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This paper presents two new methods, space mapping (SM) with prior knowledge input (PKI-D) with difference and compound space mapping-based neuromodeling. Both methods combine two powerful techniques, space mapping-based neuromodeling and PKI-D with

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INTERNATIONAL JOURNAL OF NUMERICAL MODELLING: ELECTRONIC NETWORKS, DEVICES AND FIELDS
Int. J. Numer. Model.
(2007)
Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/jnm.656
A knowledge-based neuromodeling using space mappingtechnique: Compound space mapping-based neuromodeling
Murat Simsek*
,
y
and N. Serap Sengor
Istanbul Technical University, Faculty of Electrical and Electronic Engineering, Electronics Engineering Department,Maslak, TR-34469, Istanbul, Turkey
SUMMARYThis paper presents two new methods,
space mapping
(SM)
with prior knowledge input
(PKI-D)
withdiﬀerence
and
compound space mapping-based neuromodeling
. Both methods combine two powerfultechniques, space mapping-based neuromodeling and PKI-D with diﬀerence. The knowledge-basedmodeling methods in the RF/microwave literature merge the prior knowledge about the device to bemodeled with neural network structures while a knowledge-based method, SP, focuses on reducing thecomputational burden. The main advantage of the proposed methods over these already existingknowledge-based methods are their better extrapolation capability and reduced number of training setdata. The simulation results obtained reveal that both methods decrease the cost of training and improvethe extrapolation capability and output performance of the SP-based neuromodeling. Copyright
#
2007John Wiley & Sons, Ltd.Received 6 March 2007; Revised 3 July 2007; Accepted 9 August 2007
KEY WORDS
: knowledge-based neuromodeling; artiﬁcial neural networks; space mapping
1. INTRODUCTIONNew techniques and methods that would fulﬁll the need of computationally fast and accuratemodels have been sought in modeling especially radio frequency (RF) devices, as market needsmore and more usage of these devices. To provide such techniques and methods, approach inmodeling these devices have gone through reconsideration during the past decade. Now, thetime spent during design process is as important as the eﬃciency of the models, so especiallyartiﬁcial neural network (ANN)-based structures, which are capable of nonlinear modeling andfast once the training phase is completed, have begun to be used eﬀectively even in toolboxes of design-oriented simulators [1 –3]. Nevertheless, there is a need for the improvement of these
*Correspondence to: Murat Simsek, Electronics and Telecommunication Engineering Department, Electric andElectronics Engineering Faculty, Istanbul Technical University, 80626 Maslak, Istanbul, Turkey.
y
E-mail: simsekmu@itu.edu.trCopyright
#
2007 John Wiley & Sons, Ltd.
models as the computational demand for training set is still a problem to solve. Space mapping(SM) technique as it is proposed in [4] aims at resolving this problem in general. Another aspectthat needs to be improved is ANN-based models with better extrapolation capability. As ANNmodels are black-box models, even though their interpolation capability is better than mostconventional modeling approach as look-up tables, polynomial approximations, they are highlydependent on training set. A way to overcome this ﬂaw is to make use of the already existingknowledge about the device to be modeled along with ANN-based modeling. Thus, diﬀerentknowledge-based methods as prior knowledge input (PKI), source diﬀerence (SD), priorknowledge input with diﬀerence (PKI-D) and equivalent circuit-state-space equation-neuralnetwork (EC-SSE-NN) have been proposed [5 – 8].
In this work, two new methods that focus on decreasing the training set burden andimproving the extrapolation capability are proposed. In these models, to reduce the training setburden, SM is used, whereas to implement the prior knowledge about the device to be modeled,a knowledge-based method PKI-D is integrated. The proposed methods are hybrid methods-based on PKI-D and SM-based neuromodeling (SMN), namely SMN with PKI-D method andcompound space mapping-based neuromodeling (CSMN). Both models combine theadvantages of SMN and PKI-D and improve the two aspects mentioned above. Thus, ﬁrstPKI-D and SMN will be introduced in Section 2, and then in Section 3, two proposed methodsSMN with PKI-D method and CSMN will be introduced. Both methods are inspired by the ideaof combining the advantages of SMN and PKI-D in one structure. While SMN with PKI-Ddoes this in discrete processing steps, CSMN does it in an integrated manner. In order to showthat these newly proposed methods improve the extrapolation capability while decreasing thecomputational burden of forming the training set, two diﬀerent modeling applications areintroduced and simulation results for these applications are given. The simulation results willespecially exploit the advantage of the proposed methods over SMN when extrapolationcapability is considered. Another important result will be on the number of training data neededto get the models. Further comparison of the proposed methods and the other methods are alsogiven in [7,9]. While one of the applications deals with modeling the characteristic impedance of micro-strip line, the other is on function approximation. Modeling the Branin function isconsidered to show that even though the starting point is to improve modeling of RF devices,the methods proposed can be used in diﬀerent applications where the problem can be expressedas a function approximation problem.2. A PRELIMINARY ON KNOWLEDGE-BASED METHODSThere are various approaches in modeling RF/microwave devices. While from ElectroMagnetic(EM) simulators highly accurate models can be obtained, from equivalent circuits and empiricrelations computationally eﬃcient models can be derived. Since EM simulators give almostexact solutions they are named ‘ﬁne models’ while other approaches as equivalent circuits,empiric relations are named ‘coarse models’. The best approach should be capable of deriving amodel as accurate as ‘ﬁne model’ and as computationally eﬃcient as ‘coarse model’.ANN structures for modeling RF/microwave devices have numerous advantages overtraditional modeling approaches [1]. Their main advantage is once trained with a well-selected,proper training data obtained from EM simulators, they give, in the training data interval, asgood results as the data set even for the parameter values that were not in the training set. This is
M. SIMSEK AND N. SERAP SENGORCopyright
#
2007 John Wiley & Sons, Ltd.
Int. J. Numer. Model
. (2007)DOI: 10.1002/jnm
due to their high function approximation property and generalization capability. Once thetraining phase is completed ANN models are computationally very eﬃcient as they involve verysimple computations compared with EM simulators. With these properties, ANN structures aregood candidates for model deriving, but they also have disadvantages. As ANN models formthe input/output relation just by using training data through the training phase no knowledgeabout the phenomena to be modeled is used. Hence, even though they give satisfactory resultsfor interpolation, they are poor in extrapolation. Another drawback is its accuracy is very muchdependent on the training data, so there is need of large training data with well-selected valuesover a large interval.In order to overcome these disadvantages of ANN structures and to beneﬁt from theadvantages they oﬀer, various knowledge-based methods that make use of ‘coarse model’ areproposed [1,4 – 9]. Amongst these, SM technique introduces a procedure to form a mapping
which when used with ‘coarse model’ gives accurate results while using ‘ﬁne model’ eﬃciently.While SM can be employed especially to form eﬃcient training set for ANN structures, otherknowledge-based methods as PKI, SD and PKI-D are eﬀective for implementing the alreadyexisting relations and associations about the device to be modeled. In the sequence, ﬁrst PKI-Dmethod will be introduced and then SMN will be reviewed to ease the idea behind the proposedmethods given in Section 3.
2.1. Prior knowledge input with diﬀerence method
PKI-D is a hybrid method combining the advantages of two methods, PKI and SD [7]. Itestablishes input/output relationship not only considering model inputs as inputs but alsothrough extra input obtained from ‘coarse model’ output as shown in Figure 1(a). The ANNstructure in the method is trained to comprehend the diﬀerence between the ‘ﬁne model’ and the‘coarse model’. ‘Fine model’ gives target response while ‘coarse model’ exploits approximateoutput compared to target response.In PKI-D method like SD, the ﬁnal output to be utilized is obtained by summing PKI-Doutput and ‘coarse model’ output which can be followed from Figure 1(b). While extra inputreduces model complexity and adds regional input/output relationship to PKI-D, the learningrange is narrowed since the diﬀerence between ‘ﬁne model’ output and ‘coarse model’ output is
Figure 1. PKI-D model structure: (a) training phase of PKI-D and (b) ﬁnal model of PKI-D.
NEUROMODELING USING SPACE MAPPING TECHNIQUECopyright
#
2007 John Wiley & Sons, Ltd.
Int. J. Numer. Model
. (2007)DOI: 10.1002/jnm
used as the desired output; thus, the learning process is improved. Hence, this hybrid methodcombines the advantages of both PKI and SD methods. In [7], the comparison of PKI-D withplain ANN model, SD and PKI methods are given and the results obtained there show thatPKI-D is superior to the others.
2.2. Space mapping-based neuromodeling
The aim of SMN is to ﬁnd an appropriate mapping from input space of the ‘ﬁne model’ to inputspace of the ‘coarse model’, and ANN structure establishes this mapping which is denoted by
P
ð
:
Þ
:
Once this mapping is determined the expectation is the ‘coarse model’ will generate resultsas adequate as the ‘ﬁne model’ without much computational burden. The vectors
x
c
and
x
f
represent the input parameters of the ‘coarse model’ and ‘ﬁne model’, respectively.
R
c
ð
x
c
Þ
represents the corresponding ‘coarse model’ response
R
c
and
R
f
ð
x
f
Þ
represents the correspond-ing ‘ﬁne model’ response
R
f
:
Initial base points for ‘ﬁne model’ input space are obtained fromstar distribution [1] in which optimal ‘coarse model’ point
x
n
c
is used as the mid-point value. Allsteps of the SMN method have been stated in [1]. For error evaluation, i.e. to evaluate thediﬀerence between ﬁne model output and coarse model output, the same upper bound is used inﬁnding the new
x
f
and in deciding to stop the training phase of ANN structure that establishesthe mapping between two model input spaces. In addition, the error upper bound used todetermine ANN structure has been decreased at each iteration step in order to provide SMN toconverge to optimal response of the ‘coarse model’.In general, feed-forward neural network structures as multilayer perceptron (MLP) or radialbasis function (RBF) networks are used to set up the mapping
P
ð
:
Þ
between ‘ﬁne model’ and‘coarse model’ input spaces. The formulation of feed-forward ANN structure is as follows:
y
i
¼
X
n
h
j
¼
1
a
ij
c
j
ð
w
T
j
x
f
þ
y
j
Þ ð
1
Þ
where
n
h
is the number of hidden neurons and
C
j
ðÞ
is the activation function,
w
j
weight vectorand
y
j
the bias associated with the
j
th hidden layer neuron and
a
ij
’s correspond to output layerweights. The training phase to determine the weights and biases can be formulated as anoptimization problem as follows:
½
W
;
y
;
A
¼
arg min
w
;
y
;
A
fjj½
e
T1
e
T2
e
T
N
T
jjg ð
2
Þ
where
e
i
¼
R
c
ð
y
Þ
R
f
ð
x
ð
i
Þ
f
Þ ð
3
Þ
Once the optimization problem stated by (2) is solved, the parameters
W
¼ ½
w
1
w
2
. . .
w
n
h
;
y
¼ ½
y
1
y
2
. . .
y
n
h
;
A
¼ ½
a
1
a
2
. . .
a
n
necessary to set up the mapping
P
ð
:
Þ
are determined.Thus, the mapping
P
ð
:
Þ
is constructed by ANN structure and once this phase is complete,
x
c
can be determined by
P
ð
:
Þ
using
x
f
:
The SMN response
Y
c
is obtained using ‘coarse model’response function
R
c
while the input of ‘coarse model’
x
c
is determined from
x
f
using themapping
P
ð
:
Þ
:
This model is constructed with less ‘ﬁne model’ response than conventionalneuromodel, since instead of determining an ANN structure through training which requires alarge ‘ﬁne model’ response only mapping
P
ð
:
Þ
is formed and used with ‘coarse model’. Whiledetermining
P
ð
:
Þ
through training phase, ‘coarse model’ is not adapted and this decreases cost of coarse approximation, since it is used only to form input/output relation. The steps of the SMN
M. SIMSEK AND N. SERAP SENGORCopyright
#
2007 John Wiley & Sons, Ltd.
Int. J. Numer. Model
. (2007)DOI: 10.1002/jnm
method can be followed from block on the right-hand side of Figure 4 which corresponds to theprocesses of SMN.3. PROPOSED METHODSEach of the knowledge-based methods summarized in Section 2 improve one of the two maindrawbacks of ANN-based modeling, but do not resolve both at the same time. While PKI-Dimproves the extrapolation capability [4], SMN decreases the computational burden of utilizing‘ﬁne model’. The main purpose of this section is to introduce two new methods capable of improving both. While the ﬁrst one uses both methods consecutively, the other combines theadvantages of PKI-D and SMN in one structure where the training phases of ANN structuresare carried out in the same computation loop. As mentioned above by these proposed methods,the need for ﬁne model responses is decreased as the characterization of SM ﬁne modelresponses are used only when needed. This advantage of the proposed methods can be followedfrom [9], where a comparison between diﬀerent methods is given for the characteristicimpedance modeling of micro-strip line.
3.1. SMN with PKI-D method
SMN method exploits less training data and has less computational cost than other modelingapproaches. The data that have been used in SMN to obtain a better model performance can beused in PKI-D method as a training set. For these reasons, these two diﬀerent methods will beprocessed in sequential time steps. The new hybrid learning model is called ‘SMN with PKI-D’.In this method, PKI-D exploits less number of but more relevant training data to acquire ﬁnemodel behavior, and some important aspects such as generalization, extrapolation capabilityand model accuracy also are improved compared to the other knowledge-based methods [9].First, SMN part is processed and a mapping
P
ð
:
Þ
is formed. Then, initial points and newextracted points are combined in a set which is used during the training phase of PKI-D. AsSMN extracts relevant input points for PKI-D through ﬁne model, the accuracy of PKI-D as amodel is improved compared to PKI-D model without beneﬁciating SMN. This approach in thetraining phase of PKI-D solves the problems that are related to ﬁnding appropriate training setand reducing the number of training data. Inputs and outputs for SMN with PKI-D andtraining phases are given in Figure 2. PKI-D acquires the diﬀerence
ð
D
Þ
between ﬁne and coarsemodel responses while input of both ﬁne and coarse models is the input space of originalproblem.
Y
c
;
which is applied as an input to PKI-D, helps to set relationships between
D
and
x
f
and also reduces the problem complexity during the PKI-D training phase. The ﬁnal modelobtained as the result of SMN with PKI-D approach is given in Figure 3. SMN and PKI-D runsequentially in the ﬁnal model where model input is
x
f
and model output is
Y
d
:
The output of the model obtained at the end of the training phases in SMN with PKI-D method will be aseﬀective as
D
:
All the steps of the algorithm for SMN with PKI-D are shown explicitly in Figure 4. In thisﬁgure, there are two parts; in one of them SMN process is demonstrated while in the other thetraining of PKI-D is illustrated. After the process related to SMN is completed and the mapping
P
ð
:
Þ
is formed, the process related to PKI-D starts. During the training phase of PKI-D, inputsdetermined as the result of SMN are used.
NEUROMODELING USING SPACE MAPPING TECHNIQUECopyright
#
2007 John Wiley & Sons, Ltd.
Int. J. Numer. Model
. (2007)DOI: 10.1002/jnm

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