2u(x),...)=0.(2.2.1)
The unce ain y o ˜
K(x,ω)is in oduced by a mul iplica i e noise as ollows
˜
K(x,ω)=K(x)ε(x,ω), (2.2.2)
whe e ε(x,ω)∈R×Sis a andom noise ield1,Sa andom space, ωa andom e en
(o a andom a iable).
This app oach is pa icula ly bene icial in scena ios whe e ma e ial p ope ies such
as conduc i i y, di usi i y, o elas ici y coe icien s a e no well-de ined and may
exhibi spa ial a iabili y due o inhe en andomness in he medium. By modeling
K(x)as a andom a iable, one can cap u e hese a ia ions, he eby ensu ing ha he
PINN solu ions a e capable o accoun ing o eal-wo ld unce ain ies.
Le us deno e wi h nL+1 he numbe o samples (ε1(x),...,ε
nL+1(x)), om which
we ob ain he co esponding coe icien s ˜
K (x). Then, acco ding o he gi en de ini ion
1A andom ield is a ma hema ical model used o desc ibe spa ially o empo ally a ying andom phe-
nomena. I gene alizes he concep o a s ochas ic p ocess o mul iple dimensions, allowing andomness o
be de ined o e space, ime, o bo h. A andom ield K(x,ω)is a amily o andom a iables indexed by a
poin xin a domain D⊆Rdand de ined on a p obabili y space (, F,P),whe ex ep esen s spa ial (o
spa io empo al) coo dina es in D,ω∈is a sample om he p obabili y space, and K(x,ω)is a andom
a iable o each ixed x. Fo a ixed loca ion x, he unc ion K(x,ω)desc ibes a andom a iable, while
o a ixed ealiza ion ω, he unc ion K(x,ω)desc ibes a de e minis ic unc ion o e space.
123
153 Page 6 o 19 Ope a ions Resea ch Fo um (2025) 6:153
o a MO-PINN, he numbe o ou pu neu ons has o be ixed o nL+1, and, ecalling
he i s e m o Eq. 2.1.3, he PDE esidual becomes
Rθ
PDE(xi)=1
nL+1
nL+1
=1
| θ
(xi)|2,(2.2.3)
whe e
θ
=F˜
K (x), uθ(x), ∇uθ(x), ∇2uθ(x), . . . , =1,...,nL+1.(2.2.4)
The e o e, he loss unc ion Eq. 2.1.3 can be eplaced by
L(θ) := λPDE
NPDE
NPDE
i=11
nL+1
nL+1
=1
| θ
(xi)|2+λIC
NIC
NIC
j=1
|Rθ
IC(xj)|2+λBC
NBC
NBC
k=1
|Rθ
BC(xk)|2.
(2.2.5)
By minimizing he loss unc ion in Eq. 2.1.4, a se o solu ions u(1)
θ∗(x),...,u(nL+1)
θ∗(x)
is ob ained, each co esponding o a PDE ealiza ion wi h esidual Eq. 2.2.4. The inal
p edic ion is hen gi en by hei a e age, ¯uθ∗(x).
A key ad an age o his app oach is ha i enables a di ec e alua ion o he mean
and a iance o he p edic ed solu ions, which is impe a i e o comp ehending he
impac o unce ain y on he sys em’s beha io . By aining he ne wo k on mul iple
ealiza ions o K, an ensemble o solu ions is gene a ed, he eby p o iding a p oba-
bilis ic in e p e a ion o he PDE solu ion a he han a single de e minis ic ou come.
This gua an ees he eliabili y o he solu ions, e en when PDE pa ame e s a e subjec
o a iabili y.
2.3 Hype pa ame e Op imiza ion
The pe o mance o MO-PINNs, as o any NN, is highly dependen on he choice o
hype pa ame e s, including he numbe o hidden laye s, neu ons pe laye , lea ning
a e, ac i a ion unc ions, and he numbe o ou pu neu ons. To sys ema ically iden i y
he op imal con igu a ion, a hype pa ame e op imiza ion s a egy is essen ial (see [15,
23]).
In his s udy, he hype pa ame e uning p ocess was execu ed ia a g id sea ch,
a me hod in ol ing he exhaus i e explo a ion o a manually de ined subse o he
hype pa ame e space. This p ocess is epea ed o each elemen in he subse , wi h
he objec i e o iden i ying he elemen ha minimizes he p edic ion e o . The hype -
pa ame e s conside ed include he numbe nL+1o ou pu neu ons and he ac i a ion
unc ion φ(L+1) o he ou pu laye . Fo each combina ion o hese, and o di e en
pe cen ages o e o s in he dis ibu ion o ε , he MO-PINN was ained on he ixed
da ase {xj}M ain
j=1⊆D⊆Rdminimizing he loss unc ion Eq. 2.2.5. Then, he same
123
Ope a ions Resea ch Fo um (2025) 6:153 Page 7 o 19 153
combina ions we e es ed on {xs}M es
s=1, e alua ing he pe o mance o he ne wo k
based on he mean squa ed e o (MSE) be ween he p edic ed and ue solu ions o
he PDE, i.e.,
MSE := 1
M es
M es
s=1
[u(xs)−¯uθ∗(xs)]2.
Fo each pe cen age o e o in he coe icien K, he op imal se o hype pa ame e s
was iden i ied as he combina ion ha esul ed in he lowes MSE, ensu ing ha he
model achie es high accu acy while main aining obus ness o unce ain ies.
A c i ical aspec o hype pa ame e op imiza ion in MO-PINNs is balancing accu-
acy and compu a ional e iciency. While he addi ion o mo e ou pu neu ons has been
shown o enhance unce ain y quan i ica ion by cap u ing a mo e ex ensi e ange o
po en ial solu ions, i concomi an ly inc eases he compu a ional bu den. The selec ion
o he ac i a ion unc ion in luences bo h nume ical s abili y and he exp essi eness o
he model. Fo ins ance, he employmen o smoo h ac i a ion unc ions, such as Swish,
can assis in add essing issues ela ed o g adien anishing. Con e sely, adi ional
ac i a ion unc ions, including Sigmoid, may yield supe io pe o mance when he
ou pu ange is cons ained. These conside a ions highligh he complexi y o hype -
pa ame e selec ion in MO-PINNs, emphasizing he need o a sys ema ic app oach
o op imize pe o mance while main aining compu a ional easibili y. Fo simplici y,
om now on le us deno e ¯uθ∗(x)wi h ¯u(x).
3 Nume ical Resul s
This sec ion ocuses on e alua ing he HOMO-PINN me hodology by add essing wha
happens i one conside s mo e han one noisy PDE coe icien and uses hem o ob ain
mul iple p edic ed solu ions, and i he mean o hese is somehow close o he analy ical
solu ion o he p oblem. Fo his pu pose, le us suppose he coe icien Kin Eq. 2.2.1
o be a ec ed by a non-linea noise de ined as in Eq. 2.2.2, whe e
ε (x,ω)=exp ξ (1)(ω) sin(2πx)+ξ (2)(ω) cos(8πx)+ξ (3)(ω) cos π
8x
(3.0.1)
wi h ξ (i)ha ing a Gaussian dis ibu ion wi h ixed mean μ=1 and a iance σ2∈
{0.05,0.1,0.2}.
Fo all cases, di e en numbe s o ou pu neu ons and ac i a ion unc ions o he
ou pu laye a e conside ed, while he weigh s in he loss unc ion Eq. 2.2.5 a e always
se as λPDE =1/5 and λIC =λBC =2/5·nL+1. The a ionale behind his se up is ha ,
as he numbe nL+1inc eases, he PDE esidual ends o domina e he op imiza ion,
which may lead o weak en o cemen o ini ial and bounda y condi ions. By scaling
he IC and BC con ibu ions p opo ionally o nL+1, he di e en e ms o he loss
a e kep balanced, mi iga ing po en ial con e gence issues. The speci ic a ios (1/5
and 2/5) we e de e mined empi ically o p o ide a obus ade-o be ween accu acy
and s abili y in he es cases. The hidden laye s o he HOMO-PINN ha e he same
a chi ec u e o all es s, consis ing o ou laye s o 40 neu ons each, wi h hype bolic
123
153 Page 8 o 19 Ope a ions Resea ch Fo um (2025) 6:153
angen ( anh) as ac i a ion unc ion. Fo he aining, 30,000 epochs we e conduc ed
using a lea ning a e o 3 ×10−3and employed he ADAM op imize .
The expe imen s we e execu ed on an NVIDIA GeFo ce RTX 3090 GPU wi h an
In el(R) Co e(TM) i9-9900K CPU @ 3.60GHz, 8-Co es P ocesso and 128 GB o
RAM, in Tenso Flow 2.0 using a Py hon in e p e e .
3.1 G oundwa e Flow Equa ion
The beha io o g oundwa e low is in luenced by a ious ac o s, including he
hyd aulic conduc i i y o he subsu ace ma e ials, he po osi y o he geological o -
ma ions, he p esence o na u al and a i icial echa ge and discha ge sou ces, and
he empo al a ia ions in hese ac o s. The gene al o m o he g oundwa e low
equa ion is
∇·(K∇u)+S∂u
∂ =Q >0,(3.1.1)
whe e uis he hyd aulic head, Kis he hyd aulic conduc i i y enso , Sis he s o age
coe icien , and Q ep esen s sou ces o sinks o wa e (see [5] o de ails). The equa ion
ha desc ibes he wa e ex ac ion om a poin sink in an in ini e aqui e is ob ained
by Eq. 3.1.1 speci ying he sou ce e m using Vaas he ins an aneous sou ce o sink
wa e olume loca ed in δ, which indica es he Di ac del a unc ion. Then,
∇·(K( ,x)∇u( ,x)) +S∂u( ,x)
∂ =Vaδ( )δ(x), (3.1.2)
o all x∈Rn, >0, wi h ini ial condi ions and bounda y condi ions
u( ,x)| =0=0∀x∈Rnu( ,x)|x→±∞ =0∀ >0
To nume ically sol e he p oblem Eq. 3.1.2, i is necessa y o es ima e he Di ac del a
unc ion δ, as i is no di e en iable e e ywhe e and can he e o e only be desc ibed
in e ms o dis ibu ions. A Di ac app oxima ion can be buil as
δη(x)=1
ηϕx
η,(3.1.3)
whe e o y=x/η
ϕ(y)=1
2(1+cos(π y)) i |y|≤1
0 o he wise.
F om now on, le us conside he one-dimensional spa ial case, wi h he Del a
app oxima ion Eq. 3.1.3, hen Eq. 3.1.2 becomes
−K∂2u
∂x2+S∂u
∂ =Vaδη( )δη(x), (3.1.4)
123
Ope a ions Resea ch Fo um (2025) 6:153 Page 15 o 19 153
Fig. 10 Hea map display o a ious combina ions o he conside ed hype pa ame e s (i.e., numbe o ou pu
neu ons on x-axis and ac i a ion unc ion o he ou pu laye on y-axis) o he absolu e e o be ween he
analy ical solu ion and he mean o he p edic ed solu ions, as de ined in Eq. 3.1.7
Fig. 11 Compa ison boxplo o numbe o ou pu neu ons (on he x-axis), while he y-axis e e s o he
absolu e e o de ined in Eq. 3.1.7. The line in each box ep esen s he median, whe eas he di e en colo s
e e o he choice o ac i a ion unc ion
123
153 Page 16 o 19 Ope a ions Resea ch Fo um (2025) 6:153
side ing he same h ee di e en pe cen ages o noise based on he a iance o he ε
sampling, i.e., σ2∈{0.05,0.1,0.2}. Then, he PDE esidual in Eq. 2.2.3 becomes
Rθ
PDE(xi)=1
nL+1
nL+1
=1
|˜
K u(xi)−2sin(πxi)cos2(π xi)+sin3(π xi)|2.
Figu e 8illus a es ha he analy ical solu ion, as de ined in Eq. 3.2.2, is cons ained
o he in e al [−1,1], he eby sugges ing ha anh ep esen s he op imal choice o
ac i a ion unc ion o a PINN wi h one ou pu neu on, i.e., nL+1=1. As explained
in Sec ion 2.3, a g id sea ch has been conduc ed on all possible combina ions o he
hype pa ame e s, speci ically he numbe o ou pu neu ons nL+1∈{3,5,10}and
he ac i a ion unc ion o he ou pu laye φ(L+1)∈{linea , anh, ELU}, in o de o
e alua e he pe o mance o each combina ion. Figu e9shows he esul s ob ained
wi h ELU as ac i a ion unc ion o he ou pu laye and 10 as he numbe o ou pu
neu ons. As can be no iced, despi e he single p edic ed solu ions co esponding o he
coe icien ˜
K wi h a iance 0.2 being inaccu a e, hei mean is close o he analy ical
solu ion. Indeed, al hough he mos app op ia e ac i a ion unc ion o his p oblem,
i.e., anh, has op imal esul s, Fig. 10 displays ha he o he wo chosen ac i a ion
unc ions imp o e hei accu acy when he numbe o ou pu neu ons inc eases. In
suppo o his, Fig. 11 highligh s he i ele ance o he choice o he ac i a ion unc ion
when nL+1g ows. While, as can be seen in Fig. 12, he choice o he ac i a ion unc ion
is c ucial o he s abili y o he p oblem when he coe icien is a ec ed by a 20%
Fig. 12 Compa ison boxplo o noise pe cen ages on he coe icien Kin Eq. 3.2.1 (on he x-axis), while
he y-axis e e s o he absolu e e o de ined in Eq. 3.1.7. The line in each box ep esen s he median,
whe eas he di e en colo s e e o he choice o ac i a ion unc ion
123
Ope a ions Resea ch Fo um (2025) 6:153 Page 17 o 19 153
Fig. 13 E o s o a ious combina ions o he conside ed hype pa ame e s, i.e., numbe o ou pu neu ons
and ac i a ion unc ion o he ou pu laye (in di e en colo s). The x-axis indica es he noise pe cen ages
on he coe icien Ko he conside ed PDE, while he y-axis he espec i e absolu e e o s
noise. In his case, using anh (in o ange in Fig. 13) is necessa y o signi ican ly lowe
he app oxima ion e o on he solu ion.
4 Conclusions
In his s udy, a no el app oach, HOMO-PINNs, has been de eloped and p esen ed in
o de o enhance he accu acy o solu ions o PDEs unde unce ain y. This me hod-
ology in eg a es hype pa ame e op imiza ion wi h UQ o iden i y he mos e ec i e
hype pa ame e s o PINNs. The expe imen al indings on bo h g oundwa e low
and Poisson equa ions demons a e he obus ness o HOMO-PINNs in handling a -
ious deg ees o unce ain y on he s ochas ic coe icien K(x). The nume ical esul s
obse ed ha highe a iance in K(x) esul s in inc eased solu ion dispe sion, necessi-
a ing mo e ou pu neu ons o accu a ely cap u e unce ain y. Con e sely, he selec ion
o app op ia e ac i a ion unc ions has been shown o signi ican ly mi iga e he ins a-
bili y in oduced by andom luc ua ions in K(x). I was he e o e obse ed ha he
selec ion o he numbe o ou pu neu ons and he choice o ac i a ion unc ions impac
he pe o mance o he NN. I is wo h no ing ha ou indings imply ha , in ins ances
whe e an e oneous ac i a ion unc ion is employed, inc easing he numbe o ou pu
neu ons can acili a e he p ese a ion o he accu acy o he p edic ed solu ion. The
mean p edic ion o MO-PINNs demons a es esilience o high unce ain y, con in-
gen on he implemen a ion o an op imized hype pa ame e s a egy. This adap abili y
unde sco es he necessi y o op imizing hype pa ame e s ailo ed o he dis inc i e
123
153 Page 18 o 19 Ope a ions Resea ch Fo um (2025) 6:153
cha ac e is ics o each p oblem, he eby illus a ing he in e play be ween hype pa-
ame e selec ions and he accu acy o solu ions ac oss a ying le els o coe icien
noise.
Acknowledgemen s We would like o hank he e iewe s o hei aluable commen s, which helped
o imp o e ou pape . S. Cuomo also acknowledges GNCS-INdAM and he UMI-TAA, UMI-AI esea ch
g oups. This wo k has been suppo ed by he p ojec : PNRR Cen o Nazionale HPC, Big Da a e Quan-
um Compu ing,(CN_00000013)(CUP: E63C22000980007), unde he NRRP MUR p og am unded by
he Nex Gene a ionEU. S. Cuomo has been pa ially suppo ed by he I alian PRIN p ojec Nume ical
Op imiza ion wi h Adap i e accu acy and applica ions o machine lea ning (CUP: E53D23007690006),
P oge i di Rice ca di In e esse Nazionale 2022.
Au ho Con ibu ions M.D. and L.P. w o e he o iginal d a , cu a ed he concep ualiza ion o he me hodol-
ogy, so wa e, and alida ion o he esul s. A.L. and S.C. supe ised he me hodology and concep ualiza ion,
by also pa icipa ing in he alida ion and isualiza ion.
Funding Open access unding p o ided by Uni e si à degli S udi di Napoli Fede ico II wi hin he CRUI-
CARE Ag eemen .
Da a A ailabili y No da ase s we e gene a ed o analysed du ing he cu en s udy.
Decla a ions
Compe ing In e es s The au ho s decla e no compe ing in e es s.
Open Access This a icle is licensed unde a C ea i e Commons A ibu ion 4.0 In e na ional License, which
pe mi s use, sha ing, adap a ion, dis ibu ion and ep oduc ion in any medium o o ma , as long as you gi e
app op ia e c edi o he o iginal au ho (s) and he sou ce, p o ide a link o he C ea i e Commons licence,
and indica e i changes we e made. The images o o he hi d pa y ma e ial in his a icle a e included
in he a icle’s C ea i e Commons licence, unless indica ed o he wise in a c edi line o he ma e ial. I
ma e ial is no included in he a icle’s C ea i e Commons licence and you in ended use is no pe mi ed
by s a u o y egula ion o exceeds he pe mi ed use, you will need o ob ain pe mission di ec ly om he
copy igh holde . To iew a copy o his licence, isi h p://c ea i ecommons.o g/licenses/by/4.0/.
Re e ences
1. Al Sa wan A, Song C, Waheed UB (2021) Is i ime o swish? Compa ing ac i a ion unc ions in
sol ing he Helmhol z equa ion using PINNs, In: 82nd EAGE Annual Con e ence & Exhibi ion,
Eu opean Associa ion o Geoscien is s & Enginee s. pp 1–5
2. Bha adwaja B, Nabian MA, Sha ma B, Choudh y S, Alanka A (2022) Physics-in o med machine
lea ning and unce ain y quan i ica ion o mechanics o he e ogeneous ma e ials. In eg Ma e Manu-
ac Inno 11:607–627
3. Cai S, Mao Z, Wang Z, Yin M, Ka niadakis GE (2021) Physics-in o med neu al ne wo ks (PINNs) o
luid mechanics: a e iew. Ac a Mech Sin 37:1727–1738
4. Chang W, Yuchen F, Yongqing Z, Xin L, Chaoqun Z, Heyang W (2023) Mul i-ou pu physics-in o med
neu al ne wo ks model based on he Runge-Ku a me hod. Chinese J Theo e Appl Mech 55:2405–2416
5. Cuomo S, Rosa M, Giampaolo F, Izzo S, Cola VS (2023) Sol ing g oundwa e low equa ion using
physics-in o med neu al ne wo ks. Compu Ma h Appl 145:106–123
6. Cuomo S, Rosa M, Piccialli F, Pompameo L (2024) Railway sa e y h ough p edic i e e ical dis-
placemen analysis using he PINN-EKF syne gy. Ma h Compu Simul 223:368–379
7. Ryck T, Jag ap AD, Mish a S (2024) E o es ima es o physics-in o med neu al ne wo ks app oxi-
ma ing he Na ie -S okes equa ions. IMA J Nume Anal 44:83–119
123
Ope a ions Resea ch Fo um (2025) 6:153 Page 19 o 19 153
8. Escapil-Inchauspé P, Ruz GA (2023) Hype -pa ame e uning o physics-in o med neu al ne wo ks:
applica ion o Helmhol z p oblems. Neu ocompu ing 561:126826
9. Fokas A (2004) Bounda y- alue p oblems o linea PDEs wi h a iable coe icien s. P oceedings o
he Royal Socie y o London. Se ies A: Ma h, Phys Eng Sci 460:1131–1151
10. G aham IG, Kuo FY, Nuyens D, Scheichl R, Sloan IH (2011) Quasi-Mon e Ca lo me hods o ellip ic
PDEs wi h andom coe icien s and applica ions. J Compu Phys 230:3668–3694
11. Hao TT, Yan WJ, Chen JB, Sun TT, Yuen KV (2024) Mul i-ou pu mul i-physics-in o med neu al
ne wo k o lea ning dimension- educed p obabili y densi y e olu ion equa ion wi h unknown spa io-
empo al-dependen coe icien s. Mech Sys Signal P ocess 220:111683
12. Jag ap AD, Kawaguchi K, Ka niadakis GE (2020) Adap i e ac i a ion unc ions accele a e con e gence
in deep and physics-in o med neu al ne wo ks. J Compu Phys 404:109136
13. Khoo Y, Lu J, Ying L (2021) Sol ing pa ame ic PDE p oblems wi h a i icial neu al ne wo ks. Eu J
Appl Ma h 32:421–435
14. Kingma DP, Ba J (2014) Adam: a me hod o s ochas ic op imiza ion. a Xi :1412.6980
15. Le DK, Guo M, Yoon JY (2023) Hype pa ame e op imiza ion o physics-in o med neu al ne wo ks
u ilizing gene ic algo i hm. A ailable a SSRN 4590874
16. Li J, Long X, Deng X, Jiang W, Zhou K, Jiang C, Zhang X (2024) A p incipled dis ance-awa e
unce ain y quan i ica ion app oach o enhancing he eliabili y o physics-in o med neu al ne wo k.
Reliab Eng Sys Sa e y 245:109963
17. Li W, Lee KM (2021) Physics in o med neu al ne wo k o pa ame e iden i ica ion and bounda y
o ce es ima ion o complian and biomechanical sys ems. In J In ell Robo Appl 5:313–325
18. Liu W, Liu Y, Li H, Yang Y (2023) Mul i-ou pu physics-in o med neu al ne wo k o one-and wo-
dimensional nonlinea ime dis ibu ed-o de models. Ne w He e ogen Media 18
19. Qian Y, Zhang Y, Huang Y, Dong S (2023) E o analysis o physics-in o med neu al ne wo ks o
app oxima ing dynamic PDEs o second o de in ime. a Xi :2303.12245
20. Raissi M, Pe dika is P, Ka niadakis GE (2019) Physics-in o med neu al ne wo ks: a deep lea ning
amewo k o sol ing o wa d and in e se p oblems in ol ing nonlinea pa ial di e en ial equa ions.
J Compu Phys 378:686–707
21. Ramachand an P, Zoph B, Le QV (2017) Sea ching o ac i a ion unc ions. a Xi :1710.05941
22. Sha ma P, E ans L, Tindall M, Ni hia asu P (2023) Hype pa ame e selec ion o physics-in o med neu-
al ne wo ks (PINNs)-applica ion o discon inuous hea conduc ion p oblems. In: Pa B (ed) Nume ical
Hea T ans e . Fundamen als, pp 1–15
23. Wang Y, Han X, Chang CY, Zha D, B aga-Ne o U, Hu X (2023) Au o-PINN: unde s anding and
op imizing physics-in o med neu al a chi ec u e. a Xi :2205.13748 2
24. Yang M, Fos e JT (2022) Mul i-ou pu physics-in o med neu al ne wo ks o o wa d and in e se
PDE p oblems wi h unce ain ies. Compu Me hods Appl Mech Eng 402:115041
25. Zhao S, Peng Y, Zhang Y, Wang H (2022) Pa ame e es ima ion o powe elec onic con e e s wi h
physics-in o med machine lea ning. IEEE T ans Powe Elec on 37:11567–11578
Publishe ’s No e Sp inge Na u e emains neu al wi h ega d o ju isdic ional claims in published maps
and ins i u ional a ilia ions.
123
