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APUSNCURSINRSM.2017.8072529

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Machine Learning Based Method of Moments
(ML-MoM)
He Ming Yao and Li Jun Jiang*
Yu Wei Qin
Department of Electrical and Electronic Engineering
The University of Hong Kong
Hong Kong
yaohmhk@hku.hk, jianglj@hku.hk*
Department of Electrical and Computer Engineering
Carnegie Mellon University
Pittsburgh, PA, USA
yuweiq@andrew.cmu.edu
The popular cloud computing platform ? Amazon Web Service
(AWS) is utilized to support the computation [7-8].
Abstract?This paper proposes a novel method by rethinking
the method of moments (MoM) solving process into a machine
learning training process. Based on the artificial neural network
(ANN), the conventional MoM matrix is treated as the training
data set, based on which machine learning training process
becomes conventional linear algebra MoM solving process. The
trained result is the solution of MoM. The multiple linear
regression (MLR) is utilized to train the model. Amazon Web
Service (AWS) is used as the computations platform to utilize the
existing hardware and software resources for machine learning.
To verify the feasibility of the proposed new machine learning
based method of moments (ML-MoM), we choose the static
parasitic capacitance extraction and dynamic electromagnetics
scattering as examples. The proposed novel idea opens a new
gateway between conventional computational electromagnetics
and machine learning algorithms with various application
potentials.
II. ML-MOM TRAINING PROCESS
The equation (1) is a conventional MoM matrix equation
based on the static integral equation [9-10], where the charge
density ? is discretized using the pulse basis function and the
corresponding excitation vector is represented by ?.
S11
[ ?
SM1
or
(1)
???= ?
where M is the number of field point, N is the unknown number
for vector ?, and the vector V has N terms. The elastance matrix
S has the dimension M by N. While S, ? and V are real numbers
in static field problem, they are complex for dynamic problems.
The elastance element Smn could be tested using Galerkin or
point correlation methods while the Green?s function is used as
the integration kernel for Smn. When M=N, Eqn (1) is
conventional MoM matrix system.
Training 1
Training i
V1
?
?
.
.
.
.
.
.
S12
.
.
.
S11
?1 ? 2
.
.
.
?N
S1N
Si1
Si 2
?N
.
.
.
?1 ? 2
?N
.
.
.
?2
.
.
.
?1
VM
?
.
.
.
.
.
.
As today?s most popular research field, machine learning
(ML) aims at learning from known information and make
predictions upon unknown data [5]. Due to large quantities of
applications, machine learning technologies have enormous
computing resources ranging from software algorithms to
hardware platforms, which potentially could benefit other
scientific computation fields. Hence, these sources motivate us
to build the direct connection between the machine learning and
MoM so that we could use ML resources to serve CEM.
Training M
Vi
S iN
SM1
.
.
.
The method of moments (MoM) [1] is a popular
computational electromagnetics (CEM) method applicable to
parasitic capacitance extractions, electromagnetic scattering and
radiation, signal and power integrity [2-4], etc.
.
.
.
.
.
.
I. INTRODUCTION
.
.
.
.
.
.
Keywords? MoM; Machine Learning; Artificial Neural
Network; Capacitance Extraction; Electromagnetic Scattering
SM 2
S MN
S
Fig. 1. Artificial nerual network representation of the machine learning based
MoM (ML-MoM). ?? is the unknown, ?? , is the excitation of the MoM
equation, and ??? is the matrix coefficient of MoM.
In this paper, the conventional MoM is reinterpreted into a
machine learning process using the artificial neural network
(ANN). Different from previous ANN curve fitting, by this
method, the conventional MoM solving process is transformed
into an ANN training process. Consequently, we can utilize
multiple linear regression (MLR) to train the model, which is
typical training process in machine learning regime [6]. The
trained model is the final solution of MOM. The proposed
approach is used to solve both static parasitic capacitance
extractions and dynamic electromagnetics scattering problem.
978-1-5386-3284-0/17/$31.00 �17 IEEE
S1N ?1
V1
? ][ ? ] = [ ? ]
SMN ?N
VM
?
?
?
Our novel approach begin by reconstructing MoM into a
training model, as shown in Fig. 1. That is a single layer
perceptron (neural network structure). While the matrix element
Smn is used as the input training data, the charge density unknown
?? is used as the weighting coefficient. The output of the neural
network is the potential Vm. Thus, solving MoM becomes a new
process: feeding Smn to train the model. After training, the
accurate estimation of the weighting coefficient ?? can be
973
AP-S 2017
Highly developed machine learning algorithms and
computation platform provide software and hardware supports
to realize our novel approach. We use the multiple linear
regression (MLR) to train our model to achieve the MOM results.
By utilizing the following cost function, MLR intrinsically
decreases the least square error:
1
2
??? = arg ??? ??
?=1 (?? ? ? ? ?? )
Capacitance (pF)
(a)
50
100
150
200
250
Angle ? (degrees)
300
350
400
(b)
ACKNOWLEDGMENT
REFERENCES
[1]
R. F. Harrington, Field Computation by Method of Moment, Macmillan,
New York, 1968.
[2] C. Craey, B. Parvais, and X. Dardenne, ?MoM Simulation of Signal-toNoise Patterns in Infinite and Finite Receiving Antenna Arrays,? IEEE
Trans. Antennas Propag, vol. 52, no. 12,pp. 3245-3256, Dec. 2005
[3] P. D. Mehta and S. B. Chakrabarty, ?Electrical capacitance of dielectric
coated metallic parallel piped and closed cylinder isolated infree space,?
J Electrostat, vol. 71, no. 71, pp. 756-762, Aug. 2013.
[4] A. I. Mackenzie, S. M. Rao, M. E. Baginski, ?Electromagnetic Scattering
From Arbitrarily Shaped Dielectric Bodies Using Paired Pulse Vector
Basis Functions and Method of Moments,? IEEE Trans. Antennas Propag,
vol. 57, no. 7, pp. 2076-2083, May. 2009
[5] T. Mitchell, Machine Learning, McGraw Hill, 1997.
[6] C. M. Bishop, Pattern Recognition and Machine Learning. Springer, Aug.
2006.
[7] J. Hugunin, ?The Python Matrix Object: Extending Python for Numerical
Computation,? Proceedings of the Third Python Workshop, Reston, Va.,
Dec. 1995.
[8] http://docs.aws.amazon.com/gettingstarted/latest/awsgsg-intro/gsg-awsintro.html
[9] W. C. Chew and L. Jiang, "A Complete Variational Method for
Capacitance Extractions," Prog Electromagn Res, vol. 56, pp. 19-32, Jan.
2006.
[10] A. G. Papagiannakis, ?Application of a point-matching MoM reduced
scheme to scattering from finite cylinders?. IEEE Transactions Microw.
Theory Tech, vol. 45, no. 9, pp. 1545-1553, Aug. 1997.
2.54
2.52
2.50
2.48
2.46
MOM
ML-MOM, M=1008
ML-MOM, M=2268
2000
0
0
This work was supported in part by the Research Grants
Council of Hong Kong (GRF 17207114 and GRF 17210815),
NSFC 61271158, and Hong Kong UGC AoE/P?04/08.
2.56
1500
1
a
In this paper, by combining MoM and ANN, we reconstruct
conventional MoM into novel machine learning based method
of moment (ML-MoM), which is solved by a standard machine
learning training process-MLR in ML field. The novel MLMoM is discussed by analyzing its application on two numerical
examples, which indicates the feasibility of our new method.
2.58
N
x
2
IV. CONCLUSION
2.60
1000
?=?
3
(a)
Power planes as shown in Fig. 2. is used as the first example
to extract capacitances. Three coupling plates (Vdd, Vcc, and
GND) are located at two layers. The ground plate has a
dimension of 12� mm2, and the distance between two layers
is 0.2mm. The conventional MoM and the newly proposed MLMoM based on MLR training are compared in the capacitance
extraction. It clearly states the efficiency and feasibility of MLMoM.
2.42
?
4
Fig. 3. ML-MoM Current density produced by scattering on metal cylinder.
(a) TMz uniform plane wave incident on a circular Metal cylinder, (b) Current
distribution of ML-MoM with N=360 and M=720
III. NUMERICAL EXAMPLES
2.44
5
?
z
We choose Amazon Web Service (AWS) ? a cloudcomputing service suite as machine learning hardware platform,
which offers a wide range of machines and softwares with
ranges of configurations. The machine used is m4 instance, the
latest generation of General Purpose instance offered by AWS
and configured with 16 vCPUs and 64 GB memories. Extra 200
GB volume is attached, to verify validation of a large scale of
matrix sizes.
500
y
z
In the static field, all data in equation (1) are real numbers.
based on which our model can be easily trained by MLR.
However, when the data in equation (1) is complex for dynamic
fields, we can still train our model to achieve MOM solutions.
But the real and imaginary parts of the matrix have to be
processed specially to make them into logitimate new data
groups before the training.
0
6
(2)
2
?
cylinder of radius a=1m with wavelength ?=a/2. The
circumference is divided into 360 segments as N for both
methods and 720 segments as M for ML-MoM. The results of
conventional MoM and ML-MoM based on MLR training are
compared in current density. With the same N, ML-MoM can
offer better result.
Current Density (mA/m)
obtained. According to experience in ML, more training data
will provide better accuracy in the final solution. Hence, when
the new algorithm is used, more field points than source points
are needed, which means M>N. Hence, the MoM solving
process becomes a machine learning training process.
2500
(b)
Fig. 2. Power-ground plane structure for capacitance extraction. (a) Powerground plane structure vertical view and plane view, (b) Capacitance
comparison between MoM and ML-MoM. N is the number of charge
discretization, and M is the field testing point number.
The PEC cylinder as shown in Fig. 3. is used as the example
for the dynamic field. A TMz uniform wave incident upon the
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