electronics
Article
Extraction of Interconnect Parasitic Capacitance Matrix Based
on Deep Neural Network
Yaoyao Ma 1,2,3 , Xiaoyu Xu 4 , Shuai Yan 4 , Yaxing Zhou 4 , Tianyu Zheng 4 , Zhuoxiang Ren 4,5, *
and Lan Chen 1,3
1
2
3
4
5
*
Citation: Ma, Y.; Xu, X.; Yan, S.; Zhou,
Y.; Zheng, T.; Ren, Z.; Chen, L.
Extraction of Interconnect Parasitic
Capacitance Matrix Based on Deep
Neural Network. Electronics 2023, 12,
1440. https://doi.org/10.3390/
Institute of Microelectronics of the Chinese Academy of Sciences, Beijing 100029, China
University of Chinese Academy of Sciences, Beijing 100049, China
Beijing Key Laboratory of Three-Dimensional and Nanometer Integrated Circuit Design Automation Technology,
Beijing 100029, China
Institute of Electrical Engineering, Chinese Academy of Sciences, Beijing 100190, China
Group of Electrical and Electronic Engineering of Paris, Sorbonne Université, Université Paris-Saclay,
CentraleSupélec, CNRS, 75005 Paris, France
Correspondence: zhuoxiang.ren@upmc.fr
Abstract: Interconnect parasitic capacitance extraction is crucial in analyzing VLSI circuits’ delay
and crosstalk. This paper uses the deep neural network (DNN) to predict the parasitic capacitance
matrix of a two-dimensional pattern. To save the DNN training time, the neural network’s output
includes only coupling capacitances in the matrix, and total capacitances are obtained by summing
corresponding predicted coupling capacitances. In this way, we can obtain coupling and total
capacitances simultaneously using a single neural network. Moreover, we introduce a mirror flip
method to augment the datasets computed by the finite element method (FEM), which doubles the
dataset size and reduces data preparation efforts. Then, we compare the prediction accuracy of
DNN with another neural network ResNet. The result shows that DNN performs better in this case.
Moreover, to verify our method’s efficiency, the total capacitances calculated from the trained DNN
are compared with the network (named DNN-2) that takes the total capacitance as an extra output.
The results show that the prediction accuracy of the two methods is very close, indicating that our
method is reliable and can save the training workload for the total capacitance. Finally, a solving
efficiency comparison shows that the average computation time of the trained DNN for one case is
not more than 2% of that of FEM.
Keywords: interconnect wire; parasitic capacitance matrix; data augmentation; DNN; ResNet
electronics12061440
Academic Editors: Andrea Prati, Luis
Javier García Villalba and Vincent
A. Cicirello
Received: 22 February 2023
Revised: 15 March 2023
Accepted: 16 March 2023
Published: 17 March 2023
Copyright: © 2023 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
1. Introduction
With the shrinking of the feature size, the interconnect wires’ delay caused by the
parasitic parameters has exceeded the gate delay and becomes the main part of the chip’s
total delay [1]. Additionally, parasitic parameters of interconnect wires also cause crosstalk
between wires, which may lead to confusion in the transmitted signal logic. Parasitic
parameters consist of resistance, capacitance, and inductance; among them, parasitic
capacitance plays a significant role owing to its extraction complexity and critical influence
on signal integrity analysis [2,3].
In advanced process technology nodes, the accuracy requirement of the parasitic
capacitance extraction is that the relative error should be less than 5% [4]. The parasitic
capacitance extraction methods can be divided into two categories: the field solver method
and the pattern-matching method [5]. The field solver method solves maxwell equations
through traditional numerical methods, such as floating random walk (FRW) [6], finite
element method (FEM) [7], and boundary element method (BEM) [8,9], etc. It has high
accuracy but suffers high calculation costs for full chip extraction. The pattern-matching
method first segments the layout and obtains the interconnect wires’ patterns; then matches
Electronics 2023, 12, 1440. https://doi.org/10.3390/electronics12061440
https://www.mdpi.com/journal/electronics
Electronics 2023, 12, 1440
2 of 18
the corresponding pattern in the pattern library; in the end, calculates each pattern’s
of is19
through pre-built capacitance models that are stored in the pattern library.3 It
fast and is often used for parasitic extraction at the full chip level. Figure 1 shows the flow
of the pattern-matching method. The capacitance extraction dimension changes from 1D,
2D,
and 2.5D
to1D,
3D 2D,
and and
the extraction
is improved
accordingly
[10]. Directly
using
changes
from
2.5D to 3Daccuracy
and the extraction
accuracy
is improved
accordingly
the
3D
model
will
cause
huge
memory
and
time
cost
for
the
full-chip
extraction
[11].
The
[10]. Directly using the 3D model will cause huge memory and time cost for the full-chip
2.5D
method[11].
is anThe
alleviation
of 3D is
method,
which considers
the 3D which
effect by
a combination
extraction
2.5D method
an alleviation
of 3D method,
considers
the 3D
ofeffect
two by
orthogonal
2D patterns
It achieves
a good balance
betweenaaccuracy
and
a combination
of two[12].
orthogonal
2D patterns
[12]. It achieves
good balance
efficiency
is usedand
in several
commercial
electronic
design commercial
automation (EDA)
tools,design
such
between and
accuracy
efficiency
and is used
in several
electronic
as
StarRC [13](EDA)
and Calibre
xRC as
[14].
automation
tools, such
StarRC [13] and Calibre xRC [14].
Electronics 2023, 12, x FOR PEER REVIEW
capacitance
Figure1.1.The
Theflow
flowofofparasitic
parasiticcapacitance
capacitanceextraction
extractionusing
usingthe
thepattern-matching
pattern-matchingmethod
method[3,15].
[3,15].
Figure
With
Withthe
thecontinuous
continuousdevelopment
developmentofofbig
bigdata
datatechnology
technologyand
andhigh-performance
high-performance
computing,
computing,machine
machinelearning
learningand
anddeep
deeplearning
learninghave
havebeen
beenapplied
appliedtotomany
manyfields
fields[16].
[16].In
In
addition
additionto
tocommon
commonfields
fieldssuch
suchas
asimage
imageanalysis
analysisand
andnatural
naturallanguage
languageprocessing
processing[17],
[17],
they
been
implemented
in the
electronic
and electrical
engineering
[18–20].
theyhave
havealso
also
been
implemented
infield
the of
field
of electronic
and electrical
engineering
Some
researchers
use
machine
learning
and
deep
learning
to
solve
the
problem
of
parasitic
[18,19,20]. Some researchers use machine learning and deep learning to solve the problem
parameter
extraction.
Kasai
et al. [21]
employed
multilayer
perception
(MLP)
to extract
the
of parasitic
parameter
extraction.
Kasai
et al. [21]
employed
multilayer
perception
(MLP)
total
capacitance
and
partial
coupling
capacitances
of
3D
interconnect
wires.
The
dimension
to extract the total capacitance and partial coupling capacitances of 3D interconnect wires.
ofThe
thedimension
MLP output
is only
one.layer
Hence,
it needs
train multiple
of layer
the MLP
output
is only
one.toHence,
it needs neural
to trainnetworks
multiple
toneural
predict
the totaltocapacitance
capacitances
of thecapacitances
same pattern.
Besides,
networks
predict theand
totalcoupling
capacitance
and coupling
of the
same
its
prediction
error
for
total
capacitance
was
not
very
good
since
some
errors
exceeded
pattern. Besides, its prediction error for total capacitance was not very good since some
10%.
Liexceeded
et al. [22]10%.
combined
the[22]
adjacent
capacitance
formulas
throughformulas
the Levenberg–
errors
Li et al.
combined
the adjacent
capacitance
through
Marquardt
least
square
method,
reducing
the
number
of
capacitance
formulas
in capacitance
the pattern
the Levenberg–Marquardt least square method, reducing the number of
library.
It also
classifierItfor
automatic
pattern
a deep pattern
neural
formulas
in constructed
the pattern alibrary.
also
constructed
a matching
classifier through
for automatic
network
(DNN),
which
improved
pattern-matching
efficiency.
Yang
et
al.
[11]
leveraged
a
matching through a deep neural network (DNN), which improved pattern-matching
convolutional neural network (CNN), ResNet-34, to extract parasitic capacitances of 2D
efficiency. Yang et al. [11] leveraged a convolutional neural network (CNN), ResNet-34,
interconnect wires. For the total capacitance and coupling capacitances, this work trains
to extract parasitic capacitances of 2D interconnect wires. For the total capacitance and
two separate ResNet networks for prediction. Furthermore, it also compared the prediction
coupling capacitances, this work trains two separate ResNet networks for prediction.
performance of the CNN with the MLP, showing that the accuracy is higher than the MLP.
Furthermore, it also compared the prediction performance of the CNN with the MLP,
Abouelyazid et al. [4] focused on the interconnect pattern containing metal connectivity and
showing that the accuracy is higher than the MLP. Abouelyazid et al. [4] focused on the
trained a DNN model to predict one specific coupling capacitance. Abouelyazid et al. [5]
interconnect pattern containing metal connectivity and trained a DNN model to predict
took into account the trapezoidal, wire thickness, dielectric layer thickness, et al. into
one specific coupling capacitance. Abouelyazid et al. [5] took into account the trapezoidal,
the model variation and employed DNN and support vector regression (SVN) to predict
wire thickness, dielectric layer thickness, et al. into the model variation and employed
one specific coupling capacitance of 2D patterns, respectively. Abouelyazid et al. [23]
DNN and support vector regression (SVN) to predict one specific coupling capacitance of
2D patterns, respectively. Abouelyazid et al. [23] employed DNN to predict the parasitic
capacitance matrix. For total capacitances and coupling capacitances in the matrix, this
work utilized two separate DNN networks for prediction. Moreover, a hybrid extraction
flow was proposed. It can identify the accuracy of three extraction methods, field-solver,
Electronics 2023, 12, 1440
3 of 18
employed DNN to predict the parasitic capacitance matrix. For total capacitances and
coupling capacitances in the matrix, this work utilized two separate DNN networks for
prediction. Moreover, a hybrid extraction flow was proposed. It can identify the accuracy of
three extraction methods, field-solver, rule-based, and DNN-based, and choose the fastest
extraction method to meet the user’s required accuracy.
In the parasitic capacitance matrix, the total capacitance on the matrix diagonal equals
the sum of the remaining coupling capacitances in the same row. Using this constraint,
we train only one DNN model to predict the capacitance matrix in this study. The neural
network’s output includes only coupling capacitances in the capacitance matrix, and the
total capacitances are obtained by summing corresponding predicted coupling capacitances.
The trained DNN model can be used as a capacitance model in the pattern-matching
method shown in Figure 1. The main contributions of this work are as follows. First, we
propose a mirror flip method to realize data augmentation. Second, we use the DNN
to predict the parasitic capacitance matrix. The neural network output includes merely
coupling capacitances in the matrix. The total capacitance is obtained by calculating the
sum of the coupling capacitances. Table 1 shows the comparison of our work with other
state-of-the-art works.
Table 1. The comparison among different capacitance extraction methods, including our works.
Property
Abouelyazid
et al. [4]
Abouelyazid
et al. [5]
Prediction content
One coupling
capacitance
One coupling
capacitance
Pattern dimension
Data augmentation
2D
No
Type of the method
DNN
2D
No
DNN,
SVN
Number of neural
networks
1
Kasai et al. [21]
Abouelyazid
et al. [23]
This Work
Capacitance
matrix
Capacitance
matrix
3D
No
2D
Yes
3D
No
One total
capacitance and
its corresponding
coupling
capacitances
2D
No
MLP
ResNet
DNN
DNN
1
2
2
1
One capacitance
1
Yang et al. [11]
The rest of the paper is organized as follows. Section 2 first recalls the definition
of the parasitic capacitance matrix. Then it introduces the data acquisition flow, data
augmentation method, and the DNN structure used in the work. Finally, the densitybased data representation for the ResNet input and the ResNet structure used in this
work are described in detail. In Section 3, we first utilize DNN to predict the parasitic
capacitance matrix of a 2D pattern. Then, ResNet is employed to predict the parasitic
capacitance matrix for comparison. Then, to further verify our method, the calculated total
capacitances from the trained DNN are compared with another network (named DNN-2)
that takes the total capacitance as the output. Then, the prediction performance of DNN
under different training set sizes is tested. Then, the solving efficiency of the trained DNN
and the FEM is compared. In the end, we apply our method to a new pattern to verify
its feasibility. Section 4 discusses the experimental results and future works. Section 5
concludes this work.
2. Materials and Methods
2.1. Parasitic Capacitance Matrix
For multi-conductor interconnect systems, capacitance extraction means calculating
the capacitance matrix between those conductors. The coupling capacitance Cij between
conductor i and j satisfies Equation (1). Vij represents the potential difference between i
and j, and Q j denotes the total charges on the surface of conductor j.
Cij =
Qj
( j 6= i)
Vij
(1)
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Electronics 2023, 12, 1440
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For an electrostatic system containing 𝑛𝑛 conductors, the capacitance matrix is
defined as Equation (2), where 𝑞𝑞𝑖𝑖 (𝑖𝑖 = 1,2, … , 𝑛𝑛) denotes the total charges on the
For an𝑖𝑖.electrostatic
system of
containing
n conductors,
capacitance
matrix
is defined
conductor
𝑢𝑢𝑖𝑖 is the potential
conductor
𝑖𝑖. 𝐶𝐶𝑖𝑖𝑖𝑖 on thethe
matrix
diagonal
represents
the
as
Equation
(2),
where
q
i
=
1,
2,
.
.
.
,
n
denotes
the
total
charges
on
the
conductor
(
)
i
i is
total capacitance or self-capacitance of conductor 𝑖𝑖, whose value is equal to the sum i.ofuthe
the
potential
of
conductor
i.
C
on
the
matrix
diagonal
represents
the
total
capacitance
or
ii
remaining coupling capacitances 𝐶𝐶𝑖𝑖𝑖𝑖 (𝑗𝑗 = 1,2, … , 𝑛𝑛, 𝑗𝑗 ≠ 𝑖𝑖) in the same row.
self-capacitance of conductor i, whose value is equal to the sum of the remaining coupling
The method to calculate the capacitance matrix is roughly as follows. The potential
capacitances Cij ( j = 1, 2, . . . , n, j 6= i ) in the same row.
of conductor 𝑖𝑖 is set as 𝑢𝑢𝑖𝑖 = 1 , and the potential of other conductors is 𝑢𝑢𝑗𝑗 =
The method to calculate the capacitance matrix is roughly as follows. The potential of
0 (𝑗𝑗 = 1,2, … , 𝑛𝑛, 𝑗𝑗 ≠ 𝑖𝑖) , then 𝑉𝑉𝑖𝑖𝑖𝑖 = 1 . The 𝐶𝐶𝑖𝑖𝑖𝑖 can be obtained by computing the total
conductor i is set as u = 1, and the potential of other conductors is u = 0 ( j = 1, 2, . . . , n, j 6= i ),
charges on the surfacei of conductor 𝑗𝑗. In addition, since the totalj capacitance of conductor
then V = 1. The C can be obtained by computing the total charges on the surface of
𝑖𝑖 is the ijcapacitance ofij this conductor to the ground, and the potential difference between
conductor j. In addition, since the total capacitance of conductor i is the capacitance of
them is also 1. Hence, the total capacitance 𝐶𝐶𝑖𝑖𝑖𝑖 equals the charges on the surface of
this conductor to the ground, and the potential difference between them is also 1. Hence,
conductor 𝑖𝑖 . Repeating this process, the entire capacitance matrix can be determined.
the total capacitance Cii equals the charges on the surface of conductor i. Repeating this
Moreover,
since
thecapacitance
capacitancematrix
between
conductors isMoreover,
a single value,
process, the
entire
cantwo
be determined.
since the
the capacitance
capacitance
matrix
is
symmetric.
between two conductors is a single value, the capacitance matrix is symmetric.
𝑞𝑞1  𝐶𝐶11 𝐶𝐶12
q1 𝑞𝑞2
C𝐶𝐶1121 C𝐶𝐶1222
�
 q2�⋮ � =
C
   21⋮ C22⋮
 .. 
.. 𝑛𝑛1 𝐶𝐶.. 𝑛𝑛2
𝑞𝑞𝑛𝑛= 
.
 𝐶𝐶
.
.
qn
Cn1 Cn2

…
.…
..
. .⋮.
..
…
.
𝐶𝐶1𝑛𝑛 𝑢𝑢1 
C2𝑛𝑛
1n 𝑢𝑢u
𝐶𝐶
21
� � 
C⋮2n�
⋮u2 
.. 𝑢𝑢𝑛𝑛.. 
𝐶𝐶𝑛𝑛𝑛𝑛
.  . 
Cnn un
(2)
(2)
...
2.2. Dataset Acquisition
2.2. Dataset
Acquisition
In the pattern-matching
method, 2.5D extraction is a way that considers the 3D effect
the pattern-matching
method, 2.5D
extraction
is a way that
theis3D
effect
with In
a combination
of two orthogonal
2D patterns.
Moreover,
eachconsiders
2D pattern
defined
with
combination of
orthogonal
patterns.
Moreover,
2D patternon
is each
defined
by
by
thea combination
of two
different
metal 2D
layers
and the
number each
of conductors
layer
the
combination
of
different
metal
layers
and
the
number
of
conductors
on
each
layer
[11].
[11]. Figure 2 shows a 2D pattern studied in this work. It contains three metal layers, layer
shows
a 2D
studied
in this work.
It contains
three
metal layers,
𝑘𝑘Figure
, layer2 m,
layer
𝑛𝑛 , pattern
and a large
substrate
ground.
Each layer
contains
one orlayer
morek,
layer m, layerconductors.
n, and a largeThe
substrate
ground.
Eachand
layerconductors
contains one
more interconnect
interconnect
substrate
ground
areorembedded
in the
conductors.
The substrate
ground andisconductors
are embedded
in the dielectric,
whose
dielectric,
whose
relative permittivity
3.9. The studied
pattern contains
five conductors
relative
is 3.9. Conductor
The studied1pattern
contains
five conductors
and one
substrate
and
one permittivity
substrate ground.
is always
in the center,
and its central
position
𝑥𝑥1
ground.
Conductor
1
is
always
in
the
center,
and
its
central
position
x
is
a
constant,
which
1
is a constant, which equals 0. Hence, there are nine geometric variables in total, including
equals
0. Hence,
thereand
are nine
geometric
variablesnamely
in total,𝑥𝑥including
the center positions
the
center
positions
widths
of conductors,
2 , 𝑥𝑥3 , 𝑥𝑥4 , 𝑥𝑥5 , and 𝑤𝑤1 , 𝑤𝑤2 ,
and
widths
of
conductors,
namely
x
,
x
,
x
,
x
,
and
w
,
w
,
w
,
w
,
w
2 3 4 5
2
3
5.
1
4
𝑤𝑤3 , 𝑤𝑤4 , 𝑤𝑤5 .
Figure 2.
2. The
The illustration
illustration of
of the
the studied
studied pattern
pattern and
and its
its geometry
geometry variables.
variables.
Figure
The capacitance
thisthis
pattern
is defined
in Equation
(3). It ignores
substrate
The
capacitancematrix
matrixof of
pattern
is defined
in Equation
(3). It the
ignores
the
ground’s
total
capacitance
and
only
considers
the
coupling
capacitances
between
the
substrate ground’s total capacitance and only considers the coupling capacitances
substrate ground and other conductors.
between the substrate ground and other conductors.


𝐶𝐶15
𝐶𝐶16
C𝐶𝐶1111 C𝐶𝐶1212 C𝐶𝐶13
𝐶𝐶14
16
15 C
13 C
14 C
⎡
⎤
C𝐶𝐶2121 C𝐶𝐶2222 C𝐶𝐶23
𝐶𝐶25
𝐶𝐶26
𝐶𝐶24
26 
25 C
23 C
24 C
⎢

⎥
C𝐶𝐶3131 C𝐶𝐶3232 C𝐶𝐶33

𝐶𝐶35
𝐶𝐶36
C 𝐶𝐶
==
(3)
(3)
𝐶𝐶34
36 
35 C
33 C
34 C
⎥
⎢
C

𝐶𝐶44
𝐶𝐶45
𝐶𝐶46
⎢𝐶𝐶4141 C𝐶𝐶4242 C𝐶𝐶43
43 C
44 C
45 C
46 ⎥
𝐶𝐶54
𝐶𝐶55
𝐶𝐶56
⎣𝐶𝐶5151 C𝐶𝐶5252 C𝐶𝐶53
C
53 C
54 C
55 C
56 ⎦
The acquisition of datasets is a crucial step in deep learning. In order to achieve broad
coverage of the sampling range, we employ the Latin hypercube sampling (LHS) [24]
The acquisition of datasets is a crucial step in deep learning. In ord
coverage of the sampling range, we employ the Latin hypercube s
method. The studied pattern contains nine geometric variables. Suppo
Electronics 2023, 12, 1440
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of data for each parameter, the flow of LHS is roughly as follows.
Firs
𝑁𝑁 values within the range of each geometric variable. Then randomly
results,
then
a dataset
with 𝑁𝑁
×geometric
9 size is
obtained.
After
obtaining
method. The
studied
pattern contains
nine
variables.
Suppose
sampling
N sets the p
Electronics 2023, 12, x FOR PEER REVIEW
6 of 19
of data for each parameter, the flow of LHS is roughly as follows. First, randomly sample
results, we then construct simulation models and utilize the FEM p
N values within the range of each geometric variable. Then randomly match the sampling
their
matrix.
Toissum
up,After
theobtaining
flow ofthethe
dataset
acquisition
results,capacitance
then a dataset with
N × 9 size
obtained.
parameter
sampling
The
acquisition
of datasets
is a crucial
step
inutilize
deep the
learning.
In order to
to compute
achieve broad
results,
we
then
construct
simulation
models
and
FEM
program
their
3, and the above process is conducted through the FEM software
EM
coverage
of matrix.
the sampling
the Latin
hypercube
sampling
(LHS)
[24]
capacitance
To sum range,
up, thewe
flowemploy
of the dataset
acquisition
is shown
in Figure
3, and
by
the The
authors’
team [25],
whose
modeling
and
solving
resultby
method.
studied
pattern
contains
nine
geometric
variables.
Suppose
sampling
𝑁𝑁interfaces
sets
the above
process
is conducted
through
the
FEM
software
EMPbridge
developed
the
of
data for
each[25],
parameter,
the flow of
LHS
is roughly
follows.are
First,
randomly
sample
authors’
team
whose modeling
and
solving
result as
interfaces
shown
in Figure
4.
4.
𝑁𝑁 values within the range of each geometric variable. Then randomly match the sampling
results, then a dataset with 𝑁𝑁 × 9 size is obtained. After obtaining the parameter sampling
results, we then construct simulation models and utilize the FEM program to compute
their capacitance matrix. To sum up, the flow of the dataset acquisition is shown in Figure
3, and the above process is conducted through the FEM software EMPbridge developed
by the authors’ team [25], whose modeling and solving result interfaces are shown in Figure
4.
Figure 3. The flow of datasets acquisition.
Figure
3. The
datasets
Figure
3. flow
Theofflow
of acquisition.
datasets
(a) Model building interface
acquisition.
(b) Solving result interface
Figure 4.
4. The
The model
model building
building and
and solving
solving result
result interfaces
interfaces of
of EMPbridge.
EMPbridge.
Figure
2.3. Data Augmentation
(a) Model
The size of datasets also has a crucial impact on the deep learning prediction effect.
When the size is small, due to a lack of sufficient supervision information, it is easy to lead
to poor generalization
of over-fitting
is prone
building
interface ability of the model, and the phenomenon(b)
Solving result
interfac
to occur [26]. However, the acquisition of enough datasets, such as with the FEM
simulation,
often
accompanied
by heavy
time and labor
Data augmentation
is a
Figure 4. isThe
model
building
and solving
resultcosts.
interfaces
of EMPbridge.
2.3. Data Augmentation
The size of datasets also has a crucial impact on the deep learning predictio
When the size is small, due to a lack of sufficient supervision information, it is eas
Electronics 2023, 12, 1440
6 of 18
to poor generalization ability of the model, and the phenomenon of over-fitting
to occur [26]. However, the acquisition of enough datasets, such as with the FEM
tion, is often accompanied by heavy time and labor costs. Data augmentation is a
2.3. Data Augmentation
alleviate this problem. It can expand the original data by setting a transformat
The size of datasets also has a crucial impact on the deep learning prediction effect.
without additional computation. Common transformation operations
include r
Electronics 2023, 12, x FOR PEER REVIEW
7 of 19
When the size is small, due to a lack of sufficient supervision information, it is easy
to
mirror
flip,generalization
cropping, etc.
[27].of the model, and the phenomenon of over-fitting is
lead
to poor
ability
work
mirror flip
method,
as shown
inwith
Figure
5, to augm
prone This
to occur
[26]. introduces
However, theaacquisition
of enough
datasets,
such as
the FEM
simulation,
is
often
accompanied
by
heavy
time
and
labor
costs.
Data
augmentation
a
way
to
alleviate
this
problem.
It
can
expand
the
original
data
by
setting
a
transformation
tasets. After flipping, the geometric variables of conductors change as is
follows.
T
way
to
alleviate
this
problem.
It
can
expand
the
original
data
by
setting
a
transformation
rule
without
additional
computation.
Common
transformation
operations
include
tion and width of conductor 1 are unchanged; The widths of conductors 4 and 5
rule without
additional
computation.
Common transformation operations include rotation,
rotation,
mirror
flip, cropping,
etc. [27].
changed,
but
their
horizontal
positions
are multiplied by −1; For conductors 2 and
mirror
flip,
cropping,
etc. [27].
This
work
introduces
a mirror flip method, as shown in Figure 5, to augment
widths
are directly
exchanged,
horizontal
coordinates
are
firstdatasets.
multiplied
by
This After
work
introduces
a mirror
flip and
method,
as of
shown
in Figure
5, to augment
datasets.
flipping,
the
geometric
variables
conductors
change
as follows.
The
After
the geometric
variables
of conductors
change
as of
follows.
The position
thenflipping,
exchanged.
transformation
rule of
parasitic
capacitances
is shown
position
and width
ofThe
conductor
1 are unchanged;
The
widths
conductors
4 and
5 and
are in Fig
width
of
conductor
1
are
unchanged;
The
widths
of
conductors
4
and
5
are
unchanged,
̂
unchanged,
their
horizontal
positions
multiplied
−1;and
For conductors
3,
representsbut
the
original
value
beforearethe
mirror by
flip,
𝐶 denotes2 and
thebut
value of
their
horizontal
positions
are
multiplied
by
−
1;
For
conductors
2
and
3,
their
widths
are
ditheir
widths
are
directly
exchanged,
and
horizontal
coordinates
are
first
multiplied
by
−1
capacitance matrix.
rectly exchanged, and horizontal coordinates are first multiplied by −1 and then exchanged.
and then exchanged. The transformation rule of parasitic capacitances is shown in Figure
The
rule of parasitic
capacitances
is flip,
shown
6. C
the
6.
𝐶𝐶 transformation
represents the original
value before
the mirror
andin𝐶𝐶̂Figure
denotes
therepresents
value of the
original
value before
the mirror flip, and Ĉ denotes the value of the new capacitance matrix.
new
capacitance
matrix.
Figure
5.
The
mirror
flip illustration
of thepattern.
studied pattern.
Figure
Figure5.
5. The
Themirror
mirrorflip
flip illustration
illustration of
of the
the studied
studied pattern.
Figure 6. The transformation rule between the original data and the new data.
According to the transformation rule, a new set of data can be obtained, and the
datasets are doubled in size. It is worth remembering that this data augmentation method
isFigure
also applicable
to other similar
patterns
athe
transformation
isthe
determined.
6. 6.
The
transformation
rule between
the once
original
data
and thedata
newrule
data.
Figure
The
transformation
rule
between
original
and
new data.
2.4. Prediction of Parasitic Capacitance Matrix Based on the DNN
In the parasitic capacitance matrix, the total capacitance equals the sum of the
remaining coupling capacitances in the same row. With this information, the DNN
Electronics 2023, 12, 1440
7 of 18
According to the transformation rule, a new set of data can be obtained, and the
datasets are doubled in size. It is worth remembering that this data augmentation method
is also applicable to other similar patterns once a transformation rule is determined.
Electronics 2023, 12, x FOR PEER REVIEW
8 of 1
2.4. Prediction of Parasitic Capacitance Matrix Based on the DNN
In the parasitic capacitance matrix, the total capacitance equals the sum of the re-
capacitance
is indirectly
obtained
by summing
coupling
Th
maining
coupling
capacitances
in the same
row. With the
this predicted
information,
the DNNcapacitances.
network
output
this work,variables,
and the total
capacitance
is indiinput includes
of DNNonly
is acoupling
vector capacitances
containing in
geometric
and
the output
is a vecto
rectly
obtained
summing
the predicted coupling capacitances. The input of DNN is a
containing
allby
coupling
capacitances.
vectorTaking
containing
variables,
the outputits
is aparasitic
vector containing
all coupling
the geometric
studied pattern
as and
an example,
capacitance
matrix has bee
capacitances.
given in Equation (3). Due to the matrix’s symmetric property, there are 15 non-repetitiv
Taking the studied pattern as an example, its parasitic capacitance matrix has been
coupling capacitances in total, which are 𝐶𝐶12 , 𝐶𝐶13 , 𝐶𝐶14 , 𝐶𝐶15 , 𝐶𝐶16 , 𝐶𝐶23 , 𝐶𝐶24 , 𝐶𝐶25 , 𝐶𝐶26
given in Equation (3). Due to the matrix’s symmetric
property, there are 15 non-repetitive
𝐶𝐶
,
𝐶𝐶
,
𝐶𝐶
,
𝐶𝐶
,
𝐶𝐶
,
and
𝐶𝐶
.
Therefore,
as
the input dimensio
34
35
36
45
46
56
coupling capacitances in total, which are C12 , C13 , C14 , shown
C15 , C16 ,in
C23Figure
, C24 , C7,
25 , C26 , C34 , C35 ,
of
DNN
is
nine,
that
is,
all
geometric
variables;
the
output
dimension
C36 , C45 , C46 , and C56 . Therefore, as shown in Figure 7, the input dimensionisoffifteen,
DNN isincludin
all coupling
capacitances.
Thethebatch
(BN)including
layer could
mitigate th
nine,
that is, all geometric
variables;
outputnormalization
dimension is fifteen,
all coupling
capacitances.
The batch
normalization
layer
the vanishing
vanishing gradient
problem
since it(BN)
shifts
thecould
inputmitigate
data distribution
togradient
the unsaturate
problem
the input
data distribution
to the
the activation
region since
of theit shifts
activation
function
[28]. Hence,
weunsaturated
add a BNregion
layerofbefore
each activatio
function
[28].
Hence,
we
add
a
BN
layer
before
each
activation
function
the DNN (4) is th
function in the DNN structure, although it is not shown in Figure 7.inEquation
structure, although it is not shown in Figure 7. Equation (4) is the loss function, which is
loss function, which is defined by the mean square error (MSE) between the predicte
defined by the mean square error (MSE) between the predicted value and the reference
value and the reference value in the datasets. In Equation (4), 𝑁𝑁 represents the size o
value in the datasets. In Equation (4), N represents the size of datasets, and K denotes the
datasets,
and 𝐾𝐾 of
denotes
the which
outputis dimension
DNN,
whichvalue,
is fifteen
here.
𝐶𝐶𝑖𝑖𝑖𝑖 i
0
output
dimension
the DNN,
fifteen here.ofCijthe
is the
predicted
and C
ij
′
the
predicted
value,
and
𝐶𝐶
represents
the
reference
value.
represents the reference value.𝑖𝑖𝑖𝑖
𝑁𝑁
𝐾𝐾
𝑖𝑖=1
𝑗𝑗=1
2
1 N 11 K 1
𝑀𝑀𝑀𝑀𝑀𝑀
= �
|𝐶𝐶ij0 𝑖𝑖𝑖𝑖 − 𝐶𝐶𝑖𝑖𝑖𝑖′ |2
MSE =
Cij�
−C
∑
∑
N i=1 𝑁𝑁
K j=1 𝐾𝐾
(4)
(4
Figure7. 7.
The
illustration
of DNN
structure.
Figure
The
illustration
of DNN
structure.
2.5.
Data
Representation
2.5.Density-Based
Density-Based
Data
Representation
For comparison, we employ the convolutional neural network ResNet to predict the
For comparison, we employ the convolutional neural network ResNet to predict th
capacitance matrix. A density-based data representation method [11,29] is used to describe
capacitance matrix. A density-based data representation method [11,29] is used t
the geometric characteristics of the metal layer and represents the input of ResNet. This
describe
geometric
characteristics
of the[23].
metal
layer and
method
canthe
effectively
describe
the layout feature
Furthermore,
it isrepresents
suitable forthe
the input o
ResNet.
This
method
can
effectively
describe
the
layout
feature
[23].
Furthermore,
it i
feature extraction operation performed by the convolutional neural network. In this way,
suitable
for placement
the feature
operation
performed
by the convolutional neura
the
conductor
of a extraction
metal layer can
be represented
as a vector.
network. In this way, the conductor placement of a metal layer can be represented as
vector.
The extraction window width 𝑊𝑊 determines the extraction range of the pattern i
the horizontal direction and the variation range of the geometric variables. A simulatio
experiment is conducted to determine the extraction window width 𝑊𝑊 of the studie
pattern. In the simulation, conductor 1 and conductor 3 take the minimum width unde
Electronics 2023, 12, 1440
8 of 18
The extraction window width W determines the extraction range of the pattern in
the horizontal direction and the variation range of the geometric variables. A simulation
experiment is conducted to determine the extraction window width W of the studied
pattern. In the simulation, conductor 1 and conductor 3 take the minimum width under
Electronics 2023, 12, x FOR PEER REVIEW
the process. Conductor 4 and conductor 5 take a very large width. Then move conductor 39 of 19
away from conductor 1 until the coupling capacitance between them is less than 1% of the
total capacitance of conductor 1. The distance between conductor 1 and conductor 3 at this
time multiplied by two is the width of the extraction window [5,11].
cells atOnce
equal
To accurately
the cell width
theintervals.
extraction window
size W isdistinguish
determined, different
the flow ofgeometries,
this data representation
should
notisbeasmore
thanFirst,
half the
the extraction
minimumwindow
spacingsize
𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚
betweeninto
twoL interconnects
method
follows.
is divided
cells at equal [11].
Therefore,
study,distinguish
we let thedifferent
cell width
equal half
thewidth
minimum
and the
intervals. in
To this
accurately
geometries,
the cell
shouldspacing,
not be more
than
half
the
minimum
spacing
s
between
two
interconnects
[11].
Therefore,
in
this
number of cells 𝐿𝐿 is determined by
minEquation (5) [23].
study, we let the cell width equal half the minimum spacing, and the number of cells L is
𝑊𝑊
determined by Equation (5) [23]. 𝐿𝐿 =
(5)
0.5 × W
𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚
L=
(5)
0.5 × smin
Second, taking one layer as an example, a one-dimensional vector with a size of 𝐿𝐿 is
takingvalue
one layer
as an example,
a one-dimensional
vector with
a size
of Lcell
is area
created.Second,
The vector
is determined
by the
ratio of the conductor
area
to the
created. The vector value is determined by the ratio of the conductor area to the cell area at
at each position. When the conductor occupies the entire cell area, the vector value is 1;
each position. When the conductor occupies the entire cell area, the vector value is 1; when
when the conductor occupies part of a cell, the vector value is taken as the ratio of the
the conductor occupies part of a cell, the vector value is taken as the ratio of the occupied
occupied
area
the The
cell illustration
area. The illustration
of the density-based
dataofrepresentation
of
area to the
celltoarea.
of the density-based
data representation
one layer is
oneshown
layerin
isFigure
shown8.in Figure 8.
Figure
8. 8.
The
density-based
representation
one
metal
layer.
Figure
The
density-based data
data representation
ofof
one
metal
layer.
PredictionofofParasitic
Parasitic Capacitance
Capacitance Matrix
onon
ResNet
2.6.2.6.
Prediction
MatrixBased
Based
ResNet
ResNet was proposed by He [30] in 2015, which solves the network degradation
ResNet was proposed by He [30] in 2015, which solves the network degradation
problem of deep networks. In this study, we modify the convolution kernel size of ResNet
problem
ofinput
deepdata
networks.
this
wethe
modify
thecapacitance
convolution
kernel
of ResNet
to fit our
and thenInuse
it study,
to predict
parasitic
matrix.
Thesize
network
to structure
fit our input
data
and
then
use
it
to
predict
the
parasitic
capacitance
matrix.
ResNet-18 used in this work is shown in Figure 9, including 17 two-dimensional The
network
structure
ResNet-18
used in this
work
is shown
Figure
9,7including
17 twoconvolutional
layers,
one fully connected
layer,
and two
poolingin
layers.
“1 ×
conv, 64, /2”
dimensional
layers,
fullyofconnected
layer, layer
and two
in the figureconvolutional
indicates that the
outputone
channel
the convolutional
is 64,pooling
the size oflayers.
kernel
is 1 ×
7, andindicates
the stride that
is 2. The
line in the
is called
“1 the
× 7convolution
conv, 64,/2”
in the
figure
theconnection
output channel
of figure
the convolutional
a shortcut.
The
solid
indicates that kernel
the input
output
of the
residual
are
layer
is 64, the
size
of line
the convolution
is and
1 × 7,
and the
stride
is 2.module
The connection
directly
connected.
The
dotted
line
means
the
input
and
output
of
the
residual
module
are
line in the figure is called a shortcut. The solid line indicates that the input and output
of
inconsistent in size and cannot be directly connected. Hence a 1 × 1 convolutional kernel is
the residual module are directly connected. The dotted line means the input and output
added, which makes the input and output the same size. A detailed illustration of these
of two
the residual
are inconsistent
in size
cannot
be directly
connected.
Hence a
shortcutsmodule
is also given
in Figure 9. The
lossand
function
definition
of the
ResNet is the
1 ×same
1 convolutional
kernel is added, which makes the input and output the same size. A
as Equation (4).
detailed illustration of these two shortcuts is also given in Figure 9. The loss function
definition of the ResNet is the same as Equation (4).
Electronics 2023,
2023, 12,
12, 1440
Electronics
x FOR PEER REVIEW
9 of10
18 of 19
Figure9.9.The
Theneural
neuralnetwork
network
structure
of the
modified
ResNet-18.
(a) The
concrete
structure
Figure
structure
of the
modified
ResNet-18.
(a) The
concrete
structure
of theof the
dotted
line;
(b)
The
concrete
structure
of
the
solid
line.
dotted line; (b) The concrete structure of the solid line.
3.3.Experiments
Experimentsand
andResults
Results
The
research
case
this
experiment
hashas
been
introduced
in Section
2.2. It
hasIt nine
The research caseinin
this
experiment
been
introduced
in Section
2.2.
has nine
geometric variables. Equation (3) defines its parasitic capacitance matrix, which includes
geometric variables. Equation (3) defines its parasitic capacitance matrix, which includes
fifteen non-repetitive coupling capacitances. In this study, only the coupling capacitances
fifteen non-repetitive coupling capacitances. In this study, only the coupling capacitances
are considered during the deep learning training, and the total capacitance is calculated by
are considered during the deep learning training, and the total capacitance is calculated
summing up coupling capacitances. The structure and geometric variables’ sampling range
bythe
summing
upthe
coupling
capacitances.
The structure
and
variables’
sampling
of
case meet
55 nm process
requirements.
Moreover,
thisgeometric
work is also
applicable
to
rangeprocess
of thetechnology.
case meet Table
the 55
nm the
process
requirements.
Moreover,
work
is also
other
2 lists
first four
layers’ process
criterionthis
of the
55 nm,
applicable
to
other
process
technology.
Table
2
lists
the
first
four
layers’
process
criterion
which includes the interconnect wires’ thickness, the minimum wire width wmin , and the
of the 55 nm,
which
interconnect
wires’
thickness,
theresearch
minimum
wire
width
minimum
spacing
sminincludes
betweenthe
two
wires of each
metal
layer. This
selects
the
𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚 , and the
minimum
𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚
between
of neach
metal
layer.
combination
of layer
2, layer spacing
3, and layer
4, that
is, k =two
2, mwires
= 3, and
= 4 in
Figure
2. This
research selects the combination of layer 2, layer 3, and layer 4, that is, 𝑘𝑘 = 2, 𝑚𝑚 = 3, and
Table
Multilayer
interconnect 55 nm process standard (first four metal layers) [11].
𝑛𝑛 = 42. in
Figure 2.
Layer
Thickness (µm)
wmin (µm)
smin (µm)
Table 2. Multilayer interconnect 55 nm process standard
(first four metal layers)
[11].
1
Layer
2
13
4
2
3
4
0.1
Thickness
0.16 (𝝁𝝁𝝁𝝁)
0.2
0.1
0.2
0.16
0.2
0.2
0.054
𝒘𝒘0.081
𝒎𝒎𝒎𝒎𝒎𝒎 (𝝁𝝁𝝁𝝁)
0.09
0.054
0.09
0.081
0.09
0.09
0.108
𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚0.08
(𝝁𝝁𝝁𝝁)
0.09
0.108
0.09
0.08
0.09
0.09
Electronics
FOR PEER REVIEW
Electronics 2023,
2023, 12,
12, x1440
1110ofof19
18
As shown
thethe
extraction
window
width
is taken
as W as
= 56
, where
As
shown in
inFigure
Figure10,
10,
extraction
window
width
is taken
𝑊𝑊×
=s56
𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚 ,
min×
smin = 𝑠𝑠0.09
µm.
For
a
higher
precision
in
reference
values,
the
solving
domain
width
of
where
=
0.09
𝜇𝜇𝜇𝜇.
For
a
higher
precision
in
reference
values,
the
solving
domain
𝑚𝑚𝑚𝑚𝑚𝑚
numerical
computation
is taken as is
10 taken
µm. Once
the𝜇𝜇𝜇𝜇
window
W is determined,
. Once width
width
of numerical
computation
as 10
the window
width 𝑊𝑊 the
is
ResNet
input
data
dimension
L
of
one
metal
layer
can
be
computed
using
Equation
(5),
and
determined, the ResNet input data dimension 𝐿𝐿 of one metal layer can be computed
the result
is L =(5),
112.
Since
three
metal
layers,
theare
input
data
sizelayers,
of the the
ResNet
is
using
Equation
and
the there
resultare
is 𝐿𝐿
= 112.
Since
there
three
metal
input
3
×
112.
data size of the ResNet is 3 × 112.
Figure 10.
10. The
Theillustration
illustration of
of extraction
extraction window
window width
width and
and the
the solving
solving domain
domain width.
width.
Figure
Traditional multilayer
thethe
Manhattan
structure
[31],[31],
that
Traditional
multilayer interconnect
interconnectrouting
routingfollows
follows
Manhattan
structure
is,
placing
adjacent
layers’
interconnects
orthogonally
to
each
other.
In
this
way,
the
that is, placing adjacent layers’ interconnects orthogonally to each other. In this way, the
crosstalk
caused
by
coupling
capacitances
can
be
minimized
[32].
The
studied
pattern
crosstalk caused by coupling capacitances can be minimized [32]. The studied pattern
shown in
in Figure
geometry
from
thethe
layout.
Layers
k and
shown
Figure 22 isisaacross-section
cross-sectionview
viewofofa small
a small
geometry
from
layout.
Layers
𝑘𝑘
n
are
regarded
as
horizontal
routing
here,
and
the
three
conductors
in
the
middle
layer
are
and 𝑛𝑛 are regarded as horizontal routing here, and the three conductors in the middle
regarded
as verticalas
page
routing.
, w2 , and𝑤𝑤w,3 𝑤𝑤
represent
the width of the interconnects.
layer
are regarded
vertical
pagew1routing.
1
2 , and 𝑤𝑤3 represent the width of the
Hence,
we
let
the
maximum
variation
range
of
w
,
w
,
and
w not exceed ten times the
2
1 range
interconnects. Hence, we let the maximum variation
of 𝑤𝑤31 , 𝑤𝑤2 , and 𝑤𝑤3 not exceed
minimum width. Finally, the sampling range of each geometric variable is listed in Table 3.
ten times the minimum width. Finally, the sampling range of each geometric variable is
listed in Table 3.
Table 3. Parameters’ sampling range.
Table 3.Parameter
Parameters’ sampling
range. Range
Sampling
Parameter
Sampling Range
x2
[1.0, 2.05]
w2
[0.09,
0.9]
Parameter
Sampling
Range
Parameter
Sampling
Range
[−2.05,
−1.0]
𝑤𝑤2w3
𝑥𝑥2 x3
[1.0,
2.05]
[0.09,[0.09,
0.9] 0.9]
x
[−1.0, 1.0]
[0.081, 3.0]
𝑤𝑤3w4
𝑥𝑥3 4
[−2.05, −1.0]
[0.09, 0.9]
x5
[−1.0, 1.0]
w5
[0.09, 3.0]
𝑤𝑤4 𝑥𝑥4 w1
[−1.0,
1.0]
[0.081, 3.0]
[0.09,
0.9]
𝑤𝑤5
𝑥𝑥5
[−1.0, 1.0]
[0.09, 3.0]
𝑤𝑤
[0.09, 0.9]
The 1deep learning framework used in the study is PyTorch. The hardware configuration
as follows:
CPU
Intel Core used
i7-12700KF,
NVIDIA
RTX3080Ti,
16GB
Theisdeep
learning
framework
in theGPU
study
is PyTorch.
The with
hardware
RAM. For the is
studied
pattern,
25,000
of i7-12700KF,
samplings were
and the reference
configuration
as follows:
CPU
Intelsets
Core
GPUgenerated,
NVIDIA RTX3080Ti,
with
capacitance
values
were
computed
using
theofFEM
solver using
16GB
RAM.matrix
For the
studied
pattern,
25,000
sets
samplings
were EMPbridge
generated, software.
and the
Finally, 50,000
sets of data
werevalues
obtained
after
using the using
data augmentation
method
dereference
capacitance
matrix
were
computed
the FEM solver
using
scribed
in
Section
2.2.
The
total
datasets
were
randomly
split
into
three
parts:
80%
of
the
EMPbridge software. Finally, 50,000 sets of data were obtained after using the data
sampling wasmethod
used asdescribed
the training
set, 10%2.2.
was
used
the validation
set, and split
10% into
was
augmentation
in Section
The
totalasdatasets
were randomly
used
as
the
testing
set.
Too-small
coupling
capacitances
are
usually
not
considered
in
the
three parts: 80% of the sampling was used as the training set, 10% was used as the
extraction [11,23].
thiswas
work,
all as
coupling
capacitances
are considered
the neural
validation
set, andIn
10%
used
the testing
set. Too-small
couplingduring
capacitances
are
network
training
stage.
However,
in
the
testing
stage,
the
coupling
capacitances
whose
usually not considered in the extraction [11,23]. In this work, all coupling capacitances are
values are less
thanthe
1%neural
of the corresponding
total
capacitance
arein
specially
treated.
When
considered
during
network training
stage.
However,
the testing
stage,
the
calculating
the
predicted
total
capacitance,
all
coupling
capacitances
are
included
in
the
calcoupling capacitances whose values are less than 1% of the corresponding total
culation.
When
evaluating
the
prediction
accuracy
of
the
coupling
capacitances,
the
small
capacitance are specially treated. When calculating the predicted total capacitance, all
values arecapacitances
not considered
order toinevaluate
only theWhen
accuracy
of significant
coupling
coupling
are in
included
the calculation.
evaluating
the prediction
capacitances.
Since
one
coupling
capacitance
corresponds
to
two
total
capacitances,
such
accuracy of the coupling capacitances, the small values are not considered in order
to
as Cij , which contributes to total capacitances Cii and Cjj , the error of Cij is not evaluated
evaluate only the accuracy of significant coupling capacitances. Since one coupling
only when both Cij /Cii and Cij /Cjj are less than 1%.
capacitance corresponds
to two total capacitances, such as 𝐶𝐶𝑖𝑖𝑖𝑖 , which contributes to total
Electronics 2023, 12, 1440
11 of 18
3.1. Experiment Results
The hyperparameters of the neural network have a great influence on the prediction
effect. Different combinations of hyperparameters may cause a significant difference in
the prediction effect of the model. For scenarios with few kinds of hyperparameters, the
grid search method is commonly used. However, the DNN has many hyperparameters
needed to be tuned, and the grid search method will become time-consuming. To alleviate
this problem, we utilize an automated hyperparameter tuning method, the tree-structured
parzen estimators (TPE) method [33]. TPE is based on the Bayesian optimization algorithm,
and it can select the next set of hyperparameters according to the evaluation results of the
existing hyperparameter combinations [34]. The TPE method can find the better or best
choice among all hyperparameter combinations within the maximum evaluation number.
For the DNN training, the optimizer is Adam [35]. Weights and biases are initialized
by the Xavier initialization [36]. The remaining hyperparameters are determined with TPE,
and their search space is shown in Table 4. The maximum evaluation number is 100. The
objective function of the TPE is the predicted coupling capacitances’ average relative error
of the trained neural network on the testing set, and the procedure is implemented using
the Hyperopt python package [37]. In the end, the hyperparameter combination with better
performance for this studied pattern is as follows. The activation function is Swish; the
batch size is 32; the learning rate is 10−4 ; the number of hidden layers is 4, and each hidden
layer has 500 neurons.
Table 4. Hyperparameters’ space for TPE.
Hyperparameter
Search Range
Activation function
Batch size
Learning rate
Number of hidden layers
Neurons in each hidden layer
ReLU [38], Tanh, Swish [39]
32, 64, 128, 256, 512
10−2 , 10−3 , 10−4 ,10−5
1, 2, 3, 4, 5, 6
50, 100, 200, 300, 400, 500, 600
Additionally, we experimented with ResNet-18 and ResNet-34 to predict their respective capacitances. Both employ the Adam optimizer. Since their hyperparameters needed
to be tuned only for batch size and learning rate, hence we utilized the grid search method
to find the better combination. Batch size is enumerated in {32,64,128,256,512}; the learning
rate is enumerated in {10−2 ,10−3 ,10−4 ,10−5 }. Eventually, ResNet-18 has better performance
than ResNet-34, and its hyperparameter combination is as follows. The batch size is set to
32, and the learning rate is 10−3 .
Tables 5 and 6 are the prediction results obtained by those two neural networks under
1000 epochs, and their testing sets are consistent. The predicted total capacitances are
indirectly obtained by summing the predicted coupling capacitances. We can see that DNN
performs better both in the coupling capacitance and total capacitance prediction in this
research case. Furthermore, the training time of DNN is only 31% of ResNet-18. In addition,
those two networks’ prediction accuracy on the total capacitance all meets the accuracy
requirements (<5%) in the advanced process criterion, showing the computation method of
total capacitance introduced in this work is reliable.
Table 5. The predictive performance of different neural networks on the coupling capacitance (the
relative error is only evaluated for coupling capacitances greater than 1% of the corresponding total
capacitance).
Network
Training
Time (h)
Average
Relative Error
Maximum
Relative Error
Ratio
(>5%)
DNN
ResNet-18
0.89
2.90
0.12%
0.21%
10.66%
10.69%
0.01%
0.05%
Network
DNN
ResNet-18
Training
Time (h)
0.89
2.90
Average
Relative Error
0.12%
0.21%
Maximum
Relative Error
10.66%
10.69%
Ratio
(>5%)
0.01%
0.05%
Electronics 2023, 12, 1440
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Table 6. The predictive performance of different neural networks on the total capacitance.
Average of differentMaximum
Ratio
Table 6. The predictive performance
neural networks on the total
capacitance.
Network
Network
DNN
ResNet-18
DNN
ResNet-18
Relative Error
Relative Error
Average
Maximum
0.05%
2.76%
Relative Error
Relative Error
0.05% 0.05%
4.45% 2.76%
0.05%
4.45%
(>5%)
0
0
Ratio
(>5%)
0
0
Figure 11 shows the relative error distribution of two neural networks for the
coupling capacitance and the total capacitance, respectively. The larger errors of coupling
Figure 11 shows the relative error distribution of two neural networks for the coupling
capacitance usually occur at lower capacitance values. In addition, the total capacitance
capacitance and the total capacitance, respectively. The larger errors of coupling capacierror
ofat
DNN
more concentrated
that of
tancedistribution
usually occur
loweriscapacitance
values. Inthan
addition,
theResNet-18.
total capacitance error
distribution of DNN is more concentrated than that of ResNet-18.
(a) DNN
(b) ResNet-18
Figure
11. The
Therelative
relative
distribution
of thecoupling
predicted
coupling
and total
Figure 11.
errorerror
distribution
of the predicted
capacitance
and capacitance
total capacitance:
capacitance:
DNN; (b) ResNet-18.
(a) DNN; (b)(a)
ResNet-18.
3.2.The
TheComparison
Comparison of
forfor
Predicting
TotalTotal
Capacitance
UsingUsing
the DNN
3.2.
of Two
TwoMethods
Methods
Predicting
Capacitance
the DNN
In this study, the total capacitance is obtained indirectly by calculating the sum of
coupling capacitances. To compare this method with the way that directly predicts the total
capacitance using DNN, we constructed another DNN model called DNN-2. The input
of this neural network is consistent with the DNN structure introduced in Section 2.4, but
Electronics 2023, 12, 1440
In this study, the total capacitance is obtained indirectly by calculating the sum of
coupling capacitances. To compare this method with the way that directly predicts the
total capacitance using DNN, we constructed another DNN model called DNN-2. The
13 of 18
input of this neural network is consistent with the DNN structure introduced in Section
2.4, but the output dimension of DNN-2 is five, corresponding to the five total
capacitances of this research case. The training sets, validation sets, and testing sets are
the outputwith
dimension
DNN-23.1.
is five,
to the five total capacitances
of this
consistent
those inofSection
The corresponding
approach of hyperparameters’
settings of DNNresearch
case.
The
training
sets,
validation
sets,
and
testing
sets
are
consistent
with
those
in
2 is the same as those of the DNN introduced in Section 3.1.
Section
3.1.
The
approach
of
hyperparameters’
settings
of
DNN-2
is
the
same
as
those
of
Finally, after trial and error, the hyperparameters’ combination with better
the DNN introduced
in Section
3.1.
performance
is as follows.
The activation
function is Tanh. The learning rate is 10−3 . The
Finally,
after
trial
and
error,
the
hyperparameters’
combination
betterinperforbatch size is 64. The number of hidden layers is 3, and the
number ofwith
neurons
each
−3 . The batch
mance
is
as
follows.
The
activation
function
is
Tanh.
The
learning
rate
is
10
hidden layer is 500. The prediction results of DNN-2 are shown in Table 7. At the same
size the
is 64.
The number
of hidden
3, and
the number
neurons
hidden
time,
prediction
results
of DNNlayers
on theistotal
capacitance
in of
Section
3.1 in
areeach
given
as a
layer
is
500.
The
prediction
results
of
DNN-2
are
shown
in
Table
7.
At
the
same
time,
the
comparison. From the table, we can see that the predictive performance of the two DNN
prediction
results
of
DNN
on
the
total
capacitance
in
Section
3.1
are
given
as
a
comparison.
models is very close. Therefore, the method used in this study is reliable. Moreover, the
From also
the table,
we can
the predictive
the for
twopredicting
DNN models
result
indicates
thatsee
ourthat
method
can saveperformance
the trainingofcost
totalis
very close. Therefore, the method used in this study is reliable. Moreover, the result also
capacitance.
indicates that our method can save the training cost for predicting total capacitance.
Table 7. The prediction results of two methods using DNN for total capacitance.
Table 7. The prediction results of two methods using DNN for total capacitance.
Training
Average
Maximum
Training
Average
Maximum
Relative Error
Relative Error
Method Time (h)
Time (h)
Relative Error
Relative Error
DNN
0.89
0.05%
2.76%
DNN
0.89
0.05%
DNN-2
0.35
0.05%
4.96% 2.76%
Method
DNN-2
0.35
0.05%
4.96%
Ratio
(>5%) Ratio
(>5%)
0
0
0
0
3.3. The Predictive Performance of DNN under Different Training Set Sizes
3.3. The Predictive Performance of DNN under Different Training Set Sizes
This subsection tests the predictive performance of DNN under different training set
Thisvalidation
subsectionset
tests
the
predictive
performance
of as
DNN
under 3.1.
different
training
set
sizes. The
and
testing
set are
still the same
in Section
The ratio
of the
sizes.
The
validation
set
and
testing
set
are
still
the
same
as
in
Section
3.1.
The
ratio
of
the
training set size gradually increases from 10% to 80%; that is, the size of training sets varies
training set size gradually increases from 10% to 80%; that is, the size of training sets varies
from 5000 to 40,000. The structure of DNN is also the same as in Section 3.1. Finally, the
from 5000 to 40,000. The structure of DNN is also the same as in Section 3.1. Finally, the
average relative error and the ratio of errors greater than 5% of each trained model on the
average relative error and the ratio of errors greater than 5% of each trained model on the
testing set are drawn in Figure 12. It can be seen that both the average relative error and
testing set are drawn in Figure 12. It can be seen that both the average relative error and
the ratio (>5%) overall decrease with the increase of the training set size. Moreover, when
the ratio (>5%) overall decrease with the increase of the training set size. Moreover, when
the size of training sets is more than 20,000, the change trend in both factors slows down.
the size of training sets is more than 20,000, the change trend in both factors slows down.
(a) Coupling capacitance
(b) Total capacitance
Figure
of average
averagerelative
relative
error
of errors
greater
than
5%different
under
Figure12.
12. Distribution
Distribution of
error
andand
the the
ratioratio
of errors
greater
than 5%
under
different
sizes
of
training
sets:
(a)
Coupling
capacitance;
(b)
Total
capacitance.
sizes of training sets: (a) Coupling capacitance; (b) Total capacitance.
3.4. The Comparison of Solving Time between the FEM and the Trained DNN
To compare the solving efficiency between the trained DNN and the FEM, we select
100 sets of data in the testing set and compute their parasitic capacitance matrix with the
trained DNN and FEM, respectively. Since both the trained DNN and FEM need to process
the input data in advance (i.e., FEM needs to generate mesh, and the trained DNN needs
3.4. The Comparison of Solving Time between the FEM and the Trained DNN
Electronics 2023, 12, 1440
To compare the solving efficiency between the trained DNN and the FEM, we select
14 of 18
100 sets of data in the testing set and compute their parasitic capacitance matrix with the
trained DNN and FEM, respectively. Since both the trained DNN and FEM need to
process the input data in advance (i.e., FEM needs to generate mesh, and the trained DNN
to construct
the input
vector),
the data
preparation
time time
is ignored
here,here,
and and
onlyonly
the
needs
to construct
the input
vector),
the data
preparation
is ignored
solving
time
is
counted.
The
programming
language
of
DNN
is
Python,
and
it
uses
GPU
the solving time is counted. The programming language of DNN is Python, and it uses
for computation.
The programming
language
of the of
FEM
C++.isFinally,
average
GPU
for computation.
The programming
language
theisFEM
C++. the
Finally,
the
computation time of those two methods for one case is shown in Table 8. The solving time
average computation time of those two methods for one case is shown in Table 8. The
of the trained DNN takes only 2% of FEM.
solving time of the trained DNN takes only 2% of FEM.
Table 8. The average computation time of the FEM and the trained DNN for one testing case.
Table 8. The average computation time of the FEM and the trained DNN for one testing case.
Method
Method
FEM
FEM
Thetrained
trained DNN
DNN
The
Time (ms)
(ms)
Time
759.49
759.49
14.64
14.64
3.5. The
The Verification
Verification of
of aa New
New Pattern
Pattern
3.5.
To further
verify our
our method,
method, we
we experiment
experiment with
with aa new
new pattern
pattern called
called pattern-2,
pattern-2,
To
further verify
whose
geometry
and
parameters
are
shown
in
Figure
13.
In
pattern-2,
conductor
is
whose geometry and parameters are shown in Figure 13. In pattern-2, conductor 11 is
centered, conductor
conductor22and
andconductor
conductor
4 are
always
right
of center
the center
centered,
4 are
always
on on
thethe
right
sideside
of the
line, line,
and
and
conductor
3
and
conductor
5
are
always
to
the
left
of
the
center
line.
We
choose
the
conductor 3 and conductor 5 are always to the left of the center line. We choose the same
same
metal
layers
as
the
pattern
in
Figure
2,
but
each
layer
contains
a
different
number
of
metal layers as the pattern in Figure 2, but each layer contains a different number of
conductors.
Subsequently,
we
built
a
new
DNN
network
called
DNN-3.
Since
the
geometric
conductors. Subsequently, we built a new DNN network called DNN-3. Since the
variables and
the total
conductors
of pattern-2
the same
pattern
in
geometric
variables
andnumber
the totalofnumber
of conductors
of are
pattern-2
areas
thethe
same
as the
Figure
2,
the
input
and
output
layer
used
for
DNN-3
is
the
same
as
in
Figure
7.
pattern in Figure 2, the input and output layer used for DNN-3 is the same as in Figure 7.
Figure 13. The
The illustration of pattern-2 and its geometry variables.
The metal
metal layer
layer selected
selected in
in Pattern-2
Pattern-2 is
is the
the same
same as
as the
the pattern
pattern in
in Figure
Figure 2;
2; hence
hence their
their
The
extraction
window
width
is
also
the
same.
The
sampling
range
of
each
geometric
parameter
extraction window width is also the same. The sampling range of each geometric
in pattern-2
shown in
utilized
field solver
EMPBridge
to generate
parameter
inispattern-2
is Table
shown9.inWe
Table
9. Wethe
utilized
the field
solver EMPBridge
to
25,000
sets
of
data
and
used
the
data
augmentation
method
introduced
in
2.3
generate 25,000 sets of data and used the data augmentation method introducedSection
in Section
to augment
thethe
datasets,
and
finally
obtained
50,000
the
2.3
to augment
datasets,
and
finally
obtained
50,000sets
setsofofdata.
data.AAtotal
total of
of 80%
80% of
of the
datasets were
were used
used as
as training
training sets,
sets, 10%
10% were
were used
used as
as verification
verification sets,
sets, and
and 10%
10% were
were used
used
datasets
as testing sets.
as testing sets.
Table 9.
Table
9. Parameters’
Parameters’ sampling
sampling range
range of
of pattern-2.
pattern-2.
Parameter
Sampling Range
Parameter
Sampling Range
x2
x3
x4
x5
w1
[1.04, 1.52]
[−1.52, −1.04]
[1.05, 1.52]
[−1.52, −1.05]
[0.09, 0.9]
w2
w3
w4
w5
-
[0.081, 2.0]
[0.081, 2.0]
[0.09, 2.0]
[0.09, 2.0]
-
The approach of hyperparameters’ settings of DNN-3 is the same as those of the DNN
introduced in Section 3.1. In the end, the better hyperparameter combination obtained with
TPE is as follows. The activation function is Swish. The batch size is 64. The learning rate
Electronics 2023, 12, 1440
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is 10−4 ; The number of hidden layers is 2, and each hidden layer has 500 neurons. The
prediction performance of DNN-3 on the testing set is shown in Table 10. It can be seen
that our method performs well in the new example, and the average relative error of the
coupling capacitance is only 0.10%, the average relative error of the total capacitance is only
0.04%, and the prediction accuracy of the total capacitance all meets advanced technology
accuracy requirements (errors < 5%).
Table 10. The predictive performance of DNN-3 on the coupling capacitances and total capacitances.
Property
Coupling
Capacitance
Total capacitance
Training
Time (h)
Average
Relative Error
Maximum
Relative Error
Ratio
(>5%)
0.32
0.10%
5.04%
0.00%
-
0.04%
0.37%
0
4. Discussion
This study uses the DNN to predict the parasitic capacitance matrix of a 2D pattern.
The network output only includes coupling capacitances. The total capacitance is obtained
by calculating the sum of the corresponding coupling capacitances. Moreover, a data
augmentation method is employed, doubling the dataset’s size. For comparison, this study
also utilized ResNet to predict the parasitic capacitance matrix. The above experiments can
derive the following discussions.
•
•
•
•
•
•
Data augmentation technique can obtain new data through the original dataset without
additional computation. Section 2.2 introduces a mirror flip method that doubles the
size of datasets size and reduces the data preparation effort.
In this paper, we train only one DNN model to predict the capacitance matrix. The
neural network’s output includes only coupling capacitances in the capacitance matrix, and the total capacitances are obtained by summing corresponding predicted
coupling capacitances. For the research case, experimental results in Table 6 show that
the prediction accuracy of the total capacitance obtained by this method meets the
advanced process requirements (<5%).
The comparison in Table 7 indicates that the predictive performance of our method
with the method that directly predicts the total capacitance using DNN is very close.
Moreover, the training time of our method is 0.89 h, and that of DNN-2 is 0.35 h.
Therefore, our method only takes about 72% of the total training time but can simultaneously predict the total capacitances and coupling capacitances. In addition, the
hyperparameters’ tuning of DNN-2 is time-consuming, and our method also saves
this process. The above results indicate that the method introduced in this work is
reliable and can save the training cost for predicting total capacitance.
In [11], ResNet is used to predict the total capacitance and coupling capacitances,
which shows good precision. To compare this kind of neural network’s prediction
performance with DNN for this research case, we also utilized the ResNet to predict the
capacitance matrix. The results show that the performance of DNN is better than the
ResNet for this research case, and the training time of DNN is only 31% of ResNet-18.
We tested the prediction effect of DNN under different dataset sizes. With the increase
in training data size, the average relative error and the ratio of errors greater than 5%
of trained models gradually decrease. Further, when the training data size is greater
than 20,000, the changing trend of the average relative error and the ratio (>5%) tends
to slow down.
In addition, the calculation efficiency of the FEM and the trained DNN are compared.
The average computing time of the trained DNN for one case is only 2% of that of FEM,
which shows that the trained DNN has good efficiency. Furthermore, we validate
our method to another pattern called pattern-2. The results show that our method
performs well on the new pattern, indicating its feasibility.
Electronics 2023, 12, 1440
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•
•
This experiment only considers horizontal changes in the interconnect geometry.
However, in real manufacturing, there will be more process variations. For example,
the permittivity of the dielectric layer, interconnect wire’s thickness, etc. Therefore,
in the future, the parasitic capacitance prediction will add more variables to the
consideration.
When there are corners or connections in the interconnect wires [4,40], the two-dimensional
analysis is no longer applicable. We prepare to solve the parasitic capacitance prediction of this kind of problem in the next step. The neural network’s input data
representation is critical, and one approach is using the density-based data representation of the interconnect’s top view [23].
5. Conclusions
This study used DNN to predict the parasitic capacitance matrix of a two-dimensional
pattern. The neural network’s output includes only coupling capacitances in the capacitance matrix, and the total capacitances are obtained by summing corresponding coupling
capacitances, thus saving the cost of training total capacitance. For the research case in
this study, experimental results show that the average relative error of the predicted total
capacitance based on DNN in the testing set is only 0.05%, and the average relative error
of the predicted coupling capacitance is only 0.12%, showing good prediction accuracy.
Furthermore, a data augmentation method was introduced, reducing the data preparation
effort. Our study can be used to construct capacitance models in the pattern-matching
method.
Author Contributions: Conceptualization, Y.M., X.X., S.Y. and Z.R.; Data curation, Y.M., Y.Z. and
T.Z.; Formal analysis, Y.M., X.X., S.Y. and Z.R.; Funding acquisition, X.X., S.Y. and Z.R.; Investigation,
S.Y. and L.C.; Methodology, Y.M., X.X. and S.Y.; Project administration, Z.R.; Resources, Y.M., X.X.,
Y.Z., T.Z. and Z.R.; Software, Y.M., S.Y., Y.Z., T.Z. and Z.R.; Supervision, X.X., S.Y., Z.R. and L.C.;
Validation, Y.M.; Visualization, Y.M.; Writing—original draft, Y.M.; Writing—review & editing, X.X.,
S.Y., and Z.R. All authors have read and agreed to the published version of the manuscript.
Funding: This research was funded by The Institute of Electrical Engineering, CAS (Y960211CS3,
E155620101, E139620101).
Data Availability Statement: The data that support the findings of this study are available from the
corresponding author upon reasonable request.
Conflicts of Interest: The authors declare no conflict of interest.
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