I
AE
S
I
n
t
e
r
n
at
ion
al
Jou
r
n
al
of
Ar
t
if
icial
I
n
t
e
ll
ig
e
n
c
e
(
I
J
-
AI
)
Vol.
14
,
No.
5
,
Oc
tober
2025
,
pp
.
3744
~
3756
I
S
S
N:
2252
-
8938
,
DO
I
:
10
.
11591/i
jai
.
v
14
.i
5
.
pp
37
44
-
3756
3744
Jou
r
n
al
h
omepage
:
ht
tp:
//
ij
ai
.
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06
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K
e
y
w
o
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d
s
:
B
a
c
kpr
opa
ga
ti
on
Ga
s
f
low
Ne
ur
a
l
ne
twor
ks
Nonlinea
r
s
olver
R
e
s
e
r
voir
s
im
ulation
Th
i
s
i
s
a
n
o
p
en
a
c
ces
s
a
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t
i
c
l
e
u
n
d
e
r
t
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e
CC
B
Y
-
SA
l
i
ce
n
s
e.
C
or
r
e
s
pon
din
g
A
u
th
or
:
Adr
ianto
Doc
tor
a
l
P
r
ogr
a
m
of
P
e
tr
oleum
E
nginee
r
ing
,
F
a
c
ult
y
of
M
ini
ng
a
nd
P
e
t
r
oleum
E
nginee
r
ing
I
ns
ti
tut
T
e
knologi
B
a
ndung
Ga
ne
s
ha
S
tr
e
e
t
No.
10,
B
a
ndung
-
40132,
I
ndone
s
ia
E
mail:
a
dr
ianto
.
it
b@gmail
.
c
om
1.
I
NT
RODU
C
T
I
ON
F
os
s
il
f
ue
ls
,
pa
r
ti
c
ular
ly
thos
e
or
igi
na
ti
ng
f
r
om
h
ydr
oc
a
r
bon
(
oil
a
nd
ga
s
)
,
r
e
main
a
s
the
pr
im
a
r
y
global
s
our
c
e
of
e
ne
r
gy
de
s
pit
e
the
r
is
ing
us
e
of
r
e
ne
wa
ble
r
e
s
our
c
e
s
[
1]
–
[
3]
.
T
o
e
f
f
e
c
ti
ve
ly
de
ve
lo
p
the
oil
a
nd
ga
s
f
ield,
we
ne
e
d
to
unde
r
s
tand
how
f
lui
ds
move
thr
ough
the
por
ous
medium
in
the
s
ubs
ur
f
a
c
e
.
I
n
a
ddi
ti
on,
the
knowle
dge
of
f
lui
d
f
low
in
po
r
o
us
media
a
ls
o
plays
a
n
im
por
tant
r
ole
in
the
s
tudy
of
gr
oundwa
ter
[
4]
–
[
6]
,
ge
other
mal
e
ne
r
gy
[
7]
,
[
8
]
,
a
nd
C
O
2
s
e
que
s
tr
a
ti
on
[
9
]
–
[
11]
.
R
e
s
e
r
voir
s
im
ulati
on
is
one
of
the
f
ields
in
pe
tr
oleum
r
e
s
e
r
voir
e
ngin
e
e
r
ing
w
he
r
e
numer
ica
l
c
omput
e
r
pr
ogr
a
ms
a
r
e
us
e
d
to
pr
e
dict
f
lui
d
f
low
be
ha
vior
withi
n
oil
a
nd
ga
s
r
e
s
e
r
voir
s
.
S
ince
t
his
model
c
ontains
unc
e
r
tainti
e
s
,
the
phys
ica
l
pr
op
e
r
ti
e
s
of
r
oc
ks
a
nd
f
lui
ds
,
a
nd
ini
t
ial
r
e
s
e
r
voir
c
ondit
ions
c
a
n
be
a
djus
ted
a
s
long
a
s
they
a
r
e
withi
n
r
e
a
s
ona
ble
e
nginee
r
ing
li
mi
ts
unti
l
a
p
r
oduc
ti
on
his
tor
y
matc
h
is
a
c
hieve
d
[
12]
–
[
14]
.
A
good
his
tor
y
m
a
tch
im
pli
e
s
that
the
r
e
s
e
r
voir
model
c
a
n
pr
e
dict
the
f
utu
r
e
be
h
a
viour
of
hydr
oc
a
r
bon
p
r
oduc
ti
on.
T
he
r
e
f
o
r
e
,
r
e
s
e
r
voir
s
im
ulation
c
a
n
a
s
s
i
s
t
the
de
c
is
ion
-
m
a
king
pr
oc
e
s
s
in
r
e
s
e
r
voir
mana
ge
ment
a
nd
de
ve
lopm
e
nt
a
s
it
c
a
n
e
s
ti
mate
oil
a
nd/or
ga
s
r
e
c
ove
r
y
unde
r
va
r
ious
pr
o
duc
ti
on
s
c
e
na
r
ios
[
15]
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
nt
J
Ar
ti
f
I
ntell
I
S
S
N:
2252
-
8938
B
ac
k
pr
opagati
on
ne
ur
al
ne
tw
or
k
s
for
s
olving
gas
fl
ow
e
quati
ons
in
por
ous
me
dia
(
A
dr
iant
o)
3745
Due
to
the
nonli
ne
a
r
it
y
of
the
s
ys
tem
of
e
qua
ti
ons
in
a
r
e
s
e
r
voir
s
im
ulation
,
s
olvi
ng
it
a
c
c
ur
a
tely
a
nd
e
f
f
icie
ntl
y
is
a
c
ha
ll
e
ng
ing
tas
k.
A
r
e
s
e
r
voir
mod
e
l
is
a
s
im
pli
f
ica
ti
on
o
f
the
p
r
ope
r
ti
e
s
o
f
r
oc
ks
,
f
l
uids
,
a
nd
their
int
e
r
a
c
ti
ons
a
s
we
ll
a
s
the
f
lui
d
f
low
mec
ha
ni
s
m
in
the
r
e
s
e
r
voir
dur
ing
p
r
oduc
ti
on.
R
e
s
e
r
voir
g
e
ometr
y
c
a
n
be
modele
d
with
va
r
ious
gr
id
s
ha
pe
s
,
s
uc
h
a
s
r
e
gular
c
a
r
tes
ian,
r
e
c
ti
li
ne
a
r
,
c
ur
vil
inea
r
,
or
uns
tr
uc
tur
e
d.
R
e
s
e
r
voir
f
lui
ds
c
ould
be
r
e
pr
e
s
e
nted
by
blac
k
-
oil
or
c
ompos
it
ional
models
,
de
pe
nding
on
the
nu
mber
of
pha
s
e
s
a
nd
the
number
of
c
omponents
the
modele
r
is
c
onc
e
r
ne
d
wi
th.
T
he
pr
oduc
ti
on
pe
r
iod
is
divi
de
d
int
o
s
e
ve
r
a
l
ti
mes
teps
ba
s
e
d
on
the
li
mi
t
o
f
a
c
c
ur
a
c
y
to
be
a
c
hieve
d.
W
e
f
ound
that
the
mor
e
c
ompl
e
x
the
r
e
s
e
r
voir
numer
ica
l
model
a
nd
the
higher
the
e
xpe
c
ted
a
c
c
ur
a
c
y,
the
g
r
e
a
ter
the
c
os
t
a
nd
c
omput
a
ti
o
na
l
ti
me
to
s
olve
i
t.
T
hus
,
ou
r
s
tudy
a
im
s
to
de
ve
lo
p
a
ne
w
a
ppr
oa
c
h
to
a
c
c
ur
a
tely
s
olve
the
nonli
ne
a
r
e
qua
ti
o
ns
f
ound
in
oil
/gas
r
e
s
e
r
voir
s
im
ulations
.
Ar
ti
f
icia
l
ne
u
r
a
l
ne
twor
ks
(
AN
N)
ha
ve
be
e
n
wide
ly
us
e
d
s
uc
c
e
s
s
f
ull
y
in
modeling
nonli
ne
a
r
it
ies
in
many
f
ields
[
16]
–
[
18]
.
Among
AN
Ns
,
ba
c
kpr
opa
g
a
ti
on
ne
ur
a
l
ne
twor
ks
(
B
P
NN
)
a
r
e
the
mos
t
popula
r
ly
us
e
d
ne
ur
a
l
ne
twor
k
models
.
B
P
NN
is
ba
s
ica
ll
y
a
f
e
e
df
or
wa
r
d
ne
twor
k
tr
a
ined
us
ing
the
e
r
r
or
gr
a
dient
c
a
lcula
ti
on,
whic
h
is
c
a
ll
e
d
ba
c
kwa
r
d
pa
s
s
.
T
he
B
P
NN
ne
twor
k
c
a
n
lea
r
n
a
nd
r
e
membe
r
e
xtens
ive
input
-
output
mapping
r
e
lations
without
r
e
quir
ing
p
r
ior
knowle
dge
of
t
he
mathe
matica
l
e
qua
ti
on
that
de
f
ines
thes
e
r
e
lat
ions
hips
.
T
he
lea
r
ning
r
ule
e
mpl
oys
the
s
tee
pe
s
t
de
s
c
e
nt
method,
uti
li
z
ing
ba
c
kpr
opa
ga
ti
on
to
a
djus
t/
mo
dif
y
the
we
ight
to
r
e
duc
e
the
s
um
of
s
qua
r
e
d
e
r
r
or
s
.
T
h
is
f
e
a
tur
e
make
s
B
P
NN
popular
f
or
pr
e
dicting
c
ompl
e
x
nonli
ne
a
r
s
ys
tems
.
B
e
s
ide
s
B
P
NN
,
other
AN
N
methods
ha
ve
a
l
s
o
be
e
n
us
e
d
in
f
lui
d
f
low
s
tudi
e
s
.
I
s
ka
nda
r
a
nd
Kur
ihar
a
[
19]
uti
li
z
e
long
s
hor
t
-
ter
m
memor
y
(
L
S
T
M
)
to
f
o
r
e
c
a
s
t
the
pr
oduc
e
d
oi
l,
CO
2
a
nd
wa
ter
d
ur
ing
the
c
a
r
bon
c
a
ptur
e
,
uti
li
ty
,
a
nd
s
tor
a
ge
(
C
C
US)
ope
r
a
ti
ons
.
Z
ha
ng
e
t
al
.
[
20]
s
uc
c
e
s
s
f
ull
y
c
ombi
ne
d
B
a
ye
s
ian
M
a
r
kov
c
ha
in
M
onte
C
a
r
lo
(
M
C
M
C
)
a
nd
L
S
T
M
to
a
s
s
is
t
the
hi
s
tor
y
matc
hing
pr
oc
e
s
s
a
n
d
c
a
ptur
e
s
ubs
ur
f
a
c
e
unc
e
r
tainty
in
the
10th
S
P
E
c
ompar
a
ti
v
e
model.
I
n
the
wor
k
by
Z
ha
ng
e
t
al
.
[
21]
,
r
e
c
ur
r
e
nt
ne
ur
a
l
ne
twor
k
(
R
NN
)
,
L
S
T
M
,
a
nd
ga
ted
r
e
c
ur
r
e
nt
uni
t
(
GR
U)
c
a
n
a
c
c
ur
a
tely
a
nd
e
f
f
e
c
ti
ve
ly
pr
e
dict
r
e
s
e
r
voir
outf
low
in
wa
ter
r
e
s
our
c
e
s
.
L
i
e
t
al
.
[
22]
pr
opos
e
d
a
de
e
p
ne
ur
a
l
ne
two
r
k
(
DN
N)
-
ba
s
e
d
r
e
s
e
r
voir
s
im
ulator
f
or
hydr
a
uli
c
f
r
a
c
tur
ing
a
nd
va
li
da
ted
by
s
im
ulating
3D
s
ynthetic
model
a
nd
unc
onve
nti
on
a
l
f
ield
.
S
a
ntos
e
t
al
.
[
23
]
p
r
opos
e
d
a
3D
c
onvolut
ional
ne
ur
a
l
ne
twor
k
(
C
NN
)
that
is
a
ble
to
p
r
e
dict
f
lui
d
f
l
ow
in
3D
digi
tal
r
oc
k
i
mage
s
.
T
he
s
e
methods
e
xc
e
l
in
te
r
ms
of
c
omput
a
ti
ona
l
t
im
e
be
c
a
us
e
they
a
r
e
s
tand
-
a
lone
models
a
nd
a
r
e
not
a
ppli
e
d
to
s
olve
nonli
ne
a
r
di
s
c
r
e
te
e
qua
ti
ons
in
r
e
s
e
r
voir
s
im
ulation.
How
e
ve
r
,
the
a
f
or
e
mentioned
ne
ur
a
l
ne
twor
k
a
ppr
oa
c
he
s
ne
c
e
s
s
it
a
te
a
s
ubs
tantial
qua
nti
ty
of
his
tor
ica
l
da
ta
to
f
or
e
c
a
s
t
a
li
mi
ted
tempo
r
a
l
s
pa
n,
a
nd
a
r
e
c
ompar
a
ti
ve
ly
les
s
e
f
f
e
c
ti
ve
in
p
r
ovidi
ng
a
n
e
xplana
ti
on
o
f
t
he
f
l
uid
f
low
phe
nomena
oc
c
ur
r
ing
withi
n
the
r
e
s
e
r
voir
.
R
a
is
s
i
e
t
al
.
[
24
]
int
r
oduc
e
d
phys
ics
-
inf
or
med
ne
ur
a
l
ne
twor
ks
(
P
I
NN
)
whic
h
uti
li
s
e
s
ne
ur
a
l
ne
twor
ks
that
a
tt
e
mpt
to
obtain
c
onti
nuous
s
olut
io
ns
of
pa
r
ti
a
l
dif
f
e
r
e
nti
a
l
e
qua
ti
on
s
(
P
DE
)
by
incor
por
a
ti
ng
the
phys
ics
of
ini
ti
a
l
a
nd
bounda
r
y
c
ondit
ions
a
s
the
los
s
f
unc
ti
on
.
I
hunde
a
nd
Olor
ode
[
25]
de
m
ons
tr
a
ted
that
P
I
NN
c
a
n
incor
po
r
a
te
phys
ica
l
c
ons
tr
a
int
s
wi
thout
s
igni
f
ica
ntl
y
r
e
duc
ing
the
a
c
c
ur
a
c
y
of
c
omp
os
it
ional
model,
but
r
e
qui
r
e
up
to
mi
ll
ions
o
f
unique
da
t
a
in
their
s
tudi
e
s
.
Ha
n
e
t
al
.
[
26]
pr
opos
e
d
the
phys
ics
-
inf
or
med
ne
ur
a
l
ne
twor
k
ba
s
e
d
on
domain
de
c
o
mpos
it
ion
(
P
I
NN
-
DD
)
whic
h
s
uc
c
e
s
s
f
ull
y
s
olved
the
pr
oblem
of
lar
ge
-
s
c
a
le
r
e
s
e
r
voir
s
im
ulation
with
li
mi
ted
da
ta,
but
a
t
a
high
c
omput
a
ti
ona
l
ti
me
a
nd
c
os
t.
Z
h
a
ng
[
27]
c
a
me
up
with
phys
ics
-
inf
or
med
de
e
p
c
onvolut
ion
a
l
ne
ur
a
l
ne
twor
k
(
P
I
DC
NN
)
,
whic
h
is
mor
e
e
f
f
ic
ient
than
f
ull
y
c
onne
c
ted
ne
ur
a
l
ne
twor
ks
,
a
s
2D
va
r
iabl
e
s
may
be
tr
e
a
ted
a
s
im
a
ge
s
.
T
he
pr
im
a
r
y
c
on
s
tr
a
int
of
P
I
DC
NN
is
it
s
r
e
s
tr
iction
to
s
tr
uc
tur
e
d
gr
ids
,
while
the
method
f
or
r
e
pr
e
s
e
nti
ng
f
e
a
tur
e
s
s
e
t
up
on
uns
tr
uc
tur
e
d
gr
ids
in
a
n
im
a
ge
-
li
ke
f
or
mat
r
e
mai
ns
unc
lea
r
.
A
P
I
NN
us
ing
a
c
a
pa
c
it
a
nc
e
r
e
s
is
tanc
e
model
(
C
R
M
)
wa
s
de
ve
loped
by
M
a
nigl
io
e
t
al
.
[
28]
to
f
or
e
c
a
s
t
oil
pr
oduc
ti
on
in
r
e
s
e
r
voir
s
with
wa
ter
f
loodi
ng,
e
li
mi
na
ti
ng
the
ne
e
d
f
or
a
3D
model
a
nd
e
ns
ur
i
ng
c
ons
is
tenc
y
in
p
r
oduc
ti
on
da
ta.
W
hil
e
AN
N
a
nd
the
c
ombi
na
ti
on
of
s
ome
methods
wi
th
other
types
of
ne
u
r
a
l
ne
two
r
ks
ha
ve
be
e
n
de
ve
loped,
c
on
ve
nti
ona
l
r
e
s
e
r
voir
s
im
ulation
is
s
ti
ll
r
e
quir
e
d
f
or
va
r
ious
typ
e
s
of
r
e
s
e
r
voir
mana
ge
ment
pr
oblems
s
uc
h
a
s
opti
mi
z
a
ti
on
of
inf
i
ll
dr
il
li
ng
c
a
mpaign
a
nd
a
ppli
c
a
ti
on
o
f
a
ppr
o
pr
iate
e
nha
nc
e
d
oil
r
e
c
ove
r
y
(
E
OR
)
methods
.
T
his
s
tudy
a
im
s
to
ha
r
ne
s
s
the
nonli
ne
a
r
it
y
modeling
c
a
pa
bil
it
ies
of
B
P
NN
to
be
us
e
d
a
s
a
ne
w
a
lt
e
r
na
ti
ve
nonli
ne
a
r
s
olver
in
r
e
s
e
r
voir
s
im
ulati
on.
T
his
pa
pe
r
is
or
ga
nize
d
a
s
f
oll
ows
:
i
n
s
e
c
ti
on
2,
the
gove
r
ning
e
qua
ti
ons
f
or
the
1D
ga
s
f
low
in
po
r
ous
medium,
dis
c
r
e
t
iza
ti
on,
a
nd
wor
kf
lows
f
or
Ne
wton
method
a
nd
ne
ur
a
l
ne
twor
k
a
ppr
oa
c
h
is
p
r
e
s
e
nted.
S
e
c
ti
on
3
c
ontains
the
r
e
s
ult
s
of
the
ne
ur
a
l
ne
twor
k
a
ppr
oa
c
h
to
s
olvi
ng
one
-
dim
e
ns
ional
ga
s
f
low
in
a
por
ous
medium
,
va
li
da
ted
by
c
ompar
ing
the
s
olut
ions
obtaine
d
f
or
the
N
e
wton
method.
T
he
e
f
f
e
c
t
o
f
l
e
a
r
ning
r
a
te
pa
r
a
mete
r
s
on
the
c
omput
a
ti
on
ti
me
a
nd
the
number
of
it
e
r
a
ti
ons
is
inves
ti
ga
ted
on
bo
th
homo
ge
ne
ous
a
nd
he
ter
oge
ne
ous
model.
S
e
c
ti
on
4
s
umm
a
r
ize
s
the
pe
r
f
or
manc
e
of
the
pr
opos
e
d
ne
ur
a
l
ne
twor
k
-
ba
s
e
d
nonli
ne
a
r
s
olve
r
in
r
e
s
e
r
voir
s
im
ulation
a
nd
pr
ovides
s
ome
s
ugge
s
ti
ons
f
or
f
utur
e
r
e
s
e
a
r
c
h.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
S
N
:
2252
-
8938
I
nt
J
Ar
ti
f
I
ntell
,
Vol.
14
,
No.
5
,
Oc
tober
20
25
:
37
44
-
3756
3746
2.
M
E
T
HO
D
2.
1
.
M
od
e
l
d
e
s
c
r
ip
t
ion
F
igur
e
1
il
lus
tr
a
tes
the
one
-
dim
e
ns
ional
model
th
e
s
ingl
e
-
pha
s
e
ga
s
f
low
in
a
po
r
ous
medium
.
At
ini
ti
a
l
c
ondit
ions
,
a
ll
c
e
ll
s
a
r
e
a
s
s
umed
to
ha
ve
the
identica
l
pr
e
s
s
ur
e
of
5
,
000
ps
i.
T
he
Dir
ichle
t
c
o
ndit
ion
is
a
s
s
igned
a
t
the
lef
t
by
ke
e
ping
the
p
r
e
s
s
ur
e
c
ons
ta
nt
,
whe
r
e
a
s
the
Ne
umann
c
ondit
ion
of
no
-
f
low
bo
unda
r
y
is
a
s
s
igned
a
t
the
r
ight
.
Othe
r
model
pa
r
a
mete
r
s
whic
h
r
e
late
d
to
the
s
im
ulation
a
r
e
s
umm
a
r
ize
d
in
T
a
bl
e
1
.
F
igur
e
1
.
A
one
-
dim
e
ns
ional
model
of
ga
s
f
low
in
a
por
ous
medium
T
a
ble
1.
M
ode
l
pa
r
a
mete
r
s
P
a
r
a
me
te
r
s
V
a
lu
e
T
e
mpe
r
a
tu
r
e
200
I
ni
ti
a
l
pr
e
s
s
ur
e
(
ps
i)
5
,
000
C
1 (
%
)
100
P
e
r
me
a
bi
li
ty
(
md)
1, 10
-
20
(
he
te
r
oge
ne
ous
)
P
or
os
it
y (
f
r
a
c
ti
on)
0.1, 0.1
-
0.2 (
he
te
r
oge
ne
ous
)
D
is
ta
nc
e
(
f
t)
1
,
000
D
ur
a
ti
on (
da
y)
6.9
Δ
0.0069
I
n
one
dim
e
ns
ion
(
li
ne
a
r
f
low)
,
the
e
qua
ti
ons
r
uli
ng
the
s
ingl
e
-
pha
s
e
ga
s
f
low
in
a
por
ous
medium
is
a
s
(
1)
:
(
)
=
1
(
)
(
1)
whe
r
e
is
ga
s
vis
c
os
it
y;
is
ga
s
de
viation
f
a
c
tor
;
is
pr
e
s
s
ur
e
;
is
dis
tanc
e
in
the
x
-
c
oor
dinate
;
is
ti
me;
is
dif
f
us
ivi
ty
c
oe
f
f
icie
nt,
e
qua
l
to
0
.
0
0
6
3
3
/
,
with
a
nd
is
th
e
pe
r
mea
bil
it
y
a
nd
po
r
os
it
y,
r
e
s
pe
c
ti
ve
ly.
W
e
us
e
d
the
dir
e
c
t
c
or
r
e
lation
of
P
a
pa
y
[
29
]
to
e
s
ti
mate
ga
s
de
viation
f
a
c
tor
,
Z
,
a
nd
the
c
or
r
e
l
a
ti
on
of
Gonz
a
lez
e
t
al
.
[
30]
to
e
s
ti
mate
ga
s
vis
c
os
it
y,
.
T
o
obtain
a
numer
ica
l
s
olut
ion
,
(
1
)
is
dis
c
r
e
ti
z
e
d
us
ing
the
f
ini
te
dif
f
e
r
e
nc
e
method.
T
he
lef
t
-
ha
nd
s
ide
of
(
1
)
r
e
lating
to
s
pa
c
e
is
dis
c
r
e
ti
z
e
d
us
ing
c
e
nter
e
d
f
ini
te
dif
f
e
r
e
nc
e
,
whe
r
e
a
s
the
r
ight
-
ha
nd
s
ide
r
e
lating
to
ti
me
us
e
s
f
or
wa
r
d
f
ini
te
dif
f
e
r
e
nc
e
,
thus
r
e
s
ult
ing
in
the
r
e
s
idual
in
(
2
)
:
=
−
1
+
1
−
β
i
+
1
+
+
1
+
1
(
2)
with
c
oe
f
f
icie
nts
,
a
nd
de
f
ined
a
s
(3
-
5)
:
=
(
(
)
−
1
+
1
+
4
(
)
+
1
−
(
)
+
1
+
1
)
4
Δ
2
(
3
)
=
2
(
)
+
1
2
+
1
(
1
+
1
−
+
1
)
(
4
)
=
(
−
(
)
−
1
+
1
+
4
(
)
+
1
+
(
)
+
1
+
1
)
4
Δ
2
(
5
)
F
igur
e
2
s
hows
the
dis
tr
ibut
ion
o
f
r
oc
k
pr
ope
r
t
ies
to
s
e
e
the
e
f
f
e
c
t
of
r
e
s
e
r
voir
he
ter
oge
ne
it
y
on
s
olver
pe
r
f
or
manc
e
.
T
he
pe
r
mea
bil
it
y
dis
tr
i
buti
on
is
pr
e
s
e
nted
in
F
igur
e
2(
a
)
,
while
the
por
os
it
y
dis
tr
ibut
i
on
is
s
hown
in
F
igur
e
2(
b
)
.
W
e
us
e
d
uni
f
or
m
dis
tr
ibut
ion
to
ge
ne
r
a
te
pe
r
mea
bil
it
y
with
a
r
a
nge
of
10
to
20
m
il
idar
c
ies
,
a
nd
por
os
it
y
with
a
r
a
nge
of
0
.
1
to
0.
2
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
nt
J
Ar
ti
f
I
ntell
I
S
S
N:
2252
-
8938
B
ac
k
pr
opagati
on
ne
ur
al
ne
tw
or
k
s
for
s
olving
gas
fl
ow
e
quati
ons
in
por
ous
me
dia
(
A
dr
iant
o)
3747
(
a
)
(
b)
F
igur
e
2.
Dis
tr
ibut
ion
of
(
a
)
pe
r
mea
bil
it
y
in
10
(
to
p)
,
25
(
mi
dd
le)
,
a
nd
50
(
bott
om
)
gr
ids
,
a
nd
(
b)
por
os
it
y
in
10
(
top)
,
25
(
mi
ddle
)
,
a
nd
50
(
bott
om)
gr
ids
in
he
t
e
r
oge
ne
ous
models
2.
2
.
Ne
wt
on
m
e
t
h
o
d
T
he
Ne
wton
-
R
a
phs
on
(
N
-
R
)
method
is
one
of
the
mos
t
f
r
e
que
ntl
y
us
e
d
a
nd
e
f
f
icie
nt
tec
hniques
in
s
olvi
ng
s
ys
tems
of
e
qua
ti
ons
f
ound
in
mathe
matic
a
l
a
nd
e
nginee
r
ing
pr
ob
lems
.
T
he
N
-
R
method
s
e
a
r
c
he
s
f
or
the
r
oots
of
a
n
e
qua
ti
on
us
ing
the
tange
nt
li
ne
o
f
a
c
u
r
ve
it
e
r
a
ti
ve
ly
unti
l
it
a
ppr
oa
c
he
s
the
s
olut
ion.
T
he
r
e
s
idual
in
(
2
)
c
a
n
be
w
r
it
ten
in
matr
ix
f
o
r
m
a
s
=
,
a
s
s
tate
d
in
(
6
).
[
−
0
0
⋯
0
0
0
+
1
−
+
1
+
1
0
⋯
0
0
0
⋮
⋮
⋮
⋮
⋱
⋮
⋮
⋮
0
0
0
0
⋯
−
1
−
−
1
−
1
0
0
0
0
⋯
0
−
+
]
×
[
+
1
⋮
−
1
]
=
[
0
0
0
0
]
(
6)
T
o
upda
te
t
he
s
olut
ion
a
t
e
a
c
h
it
e
r
a
ti
on
,
the
s
olut
i
on
a
t
the
pr
e
vious
it
e
r
a
ti
on
a
nd
the
inver
s
e
of
the
J
a
c
obian
matr
ix
a
r
e
r
e
quir
e
d
in
(
7
)
.
+
1
=
−
−
1
(
)
(
7)
T
he
J
a
c
obian
matr
ix
is
a
c
oll
e
c
ti
on
of
a
ll
the
f
i
r
s
t
p
a
r
ti
a
l
de
r
ivatives
of
the
r
e
s
idual
in
(
8
)
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
S
N
:
2252
-
8938
I
nt
J
Ar
ti
f
I
ntell
,
Vol.
14
,
No.
5
,
Oc
tober
20
25
:
37
44
-
3756
3748
=
[
1
1
⋯
1
⋮
⋱
⋮
1
⋯
]
(
8)
2.
3.
B
ac
k
p
r
op
agat
ion
n
e
u
r
al
n
e
t
wor
k
s
W
e
r
bos
[
31
]
f
i
r
s
t
pr
opos
e
d
the
us
e
of
c
ha
in
r
u
l
e
s
to
s
ys
tema
ti
c
a
ll
y
c
a
lcula
te
gr
a
dients
in
ne
ur
a
l
ne
twor
ks
,
whic
h
is
the
ba
s
ic
c
onc
e
pt
in
ba
c
kpr
opa
ga
ti
on.
L
a
ter
,
R
umelha
r
t
e
t
al
.
[
32]
popular
ize
d
the
ba
c
kpr
opa
ga
ti
on
a
lgor
it
hm
by
us
ing
it
to
tr
a
in
n
e
ur
a
l
ne
twor
ks
with
mul
ti
ple
laye
r
s
.
Gouliana
s
e
t
al
.
[
33]
int
r
oduc
e
d
a
method
to
s
olve
a
s
ys
tem
of
nonli
ne
a
r
a
lgebr
a
ic
e
qua
ti
ons
by
us
ing
the
ba
c
kpr
opa
ga
ti
on
method.
F
o
r
e
xa
mpl
e
,
f
or
s
ys
tem
of
e
qua
ti
ons
with
e
qua
ti
ons
a
nd
unknown
va
r
iable
s
a
s
(
9)
:
1
(
)
=
1
(
1
,
2
,
…
,
)
=
11
11
(
)
+
12
12
(
)
+
⋯
+
1
,
1
1
1
(
)
−
1
=
0
2
(
)
=
2
(
1
,
2
,
…
,
)
=
21
21
(
)
+
22
22
(
)
+
⋯
+
2
,
2
2
2
(
)
−
2
=
0
⋮
(
)
=
(
1
,
2
,
…
,
)
=
1
1
(
)
+
2
2
(
)
+
⋯
+
,
(
)
−
=
0
(
9)
In
(
9
)
a
b
ov
e
i
s
a
na
lo
go
u
s
t
o
r
e
s
i
du
a
l
(
2
)
.
S
o,
f
or
e
x
a
m
pl
e
,
th
e
r
e
s
i
du
a
l
f
or
th
e
s
e
c
o
nd
gr
i
d
c
a
n
be
wr
i
tt
e
n
i
n
(
10)
:
2
(
1
,
2
,
3
)
=
2
(
1
,
2
,
)
(
10)
He
nc
e
,
we
ha
ve
the
f
o
ll
owing
(
11
)
:
2
(
)
1
+
2
(
)
2
+
2
(
)
3
=
21
21
(
)
+
22
22
(
)
+
23
23
(
)
(
11)
whe
r
e
21
(
)
=
2
(
)
1
;
22
(
)
=
2
(
)
2
;
23
(
)
=
2
(
)
3
a
nd
21
=
22
=
23
=
1
.
A
ne
ur
a
l
ne
twor
k
a
r
c
hit
e
c
tur
e
with
f
our
laye
r
s
c
a
n
be
f
or
med
a
s
s
hown
in
F
igur
e
3
.
T
he
we
ight
va
lue
c
onne
c
ti
ng
the
f
ir
s
t
a
nd
s
e
c
ond
laye
r
s
is
u
pda
ted
it
e
r
a
ti
ve
ly
to
ge
t
the
va
lue
that
is
c
los
e
s
t
to
the
s
olut
ion.
T
he
n
,
the
we
ight
that
c
onne
c
ts
laye
r
2
to
laye
r
3
r
e
pr
e
s
e
nts
the
f
unc
ti
on
(
)
.
W
hil
e
the
we
ight
on
laye
r
3
to
laye
r
4
p
uts
the
c
oe
f
f
icie
nt
in
the
e
qua
ti
on.
T
he
r
e
s
idual
e
qua
ti
on
in
(
2
)
doe
s
not
a
dd
a
f
ixed
c
ons
tant
s
o
that
thi
s
ne
ur
a
l
ne
twor
k
model
doe
s
no
t
include
b
ias
in
the
las
t
laye
r
o
r
o
ther
laye
r
s
.
T
he
it
e
r
a
ti
on
pr
oc
e
s
s
to
upda
te
the
s
olut
ion
is
c
a
r
r
ied
out
with
t
h
e
f
oll
owing
(
12)
:
+
1
=
−
(
)
∑
(
)
(
)
=
1
(
12)
with
(
)
is
the
a
da
pti
ve
lea
r
ning
r
a
te
pa
r
a
mete
r
a
t
e
a
c
h
it
e
r
a
ti
on
in
(
13
)
:
(
)
<
2
∑
(
(
)
)
2
=
1
(
13)
T
he
a
bove
s
ys
tem
of
nonli
ne
a
r
e
qua
ti
ons
is
s
im
il
a
r
to
the
r
e
s
idual
e
qua
ti
ons
f
ound
in
r
e
s
e
r
voir
s
im
ulations
.
One
ne
ur
on
in
laye
r
1
ha
s
a
f
ixed
va
lue
of
1.
I
n
laye
r
2
,
thi
s
va
lue
is
the
pr
e
s
s
ur
e
s
olut
ion
that
will
be
s
olved
f
or
e
a
c
h
gr
id
s
o
the
nu
mber
of
ne
ur
ons
will
de
pe
nd
on
the
number
of
gr
ids
in
the
mo
de
l.
T
he
ne
ur
ons
in
laye
r
3
r
e
p
r
e
s
e
nt
the
th
r
e
e
ter
ms
in
t
he
lef
t
-
ha
nd
s
e
gment
of
the
r
e
s
idual
e
qua
ti
on
.
W
hil
e
the
ne
ur
on
in
laye
r
4
is
the
output
laye
r
whos
e
va
lue
is
e
xpe
c
ted
to
be
c
los
e
to
z
e
r
o.
T
a
ble
2
s
hows
the
n
umber
of
ne
ur
ons
in
e
a
c
h
ne
twor
k
laye
r
.
T
a
ble
2
.
Numbe
r
o
f
ne
ur
ons
in
e
a
c
h
ne
twor
k
laye
r
N
umbe
r
of
gr
id
s
N
umbe
r
of
ne
ur
ons
L
a
ye
r
1
L
a
ye
r
2
L
a
ye
r
3
L
a
ye
r
4
10
1
10
30
10
25
1
25
75
25
50
1
50
150
50
Evaluation Warning : The document was created with Spire.PDF for Python.
I
nt
J
Ar
ti
f
I
ntell
I
S
S
N:
2252
-
8938
B
ac
k
pr
opagati
on
ne
ur
al
ne
tw
or
k
s
for
s
olving
gas
fl
ow
e
quati
ons
in
por
ous
me
dia
(
A
dr
iant
o)
3749
F
igur
e
3
.
B
P
NN
f
o
r
s
olvi
ng
nonli
ne
a
r
e
qua
ti
ons
[
3
3]
3.
RE
S
UL
T
S
AN
D
DI
S
CU
S
S
I
ON
3.
1.
M
od
e
l
va
li
d
at
ion
W
e
va
li
da
te
d
the
s
im
ulation
r
e
s
ult
s
by
c
ompar
ing
the
pr
e
s
s
ur
e
dis
tr
ibut
ion
s
olut
ion
f
r
om
the
c
a
lcula
ti
on
us
ing
the
B
P
NN
a
ppr
oa
c
h
with
the
s
o
lut
ion
f
r
om
the
Ne
wton
method.
C
ons
tant
pr
e
s
s
ur
e
a
t
the
lef
t
bounda
r
y
c
ondit
ion
a
nd
no
f
low
a
t
the
r
ight
bo
unda
r
y
c
ondit
ion
lea
d
to
the
pr
e
s
s
ur
e
dis
tr
ibut
ion
p
r
of
il
e
a
s
s
e
e
n
in
F
igur
e
4
.
B
e
c
a
us
e
the
pe
r
mea
bil
it
y
of
the
he
ter
oge
ne
ous
model
(
10
-
20
md)
is
r
e
latively
higher
than
the
homogene
ous
model
(
1
md)
,
the
pr
e
s
s
ur
e
dr
op
in
the
he
ter
oge
ne
ous
model
is
much
f
a
s
ter
.
Highe
r
pe
r
mea
bil
i
ty
f
a
c
il
it
a
tes
the
mor
e
e
f
f
icie
nt
moveme
nt
of
f
lui
ds
th
r
ough
the
r
oc
k
matr
ix
[
34]
,
[
3
5]
.
T
his
indi
c
a
tes
that
f
lui
d
c
a
n
be
quickly
e
xt
r
a
c
ted
f
r
om
t
he
r
e
s
e
r
voir
upon
the
be
ginni
ng
of
p
r
oduc
ti
on,
r
e
s
ult
ing
in
a
mor
e
r
a
pid
de
c
r
e
a
s
e
in
pr
e
s
s
ur
e
.
At
the
e
nd
of
the
s
im
ulation
ti
me
of
6.
9
da
ys
,
the
pr
e
s
s
ur
e
a
t
the
r
ight
e
nd
dr
ops
to
4987
ps
i
a
n
d
4962
ps
i
f
o
r
the
ho
mogene
ous
a
nd
he
ter
oge
ne
ous
models
,
r
e
s
pe
c
ti
ve
ly.
F
igur
e
4
s
hows
that
the
pr
e
s
s
ur
e
s
olut
ions
on
e
a
c
h
gr
id
a
r
e
in
go
od
a
gr
e
e
ment
be
twe
e
n
the
pr
e
s
s
ur
e
point
s
f
r
om
t
he
B
P
P
N
a
ppr
oa
c
h
a
nd
f
r
om
Ne
wton's
method
in
both
the
homogene
ous
model
i
n
F
igur
e
4(
a
)
a
nd
the
he
ter
oge
ne
ous
model
in
F
igur
e
4(
b)
.
T
he
li
ne
s
de
note
the
s
olut
ion
of
the
Ne
wton
method,
while
the
s
ymbol
s
de
note
the
s
olut
ion
of
the
B
P
NN
method
.
(
a
)
(
b)
F
igur
e
4
.
P
r
e
s
s
ur
e
dis
tr
ibut
ion
in
(
a
)
homogene
ous
model
a
nd
(
b)
he
ter
oge
ne
ous
model
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
S
N
:
2252
-
8938
I
nt
J
Ar
ti
f
I
ntell
,
Vol.
14
,
No.
5
,
Oc
tober
20
25
:
37
44
-
3756
3750
F
igur
e
5
s
hows
the
pr
e
s
s
ur
e
e
r
r
or
incr
e
a
s
ing
ove
r
ti
me
in
both
the
homogene
ous
model
in
F
igur
e
5
(
a
)
a
nd
the
he
ter
oge
ne
ous
model
in
F
igu
r
e
5(
b)
.
T
his
may
be
due
to
the
tr
unc
a
ti
on
e
r
r
or
of
the
f
ini
te
dif
f
e
r
e
nc
e
method
gr
owing
a
s
ti
me
pr
ogr
e
s
s
e
s
[
36]
,
[
37
]
.
M
or
e
ove
r
,
f
ini
te
pr
e
c
is
ion
in
n
umer
ica
l
c
omput
a
ti
ons
c
a
us
e
s
r
ound
-
of
f
e
r
r
or
s
.
T
he
s
e
s
mall
inac
c
ur
a
c
ies
c
a
n
a
c
c
umul
a
te
ove
r
a
long
s
im
ula
ti
on
a
nd
a
f
f
e
c
t
s
olut
ion
a
c
c
ur
a
c
y
[
38
]
,
[
39]
.
T
o
ove
r
c
ome
thi
s
,
s
ome
r
e
s
e
a
r
c
he
r
s
us
ua
ll
y
us
e
high
-
or
de
r
f
ini
te
dif
f
e
r
e
nc
e
s
[
40]
,
[
41]
or
us
e
s
maller
ti
me
s
teps
dur
ing
c
r
i
ti
c
a
l
pa
r
ts
of
the
s
olut
ion
[
42
]
.
How
e
ve
r
,
it
s
hows
that
the
magnitude
of
the
pr
e
s
s
ur
e
e
r
r
or
is
ve
r
y
s
mall
with
maximum
a
bs
olut
e
e
r
r
or
(
M
AE
)
of
2
.
41
×
10
−
7
ps
i
a
nd
4
.
18
×
10
−
8
ps
i
in
the
homogene
ous
a
nd
he
ter
oge
ne
o
us
models
.
T
his
ind
ica
tes
that
B
P
NN
pr
oduc
e
s
a
n
a
c
c
ur
a
te
s
olut
ion
in
p
r
e
dicting
the
p
r
e
s
s
ur
e
dis
tr
ibut
ion
in
thi
s
model.
(
a
)
(
b)
F
igur
e
5
.
Abs
olut
e
e
r
r
o
r
o
f
pr
e
s
s
ur
e
s
be
twe
e
n
NN
s
olver
a
nd
Ne
wton
method
in
(
a
)
homogene
ous
m
ode
l
a
nd
(
b)
he
ter
oge
ne
ous
model
3.
2.
Num
b
e
r
of
it
e
r
at
ion
s
T
o
a
c
hieve
a
n
e
f
f
icie
nt
B
P
NN
c
omput
a
ti
on
,
Gou
li
a
na
s
[
33]
r
e
c
omm
e
nds
s
e
tt
ing
the
lea
r
ning
r
a
te
be
twe
e
n
0
a
nd
2.
T
he
r
e
f
or
e
,
w
e
c
ompar
e
d
the
le
a
r
ning
r
a
te
pa
r
a
mete
r
whic
h
is
a
hype
r
pa
r
a
mete
r
in
ne
ur
a
l
ne
twor
ks
with
va
lues
f
r
om
0.
1
to
0.
9
with
a
r
a
n
ge
of
0.
2
.
F
igur
e
6
s
hows
the
pe
r
f
or
manc
e
o
f
th
e
B
P
NN
s
olver
a
s
indi
c
a
ted
by
the
a
ve
r
a
ge
it
e
r
a
ti
ons
a
ga
ins
t
the
a
da
pti
ve
lea
r
ning
r
a
te
pa
r
a
mete
r
on
the
ho
mo
ge
ne
ous
model
in
F
igur
e
6(
a
)
a
nd
the
he
ter
oge
ne
ous
mo
d
e
l
in
F
igur
e
6(
b
)
.
I
t
s
hows
that
the
i
ter
a
ti
ons
r
e
q
uir
e
d
to
obtain
the
s
olut
ion
on
e
a
c
h
ti
mes
tep
incr
e
a
s
e
by
us
ing
a
f
ine
r
gr
id
.
I
nc
r
e
a
s
ing
the
r
e
s
olut
ion
of
the
gr
id
r
e
s
ult
s
in
a
g
r
e
a
ter
number
of
unknowns
while
s
olvi
ng
is
s
ue
s
,
thus
incr
e
a
s
e
s
the
s
ize
a
nd
c
ompl
e
xit
y
of
the
s
ys
tem
of
e
qua
ti
ons
a
s
we
ll
a
s
t
he
dim
e
ns
ion
J
a
c
obian
matr
ix
.
Nonlinea
r
it
ies
in
the
gove
r
ning
e
qua
ti
ons
may
be
higher
on
f
iner
g
r
ids
.
T
he
r
e
f
or
e
,
a
dd
r
e
s
s
ing
thes
e
nonli
ne
a
r
it
ies
may
ne
c
e
s
s
it
a
te
mor
e
it
e
r
a
ti
ons
,
a
s
t
he
a
ppr
oa
c
h
mus
t
c
onti
nuous
ly
li
ne
a
r
ize
a
nd
r
e
s
olve
the
s
ys
tem.
M
or
e
ove
r
,
he
ter
oge
ne
ous
models
a
ls
o
a
r
e
mor
e
li
ke
ly
to
r
e
quir
e
a
lar
ge
r
number
of
it
e
r
a
ti
ons
.
T
he
he
ter
oge
ne
ous
model
with
50
gr
ids
r
e
quir
e
the
hi
ghe
s
t
number
of
it
e
r
a
ti
ons
a
mong
other
models
.
T
his
may
be
c
a
us
e
phys
ica
l
va
r
iable
s
li
ke
pe
r
mea
bil
it
y
a
nd
por
os
it
y
e
xhibi
t
loca
li
z
e
d
va
r
iabili
ty
,
r
e
s
ult
ing
in
a
mo
r
e
c
ompl
e
x
s
e
t
of
e
qua
ti
ons
to
s
olve.
S
igni
f
ica
nt
va
r
i
a
ti
ons
in
thes
e
c
ha
r
a
c
ter
is
ti
c
s
mi
ght
lea
d
to
s
ha
r
p
gr
a
dients
that
r
e
quir
e
mor
e
it
e
r
a
ti
ons
[
43]
.
I
n
a
ddit
ion,
e
r
r
or
s
a
r
is
ing
a
t
loca
ti
ons
with
majo
r
pr
ope
r
ty
dif
f
e
r
e
nc
e
s
(
e
.
g.
,
bounda
r
ies
be
twe
e
n
high
-
a
nd
low
-
p
e
r
mea
bil
it
y
z
one
s
)
c
ould
pr
opa
ga
te
a
c
r
os
s
the
s
olut
ion
domain,
lea
ding
to
a
ddit
ional
it
e
r
a
ti
ons
f
o
r
c
or
r
e
c
ti
ons
[
44]
,
[
45]
.
I
n
e
a
c
h
s
im
ul
a
t
io
n
r
un,
we
s
e
t
a
t
ol
e
r
a
n
c
e
of
a
b
s
ol
ut
e
m
a
x
im
u
m
e
r
r
or
of
10
−
6
p
s
i,
s
o
th
a
t
t
he
i
te
r
a
ti
on
s
o
f
th
e
N
e
wt
on
a
n
d
B
P
N
N
m
e
t
ho
d
s
wi
ll
c
o
nt
in
ue
t
o
r
u
n
u
nt
il
t
he
y
r
e
a
c
h
th
a
t
li
m
it
.
F
i
gu
r
e
7
s
h
o
w
s
t
he
d
e
c
r
e
a
s
e
i
n
r
e
s
i
du
a
l
e
r
r
or
a
s
i
t
e
r
a
ti
on
s
pr
o
gr
e
s
s
f
o
r
t
he
h
omo
g
e
n
e
o
u
s
m
od
e
l
i
n
F
i
gu
r
e
7
(
a
)
a
nd
t
h
e
he
t
e
r
o
ge
n
e
o
u
s
mo
de
l
in
F
ig
ur
e
7(
b
)
.
I
t
s
ho
w
s
t
ha
t
B
P
N
N
r
e
qu
ir
e
s
mor
e
it
e
r
a
t
io
n
s
th
a
n
Ne
wt
on
m
e
t
ho
d
in
a
ll
c
a
s
e
s
.
A
n
in
ter
e
s
t
in
g
f
i
nd
in
g
f
r
om
F
i
gur
e
7
i
s
t
h
a
t
th
e
N
e
wt
on
m
e
th
od
r
e
qu
ir
e
s
f
e
w
e
r
it
e
r
a
ti
on
s
a
t
th
e
b
e
gi
nni
ng
of
t
he
s
im
ul
a
t
io
n
a
n
d
t
h
e
n
in
c
r
e
a
s
e
s
.
T
h
e
v
ic
e
v
e
r
s
a
i
s
ob
s
e
r
ve
d
f
or
B
P
N
N,
wh
e
r
e
mor
e
i
te
r
a
ti
on
s
a
r
e
r
e
q
uir
e
d
a
t
t
he
be
gi
nn
in
g
of
th
e
s
im
u
l
a
t
io
n
bu
t
d
e
c
r
e
a
s
e
a
s
th
e
s
im
ul
a
t
io
n
pr
ogr
e
s
s
e
s
.
T
hi
s
m
a
y
be
d
u
e
to
t
he
a
b
il
i
ty
of
B
P
NN
t
o
l
e
a
r
n
th
e
p
a
t
te
r
n
of
p
r
e
s
s
ur
e
dr
o
p
s
o
th
a
t
f
e
w
e
r
i
te
r
a
ti
on
s
a
r
e
r
e
q
uir
e
d
a
t
t
he
e
n
d
o
f
s
im
ul
a
t
io
n.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
nt
J
Ar
ti
f
I
ntell
I
S
S
N:
2252
-
8938
B
ac
k
pr
opagati
on
ne
ur
al
ne
tw
or
k
s
for
s
olving
gas
fl
ow
e
quati
ons
in
por
ous
me
dia
(
A
dr
iant
o)
3751
(
a
)
(
b)
F
igur
e
6
.
N
umber
s
o
f
it
e
r
a
ti
on
a
t
s
e
ve
r
a
l
a
da
pti
ve
l
e
a
r
ning
r
a
te
pa
r
a
mete
r
s
(
AL
R
P
)
in
(
a
)
homogene
ous
model
a
nd
(
b)
he
ter
oge
ne
ous
mo
de
l
(
a
)
(
b)
F
igur
e
7
.
T
he
de
c
r
e
a
s
e
in
r
e
s
iduals
a
t
e
a
c
h
it
e
r
a
ti
o
n
f
or
a
ll
c
a
s
e
s
tudi
e
s
of
(
a
)
homogene
ous
model
a
n
d
(
b)
he
ter
oge
ne
ous
model
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
S
N
:
2252
-
8938
I
nt
J
Ar
ti
f
I
ntell
,
Vol.
14
,
No.
5
,
Oc
tober
20
25
:
37
44
-
3756
3752
3.
3.
Com
p
u
t
at
io
n
t
im
e
C
omput
a
ti
on
ti
me
indi
c
a
tes
the
e
f
f
icie
nc
y
of
a
n
onli
ne
a
r
s
olver
in
r
e
a
c
hing
a
s
olut
ion.
F
igur
e
8
s
hows
the
e
f
f
e
c
t
of
the
lea
r
ning
r
a
te
pa
r
a
mete
r
on
the
c
omput
a
ti
on
ti
me
to
r
e
a
c
h
a
c
onve
r
ge
d
s
olut
io
n
f
or
the
homogene
ous
model
in
F
igur
e
8
(
a
)
a
nd
the
he
ter
o
ge
ne
ous
model
in
F
igu
r
e
8
(
b)
.
It
s
hows
that
the
e
f
f
icie
nc
y
of
the
B
P
NN
method
is
a
f
f
e
c
ted
by
the
lea
r
ning
r
a
t
e
whic
h
is
s
e
t
on
the
ne
twor
k
.
I
n
ge
ne
r
a
l,
a
lea
r
nin
g
r
a
te
of
0.
1
r
e
qui
r
e
s
the
mos
t
c
omput
a
ti
on
ti
me
c
ompar
e
d
to
the
other
s
in
a
ll
c
a
s
e
s
.
T
he
c
omput
a
ti
on
t
im
e
will
de
c
r
e
a
s
e
a
s
the
lea
r
ning
r
a
te
incr
e
a
s
e
s
unti
l
a
c
e
r
tain
point
whe
r
e
incr
e
a
s
ing
the
lea
r
ning
r
a
te
wil
l
inc
r
e
a
s
e
the
c
omput
a
ti
on
ti
me.
T
he
lea
r
ning
r
a
te
de
ter
mi
ne
s
t
h
e
s
ize
of
the
s
teps
take
n
in
the
t
r
a
ini
ng
pr
oc
e
s
s
of
t
he
ne
ur
a
l
ne
twor
k,
e
s
pe
c
ially
in
mi
n
im
izing
the
los
s
f
un
c
ti
on
[
46]
.
A
s
mall
lea
r
ning
r
a
te
c
a
n
a
void
th
e
r
is
k
of
ove
r
s
hooti
ng
but
c
ons
e
que
ntl
y
take
s
mor
e
ti
me.
W
hil
e
,
a
lar
ge
lea
r
ning
r
a
te
c
a
n
s
pe
e
d
up
the
c
on
ve
r
ge
nc
e
r
a
te
but
ha
s
the
r
is
k
of
ove
r
s
hooti
ng
.
I
n
the
homog
e
ne
ous
model,
it
wa
s
f
ound
that
the
opti
mum
lea
r
ning
r
a
te
is
1.
1,
1.
3
,
a
nd
1.
9
f
o
r
the
number
of
gr
ids
10
,
2
5
a
nd
50
.
W
hil
e
in
the
he
ter
oge
ne
ous
model,
the
opti
mum
lea
r
ning
r
a
te
is
a
t
1.
1
,
1
.
7
,
a
nd
1
.
9
f
o
r
the
number
of
gr
ids
10
,
25
,
a
nd
50.
(
a
)
(
b)
F
igur
e
8
.
C
omput
a
ti
on
t
im
e
f
o
r
(
a
)
homogene
ous
model
a
nd
(
b
)
he
ter
oge
ne
ous
model
T
a
ble
3
s
hows
that
B
P
NN
is
not
a
s
e
f
f
icie
nt
a
s
Ne
wton
method
in
s
olvi
ng
the
s
ingl
e
-
pha
s
e
ga
s
f
low
in
thi
s
s
tudy.
On
homogene
ous
models
,
the
c
om
putation
ti
me
of
B
P
NN
is
a
bout
double
that
o
f
Ne
wton
method.
M
e
a
nwhile,
the
dif
f
e
r
e
nc
e
in
c
omput
a
ti
on
ti
me
incr
e
a
s
e
s
non
-
li
ne
a
r
ly
in
the
he
ter
oge
ne
ou
s
model.
T
he
di
f
f
e
r
e
nc
e
is
a
bout
1
.
5
ti
mes
in
the
he
ter
oge
n
e
ous
model
with
10
g
r
ids
,
while
it
is
10
ti
mes
in
t
he
model
with
50
gr
ids
.
T
he
inef
f
icie
nc
y
of
B
P
NN
may
be
d
ue
to
the
na
tur
e
o
f
B
P
NN
whic
h
invol
ve
s
f
o
r
wa
r
d
pa
s
s
a
nd
Evaluation Warning : The document was created with Spire.PDF for Python.
I
nt
J
Ar
ti
f
I
ntell
I
S
S
N:
2252
-
8938
B
ac
k
pr
opagati
on
ne
ur
al
ne
tw
or
k
s
for
s
olving
gas
fl
ow
e
quati
ons
in
por
ous
me
dia
(
A
dr
iant
o)
3753
ba
c
kwa
r
d
pa
s
s
s
o
that
the
c
a
lcula
ti
on
is
e
xpe
ns
ive
[
47]
.
M
or
e
ove
r
,
the
f
low
model
in
thi
s
s
tudy
is
r
e
latively
s
im
ple
s
ince
it
is
only
one
pha
s
e
a
nd
invol
ve
s
onl
y
a
bout
a
doz
e
n
gr
ids
w
he
r
e
Ne
wton
method
is
k
nown
f
or
it
s
e
f
f
icie
nc
y
on
s
mall
pr
oblem
d
im
e
ns
ions
[
48]
.
W
e
e
xa
mi
ne
d
c
omput
a
ti
ona
l
pe
r
f
or
manc
e
a
t
va
r
i
ous
Δ
t
va
lues
:
0.
0138s
,
0.
0069
s
,
a
nd
0.
000345s
.
R
e
duc
ing
time
-
s
tep
s
ize
s
(
Δ
)
incr
e
a
s
e
c
omput
a
ti
o
na
l
c
os
ts
f
or
bo
th
methods
a
c
r
os
s
a
ll
models
a
n
d
gr
id
r
e
s
olut
ions
.
F
or
e
xa
mpl
e
,
in
the
homogene
ous
mo
de
l
with
50
gr
ids
,
the
Ne
wton
method
take
s
30
.
5
s
e
c
onds
f
or
Δ
t
=
0
.
0
1
3
8
s
,
45.
5
s
e
c
onds
f
or
Δ
t
=
0
.
0069s
,
a
nd
69.
9
s
e
c
onds
f
or
Δ
t
=
0
.
0
0
0
3
4
5
s
.
I
nc
r
e
a
s
ing
the
number
of
g
r
ids
s
igni
f
ica
ntl
y
incr
e
a
s
e
s
c
omput
a
ti
ona
l
ti
me
f
o
r
both
methods
.
T
his
e
f
f
e
c
t
is
mor
e
pr
o
nounc
e
d
in
the
B
P
NN
a
ppr
oa
c
h.
F
or
ins
tanc
e
,
in
the
homogene
ous
model
with
Δ
t
=
0
.
0
1
3
8
s
,
B
P
NN
take
s
4.
3
s
e
c
onds
f
or
10
g
r
ids
but
178.
9
s
e
c
onds
f
or
50
gr
ids
.
T
he
pe
r
f
or
manc
e
ga
p
be
twe
e
n
Ne
wton
a
nd
B
P
NN
is
mor
e
pr
onounc
e
d
in
models
with
highe
r
gr
id
r
e
s
olut
ions
a
nd
s
maller
ti
me
s
teps
.
F
or
e
xa
mpl
e
,
in
the
he
ter
oge
ne
ous
model
with
50
gr
ids
a
nd
Δ
t
=
0
.
006
9
s
,
the
Ne
wton
method
take
s
90.
4
s
e
c
onds
c
ompar
e
d
to
610.
9
s
e
c
onds
f
o
r
B
P
NN
.
T
he
c
omput
a
ti
ona
l
ti
m
e
s
f
or
the
he
ter
oge
ne
ous
model
a
r
e
typi
c
a
ll
y
lon
ge
r
than
thos
e
f
or
the
homogene
ous
one
,
indi
c
a
ti
ng
the
gr
e
a
ter
c
ompl
e
xit
y
invol
ve
d
in
s
olvi
ng
he
ter
oge
ne
o
us
gr
ids
.
F
or
e
xa
mpl
e
,
a
t
Δ
=
0
.
0138
a
nd
50
g
r
ids
,
the
Ne
wton
method
r
e
quir
e
s
30.
5
s
e
c
onds
f
or
the
homogene
ous
model
a
nd
63.
3
s
e
c
onds
f
or
the
he
ter
oge
ne
ous
model.
I
n
a
ddit
ion
,
the
dif
f
e
r
e
nc
e
in
c
omput
a
ti
on
ti
me
a
ls
o
looks
nonli
ne
a
r
a
s
s
hown
in
T
a
ble
4
.
T
a
ble
3
.
C
omput
a
ti
on
ti
me
f
or
B
P
NN
a
ppr
oa
c
h
a
n
d
Ne
wton
method
M
ode
l
C
omput
a
ti
ona
l
ti
me
(
s
e
c
onds
)
Δ
=
0
.
0138
s
Δ
=
0
.
0069
s
(
ba
s
e
c
a
s
e
)
Δ
=
0
.
000345
s
N
e
w
to
n
B
P
N
N
N
e
w
to
n
B
P
N
N
N
e
w
to
n
B
P
N
N
H
omoge
ne
ous
10 gr
id
s
2.3
4.3
3.8
7.9
7.3
16.7
25
gr
id
s
8.9
23.5
14.8
30.7
24.5
66.9
50 gr
id
s
30.5
178.9
45.5
172.3
69.9
518
H
e
te
r
oge
ne
ous
10 gr
id
s
3.3
6.1
6.2
8.1
12.6
19.5
25 gr
id
s
15.1
76.1
22.5
73.2
36.6
198.4
50 gr
id
s
63.3
547.99
90.4
610.9
104.1
496.6
T
a
ble
4
.
R
e
lative
c
omput
a
ti
ona
l
ti
me
of
B
P
NN
wit
h
Ne
wton
method
M
ode
l
Δ
=
0
.
0138
s
Δ
=
0
.
0069
s
(
ba
s
e
c
a
s
e
)
Δ
=
0
.
000345
s
H
omoge
ne
ous
10 gr
id
s
1.8
2
2.2
25 gr
id
s
2.6
2
2.7
50 gr
id
s
5.8
3.7
7.4
H
e
te
r
oge
ne
ous
10 gr
id
s
1.8
1.3
1.5
25 gr
id
s
5
3.2
5.4
50 gr
id
s
8.6
6.7
4.7
4.
CONC
L
USI
ON
T
his
r
e
s
e
a
r
c
h
pr
e
s
e
nts
a
nove
l
method
f
or
s
olvi
ng
nonli
ne
a
r
e
qua
ti
ons
in
r
e
s
e
r
voir
s
im
ulation
ba
s
e
d
on
B
P
NN
.
T
he
s
tudy
s
uc
c
e
s
s
f
ull
y
de
mons
tr
a
tes
the
s
olver
's
a
bil
it
y
to
p
r
oduc
e
highl
y
a
c
c
ur
a
te
s
olut
ions
,
whic
h
we
r
e
ve
r
if
i
e
d
a
ga
ins
t
the
c
las
s
ic
Ne
wton
method.
T
he
pr
e
s
s
ur
e
s
olut
ions
a
c
hieve
d
a
M
AE
of
only
2
.
41
×
10
−
7
ps
i
a
nd
4
.
18
×
10
−
8
ps
i
f
o
r
homogene
ous
models
a
nd
he
t
e
r
oge
ne
ous
models
,
r
e
s
pe
c
ti
ve
ly.
How
e
ve
r
,
a
de
tailed
pe
r
f
o
r
manc
e
a
na
lys
is
r
e
ve
a
ls
that
the
B
P
NN
s
olver
,
in
it
s
c
ur
r
e
nt
f
o
r
m,
is
les
s
c
omput
a
ti
ona
ll
y
e
f
f
icie
nt
than
the
Ne
wton
method
f
or
the
pr
ob
lems
s
tudi
e
d.
R
e
ga
r
ding
c
omput
a
ti
on
ti
me,
the
B
P
NN
a
ppr
oa
c
h
wa
s
a
ppr
oxim
a
tely
twice
a
s
s
low
f
or
homogene
ous
models
.
T
his
pe
r
f
o
r
manc
e
ga
p
wide
ne
d
non
-
li
ne
a
r
ly
f
or
mo
r
e
c
ompl
e
x,
he
ter
oge
ne
ous
models
,
whe
r
e
the
B
P
NN
s
olver
wa
s
be
twe
e
n
1.
5
to
10
ti
mes
s
lowe
r
,
de
pe
nding
on
the
gr
id
r
e
s
olut
ion.
T
he
nu
mber
of
gr
ids
,
r
oc
k
he
ter
oge
ne
it
y
,
a
nd
the
a
da
pti
v
e
lea
r
ning
r
a
te
pa
r
a
mete
r
he
a
vil
y
inf
luenc
e
the
s
olver
's
e
f
f
ici
e
nc
y.
T
he
s
im
ulatio
n
r
e
s
ult
s
s
howe
d
that
while
th
e
B
P
NN
method
r
e
quir
e
d
mo
r
e
it
e
r
a
ti
ons
ove
r
a
ll
,
it
e
xhibi
ted
a
lea
r
ning
be
ha
vior
;
the
i
ter
a
ti
on
ne
e
de
d
in
e
a
c
h
ti
me
s
tep
de
c
r
e
a
s
e
d
ove
r
the
s
im
ulation
ti
me,
in
c
ont
r
a
s
t
to
the
Ne
wton
method
.
T
h
is
s
ugge
s
ts
that
the
ne
twor
k
a
da
p
ts
to
the
s
olut
ion's
pa
tt
e
r
n
ove
r
ti
me
.
T
he
s
e
lec
ti
on
of
a
n
opti
mal
lea
r
ning
r
a
te
wa
s
a
l
s
o
c
r
it
i
c
a
l,
a
s
it
s
igni
f
ica
ntl
y
im
pa
c
ted
c
onve
r
ge
nc
e
s
pe
e
d
a
nd
c
omput
a
ti
ona
l
c
os
t.
F
ur
ther
r
e
s
e
a
r
c
h
s
hould
tes
t
th
e
B
P
NN
method
on
mor
e
c
ompl
e
x
f
lui
d
models
a
nd
inves
t
igate
dif
f
e
r
e
nt
ne
twor
k
a
r
c
hit
e
c
tu
r
e
s
a
nd
opti
mi
z
a
ti
ons
to
im
pr
ove
it
s
e
f
f
icie
nc
y
f
or
lar
ge
r
s
im
ulations
.
Evaluation Warning : The document was created with Spire.PDF for Python.