Deep Learning for Inverting Borehole Resistivity Measurements.

Deep Lea ning o In e ing
Bo ehole Resis i i y
Measu emen s
Jon Ande Ri e a Gonz´alez
Supe ised by Da id Pa do and Elisabe e Albe di
No embe 2022
(cc)2022 JON ANDER RIVERA GONZALEZ (cc by 4.0)
Deep Lea ning o In e ing
Bo ehole Resis i i y
Measu emen s
Jon Ande Ri e a Gonz´alez
Supe ised by Da id Pa do and Elisabe e Albe di
No embe 2022
This disse a ion has been possible wi h he suppo o he Uni e si y o he
Basque Coun y (UPV/EHU) g an No. PIF18/017; he BCAM “Se e o Ochoa”
acc edi a ion o excellence (SEV-2017-0718); he Basque Go e nmen h ough he
BERC 2018-2021 p og am; and he Consolida ed Resea ch G oup MATHMODE
(IT1294-19;IT1456-22) gi en by he Depa men o Educa ion.
i
Acknowledgemen s
The las ou yea s o my li e ha e been an ad en u e o lea ning and pe sonal
g ow h. In his jou ney, I ha e been o una e o ha e Da id Pa do as my su-
pe iso . The day I me him I was su p ised by he na u alness wi h which he
ea ed me. F om hen on, I go o know he es o his mul iple i ues: wo k
discipline (always a ailable o any ques ions o que ies), en husiasm o esea ch
(always looking o new challenges and new p oblems o sol e), and in ui ion– he
one ha imp esses me he mos . I will be e e nally g a e ul o him and he knows
ha , whe e e he is, he will always ha e a iend he e.
I would also like o hank my o he supe iso Elisabe e Albe di o all he
wo k she has done o help me. I would like o hank he o he us she has
placed in me when making me pa o he p ojec s and o all he ad ice she has
gi en me du ing hese ou yea s. Finally, she has augh me o manage wi hin
he Uni e si y and she has gi en me he oppo uni y o each some classes and
see how hings look om he o he side o he class oom.
I am deeply g a e ul o Mos a a Shah ia i o all his help. He is a lo ely pe son.
F om he i s day I came o he g oup, he welcomed me and had he pa ience o
help me day a e day. He augh me e e y hing he knew and helped me build
he ounda ion o wha oday is my Disse a ion. I also wan o hank him o
he oppo uni y he ga e me o go o Aus ia o collabo a e wi h him. I lea ned
a lo om ha expe ience. I expec o con inue seeing each o he in he coming
yea s.
I would like o exp ess my g a i ude o Ja ie Omella o e e y hing he has
done o me. I ha e sha ed many hou s o wo k wi h him and he augh me new
hings. He was always a ailable o help and oge he we managed o ge he job
done. I ha e seen ew expe s ad anced p og amme s like him and I hank him
o ansmi ing all ha knowledge o me.
I wish o hank all my colleagues om he MATHMODE g oup and BCAM. Es-
pecially, I would like o hank Judi Mu˜noz-Ma u e and Ana Fe nandez-Na amuel.
The i s one o helping me wi h bu eauc acy ques ions; he second one o sha -
ing wi h me he en iching expe iences while de eloping ou Disse a ions.
Finally, I wan o hank my amily and iends o hei suppo . In pa icula ,
I wan o o e my e e nal g a i ude o he h ee people ha ha e been wi h me all
his way long. Fi s , o my pa en s Ma i Ca men and Michel. Thank you o all
you uncondi ional suppo and lo e, o eaching me he alues ha ha e made
ii
Acknowledgemen s
me who I am, and o encou aging me o pu sue a Ph.D deg ee. Las bu no
leas , o my pa ne Amaia. She has consis en ly suppo ed me when I needed
i : she was he se eni y when I was s essed and he encou agemen when I was
us a ed. Thank you o us ing me and o ne e ceasing o emind me ha
I was capable o doing i . Zaila bada ez da ezinezkoa, zaila bada badago lo zea.
iii

Abs ac
The Ea h’s subsu ace is o med by di e en ma e ials, mainly po ous ocks pos-
sibly con aining mine als and illed wi h sal y wa e and/o hyd oca bons. The
o ma ions ha hese ma e ials c ea e a e o en i egula , appea ing geome ically
ab up o ms wi h di e en p ope ies ha a e mixed wi hin he same laye .
One o he main objec i es in geophysics is o de e mine he pe ophysical
p ope ies o he Ea h’s subsu ace. In his way, companies can disco e hyd o-
ca bon ese oi s and maximize he p oduc ion, and de e mine op imal loca ions
o hyd ogen s o age o CO2-seques a ion. To achie e hese goals, companies
o en eco d elec omagne ic measu emen s using Logging While D illing (LWD)
ins umen s, which a e able o eco d da a while d illing. The eco ded da a is
p ocessed o p oduce a map o he Ea h’s subsu ace. Based on he econs uc ed
Ea h model, he ope a o adjus s he well ajec o y in eal- ime o u he ex-
plo e exploi a ion a ge s, including oil and gas ese oi s, and o maximize he
pos e io p oduc i i y o he a ailable ese es. This eal- ime adjus men ech-
nique is called geos ee ing.
Nowadays, geos ee ing plays an essen ial ole in geophysics. Howe e , i e-
qui es he capabili y o sol ing in e se p oblems in eal ime. This is challenging
since in e se p oblems a e o en ill-posed.
The e exis mul iple adi ional me hods o sol e in e se p oblems, mainly,
g adien -based o s a is ics-based me hods. Howe e , hese me hods ha e se e e
limi a ions. In pa icula , hey o en need o compu e he o wa d p oblem hun-
d eds o imes o each se o measu emen s, which is compu a ionally expensi e
in h ee-dimensional (3D) p oblems.
To o e come hese limi a ions, we p opose he use o Deep Lea ning (DL)
echniques o sol e in e se p oblems. Al hough he aining s age o a Deep
Neu al Ne wo k (DNN) may be ime-consuming, a e he ne wo k is p ope ly
ained, i can o ecas he solu ion in a ac ion o a second, acili a ing eal-
ime geos ee ing ope a ions. In he i s pa o his disse a ion, we in es iga e
app op ia e loss unc ions o ain a DNN when dealing wi h an in e se p oblem.
Addi ionally, o p ope ly ain a DNN ha app oxima es he in e se solu ion,
we equi e a la ge da ase con aining he solu ion o he o wa d p oblem o
many di e en Ea h models. To c ea e such da ase , we need o sol e a Pa ial
Di e en ial Equa ion (PDE) housands o imes. Building a da ase may be
ime-consuming, especially o wo and h ee-dimensional p oblems since sol ing
i
Abs ac
PDEs using adi ional me hods, such as he Fini e Elemen Me hod (FEM), is
compu a ionally expensi e. Thus, we wan o educe he compu a ional cos o
building he da abase needed o ain he DNN. Fo his, we p opose he use o
e ined Isogeome ic Analysis ( IGA) me hods.
In addi ion, we explo e he possibili y o using DL echniques o sol e PDEs,
which is he main compu a ional bo leneck when sol ing in e se p oblems. Ou
main goal is o de elop a as o wa d simula o o sol ing pa ame ic PDEs. As
a i s s ep, in his disse a ion we analyze he quad a u e p oblems ha appea
while sol ing PDEs using DNNs and p opose di e en in eg a ion me hods o
o e come hese limi a ions.
Resumen
El subsuelo e es e es ´a o mado po di e en es ma e iales, p incipalmen e po
ocas po osas que posiblemen e con ienen mine ales y es ´an ellenas de agua sa-
lada y/o hid oca bu os. Po lo gene al, las o maciones que c ean es os ma e iales
son i egula es y con ma e iales de di e en es p opiedades mezclados en el mismo
es a o.
Uno de los p incipales obje i os en geo ´ısica es de e mina las p opiedades
pe o ´ısicas del subsuelo de la Tie a. De es e modo, las compa˜n´ıas pueden de-
e mina la localizaci´on de las ese as de hid oca bu os pa a maximiza su p o-
ducci´on o descub i localizaciones ´op imas pa a el almacenamien o de hid ´ogeno o
el dep´osi o de CO2. Pa a es e p op´osi o, las compa˜n´ıas egis an mediciones elec-
omagn´e icas u ilizando he amien as de Medici´on Du an e Pe o aci´on (MDP),
las cuales son capaces de ecaba da os mien as se lle a a cabo el p oceso de
p ospecci´on. Los da os ob enidos se p ocesan pa a p oduci un mapa del sub-
suelo de la Tie a. Bas´andose en el mapa gene ado, el ope ado ajus a en iempo
eal la ayec o ia de la he amien a de p ospecci´on pa a segui explo ando ob-
je i os de explo aci´on, incluidos los yacimien os de pe ´oleo y gas, y maximiza
la pos e io p oduc i idad de las ese as disponibles. Es a ´ecnica de ajus e en
iempo eal se denomina geo-na egaci´on.
Hoy en d´ıa, la geo-na egaci´on desempe˜na un papel esencial en geo ´ısica. Sin
emba go, equie e la esoluci´on de p oblemas in e sos en iempo eal. Es o supone
un e o, ya que los p oblemas in e sos suelen es a mal plan eados.
Exis en m´ul iples m´e odos adicionales pa a esol e los p oblemas in e sos,
p incipalmen e, los m´e odos basados en el g adien e o en la es ad´ıs ica. Sin
emba go, es os m´e odos ienen g a es limi aciones. En pa icula , a menudo
necesi an calcula el p oblema in e so cien os de eces pa a cada conjun o de
mediciones, lo que es compu acionalmen e ca o en p oblemas idimensionales
(3D).
Pa a supe a es as limi aciones, p oponemos el uso de ´ecnicas de Ap endizaje
P o undo (AP) pa a esol e los p oblemas in e sos. Aunque la e apa de en e-
namien o de una Red Neu onal P o unda (RNP) puede eque i mucho iempo,
una ez que la ed es ´a co ec amen e en enada puede p edeci la soluci´on en
una acci´on de segundo, acili ando las ope aciones de geo-na egaci´on en iempo
eal. En la p ime a pa e de es a esis, in es igamos las unciones de p´e dida
ap opiadas pa a en ena una RNP cuando se a a de un p oblema in e so.
i
Resumen
Adem´as, pa a en ena adecuadamen e una RNP que se ap oxime a la soluci´on
in e sa, necesi amos un g an conjun o de da os que con enga la soluci´on del
p oblema di ec o pa a muchos modelos e es es di e en es. Pa a c ea dicho
conjun o de da os, necesi amos esol e una Ecuaci´on en De i adas Pa ciales
(EDPs) miles de eces. La c eaci´on de un conjun o de da os puede lle a mucho
iempo, especialmen e pa a los p oblemas bidimensionales y idimensionales, ya
que la esoluci´on de la EDPs median e m´e odos adicionales, como el M´e odo de
Elemen os Fini os (MEF), es compu acionalmen e ca o. Po lo an o, que emos
educi el cos e compu acional de la cons ucci´on de la base de da os necesa ia
pa a en ena la RNP. Pa a ello, p oponemos el uso de m´e odos de An´alisis Iso-
geom´e ico e inado (AIG ).
Adem´as, explo amos la posibilidad de u iliza ´ecnicas de AP pa a esol e
EDPs, que es la limi aci´on compu acional p incipal al esol e p oblemas in e -
sos. Nues o obje i o p incipal es desa olla un simulado ´apido pa a esol e
EDPs pa am´e icas. Como p ime paso, en es a esis analizamos los p oblemas
de cuad a u a que apa ecen al esol e EDPs u ilizando RNPs y p oponemos
di e en es m´e odos de in eg aci´on pa a supe a es as limi aciones.
ii
LIST OF FIGURES
2.10 Analy ical solu ion s DNN p edic ed solu ion e alua ed o e he
es da ase using he wo-s ep based loss unc ion. . . . . . . . . 23
2.11 Example B.1. E olu ion o he di e en e ms o he Encode -
Decode loss unc ion gi en by Equa ion 2.25 wi hou egula iza ion. 26
2.12 Example B.1. E olu ion o he di e en e ms o he Encode -
Decode loss unc ion gi en by Equa ion 2.25 wi h he egula iza-
ion e m p esc ibed by Equa ion 2.17. . . . . . . . . . . . . . . . 27
2.13 Geosignal c oss-plo s o he Example B.1 wi hou egula iza ion
o he es da ase . Fi s ow: C oss-plo s o ype 1. Second ow:
C oss-plo s o ype 2. Fi s column: A enua ion. Second column:
Phase. ................................. 29
2.14 C oss-plo s o ype 4 o Example B.1 wi hou egula iza ion o
he aining da ase ( i s column), and wi h egula iza ion o he
aining da ase (second column) and he es da ase ( hi d col-
umn). Fi s ow: dis ance o he uppe laye . Second ow: dis ance
o he lowe laye . Thi d ow: esis i i y o uppe laye . Fou h
ow: esis i i y o lowe laye . Fi h ow: esis i i y o cen al laye . 32
2.15 Fo ma ion o model p oblem I. . . . . . . . . . . . . . . . . . . . . 33
2.16 In e ed o ma ion o model p oblem I using he in e sion s a egy
o Example B.2, i.e., wi h inpu measu emen s co esponding o 65
logging posi ions pe sample. . . . . . . . . . . . . . . . . . . . . . 34
2.17 Model p oblem I. Compa ison be ween F˝Iand F˝Iθ˚using he
in e sion s a egy o Example B.2, i.e., wi h inpu measu emen s
co esponding o 65 logging posi ions pe sample. . . . . . . . . . 35
2.18 In e ed o ma ion o model p oblem I using he in e sion s a egy
o Example B.1, i.e., wi h inpu measu emen s co esponding o
one logging posi ion pe sample. . . . . . . . . . . . . . . . . . . . 37
2.19 Model p oblem I. Compa ison be ween F˝Iand F˝Iθ˚wi hou
egula iza ion using he Encode -Decode loss unc ion and he
in e sion s a egy o Example B.1, i.e., wi h inpu measu emen s
co esponding o one logging posi ion pe sample. . . . . . . . . . 38
2.20 Model p oblem I. Compa ison be ween F˝Iand F˝Iθ˚using
he wo-s ep based loss unc ion wi hou egula iza ion and he
in e sion s a egy o Example B.1, i.e., wi h inpu measu emen s
co esponding o one logging posi ion pe sample. . . . . . . . . . 39
2.21 Model p oblem I. Compa ison be ween F˝Iand F˝Iθ˚wi h
egula iza ion using he in e sion s a egy o Example B.1, i.e.,
wi h inpu measu emen s co esponding o one logging posi ion
pe sample. .............................. 40
xi

LIST OF FIGURES
2.22 Model p oblem I. Compa ison be ween F˝Iand Fφ˚˝Iwi h
egula iza ion using he in e sion s a egy o Example B.1, i.e.,
wi h inpu measu emen s co esponding o one logging posi ion
pe sample. .............................. 41
2.23 Model p oblem I. Compa ison be ween F˝Iand Fφ˚˝Iθ˚wi h
egula iza ion using he in e sion s a egy o Example B.1, i.e.,
wi h inpu measu emen s co esponding o one logging posi ion
pe sample. .............................. 42
2.24 Model p oblem 2. Compa ison be ween ac ual and p edic ed o -
ma ions wi h egula iza ion using he in e sion s a egy o Exam-
ple B.1, i.e., wi h inpu measu emen s co esponding o one logging
posi ionpe sample. ......................... 43
2.25 Model p oblem 2. Compa ison be ween F˝Iand F˝Iθ˚wi h
egula iza ion using he in e sion s a egy o Example B.1, i.e.,
wi h inpu measu emen s co esponding o one logging posi ion
pe sample. .............................. 44
2.26 Model p oblem 2. Compa ison be ween F˝Iand Fφ˚˝Iwi h
egula iza ion using he in e sion s a egy o Example B.1, i.e.,
wi h inpu measu emen s co esponding o one logging posi ion
pe sample. .............................. 45
2.27 Model p oblem III, ajec o y 1. Compa ison be ween ac ual and
p edic ed o ma ions and he co esponding coaxial logs wi h egu-
la iza ion using he in e sion s a egy o Example B.1, i.e., wi h in-
pu measu emen s co esponding o one logging posi ion pe sample. 47
2.28 Model p oblem III, T ajec o y 2. Compa ison be ween ac ual and
p edic ed o ma ions and he co esponding coaxial logs wi h egu-
la iza ion using he in e sion s a egy o Example B.1, i.e., wi h in-
pu measu emen s co esponding o one logging posi ion pe sample. 48
3.1 A schema ic LWD ins umen wi h wo ansmi e s and wo e-
cei e s loca ed symme ically a ound he ool cen e . . . . . . . . 53
3.2 Example o he Hpcu lq ˆ H1space o a 2.5D o mula ion dis-
c e ized by Cp´1Isogeome ic Analysis (IGA) wi h uni o m 8 ˆ8
elemen s in Ωx,z, polynomial deg ee p“4, and con inui y k“3.
The uni a ia e basis unc ions o Hx, Hy, and Hza e shown in blue,
ed, and pu ple, espec i ely. Thin g ay lines in he mesh skele on
deno e he high-con inui y elemen in e aces. . . . . . . . . . . . 55
x
LIST OF FIGURES
3.3 Hpcu lqˆH1 IGA space in Ωx,z, associa ed wi h he 8ˆ8 domain
o Figu e 3.2 wi h p“4 and k“3, a e one le el o symme -
ic pa i ioning by he IGA disc e iza ion ha esul s in 4 ˆ4
mac oelemen s. IGA educes he con inui y o basis unc ions
by k´1 deg ees ac oss he mac oelemen sepa a o s ( he low-
con inui y bases a e shown in black). Thin g ay lines in he mesh
skele on deno e he high-con inui y elemen in e aces, while hick
black lines illus a e he mac oelemen bounda ies. We e e o he
e ical and ho izon al sepa a o s as “ s” and “hs”, espec i ely. . 57
3.4 A d awing o he compu a ional domain Ωx,z and he ool ajec-
o y. The cen al subg id bounded by a magen a box is composed
o a se o ine elemen s loca ed in he p oximi y o he logging
ins umen . The emaining elemen s g ow smoo hly in size un il
eaching he bounda y. . . . . . . . . . . . . . . . . . . . . . . . . 59
3.5 Nume ical e o s when compu ing a enua ion a io, A, and phase
di e ence, P, in a homogeneous medium using IGA and IGA dis-
c e iza ions, ob ained by a 64 ˆ64 elemen mesh wi h di e en
elemen sizes hand polynomial deg ees p. ............. 61
3.6 Compa ison o he decay o he nume ical and analy ical coaxial
magne ic ields o some Fou ie modes, ob ained in a g id o 64ˆ64
elemen s wi h h“0.025 m and di e en polynomial deg ees. . . . 62
3.7 Compu a ional cos in e ms o FLOPs and ime o sol ing a 2.5D
bo ehole esis i i y p oblem pe logging posi ion pe Fou ie mode.
We es IGA disc e iza ions wi h wo di e en g ids o 64ˆ64 and
128 ˆ128 elemen s. The compu a ional imes co espond o he
use o pa allel sol e PARDISO using wo h eads. . . . . . . . . . 63
3.8 Model p oblem wi h a cons an dip angle o 80˝passing h ough
a geological aul and h ee di e en ma e ials (well ajec o y is
highligh ed by a ed dashed line). Dimensions a e in me e s. . . . 65
3.9 Appa en esis i i ies based on he a enua ion a io, ρA, and phase
di e ence, ρP, o he i s model p oblem, compa ed wi h he
eal (exac ) esis i i y, ρe. We ob ain he esul s using a IGA
disc e iza ion wi h 64ˆ64 elemen s, p“4, and 8ˆ8 mac oelemen s. 66
3.10 Second model p oblem wi h wo geological aul s and inclined lay-
e s. The ool ajec o y ( ed dashed line) has di e en dip angles
and passes h ough sands one (yellow), oil-sa u a ed (g ay), and
wa e -sa u a ed (g een) laye s. Dimensions a e in me e s. . . . . . 66
x i
LIST OF FIGURES
3.11 Appa en esis i i ies based on he a enua ion a io, ρA, and phase
di e ence, ρP, o he second model p oblem, compa ed wi h he
eal (exac ) esis i i y, ρe. We employ a IGA disc e iza ion wi h
64 ˆ64 elemen s, p“4, and 8 ˆ8 mac oelemen s. . . . . . . . . . 67
3.12 Va ying pa ame e s a each logging posi ion when p oducing he
aining da ase o DL in e sion. . . . . . . . . . . . . . . . . . . 67
3.13 (a) A enua ion a io s. phase di e ence, and (b) appa en e-
sis i i y based on a enua ion s. appa en esis i i y based on
phase, ob ained o he 100,000 Ea h models. We use IGA dis-
c e iza ion wi h 64 ˆ64 elemen s, p“4, and 8 ˆ8 mac oelemen s
o gene a ing he da abase. . . . . . . . . . . . . . . . . . . . . . 69
4.1 Ske ch o he a qui ec u e o uNN ................... 77
4.2 Loss e olu ion o he aining p ocess o ou wo model p oblems. 78
4.3 Exac s app oxima e Ri z me hod solu ions o model p oblem 1
using ou elemen s o e alua ing FRp qand a Neu al Ne wo k
(NN)wi h31weigh s. ........................ 79
4.4 Exac s app oxima e Ri z me hod solu ions o model p oblem 2
using en elemen s o e alua ion o FRp qand a NN wi h 31 weigh s. 80
4.5 Exac (uexac “0) and app oxima ed solu ion o a p oblem gi en
by Eq. (4.27) and sol ed wi h he Leas Squa e (LS) me hod. . . 80
4.6 Neu al Ne wo k app oxima ion uNN and i s piecewise-linea ele-
men app oxima ion u˚
NN,4....................... 82
4.7 Black poin s (do s) co espond o he o iginal (a) aining/(b) al-
ida ion pa i ions and blue poin s (ci cles) a e he poin s added by
he e inemen pe o med in he i s and hi d elemen s. . . . . . 83
4.8 Loss e olu ion o he aining p ocess o ou wo model p oblems
when we use a piecewise-linea app oxima ion o he NN. . . . . . 89
4.9 Ri z me hod solu ion when we use a piecewise-linea app oxima-
ion o he NN o sol e he p oblem. . . . . . . . . . . . . . . . . 89
4.10 Loss e olu ion o he aining p ocess o ou wo model p oblems
when using adap i e in eg a ion. . . . . . . . . . . . . . . . . . . . 90
4.11 Ri z me hod solu ion when using adap i e in eg a ion. . . . . . . 91
4.12 The solu ion and aining in o ma ion o Expe imen 1 wi hou
egula iza ion.............................. 92
4.13 The solu ion and aining in o ma ion o Expe imen 1 wi h eg-
ula iza ion. .............................. 93
4.14 The solu ion and aining in o ma ion o Expe imen 2 wi hou
egula iza ion.............................. 94
x ii
LIST OF FIGURES
4.15 The solu ion and aining in o ma ion o Expe imen 2 wi h eg-
ula iza ion. .............................. 95
x iii
Lis o Tables
2.1 Ca ego ies o geophysical a iables: ypes Ao B. We apply a
di e en escaling o each o hem. . . . . . . . . . . . . . . . . . 15
2.2 R2 ac o s o c oss-plo s o ype 1 and 2 and Examples B.1 and
B.2, wi h and wi hou egula iza ion, o aining and es da ase s.
Numbe s below 0.96 a e ma ked in bold ace. . . . . . . . . . . . . 30
2.3 R2 ac o s o c oss-plo s o ype 3 and Examples B.1 and B.2,
wi h and wi hou egula iza ion, o he es da ase . . . . . . . . 31
3.1 Compu a ional cos o he 2.5D bo ehole esis i i y measu emen s
pe logging posi ion pe Fou ie mode. We epo he solu ion ime
and FLOPs when using Cp´1IGA, IGA wi h 8ˆ8 mac oelemen s,
and C0FEM wi h he same numbe o elemen s and polynomial
deg ee. The compu a ional imes co espond o he use o pa allel
sol e PARDISO using wo h eads. . . . . . . . . . . . . . . . . . 64
3.2 Va ying pa ame e s employed o gene a e he aining da ase o
DLin e sion. ............................. 68
4.1 Loss alues o he exac solu ion FRpuexac q, op imum piecewise-
linea solu ion FRp˜u˚
NN,¨q( o a ou and a en equidis an elemen
pa i ion), and piecewise-linea solu ion FRpu˚
NN,¨qusing a DNN. . 88
xix

1 In oduc ion
1.1 Mo i a ion and Li e a u e Re iew
The Ea h’s subsu ace is o med by di e en ma e ials, mainly po ous ocks
con aining mine als and illed wi h sal y wa e and/o hyd oca bons. The o -
ma ions ha hese ma e ials c ea e a e o en i egula , appea ing ab up o ms
wi h peaks o b eaks. Fu he mo e, each o he se e al laye s ha compose he
Ea h is composed o a ious ma e ials wi h di e en ma e ial p ope ies. Figu e
1.1 shows an example o a lamina subsu ace o ma ion.
Figu e 1.1: Ea h subsu ace o ma ion wi h di e en laye s. Pho o aken in
Sopela (Biscay, Spain).
Se e al ields demand a map o he subsu ace in o de o ca y ou hei
ac i i ies, needed o : (a) minimize ea hquake-induced damage, (b) enhance he
p oduc ion o geo he mal ene gy, (c) s o e di e en ma e ials such as hyd ogen
in subsu ace ese oi s, and (d) maximize hyd oca bon eco e y.
In his las applica ion, companies o en eco d elec omagne ic (EM) measu e-
men s using a Logging While D illing (LWD) ins umen . These ools inco po-
a e di e en ansmi e s ha gene a e an EM ield. In he same way, se e al
ecei e s a e placed along he ool in o de o ecei e he emi ed wa e a e e-
bounding in he su oundings o he bo ehole. Depending on he ma e ials and/o
1
1 In oduc ion
he o ma ion o he su oundings, he ecei ed wa es exhibi di e en p ope ies.
Figu e 1.2 shows an example o a con en ional LWD ins umen .
Tx1,1 Tx1,2
Tx2,1 Tx2,2
Rx1Rx2
0.2032 m
0.8128 m,2 MHz
2.4384 m,0.25 MHz
Con en ional LWD
Tx
Rx1
Rx2
12 m,24 kHz
25 m,2 kHz
Deep azimu hal
Figu e 1.2: Example o a con en ional LWD ins umen . This ool is equipped
wi h a pai o ecei e s ( ed) and wo pai s o ansmi e s (black).
These ools ha e gained impo ance in he oil and gas indus y in he las
decades due o hei capabili y o eco d logging da a du ing d illing. The
eco ded da a is p ocessed o p oduce a map o he Ea h’s subsu ace nea by
he well. Based on he econs uc ed Ea h model, he ope a o adjus s he
well- ajec o y in eal- ime o u he explo e exploi a ion a ge s, including oil
and gas ese oi s, and o maximize he pos e io p oduc i i y o he a ailable
ese es. This eal- ime na iga ion echnique is called geos ee ing. As a con-
sequence o he emendous p oduc i i y inc ease achie ed wi h his echnique,
nowadays geos ee ing plays an essen ial ole in he oil and gas indus y [40].
The main di icul y one aces when dealing wi h geos ee ing p oblems is o
ob ain a map o he Ea h’s subsu ace. We mus sol e he ollowing in e se
p oblem: gi en he measu emen s M eco ded by he ool and he well ajec-
o y T, we wan o ob ain he subsu ace p ope ies ρ. In con as , he o wa d
p oblem is he one ha gi en he subsu ace p ope ies ρand he well ajec o y
T, i p oduces he measu emen s M eco ded by he ool. Figu e 1.3 p esen s a
schema ic desc ip ion o he o wa d and in e se p oblems.
Un o una ely, adi ional in e sion me hods ha e se e e limi a ions, which
o ce geophysicis s o con inuously look o new solu ions o his p oblem (see,
e.g., [28, 41, 49, 64, 99, 128, 131, 156]). In pa icula , in e se p oblems a e no
well-de ined, ha is, he e may exis mul iple ou pu s o a gi en inpu [141,
145]. G adien -based me hods equi e simula ing he o wa d p oblem dozens o
imes o each se o measu emen s. Mo eo e , hese me hods also es ima e he
de i a i es o he measu emen s wi h espec o he in e sion a iables, which is
o en challenging and ime consuming [141]. To alle ia e he high compu a ional
cos s associa ed wi h hese in e sion me hods, simpli ied 1.5-dimensional (1.5D)
me hods a e common (see, e.g., [64, 99, 133]). Fo he in e sion o bo ehole
esis i i y measu emen s, an al e na i e is o apply s a is ics-based me hods [53,
2
1 In oduc ion
Subsu ace
p ope ies ρ
+
Well ajec o y T
Measu emen s M
Measu emen s M
+
Well ajec o y T
Subsu ace
p ope ies ρ
F
I
Fo wa d:
In e se:
Figu e 1.3: Schema ic desc ip ion o o wa d and in e se p oblems.
85, 147]. The s a is ical me hods also pe o m o wa d simula ions hund eds o
imes o each se o measu emen s. Bo h g adien and s a is ics-based me hods
only e alua e he in e se ope a o . Thus, he en i e in e sion p ocess is epea ed
a each new logging posi ion.
Deep Lea ning (DL) echniques seem ap opia e o o e come he limi a ions
o adi ional me hods while sol ing in e sion p oblems. The la ge amoun o
esea ch a icles and indus ial applica ions o DL algo i hms in di e en a eas –
compu e ision [82], speech ecogni ion [4, 5, 154], biome ics [16], sel -d i ing
ca s [54, 114], and heal hca e [43, 108] o men ion a ew – a e exponen s o hei
high pe o mance and capabili y o sol e all kind o p oblems.
In addi ion, in ecen yea s he e ha e been signi ican ad ances in he ield o
DL, wi h he appea ance o Residual Neu al Ne wo ks (RNNs) [59], which p e en
g adien degene a ion du ing he aining s age, and Encode -Decode (sequence-
o-sequence) Deep Neu al Ne wo ks (DNNs), which ha e imp o ed he DL wo k
capabili y in compu e ision applica ions [10]. Due o he high demand om
indus y o use DNNs, dedica ed lib a ies and packages such as Tenso low [82],
Ke as [29], and Py o ch [103] ha e been de eloped. These lib a ies acili a e he
use o DNNs ac oss di e en indus ial applica ions [39, 66, 83, 117, 132, 144,
157]. All hese ad ances make DNNs one o he mos powe ul and as -g owing
A i icial In elligence (AI) ools p esen ly. The i s main con ibu ion o his
disse a ion is o design a as in e sion me hod using DL echniques
o sol e bo ehole measu emen p oblems ha allows he applica ion o
geos ee ing echniques.
Howe e , DNNs also ace impo an challenges when applied o he in e sion
o bo ehole esis i i y p oblems. In pa icula , he aining s age can be ime-
3
1 In oduc ion
consuming. Howe e , his is an o line cos incu ed du ing he aining s age.
Then, a e he ne wo k is p ope ly ained, i can o ecas a solu ion in a ac-
ion o a second [131]. This ea u e allows eal- ime in e sion, which acili a es
geos ee ing ope a ions. Ano he limi a ion o DNN is ha hey equi e a la ge
da ase (also known as g ound u h). In ou case, i consis s o he solu ion o
he o wa d p oblem o di e en Ea h models [60, 130, 131].
To gene a e he da abase o DL in e sion, we mus sol e he o wa d p ob-
lem. Ou o wa d p oblem – simula ion o bo ehole measu emen s – is go e ned
by Pa ial Di e en ial Equa ions (PDEs). In ou case, we conside esis i i y
measu emen s go e ned by a se o ou ime-dependen i s -o de PDE named
Maxwell’s equa ions [45]. He e, knowing some elec ical p ope ies (i.e., elec i-
cal conduc i i ies o he subsu ace ma e ials), we can ob ain he co esponding
elec ic and magne ic ields (i.e., eco ded measu emen s).
We sol e he o wa d p oblem using nume ical simula ion me hods such as he
Fini e Elemen Me hod (FEM) [6, 11, 58, 69, 98, 115, 133] o he Fini e Di e ence
Me hod (FDM) [35, 36, 77, 140]. Mo eo e , we need o op imally sample he
pa ame e space desc ibing ele an Ea h models. This p ocess may be ime-
consuming, especially o wo and h ee-dimensional p oblems. In hose cases,
i is common o educe he Ea h model dimensionali y o wo o one spa ial
dimensions using a Fou ie o a Hankel ans o m. These ans o ma ions lead
o he so-called 2.5D [3, 48, 92, 101, 135] and 1.5D [12, 99, 129] o mula ions,
espec i ely. In pa icula , 1.5D simula ions a e inaccu a e when dealing wi h
geological aul s.
Gale kin me hods a e e ec i e o simula ing well-logging p oblems (see, e.g., [24,
27, 84, 95, 97, 120, 146]). Isogeome ic Analysis (IGA), in oduced by [61], is a
widely used Gale kin me hod o sol ing PDEs. IGA has been success ully em-
ployed in a ious EM [20, 21, 93, 137, 138] and geo echnical [56, 134] applica ions.
IGA uses spline basis unc ions in oduced in Compu e -Aided Design (CAD) as
basis unc ions o FEM. These basis unc ions exhibi high con inui y (up o
Cp´1, being p he polynomial o de o spline bases) ac oss he elemen in e aces.
When compa ing IGA and FEM, he o me p o ides smoo he solu ions o
wa e p opaga ion p oblems wi h a lowe numbe o unknowns [31, 61]. Howe e ,
in con as o he minimal in e connec ion o elemen s in FEM, high-con inui y
IGA disc e iza ions s eng hen he in e connec ion be ween elemen s, leading o
an inc ease o he cos o ma ix LU ac o iza ion pe deg ee o eedom when
using spa se di ec sol e s [30]. In o de o a oid his deg ada ion and also bene i
om he ecu si e pa i ioning capabili y o mul i on al di ec sol e s, [47] de el-
oped a new me hod called e ined Isogeome ic Analysis ( IGA). This disc e iza-
ion echnique conse es desi able p ope ies o high-con inui y IGA disc e iza-
ions, while i pa i ions he compu a ional domain in o blocks o mac oelemen s
4
2 Sol ing In e se P oblems using Deep Lea ning
ecei e is signi ican ly la ge han ha o he p e iously conside ed LWD ins u-
men . I also employs il ed ecei e s ha a e sensi i e o he p esence o bed
bounda ies. We eco d se e al measu emen s wi h his logging ins umen : (a)
he a enua ion and phase di e ences, deno ed as deep coaxial, compu ed using
Equa ion (2.3) wi h H2
zz “1, and (b) he a enua ion and phase di e ences o a
di ec ional measu emen exp essed as:
Geosignal “ln Hzz ´Hzx
Hzz `Hzx “ln |Hzz ´Hzx |
|Hzz `Hzx |
loooooooomoooooooon
ˆ20 logpeq“:a enua ion pdBq
`ipphpHzz ´Hzxq´phpHzz `Hzxqq
looooooooooooooooooooomooooooooooooooooooooon
ˆ180
π“:phase di e ence (deg ee)
.
(2.4)
These measu emen s exhibi a discon inui y as a unc ion o he dip angle a 90
deg ees. Indeed, such discon inui y is essen ial in he measu emen s i one wan s
o disce n be ween op and bo om o he logging ins umen (see Figu e 2.4).
𝑇 𝑥
𝑅𝑥𝑇 𝑥
𝑅𝑥
𝑇 𝑥
𝑅𝑥𝑇 𝑥
𝑅𝑥
100 𝛺⋅𝑚
1𝛺⋅𝑚
100 𝛺⋅𝑚
𝐷
𝐶
𝐵
𝐴
Figu e 2.4: Illus a ion wi h ou logging ajec o ies. By symme y, measu e-
men s eco ded wi h ajec o ies A and D a e iden ical. The same
occu s wi h ajec o ies B and C. I hese measu emen s a e con in-
uous wi h espec o he dip angle, hen hey coincide a 90 deg ees,
which disables he possibili y o iden i ying i a nea by bed bounda y
is loca ed abo e o below he logging ins umen .
Fo ou bo ehole esis i i y applica ions, we conside a ze o- hickness bo ehole
embedded in a h ee-laye medium (see Figu e 2.5). A common p ac ice in he
ield is o cha ac e ize his medium wi h se en pa ame e s, as desc ibed in Figu e
2.5. In his wo k, o simpli y he p oblem, we conside only i e o hem by
es ic ing he sea ch o iso opic o ma ions (ρ “ρh) wi h ze o dip angle (β“
0), as illus a ed in Figu e 2.6. Thus, np“5.
11

2 Sol ing In e se P oblems using Deep Lea ning
Bo ehole
𝑑𝑢
𝑑𝑙
𝛽
𝜌𝑙
𝜌ℎ
𝜌𝑣
𝜌𝑢
Figu e 2.5: Well ajec o y in a 1D medium. The black ci cle indica es he las
ajec o y posi ion. ρhand ρ a e he ho izon al and e ical esis-
i i ies o he hos laye co esponding o he inal logging posi ion,
espec i ely. ρuand ρla e he esis i i y alues o he uppe and lowe
laye s o he hos laye , espec i ely. duand dlshow he dis ance om
he inal logging posi ion o he uppe and lowe bed bounda ies, e-
spec i ely.
𝑑𝑢
𝑑𝑙
𝜌𝑢
𝜌𝑙
𝜌ℎ
(a) Example B.1: ajec o y wi h
1 logging posi ions
𝐭
𝑑𝑢
𝑑𝑙
𝜌𝑢
𝜌𝑙
𝜌ℎ
(b) Example B.2: ajec o y wi h
65 logging posi ions
Figu e 2.6: Model p oblems co esponding o examples B.1 and B.2, espec i ely.
In his example, we conside wo cases (see Figu e 2.6) acco ding wi h he
di e en numbe s o logging posi ions we conside pe da a sample.
2.1.5.1 Example B.1: One Logging Posi ion
In his case, each ajec o y consis s o a single logging posi ion. The e o e, o
each sample, we eco d six eal numbe s ( h ee a enua ions and h ee phases),
i.e., nm“6. A each logging posi ion, he ajec o y is desc ibed by one numbe :
he ajec o y dip angle. Thus, n “1.
12
2 Sol ing In e se P oblems using Deep Lea ning
2.1.5.2 Example B.2: Six y-Fi e Logging Posi ions
In his case, he logging ajec o y o each sample is o med by 65 logging posi ions
wi h a logging s ep size o 0.3048 m(see [130, 131] o u he de ails). Thus,
o each Ea h model p, we pa ame ize mwi h 6 ˆ65 “390 eal numbe s
(nm“390). Fo his example, we assume ha he a ia ion o he dip angle a
a gi en logging posi ion wi h espec o he p e ious one is cons an . We deno e
ha cons an dip angle a ia ion as α . Then, a he i- h logging posi ion, he
ajec o y dip angle is αi“αini ` pi´1qα , whe e αini is he ini ial dip angle.
Hence, we ha e n “2.
2.2 Da a Space and G ound T u h
In his wo k, we employ a DNN o app oxima e he disc e e in e se ope a o
I. Gi en a supe ised da abase o n-pai s pmi,Ip i,miqq,i“1, ..., n, he DNN
builds an app oxima ion o he unknown unc ion I. This sec ion desc ibes he
cons uc ion o he supe ised da abase.
We i s selec he numbe o samples, n, and wo subspaces o Rnpand
Rn , espec i ely. Then, we selec he nsamples in hose subspaces, namely,
pp 1,p1q, ..., p n,pnqq. To each o hese samples, we apply he ope a o F. Tha
is, we compu e pFp 1,p1q, ..., Fp n,pnqq. Finally, he n-pai s pmi,Ip i,miqq :“
pFp i,piq,piq,i“1, ..., n o m ou supe ised da abase.
We deno e by TPRn ˆn o he se o all ajec o y samples p 1, ..., nq. In
o he wo ds, Tis a ma ix wi h ibeing i s i- h column. Simila ly, we de ine
M“ pm1, ..., mnq P Rnmˆnand P“ pp1, ..., pnq P Rnpˆn.
Example A: Simple model p oblem wi h known analy ical solu ion. We selec
n“103uni o mly spaced samples wi hin he subspace ´33,33s Ă R.
Example B: In e sion o bo ehole esis i i y measu emen s. We selec n“
106. Then, o he i e pa ame e s desc ibed in Sec ion 2.1.5, we selec andom
samples o he ollowing escaled a iables o e he co esponding in e als o m-
ing a subspace o R5:
logpρlq,logpρuq,logpρhq P 0,3s
logpdlq,logpduq P ´2,1s.(2.5)
We conside a bi a y high-angle ajec o ies. Fo each model p oblem, we
andomly selec he ajec o y pa ame e s wi hin he ollowing in e als:
αini P 830,970s
α P ´0.0450,0.0450s (only o Example B.2).(2.6)
13
2 Sol ing In e se P oblems using Deep Lea ning
2.3 Da a P ep ocessing
No a ion. Fo each ou pu pa ame e o Fand I, we deno e by x“ px1, ..., xnq
he n-samples associa ed wi h ha pa ame e . These xia e eal scala alues
o i“1, ..., n. Fo example, in he bo ehole esis i i y example, each a iable x
con ains nsamples o each pa icula geophysical quan i y such as esis i i ies,
dis ances, o gi en measu emen s (a enua ions, phases, e c.). Each dimension
co esponds o a pa icula alue (sample) o ha a iable, o example, he
geosignal a enua ion eco ded a a speci ic logging posi ion. F om he algeb aic
poin o iew, he a iable xdeno es a ow o ei he ma ix Mo P.
Da a p ep ocessing algo i hm. This algo i hm consis s o h ee s eps.
1. Loga i hmic change o coo dina es. We in oduce he ollowing change
o a iables:
Rlnpxq:“ pln x1, ..., ln xnq.(2.7)
Fo some geophysical a iables (e.g., esis i i y), his change o a iables
ensu es ha equal-size ela i e e o s co espond o simila -size absolu e
e o s. Thus, his change o a iables allows us o pe o m local (wi hin a
a iable) compa isons.
2. Remo e ou lie samples. In p ac ice, o en ou lie measu emen s a e
p esen in he sample da abase. These ou lie s appea due o measu emen
e o o he physics o he p oblem. Fo example, in bo ehole esis i i y
measu emen s, some appa en esis i i y measu emen s app oach in ini y,
p oducing “ho ns” in he logs. When ou lie measu emen s exis s in any
pa icula a iable o he i- h sample xi, hen he en i e sample should be
emo ed. O he wise, ou lie measu emen s a ec he en i e minimiza ion
p oblem, leading o poo nume ical esul s. The emo al p ocess may be
au oma ed using s a is ical indica o s, o decided by he use based on a p i-
o i physical knowledge abou he p oblem. We ollow his second app oach
in his wo k.
3. Linea change o coo dina es. We now in oduce a linea escaling
mapping in o he in e al 0.5,1.5s. We selec his in e al since i has uni
leng h and he mean o a no mal (o a uni o m) dis ibu ion a iable xis
equal o one. Le xmin :“minixi,xmax :“maxixi. We de ine
Rlinpxq:“ˆx1´xmin
xmax ´xmin `0.5, ..., xn´xmin
xmax ´xmin `0.5˙,(2.8)
whe e he limi s xmin and xmax a e ixed o all possible app oxima ions xapp.
This change o a iables allows us o pe o m a global compa ison be ween
14
2 Sol ing In e se P oblems using Deep Lea ning
e o s co esponding o di e en a iables since hey all ake alues o e he
same in e al.
Rema k: xmin and xmax could also be selec ed based on he physically alid
in e al o each pa icula a iable a he han on he aining samples.
Va iables classi ica ion. We ca ego ize each inpu and ou pu geophysical a i-
able xin o wo ypes: ei he linea (A) o log-linea (B). When necessa y, we shall
indica e ha a pa icula a iable belongs o a speci ic ca ego y by adding he
co esponding symbol as subindex o he a iable, e.g., xA. Table 2.1 desc ibes
he domain o hose a iables as well as he escaling employed o each o hem.
Va iables o ype Aonly equi e a global escaling while hose o ype B equi e
bo h a local and a global change o a iables.
Geophysical Va iables Ca ego y Domain Rescaling
Angles, a enua ions, ARnRlinpxq
phases, and geosignals
Appa en esis i i ies, Bpa, 8qnRlinpRlnpxqq
esis i i ies, and dis ances aą0
Table 2.1: Ca ego ies o geophysical a iables: ypes Ao B. We apply a di -
e en escaling o each o hem.
Fo simplici y, we deno e by R he esul o he abo e escalings, i.e., RpxAq:“
RlinpxAq, and RpxBq:“RlinpRlnpxBqq. In gene al, gi en a a iable x(o ca ego y
Ao B), we ep esen xR:“Rpxq. Gi en a ma ix XPRnxˆn, we abuse no a ion
and deno e by XR:“RpXq P Rnxˆn o he ma ix ha esul s om applying
ope a o R ow-wise.
Rema k: Subs i u ing in Equa ion 2.7 he na u al loga i hm by he base en
loga i hm does no a ec he de ini ion o R. Resul s a e iden ical.
2.4 No ms and E o s
We i s in oduce bo h he ec o and he ma ix no ms ha we use du ing he
aining p ocess.
No ms. We in oduce a no m ||¨||Xassocia ed wi h he a iable x. In gene al,
we employ he l1o l2 ec o no ms and, o ma ices, he l1and F obenius no ms.
15
2 Sol ing In e se P oblems using Deep Lea ning
Absolu e and ela i e e o s. Le xapp “ pxapp
1, ..., xapp
nqbe an app oxima ion
o x. We de ine he absolu e e o Aebe ween xapp and xin he ||¨||Xno m as
AX
epxapp,xq:“ ||xapp ´x||X.(2.9)
This e o measu e has limi ed use since i is challenging o selec an absolu e
e o h eshold ha dis inguishes be ween a good and a bad quali y app oxima-
ion. To o e come his issue, p ac i ione s o en employ ela i e e o s. We de ine
he ela i e e o Rein pe cen be ween xapp and xin he ||¨||Xno m as:
RX
epxapp,xq:“100||xapp ´x||X
||x||X
.(2.10)
E o con ol. Fo a a iable xand i s app oxima ion xapp, we wan o con ol
he ela i e e o o he escaled a iable, ha is:
RX
epxapp
R,xRq.(2.11)
The alue B“ ||xR||Xis expec ed o be simila o all a iables x. Thus:
ÿ
xR
AX
epxapp
R,xRq “ ÿ
xR||xapp
R´xR||X«Bÿ
xR
||xapp
R´xR||X
||xR||X“B
100 ÿ
xR
RX
epxapp
R,xRq.
(2.12)
The e o e, he minimum o he i s and las e ms o he abo e equa ion coincide.
2.5 DNN A chi ec u es
To app oxima e he o wa d and in e se p oblems, we use DNN a chi ec u es
based on esidual- ype blocks [59, 110] wi h con olu ional ope a o s [60, 67, 74,
151]. This wo k does no discuss op imal da a sampling echniques no he
decision-making o he op imal selec ion o DNN a chi ec u es [89, 109]. In he
ollowing, we i s de ine he main ope a o s o ou DNN a chi ec u es, ollowed
by a desc ip ion o he o wa d and in e se DNN a chi ec u es.
We deno e by N o ou nonlinea ac i a ion unc ion. In ou case, we em-
ploy he ec i ied linea uni (ReLU), de ined componen -wise o each en y x
as maxp0, xq[60]. We now in oduce a 1D con olu ional ope a o Cc,k
ψ, whe e
cis he il e size (ou pu dimensionali y), k he ke nel size, and ψ he weigh s
[59, 60]. Then, we de ine he ollowing block:
Bc,k
ψ:“´N˝Cc,k
ψ1˝N˝Cc,k
ψ2`Cc,k
ψ3¯,(2.13)
whe e now ψ“ pψ1, ψ2, ψ3qa e all weigh s associa ed o block Bc,k
ψ.
16

2 Sol ing In e se P oblems using Deep Lea ning
2.5.1 Fo wa d P oblem DNN A chi ec u e
Each inpu sample has dimension np`n , and con ains he a iables ep esen ing
he ma e ial p ope ies and he ajec o y. We de ine ou DNN a chi ec u e as:
FR,φ :“N˝Cc6,1
φ6˝L˝U˝Bc5,3
φ5˝U˝Bc4,3
φ4˝¨¨¨˝U˝Bc1,3
φ1,(2.14)
whe e
•Uis a 1D upsampling ope a o wi h upsampling ac o equal o wo (using
he Tenso Flow ou ine upsampling1D [2, 102, 142]).
•Lis a bilinea esampling ope a o wi h esampling ac o equal o he
numbe o logging posi ions [102, 142], i.e., 1 o Example B.1 and 65 o
Example B.2.
•ci:“40i, o i“1,¨¨¨ ,5 and c6“n1
m“6, whe e n1
mis he numbe o
e alua ed measu emen s pe logging posi ion.
•φ“ φi:i“1,¨¨¨ ,6uis a se o all weigh s associa ed o he o wa d
DNN a chi ec u e.
Lexpands (in case o 65 logging posi ions) o sh inks (in case o one logging
posi ion) i s inpu dimension. The ou pu o he men ioned bilinea esampling
is a ma ix in which i s i s dimension is equal o he numbe o logging posi-
ions [102, 142]. All he esampling ope a o s conside ed in he Equa ion 2.14
aise/sh ink he dimension o hei inpu g adually o a oid missing in o ma ion
due o a sudden dimension change. The ou pu is a ma ix o dimension pnl, n1
mq,
whe e nlis he numbe o logging posi ions.
2.5.2 In e se P oblem DNN A chi ec u e
The inpu o he DNN is a ma ix o dimension pnl, n1
m`2q, whe e nlis he numbe
o logging posi ions. The i s wo columns o he a o emen ioned ma ix a e he
sine and cosine o he ajec o y dip angle a each logging posi ion. Analogously
o he o wa d p oblem, we conside he ollowing a chi ec u e:
IR,θ :“N˝Dnp
θ7˝S˝Bc6,3
θ6˝Bc5,3
θ5˝Bc4,3
θ4˝¨¨¨˝Bc1,3
θ1,(2.15)
•Dn
θis a ully-connec ed laye wi h nbeing i s numbe o uni s and θi s
weigh s [2, 60].
•Sis a la ening laye ha ecei es a 2D ma ix and ou pu s a 1D ec o
[2].
17
2 Sol ing In e se P oblems using Deep Lea ning
•ci:“40i, o i“1,¨¨¨ ,5.
•θ“ θi:i“1,¨¨¨ ,7uis a se o all he weigh s associa ed o each block
and laye
Dnp
θ7pe o ms he ul ima e ea u e ex ac ion and down-sampling. The ou pu
o his DNN is a ec o consis ing o ma e ial p ope ies.
2.6 Loss Func ion
In his sec ion, we conside a se o weigh s θPΘ and a unc ion IR,θ ha depends
upon he selec ed DNN a chi ec u e. Then, we in oduce a loss unc ion LpIR,θq.
We de ine he minimize o he loss unc ion o e all possible weigh se s θas:
IR,θ˚:“a g min
θPΘLpIR,θq.(2.16)
Func ion Iθ˚:“R´1˝IR,θ˚˝Ris he inal DNN app oxima ion o I. In he
ollowing, we analyze he ad an ages and limi a ions associa ed wi h he use o
di e en loss unc ions.
2.6.1 Da a Mis i
A simple loss unc ion based on he da a mis i is gi en by:
LpIR,θq:“ ||IR,θpTR,MRq´PR||P.(2.17)
In he abo e equa ion, symbol ||¨||Pindica es l1o F obenius no ms in oduced
in Sec ion 2.4.
Example A: Model p oblem wi h known analy ical solu ion. In his exam-
ple, np“1. Thus, ma ix no ms educe o ec o no ms. Figu e 2.7 illus a es
he esul s we ob ain using he l1and l2no ms, espec i ely. These disappoin -
ing esul s a e expec ed. In he case o l2-no m, o a su icien ly lexible DNN
a chi ec u e he exac solu ion is Iθ˚“0. We wan o minimize
ÿ
iPIpIR,θpmiq´piq2,(2.18)
whe e I“ 1, ..., nudeno es he aining da ase . Fo e e y sample o he o m
pmi,?miq, he e exis s ano he one pmi,´?miq, which is sa is ied in ou da ase
18
2 Sol ing In e se P oblems using Deep Lea ning
by cons uc ion (see Sec ion 2.2). Then, o a speci ic sample mi, he solu ion
ha minimizes he loss mus sa is y
pIR,θpmiq´?miq2`pIR,θpmiq´p´?miqq2.(2.19)
Taking he de i a i e o Eq. (2.19) wi h espec o IR,θpmiqand equaling i o
ze o, we ob ain
4¨IR,θpmiq “ 0.(2.20)
Thus, o any sample mi, he unc ion is minimized when he app oxima ed alue
is IR,θpmiq “ 0. We hus conclude ha o l2-no m, he app oxima ed solu ion
mus be Iθ˚“0.
In he case o l1-no m, any solu ion in be ween he wo squa e oo b anches
is alid. We wan o minimize
ÿ
iPI|IR,θpmiq´pi|,(2.21)
whe e I“ 1, ..., nudeno es he aining da ase . Then, o a speci ic sample mi
he solu ion ha minimizes he loss mus sa is y
|IR,θpmiq´?mi|`|IR,θpmiq´p´?miq|.(2.22)
By analyzing each possible case, we can exp ess Eq. (2.22) as ollows
$
&
%
´2¨IR,θpmiq,i IR,θpmiqă´?mi,
2?mi,i ´?miďIR,θpmiq ď ?mi,
2¨IR,θpmiq,i IR,θpmiq ą ?mi.
(2.23)
We see ha he loss unc ion o he exac solu ion a ains i s minimum a e e y
poin Iθ˚pmiq P ´?mi,?mis. Mo eo e , ou nume ical solu ions in Figu e 2.7
con i m hese simple ma hema ical obse a ions. Thus, he da a mis i loss unc-
ion is unsui able o in e sion pu poses.
2.6.2 Mis i o he Measu emen s
To o e come he a o emen ioned limi a ion, we conside he ollowing loss unc-
ion ha measu es he mis i o he measu emen s (see [68]):
LpIR,θq:“ }pFR˝IR,θqpTR,MRq´MR}M,(2.24)
whe e FR:“R˝F˝R´1, and || ¨ ||Mindica es a ma ix no m o he ype
in oduced in Sec ion 2.4.
19
2 Sol ing In e se P oblems using Deep Lea ning
0 500 1,000
−33
0
33
P edic ed
Real
𝑚
𝜃∗(𝑚)
(a) }¨}1-no m
0 500 1,000
−33
0
33
P edic ed
Real
𝑚
𝜃∗(𝑚)
(b) }¨}2-no m
Figu e 2.7: Analy ical solu ion s DNN p edic ed solu ion e alua ed o e he es
da ase using he loss unc ion based on he da a mis i .
0 500 1,000
−33
0
33
P edic ed
Real
𝑚
𝜃∗(𝑚)
(a) }¨}1-no m
0 500 1,000
−33
0
33
P edic ed
Real
𝑚
𝜃∗(𝑚)
(b) }¨}2-no m
Figu e 2.8: Analy ical solu ion s DNN p edic ed solu ion e alua ed o e he es
da ase using he loss unc ion based on he measu emen s mis i .
20
2 Sol ing In e se P oblems using Deep Lea ning
0 400 800
10−2
Epoch n .
Loss
aining
alida ion
(a) }FR,φpTR,PRq ´ MR}M
0 400 800
10−2
Epoch n .
Loss
aining
alida ion
(b) }pFR,φ ˝IR,θqpTR,MRq ´ MR}M
0 400 800
10−1
Epoch n .
Loss
aining
alida ion
(c) ||IR,θpTR,MRq ´ PR||P
0 400 800
10−1
Epoch n .
Loss
aining
alida ion
(d) To al Loss
Figu e 2.12: Example B.1. E olu ion o he di e en e ms o he Encode -
Decode loss unc ion gi en by Equa ion 2.25 wi h he egula iza ion
e m p esc ibed by Equa ion 2.17.
27

2 Sol ing In e se P oblems using Deep Lea ning
we can sa ely omi hese c oss-plo s. O he wise, c oss-plo s display in e es ing
in o ma ion beyond wha R2p o ides.
The p ope in e p e a ion o he c oss-plo s (o al e na i ely, R2 ac o s) is o
u mos impo ance. C oss-plo s o ype 1 (Equa ion 2.281) indica e how well he
o wa d unc ion is app oxima ed o e he gi en da ase . The c oss-plo s o ype
2 (Equa ion 2.282) display how well he composi ion o he p edic ed o wa d
and in e se mappings app oxima e he iden i y. These wo ypes o c oss-plo s
o en deli e high R2 ac o s, since he co esponding app oxima ions a e di ec ly
buil in o he Encode -Decode loss unc ion gi en by Equa ion 2.25. Table 2.2
con i ms hose heo e ical p edic ions o he mos pa .
An in-dep h inspec ion o Table 2.2 e eals ha o he he geosignal measu e-
men s (bo h a enua ion and phase) co esponding o he Example B.1 wi hou
egula iza ion, he c oss-plo s 2 exhibi signi ican ly be e R2 ac o s han hose
co esponding o he c oss-plo s 1. Figu e 2.13 shows he co esponding c oss-
plo s. The an i-diagonal g ey line shown in c oss-plo s o ype 1 co esponds o
dip angles o he logging ins umen ha a e close o 90 deg ees. A ha an-
gle, he geosignal is discon inuous. Thus, i is no p ope ly app oxima ed ia
DL algo i hms, which app oxima e con inuous unc ions. C oss-plo s o ype 2
seem o ix ha issue by deli e ing highe R2 ac o s and appa en ly nice ig-
u es. Howe e , hey ampli y he p oblem. In eali y, he DL app oxima ion o
he in e se ope a o is in e ing an inco ec o wa d app oxima ion. Nume ical
esul s below illus a e his p oblem.
Ob aining high R2 ac o s associa ed o c oss-plo s o ype 3 (Equa ion 2.283) is
a challenging ask as we discuss in Rema k A o Sec ion 2.6. Equa ion 2.27 shows
a simple example in which c oss-plo s o ype 1 and 2 deli e pe ec R2ma ks
and esul s, while c oss-plo s o ype 3 a e disas ous. This is also he si ua ion
ha occu s in Example B.2. (see Table 2.3). While he o iginal aining da ase
is based on 1D Ea h models, he one ob ained a e he p edic ed DNN in e sion
is a piecewise 1D Ea h model, o which Fφ˚is un ained o . When his occu s,
he aining da abase should be upg aded, ei he by inc easing he space o he
da a samples o by selec ing a di e en pa ame e iza ion (e.g., measu emen s) o
each sample. In ou case, we choose o pa ame ize each sample independen ly
( he la e s a egy) and we mo e o Example B.1.
Table 2.3 shows mixed esul s o he Example B.1. Resul s wi hou egula -
iza ion a e un ema kable wi h he geosignal o ecas s showing poo esul s. The
DNN in e se app oxima ion accu a ely in e s o he ou come p edic ed by he
DNN o wa d app oxima ion. Ne e heless, since he DNN p edic s solu ions
a om he ue o wa d unc ion, he p edic ions a e poo . Again, his poo
o ecas ing occu s because he DNN in e se app oxima ion encoun e s subsu -
ace models o which he o wa d DNN app oxima ion is un ained. As a esul ,
28
2 Sol ing In e se P oblems using Deep Lea ning
0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00
G ound T u h
0.75
0.50
0.25
0.00
0.25
0.50
0.75
1.00
P edic ed
2= 0.9531
A en-Geosignal
−1 0 1
−1
0
1
G ound u h
P edic ed alue
0.75 0.50 0.25 0.00 0.25 0.50 0.75
G ound T u h
0.75
0.50
0.25
0.00
0.25
0.50
0.75
P edic ed
2= 0.9487
Phase-Geosignal
−1−0.5 0 0.5 1
−1
−0.5
0
0.5
1
G ound u h
P edic ed alue
0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00
G ound T u h
0.75
0.50
0.25
0.00
0.25
0.50
0.75
1.00
P edic ed
2= 0.9999
A en-Geosignal_FI
−1 0 1
−1
0
1
G ound u h
P edic ed alue
0.75 0.50 0.25 0.00 0.25 0.50 0.75
G ound T u h
0.75
0.50
0.25
0.00
0.25
0.50
0.75
P edic ed
2= 0.9999
Phase-Geosignal_FI
−1−0.5 0 0.5 1
−1
−0.5
0
0.5
1
G ound u h
P edic ed alue
Figu e 2.13: Geosignal c oss-plo s o he Example B.1 wi hou egula iza ion
o he es da ase . Fi s ow: C oss-plo s o ype 1. Second ow:
C oss-plo s o ype 2. Fi s column: A enua ion. Second column:
Phase.
29
2 Sol ing In e se P oblems using Deep Lea ning
C oss-plo s 1
A en. A en. A en. Phase Phase Phase
R2 ac o s LWD Deep Deep LWD Deep Deep
Coaxial Coaxial Geosignal Coaxial Coaxial Geosignal
Example B.1
T aining 0.9997 0.9992 0.9509 0.9996 0.9994 0.9468
Tes 0.9995 0.9984 0.9531 0.9990 0.9991 0.9487
Wi hou Reg.
Example B.1
T aining 0.9998 0.9998 0.9897 0.9998 0.9998 0.9893
Tes 0.9998 0.9998 0.9893 0.9998 0.9998 0.9890
Wi h Reg.
Example B.2
T aining 0.9959 0.9975 0.9872 0.9954 0.9980 0.9853
Tes 0.9924 0.9960 0.9775 0.9920 0.9974 0.9765
Wi hou Reg.
C oss-plo s 2
A en. A en. A en. Phase Phase Phase
R2 ac o s LWD Deep Deep LWD Deep Deep
Coaxial Coaxial Geosignal Coaxial Coaxial Geosignal
Example B.1
T aining 0.9997 0.9995 0.9998 0.9999 0.9996 0.9999
Tes 0.9997 0.9994 0.9999 0.9999 0.9996 0.9999
Wi hou Reg.
Example B.1
T aining 0.9971 0.9980 0.9779 0.9970 0.9979 0.9798
Tes 0.9970 0.9979 0.9785 0.9970 0.9978 0.9803
Wi h Reg.
Example B.2
T aining 0.9931 0.9958 0.9800 0.9933 0.9967 0.9821
Tes 0.9890 0.9930 0.9701 0.9881 0.9944 0.9720
Wi hou Reg.
Table 2.2: R2 ac o s o c oss-plo s o ype 1 and 2 and Examples B.1 and B.2,
wi h and wi hou egula iza ion, o aining and es da ase s. Num-
be s below 0.96 a e ma ked in bold ace.
30
2 Sol ing In e se P oblems using Deep Lea ning
bo h he o wa d and in e se DDN app oxima ions depa s ongly om he ue
solu ions. In o he wo ds, he in e se can only comply wi h hei composi ion o
be close o he iden i y, which is no obus o deli e accu a e and physically
ele an app oxima ions.
C oss-plo s 3
A en. A en. A en. Phase Phase Phase
R2 ac o s LWD Deep Deep LWD Deep Deep
Coaxial Coaxial Geosignal Coaxial Coaxial Geosignal
Example B.1
Wi hou Reg. 0.9468 0.7406 0.0013 0.9383 0.9116 0.0167
Wi h Reg. 0.9971 0.9979 0.9807 0.9969 0.9979 0.9856
Example B.2
Wi hou Reg. 0.5721 0.8383 0.0253 0.4546 0.8611 0.0284
Wi h Reg. 0.9010 0.9701 0.5901 0.8621 0.9618 0.5877
Table 2.3: R2 ac o s o c oss-plo s o ype 3 and Examples B.1 and B.2, wi h
and wi hou egula iza ion, o he es da ase .
To pa ially alle ia e he abo e p oblem, we en ision h ee possible solu ions.
Fi s , we can inc ease he aining da ase . This op ion is ime-consuming and
o en impossible o achie e in p ac ice. Fo example, he ein, we al eady employ
1,000,000 samples. Second, we can include egula iza ion. Resul s wi h egula -
iza ion a e o high quali y (see Table 2.3). Howe e , he egula iza ion e m may
hide al e na i e physical solu ions o he in e se p oblem. Thus, he egula iza-
ion diminishes he abili y o pe o m unce ain y quan i ica ion. Simila ly, i
may induce on he use excessi e con idence in he esul s. A hi d op ion is o
conside he wo-s ep based loss unc ion gi en by Equa ion 2.26. Following his
app oach, we i s adjus he o wa d DNN app oxima ion be o e aining he
DNN in e se app oxima ion. Fixing he o wa d DNN o en p o ides a p ope
o ecas e en in a eas wi h a lowe a e o aining samples be o e p oducing a
DNN app oxima ion ha app oxima es he in e se o he DNN o wa d app oxi-
ma ion. Following his wo-s ep app oach wi hou egula iza ion, we ob ain high
R2 ac o s o c oss-plo s o ype 3: abo e 0.95 o he geosignal a enua ion and
phase, and abo e 0.99 o he emaining measu emen s.
Finally, he R2 ac o s o he c oss-plo s o ype 4 do no e lec on he accu acy
o he DNN algo i hm, bu a he on he na u e o he in e se p oblem a hand.
Low R2 ac o s indica e he e exis mul iple solu ions. A egula iza ion e m
(e.g., Equa ion 2.17) inc eases he R2indica o . Figu e 2.14 clea ly illus a es
his ac . Howe e , i is misleading o conclude ha esul s wi hou egula iza ion
31
2 Sol ing In e se P oblems using Deep Lea ning
a e always wo se. They may simply exhibi a di e en (bu s ill alid) solu ion
o he in e se p oblem.
2.0 1.5 1.0 0.5 0.0 0.5 1.0
G ound T u h
2.0
1.5
1.0
0.5
0.0
0.5
1.0
P edic ed
2= 0.0227
d_u
−2−1 0 1
−2
−1
0
1
G ound u h
P edic ed alue
2.0 1.5 1.0 0.5 0.0 0.5 1.0
G ound T u h
2.0
1.5
1.0
0.5
0.0
0.5
1.0
P edic ed
2= 0.7488
d_u
−2−1 0 1
−2
−1
0
1
G ound u h
P edic ed alue
2.0 1.5 1.0 0.5 0.0 0.5 1.0
G ound T u h
2.0
1.5
1.0
0.5
0.0
0.5
1.0
P edic ed
2= 0.7488
d_u
−2−1 0 1
−2
−1
0
1
G ound u h
P edic ed alue
2.0 1.5 1.0 0.5 0.0 0.5 1.0
G ound T u h
2.0
1.5
1.0
0.5
0.0
0.5
1.0
P edic ed
2= 0.0000
d_l
−2−1 0 1
−2
−1
0
1
G ound u h
P edic ed alue
2.0 1.5 1.0 0.5 0.0 0.5 1.0
G ound T u h
2.0
1.5
1.0
0.5
0.0
0.5
1.0
P edic ed
2= 0.7466
d_l
−2−1 0 1
−2
−1
0
1
G ound u h
P edic ed alue
2.0 1.5 1.0 0.5 0.0 0.5 1.0
G ound T u h
2.0
1.5
1.0
0.5
0.0
0.5
1.0
P edic ed
2= 0.7466
d_l
−2−1 0 1
−2
−1
0
1
G ound u h
P edic ed alue
0.0 0.5 1.0 1.5 2.0 2.5 3.0
G ound T u h
0.0
0.5
1.0
1.5
2.0
2.5
3.0
P edic ed
2= 0.1876
ho_u
0123
0
1
2
3
G ound u h
P edic ed alue
0.0 0.5 1.0 1.5 2.0 2.5 3.0
G ound T u h
0.0
0.5
1.0
1.5
2.0
2.5
3.0
P edic ed
2= 0.9732
ho_u
0123
0
1
2
3
G ound u h
P edic ed alue
0.0 0.5 1.0 1.5 2.0 2.5 3.0
G ound T u h
0.0
0.5
1.0
1.5
2.0
2.5
3.0
P edic ed
2= 0.9706
ho_u
0123
0
1
2
3
G ound u h
P edic ed alue
0.0 0.5 1.0 1.5 2.0 2.5 3.0
G ound T u h
0.0
0.5
1.0
1.5
2.0
2.5
3.0
P edic ed
2= 0.0643
ho_l
0123
0
1
2
3
G ound u h
P edic ed alue
0.0 0.5 1.0 1.5 2.0 2.5 3.0
G ound T u h
0.0
0.5
1.0
1.5
2.0
2.5
3.0
P edic ed
2= 0.9742
ho_l
0123
0
1
2
3
G ound u h
P edic ed alue
0.0 0.5 1.0 1.5 2.0 2.5 3.0
G ound T u h
0.0
0.5
1.0
1.5
2.0
2.5
3.0
P edic ed
2= 0.9717
ho_l
0123
0
1
2
3
G ound u h
P edic ed alue
0.0 0.5 1.0 1.5 2.0 2.5 3.0
G ound T u h
0.0
0.5
1.0
1.5
2.0
2.5
3.0
P edic ed
2= 0.1508
ho_h
0123
0
1
2
3
G ound u h
P edic ed alue
0.0 0.5 1.0 1.5 2.0 2.5 3.0
G ound T u h
0.0
0.5
1.0
1.5
2.0
2.5
3.0
P edic ed
2= 0.8950
ho_h
0123
0
1
2
3
G ound u h
P edic ed alue
0.0 0.5 1.0 1.5 2.0 2.5 3.0
G ound T u h
0.0
0.5
1.0
1.5
2.0
2.5
3.0
P edic ed
2= 0.8910
ho_h
0123
0
1
2
3
G ound u h
P edic ed alue
Figu e 2.14: C oss-plo s o ype 4 o Example B.1 wi hou egula iza ion o
he aining da ase ( i s column), and wi h egula iza ion o he
aining da ase (second column) and he es da ase ( hi d col-
umn). Fi s ow: dis ance o he uppe laye . Second ow: dis ance
o he lowe laye . Thi d ow: esis i i y o uppe laye . Fou h ow:
esis i i y o lowe laye . Fi h ow: esis i i y o cen al laye .
32

2 Sol ing In e se P oblems using Deep Lea ning
2.8.3 In e sion o Realis ic Syn he ic Models
We now conside h ee ealis ic syn he ic examples o assess he pe o mance o
he in e sion p ocess. In e ms o log accu acy, we obse e quali a i ely simila
esul s o he a enua ion and phase logs. Thus, in he ollowing we only display
he a enua ion logs and omi he phase logs.
2.8.3.1 Model P oblem I
Figu e 2.15 desc ibes a well ajec o y in a syn he ic model p oblem. The model
has a esis i e laye wi h a wa e -bea ing laye unde nea h, and exhibi s wo
geological aul s.
0 50 100 150 200 250 300 350 400 450 500
45
50
55
60
HD (𝑚)
TVD (𝑚)
Figu e 2.15: Fo ma ion o model p oblem I.
Fo he DNNs p oduced wi h he Example B.2 (wi h inpu measu emen s co e-
sponding o 65 logging posi ions pe sample), Figu e 2.16 shows he co esponding
in e ed models using he Encode -Decode DNN wi h and wi hou egula iza-
ion. Resul s show inaccu a e in e sion esul s, specially o he case wi hou
egula iza ion. Mo eo e , he p edic ed logs a e a om he ue logs, as Fig-
u e 2.17, and as expec ed om c oss-plo s 3 (see Table 2.3). The DNN in e sion
esul s a e piecewise 1D models. Howe e , he DNN app oxima ion only ains
wi h 1D models, no o piecewise 1D models, which explains he poo app oxi-
ma ions hey deli e (see Rema k A on Sec ion 2.6).
In he emainde o his sec ion, we es ic o DNNs p oduced wi h Example
B.1. Tha is, we pa ame ize all obse a ions a one loca ion using in o ma ion
om ha loca ion alone. Figu e 2.18 shows he co esponding in e ed mod-
els. Fo he case o he Encode -Decode loss unc ion wi hou egula iza ion,
we obse e in Figu e 2.18a an in e ed model ha is comple ely di e en om
he o iginal one. The co esponding logs (see Figu e 2.19) a e also inaccu a e, as
an icipa ed by he c oss-plo s esul s o ype 3 shown in he p e ious subsec ion.
When conside ing he wo-s ep based loss unc ion wi hou egula iza ion, he
33
2 Sol ing In e se P oblems using Deep Lea ning
0 50 100 150 200 250 300 350 400 450 500
45
50
55
60
HD (𝑚)
TVD (𝑚)
(a) Wi hou egula iza ion
0 50 100 150 200 250 300 350 400 450 500
45
50
55
60
HD (𝑚)
TVD (𝑚)
(b) Wi h egula iza ion
Figu e 2.16: In e ed o ma ion o model p oblem I using he in e sion s a egy
o Example B.2, i.e., wi h inpu measu emen s co esponding o 65
logging posi ions pe sample.
34
2 Sol ing In e se P oblems using Deep Lea ning
0 50 100 150 200 250 300 350 400 450 500
26
28
30
◦ s ◦𝜃∗
HD (𝑚)
A . (𝑑𝐵)
(a) LWD coaxial measu emen . Wi hou egula iza ion
0 50 100 150 200 250 300 350 400 450 500
26
28
30
◦ s ◦𝜃∗
HD (𝑚)
A . (𝑑𝐵)
(b) LWD coaxial measu emen . Wi h egula iza ion
Figu e 2.17: Model p oblem I. Compa ison be ween F˝Iand F˝Iθ˚using he
in e sion s a egy o Example B.2, i.e., wi h inpu measu emen s
co esponding o 65 logging posi ions pe sample.
35
2 Sol ing In e se P oblems using Deep Lea ning
eco e ed model (see Figu e 2.18b) is s ill qui e di e en om he o iginal one.
None heless, we obse e a supe b ma ching in he logs (see Figu e 2.20), which
indica es he p esence o a di e en solu ion o he in e se p oblem. This con-
i ms ha he gi en measu emen s a e insu icien o p o ide a unique solu ion
o he in e se p oblem. Fo he case wi h egula iza ion, in e sion esul s (see
Figu e 2.18b) ma ch he o iginal model, and he co esponding logs p ope ly ap-
p oxima e he syn he ic ones, see Figu e 2.21. Figu es 2.22 and 2.23 con i m ha
ou me hodology deli e s a p ope aining o he o wa d unc ion app oxima ion
and he composi ion Fφ˚˝Iθ˚, espec i ely.
36
2 Sol ing In e se P oblems using Deep Lea ning
2.8.3.2 Model P oblem II
In his p oblem, we conside a 2.5m- hick conduc i e laye su ounded by wo
esis i e laye s. A well ajec o y wi h a dip angle equal o 87˝c osses he o -
ma ion. Figu e 2.24 displays he o iginal and p edic ed models by DL. This
example illus a es some o he limi a ions o DNNs. In his case, he Ea h
models associa ed wi h pa o he ajec o y a e ou side he model p oblems
conside ed in Sec ion 2.1, which es ic o only one laye abo e and below he
logging ajec o y. Thus, he DNN is un ained o such models, and esul s
canno be us ed in hose zones. Nume ical esul s con i m hese obse a ions.
None heless, inaccu a e in e sion esul s a e simple o iden i y by inspec ion o
he logs (Figu es 2.25 and 2.26).
0 20 40 60 80 100 120 140 160 180 200 220 240
40
42
44
46
HD (𝑚)
TVD (𝑚)
(a) Ac ual o ma ion
0 20 40 60 80 100 120 140 160 180 200 220 240
40
42
44
46
HD (𝑚)
TVD (𝑚)
(b) P edic ed o ma ion using one logging posi ion wi h egula iza ion
Figu e 2.24: Model p oblem 2. Compa ison be ween ac ual and p edic ed o -
ma ions wi h egula iza ion using he in e sion s a egy o Example
B.1, i.e., wi h inpu measu emen s co esponding o one logging po-
si ion pe sample.
43

2 Sol ing In e se P oblems using Deep Lea ning
0 20 40 60 80 100 120 140 160 180 200 220 240
26
28
30
◦ s ◦𝜃∗
HD (𝑚)
A . (𝑑𝐵)
(a) LWD coaxial measu emen
0 20 40 60 80 100 120 140 160 180 200 220 240
−49
−48.8
−48.6
◦ s ◦𝜃∗
HD (𝑚)
A . (𝑑𝐵)
(b) Deep coaxial measu emen
0 20 40 60 80 100 120 140 160 180 200 220 240
12
14
16
18
20
◦ s ◦𝜃∗
HD (𝑚)
A . (𝑑𝐵)
(c) Geosignal measu emen
Figu e 2.25: Model p oblem 2. Compa ison be ween F˝Iand F˝Iθ˚wi h eg-
ula iza ion using he in e sion s a egy o Example B.1, i.e., wi h
inpu measu emen s co esponding o one logging posi ion pe sam-
ple.
44
2 Sol ing In e se P oblems using Deep Lea ning
0 20 40 60 80 100 120 140 160 180 200 220 240
26
28
◦ s 𝜙∗◦
HD (𝑚)
A . (𝑑𝐵)
(a) LWD coaxial measu emen
0 20 40 60 80 100 120 140 160 180 200 220 240
−49
−48.5
−48
◦ s 𝜙∗◦
HD (𝑚)
A . (𝑑𝐵)
(b) Deep coaxial measu emen
0 20 40 60 80 100 120 140 160 180 200 220 240
10
15
20
◦ s 𝜙∗◦
HD (𝑚)
A . (𝑑𝐵)
(c) Geosignal measu emen
Figu e 2.26: Model p oblem 2. Compa ison be ween F˝Iand Fφ˚˝Iwi h eg-
ula iza ion using he in e sion s a egy o Example B.1, i.e., wi h
inpu measu emen s co esponding o one logging posi ion pe sam-
ple.
45
2 Sol ing In e se P oblems using Deep Lea ning
2.8.3.3 Model P oblem III
We now conside a model o ma ion exhibi ing geological aul s and wo di e en
well ajec o ies. Fo well ajec o y 1, Figu e 2.27 shows he model p oblem, log-
ging ajec o y, in e sion esul s, and coaxial a enua ion logs. In e sion esul s
a e excellen . When conside ing he second well ajec o y shown in Figu e 2.28,
we obse e good in e sion esul s excep a he p oximi y o poin s wi h ho izon-
al dis ance (HD) equals o 75m and 350m. These inaccu a e in e sion esul s
a e easily iden i ied by examina ion o he co esponding logs.
46
2 Sol ing In e se P oblems using Deep Lea ning
0 50 100 150 200 250 300 350 400 450 500
40
50
60
HD (𝑚)
TVD (𝑚)
(a) Ac ual o ma ion
0 50 100 150 200 250 300 350 400 450 500
40
50
60
HD (𝑚)
TVD (𝑚)
(b) P edic ed o ma ion
0 50 100 150 200 250 300 350 400 450 500
26
28
30
◦ s ◦𝜃∗
HD (𝑚)
A . (𝑑𝐵)
(c) LWD coaxial measu emen
Figu e 2.27: Model p oblem III, ajec o y 1. Compa ison be ween ac ual and
p edic ed o ma ions and he co esponding coaxial logs wi h eg-
ula iza ion using he in e sion s a egy o Example B.1, i.e., wi h
inpu measu emen s co esponding o one logging posi ion pe sam-
ple.
47
2 Sol ing In e se P oblems using Deep Lea ning
0 50 100 150 200 250 300 350 400 450 500
40
50
60
HD (𝑚)
TVD (𝑚)
(a) Ac ual o ma ion
0 50 100 150 200 250 300 350 400 450 500
40
50
60
HD (𝑚)
TVD (𝑚)
(b) P edic ed o ma ion
0 50 100 150 200 250 300 350 400 450 500
26.5
27
27.5
◦ s ◦𝜃∗
HD (𝑚)
A . (𝑑𝐵)
(c) LWD coaxial measu emen
Figu e 2.28: Model p oblem III, T ajec o y 2. Compa ison be ween ac ual and
p edic ed o ma ions and he co esponding coaxial logs wi h eg-
ula iza ion using he in e sion s a egy o Example B.1, i.e., wi h
inpu measu emen s co esponding o one logging posi ion pe sam-
ple.
48

3 Da abase Gene a ion using IGA
Deep Lea ning (DL) me hods a e as , bu equi e a massi e aining da ase . To
dec ease he online compu a ional ime du ing ield ope a ions, we o en p oduce
such a la ge da ase a p io i (o line) using ens o housands o simula ions o
bo ehole esis i i y measu emen s (see [75]). To gene a e he da abase o DL in-
e sion, we employ simula ion me hods o sol e Maxwell’s equa ions wi h di e en
conduc i i y dis ibu ions (Ea h models). Since 3D simula ions a e expensi e
and possibly una o dable when compu ing such la ge da abases, i is common
o educe he Ea h model dimensionali y o wo o one spa ial dimensions using
a Fou ie o a Hankel ans o m. These ans o ma ions lead o he so-called
2.5D [3, 48, 92, 101, 135] and 1.5D [12, 100, 129] o mula ions, espec i ely. 1.5D
simula ions a e inaccu a e when dealing wi h geological aul s.
In his wo k, we ocus on he e icien gene a ion o a massi e da abase using
2.5D simula ions – as a p elimina y s age o DL in e sions. We p opose he use
o e ined Isogeome ic Analysis ( IGA) disc e iza ions o gene a e da abases o
DL in e sion o 2.5D geos ee ing elec omagne ic (EM) measu emen s.
3.1 2.5D Va ia ional Fo mula ion o
Elec omagne ic Measu emen s
3.1.1 3D Wa e P opaga ion P oblem
The wo ime-ha monic cu l Maxwell’s equa ions desc ibing he 3D wa e p opa-
ga ion in an iso opic medium a e
∇ˆE`iωµH“ ´iωµM
∇ˆH“ pσ`iωεqE(3.1)
whe e Eis he elec ic ield, His he magne ic ield, iis he imagina y uni ,
σis he elec ic conduc i i y, εis he elec ic pe mi i i y, µis he magne ic
pe meabili y, ω“2π is he angula equency, wi h being he ansmi e
equency, and Mis he ime-ha monic magne ic sou ce loca ed a px0, y0, z0q
and gi en by
M“δpx´x0qδpy´y0qδpz´z0q Mx, My, Mzs P R3,(3.2)
49
3 Da abase Gene a ion using IGA
wi h δp¨q being he Di ac del a unc ion de ined as ollows:
δpx´x0q:“"8, x “x0,
0, x ‰x0.(3.3)
To inco po a e an in eg able app oxima ion o he Di ac del a unc ion, we
conside a bell-like ep esen a ion o he del a unc ion. Fo example, in he x
di ec ion, we app oxima e:
δpx´x0q « 1
α?πexp „´´x´x0
α¯2,(3.4)
whe e αis a posi i e alue.
F om Maxwell’s equa ions, we ob ain he ollowing educed wa e o mula ion
in e ms o magne ic ield H:
$
’
’
&
’
’
%
Find H“ Hx, Hy, Hzs,wi h H:ΩĂR3ÑC3,such ha :
∇ˆˆ1
σ`iωε∇ˆH˙`iωµH“ ´iωµM,in Ω,
Eˆn“0,on BΩ,
(3.5)
whe e Ωis he domain o s udy and nis he uni no mal (ou wa d) ec o on he
bounda y BΩ. We de ine Ωas a enso -p oduc box:
Ω“ΩxˆΩyˆΩz“`´Lx{2, Lx{2˘ˆ`´Ly{2, Ly{2˘ˆ`´Lz{2, Lz{2˘,
being Lx,Ly, and Lzposi i e eal cons an s.
To in oduce he weak o mula ion o his p oblem, we i s de ine he Hpcu l; Ωq-
con o ming unc ional spaces
Hpcu l; Ωq:“ W“ Wx, Wy, Wzs P pL2pΩqq3:∇ˆWP pL2pΩqq3u,
H0pcu l; Ωq:“ WPHpcu l; Ωq:Wˆn“0on BΩu.(3.6)
The Hpcu l; Ωqspace is endowed wi h he inne p oduc
pW,HqHpcu l;Ωq:“ p∇ˆW,∇ˆHqpL2pΩqq3`pW,HqpL2pΩqq3
:“żΩp∇ˆWq˚¨p∇ˆHqdΩ`żΩ
W˚¨HdΩ,(3.7)
whe e ˚is he conjuga e anspose o complex ec o space and ¨deno es he
inne p oduc .
50
3 Da abase Gene a ion using IGA
We build he weak o mula ion by mul iplying Eq. (3.5) wi h an a bi a y unc-
ion WPH0pcu l; Ωq, using G een’s o mula, and in eg a ing o e he domain
Ω. The weak o mula ion is hen
$
&
%
Find HPH0pcu l; Ωq,such ha o e e y WPH0pcu l; Ωq,
ˆ∇ˆW,1
σ`iωε∇ˆH˙pL2pΩqq3`iωµpW,HqpL2pΩqq3“ ´iωµpW,MqpL2pΩqq3
(3.8)
3.1.2 2.5D Va ia ional Fo mula ion
He ein, we ocus on he case when he ma e ial p ope ies a e homogeneous
along one spa ial di ec ion, e.g., y-axis. We deno e he domain o his case
as Ω:“ΩyˆΩx,z. We pe o m a Fou ie ans o m along he y-axis o ep e-
sen he 3D p oblem as a sequence o uncoupled 2D p oblems, one pe Fou ie
mode. In his case, we de ine he magne ic ield Has a se ies expansion using
he complex exponen ials:
H:“
`8
ÿ
β“´8
Hβexppi2πβy{Lyq,(3.9)
whe e βis he Fou ie mode numbe and Hβ““Hβ
x, Hβ
y, Hβ
z‰wi h Hβ:Ωx,z Ă
R2ÑC3.
Fou ie modes sa is y he ollowing o hogonali y ela ionships, whe e δi,j is he
K onecke del a:
1
LyżLy{2
´Ly{2
exppi2πβ1y{Lyqexppi2πβ2y{Lyqdy “δβ1,β2.(3.10)
By employing a es unc ion o he o m
W:“1
Ly
Wβexppi2πβy{Lyq,(3.11)
and de ining he Hpcu lβ;Ωx,zq-con o ming unc ional spaces
Hpcu lβ;Ωx,zq:“ Wβ“ Wβ
x, Wβ
y, Wβ
zs P pL2pΩx,zqq3:Wβ
yPH1pΩx,zq
and ∇ˆ Wβ
x, Wβ
zs P pL2pΩx,zqq2(,
H0pcu lβ;Ωx,zq:“ WβPHpcu lβ;Ωx,zq:Wβˆn“0on BΩ(.
(3.12)
51
3 Da abase Gene a ion using IGA
we build he ollowing a ia ional o mula ion om Eq. (3.5) by in eg a ing o e
Ωx,z:
$
’
’
&
’
’
%
Find H“ř`8
β“´8 Hβexppi2πβy{Lyq,HβPH0pcu l; Ωx,zq
such ha o e e y βPZand WβPH0pcu l; Ωx,zq,
ˆ∇βˆWβ,1
σ`iωε∇βˆHβ˙pL2pΩx,zqq3`iωµpWβ,HβqpL2pΩx,z qq3“ ´iωµpWβ,MβqpL2pΩx,zqq3,
(3.13)
whe e
∇βˆWβ:“«iβ 2π
Ly
Wβ
z´dWβ
y
dz ,dWβ
x
dz ´dWβ
z
dx ,dWβ
y
dx ´iβ 2π
Ly
Wβ
x ,(3.14)
and Mβis he ime-ha monic magne ic sou ce w i en in e ms o he Fou ie
ans o m as
Mβ“1
Ly
δpx´x0qδpz´z0q Mx, My, Mzsexppi2πβy0{Lyq.(3.15)
This o mula ion co esponds o he 2.5D a ia ional o mula ion p e iously
desc ibed by, e.g., [27] and [120].
Rema k 3.1. To sol e he a ia ional p oblem o Eq. (3.13), we equi e an app o-
p ia e space in Ωx,z o e which Wβ,∇ˆ Wβ
x,Wβ
zs, and ∇Wβ
ya e in eg able, i.e.,
Wβ
x,Wβ
zs P Hpcu l; Ωx,zqand Wβ
yPH1pΩx,zq. Thus, we use he Hpcu lβ;Ωx,zq
solu ion space – equi alen o he Hpcu l; Ωx,zq ˆ H1pΩx,zqmixed space – ha
ul ills he men ioned equi emen s.
3.2 Bo ehole Resis i i y Measu emen Acquisi ion
Sys em
We conside he logging-while-d illing (LWD) ins umen equipped wi h ans-
mi e s (Txi) and ecei e s (Rxj) o Figu e 3.1. This ool is sensi i e o esis i i ies
wi hin he ange 0.2„500 Ω ¨m (phase esis i i y) and 0.2„300 Ω ¨m (ampli-
ude esis i i y) unde an ope a ing equency be ween 0.1 and 2 MHz [79]. Fo
he sake o simplici y, he ein, we es ic o wo ansmi e s and wo ecei e s
symme ically loca ed a ound he ool cen e (see Figu e 3.1) a an ope a ing
equency o 2 MHz.
T iaxial logging ins umen s gene a e measu emen s o all possible o ien a-
ions o he ansmi e – ecei e pai s. We ollow he no a ion p esen ed by [34]
52
3 Da abase Gene a ion using IGA
z
x
l
l
h
h
L
L
Ωx,z
T2R2R1T1Tool T ajec o y
Figu e 3.4: A d awing o he compu a ional domain Ωx,z and he ool ajec o y.
The cen al subg id bounded by a magen a box is composed o a se o
ine elemen s loca ed in he p oximi y o he logging ins umen . The
emaining elemen s g ow smoo hly in size un il eaching he bounda y.
Fou ie modes. Due o he symme y o he media along he ydi ec ion, we only
conside βP 0, N s.
3.5 Nume ical Resul s
In his sec ion, we i s assess he accu acy o he IGA app oach in a homogeneous
medium. We also in es iga e he compu a ional e iciency o he IGA amewo k
in compa ison wi h IGA and FEM app oaches. Then, we conside wo model
p oblems consis ing o high-angle wells c ossing spa ially he e ogeneous media
wi h mul iple geological aul s. Finally, we p oduce ou syn he ic aining da ase
as a p elimina y s age o DL in e sion. In ou simula ions, we conside one
ope a ional mode o a comme cial logging ool [158] wi h lR“10.16 cm and
lT“56.8325 cm (see Figu e3.1). We selec he ee space elec ic pe mi i i y
and magne ic pe meabili y as ε“8.85ˆ10´12 F¨m´1and µ“4πˆ10´7N¨A´2,
espec i ely. We also conside a ansmi e equency “2 MHz.
3.5.1 Homogeneous Medium
We assume he logging ins umen is placed in a homogeneous medium wi h
esis i i y ρ“1{σ“100 Ω ¨m. This high- esis i i y case is nume ically mo e
challenging han low- esis i i y cases since i equi es a la ge numbe o Fou ie
59

3 Da abase Gene a ion using IGA
modes and nume ical p ecision. We conside a cube domain o leng h L“18 m
o ou i s case s udy.
3.5.1.1 Accu acy Assessmen
To assess he accu acy and selec ce ain disc e iza ion pa ame e s, we compa e
he nume ical a enua ion a io, A, and phase di e ence, P, gi en by Eq.(3.17)
and Eq.(3.18), wi h he expec ed (i.e., exac ) alues, Aeand Pe, ob ained om
ρe“1{σ. Figu e 3.5 shows he nume ical e o s, i.e., |1´A{Ae|and |1´P{Pe|,
as a unc ion o he numbe o Fou ie modes when compu ing a enua ion a io
and phase di e ence in a homogeneous medium. He ein, we selec a domain wi h
64 ˆ64 elemen s o ensu e a as nume ical solu ion o ou measu emen s. We
compa e he esul s o he high-con inui y Cp´1IGA wi h FEM and also wi h a
IGA disc e iza ion ha employs 8 ˆ8 mac oelemen s. This mac oelemen size
p o ides he as es esul s o mode a e size domains (see [47]). We conside
h ee di e en mesh sizes – h“0.025 m,0.033 m,and 0.050 m– and di e en
polynomial deg ees –p“3,4,and 5. The bes esul s co espond o h“0.025 m
(blue lines in he igu e). We also obse e ha IGA and FEM disc e iza ions
deli e lowe e o s compa ed o hei IGA coun e pa s –when he same numbe
o elemen s and polynomial deg ee a e conside ed– aking in o accoun ha IGA
p o ides solu ions wi h highe compu a ional e iciency (see Sec ion 3.5.1.2).
To in es iga e he decay o he solu ion o each Fou ie mode, we compa e
nume ical esul s wi h he analy ical 2.5D solu ion in he homogeneous medium
p esen ed by [120]. In pa icula , gi en Mzas he only nonze o componen o
he magne ic sou ce, i is possible o analy ically de e mine he coaxial magne ic
ield o each Fou ie mode as ollows:
HZZ pβq“´iωµστβz`B2τβz
Bz2,(3.34)
wi h
τβz“Mz
2π
1
Ly
K0pCRqexppi2πβy0{Lyq,(3.35)
whe e K0p¨q is he modi ied Bessel unc ion o he second kind o o de ze o, and
C“ p2πβ{Lyq2`iωµσ, (3.36)
R“apx´x0q2`pz´z0q2.(3.37)
60
3 Da abase Gene a ion using IGA
10−4
10−3
10−2
10−1
100
h=0.025 m
h=0.033 m
h=0.050 m
|1−A/Ae|
IGA
IGA
FEA
10−4
10−3
10−2
10−1
100
h=0.025 m
h=0.033 m
h=0.050 m
|1−A/Ae|
IGA
IGA
FEA
10−4
10−3
10−2
10−1
100
h=0.025 m
h=0.033 m
h=0.050 m
|1−A/Ae|
IGA
IGA
FEA
10 20 30 40 50 60 70
10−4
10−3
10−2
10−1
100
h=0.025 m
h=0.033 m
h=0.050 m
No. o Fou ie modes, N
|1−P/Pe|
IGA
IGA
FEA
(a) p“3
10 20 30 40 50 60 70
10−4
10−3
10−2
10−1
100
h=0.025 m
h=0.033 m
h=0.050 m
No. o Fou ie modes, N
|1−P/Pe|
IGA
IGA
FEA
(b) p“4
10 20 30 40 50 60 70
10−4
10−3
10−2
10−1
100
h=0.025 m
h=0.033 m
h=0.050 m
No. o Fou ie modes, N
|1−P/Pe|
IGA
IGA
FEA
(c) p“5
Figu e 3.5: Nume ical e o s when compu ing a enua ion a io, A, and phase di -
e ence, P, in a homogeneous medium using IGA and IGA disc e iza-
ions, ob ained by a 64ˆ64 elemen mesh wi h di e en elemen sizes
hand polynomial deg ees p.
Figu e 3.6 compa es he decay o he nume ical coaxial magne ic ield HZZpβq
wi h i s analy ical coun e pa o some Fou ie modes. Using a domain wi h
64 ˆ64 elemen s and h“0.025 m, we moni o he decay o he p opaga ed
wa es a dis ances wi hin he in e al 0.2,1.0sm om he ansmi e s o ensu e
ha he solu ions a bo h ecei e s p ope ly app oxima e he analy ical ones.
Resul s show ha IGA disc e iza ions deli e inc eased accu acy o all es ed
polynomial deg ees.
3.5.1.2 Compu a ional E iciency
[46, 47] p o ide heo e ical cos es ima es o sol ing H1and Hpcu lqdisc e e
spaces, espec i ely. He ein, we add hese es ima es o p edic he cos o dis-
c e izing he Hpcu l; Ωx,zqˆH1pΩx,zqspace appea ing in ou 2.5D EM p oblem.
We conclude ha he cos o LU ac o iza ion o he IGA ma ix o his com-
bined space is be ween Oppqand Opp2q imes smalle han ha o IGA. De ails
a e omi ed o he sake o simplici y.
To nume ically assess he compu a ional e iciency con i ming he a o emen-
ioned heo e ical esul s, we conside wo di e en g ids in Ωx,z wi h 64 ˆ64
and 128 ˆ128 elemen s, espec i ely. Using con inui y educ ion, we spli he
61
3 Da abase Gene a ion using IGA
0.2 0.40.60.8 1
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
β=0
β=10
β=25
β=45
Dis ance om ansmi e (m)
|HZZ|(A ·m−1)
Nume ical (IGA)
Nume ical ( IGA)
Analy ical
(a) p“3
0.2 0.40.60.8 1
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
β=0
β=10
β=25
β=45
Dis ance om ansmi e (m)
|HZZ|(A ·m−1)
Nume ical (IGA)
Nume ical ( IGA)
Analy ical
(b) p“4
0.2 0.40.60.8 1
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
β=0
β=10
β=25
β=45
Dis ance om ansmi e (m)
|HZZ|(A ·m−1)
Nume ical (IGA)
Nume ical ( IGA)
Analy ical
(c) p“5
Figu e 3.6: Compa ison o he decay o he nume ical and analy ical coaxial mag-
ne ic ields o some Fou ie modes, ob ained in a g id o 64 ˆ64
elemen s wi h h“0.025 m and di e en polynomial deg ees.
mesh symme ically in o mac oelemen s whose sizes a e powe s o wo. In his
con ex , he maximum-con inui y Cp´1IGA disc e iza ion is composed o one
mac oelemen con aining he en i e g id, while C0FEM wi h minimum con inu-
i y ac oss all elemen in e aces is composed o mac oelemen s ha con ain only
one elemen . Figu e 3.7 shows he numbe o FLOPs and ime equi ed o sol e
he bo ehole esis i i y p oblem o each Fou ie mode pe logging posi ion. We
compa e he compu a ional cos s o di e en polynomial deg ees and di e en
con inui y educ ion le els o basis unc ions. The cos o IGA eaches he min-
imum wi h 8 ˆ8 mac oelemen s almos in all cases, con i ming he heo e ical
es ima es ob ained om he esul s o [47].
Ou nume ical es s show ha o a mode a e size 2.5D p oblem, he educ ion
in he numbe o FLOPs is Oppqwi h espec o IGA. When compa ed o FEM,
IGA deli e s la ge imp o emen ac o s. These imp o emen ac o s in e ms
o FLOPs also hold in e ms o ime when pe o ming a sequen ial ac o iza ion.
In ou pa allel PARDISO sol e , we obse e a small deg ada ion o he IGA
imp o emen ac o s in e ms o imes in compa ison o hose ob ained in e ms
o FLOPs (see Table 3.1).
62
3 Da abase Gene a ion using IGA
1×1
2×2
4×4
8×8
16×16
32×32
64×64
1010
1011
1012
p=3
p=4
p=5
Cp−1IGA
C0FEA
Mac oelemen size
FLOPs
(a) FLOPs (64 ˆ64 elemen s)
1×1
2×2
4×4
8×8
16×16
32×32
64×64
10−1
100
101
p=3
p=4
p=5
Cp−1IGA
C0FEA
Mac oelemen size
Time (s)
(b) Time (64 ˆ64 elemen s)
1×1
2×2
4×4
8×8
16×16
32×32
64×64
128×128
1011
1012
1013
p=3
p=4
p=5
Cp−1IGA
C0FEA
Mac oelemen size
FLOPs
(c) FLOPs (128 ˆ128 elemen s)
1×1
2×2
4×4
8×8
16×16
32×32
64×64
128×128
100
101
102
p=3
p=4
p=5
Cp−1IGA
C0FEA
Mac oelemen size
Time (s)
(d) Time (128 ˆ128 elemen s)
Figu e 3.7: Compu a ional cos in e ms o FLOPs and ime o sol ing a 2.5D
bo ehole esis i i y p oblem pe logging posi ion pe Fou ie mode.
We es IGA disc e iza ions wi h wo di e en g ids o 64 ˆ64 and
128 ˆ128 elemen s. The compu a ional imes co espond o he use
o pa allel sol e PARDISO using wo h eads.
63
3 Da abase Gene a ion using IGA
Table 3.1: Compu a ional cos o he 2.5D bo ehole esis i i y measu emen s pe logging posi ion pe Fou ie mode.
We epo he solu ion ime and FLOPs when using Cp´1IGA, IGA wi h 8 ˆ8 mac oelemen s, and C0
FEM wi h he same numbe o elemen s and polynomial deg ee. The compu a ional imes co espond o
he use o pa allel sol e PARDISO using wo h eads.
Domain
size Polynomial
deg ee Disc e iza ion
me hod Numbe o
FLOPs Imp o emen ac o
(FLOPs) Time
(s) Imp o emen ac o
( ime)
IGA 6.56e+10 0.407
3 IGA 2.85e+10 IGA/ IGA 2.30 0.242 IGA/ IGA 1.68
FEA 1.65e+11 FEA/ IGA 5.79 0.999 FEA/ IGA 4.12
IGA 1.62e+11 0.875
64ˆ64 4 IGA 4.97e+10 IGA/ IGA 3.26 0.419 IGA/ IGA 2.09
FEA 5.56e+11 FEA/ IGA 11.19 3.061 FEA/ IGA 7.29
IGA 3.10e+11 1.645
5 GA 7.33e+10 IGA/ IGA 4.23 0.620 IGA/ IGA 2.65
FEA 1.37e+12 FEA/ IGA 18.69 6.806 FEA/ IGA 10.98
IGA 5.72e+11 3.144
3 IGA 2.24e+11 IGA/ IGA 2.55 1.423 IGA/ IGA 2.21
FEA 1.31e+12 FEA/ IGA 5.85 7.456 FEA/ IGA 5.24
IGA 1.43e+12 6.903
128ˆ128 4 IGA 3.56e+11 IGA/ IGA 4.02 2.495 IGA/ IGA 2.77
FEA 4.44e+12 FEA/ IGA 12.47 22.885 FEA/ IGA 9.17
IGA 2.82e+12 12.911
5 GA 4.97e+11 IGA/ IGA 5.67 3.305 IGA/ IGA 3.91
FEA – FEA/ IGA – – FEA/ IGA –
64

3 Da abase Gene a ion using IGA
3.5.2 He e ogeneous Media
We u he examine he accu acy o ou IGA app oxima ion o e wo syn he ic
he e ogeneous model p oblems.
3.5.2.1 One Geological Faul
We conside he model p oblem o Figu e 3.8 wi h a cons an dip angle o 80˝. We
conside he Logging While D illing (LWD) ins umen desc ibed in Sec ion 3.2
and simula e measu emen s eco ded o e 200 equally-spaced logging posi ions
h oughou he well ajec o y.
ρ=50 Ω·m
0.0
0.5
1.0
1.5
2.0
0.0 7.5 15.0
ρ=3Ω·m
ρ=1Ω·m
Figu e 3.8: Model p oblem wi h a cons an dip angle o 80˝passing h ough a
geological aul and h ee di e en ma e ials (well ajec o y is high-
ligh ed by a ed dashed line). Dimensions a e in me e s.
Figu e 3.9 shows he appa en esis i i ies based on he a enua ion a io and
phase di e ence (ρAand ρP, espec i ely). We ob ain he esul s using N “70
and a IGA disc e iza ion wi h 64ˆ64 elemen s, p“4, and 8ˆ8 mac oelemen s.
Resul s a e in good ag eemen wi h hose p esen ed by [120].
3.5.2.2 Two Geological Faul s and Inclined Laye s
Figu e 3.10 shows he second model p oblem con aining wo geological aul s
and inclined laye s. The logging ajec o y s a s om a sands one laye wi h
a esis i i y o ρ“3 Ω ¨m, and passes h ough an oil-sa u a ed laye wi h
ρ“100 Ω ¨m. The ool ajec o y also passes h ough a wa e -sa u a ed laye
wi h ρ“0.5 Ω ¨m.
In pa icula , inclined laye s p oduce he so-called s ai case app oxima ions [25].
This phenomenon occu s because he physical in e aces o he conduc i i y model
a e no aligned wi h he elemen edges. Thus, he conduc i i y pa ame e akes
di e en alues inside some elemen s o he mesh. To ackle his issue, disc e iza-
ion echniques using non i ing g ids [26, 27] a e a ailable, bu hey ha e no
been conside ed he e o simplici y.
65
3 Da abase Gene a ion using IGA
02468 10 12 14 16
100
101
102
ρA
ρP
ρe
Ho izon al dis ance (m)
Resis i i y (Ω·m)
Figu e 3.9: Appa en esis i i ies based on he a enua ion a io, ρA, and phase
di e ence, ρP, o he i s model p oblem, compa ed wi h he eal
(exac ) esis i i y, ρe. We ob ain he esul s using a IGA disc e iza-
ion wi h 64 ˆ64 elemen s, p“4, and 8 ˆ8 mac oelemen s.
0.0
3.0
4.0
5.0
6.0
8.0
9.0
11.0
0.0 30.0 60.0 90.0
5◦
5◦
1◦
ρ=3Ω·m
ρ=100 Ω·m
ρ=0.5Ω·m
Figu e 3.10: Second model p oblem wi h wo geological aul s and inclined laye s.
The ool ajec o y ( ed dashed line) has di e en dip angles and
passes h ough sands one (yellow), oil-sa u a ed (g ay), and wa e -
sa u a ed (g een) laye s. Dimensions a e in me e s.
Figu e 3.11 shows he appa en esis i i ies based on he a enua ion a ion
and phase di e ence h oughou he logging ajec o y and compa es hei alue
wi h he exac esis i i y. We simula e he esis i i ies a 1,080 logging posi ions
wi h N “70. We use a IGA disc e iza ion wi h 64 ˆ64 elemen s, p“4, and
8ˆ8 mac oelemen s.
3.5.3 Da abase Gene a ion o DL In e sion
To p oduce ou syn he ic aining da ase o DL in e sion, we conside he e o-
geneous medium con aining h ee di e en laye s and six a ying pa ame e s a
66
3 Da abase Gene a ion using IGA
0 10 20 30 40 50 60 70 80 90
100
101
102
ρA
ρP
ρe
Ho izon al dis ance (m)
Resis i i y (Ω·m)
Figu e 3.11: Appa en esis i i ies based on he a enua ion a io, ρA, and phase
di e ence, ρP, o he second model p oblem, compa ed wi h he eal
(exac ) esis i i y, ρe. We employ a IGA disc e iza ion wi h 64ˆ64
elemen s, p“4, and 8 ˆ8 mac oelemen s.
each logging posi ion, as desc ibed in Figu e 3.12 and Table 3.2. We selec h ee
di e en elec ical conduc i i ies: σc o he cen al laye , and σuand σl o he
uppe and lowe laye s, espec i ely. We assume he ool cen e is always wi hin
he middle laye and has e ical dis ances o duand dl om he uppe and lowe
laye s, espec i ely. The six h a ying pa ame e is he dip angle, ϕ, measu ed
om he e ical di ec ion.
+
Tool cen e
du
dl
Tool ajec o y
ϕ
σu
σc
σl
Figu e 3.12: Va ying pa ame e s a each logging posi ion when p oducing he
aining da ase o DL in e sion.
In he e, we c ea e a da ase o 100,000 samples and compu e he appa en esis-
i i ies ob ained om andom combina ions wi hin a gi en ange o esis i i ies
ρ“1{σP 1,100sΩ¨m (see Table 3.2). Fo gene a ing he da ase , we use wo
di e en ypes o pa alleliza ion. One pa alleliza ion is ela ed o he pa allel
67
3 Da abase Gene a ion using IGA
Table 3.2: Va ying pa ame e s employed o gene a e he aining da ase o DL
in e sion.
Va ying pa ame e s In e al
Elec ical conduc i i y o he cen al laye log10pσc) ´2,0s
Elec ical conduc i i y o he uppe laye log10pσu) ´2,0s
Elec ical conduc i i y o he lowe laye log10pσl) ´2,0s
Dis ance o he ool cen e om he uppe laye log10pdu) ´2,1s
Dis ance o he ool cen e om he lowe laye log10pdl) ´2,1s
Dip angle be ween he ool and he laye ed media ϕ 80˝,100˝s
ac o iza ion o he di ec sol e , and he o he is he i ial pa alleliza ion based
on scheduling he solu ions o independen Ea h models on o di e en p oces-
so s. Using 40 h eads, we sol e o 20 di e en Ea h models, each execu ing
o e wo h eads. Table 3.1 shows ha he equi ed ime o ma ix ac o iza-
ion o he 2.5D EM p oblem using op imal IGA disc e iza ion wi h 64 ˆ64
g id, p“4, and 8 ˆ8 mac oelemen s is abou 0.42 seconds pe Fou ie mode.
Conside ing N “70, and he addi ional ime equi ed o p e/pos p ocessing
and in e - h ead communica ions, each se o independen uns (consis s o 20
di e en Ea h models) akes abou 40 seconds. Thus, we pe o m 5,000 sequen-
ial uns o cons uc ou 100,000 samples in abou 56 hou s. To c ea e a la ge
da abase, we could execu e o e a clus e o hund eds o CPUs/ h eads, expec ing
a pe ec pa allel scalabili y. Figu e 3.13a depic s he g aphs o a enua ion a-
io, A, e sus phase di e ence, P, ob ained om he 100,000 Ea h models when
using IGA disc e iza ion o gene a ing he da abase. Since he e is a s ong
co ela ion be ween Aand P, he da a dis ibu ion on he plo ollows an almos
s aigh line. We also display in Figu e 3.13b he co ela ion be ween appa en
esis i i ies based on a enua ion a io and phase di e ence.
68
4 Quad a u e Rules when Sol ing PDEs using Deep Lea ning
Simpli ying e ms we each o
´żΩ
p∆u` qdx “0,@ PV. (4.17)
Wi h he assump ion ha p∆u` qis con inuous ha equali y can only be
sa is ied i
p∆u` qpxq “ 0xPΩ,(4.18)
and his means ha uis solu ion o (4.1).
To inish he p oo , we show ha he solu ion o (4.5) is unique. Suppose ha
u1PVand u2PVa e solu ions o (4.5),
p∇u1,∇ q “ p , q`pg, qΓN@ PV,
p∇u2,∇ q “ p , q`pg, qΓN@ PV. (4.19)
Sub ac ing he equa ions om (4.19) and selec ing “u1´u2we ob ain
żΩp∇u1´∇u2q2dx “0,(4.20)
which implies ha
∇u1pxq´∇u2pxq “ ∇pu1´u2qpxq “ 0@xPΩ.(4.21)
This implies ha pu1´u2qpxqis cons an on Ω, and wi h he bounda y condi ion
u1pxq “ 0 o xPΓDwe each o u1pxq “ u2pxq,@xPΩ.
4.2.2 Leas Squa es Me hod
Reo de ing he e ms o Eq. (4.1a) and (4.1c), we de ine:
"Gu:“u2` x PΩ,
Bu:“u1¨n´g x PΓN.(4.22)
To in oduce he Leas Squa es me hod, we de ine he unc ion FLS :VÝÑ R,
whe e he unc ion PVsa is ies he Di ichle condi ions:
FLSp q “ |pG , G q|`|pB , B qΓN|.(4.23)
We wan o minimize he unc ion FLSp qsubjec o he essen ial (Di ichle )
Bounda y Condi ions (BCs). We o en ind he minimum by aking he de i a i e
equal o ze o and ending up wi h a linea sys em o equa ions. In he con ex o
75

4 Quad a u e Rules when Sol ing PDEs using Deep Lea ning
DL, we can simply in oduce he abo e loss unc ion FLSp qdi ec ly in ou NN.
The e o e, we wan o ind
u“a g min
PV
FLSp q.(4.24)
4.3 Neu al Ne wo k Implemen a ion
We ain a NN, named uNN px;θq, wi h he ollowing a chi ec u e. We de ine
he ainable pa o ou NN wi h lea nable pa ame e s θ. We call i uθ. I is
composed by:
1. An inpu laye . This laye ecei es he da a in he o m o a nˆdma ix,
whe e nis he numbe o samples and dis he dimension o he da a.
2. One hidden dense laye wi h mneu ons and a sigmoid ac i a ion unc ion.
3. An ou pu laye ha deli e s uθ.
Then, we add non- ainable laye s o ou a qui ec u e in o de o impose Equa-
ions (4.6) o (4.23). Fo ha , we in oduce:
4. A non- ainable laye o impose he Di ichle bounda y condi ions. Fo
ha , we selec a unc ion φpxq ha sa is ies he Di ichle condi ions o he
p oblem and i s alue is nonze o e e ywhe e else [76]. In his wo k, we selec
he ollowing φpxq unc ions o 1D p oblems in he in e al Ω “ a, bs:
φpxq “ ź
xDPΓDpx´xDq.(4.25)
Then, we gene a e a new ou pu o he NN: uNN px;θq “ φpxquθpxq ha
s ongly imposes he homogeneous Di ichle bounda y condi ions.
5. A non- ainable laye o compu e he loss unc ion FRo FLS ollowing
Eqs. (4.6) o (4.23). Wi hin his laye , we e alua e he in eg als and he
de i a i es. We conside di e en quad a u e ules, being he quad a u e
poin s pa o he inpu da a o ou NN, along wi h he physical poin s o he
domain. Fo compu ing he de i a i es, we use au oma ic di e en ia ion,
excep in some speci ic cases, whe e we employ FDM. These cases a e
explici ly indica ed h oughou he ex .
76
4 Quad a u e Rules when Sol ing PDEs using Deep Lea ning
Figu e 4.1 shows a schema ic g aph o he desc ibed NN a chi ec u e. Ou
so wa e is de eloped in Py hon and we use he lib a y Tenso low 2.0.
To ain he NN, we eplace in Equa ions (4.7) o (4.24) he sea ch space V
by he mani old gene a ed by ou lea nable pa ame e s θincluded in ou NN.
The esul o he minimiza ion is a unc ion uNN px, ˜
θq, whe e ˜
θa e he op imal
lea nable pa ame e s encoun e ed as a esul o he aining. Fo simplici y, in he
ollowing we abuse no a ion and use he symbol uNN o deno e also he solu ion
uNN px, ˜
θqo ou minimiza ion p oblem.
x.
.
.uθuNN F(uNN )
Impose BC De ine loss
T ainable
hidden laye
Figu e 4.1: Ske ch o he a qui ec u e o uNN .
4.4 Quad a u e Rules
We app oxima e ou in eg als om Eqs. (4.6) and (4.23) using a quad a u e ule
o he o m
żb
a
pxqdx «
n
ÿ
i“0
ωi pxiq,(4.26)
whe e ωia e he weigh s and xia e he quad a u e poin s. Examples o quad a-
u e ules ha ollow he abo e o mula include apezoidal ule and Gaussian
quad a u e ules [90]. We classi y hese quad a u e ules in o wo g oups: (1)
hose ha only employ poin s om he in e io o he in e al; and (2) hose ha
e alua e he solu ion a one ex eme poin o mo e (a o b). In eg a ion ules
wi hin he la e g oup (e.g., he apezoidal ule) a e inadequa e o ou mini-
miza ion p oblems because he in eg and can be in ini e a he bounda y poin s
in he case o singula solu ions (e.g. model p oblem 1). Thus, we ocus on
77
4 Quad a u e Rules when Sol ing PDEs using Deep Lea ning
quad a u e ules ha only e alua e he solu ion a in e io poin s o he domain,
wi h a special ocus on Gaussian quad a u e ules.
4.4.1 Illus a ion o Quad a u e P oblems in Neu al Ne wo ks
4.4.1.1 Ri z Me hod
We conside he wo model p oblems om Sec ion 4.1. We app oxima e upxq
using he Ri z me hod. Thus, we sea ch o a NN ha minimizes he loss unc-
ional gi en by Eq. (4.6). Ou NN has one hidden laye wi h 10 neu ons (31
ainable weigh s). We use au oma ic di e en ia ion o compu e he de i a i es
and a h ee-poin Gaussian quad a u e ule o app oxima e he in eg als wi hin
each elemen . We selec he S ochas ic G adien Descen (SGD) op imize . Fo
model p oblem 1, we disc e ize ou domain wi h ou equal-size elemen s and
execu e 40,000 i e a ions du ing he op imiza ion p ocess. Fo model p oblem
2, we disc e ize ou domain wi h en equal-size elemen s and execu e 200,000
i e a ions du ing he op imiza ion p ocess.
Figu es 4.2a and 4.2b desc ibe he loss e olu ion o he aining p ocess. We
ob ain a lowe loss han he op imum loss compu ed analy ically using he exac
solu ion (i.e., FRpuexac q). This has o be due o some nume ical e o , in his
case, quad a u e e o s.
100101102103104
8
−1.54
−15
Epoch
Loss
FR(uexac )
FR(uNN )
(a) Model p oblem 1.
100101102103104105
0
−666.67
−800
Epoch
Loss
FR(uexac )
FR(uNN )
(b) Model p oblem 2.
Figu e 4.2: Loss e olu ion o he aining p ocess o ou wo model p oblems.
Figu es 4.3 and 4.4 compa e he app oxima e and exac solu ions. We obse e
a disas ous NN app oxima ion due o quad a u e e o s. Figu es 4.3b and 4.4b
show ha he g adien is (almos ) ze o a he aining (quad a u e) poin s o
he i s in e al. The e o e, his alue minimizes he nume ical app oxima ion o
78
4 Quad a u e Rules when Sol ing PDEs using Deep Lea ning
pu1, u1q. This beha io allows he app oxima ed solu ion o each la ge alues in
he i s in e al, and consequen ly maximizing he e m p , uq, and minimizing
he o al loss:
FRpuNN q “ 1
2pu1
NN , u1
NN q
looooomooooon
řqiωipu1
NN q2«0
´p , uNN q
looomooon
«8
´pg, uNN qΓN» ´8
The desc ibed quad a u e e o s can be in e p e ed as o e i ing o e he
de i a i e o he solu ion.
0 2 4 6 8 10
0
20
40
x
u
uNN
uexac
(a) Exac and app oxima e solu-
ions.
012
20
30
40
50
Gauss
poin s
x
u
(b) App oxima e solu ion in he
in e al 0,2.5s. The Gauss
quad a u e poin s co espond-
ing o he i s elemen a e in-
dica ed in blue.
Figu e 4.3: Exac s app oxima e Ri z me hod solu ions o model p oblem 1 using
ou elemen s o e alua ing FRp qand a NN wi h 31 weigh s.
4.4.1.2 Leas Squa es Me hod
We now conside he ollowing one-dimensional p oblem:
"´u2pxq “ 0xP p0,1q,
up0q “ u1p1q “ 0,(4.27)
whe e he exac solu ion is upxq “ 0. We can easily cons uc an app oxima ing
unc ion uNN ha sa is ies Eq. (4.27) a he h ee conside ed Gaussian poin s
and minimizes Eq. (4.23), while s ill being a poo app oxima ion o he exac
solu ion due o quad a u e e o s. Figu e 4.5 shows an example.
79
4 Quad a u e Rules when Sol ing PDEs using Deep Lea ning
0 2 4 6 8 10
0
50
100
150
x
u
uNN
uexac
(a) Exac and app oxima e solu-
ions.
0 0.2 0.4 0.6 0.8 1
−20
0
20
40
Gauss
poin s
x
u
uNN
uexac
(b) App oxima e solu ion in he in-
e al 0,1s. The Gauss quad a-
u e poin s co esponding o he
i s elemen a e indica ed in
blue.
Figu e 4.4: Exac s app oxima e Ri z me hod solu ions o model p oblem 2 using
en elemen s o e alua ion o FRp qand a NN wi h 31 weigh s.
0 0.25 0.5 0.75 1
0
a
uNN
uexac
Gauss
poin s
x
u
Figu e 4.5: Exac (uexac “0) and app oxima ed solu ion o a p oblem gi en by
Eq. (4.27) and sol ed wi h he Leas Squa e (LS) me hod.
80

4 Quad a u e Rules when Sol ing PDEs using Deep Lea ning
4.5 In eg al App oxima ion
We now desc ibe ou di e en me hods o imp o e he in eg al app oxima ions.
4.5.1 Mon e Ca lo In eg a ion
We conside he ollowing Mon e Ca lo in eg al app oxima ion o e a se o poin s
xiP pa, bq,
żb
a
pxqdx «pb´aq
n
n
ÿ
i“1
pxiq, xiP pa, bq @i“ 1,¨¨¨ , nu(4.28)
In he abo e, poin s xia e andomly selec ed [1]. While his me hod is use ul o
high-dimensional in eg als, o low dimensions (1D, 2D, 3D) he compu a ional
cos is high since he alue o he in eg al app oxima ion con e ges as 1{?n[150].
4.5.2 Piecewise-polynomial App oxima ion
We eplace he o iginal NN uNN by a piecewise-polynomial app oxima ion u˚
NN,¨,
whe e ¨ ep esen s he numbe o pieces o ou piecewise-linea in e pola o . This
app oxima ion can be exac ly di e en ia ed (e.q., ia FDM) and in eg a ed ( ia
a Gaussian quad a u e ule). Figu e 4.6 shows an example when we ain a NN
and we build a piecewise-linea app oxima ion o he NN wi h ou elemen s.
This me hod con ols quad a u e e o s. Howe e , i is inadequa e o high-
dimensional p oblems as we need a mesh ha is di icul o implemen and in e-
g a ion becomes ime consuming.
4.5.3 Adap i e In eg a ion
We i s conside a aining da ase o e he in e al pa, bqby aking an equidis-
an pa i ion o nelemen s. Then, we de ine he alida ion se as a global
h- e inemen o he aining da ase . Figu e 4.7 shows an example o a aining
and he co esponding alida ion da ase s. Then, o each elemen o he ain-
ing mesh (e.g., E1in Figu e 4.7), we compa e he nume ical in eg al o e ha
elemen s he sum o he in eg als o e he wo co esponding elemen s on he
alida ion da ase (in ou case, E1
1`E2
1). I he in eg al alues di e by mo e
han a s ipula ed ole ance, we h- e ine he aining elemen , and we upg ade he
alida ion da ase so i is buil as a global h- e inemen o he aining da ase .
This p ocess is desc ibed in Algo i hm 1. Figu e 4.7 also shows a aining and
he co esponding alida ion da ase a e e ining he i s and hi d elemen s.
We a e able o con ol he quad a u e e o s by adding new quad a u e poin s
o he aining da ase . Howe e , he simples way o implemen such me hod is
81
4 Quad a u e Rules when Sol ing PDEs using Deep Lea ning
012345678910
0
2
4
6
x
u
uNN
uexac
u∗
NN,4
Figu e 4.6: Neu al Ne wo k app oxima ion uNN and i s piecewise-linea elemen
app oxima ion u˚
NN,4.
Algo i hm 1: Adap i e in eg a ion me hod
Gene a e a aining da ase ;
Gene a e he co esponding alida ion da ase ;
Se ole ance and maximum i e a ion numbe imax;
while iăimax do
o j“1,¨¨¨ , n do
Compu e in eg al alues Ijo e he aining da ase o elemen s
Ej;
Compu e in eg al alues I1
j,I2
jo e he alida ion da ase o
elemen s E1
j,E2
j;
i |pI1
j`I2
jq´Ij| ą  hen
h- e ine he Ej- h elemen o he aining se ;
h- e ine he E1
j- h and E2
j- h elemen s o he alida ion se ;
else
con inue;
end
end
i“i`1;
end
82
4 Quad a u e Rules when Sol ing PDEs using Deep Lea ning
h
•••••
ab
E1E2E3E4
(a) T aining se
h/2
•••••••••
ab
E1
1E2
1E1
3E2
3
(b) Valida ion se
Figu e 4.7: Black poin s (do s) co espond o he o iginal (a) aining/(b) alida-
ion pa i ions and blue poin s (ci cles) a e he poin s added by he
e inemen pe o med in he i s and hi d elemen s.
by using meshes, which posses a limi a ion on high-dimensional in eg als. As an
al e na i e o gene a ing a mesh, one can andomly add poin s o he aining
se . This en ails di icul ies when designing an adap i e algo i hm.
In he same way ha we p opose an h-adap i e me hod, we can also wo k wi h
p-adap i i y [14] o a combina ion o hem (e.g., hp-adap i i y [38]).
4.5.4 Regula iza ion Me hods
We now in oduce a p oblem-speci ic egula ize designed o con ol he quad a-
u e e o .
In a one-dimensional se ing, we conside he in eg al unc ional FRas gi en
in (4.6), and i s app oxima ion ia a midpoin ule, ˆ
FR, gi en by
ˆ
FRpuq “b´a
N
N
ÿ
j“1ˆ1
2|u1pxjq|2´ pxjqupxjq˙`gpbqupbq`gpaqupaq.(4.29)
We no e ha g“0, excep whe e he Neumann condi ion is imposed. While we
ocus on he Ri z me hod, a simila heu is ic can be applied o he LS me hod.
We in oduce a unc ion R ha depends on he lea nable pa ame e s θo
a gi en Neu al Ne wo k uNN , such ha o any Neu al Ne wo k wi h a gi en
a chi ec u e,
|FRpuNN q´ ˆ
FRpuNN q| ă Rpθq.(4.30)
I we hen conside a loss unc ion Lgi en by
Lpθq “ ˆ
FRpuNN q`Rpθq,(4.31)
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4 Quad a u e Rules when Sol ing PDEs using Deep Lea ning
hen we may be able o imp o e he app oxima ion o he quad a u e ule, as he
loss con ains a e m ha by design con ols he quad a u e e o .
Fo simplici y, we conside only he case o a single-laye ne wo k wi h a one-
dimensional inpu , and he midpoin ule o calcula ing he in eg al o e a uni-
o m pa i ion o pa, bq. We conside a mid-poin ule as in (4.29), and de ine he
in e al leng h δ“b´a
Nand in e als Ij“`δ
2`xj, xj`δ
2˘. We es ima e he e o
o he midpoin ule o in eg a e Fas
ˇˇˇˇˇżb
a
Fpxqdx ´
N
ÿ
j“1
Fpxjqδˇˇˇˇˇ“ˇˇˇˇˇ
N
ÿ
i“1żxj`δ
2
xj´δ
2pFpxq´Fpxjqqdxˇˇˇˇˇ
ď
N
ÿ
i“1żxj`δ
2
xj´δ
2|Fpxq´Fpxjq|dx
ď
N
ÿ
i“1
max
PIj|F1p q|żxj`δ
2
xj´δ
2|x´xj|dx
“δ2
4
N
ÿ
i“1
max
PIj|F1p q|.
(4.32)
This es ima e scales as Op1
Nq o ixed F, and hus, o a la ge numbe o
in eg a ion poin s, we expec he es ima e o be su icien ly accu a e and o a oid
“o e damping” o he loss.
Wi h (4.32) in mind, we es ima e he local Lipschi z cons an s o he in eg and
as in (4.6). The nume ical es ima ion o he Lipschi z cons an s o NNs has
a ac ed a en ion, as hey o m a way o es ima ing he gene alizabili y o a
Neu al Ne wo k, and ha e been used in he aining p ocess as a way o encou age
accu a e gene aliza ion [44, 51, 125]. As we a e dealing wi h loss unc ions ha
in ol e de i a i es o he Neu al Ne wo k, we howe e need es ima es o highe
o de de i a i es o uNN . The app oach ha we employ is simila in spi i o he
wo k o [88] o ob aining apos e io i e o es ima es in PINNs.
Despi e he a i hme ic complica ions in ol ed in calcula ing R, concep ually
he idea educes o an applica ion o Taylo ’s heo em. On a single in e al o
in eg a ion Ij, we ha e ha o e e y x, he e exis s some ξxso ha
|F1pxq| “ |F1pxiq`px´xiqF2pξxq| ď |F1pxiq|` δ
2||F2||8.(4.33)
We hen ind Rusing a combina ion o local and global es ima es o he
de i a i es o he in eg and co esponding o simple poin wise e alua ions a he
in eg a ion poin s and global es ima es in ol ing he Neu al Ne wo k weigh s.
The necessa y s eps a e:
84
4 Quad a u e Rules when Sol ing PDEs using Deep Lea ning
0 2 4 6 8 10
0
2
4
x
u
uNN
uexac
(a) Model p oblem 1.
0 2 4 6 8 10
0
50
100
x
u
uNN
uexac
(b) Model p oblem 2.
Figu e 4.11: Ri z me hod solu ion when using adap i e in eg a ion.
4.6.3 Regula iza ion Me hods
We do no apply he egula iza ion me hod o model p oblem 1 as he me hod
equi es su icien egula i y in o de o p o ide he necessa y es ima es in he
calcula ion o R. Since he solu ion is singula a x“0, he necessa y Lipschi z
bounds on he in eg al unc ional canno be ob ained wi hin his amewo k.
Ins ead, we aim o demons a e ha o p oblems ha a e su icien ly egula ,
ou echnique can a oid o e i ing, and lea e open he ques ion as o how one
may adap he echnique o singula p oblems o u u e wo k. We hus conside
model p oblem 2. We p opose he loss de ined ia
Lpθq “ ˆ
FRpuNN q`Rpθq.(4.51)
Explici ly,
ˆ
FRpuNN q “ 10
N
N
ÿ
j“1
1
2|u1
NN pxjq|2´2uNN pxjq´20uNN p20q,(4.52)
whe e xj“10
N`i´1
2˘.
4.6.3.1 Expe imen 1
We conside N“50 poin s, and a single laye ne wo k wi h M“10 neu ons.
We use he Adam op imize wi h lea ning a e 10´2. We sol e model p oblem 2
wi h wo losses: wi h and wi hou egula iza ion. In bo h cases, we measu e he
me ics L,R, and ˆ
FR. Fo alida ion, we use an equidis an pa i ion o p0,10q
91

4 Quad a u e Rules when Sol ing PDEs using Deep Lea ning
wi h 49 poin s, so ha we s ill use a midpoin ule bu wi h di e en in eg a ion
poin s.
0 2 4 6 8 10
0
50
100
x
u
uNN
uexac
(a) Exac and app oxima ed solu ions.
100101102103104
0
1
2
3
·108
Epoch
Loss
Loss L
(b) E olu ion o Ldu ing aining.
100101102103104
−600
−400
−200
0
Epoch
Loss
Loss
alida ion
Loss aining
(c) E olu ion o ˆ
FRdu ing aining.
100101102103104
0
1
2
3
·108
Epoch
Loss
Loss R
(d) E olu ion o Rdu ing aining.
Figu e 4.12: The solu ion and aining in o ma ion o Expe imen 1 wi hou eg-
ula iza ion.
Figu e 4.12 shows he esul s wi hou egula iza ion. As expec ed, we see in
Figu e 4.12a ha he app oxima ion is poo due o o e i ing, which is mos
no able a ound x“0 and a ained wi hin 5000 epochs. Via he p o ided plo s
we can obse e he beginning o o e i ing in wo dis inc manne s. Fi s , we
obse e in Figu e 4.12c ha he alue o ˆ
FRe alua ed o e he alida ion da a
begins o di e ge om he alue on he aining da a, becoming appa en a
a ound 1000 epochs. We also see his beha iou e lec ed in he e olu ion o
Rin Figu e 4.12d, wi h i s mos d ama ic inc ease beginning a ound he same
92
4 Quad a u e Rules when Sol ing PDEs using Deep Lea ning
0 2 4 6 8 10
0
50
100
x
u
uNN
uexac
(a) Exac and app oxima ed solu ions.
100101102103104
−600
−400
−200
0
Epoch
Loss
Loss alida ion
Loss aining
(b) E olu ion o Ldu ing aining.
100101102103104
−600
−400
−200
0
Epoch
Loss
Loss ˆ
FR
(c) E olu ion o ˆ
FRdu ing aining.
100101102103104
0
10
20
Epoch
Loss
Loss R
(d) E olu ion o Rdu ing aining.
Figu e 4.13: The solu ion and aining in o ma ion o Expe imen 1 wi h egu-
la iza ion.
i e a ion. This apid inc ease also p o okes an inc ease in L, as seen in Figu e
4.12b. This also indica es ha e en i Ris no used as pa o he aining
p ocess, i s inc ease could be used as a me ic o iden i y o e i ing.
Figu e 4.13 desc ibes he esul s wi h egula iza ion and we obse e a di e en
beha iou . The app oxima ion is gene ally good, and we do no see any signs
o o e i ing wi hin 105epochs, as shown in Figu e 4.13a. In pa icula , he
alues o ˆ
FRa he aining and alida ion da a emain consis en in Figu e
4.13c. Th oughou Figu e 4.13 we see ha wi hin 105epochs all me ics appea
o ha e con e ged o a limi ing alue. We ob ain inal alues L« ´644.22,
ˆ
FR« ´666.07, R«24.8. We ecall ha he ue ene gy o he exac solu ion
is FRpuexac q«´666.667, which sugges s he quad a u e ule is accu a e. No ice
ha in he case wi hou egula iza ion, be o e o e i ing became appa en , R
93
4 Quad a u e Rules when Sol ing PDEs using Deep Lea ning
had al eady a ained alues o a ound 1000, which is a la ge han he alue o
Ra he ob ained solu ion when egula iza ion was used.
4.6.3.2 Expe imen 2
We now conside a smalle N. As we expec R o scale as 1
N, we an icipa e
a mo e ad e se e ec when Nis small. To iew his, we conside he same
p oblem o Expe imen 1, whe e we now selec N“20 in eg a ion poin s. We
conside M“10 neu ons and minimize ou p oblem using he Adam op imize
wi h a lea ning a e o 10´2. As be o e, we conside he cases wi h and wi hou
egula iza ion.
0 2 4 6 8 10
0
50
100
x
u
uNN
uexac
(a) Exac and app oxima ed solu ions.
100101102103
0
0.5
1
1.5
·107
Epoch
Loss
Loss L
(b) E olu ion o Ldu ing aining.
100101102103
−800
−600
−400
−200
0
Epoch
Loss
Loss
alida ion
Loss aining
(c) E olu ion o ˆ
FRdu ing aining.
100101102103
0
0.5
1
1.5
·107
Epoch
Loss
Loss R
(d) E olu ion o Rdu ing aining.
Figu e 4.14: The solu ion and aining in o ma ion o Expe imen 2 wi hou eg-
ula iza ion.
94
4 Quad a u e Rules when Sol ing PDEs using Deep Lea ning
0 2 4 6 8 10
0
50
100
x
u
uNN
uexac
(a) Exac and app oxima ed solu ions.
100101102103104
−400
−200
0
Epoch
Loss
Loss
alida ion
Loss aining
(b) E olu ion o Ldu ing aining.
100101102103104
−600
−400
−200
0
Epoch
Loss
Loss ˆ
FR
(c) E olu ion o ˆ
FRdu ing aining.
100101102103104
0
50
100
150
Epoch
Loss
Loss R
(d) E olu ion o Rdu ing aining.
Figu e 4.15: The solu ion and aining in o ma ion o Expe imen 2 wi h egu-
la iza ion.
Figu e 4.14 p esen s he loss e olu ion wi hou egula iza ion. We obse e
o e i ing, which is accompanied by di e gence o he loss on he alida ion
da ase , as well as a apid inc ease in R, wi h hese ea u es isible wi hin 5000
epochs.
Figu e 4.15 p esen s he esul s wi h egula iza ion. We obse e no signs o
o e i ing, wi h he alida ion and aining loss emaining close in Figu e 4.15b.
All me ics appea o ha e con e ged o a limi ing alue wi hin 104epochs. How-
e e , he la ge alue o Ra he ound solu ion (app oxima ely 140) has sub-
s an ially changed he op imiza ion p oblem so ha he ob ained minimize is
a om he desi ed solu ion. The inal alue o ˆ
FRis a ound ´622, which is a
om he desi ed alue o ´666.67. This expe imen highligh s he ac ha he
egula ize becomes mo e e ec i e when a la ge numbe o in eg a ion poin s a e
95
4 Quad a u e Rules when Sol ing PDEs using Deep Lea ning
used.
96

5 Conclusions and Fu u e Wo k
5.1 Conclusions
In his disse a ion, we i s ocus on he use o Deep Neu al Ne wo ks (DNNs)
o he in e sion o bo ehole esis i i y measu emen s o geos ee ing applica ions.
We analyze he s ong impac ha di e en loss unc ions ha e on he p edic ion
esul s. Fo his, we illus a e ia a simple benchma k example ha a adi ional
da a mis i loss unc ion deli e s poo esul s. As a emedy, we p opose he use o
an Encode -Decode based o a wo-s ep based loss unc ion. These app oaches
gene a e wo DNN app oxima ions: one o he o wa d unc ion and ano he
one o he in e se ope a o . Then, we apply hese wo loss unc ions in a ield
example wi h syn he ic da a, and we ob ain adequa e esul s.
To gua an ee ha he in e se DNN app oxima ion p o ides meaning ul esul s,
we need o ensu e ha he aining da ase con ains su icien samples. O he wise,
bo h o wa d and in e se DNN ope a o s may p o ide inco ec solu ions while
s ill ensu ing he composi ion o bo h ope a o s is close o he iden i y. Thus,
he app oach is highly dependen on he exis ence o a su icien ly ich aining
da ase , which acili a es he lea ning p ocess o he DNNs.
To ensu e ha he in e se DNN app oxima ion deli e s signi ican esul s, we
ind i highly bene icial o add a egula iza ion e m o he loss unc ion based
on he exis ing aining da ase . This educes he ichness we need o gua an-
ee wi hin he aining da ase s. Ne e heless, such egula iza ion e ms may
hide al e na i e easible solu ions o he in e se ope a o , which may p o ide
o e con idence in he esul s. Ano he possibili y is o conside a wo-s ep based
loss unc ion. Using his app oach, we ha e shown ha he in e se p oblem
conside ed in his wo k admi s di e en solu ions ha a e physically
easible, a ac ha was obscu ed when using he egula iza ion e m.
O he c i ical limi a ions o DNNs we encoun e in his wo k a e: (a) he lim-
i ed app oxima ion capabili ies o DNNs o ep oduce discon inuous unc ions,
(b) he need o a new da ase and ained DNN o each subsu ace pa ame iza-
ion, and (c) he poo esul s hey exhibi when hey a e e alua ed o e a sample
ha is ou side he aining da ase space. Mo e impo an ly, i is o en di icul
o iden i y he sou ce o poo esul s, which may include inadequa e selec ions
o : (i) loss unc ion, (ii) DNN a chi ec u e, (iii) egula iza ion e m, (i ) ain-
97
5 Conclusions and Fu u e Wo k
ing da ase , ( ) op imiza ion algo i hm, ( i) escaling ope a o and no ms, ( ii)
model pa ame e iza ion, ( iii) app oxima ion capabili ies o DNNs, o simply (ix)
he na u e o he p oblem due o a lack o adequa e measu emen s.
To deal wi h he a o emen ioned sou ces o e o s, we p opose a ca e ul
s ep-by-s ep e o con ol based on: (a) selec ing adequa e no ms, (b) p ope
escaling o he a iables, (c) selec ing a well sui ed loss unc ion possibly wi h a
egula iza ion e m, (d) analyzing he e olu ion o he di e en e ms o he loss
unc ion, (e) s udying mul iple c oss-plo s o di e en na u e, and ( ) pe o ming
an in-dep h assessmen o he esul s o e mul iple ealis ic es examples.
We also show i is possible o ob ain a good-quali y in e sion o geos ee ing
measu emen s wi h limi ed online compu a ional cos , hus, sui able o eal- ime
in e sion. Mo eo e , he quali y o he in e sion esul s can be apidly e alua ed
o de ec i s possible inaccu acies in he ield and selec al e na i e in e sion
me hods when needed.
As men ioned be o e, he DNN app oxima ion o he in e se ope a o is highly
dependen on he exis ence o a su icien ly ich aining da ase . In he case
o 1D laye ed o ma ions, i is o en easible o p oduce he equi ed da ase .
Howe e , o mo e complica ed cases, o example, in 2D and 3D geome ies, a
di ec ex ension may be limi ed due o he la ge numbe o in e sion a iables
and he ex emely ime-consuming p ocess o p oducing an exhaus i e da ase .
Such a la ge da abase is essen ial o laye -by-laye es ima ion o he in e ed
Ea h models, which may be used o eal- ime adjus men s o he well ajec o y
du ing geos ee ing ope a ions.
In he second pa o his disse a ion, we p opose he use o e ined Isoge-
ome ic Analysis ( IGA) disc e iza ions o gene a ing a massi e syn-
he ic da abase o Deep Lea ning (DL) in e sion o 2.5D bo ehole
elec omagne ic (EM) measu emen s. IGA deli e s compu a ional sa ings
o up o Oppqcompa ed o he high-con inui y Isogeome ic Analysis (IGA).
When compa ed o a adi ional Fini e Elemen Me hod (FEM) wi h he same
mesh size and polynomial deg ee, IGA p o ides highe imp o emen ac o s.
A he same ime, IGA p o ides su icien ly accu a e solu ions o geos ee ing
pu poses.
To c ea e a da ase o DL in e sion, we i s selec ed ce ain disc e iza ion pa-
ame e s based on he esul s o se e al homogeneous solu ions. Then, we checked
he accu acy o e homogeneous and he e ogeneous media. Finally, we gene a ed
a syn he ic da abase composed o 100,000 Ea h models wi h he co esponding
measu emen s in abou 56 hou s using a wo ks a ion equipped wi h wo CPUs.
In he las pa o his wo k, we ocus on he use o Neu al Ne wo ks (NNs)
o sol ing a Pa ial Di e en ial Equa ion (PDE). We i s illus a e ia wo
simple examples how quad a u e e o s can des oy he quali y o he
98
5 Conclusions and Fu u e Wo k
app oxima ed solu ion when sol ing PDEs using DL me hods. Fo his,
we sol e wo simple 1D p oblems based on Poisson’s equa ion using he Deep Ri z
Me hod (DRM) and a h ee-poin Gaussian quad a u e ule. Then, we p opose
ou di e en al e na i es o o e come he quad a u e p oblems, discuss
hei ad an ages and limi a ions, and illus a e hei pe o mance.
In high dimensions, Mon e Ca lo in eg a ion me hods a e he bes choice. Reg-
ula ize me hods a e ano he op ion, bu hey a e p oblem dependen and hey
need o be de i ed o each di e en a chi ec u e. Mo eo e , hey equi e u he
analysis o highly nonlinea in eg ands. Fu he mo e, hey a e limi ed only o
su icien ly smoo h in eg al unc ionals. In addi ion, mo e complex NN a chi ec-
u es (which should be needed in highe dimensions) will hinde he de i a ion o
R.
In low dimensions ( h ee o below), Mon e Ca lo in eg a ion is no compe i i e
because o i s low con e gence speed. In hese cases, adap i e in eg a ion exhibi s
as e con e gence. In he cases o piecewise-linea app oxima ion and egula iz-
e s, we a e also able o o e come he quad a u e p oblems, bu he con e gence
speed is o en slowe and he accu acy is lowe han wi h adap i e in eg a ion.
5.2 Fu u e Wo k
The e a e se e al possible u u e esea ch lines ega ding his wo k. The i s
one is o conside mo e complex Ea h models, possibly con aining geological
aul s o o he ele an subsu ace ea u es, and analyze he pe o mance o he
Encode -Decode and wo-s ep based loss unc ions.
Ano he line o esea ch consis s o educing he da ase size equi ed o sol ing
in e se bo ehole p oblems. Thus, dec easing he compu a ional cos o c ea ing
he da ase . Fo his, we use using Ac i e Lea ning echniques. Ano he op ion
is o use T ans e Lea ning echniques o highe spa ial dimensions, which can
also alle ia e da a equi emen s o ain he co esponding DNN.
Conce ning Chap e 4, one possible u u e wo k is o implemen adap i e in-
eg a ion o 2D and 3D p oblems. In he same way, he piecewise-polynomial
app oxima ion could be imp o ed by implemen ing -adap i i y o op imize he
g id.
Ul ima ely, we aim a sol ing pa ame ic PDEs using NNs. By his, we will be
able o sol e he o wa d p oblem co esponding o di e en Ea h models using
NNs. Thus, we will be able o p ope ly and e icien ly c ea e he da ase needed
o ain ou DNN ha app oxima es he solu ion o in e se bo ehole p oblems.
99
6 Main Achie emen s
6.1 Scien i ic Achie emen s
In he i s pa o his disse a ion, we in es iga e app op ia e loss unc ions o
ain a Deep Neu al Ne wo k (DNN) when dealing wi h an in e se p oblem.
In he second pa o his wo k, we p opose he use o e ined Isogeome ic
Analysis ( IGA) disc e iza ions o gene a e da abases o DL in e sion o 2.5D
geos ee ing elec omagne ic (EM) measu emen s.
In he hi d pa o his wo k, we analyze he p oblems associa ed wi h quad a-
u e ules in Deep Lea ning (DL) me hods when sol ing Pa ial Di e en ial Equa-
ions (PDEs), and we p opose se e al al e na i es o o e come quad a u e p ob-
lems.
6.2 Pee - e iewed Publica ions
6.2.1 Jou nals
2022 J. A. Ri e a, J. M. Taylo , ´
A. J. Omella and D. Pa do. On quad a-
u e ules o sol ing Pa ial Di e en ial Equa ions using Neu-
al Ne wo ks. Compu e Me hods in Applied Mechanics and Enginee ing,
2022, ol. 393, p. 114710.
h ps://doi.o g/10.1016/j.cma.2022.114710
2021 A. Hashemian, D. Ga cia, J. A. Ri e a and D. Pa do. Massi e da abase
gene a ion o 2.5 D bo ehole elec omagne ic measu emen s us-
ing e ined isogeome ic analysis. Compu e s & Geosciences, 2021,
ol. 155, p. 104808.
h ps://doi.o g/10.1016/j.cageo.2021.104808
2021 M. Shah ia i, D. Pa do, J. A. Ri e a, C. To es-Ve d´ın, A. Picon, J. Del
Se , S. Ossand´on and V. M. Calo. E o con ol and loss unc ions o
he deep lea ning in e sion o bo ehole esis i i y measu emen s.
In e na ional Jou nal o Nume ical Me hods in Enginee ing, 2021, ol.
122(6), p. 1629-1657.
h ps://doi.o g/10.1002/nme.6593
100
BIBLIOGRAPHY
[40] R. Desb andes and R. Clay on. Chap e 9 measu emen while d illing.
De elopmen s in Pe oleum Science, 38:251 – 279, 1994. (ci ed in page(s)
2)
[41] C. Dupuis and J. M. Denichou. Au oma ic in e sion o deep-di ec ional-
esis i i y measu emen s o well placemen and ese oi desc ip ion. The
Leading Edge, 34(5):504–512, 2015. (ci ed in page(s) 2)
[42] W. Ee, J. Han, and A. Jen zen. Deep Lea ning-Based Nume ical Me hods
o High-Dimensional Pa abolic Pa ial Di e en ial Equa ions and Back-
wa d S ochas ic Di e en ial Equa ions. To appea in Communica ions in
Ma hema ics and S a is ics, 5, 06 2017. (ci ed in page(s) 5)
[43] A. Es e a, A. Robicque , B. Ramsunda , V. Kulesho , M. DeP is o,
K. Chou, C. Cui, G. Co ado, S. Th un, and J. Dean. A guide o deep
lea ning in heal hca e. Na u e Medicine, 25:24–29, 2019. (ci ed in page(s)
3)
[44] M. Fazlyab, A. Robey, H. Hassani, M. Mo a i, and G. Pappas. E icien
and accu a e es ima ion o Lipschi z cons an s o deep neu al ne wo ks.
Ad ances in Neu al In o ma ion P ocessing Sys ems, 32:11427–11438, 2019.
(ci ed in page(s) 84)
[45] D. Fleisch. A s uden ’s guide o Maxwell’s equa ions. Camb idge Uni e si y
P ess, 2008. (ci ed in page(s) 4)
[46] D. Ga cia, D. Pa do, and V. M. Calo. Re ined isogeome ic analysis o luid
mechanics and elec omagne ics. Compu e Me hods in Applied Mechanics
and Enginee ing, 356:598–628, 2019. (ci ed in page(s) 5, 55, 56, 57, 61)
[47] D. Ga cia, D. Pa do, L. Dalcin, M. Paszy´nski, N. Collie , and V. M.
Calo. The alue o con inui y: Re ined isogeome ic analysis and as di-
ec sol e s. Compu e Me hods in Applied Mechanics and Enginee ing,
316:586–605, 2017. (ci ed in page(s) 4, 56, 57, 60, 61, 62)
[48] S. Ge nez, A. Bouchedda, E. Gloaguen, and D. Pa adis. Aim4 es, an open-
sou ce 2.5D ini e di e ences MATLAB lib a y o aniso opic elec ical
esis i i y modeling. Compu e s & Geosciences, 135:104401, Feb. 2020.
(ci ed in page(s) 4, 49)
[49] M. Ghasemi, Y. Yang, E. Gildin, Y. E endie , and V. M. Calo. Fas mul-
iscale ese oi simula ions using pod-deim model educ ion. Socie y o
Pe oleum Enginee s, pages 1–18, 2015. (ci ed in page(s) 2)
107

BIBLIOGRAPHY
[50] S. Goswami, C. Ani escu, S. Chak abo y, and T. Rabczuk. T ans e lea n-
ing enhanced physics in o med neu al ne wo k o phase- ield modeling o
ac u e. Theo e ical and Applied F ac u e Mechanics, 106:102447, 2020.
(ci ed in page(s) 71)
[51] H. Gouk, E. F ank, B. P ah inge , and M. J. C ee. Regula isa ion o neu al
ne wo ks by en o cing lipschi z con inui y. Machine Lea ning, 110(2):393–
416, 2021. (ci ed in page(s) 84)
[52] A. G¨une¸s Baydin, B. A. Pea lmu e , A. And eye ich Radul, and J. Ma k
Siskind. Au oma ic di e en ia ion in machine lea ning: A su ey. Jou nal
o Machine Lea ning Resea ch, 18:1–43, 2018. (ci ed in page(s) 6)
[53] J. Gunning and M. E. Glinsky. De ec ion o ese oi quali y using bayesian
seismic in e sion. Geophysics, 72(3):R37–R49, 2007. (ci ed in page(s) 3)
[54] A. Gup a, A. Anpalagan, L. Guan, and A. S. Khwaja. Deep lea ning o ob-
jec de ec ion and scene pe cep ion in sel -d i ing ca s: Su ey, challenges,
and open issues. A ay, 10:100057, 2021. (ci ed in page(s) 3)
[55] J. Hadama d. Lec u es on Cauchy’s p oblem in linea pa ial di e en ial
equa ions. Yale Uni e si y P ess, 1923. (ci ed in page(s) 7)
[56] T. Hageman, K. M. P. Fa hima, and R. de Bo s . Isogeome ic analysis o
ac u e p opaga ion in sa u a ed po ous media due o a p essu ised non-
New onian luid. Compu e s and Geo echnics, 112:272–283, Aug. 2019.
(ci ed in page(s) 4)
[57] J. Han, A. Jen zen, and W. Ee. Sol ing high-dimensional pa ial di e en ial
equa ions using deep lea ning. P oceedings o he Na ional Academy o
Sciences, 115, 07 2017. (ci ed in page(s) 5)
[58] J. R. Hause . Pa ial Di e en ial Equa ions: The Fini e Elemen Me hod.
Nume ical Me hods o Nonlinea Enginee ing Models. Sp inge Ne he -
lands, 2009. (ci ed in page(s) 4, 5)
[59] K. He, X. Zhang, S. Ren, and J. Sun. Deep esidual lea ning o image
ecogni ion. a Xi :1512.03385, 2015. (ci ed in page(s) 3, 16)
[60] C. F. Higham and D. J. Higham. Deep lea ning: An in oduc ion o
applied ma hema icians. Compu ing Resea ch Reposi o y, abs/1801.05894,
2018. (ci ed in page(s) 4, 16, 17)
108
BIBLIOGRAPHY
[61] T. J. R. Hughes, J. A. Co ell, and Y. Bazile s. Isogeome ic analysis:
CAD, ini e elemen s, NURBS, exac geome y and mesh e inemen . Com-
pu e Me hods in Applied Mechanics and Enginee ing, 194(39-41):4135–
4195, 2005. (ci ed in page(s) 4)
[62] C. Hu ´e, H. Pham, and X. Wa in. Some machine lea ning schemes o high-
dimensional nonlinea PDEs. Ma h. Compu ., 89:1547–1579, 2020. (ci ed
in page(s) 5)
[63] S. ichi Ama i. Backp opaga ion and s ochas ic g adien descen me hod.
Neu ocompu ing, 5(4):185–196, 1993. (ci ed in page(s) 6)
[64] O. Ijasana, C. To es-Ve d´ın, and W. E. P eeg. In e sion-based pe ophys-
ical in e p e a ion o logging-while-d illing nuclea and esis i i y measu e-
men s. Geophysics, 78 (6):D473–D489, 2013. (ci ed in page(s) 2, 10)
[65] A. D. Jag ap, E. Kha azmi, and G. E. Ka niadakis. Conse a i e physics-
in o med neu al ne wo ks on disc e e domains o conse a ion laws: Ap-
plica ions o o wa d and in e se p oblems. Compu e Me hods in Applied
Mechanics and Enginee ing, 365:113028, 2020. (ci ed in page(s) 71)
[66] B. Jan, H. Fa man, M. Khan, M. Im an, I. U. Islam, A. Ahmad, S. Ali,
and G. Jeon. Deep lea ning in big da a analy ics: A compa a i e s udy.
Compu e s & Elec ical Enginee ing, 75:275 – 287, 2019. (ci ed in page(s)
3)
[67] K. H. Jin, M. T. McCann, E. F ous ey, and M. Unse . Deep con olu ional
neu al ne wo k o in e se p oblems in imaging. IEEE T ansac ions on
Image P ocessing, 26(9):4509–4522, 2017. (ci ed in page(s) 16)
[68] Y. Jin, X. Wu, J. Chen, Y. Huang, e al. Using a physics-d i en deep
neu al ne wo k o sol e in e se p oblems o lwd azimu hal esis i i y mea-
su emen s. In SPWLA 60 h Annual Logging Symposium. Socie y o Pe o-
physicis s and Well-Log Analys s, 2019. (ci ed in page(s) 19)
[69] C. Johnson. Nume ical Solu ion o Pa ial Di e en ial Equa ions by he
Fini e Elemen Me hod. Do e Books on Ma hema ics Se ies. Do e Publi-
ca ions, Inco po a ed, 2012. (ci ed in page(s) 4, 73)
[70] G. Ka ypis and V. Kuma . A as and high quali y mul ile el scheme
o pa i ioning i egula g aphs. SIAM Jou nal on Scien i ic Compu ing,
20(1):359–392, 1998. (ci ed in page(s) 58)
109
BIBLIOGRAPHY
[71] E. Kha azmi, Z. Zhang, and G. E. Ka niadakis. VPINNs: Va ia ional
Physics-In o med Neu al Ne wo ks Fo Sol ing Pa ial Di e en ial Equa-
ions, 2019. (ci ed in page(s) 5, 6, 71)
[72] E. Kha azmi, Z. Zhang, and G. E. Ka niadakis. hp- pinns: Va ia ional
physics-in o med neu al ne wo ks wi h domain decomposi ion. Compu e
Me hods in Applied Mechanics and Enginee ing, 374:113547, 2021. (ci ed
in page(s) 71)
[73] R. Khodayi-Meh and M. Za lanos. Va ne : Va ia ional neu al ne wo ks
o he solu ion o pa ial di e en ial equa ions. In P oceedings o he 2nd
Con e ence on Lea ning o Dynamics and Con ol, olume 120 o P oceed-
ings o Machine Lea ning Resea ch, pages 298–307, 2020. (ci ed in page(s)
71)
[74] A. K izhe sky, I. Su ske e , and G. E. Hin on. Imagene classi ica ion
wi h deep con olu ional neu al ne wo ks. In F. Pe ei a, C. J. C. Bu ges,
L. Bo ou, and K. Q. Weinbe ge , edi o s, Ad ances in Neu al In o ma ion
P ocessing Sys ems 25, pages 1097–1105. Cu an Associa es, Inc., 2012.
(ci ed in page(s) 16)
[75] D. Kushni , N. Velke , A. Bonda enko, G. Dya lo , and Y. Dashe sky.
Real- ime simula ion o deep azimu hal esis i i y ool in 2D aul model
using neu al ne wo ks. In SPE Annual Caspian Technical Con e ence and
Exhibi ion, Oc . 2018. (ci ed in page(s) 49)
[76] I. Laga is, A. Likas, and D. Fo iadis. A i icial neu al ne wo ks o sol ing
o dina y and pa ial di e en ial equa ions. IEEE T ansac ions on Neu al
Ne wo ks, 9:987–1000, 1998. (ci ed in page(s) 76)
[77] R. J. LeVeque. Fini e di e ence me hods o o dina y and pa ial di e en ial
equa ions: s eady-s a e and ime-dependen p oblems. SIAM, 2007. (ci ed
in page(s) 4, 5)
[78] D. B. Lindell, J. N. Ma el, and G. We zs ein. Au oin : Au oma ic in eg a-
ion o as neu al olume ende ing. In P oceedings o he con e ence on
Compu e Vision and Pa e n Recogni ion (CVPR), 2021. (ci ed in page(s)
6)
[79] C. R. Liu. Theo y o Elec omagne ic Well Logging. Else ie , Ams e dam,
Ne he lands, 2017. (ci ed in page(s) 52)
110
BIBLIOGRAPHY
[80] L. O. Lose h and B. U sin. Elec omagne ic ields in plana ly laye ed
aniso opic media. Geophysical Jou nal In e na ional, 170:44–80, 2007.
(ci ed in page(s) 10, 23)
[81] L. Lu, X. Meng, Z. Mao, and G. Ka niadakis. Deepxde: A deep lea ning
lib a y o sol ing di e en ial equa ions. SIAM Re iew, 63:208–228, 2021.
(ci ed in page(s) 5)
[82] L. Lu, Y. Zheng, G. Ca nei o, and L. Yang. Deep Lea ning o Compu e
Vision: Expe echniques o ain ad anced neu al ne wo ks using Tenso -
Flow and Ke as. Sp inge , Swi ze land, 2017. (ci ed in page(s) 3)
[83] M. L¨angk is , L. Ka lsson, and A. Lou i. A e iew o unsupe ised ea u e
lea ning and deep lea ning o ime-se ies modeling. Pa e n Recogni ion
Le e s, 42:11 – 24, 2014. (ci ed in page(s) 3)
[84] Z. Ma, D. Liu, H. Li, and X. Gao. Nume ical simula ion o a mul i- equency
esis i i y logging-while-d illing ool using a highly accu a e and adap i e
highe -o de ini e elemen me hod. Ad ances in Applied Ma hema ics and
Mechanics, 4(4):439–453, 2012. (ci ed in page(s) 4)
[85] A. Malin e no and C. To es-Ve d´ın. Bayesian in e sion o DC elec ical
measu emen s wi h unce ain ies o ese oi moni o ing. In e se P ob-
lems, 16(5):1343–1356, oc 2000. (ci ed in page(s) 3)
[86] Z. Mao, A. D. Jag ap, and G. E. Ka niadakis. Physics-in o med neu al
ne wo ks o high-speed lows. Compu e Me hods in Applied Mechanics
and Enginee ing, 360:112789, 2020. (ci ed in page(s) 6, 71)
[87] X. Meng, Z. Li, D. Zhang, and G. E. Ka niadakis. Ppinn: Pa a eal physics-
in o med neu al ne wo k o ime-dependen pdes. Compu e Me hods in
Applied Mechanics and Enginee ing, 370:113250, 2020. (ci ed in page(s)
71)
[88] S. Mish a and R. Molina o. Es ima es on he gene aliza ion e o o physics
in o med neu al ne wo ks (PINNs) o app oxima ing PDEs. a Xi p ep in
a Xi :2006.16144, 2020. (ci ed in page(s) 5, 84)
[89] D. Moghadas. One-dimensional deep lea ning in e sion o elec omagne ic
induc ion da a using con olu ional neu al ne wo k. Geophysical Jou nal
In e na ional, 222(1):247–259, 2020. (ci ed in page(s) 16)
[90] P. Moin. Fundamen als o Enginee ing Nume ical Analysis. Camb idge
Uni e si y P ess, 2010. (ci ed in page(s) 77)
111
BIBLIOGRAPHY
[91] D. Mo a i. Leas -Squa es Solu ion o Linea Di e en ial Equa ions. Ma h-
ema ics, 5, 02 2017. (ci ed in page(s) 73)
[92] M. J. Nam, D. Pa do, and C. To es-Ve d´ın. Simula ion o bo ehole-
eccen e ed iaxial induc ion measu emen s using a Fou ie hp ini e-
elemen me hod. Geophysics, 78(1):D41–D52, Jan. 2013. (ci ed in page(s)
4, 49)
[93] D. M. Nguyen, A. E g a o , and J. G a esen. Isogeome ic shape op imiza-
ion o elec omagne ic sca e ing p oblems. P og ess In Elec omagne ics
Resea ch B, 45:117–146, 2012. (ci ed in page(s) 4)
[94] V. P. Nguyen, C. Ani escu, S. P. Bo das, and T. Rabczuk. Isogeome ic
analysis: An o e iew and compu e implemen a ion aspec s. Ma hema ics
and Compu e s in Simula ion, 117:89–116, 2015. (ci ed in page(s) 5)
[95] C. M. B. Nunes and C. R´egis. GEMM3D: An edge ini e elemen p og am
o 3D modeling o elec omagne ic ields and sensi i i ies o geophysical
applica ions. Compu e s & Geosciences, 139:104477, June 2020. (ci ed in
page(s) 4)
[96] G. Pang, L. Lu, and G. E. Ka niadakis. PINNs: F ac ional Physics-
In o med Neu al Ne wo ks. SIAM Jou nal on Scien i ic Compu ing,
41(4):A2603–A2626, 2019. (ci ed in page(s) 5)
[97] D. Pa do, L. Demkowicz, C. To es-Ve d´ın, and M. Paszynski. Two-
dimensional high-accu acy simula ion o esis i i y logging-while-d illing
(LWD) measu emen s using a sel -adap i e goal-o ien ed hp ini e elemen
me hod. SIAM Jou nal on Applied Ma hema ics, 66(6):2085–2106, Jan.
2006. (ci ed in page(s) 4)
[98] D. Pa do, P. J. Ma uszyk, V. Puzy e , C. To es-Ve din, M. J. Nam, and
V. M. Calo. Modeling o Resis i i y and Acous ic Bo ehole Logging Mea-
su emen s Using Fini e Elemen Me hods. Else ie , 2021. (ci ed in page(s)
4)
[99] D. Pa do and C. To es-Ve din. Fas 1D in e sion o logging-while-d illing
esis i i y measu emen s o he imp o ed es ima ion o o ma ion esis i -
i y in high-angle and ho izon al wells. Geophysics, 80 (2):E111–E124, 2014.
(ci ed in page(s) 2, 4, 10)
[100] D. Pa do and C. To es-Ve d´ın. Fas 1D in e sion o logging-while-d illing
esis i i y measu emen s o imp o ed es ima ion o o ma ion esis i i y in
112

BIBLIOGRAPHY
high-angle and ho izon al wells. Geophysics, 80(2):E111–E124, Ma . 2015.
(ci ed in page(s) 49)
[101] D. Pa do, C. To es-Ve d´ın, M. J. Nam, M. Paszynski, and V. M. Calo.
Fou ie se ies expansion in a non-o hogonal sys em o coo dina es o
he simula ion o 3D al e na ing cu en bo ehole esis i i y measu e-
men s. Compu e Me hods in Applied Mechanics and Enginee ing, 197(45-
48):3836–3849, Aug. 2008. (ci ed in page(s) 4, 49)
[102] J. A. Pa ke , R. V. Kenyon, and D. E. T oxel. Compa ison o in e pola ing
me hods o image esampling. IEEE T ansac ions on Medical Imaging,
2(1):31–39, 1983. (ci ed in page(s) 17)
[103] A. Paszke, S. G oss, S. Chin ala, G. Chanan, E. Yang, Z. DeVi o, Z. Lin,
A. Desmaison, L. An iga, and A. Le e . Au oma ic di e en ia ion in py-
o ch. In NIPS-W, 2017. (ci ed in page(s) 3)
[104] M. Paszy´nski, R. G zeszczuk, D. Pa do, and L. Demkowicz. Deep lea ning
d i en sel -adap i e hp ini e elemen me hod. In Compu a ional Science –
ICCS 2021, pages 114–121. Sp inge In e na ional Publishing, 2021. (ci ed
in page(s) 5)
[105] C. G. Pe a, O. Schenk, and M. Ani escu. Real- ime s ochas ic op imiza ion
o complex ene gy sys ems on high-pe o mance compu e s. Compu ing in
Science & Enginee ing, 16(5):32–42, 2014. (ci ed in page(s) 58)
[106] C. G. Pe a, O. Schenk, M. Lubin, and K. G¨ae ne . An augmen ed
incomple e ac o iza ion app oach o compu ing he Schu complemen
in s ochas ic op imiza ion. SIAM Jou nal on Scien i ic Compu ing,
36(2):C139–C162, 2014. (ci ed in page(s) 58)
[107] L. Piegl and W. Tille . The NURBS Book. Sp inge -Ve lag, New Yo k, NY,
2nd edi ion, 1997. (ci ed in page(s) 54)
[108] S. Pu usho ham, C. Meng, Z. Che, and Y. Liu. Benchma king deep lea ning
models on la ge heal hca e da ase s. Jou nal o Biomedical In o ma ics,
83:112–134, 2018. (ci ed in page(s) 3)
[109] V. Puzy e . Deep lea ning elec omagne ic in e sion wi h con olu ional
neu al ne wo ks. Geophysical Jou nal In e na ional, 218(2):817–832, 2019.
(ci ed in page(s) 16)
[110] T. M. Quan, D. G. C. Hildeb and, and W.-K. Jeong. Fusionne : A deep
ully esidual con olu ional neu al ne wo k o image segmen a ion in con-
nec omics, 2016. (ci ed in page(s) 16)
113
BIBLIOGRAPHY
[111] N. Rahaman, A. Ba a in, D. A pi , F. D axle , M. Lin, F. A. Hamp ech ,
Y. Bengio, and A. Cou ille. On he Spec al Bias o Neu al Ne wo ks,
2019. (ci ed in page(s) 5)
[112] M. Raissi, P. Pe dika is, and G. 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. (ci ed in page(s) 6, 71)
[113] M. Raissi, P. Pe dika is, and G. E. Ka niadakis. Physics In o med Deep
Lea ning (Pa I): Da a-d i en Solu ions o Nonlinea Pa ial Di e en ial
Equa ions. 2017. (ci ed in page(s) 5)
[114] S. Ranjan and S. Sen hamila asu. Applied Deep Lea ning and Compu e
Vision o Sel -D i ing Ca s: Build au onomous ehicles using deep neu al
ne wo ks and beha io -cloning echniques. Pack Publishing, 2020. (ci ed
in page(s) 3)
[115] J. N. Reddy. In oduc ion o he ini e elemen me hod. McG aw-Hill Edu-
ca ion, 2019. (ci ed in page(s) 4)
[116] W. Ri z. ¨
Ube eine neue me hode zu l¨osung gewisse a ia ionsp obleme
de ma hema ischen physik. Jou nal ¨u die eine und angewand e Ma he-
ma ik, 135:1–61, 1909. (ci ed in page(s) 73)
[117] J. A. Ri e a, D. Pa do, and E. Albe di. Design o loss unc ions o sol ing
in e se p oblems using deep lea ning. In Compu a ional Science – ICCS
2020, pages 158–171. Sp inge In e na ional Publishing, 2020. (ci ed in
page(s) 3)
[118] J. A. Ri e a, J. M. Taylo , ´
Angel J. Omella, and D. Pa do. On quad a u e
ules o sol ing pa ial di e en ial equa ions using neu al ne wo ks. Com-
pu e Me hods in Applied Mechanics and Enginee ing, 393:114710, 2022.
(ci ed in page(s) 5)
[119] ´
A. Rod ´ıguez-Rozas and D. Pa do. A p io i Fou ie analysis o 2.5D ini e
elemen s simula ions o logging-while-d illing (LWD) esis i i y measu e-
men s. P ocedia Compu e Science, 80:782–791, 2016. (ci ed in page(s)
58)
[120] ´
A. Rod ´ıguez-Rozas, D. Pa do, and C. To es-Ve d´ın. Fas 2.5D ini e
elemen simula ions o bo ehole esis i i y measu emen s. Compu a ional
Geosciences, 22(5):1271–1281, 2018. (ci ed in page(s) 4, 52, 53, 58, 60, 65)
114
BIBLIOGRAPHY
[121] L. Ru ho o and E. Habe . Deep Neu al Ne wo ks Mo i a ed by Pa ial
Di e en ial Equa ions. Jou nal o Ma hema ical Imaging and Vision, 62,
2020. (ci ed in page(s) 5)
[122] F. Sahli Cos abal, Y. Yang, P. Pe dika is, D. E. Hu ado, and E. Kuhl.
Physics-in o med neu al ne wo ks o ca diac ac i a ion mapping. F on ie s
in Physics, 8, 2020. (ci ed in page(s) 71)
[123] E. Samaniego, C. Ani escu, S. Goswami, V. Nguyen-Thanh, H. Guo,
K. Hamdia, X. Zhuang, and T. 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. (ci ed
in page(s) 5)
[124] A. Sa mien o, A. Cˆo es, D. Ga cia, L. Dalcin, N. Collie , and
V. Calo. Pe IGA-MF: A mul i- ield high-pe o mance oolbox o s uc u e-
p ese ing B-splines spaces. Jou nal o Compu a ional Science, 18:117–131,
2017. (ci ed in page(s) 57)
[125] K. Scaman and A. Vi maux. Lipschi z egula i y o deep neu al ne wo ks:
analysis and e icien es ima ion. In P oceedings o he 32nd In e na ional
Con e ence on Neu al In o ma ion P ocessing Sys ems, pages 3839–3848,
2018. (ci ed in page(s) 84)
[126] O. Schenk and K. G¨a ne . Sol ing unsymme ic spa se sys ems o lin-
ea equa ions wi h PARDISO. Fu u e Gene a ion Compu e Sys ems,
20(3):475–487, Ap . 2004. (ci ed in page(s) 58)
[127] O. Schenk, K. G¨a ne , and W. Fich ne . E icien spa se LU ac o iza ion
wi h le - igh looking s a egy on sha ed memo y mul ip ocesso s. BIT
Nume ical Ma hema ics, 40(1):158–176, 2000. (ci ed in page(s) 58)
[128] D. J. Sei e , S. A. Dossa y, R. E. Chemali, M. S. Bi a , A. A. Lo y,
J. L. Pi che , and M. A. Bay akda . Deep elec ical images, geosignal,
and eal- ime in e sion help guide s ee ing decisions. Socie y o Pe oleum
Enginee s, pages 1–9, 2009. (ci ed in page(s) 2)
[129] M. Shah ia i and D. Pa do. Bo ehole esis i i y simula ions o oil-wa e
ansi ion zones wi h a 1.5D nume ical sol e . Compu a ional Geosciences,
24(3):1285–1299, Ma . 2020. (ci ed in page(s) 4, 49)
115
BIBLIOGRAPHY
[130] M. Shah ia i, D. Pa do, B. Mose , and F. Sobieczky. A deep neu al ne wo k
as su oga e model o o wa d simula ion o bo ehole esis i i y measu e-
men s. P ocedia Manu ac u ing, 42:235 – 238, 2020. In e na ional Con-
e ence on Indus y 4.0 and Sma Manu ac u ing (ISM 2019). (ci ed in
page(s) 4, 13)
[131] M. Shah ia i, D. Pa do, A. Pic´on, A. Gald an, J. D. Se , and C. To es-
Ve d´ın. A deep lea ning app oach o he in e sion o bo ehole esis i i y
measu emen s. Compu a ional Geosciences, 2020. (ci ed in page(s) 2, 4,
13)
[132] M. Shah ia i, D. Pa do, J. Ri e a, C. To es-Ve d´ın, A. Picon, J. Del Se ,
S. Ossand´on, and V. Calo. E o con ol and loss unc ions o he deep
lea ning in e sion o bo ehole esis i i y measu emen s. In e na ional Jou -
nal o Nume ical Me hods in Enginee ing, 122, 2020. (ci ed in page(s) 3)
[133] M. Shah ia i, S. Rojas, D. Pa do, A. Rod ´ıguez-Rozas, S. A. Bak , V. M.
Calo, and I. Muga. A nume ical 1.5D me hod o he apid simula ion o
geophysical esis i i y measu emen s. Geosciences, 8(6):1–28, 2018. (ci ed
in page(s) 2, 4, 8, 10)
[134] S. Shah okhabadi, T. D. Cao, and F. Vahedi a d. Isogeome ic analysis
h ough B´ezie ex ac ion o he mo-hyd o-mechanical modeling o sa u-
a ed po ous media. Compu e s and Geo echnics, 107:176–188, Ma . 2019.
(ci ed in page(s) 4)
[135] J. Shen and W. Sun. 2.5-D modeling o c oss-hole elec omagne ic mea-
su emen by ini e elemen me hod. Pe oleum Science, 5(2):126–134, May
2008. (ci ed in page(s) 4, 49)
[136] K. Shukla, P. C. Di Leoni, J. Blackshi e, D. Spa kman, and G. E. Ka -
niadakis. Physics-in o med neu al ne wo k o ul asound nondes uc i e
quan i ica ion o su ace b eaking c acks. Jou nal o Nondes uc i e E al-
ua ion, 39(3):61, 2020. (ci ed in page(s) 71)
[137] A. Simona, L. Bona en u a, C. de Falco, and S. Sch¨ops. IsoGeome ic ap-
p oxima ions o elec omagne ic p oblems in axisymme ic domains. Com-
pu e Me hods in Applied Mechanics and Enginee ing, 369:113211, Sep .
2020. (ci ed in page(s) 4)
[138] R. N. Simpson, Z. Liu, R. V´azquez, and J. A. E ans. An isogeome ic
bounda y elemen me hod o elec omagne ic sca e ing wi h compa ible
B-spline disc e iza ions. Jou nal o Compu a ional Physics, 362:264–289,
June 2018. (ci ed in page(s) 4)
116