A semi-automated method for dynamic model abstraction
Kangsun Lee and Paul A. Fi-l'-i, 1:
Department of Computer Information Science and Engineering
University of Florida
Bldg. CSE, Room 301
Gainesville, FL 32611
As complex models are used in practice, modelers require tin 1. !I ways of abstracting their models. Through the use
of hierarchy, we are able to simplify and organize the complex -1. i, The problem with the hierarchical modeling
is that -- -1. i, components in each level are dependent on the next-lowest level so that we are unable to run each
level independently. We present a way to augment hierarchical modeling where abstraction can take place on two
fronts: structural and behavioral. Our approach is to use structural abstraction in order to organize the -1. i,
hierarchically, and then apply behavioral abstraction to each level in order to approximate lower level's behavior so
that it can be executed independently. The proposed abstraction method is done by semi-automatic way and gives
advantages to view and ;. .1- complex -- -l. i,- at l!!. I i I levels of abstraction.
Keywords: Abstraction, Multimodeling, Object-Oriented Modeling, System Identification
Real world dynamic -1. i,1- involve a large number of variables and interconnections, which sometimes make the
modeling process untractable. Model abstraction is a method for reducing the .!ii!.1. .11- of a simulation model
while maintaining the validity of the simulation results with respect to the question that the simulation is being used
to address.1 Computational II. !. i. and representational ... ii i are main reasons of using abstract models in
simulation2 4 and well as in programming languages. -7 Much recent research has recognized the need for multiple
levels of abstraction,3'8'9 good representations and ;,1li -I to change from one representation to another for. !I i-I
problem solving. Use of abstraction hierarchy has been standard answer to this problem.9-12 While the hierarchical
approach is sound for well-structured models defined in terms of state space functions and set-theoretic components,
selection of -- -1. 1, components in each level are dependent on the next-lowest level, since actual functional semantics
are contained in the lowest level. This means we are unable to execute each level independently.
We propose an abstraction method which can better handle multiple levels of abstraction within hierarchical
modeling framework. Our method is to construct abstraction hierarchy first and then make each level an independent
structure by approximating its lower levels behaviors so that each level can be executed and ii! .1 1 apart from
the rest of hierarchy. We define our -- -1. 1, abstraction to be one of two I- ,. structural or behavioral. i i Ii, I!
abstraction is a process of organizing the -- -I. i, hierarchically using refinement and homomorphism. Refinement
is the process of refining a model to more detailed models of the same I .- (homogeneous refinement) or different
I ,. (heterogeneous refinement), while homomorphism is a mapping that preserves the behavior of the lower-level
-1. i, under the set mappings.12 In structural abstraction, one constructs an abstraction hierarchy with simple
model I- i at first, refining them with more complex model I- ,. later. iiLi IIn I! abstraction corresponds to
this iterative 1 ..... 'n. 1 and resulting hierarchy can be used to provide multiple abstractions for the -- -I. i,
After creating the hierarchy, we ii want to isolate abstraction levels, so a level can be executed alone. This
is where the behavioral approaches are ,!1..'. -,1 Behavioral abstraction focuses only on behavioral equivalence
without structural preservation. Below the structural abstraction, each component is black-box with no detailed
internal structure. Behavioral abstraction is used to represent the black-box by approximating the behavior of lower
level -- -1. i, components. By combining structural and behavioral abstraction together, each level of abstraction is
Other author information:
P.A.F.:Email: firstname.lastname@example.org; !i 1.ir..i and Fax: 352-392-1414; WWWV home page: http://www.cise.ufl.edu/~ fishwick
K.L.: Email: email@example.com; !I. 1. ..... 352-392-1435; WWWV home page: http://rwww.cise.ufl.edu/~ kslee
independent from the lower abstraction levels, so a level can be executed apart from the rest of the hierarchy. In
depth discussions of each abstraction technique follow in the subsequent sections.
This paper is organized as follows : In section 2, we present a -- -I. i, abstraction method after examining existing
abstraction techniques. Section 3 shows how our abstraction method works in an actual example and then we
conclude this paper with future works to be achieved.
2. DYNAMIC MODEL ABSTRACTION
Models must be multi-1 i. I 1 so that different abstraction levels of the model respond to hti.l, i. II needs of the
;, ,1- 11,3,9 The important issue for the multi-layer modeling is that how to organize a series of models in a way
such that it facilitates multiple resolutions of the 0., We follow the multimodeling rn. I ,. .!.-_ -to organize
the -I. i, for this purpose. Multimodeling is a modeling process in which we model a -- -I. i, at multiple levels
of abstraction. It provides a way of structuring a .t!!. I. iI model t- I. together under one framework so that each
SI"- performs its part, and the behavior is preserved as levels are mapped.4'21'22 Since it has different model i" -
together under one structure, unlike the other homogeneous-structural abstraction, where dynamical I -r. ii- are
abstracted with only one model I- ".- and refined with the same model I- ". multimodeling can ii!I.1 a number of
abstraction perspectives and accommodate a larger I i;. of questions. Detailed discussion on multimodeling and
examples of modeling techniques can be found in Refs. 16-20.
While the multimodeling approach is sound for well-structured models defined in terms of state space functions
and set-theoretic components, selecting -1. 11, components in each level is dependent on the next-lowest level due
to hierarchical structure. This implies that we are unable to simulate each level ',.,.1, ..... il,. It is possible
to obtain output for ;,ii abstraction level but, nevertheless, the -l. it model must be executed at the lowest
level of the hierarchy, since there is where we find the actual functional semantics associated with the model.
A new definition and iit. -i!l.,1._-- is needed to better handle abstraction of -- -,. iI- and components. This is
where the behavioral abstraction approaches are i!!.1..-. ,1 By incorporating behavioral abstraction approaches
into multimodeling ,ii. !. 1. .1.._-, abstraction in multimodeling allows each level to be understood independently of
the others, so that discarding all the abstractions below, :;,i given level will still result in a complete behavioral
description of a -. i 13 The components are "l,! !: ;,.,.. with no detailed internal structure. Behavior is
described as a set of input-output pairs defining a black box. We have two approaches for -1 I f- i - -1. 11, behavior:
1 1I 11, approach : captures only the steady state -lI. 11 output value instead of a complete output trajectory.
The input value is defined to be the time integral value over the simulation trajectory.
Dynamic approach : one needs to associate time-dependent input and output f i. 1. .1 -
Though the static and dynamic approaches describe til!. i. i i allowable behaviors of the same phenomenon, abstrac-
tion techniques for the dynamic approach can be applied to static approach too. Therefore, we'll focus on dynamical
behavioral abstraction for illustrating abstraction techniques we'll use.
We denote the output of the dynamical -I. i,1 at time t by y(t) and the input by u(t). The data, defining -- -. 11,
behavior, are assumed to be collected in discrete time. At time t we have the data set
Zt = y(1), u(1), ..., y(t), u(t) (1)
A model of a dynamical -- i can be seen as a mapping from past data Zt-1 to the next output y(t):
y(t) = g(Zt-) (2)
y represents the predicted value whereas y(t) is the exact value. The problem of dynamical behavioral abstraction is
to find a mapping gt that gives good prediction in (2) using the information in a data record Zt. Parametric model
estimation in -- -1. i, 1-[. ,II I, iI..- I is the theory and art of building mathematical model of gt. Modeling the
-. -* i consists of selecting a general, parameterized mathematical representation and then tuning the parameters,
so that behavior predicted by the model coincides with measurements from the real -1. i A Parameter estimation
procedure provides a search through parameter space, effectively, to achieve a close-to optimal mapping between
the actual values of the -1. I i and the approximate abstract -. 1i1 Commonly used parameter models are ARX,
ARMAX, OE(Output Error) and BJ(Box-Jenkins). For the detailed structure of these models, see 23,25.
Neural networks have been established as a general approximation tool for fitting models from input/output
data.26 29 From the -- -'. ii identification perspective, a neural network ii ,- be considered as another candidate
model structure.24'30 The inputs are linearly combined at the nodes of the hidden 1 ,- i (s) and then -,l P i. l. 1 to a
threshold-like non-l l,. ,i i and then the procedure is repeated until the output nodes are reached. These produce the
values that should be matched to the variable y(t) in the observed pair (y(t), u(t)). Thus, behavioral abstraction by
neural networks is to find 1,,, 0, u(t)), where 0 represents the weights of the linear combinations involved in network
structure. Backpropagation, recurrent and temporal neural networks have been shown to be applicable to -- -I. i,
identification.31'13'32 On the other hand, recently introduced wavelet decomposition achieves the same ,!,L Mi of
approximation with a network of reduced size by replacing the neurons by I- I. i- ', i.e. computing units obtained
by cascading an ;I! i. transform and multidimensional wavelets.33
By integrating multimodeling techniques and behavioral techniques, we propose a method for dynamic model
abstraction. An execution will be shown in detail in the following section.
Do structural abstraction using Multimodeling
While ( there's a need to abstract a -- -1. 1, further) do
For the entire multimodeling structure
('IC. I: if a model component is within user interest;
If not // irrelevant to user interest
Box it for behavioral abstraction;
Apply one of behavioral abstraction techniques to the box
After structural abstraction, the user specifies the parts where behavioral abstraction is needed for a question
he wants to ask, and then a search process is started to find out actual model components relevant to user interest.
During behavioral abstraction, user intervention is needed to find good parameter values upon which a behavioral
abstraction technique is executed. For example, order of equation is needed for parametric model estimation technique
and number of hidden layer for neural network.
3. FULTON : steamship modeling
Consider a steam-powered propulsion ship model, named FULTON*, as shown in Fig. 1. In our -- -l. i furnace heats
water in boiler: when the fuel valve is 01'.'\ !, I flows and furnace heats; when fuel valve is CLOSi.1), no fuel flows
and furnace is at ambient temperature. Heat from the fuel is added to the water to form high-pressure steam, which
performs work by expanding against the turbine blades. To be compressed at low pressure, the steam is condensed
in a condenser, which requires the exhaust of heat from the water. There is a heat exchange is accomplished by
circulating sea water and the steam is condensed again into liquid, at which point it can be pumped back to the
3.1. Structural Abstraction of FULTON
Fig. 2 and Fig. 3 shows structural abstraction for the FULTON -l. i, Since FULTON has 4 components whose
function is distinct from others, we start with 4 Functional Block Models, L1, ..., L4.
Low level continuous models of BLOCK L2,..., L4 are defined by the laws of Ill. .i 1i i11i.; and ii. i,- conser-
1. (Li) BO1.1 i: ASS1.. 11; : defined as 1 S. I ,! iIi,- i' I.. Machine) in Fig. 3
*We name it "FULTON", after Robert Fulton, who designed and oversaw construction of the first steam-powered warship, the "USS
FULTON", for the US
Figure 1. High-level view of a shipboard steam-powered propulsion plant
ON/OFF BOILER As TRBN A CONDENSER A PUMP
ASSEMBLY ASSEMBLY ASSEMBLY
L1 L2 L3 L4
Figure 2. Top level of structurally abstracted model for FULTON
LA ---- I=0F N
Figure 3. SiL, I abstraction ofLo
Figure 3. SI i I [I I abstraction of LI
Table 1. Question-answering using the multimodel from structural abstraction
Question Model Answer
If the -- -, i i is warm, I S. !-1 System becomes cold when
how can it become cold? the water temperature equals
the ambient temperature
Why did the water start 1 S. !-2 The water was being heated and
to boil? the temperature reached 100oC
which caused the water to boil
2. (L2) TUI:1;1\ : Ar, = A, ki
3. (L3) CONDi.\Si.1: ASSi..li;lY : Awe = Ar= Awp
4. (L4) PUMP ASSi .. i;lY : Ai, = Awc kp
where, A, : Amount of steam into the turbine, Ar : Amount of steam remained after being exhausted in turbine,
ki : steam loss rate in turbine, Aw, : Amount of water in condenser assembly, Ai, : Amount of water increased in
boiler assembly by pumping water from pump assembly.
Since BOILi1.1; AiSS;i.11;1. has several states with transitions among them, we model it by 1 S.' as shown in
Fig. 3. The top most 1 S. I in Fig. 3 shows a two-state 1 S. I with input. We label this 1 S. !-1. The second label
includes a detailed representation of state "-' .1 Cold". By combining this new I S. I with I S. !-1, we create 1 S. !-2 (a
complete model of the boiling process). 1 S., 1-3 is constructed similarly. Each state is modeled as continuous models,
Mi, ..., MA6: The continuous models contained within states heating and cooling are made by combining Newton's
Law with the capacitance law .12
1. (Mi) COLD : T = a, A = Ai,, Af = 0, A, = 0.
2. (M2) HEATING : T = ki(100 T), A = Ai,, ,A = 0, A = 0.
3. (Ms3) COOLING :T = k2(100 ),A. = Ai, A, = 0, A = 0.
4. (AM4) BOILING :T=100, Aw = -k4 Ai,, Af = ks, A, = Af Aw.
5. (AMs) 0(\ .1i FLOW : same as BOILING with constraint Af = At
6. (MA) UNDi. iFLOW : T = undefined, Aw = Af = 0.
where, T (Temperature), Aw (amount of water), and Af (amount of foam on the top of water), At (maximum
steam amount of boiler), A, (amount of steam), I indicates fuel valve's position and a is the ambient temperature
of the water and ki (constants).
Now that we have specified 3 1 S. I levels, and one functional block level, we can use these models to answer
questions about the -I. in, A transition in 1 S. I ii sometimes refer to a more detailed state specification than is
available at the current level of abstraction, according to the questions. We present sample questions and answers
that can be practically derived from the presented multimodel. Given an arbitrary questions as shown in Table 1, a
model is chosen on which to base the answer. An actual natural language processing -- -I. -i does not currently exist,
therefore processing implied by Table 1 is idealized. By examining this table, we found that structural abstraction
is useful not only for model building but also for facilitating question-answering using a i. -' of abstraction.
Figure 4. The process of behavioral abstraction
3.2. Behavioral Abstraction of FULTON
Behavioral abstraction is to build a simulation metamodel, 537 that is a model of simulation model. We start with
set of input-out data pair from simulation results as shown in Fig. 4. \\ iili this prior knowledge, the method of
behavioral abstraction is to find how the inputs relate to outputs by ;1ii1 i-,_ one of abstraction techniques. We
showed several techniques to build a simulation metamodel in section 2. A mathematical metamodel is generated
by ,'!''l i- a parametric estimation technique, while neural network or wavelet network models can be generated
by training neurons or wavelons. In the following section, we'll see how one can earn benefits from behaviorally
abstracted models for given questions.
3.2.1. Question 1 : How are the four .,i". components .'. ,"
To focus on interaction between components at high level, we don't need detailed model for each. Since Boiler
component is refined into I S. I and low level continuous functional block models, we apply behavioral abstraction
technique and replace it with neural network BLOCK model as shown in Fig. 5. The shaded box represents the
behaviorally abstracted BLOCK model. Newly generated model should have the outputs which the original model
generates based on the input, but has some accuracy loss. It runs with time-saving, since we don't have to go down the
abstraction hierarchy in order to find the model's functional semantics. We experimented with ADALINE(ADAptive
1.I 1. I ,. 1ii) neural network using Matlab's neural network toolbox.38 Fig. 6 shows how closely the abstracted
model performs for this given question. The solid line represents the actual output from the original and dotted line
represents the output from further abstracted model by ,11'1 i,- the behavioral abstraction technique.
3.2.2. Question 2 : II !,,, is the behavior trajectory of the Boiler over .. "!
To refine a certain component, details of other components could be omitted as long as the resulting model provides
the same outputs with a certain accuracy confidence. We search model components irrelevant to the question, and
box it for behavioral abstraction. To answer question 2 I! i. !!, ,, we abstract other components except boiler into
one unit as shown in Fig. 7. The turbine, condenser and pump components are abstracted into one black-box model,
but still provide the input to the boiler component with an allowable inaccuracy. We experimented this by ARX
model using Matlab's -i. 11i identification toolbox.23 Result from this newly abstracted model is good enough to
see almost overlapped line as shown in Fig. 8.
3.2.3. Question 3: II !,,, is the work trajectory according to fuel 1'.... / ,/
The work trajectory is the output of the FULTON model. Since the question only concerns the behavior of the
entire FULTON model without structural details, all components can be fully abstracted into one BLOCK model.
Fig. 9 shows this model. Abstraction results and loss of accuracy are shown in Fig. 10.
---- --------------- ---------
\ov Z=OA ."/=of Xa
. . . . .
- - - - -
-A M 5g
I ----- -----
Figure 5. Behavioral Abstraction for question 1
Abstraction by ADALINE(delay=2, learning rate = 0 4)
5 10 15 20 25 30 35 40 45 50
(a) Result from Behavioral abstraction
(b) Abstraction Error
Figure 6. Behavioral Abstraction Result for question 1
4. CONCLUSIONS AND FUTURE WORK
We have presented a dynamic model abstraction method. Our approach is to use structural abstraction to organize
the whole -- -I. i-, hierarchically with simple -I. I I |" first, and then graduate to more complex model I- I" -
The multimodeling method is used for this purpose. Below the structural abstraction, each component is black-box
with no detailed internal structure. Behavioral abstraction is used to represent those black-boxes which approximate
the behavior of the -I. I, components with various abstraction techniques discussed in section 2. By combining
structural and behavioral abstraction together, each level of abstraction is independent from the lower abstraction
levels, so a level can be executed apart from the rest of hierarchy. We showed how this method can provide benefits
Figure 7. Behavioral Abstraction for question 2
Abstration by ARX model(poly=zeros=1, delay=0)
20 25 30
(a) Result from Behavioral abstraction
0 5 10 15 20 25 30 35 40 45 50
(b) Abstraction Error
Figure 8. Behavioral Abstraction Result for question 2
by question-answering examples to the -I, i, Unnecessary details could be It !1III omitted and answers were
given with time-saving way by not having to simulate at the lowest level. We're developing MOOS,. . liii L.11, li!
Object-Oriented Simulation Environment),39 where the proposed abstraction method is being implemented. MOOSE
models are constructed using a graphical user interface which begins with the user i i"f- i1,- an object oriented class
hierarchy by following 001'. "(Object-Oriented P1! - I. ,1. l 1 `1. This procedure takes advantage of structural
abstraction. For exploiting behavioral model abstraction, we provide three basic techniques(parametric estimation,
neural network, wavelet network) and allow the user to choose which they would like.
Having proposed framework to organize models according to abstraction level, our question is how we can select
an optimal abstraction level under a certain time and accuracy constraints. For applications with significant time
constraints, we 1i ,- need to use simple models that are computationally less complex even at the expense of reduced
accuracy. The optimal model for this application is the model that maximizes the tradeoff between model accuracy
and model cost. We're -1rl- ii, a method that selects, from the candidate models, a model that is accurate and
computable within a time constraint and planning to add this i1, ,1,;l. to MOOS.
ON/OFF BOILER As TURBINE 4" CONDENSER Ac PUMP Am
ASSEMBLY ASSEMBLY ASSEMBLY
[ L1 L2 L3 L4
neural network model
Figure 9. Behavioral abstraction for question 3
Abstraction by ADALINE(delay=2, learning rate= 0 4
5 10 15 20 25 30 35 40 45 50
(a) Result from behavioral abstraction
5 10 15 20 25 30
(b) Abstraction error
Figure 10. Behavioral abstraction result for question 3
ACKNOWLEDGE EN IENTS
We would like to thank the following funding sources that have contributed towards our study of modeling and
implementation of a multimodeling simulation environment for ;, i, 1 -i- and planning: (1) Rome Laboratory, (;, Iti--
Air Force Base, New York under contract 1 1i.1 '2-95-C-0267 and grant 1 ;111 .1'-95-1-0031; (2) Department of the
Interior under grant 14-45-0009-1544-154 and the (3) National Science Foundation Engineering Research Center
(i.1l i ) in Particle Science and T 1 i! .1.. at the U1 i i i of Florida (with Industrial Partners of the 1. i 1:) under
1. F. K. Frantz, "A Taxonomy of Model Abstraction Techniques," Proceedings of the I!','. II ,.,. Simulation
C.,'- ,.. ..... pp. 1413-1420, 1995.
2. P. A. I .i-!- I. !: "A T ,i.. ...-ii for Process Abstraction in Simulation Modeling," in IEEE International C.f-. -
ence on S.,,. .... Man and C,!1., i 1'. *, vol. 1, pp. 144 151, (Alexandria, Virginia), October 1987.
3. P. A. I i-I !:I "Abstraction Level Traversal in Hierarchical Modeling," in Modelling and Simulation Method-
ology: Knowledge S.,.. ... Paradigms, B. P. Zeigler, M. 1.i! and T. Oren, eds., pp. 393 429, 1.!- I r North
4. B. P. Zeigler, "Towards a Formal Theory of Modelling and Simulation: ,ini ,i.- Preserving Morphisms,"
Journal of the Association for Computing Machinery 19(4), pp. 742 764, 1972.
5. M. G. Valdis Berzins and D. Naumann, "Abstraction-Based Software Development," Communications of the
.I .1 29(5), pp. 402-415, 1986.
6. G. Booch, Object Oriented Design with applications, The B. i ii i,! /Cummings Publishing C '!!!il i! Inc., 1991.
7. R. W. Sebesta, Concepts of Programming Languages, The B. i ii1 i /Cummings Publishing C '!i i* i.i, Inc., 1992.
8. B. Falkenhainer and K. D. Forbus, "Compositional modeling: finding the right model for the job," IA 't '. l
Intelligence 51, pp. 95-143, 1991.
9. P. K. Davis and R. Hillestad, "A i. i, Di- i,. i., and the ('I h11 II, "- of Crossing Levels of Resolution
When Designing and Connecting Models," in Proceedings of AI, Simulation and Planning in High Autonomous
S , .. ,'. I ), pp. 180-188, 1993.
10. J. R. Hobbs, "Granularity," Proceedings of the Seventh National C,,.f, .- .. on A, 1t ',,l Intelligence 1985.
11. C.-J. Luh and B. P. Zeigler, "Abstraction Morphisms for Task Planning and Execution," in Proceedings of AI,
Simulation and Planning in High Autonomous S, .i'. I ,'.1), pp. 50-59, 1991.
12. P. A. I i-l!'- i !: Simulation Model Design and Execution: Building Digital Worlds, Prentice Hall, 1995.
13. P. A. i ,-!- I: and K. Lee, "Two Methods for Exploiting Abstraction in Systems," AI, Simulation and Planning
in High Autonomous S.,!i' '." pp. 257-264, 1996.
14. P. A. I i,-l.- I!: "AT i ,,.. ..! for Simulation Modeling Based on a Computational Framework," HE l. i... ..*.
on IE Research, Accepted May 1996 1996.
15. P. A. I i-l!- i !:, "Toward a Convergence of Systems and Software F'! _i!. i_ IEEE 1, 1...... .'.... on S./." "...
Man and Cd ,. .i ,'. 1996. Submitted May 1996.
16. P. A. I i-l- i !: "Heterogeneous Decomposition and Coupling for Combined Modeling," in 1991 II .'. Simu-
lation C-4.'f,. .... pp. 1199 1208, (Phoenix, AZ), December 1991.
17. P. A. I --Il- !: and B. P. Zeigler, "A Multimodel V1. Ill ..l. I..- for Qualitative Model F-_... ii_ .I .I
L ...-.i... .... on Modeling and Computer Simulation 2(1), pp. 52-81, 1992.
18. P. A. 1 i-li- 1 !: "An Integrated Approach to System Modelling using a Synthesis of Artificial Intelligence,
Software Engineering and Simulation Methodologies," .Ii .f ......; .i.... on Modeling and Computer Simulation
19. P. A. I i-lr- i !:, "A Simulation Environment for Multimodeling," Discrete Event D, ...... i' S.i ... !.... j and
Applications 3, pp. 151-171, 1993.
20. P. A. I i-l!- i !: H. N i i i! i![ J. i, I !:!I. 1 and A. Bonarini, "A multimodel approach to reasoning and simu-
lation," IEEE I ...-.. ; .... on ,i- ..I... Man and C,.I., i ., ,"', (10), pp. 1433-1449, 1994.
21. P. A. i i-l!- !: "The Role of Process Abstraction in Simulation," IEEE I on....... .. on t.! ..... Man and
C.. ,, ,*", 18, pp. 18 39, January/February 1988.
22. B. P. Zeigler, Object Oriented Simulation with Hierarchical, Modular Models: Intelligent Agents and Endomor-
phic !. ,.i. Academic Press, 1990.
23. z," .... ,! 1. iT; ,,!'.. Toolbox, The MathWorks, Inc., 1991.
24. L. L.itl,-! and T. Soderstrom, I /.. .,, and Practice of Recursive 1l, !'tr; .*',. MIT Press, Cambridge, Mass,
25. K.C. Tan, Y. Li, D.J. '1!, i i -Smith and K.C. I!i > ii, ".; -i. i, Identification and Linearisation Using Genetic
Algorithms with Simulated Annealing," Proc. First IEE/IEEE Int. C,-f on GA in Eng. Syst.: Innovations
and Appl. pp. 164-169, 1995.
26. G. Cynbenko, "Approximation by Superposition of a Sigmoidal Function," Mathematics of control, signals and
.,,i ..... 2, pp. 303-314, 1989.
27. S. M. Carrol and B. W. Disckinson, "Construction of Neural Nets using the Radon Transform," in IJ( .\.~ 1989.
28. Z. Tang, C. de Almeida, and P. A. I ,- I i !:I 1 11i.- Series Forecasting using Neural Networks vs. Box-Jenkins
11,,i ,,l1 1,,. ," Simulation 57, pp. 303-310, November 1991.
29. Z. Tang and P. A. i -!,- ., !: .. 1-1 forward Neural Nets as Models for Time Series Forecasting," ORSA Journal
of Computing 5(4), pp. 374-386, 1993.
30. A. R. Barron, -"Si ii-i1, I! Properties of Artificial Neural Networks," Proceedings of the 28th IEEE C..- f. ..
on Decision and Control, pp. 280-285, 1989.
31. D. E. Rumelhart, G. E. Hinton, and R. W.\\ i !ii in-, Learning Internal Representations by Error Propagation, In
parallel Distributed Processing: Explorations in the Microstructure of Cognition, Cambridge, MA:MIT Press,
32. P. M. Mills, M. O. Tade, and A. Y. ... .,- "Identification and Control Using a Hybrid Reinforcement Learning
System," International Journal in Computer Simulation 5, pp. 109-126, 1995.
33. Q. Zhang and A. Benveniste, "\\ ,. !. i Networks," IEEE transactions on Neural Networks 3(6), 1992.
34. F. J. K. W. Edward Gettys and M. J. I:..- ., Physics, McGraw-Hill, 1989.
35. D. Caughlin, ". I...1. I Abstraction Via Solution of A General Inverse Problem to Define a Metamodel," s!.
transactions of Computer Simulation, Submitted Sept. 1996.
36. H. Pierreval, "A Metamodeling Approach Based on Neural Networks," International Journal in Computer
Simulation 6, pp. 365-378, 1996.
37. J. Signrandt and M. Smith, ".'\.iih!. System Identification using Wavelets," CSC(Centre for S.,i' ..ri and
Control) report 1996.
38. Neural Network Toolbox, The MathWorks, Inc., 1992.
39. T. G. Robert Cubert and P. A. I i-l!- 1: ".10OOS architecture of an object-oriented multimodeling," SPIE
40. P. A. I i-l!,- 1. !: "I. .1 l tii Object-Oriented Design for P!i -1; A1 Modeling," .li .f transactions on Modeling and
Computer Simulation, Submitted July 1996.