A ailable online a www.sciencedi ec .com
S uc u al In eg i y P ocedia 00 (2023) 000–000
www.else ie .com/loca e/p ocedia
F ac u e, Damage and S uc u al Heal h Moni o ing
Physics-In o med Neu al Ne wo ks o Mul iscale La ge
De o ma ion Analysis o Me ama e ials
Haolin Lia,∗, Zah a Sha i Khodaeia, Michal Ko oulb, M.H. Aliabadia
aDepa men o Ae onau ics, Impe ial College London, London, SW7 2AZ, UK
bIns i u e o Solid Mechanics, Mecha onics and Biomechanics, B no Uni e si y o Technology, Facul y o Mechanical Enginee ing, B no, Czech
Republic
Abs ac
Physics-in o med neu al ne wo ks (PINNs) ha e ecen ly eme ged as a p omising al e na i e o adi ional nume ical me hods o
sol ing solid mechanics p oblems. In his wo k, we p opose a no el PINN a chi ec u e designed o homogenisa ion p oblems o
me ama e ials unde la ge de o ma ion. The a chi ec u e inco po a es pe iodic unc ions o ensu e exac ly imposed bounda y con-
di ions and employs an ene gy-based loss o e icien aining. Th ee ep esen a i e me ama e ial s uc u es—oc e uss, gy oid,
and spindoid—a e selec ed as case s udies. The esul s demons a e ha he p oposed PINN achie es accu acy compa able o ini e
elemen analysis (FEA), while o e ing imp o ed compu a ional e iciency o high- olume- ac ion s uc u es. Beyond accu acy
and speed, he mesh ee na u e and lexibili y o PINNs p o ide clea ad an ages, highligh ing hei po en ial as a scalable ool o
modelling complex ma e ials.
©2023 The Au ho s. Published by Else ie B.V.
This is an open access a icle unde he CC BY-NC-ND license (h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0/)
Pee - e iew unde esponsibili y o P o esso Fe i Aliabadi.
Keywo ds: Physics in o med neu al ne wo k; Homogenisa ion; Me ama e ials; Mul iscale Analysis
1. In oduc ion
Physics-in o med neu al ne wo ks (PINNs) ha e seen apid de elopmen in ecen yea s Raissi e al. [2019], Ka ni-
adakis e al. [2021]. Thei applica ion in solid mechanics has also begun o eme ge. As a new mesh ee me hod, PINNs
di e no only om mesh-based app oaches such as he ini e elemen me hod (FEM), bu also om o he mesh ee
app oaches like adial basis unc ion (RBF) me hods Li and Liu [2002] o he bounda y elemen me hod Aliabadi
[2002], owing o hei global implemen a ion and app oxima ion na u e. Compa ed wi h hese me hods, PINNs as
PDE sol e s o solid mechanics o e se e al ad an ages: a) hey do no equi e p esc ibed meshes, which a e o en
expensi e and sensi i e o gene a e in FEM; b) hey p o ide inhe en ly smoo h and di e en iable app oxima ions o
he solu ion ield, unlike mos adi ional nume ical me hods ha allow only limi ed de i a i e o de s, es ic ing hei
∗Co esponding au ho . Tel.: +0-000-000-0000 ; ax: +0-000-000-0000.
E-mail add ess: [email p o ec ed]
2210-7843 ©2023 The Au ho s. Published by Else ie B.V.
This is an open access a icle unde he CC BY-NC-ND license (h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0/)
Pee - e iew unde esponsibili y o P o esso Fe i Aliabadi.
PREPRINT
2Au ho name /S uc u al In eg i y P ocedia 00 (2023) 000–000
use in high-o de p oblems; c) hey e ain all he s eng hs o neu al ne wo ks used in lea ning asks, gi ing PINNs
s ong po en ial o la ge-scale simula ions and in e se p oblems.
Thanks o hese ad an ages, he use o PINNs o sol e solid mechanics p oblems has become inc easingly popula
in bo h academia and enginee ing Haghigha e al. [2021], Bai e al. [2023], Hu e al. [2024], Wang e al. [2024]. A sys-
ema ic in es iga ion in Haghigha e al. [2021] has demons a ed he easibili y o applying PINNs o such p oblems
using he o iginal PINN o mula ion. Howe e , hei esul s also highligh se e al limi a ions, including low e i-
ciency and educed accu acy compa ed wi h adi ional nume ical me hods. To add ess hese challenges, esea che s
ha e p oposed di e en s a egies. One app oach is o e o mula e he loss unc ion om he s ong o m PDE o a
weak o m, o en h ough ene gy-based loss minimisa ion. Al hough he ene gy unc ional is ma hema ically equi -
alen o he s ong o m PDE, his weak o mula ion is easie o implemen and shows as e con e gence. No able
examples include he Deep Ene gy App oach Samaniego e al. [2020] and he Deep Ri z Me hod Yu e al. [2018]. An-
o he app oach exploi s he speci ic ea u es o solid mechanics PDEs, which o en in ol e complex geome ies. He e,
geome ic in o ma ion is in eg a ed in o he PINN a chi ec u e o solu ion p ocess, leading o geome y-awa e deep
lea ning me hods. Examples include XPINN Jag ap and Ka niadakis [2020], PINNs wi h exac bounda y condi ion
en o cemen Wang e al. [2023], and Fini e-PINN Li e al. [2024].
Among ypical solid mechanics p oblems, he so-called cell p oblem, o homogenisa ion p oblem, plays a cen al
ole Cha alambakis [2010], Li e al. [2023a]. I p o ides an e ec i e way o e alua e ma e ial p ope ies om complex
mic os uc u es and se es as a b idge be ween mic o- and mac o-scale modelling. Wi h he de elopmen o ad anced
ma e ials such as composi es and me ama e ials, homogenisa ion has become inc easingly impo an in mee ing bo h
academic and enginee ing demands. FEM emains he mos widely used sol e o homogenisa ion, bu i also su e s
om se e al d awbacks, including he di icul y o meshing highly complex geome ies and he high compu a ional
cos o 3D analyses Li e al. [2023b,2022,2025].
In his con ex , we employ he eme ging PINN app oach o o e come he challenges aced by FEM in homogeni-
sa ion p oblems. We p opose a no el neu al ne wo k a chi ec u e ailo ed o he equi emen s o homogenisa ion.
Ou s udy ocuses on me ama e ials, which ypically exhibi complex mic os uc u es, and add esses homogenisa ion
unde la ge de o ma ions, a c i ical ac o in p edic ing hei mechanical p ope ies. The p oposed me hod and case
s udies a e p esen ed in he ollowing sec ions.
2. Me hodology
2.1. Homogenisa ion o me ama e ial s uc u es
The go e ning cell (homogenisa ion) p oblem o la ge de o ma ions is de ined as:
∇·P(x)=0,∀x∈Ω
P(x)=C:F(x),∀x∈Ω
x∼x+L
(1)
whe e Pis he i s Piola–Ki chho s ess enso , Cis he elas ic cons i u i e enso , and Fis he de o ma ion g adien :
F(x)=∇(x+u(x)) (2)
whe e uis he displacemen ield. In he cell p oblem, xis pe iodic o e he domain wi h pe iod L, as deno ed by
Eq. (1).
A schema ic o he cell p oblem is shown in Fig. 1. The e ec i e domain/geome y o he me ama e ial s uc u e
is deno ed by Ω. This wo k employs a single-phase me ama e ial as he s udy objec , so Cis cons an in Eq. (1). To
sol e he PDE sys em in Eq. (1), he ollowing bounda y condi ions a e equi ed:
1
VZΩ
F(x)dV =¯
F,∀x∈Ω
n·P(x)=0,∀x∈∂Ω
(3)
PREPRINT
Au ho name /S uc u al In eg i y P ocedia 00 (2023) 000–000 3
Fig. 1. Pe iodic domain in homogenisa ion p oblems.
whe e ¯
Fdeno es he applied a e age s ain and ∂Ω ep esen s he domain bounda y, Vis he olume o he domain.
The i s ela ion in Eq. (4) p esc ibes he mac o s ain applied o he cell, and he second imposes homogeneous
Neumann bounda y condi ions.
The objec i e o he cell p oblem is o ob ain he homogenised e ec i e i s Piola–Ki chho s ess:
¯
P=1
VZΩ
P(x)dV,∀x∈Ω(4)
2.2. Hype elas ici y model
This wo k implemen s he la ge-de o ma ion o mula ion ia he Comple e Lag angian o mula ion, by in oducing
hype elas ici y.
The unc ional associa ed wi h Eq. 1de ining he po en ial ene gy is:
H(X)=ZΩ
Ψ(X(x)) dV (5)
whe e Xdeno es he de o med con igu a ion:
X(x)=x+u(x)(6)
and Ψ ep esen s he s ain ene gy. Minimising Eq. (5) is equi alen o sol ing he s ong- o m PDE in Eq. (1).
In his wo k, he s ain ene gy Ψis based on he Neo-Hookean hype elas ic model:
Ψ=C10
2¯
I1−3+1
D1
(J−1)2,(7)
whe e
•C10 is a ma e ial cons an ha go e ns he de ia o ic (shea ) esponse o he ma e ial, ela ed o he shea
modulus by µ=2C10.
•D1is a ma e ial cons an ha go e ns he olume ic esponse (comp essibili y), ela ed o he bulk modulus by
K=2
D1.
•¯
I1=J−2/3I1is he i s de ia o ic in a ian o he igh Cauchy–G een enso ,
•I1= (C), wi h C=FTF he igh Cauchy–G een enso ,
•J=de (F) is he Jacobian o he de o ma ion g adien F.
PREPRINT
4Au ho name /S uc u al In eg i y P ocedia 00 (2023) 000–000
2.3. PINN model o Homogenisa ion
PINN app oxima es he solu ion ield Xby a ully connec ed neu al ne wo k. In his wo k, we p opose a neu al
ne wo k a chi ec u e ailo ed o homogenisa ion p oblems:
X(x)=¯
F:L+NN sin x·2π
L!,cos x·2π
L!;θ!,(8)
whe e θdeno es all ainable pa ame e s in NN. This a chi ec u e p ojec s he neu al ne wo k app oxima ion in o a
pe iodic space, consis en wi h he physical se ing. As a esul , he bounda y condi ion in Eq. 5is exac ly sa is ied
once he mac oscopic s ain ¯
Fis p esc ibed.
The ene gy-based loss, de ined om Eq. 6, is hen employed o sol e he PDE sys em. The op imisa ion p oblem
is exp essed as:
min
θ
L(x)=ZΩ
Ψ(x;θ)dV,(9)
The p oposed neu al ne wo k a chi ec u e inhe en ly en o ces he bounda y condi ions in Eq. 2. The e o e, no addi-
ional bounda y-condi ion loss e m is equi ed.
3. Case s udy
We employ h ee ypical me ama e ial s uc u es o ou case s udies: oc e - uss me ama e ials, gy oid me ama e-
ials, and spindoid me ama e ials. The oc e uss is a classic la ice o in e connec ed s u s; i is s e ch-domina ed,
o e ing high s i ness- o-weigh e iciency and long se ing as a benchma k o ligh weigh s uc u al design. The
gy oid is a iply pe iodic minimal su ace wi h a smoo h, laby in h-like geome y; i p o ides iso opic s i ness and
a high su ace- o- olume a io, making i a ep esen a i e su ace-based me ama e ial. Spindoids a e a newe class
wi h complex, i egula pa e ns; hey enable unusual aniso opy and non-linea de o ma ion, exhibi ing p ope ies
no ound in adi ional la ices o TPMS s uc u es.
We selec ed hese h ee because hey a e ypical ye dis inc : he oc e uss (s u -based), he gy oid (su ace-
based), and he spindoid (i egula pa e ns). They also span di e en olume ac ions and ha e s ikingly di e en
appea ances, making hem ideal ep esen a i es o compa a i e s udy. The geome ies o he h ee me ama e ials a e
shown in Fig. 2.
Fig. 2. Me ama e ial s uc u es: (a) oc e - uss me ama e ial; (b) gy oid me ama e ial; (c) spindoid me ama e ial.
Fo he implemen a ion, we use he colloca ion me hod o e alua e he in eg al in he loss unc ion, Eq. 9. The
colloca ion poin s a e uni o mly dis ibu ed in he 3D uni cell wi h a esolu ion o 256 ×256 ×256. Because he
h ee s uc u es ha e di e en olume ac ions, he numbe s o colloca ion poin s a e 492,616 o he oc e uss,
2,261,719 o he gy oid, and 5,163,526 o he spindoid, espec i ely. To alida e he esul s, we use ini e elemen
analysis (FEA) as he e e ence; he ini e elemen models o he h ee s uc u es a e also shown in Fig. 3. No e ha
PREPRINT
Au ho name /S uc u al In eg i y P ocedia 00 (2023) 000–000 5
he Neo-Hookean hype elas ici y model used in he PINN (Eq. 7) is consis en wi h he one used in Abaqus, which
acili a es alida ion.
Fig. 3. Fini e-elemen models o me ama e ial s uc u es: (a) oc e - uss me ama e ial; (b) gy oid me ama e ial; (c) spindoid me ama e ial.
3.1. Single-axis s ain de o ma ion
Fi s , we apply a mac oscopic ensile s ain pu ely in he z( e ical) di ec ion as an example o single-axis s ain.
The esul ing homogenised s esses a e gi en in Table.1. I is seen om he able ha The PINN achie es accu acy
compa able o he FEM esul s. The de o med con igu a ions a e illus a ed in Fig. 4as a showcase.
Pxx Pyy Pzz Pxy Pxz Pyz
Oc e T uss FEM 0.087 0.087 1.344 0.0137 0.223 0.223
PINN 0.089 0.089 1.344 0.0101 0.223 0.224
Gy oid FEM 0.327 0.327 4.712 0.134 0.720 0.720
PINN 0.328 0.328 4.724 0.137 0.721 0.723
Spindoid FEM 1.153 1.153 17.740 0.292 0.846 0.846
PINN 1.152 1.157 17.788 0.292 0.845 0.850
Table 1. Placeholde cap ion o a able wi h h ee ows and se en columns.
3.2. Mul iple-axis s ain de o ma ion
Second, we apply andom mac oscopic s ains in all h ee axes, including bo h no mal and shea componen s. The
homogenised s esses a e epo ed in Table.2. Again, he PINN yields esul s compa able wi h he FEM e e ence.
The de o med con igu a ions a e shown in Fig. 5 o illus a ion.
Pxx Pyy Pzz Pxy Pxz Pyz
Oc e T uss FEM 1.170 1.210 0.722 0.124 0.183 0.162
PINN 1.175 1.217 0.743 0.125 0.184 0.163
Gy oid FEM 1.845 2.753 3.385 2.133 2.153 2.905
PINN 1.890 2.742 3.399 2.142 2.154 2.911
Spindoid FEM 4.728 20.134 12.134 7.747 7.602 6.112
PINN 4.738 20.277 12.135 7.749 7.609 6.116
Table 2. Placeholde cap ion o a able wi h h ee ows and se en columns.
PREPRINT
6Au ho name /S uc u al In eg i y P ocedia 00 (2023) 000–000
Fig. 4. De o med con igu a ions unde single-axis s ain: (a) oc e - uss me ama e ial; (b) gy oid me ama e ial; (c) spindoid me ama e ial. The colo
ep esen s he displacemen dis ibu ions.
Fig. 5. De o med con igu a ions unde andom mul i-axial s ain: (a) oc e - uss me ama e ial; (b) gy oid me ama e ial; (c) spindoid me ama e ial.
The colo ep esen s he displacemen dis ibu ions.
4. Discussion
The esul s demons a e ha he p oposed PINN me hod achie es accu acy compa able o ha o FEM ac oss bo h
single-axis and mul iple-axis s ain cases. Fo all h ee me ama e ial s uc u es, he homogenised s esses p edic ed by
PINN closely ma ch hose ob ained om FEM, alida ing he co ec ness o he o mula ion and he implemen a ion
o he ene gy-based app oach.
In e ms o e iciency, he compu ing pe o mance o PINN and FEM shows di e en beha iou depending on he
s uc u al olume ac ion (ϕ). Fo he low- olume- ac ion oc e uss (V =0.029), he compu a ional imes a e
compa able (PINN ≈8 min; FEM ≈10 min). Fo he gy oid (V =0.140), he gap widens (PINN ≈15 min; FEM ≈
47 min). Fo he high- olume- ac ion spindoid (V =0.307), he PINN me hod becomes signi ican ly as e (PINN
≈23 min; FEM ≈296 min). This di e ence a ises because FEM’s compu a ional cos g ows non-linea ly wi h he
numbe o mesh elemen s, whe eas PINN exhibi s only mino sensi i i y o he inc ease in colloca ion poin s.
The e ec i eness o he p oposed PINN model can be a ibu ed o wo key design choices. Fi s , he adop ion o an
ene gy-based loss e m enables as e con e gence compa ed o he con en ional s ong- o m PDE loss. Second, he
PREPRINT
Au ho name /S uc u al In eg i y P ocedia 00 (2023) 000–000 7
ne wo k a chi ec u e en o ces bounda y condi ions exac ly, so he op imisa ion ocuses solely on minimising s ain
ene gy. This elimina es he balance p oblem among mul iple compe ing loss e ms ha o en complica es s anda d
PINN aining.
Beyond accu acy and speed, he PINN me hod p o ides addi ional ad an ages. I na u ally suppo s pa allel com-
pu ing, which is pa icula ly use ul in enginee ing applica ions. I also a oids he need o mesh gene a ion, he eby
simpli ying p ep ocessing and imp o ing scalabili y. Mo eo e , he amewo k can be eadily ex ended o o he p ob-
lem classes, such as his o y-dependen ma e ial models, by le e aging ans e lea ning. In con as , FEM compu a-
ions mus be es a ed om sc a ch when p oblem se ings a e modi ied.
O e all, hese esul s sugges ha he p oposed PINN app oach is no only a iable al e na i e o FEM bu also
o e s unique ad an ages in lexibili y and scalabili y, making i a p omising ool o u u e s udies o complex me a-
ma e ial sys ems.
5. Conclusion
This wo k has p esen ed a physics-in o med neu al ne wo k amewo k designed o homogenisa ion o me a-
ma e ials subjec o la ge de o ma ion. By embedding pe iodici y di ec ly in o he neu al ne wo k a chi ec u e and
adop ing an ene gy-based loss, he me hod en o ces bounda y condi ions exac ly and a oids balancing mul iple loss
e ms. Compa a i e s udies wi h ini e elemen analysis show ha he p oposed PINN achie es simila accu acy, while
demons a ing as e pe o mance in complex, high- olume- ac ion s uc u es such as spindoids. Addi ional bene i s
o he PINN app oach include i s mesh ee o mula ion, ease o pa allelisa ion, and adap abili y o ex ended p oblem
classes h ough ans e lea ning. Taken oge he , hese esul s sugges ha PINNs p o ide an e ec i e and e sa ile
al e na i e o con en ional sol e s, opening new possibili ies o compu a ional mechanics and me ama e ial design.
Da a a ailabili y
Rega ding he compu a ional p ocedu es see Li, H.. (2025). Da a o “Physics-In o med Neu al Ne wo ks o
Mul iscale La ge De o ma ion Analysis o Me ama e ials” (1.0.0). Zenodo. h ps://doi.o g/10.5281/zenodo.17232199
Acknowledgemen s
The au ho s acknowledge he suppo s by he p ojec BAANG – ”Building Ac ions in Sma A ia ion wi h En i-
onmen al Gains” unded by he Eu opean Union P og amme Ho izon Eu ope unde g an ag eemen no. 101079091.
Re e ences
Mazia Raissi, Pa is Pe dika is, and Geo ge E Ka niadakis. 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. Jou nal o Compu a ional physics, 378:686–707, 2019.
Geo ge Em Ka niadakis, Ioannis G Ke ekidis, Lu Lu, Pa is Pe dika is, Si an Wang, and Liu Yang. Physics-in o med machine lea ning. Na u e
Re iews Physics, 3(6):422–440, 2021.
Shao an Li and Wing Kam Liu. Mesh ee and pa icle me hods and hei applica ions. Applied Mechanics Re iews, 55(1):1–34, 2002.
M. H. Aliabadi. The bounda y elemen me hod, olume 2: applica ions in solids and s uc u es, olume 2. John Wiley & Sons, 2002.
Ehsan Haghigha , Mazia Raissi, Ad ian Mou e, Hec o Gomez, and Ruben Juanes. A physics-in o med deep lea ning amewo k o in e sion
and su oga e modeling in solid mechanics. Compu e Me hods in Applied Mechanics and Enginee ing, 379:113741, 2021.
Jinshuai Bai, Timon Rabczuk, Ashish Gup a, Lai h Alzubaidi, and Yuan ong Gu. A physics-in o med neu al ne wo k echnique based on a modi ied
loss unc ion o compu a ional 2d and 3d solid mechanics. Compu a ional Mechanics, 71(3):543–562, 2023.
Hao eng Hu, Lehua Qi, and Xujiang Chao. Physics-in o med neu al ne wo ks (pinn) o compu a ional solid mechanics: Nume ical amewo ks
and applica ions. Thin-Walled S uc u es, 205:112495, 2024.
Lu Wang, Guangyan Liu, Guanglun Wang, and Kai Zhang. M-pinn: A mesh-based physics-in o med neu al ne wo k o linea elas ic p oblems in
solid mechanics. In e na ional jou nal o nume ical me hods in enginee ing, 125(9):e7444, 2024.
Es eban Samaniego, Cosmin Ani escu, Somda a Goswami, Vien Minh Nguyen-Thanh, Hongwei Guo, Khade Hamdia, Xiaoying Zhuang, and Ti-
mon Rabczuk. An ene gy app oach o he solu ion o pa ial di e en ial equa ions in compu a ional mechanics ia machine lea ning: Concep s,
implemen a ion and applica ions. Compu e Me hods in Applied Mechanics and Enginee ing, 362:112790, 2020.
PREPRINT
8Au ho name /S uc u al In eg i y P ocedia 00 (2023) 000–000
Bing Yu e al. The deep i z me hod: a deep lea ning-based nume ical algo i hm o sol ing a ia ional p oblems. Communica ions in Ma hema ics
and S a is ics, 6(1):1–12, 2018.
Ameya D Jag ap and Geo ge Em Ka niadakis. Ex ended physics-in o med neu al ne wo ks (xpinns): A gene alized space- ime domain decompo-
si ion based deep lea ning amewo k o nonlinea pa ial di e en ial equa ions. Communica ions in Compu a ional Physics, 28(5), 2020.
Jiaji Wang, YL Mo, Bassam Izzuddin, and Chul-Woo Kim. Exac di ichle bounda y physics-in o med neu al ne wo k epinn o solid mechanics.
Compu e Me hods in Applied Mechanics and Enginee ing, 414:116184, 2023.
Haolin Li, Yuyang Miao, Zah a Sha i Khodaei, and MH Aliabadi. Fini e-pinn: A physics-in o med neu al ne wo k a chi ec u e o sol ing solid
mechanics p oblems wi h gene al geome ies. A ailable a SSRN 5066837, 2024.
Nicolas Cha alambakis. Homogeniza ion echniques and mic omechanics. a su ey and pe spec i es. Applied Mechanics Re iews, 63(3), 2010.
Haolin Li, Zah a Sha i Khodaei, and MH Fe i Aliabadi. F -based mul iscale scheme o homogenisa ion o he e ogeneous pla es including
damage and ailu e. Compu e Me hods in Applied Mechanics and Enginee ing, 416:116369, 2023a.
Haolin Li, Zah a Sha i Khodaei, and MH Fe i Aliabadi. Mul iscale modelling o ma e ial deg ada ion and ailu e in plain wo en composi es: A
no el app oach o eliable p edic ions enabled by me a-models. Composi es Science and Technology, 233:109910, 2023b.
Haolin Li, Oma Baca eza, Zah a Sha i Khodaei, and MH Fe i Aliabadi. P obabilis ic mul i-scale design o 2d plain wo en composi es consid-
e ing meso-scale unce ain ies. Composi e S uc u es, 300:116099, 2022.
Haolin Li, Zah a Sha i Khodaei, and MH Aliabadi. F -based homogenisa ion o e icien concu en mul iscale modelling o hin pla e s uc u es.
Compu a ional Mechanics, 75(2):637–653, 2025.
PREPRINT
