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Constructing Low-Dimensional Dynamical Systems of Nonlinear Partial Differential Equations Using Optimization Jun Shuai* and Xuli Han Central South University/ School of Mathematics and Statistics, Changsha, China *Corresponding author, Email: [email protected], [email protected]
Abstract—A new approach using optimization technique for constructing low-dimensional dynamical systems of nonlinear partial differential equations (PDEs) is presented. After the spatial basis functions of the nonlinear PDEs are chosen, spatial basis functions expansions combined with weighted residual methods are used for time/space separation and truncation to obtain a high-dimensional dynamical system. Secondly, modes of lower-dimensional dynamical systems are obtained by linear combination from the modes of the high-dimensional dynamical systems (ordinary differential equations) of nonlinear PDEs. An error function for matrix of the linear combination coefficients is derived, and a simple algorithm to determine the optimal combination matrix is also introduced. A numerical example shows that the optimal dynamical system can use much smaller number of modes to capture the dynamics of nonlinear partial differential equations. Index Terms—Nonlinear Partial Differential Equations; Spatial Basis Function Expansions; Dynamical System; Error Functions; Optimization
I.
INTRODUCTION
Many problems in science and engineering are reduced to a set of nonlinear partial differential equations (PDEs) [1] through a process of mathematical modeling. Obtain a good model for the processes at a reasonable cost and accuracy is important for the industry, particularly in practical application. Such a model is necessary for fundamental analysis and control-related issues, including design, simulation, diagnosis and supervision. Generally, two kinds of methods are mainly used to model the dynamics of nonlinear PDEs: conventional discretization approaches and advanced methods based on spatial basis function expansions [2, 3]. Conventional discretization approaches such as finite difference method (FDM) and finite element method (FEM), which require the definition of a mesh (domain discretisation) where the functions are approximated locally and higher-order schemes are necessary for more accurate approximation of the spatial derivatives. Thus, these methods often lead to high-order dynamical models of nonlinear PDEs. The advanced methods based on spatial basis function expansions such as spectral methods [2, 4-6] can derive a low-order ODE system to model the dynamics of nonlinear PDE. However, these general spatial basis © 2013 ACADEMY PUBLISHER doi:10.4304/jnw.8.11.2520-2526
functions are not optimal in the sense that the dimensional ODE systems to model the dynamics of PDEs are not lowest [7]. Based on spatial basis function expansions, more studies are carried out for constructing the lowdimensional dynamical systems of the nonlinear PDEs. Approximate inertial manifold (AIM) [8-10] is an attracting method to find the low-dimensional dynamical systems of nonlinear PDE. Meanwhile, A methodology defining the basic spatial patterns taking into account the temporal dynamics of the reduced model was first derived by Hasselmann [11], which is called principal interaction patterns (PIPs). Based on the idea of Hasselmann, a general algorithm for reducing a highdimensional Fourier-Galerkin approximation to a lowdimensional system was derived by Kwasniok [12-14]. However, the variational principle in the algorithm for PIPs reduces to a high-dimensional nonlinear minimization problem. If the nonlinear partial differential equations have complex nonlinear structure, the computation cost is quite large and the solution algorithm for the nonlinear minimization may not be numerically implemented. Then, the approached constructing optimal low-dimensional dynamical systems of nonlinear partial differential equations using PIPs have numerically difficulties in practical applications. Wu [15] also introduce an useful theory constructing optimal lowdimensional dynamical system based on known database or directly from PDEs. Unfortunately, these methods lead to low-dimensional dynamical ODE systems require much computation cost. It also restricts the practical application of these methods to model the more complex nonlinear PDEs. Another common methods used for obtaining the lowdimensional dynamical systems for nonlinear partial differential equations is Kurhunen-loeve (KL) decomposition, which is also referred to as principal component analysis (PCA) [16], empirical orthogonal function analysis, or proper orthogonal decomposition (POD). KL decomposition is a popular approach to find the low-dimensional principal spatial structures from the measured data. In order to obtain the empirical eigenfunctions (EEFs), a basic assumption is made that the data are fully representative of the temporal progress of a nonlinear PDE. However, the number and locations
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of sensors in the practical processes actually determine the number of the EEFs from data [17]. As only inputoutput data are used, the order and the accuracy of the model are limited because of the availability of sensors and actuators that result in a limited number of EEFs. Furthermore, KL decomposition is a linear model reduction method (i.e., linear projection and linear reconstruction). It may not be very suitable to model the nonlinear dynamics efficiently. This is because KL decomposition produces a linear approximation to the original nonlinear problem, which may not guarantee the assumption that minor components contain important dynamical information from nonlinear PDEs. In this note, a new approach based on spatial basis function expansions for constructing low-dimensional dynamical systems of nonlinear partial differential equations (PDEs) using optimization technique [18] is presented. After selecting proper spatial basis functions, high-dimensional spectral based ODE systems are derived by basis function expansion and weighted residual methods (WRM). Thus, optimal dynamical systems are obtained by linear combination from the modes of spectral based ODE systems of nonlinear PDEs, while the combination matrix is obtained by optimization for an energy error function. A simple algorithm based on particle swarm optimization (PSO) [19-21] to determine the optimal combination matrix is also introduced. The numerical example shows that the optimal dynamical systems require much smaller number of modes to capture the dynamics of nonlinear PDEs.
To precisely characterize the nonlinear PDE systems considered in this work, we formulate the PDE of Eq. (1) as an infinite dimensional system in the Hilbert space with the inner product:
w1 , w2 w1 x w2 x dx
(4)
where w1 , w2 are two arbitrary functions of Hilbert space . Spatial basis function expansions combined with weighted residual methods, are among the most extensively used methods for the model reduction of PDEs. Such a combination has been shown to provide very accurate approximations of sufficiently smooth solutions. It is well known that a continuous function can be approximated using Fourier series [22]. Based on this principle, the spatio-temporal variable u ( x, t ) of the PDE can be expanded by a set of spatial trial functions {i ( x)}i1 as follows:
u x, t ui t i x
(5)
i 1
where ui (t ) denotes the corresponding time coefficient of i ( x) , {i ( x)}i1 denotes the infinite set of the of PDE (1). Similar eigenfunctions of linear operator to the Fourier series, only the first N modes of the expansion (5) will be retained in practice N
II.
uN x, t ui t i x
DYNAMICAL SYSTEMS OF NONLINEAR PDES
We start out with the following nonlinear partial differential equation u x, t t
u x, t u x, t F u x, t , , x
, x, t
(1)
B1u , t C1 B2 u , t C2
u , t x u , t x
Thus, the spatio-temporal variable is separated into a set of spatial BFs and the temporal variables. In the WRM, the residual equation of the model (1) generated from the truncated expansion (6) can be expressed as RN
subject to the boundary conditions: D1
(2) D2
u N x, t t
u N x, t
u x, t F u N x, t , N , x
, x, t
(7)
which is made in the sense that
RN ,i 0,
and initial condition: u x,0 u0 x
(6)
i 1
i 1, 2,
,N
(8)
(3)
where {i ( x)}iN1 is a set of weighting functions to be
where u ( x, t ) denotes the spatio-temporal state variable,
chosen. If the weighting functions {i ( x)}iN1 are chosen
x , is the spatial coordinate, t 0, is the
time variable. is a linear operator involves linear spatial derivatives on the state variable. u ( x, t ) F (u ( x, t ), , , x, t ) is a nonlinear function which x contains spatial derivatives of the state variable. B1 , C1 , D1 , B2 , C2 , D2 are constants and u0 ( x) is a function of x .
to be the basis functions {i ( x)}iN1 then the method is called the Galerkin Method. This is an easy way to obtain an ODE model from Eq. (8) for ui (t ) as follows: ui t i ui t fi u t , t
where u(t ) u1 (t ), u2 (t ),
(9)
, uN (t ) ; T
i , i 1, 2 , N denote the eigenvalues of the operator , fi (u(t ), t ) denote the nonlinear terms. The series of
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dynamical systems can be rewritten in a general form as follows: u t Lu t f u (t ), t
where L diag (1 , 2 ,
(10)
, N ) ;
f (u(t ), t ) f1 (u(t ), t ), f 2 (u(t ), t ),
, f N (u(t ), t )
T
Optimal low-dimensional dynamical systems of nonlinear PDEs Dynamical systems (10) of nonlinear PDE often have high orders. New lower-dimensional dynamical systems with low computation cost have to be derived. A. New Low-dimensional Dynamical Systems Let the ith modes of new dynamical system are the linear combination of the modes of dynamical system (10) as follows: vi t Wi1u1 t Wi 2u2 t
WiN uN t (11)
where i 1, 2, , M (M N ) and Wi1 ,Wi 2 , ,WiN denotes the corresponding combination coefficient of u1 (t ), u2 (t ), , uN (t ) . Eq. (11) can be rewritten in a general vector form as follows: v t W T u t
(12)
where
v t v1 t , v2 t ,
, vM t ;
u t u1 t , u2 t ,
, uN t ;
T
T
WM 1 W11 W21 W W22 WM 2 . W 12 WMN W1N W2 N Thus, combining with Eq. (10) and Eq. (12), the following dynamical system (13) is derived
v t Kv t g v t , t
(13)
system and the low-dimensional dynamical system is introduced by M 2 2 N Error ui t dt vi t dt 0 0 i 1 i 1
2
(16)
Combining the vector expressions of u (t ) , v(t ) with the Eq. (12) , the error (16) can be strictly derived as follows:
u t u t v t v t dt u t u t u t WW u t dt u t I WW u t dt .
Error W
T
2
T
0
T
T
2
T
(17)
0
2
T
T
0
where u(t ) [u1 (t ), u2 (t ), , uN (t )]T , W is the combination matrix in Eq. (12) which is assumed to be column orthogonal. C. Optimization The optimal low-dimensional dynamical system (13) can be obtained, while the combination matrix W is determined by minimizing the error function (17). To evaluate the error function, a temporal interval [0, Tmax ] is considered for (17) and approximated by a finite sum, where the max integral time Tmax is selected as free parameter. The temporal integral interval [0, Tmax ] generally contains the processes from initial state to steady state of the nonlinear PDE. The restricted condition for the minimizing problem (17) is that the combination matrix should be column orthogonal. Thus, the error functions for combination matrix can be approximated as follows. Since Error W
u t I WW u t dt Tmax
2
T
0
2
where K W T LW ; g (v(t ), t ) W T f (u(t ), t ) .
ti ti 1 u ti I WW T u ti , i we consider approximate error
B. Error Functions for Combination Matrix Assuming the dimension N of ODE system (10) is large enough that it can capture the almost all dynamics of the nonlinear PDE (1). Let the energy function of the ODE system (10) as follows:
E W ti ti 1 u ti I WW T u ti i
N
u t dt 0
i 1
2
i
(14)
Similarly, the energy functions of the low-dimensional dynamical system (13) can be given as follows: M
v t dt 0
i 1
2
i
(15)
To obtain a low-dimensional ODE system which can capture the dynamics of the nonlinear PDE, an error function that measures energy error between the ODE © 2013 ACADEMY PUBLISHER
2
(18)
subject to W (:, i)T W (:, i) ki ij . The optimization of E (W ) (18) mainly contains several steps. Firstly, the fourth-order Runge-Kutta method are used to calculate the ODE system (10), thus ti , u (ti ) of Error (18) are known. Secondly, the particle swarm optimization (PSO) algorithm [19-21] is used to optimize the restricted error functions (18). Particle swarm optimization (PSO) is a recently invented high-performance stochastic algorithm introduced in the middle of the 1990s. This algorithm is developed from the theory of artificial life and evolutionary computation and is used to solve
JOURNAL OF NETWORKS, VOL. 8, NO. 11, NOVEMBER 2013
optimization problems. This technique is akin to the other stochastic methods performing a global search in the parameter space without getting trapped in local minima. A key characteristic of the PSO is that the algorithm itself is highly robust. This method may offer different routes through the problem hyperspace than other methods such as evolutionary algorithms, ant colony optimization, simulated annealing, genetic algorithms (GA) [23], and so on. PSO [24] is similar to GA or evolutionary algorithms in several ways, but is much simpler than the rules of GA. This method does not have the “cross” (crossover) and “variability” (mutation) operation of a GA. Thus, this condition implies that PSO requires less computational bookkeeping and generally only a few lines of code. For applications in practical industrial engineering, a simpler algorithm is generally appealing. The calculations for the minimization of the error functions Q in the current study have established that the PSO is an effective optimization method. In PSO for the minimization of Q , positions of particles are candidate solutions to the Nk -dimensional problem, and the moves of the particles are regarded as the search processes of better solutions. The position of the i th particle is represented by xi ( xi 1 , xi 2 , , xi Nk ) , and its velocity is represented by vi (vi 1 , vi 2 , , vi Nk ) . During the search process, the particle successively adjusts its position according to two factors, namely their personal best position and the global best position. The personal best position Pi ( Pi 1 , Pi 2 , , Pi Nk ) is the best previous position of the i th particle that gives the best fitness value. The best particle among all the particles in the population is represented by Pg ( Pg 1 , Pg 2 , , Pg Nk ) . Each particle is updated by the following two best values, Pi and Pg during its iteration. The particle updates its velocity and positions as follows: vi d new wvi d old c1r1 Pi d xi d c2 r2 Pi d xi d
xi d new xi d old vi d new
(19)
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singular value decomposition for W . Let the singular value decomposition of W is W USV T , where U , S are orthogonal and diagonal matrix respectively. Thus combination matrix W US is orthogonal which the minimal error is retained by WW T USV T VSU T WW T . III.
A NUMERICAL EXAMPLE
In this section, the optimal low-dimensional dynamic system of nonlinear PDEs is constructed for a simple PDE model. Note that this example serves an illustration of the proposed methodology rather than a demonstration of the model reduction of a very large and complex system. In order to evaluate the optimal low-dimensional dynamical systems for nonlinear PDEs, the Burgers Equation is studied. The Burgers equation is a onedimensional spatial model of a variety of threedimensional physical phenomena, greatly simplifying the problems while retaining many of the complex behavior characteristics. This equation contains nonlinear convection and diffusion terms and retains many of the interesting features of the Navier-Stokes equation. The governing equation may be written as u x, t t
u
u x, t x
2 u x, t
(21)
x 2
where the viscosity is 0.3 and (21) subjects to the following boundary condition: u 0, t u 1, t 0
(22)
u x,0 sin x
(23)
and initial condition:
The numerical solution of the Burgers equations is obtained using the Runge-Kuta methods with Finite Difference discretization at the space and time domain. The approximation solution of Burgers equations in the example are given in the Fig. 1. 3
(20)
2 1
where: w is an inertia weight, c1 and c2 are two positive learning factors, and r1 and r2 are two random numbers uniformly distributed in the range between 0 and 1. A time parameter in Eq. (20) determines the different flying time for each particle. Eq. (19) is used to update the velocity according to its previous velocity and the distances of its current position from Pi and Pg . Then, the particle flies toward a new position according to Eq. (20).The principle and details of PSO algorithm is given in Ref [19]. For the optimization of (18) by PSO directly, the obtained initial combination matrix W is not orthogonal. To keep the minimization of the error function (18), a satisfied combination matrix W can be derived by
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0 -1 -2 20
150
15
100
10
Discretization locations
50
5 0
0
Simulation time
Figure 1. The approximation solution of the Burgers equations
To obtain a high dimensional ODE system (9), the spatial basis functions and the weighting functions of Burgers equation (21) are selected as
x i
2 sin(i x), i 1, 2,
,
(24)
The first four spatial basis functions given by the Eq. (2) are shown in the following Fig. 2.
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1.5
The 1st mode of initial dynamical ODE system
1
1.2 0.5
1 0
0.8
-0.5
0.6
-1
-1.5
0
0.1
0.2
0.3
0.4
0.5 0.6 Spatial domain
0.7
0.8
0.9
0.4
1
Figure 2. The first four spatial basis functions
0.2
0
0
0.5
1
1.5
2
2.5
3
3
Figure 6. The 1st mode of initial dynamical ODE system
2 1 0
1
-1
The 1st mode of the 2-order optimal dynamical system The 2nd mode of the 2-order optimal dynamical system
0.8
-2 20
150
15
0.6
100
10
Discretization location
50
5 0
0
0.4
Simulation time 0.2
Figure 3. The approximation by 10-order dynamical system
0 -0.2 -0.4 -0.6 -0.8
0.1
-1 0
-0.1
0
0.5
1
1.5
2
2.5
3
Figure 7. The 1st and 2nd modes of the 2-order optimal dynamical system
-0.2 20 15
150 10
100 5
1
50 0
The 1st mode of the initial dynamical ODE system The 2nd mode of the initial dynamical ODE system
0.8
0
0.6
Figure 4. The approximation error of the 10-order dynamical system
0.4 0.2 0
Combining basis functions expansion (5) based on (24) and Galerkin method, an initial 10-order dynamical ODE system has been derived. Synthesis of the temporal output of 10-order dynamical ODE system together with the spatial basis functions (24), the approximation of the Burgers equation is given in Fig. 3. The approximation error of the 10-order dynamical system is given in the Fig. 4. The temporal integral interval is [0,3] and the energy value of the 10-order dynamical ODE system is 0.0811. To evaluate the optimal low dimensional dynamical systems of Burgers equations (21), 10 1 and 10 2 combination matrices are obtained from the optimization for error in (18) by the standard PSO algorithm and singular value decomposition. Thus, 1order and 2-order optimal dynamical systems have been derived by linear combination (12).
-0.2 -0.4 -0.6 -0.8 -1
0
0.5
1
1.5
2
2.5
3
Figure 8. The 1st and 2nd modes of initial dynamical ODE system
0.09 Energy of inital ODE Energy of optimal ODE
0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01
0
0
-0.2 The 1st mode of 1-order optimal dynamical system
1 mode
2 mode
-0.4
-0.6
Figure 9. Comparison for the energy value of the optimal dynamical system and initial ODE system with the same modes
-0.8
-1
-1.2
-1.4
0
0.5
1
1.5
2
2.5
3
Figure 5. The 1st mode of the 1-order optimal dynamical system
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The 1st mode of 1-ordr optimal dynamical system and the 1st mode of initial 10-order dynamical ODE system are shown in Fig. 5 and Fig. 6, respectively. Similarly, the 1st and 2nd modes of 2-order optimal dynamical systems
JOURNAL OF NETWORKS, VOL. 8, NO. 11, NOVEMBER 2013
and that of initial 10-order dynamical ODE system are also shown in Fig. 7 and Fig. 8. As shown in Fig. 9, the energy value of the obtained 1order and 2-order optimal dynamical systems are 0.0809 and 0.0810, where that of the initial ODE systems with the same modes are 0.0312 and 0.0555 respectively. For the purpose of control, energy is the representative of the dynamics of systems. In Fig. 5, the optimal dynamical system with the 1 or 2 modes have almost the all energy of the initial ODE system. In other words, the optimal dynamical system can capture the dynamics of the nonlinear PDE perfectly. IV.
CONCLUSIONS
In this note, a new approach based on spatial basis function expansions for constructing low-dimensional dynamical systems of nonlinear partial differential equations (PDEs) is presented. After choosing the proper spatial basis functions, spatial basis functions expansions combined with weighted residual methods are used for time/space separation and truncation. Optimal dynamical systems are obtained by linear combination from the modes of spectral based ODE systems of nonlinear PDEs, while the combination matrix is obtained by optimization for a energy error function. A simple algorithm based on particle swarm optimization (PSO) to determine the optimal combination matrix is also introduced. The numerical example shows that the optimal dynamical systems require much smaller number of modes to capture the dynamics of nonlinear PDEs. ACKNOWLEDGEMENT Financial support from National Natural Science Foundation of China (11271376) is gratefully acknowledged. REFERENCE [1] J. Liu, M. Li, and F. He (2012), "A Novel Inpainting Model for Partial Differential Equation Based on Curvature Function," Journal of Multimedia, vol. 7(3), pp. 239-246. [2] J. P. Boyd, Chebyshev and Fourier spectral methods: Dover Pubns, 2000. [3] M. Jiang and H. deng (2012), "Optimal combination of spatial basis functions for the model reduction of nonlinear distributed parameter systems," Communications in Nonlinear Science and Numerical Simulation, vol. 17pp. 5240-5248. [4] e. a. Canuto. C., Spectral methods in fluid dynamics. New York: Springer-Verlag, 1988. [5] D. Gottlieb and S. A. Orszag, Numerical analysis of spectral methods: theory and applications: Society for industrial and applied mathematics, 1993. [6] H. Deng, H. X. Li, and G. Chen (2005), "Spectralapproximation-based intelligent modeling for distributed thermal processes," Control Systems Technology, IEEE Transactions on, vol. 13(5), pp. 686-700. [7] H. Deng, M. Jiang, and C.-Q. Huang (2012), "New spatial basis functions for the model reduction of nonlinear distributed parameter systems," Journal of Process Control, vol. 22(2), pp. 404-411.
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[8] A. Adrover, G. Continillo, S. Crescitelli, M. Giona, and L. Russo (2002), "Construction of approximate inertial manifold by decimation of collocation equations of distributed parameter systems," Computers & Chemical Engineering, vol. 26(1), pp. 113-123. [9] P. D. Christofides and P. Daoutidis (1997), "FiniteDimensional Control of Parabolic PDE Systems Using Approximate Inertial Manifolds* 1," Journal of Mathematical Analysis and Applications, vol. 216(2), pp. 398-420. [10] P. D. Christofides (1998), "Robust control of parabolic PDE systems," Chemical Engineering Science, vol. 53(16), pp. 2949-2965. [11] K. Hasselmann (1988), "PIPs and POPs: The reduction of complex dynamical systems using principal interaction and oscillation patterns," J. Geophys. Res, vol. 93pp. 1101511021. [12] F. Kwasniok (1996), "The reduction of complex dynamical systems using principal interaction patterns," Physica D: Nonlinear Phenomena, vol. 92(1-2), pp. 28-60. [13] F. Kwasniok (1997), "Optimal Galerkin approximations of partial differential equations using principal interaction patterns," Physical Review E, vol. 55(5), pp. 5365-5375. [14] F. Kwasniok (2004), "Empirical low-order models of barotropic flow," Journal of the atmospheric sciences, vol. 61(2), pp. 235-245. [15] H. Zhao and C. Wu (1998), "Construction of optimal truncated low-dimensional dynamical systems directly from PDEs," Communications in Nonlinear Science and Numerical Simulation, vol. 3(1), pp. 1-5. [16] Y. Lei, H. Lai, and Q. Li (2012), "Geometric Features of 3D Face and Recognition of It by PCA," Journal of Multimedia, vol. 6(2), pp. 207-216. [17] M. Jiang and H. deng (2013), "Improved Empirical Eigenfunctions Based Model Reduction for Nonlinear Distributed Parameter Systems," Industrial & Engineering Chemistry Research, vol. 52(2), pp. 934-940. [18] I. Bonis and C. Theodoropoulos (2012), "Model reductionbased optimization using large-scale steady-state simulators," Chemical Engineering Science, vol. 69(1), pp. 69-80. [19] N. Yusup, A. M. Zain, and S. Z. M. Hashim (2012), "Overview of PSO for Optimizing Process Parameters of Machining," Procedia Engineering, vol. 29(0), pp. 914923. [20] G. Cui, L. Qin, S. Liu, Y. Wang, X. Zhang, and X. Cao (2008), "Modified PSO algorithm for solving planar graph coloring problem," Progress in Natural Science, vol. 18(3), pp. 353-357. [21] H. Shinzawa, J.-H. Jiang, M. Iwahashi, I. Noda, and Y. Ozaki (2007), "Self-modeling curve resolution (SMCR) by particle swarm optimization (PSO)," Analytica Chimica Acta, vol. 595(1–2), pp. 275-281. [22] D. G. Zill, & Cullen, M. R., Differential equations with boundary-value problems (5th ed.). vol. Brooks/Cole Thomson Learning. Australia: Pacific Grove, CA, 2001. [23] L. D. M. SON J S, KIM I S (2004), "A study on genetic algorithm to select architecture of a optimal neural network in the hot rolling process," Journal of Materials Processing Technology, pp. 153-156. [24] Z. WANG, X. ZHAO, B. WAN, and J. XIE (2013), "Research of BP Neural Network based on Improved Particle Swarm Optimization Algorithm," Journal of Networks, vol. 8(4), pp. 947-954.
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Jun Shuai received the B. Eng. degree from the Academy of Equipment Command & Technology (General Armament Department), Beijing, China, in 1997. Currently, he is pursuing the Ph.D. degree in school of Mathematics and Statistics at the Central South University of China. He is a senior engineer at Software Park in Changsha, China. His research interests include Dynamical systems, Optimization and Signal Analyzing and Processing.
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JOURNAL OF NETWORKS, VOL. 8, NO. 11, NOVEMBER 2013
JOURNAL OF NETWORKS, VOL. 8, NO. 11, NOVEMBER 2013
Constructing Low-Dimensional Dynamical Systems of Nonlinear Partial Differential Equations Using Optimization Jun Shuai* and Xuli Han Central South University/ School of Mathematics and Statistics, Changsha, China *Corresponding author, Email: [email protected], [email protected]
Abstract—A new approach using optimization technique for constructing low-dimensional dynamical systems of nonlinear partial differential equations (PDEs) is presented. After the spatial basis functions of the nonlinear PDEs are chosen, spatial basis functions expansions combined with weighted residual methods are used for time/space separation and truncation to obtain a high-dimensional dynamical system. Secondly, modes of lower-dimensional dynamical systems are obtained by linear combination from the modes of the high-dimensional dynamical systems (ordinary differential equations) of nonlinear PDEs. An error function for matrix of the linear combination coefficients is derived, and a simple algorithm to determine the optimal combination matrix is also introduced. A numerical example shows that the optimal dynamical system can use much smaller number of modes to capture the dynamics of nonlinear partial differential equations. Index Terms—Nonlinear Partial Differential Equations; Spatial Basis Function Expansions; Dynamical System; Error Functions; Optimization
I.
INTRODUCTION
Many problems in science and engineering are reduced to a set of nonlinear partial differential equations (PDEs) [1] through a process of mathematical modeling. Obtain a good model for the processes at a reasonable cost and accuracy is important for the industry, particularly in practical application. Such a model is necessary for fundamental analysis and control-related issues, including design, simulation, diagnosis and supervision. Generally, two kinds of methods are mainly used to model the dynamics of nonlinear PDEs: conventional discretization approaches and advanced methods based on spatial basis function expansions [2, 3]. Conventional discretization approaches such as finite difference method (FDM) and finite element method (FEM), which require the definition of a mesh (domain discretisation) where the functions are approximated locally and higher-order schemes are necessary for more accurate approximation of the spatial derivatives. Thus, these methods often lead to high-order dynamical models of nonlinear PDEs. The advanced methods based on spatial basis function expansions such as spectral methods [2, 4-6] can derive a low-order ODE system to model the dynamics of nonlinear PDE. However, these general spatial basis © 2013 ACADEMY PUBLISHER doi:10.4304/jnw.8.11.2520-2526
functions are not optimal in the sense that the dimensional ODE systems to model the dynamics of PDEs are not lowest [7]. Based on spatial basis function expansions, more studies are carried out for constructing the lowdimensional dynamical systems of the nonlinear PDEs. Approximate inertial manifold (AIM) [8-10] is an attracting method to find the low-dimensional dynamical systems of nonlinear PDE. Meanwhile, A methodology defining the basic spatial patterns taking into account the temporal dynamics of the reduced model was first derived by Hasselmann [11], which is called principal interaction patterns (PIPs). Based on the idea of Hasselmann, a general algorithm for reducing a highdimensional Fourier-Galerkin approximation to a lowdimensional system was derived by Kwasniok [12-14]. However, the variational principle in the algorithm for PIPs reduces to a high-dimensional nonlinear minimization problem. If the nonlinear partial differential equations have complex nonlinear structure, the computation cost is quite large and the solution algorithm for the nonlinear minimization may not be numerically implemented. Then, the approached constructing optimal low-dimensional dynamical systems of nonlinear partial differential equations using PIPs have numerically difficulties in practical applications. Wu [15] also introduce an useful theory constructing optimal lowdimensional dynamical system based on known database or directly from PDEs. Unfortunately, these methods lead to low-dimensional dynamical ODE systems require much computation cost. It also restricts the practical application of these methods to model the more complex nonlinear PDEs. Another common methods used for obtaining the lowdimensional dynamical systems for nonlinear partial differential equations is Kurhunen-loeve (KL) decomposition, which is also referred to as principal component analysis (PCA) [16], empirical orthogonal function analysis, or proper orthogonal decomposition (POD). KL decomposition is a popular approach to find the low-dimensional principal spatial structures from the measured data. In order to obtain the empirical eigenfunctions (EEFs), a basic assumption is made that the data are fully representative of the temporal progress of a nonlinear PDE. However, the number and locations
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of sensors in the practical processes actually determine the number of the EEFs from data [17]. As only inputoutput data are used, the order and the accuracy of the model are limited because of the availability of sensors and actuators that result in a limited number of EEFs. Furthermore, KL decomposition is a linear model reduction method (i.e., linear projection and linear reconstruction). It may not be very suitable to model the nonlinear dynamics efficiently. This is because KL decomposition produces a linear approximation to the original nonlinear problem, which may not guarantee the assumption that minor components contain important dynamical information from nonlinear PDEs. In this note, a new approach based on spatial basis function expansions for constructing low-dimensional dynamical systems of nonlinear partial differential equations (PDEs) using optimization technique [18] is presented. After selecting proper spatial basis functions, high-dimensional spectral based ODE systems are derived by basis function expansion and weighted residual methods (WRM). Thus, optimal dynamical systems are obtained by linear combination from the modes of spectral based ODE systems of nonlinear PDEs, while the combination matrix is obtained by optimization for an energy error function. A simple algorithm based on particle swarm optimization (PSO) [19-21] to determine the optimal combination matrix is also introduced. The numerical example shows that the optimal dynamical systems require much smaller number of modes to capture the dynamics of nonlinear PDEs.
To precisely characterize the nonlinear PDE systems considered in this work, we formulate the PDE of Eq. (1) as an infinite dimensional system in the Hilbert space with the inner product:
w1 , w2 w1 x w2 x dx
(4)
where w1 , w2 are two arbitrary functions of Hilbert space . Spatial basis function expansions combined with weighted residual methods, are among the most extensively used methods for the model reduction of PDEs. Such a combination has been shown to provide very accurate approximations of sufficiently smooth solutions. It is well known that a continuous function can be approximated using Fourier series [22]. Based on this principle, the spatio-temporal variable u ( x, t ) of the PDE can be expanded by a set of spatial trial functions {i ( x)}i1 as follows:
u x, t ui t i x
(5)
i 1
where ui (t ) denotes the corresponding time coefficient of i ( x) , {i ( x)}i1 denotes the infinite set of the of PDE (1). Similar eigenfunctions of linear operator to the Fourier series, only the first N modes of the expansion (5) will be retained in practice N
II.
uN x, t ui t i x
DYNAMICAL SYSTEMS OF NONLINEAR PDES
We start out with the following nonlinear partial differential equation u x, t t
u x, t u x, t F u x, t , , x
, x, t
(1)
B1u , t C1 B2 u , t C2
u , t x u , t x
Thus, the spatio-temporal variable is separated into a set of spatial BFs and the temporal variables. In the WRM, the residual equation of the model (1) generated from the truncated expansion (6) can be expressed as RN
subject to the boundary conditions: D1
(2) D2
u N x, t t
u N x, t
u x, t F u N x, t , N , x
, x, t
(7)
which is made in the sense that
RN ,i 0,
and initial condition: u x,0 u0 x
(6)
i 1
i 1, 2,
,N
(8)
(3)
where {i ( x)}iN1 is a set of weighting functions to be
where u ( x, t ) denotes the spatio-temporal state variable,
chosen. If the weighting functions {i ( x)}iN1 are chosen
x , is the spatial coordinate, t 0, is the
time variable. is a linear operator involves linear spatial derivatives on the state variable. u ( x, t ) F (u ( x, t ), , , x, t ) is a nonlinear function which x contains spatial derivatives of the state variable. B1 , C1 , D1 , B2 , C2 , D2 are constants and u0 ( x) is a function of x .
to be the basis functions {i ( x)}iN1 then the method is called the Galerkin Method. This is an easy way to obtain an ODE model from Eq. (8) for ui (t ) as follows: ui t i ui t fi u t , t
where u(t ) u1 (t ), u2 (t ),
(9)
, uN (t ) ; T
i , i 1, 2 , N denote the eigenvalues of the operator , fi (u(t ), t ) denote the nonlinear terms. The series of
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dynamical systems can be rewritten in a general form as follows: u t Lu t f u (t ), t
where L diag (1 , 2 ,
(10)
, N ) ;
f (u(t ), t ) f1 (u(t ), t ), f 2 (u(t ), t ),
, f N (u(t ), t )
T
Optimal low-dimensional dynamical systems of nonlinear PDEs Dynamical systems (10) of nonlinear PDE often have high orders. New lower-dimensional dynamical systems with low computation cost have to be derived. A. New Low-dimensional Dynamical Systems Let the ith modes of new dynamical system are the linear combination of the modes of dynamical system (10) as follows: vi t Wi1u1 t Wi 2u2 t
WiN uN t (11)
where i 1, 2, , M (M N ) and Wi1 ,Wi 2 , ,WiN denotes the corresponding combination coefficient of u1 (t ), u2 (t ), , uN (t ) . Eq. (11) can be rewritten in a general vector form as follows: v t W T u t
(12)
where
v t v1 t , v2 t ,
, vM t ;
u t u1 t , u2 t ,
, uN t ;
T
T
WM 1 W11 W21 W W22 WM 2 . W 12 WMN W1N W2 N Thus, combining with Eq. (10) and Eq. (12), the following dynamical system (13) is derived
v t Kv t g v t , t
(13)
system and the low-dimensional dynamical system is introduced by M 2 2 N Error ui t dt vi t dt 0 0 i 1 i 1
2
(16)
Combining the vector expressions of u (t ) , v(t ) with the Eq. (12) , the error (16) can be strictly derived as follows:
u t u t v t v t dt u t u t u t WW u t dt u t I WW u t dt .
Error W
T
2
T
0
T
T
2
T
(17)
0
2
T
T
0
where u(t ) [u1 (t ), u2 (t ), , uN (t )]T , W is the combination matrix in Eq. (12) which is assumed to be column orthogonal. C. Optimization The optimal low-dimensional dynamical system (13) can be obtained, while the combination matrix W is determined by minimizing the error function (17). To evaluate the error function, a temporal interval [0, Tmax ] is considered for (17) and approximated by a finite sum, where the max integral time Tmax is selected as free parameter. The temporal integral interval [0, Tmax ] generally contains the processes from initial state to steady state of the nonlinear PDE. The restricted condition for the minimizing problem (17) is that the combination matrix should be column orthogonal. Thus, the error functions for combination matrix can be approximated as follows. Since Error W
u t I WW u t dt Tmax
2
T
0
2
where K W T LW ; g (v(t ), t ) W T f (u(t ), t ) .
ti ti 1 u ti I WW T u ti , i we consider approximate error
B. Error Functions for Combination Matrix Assuming the dimension N of ODE system (10) is large enough that it can capture the almost all dynamics of the nonlinear PDE (1). Let the energy function of the ODE system (10) as follows:
E W ti ti 1 u ti I WW T u ti i
N
u t dt 0
i 1
2
i
(14)
Similarly, the energy functions of the low-dimensional dynamical system (13) can be given as follows: M
v t dt 0
i 1
2
i
(15)
To obtain a low-dimensional ODE system which can capture the dynamics of the nonlinear PDE, an error function that measures energy error between the ODE © 2013 ACADEMY PUBLISHER
2
(18)
subject to W (:, i)T W (:, i) ki ij . The optimization of E (W ) (18) mainly contains several steps. Firstly, the fourth-order Runge-Kutta method are used to calculate the ODE system (10), thus ti , u (ti ) of Error (18) are known. Secondly, the particle swarm optimization (PSO) algorithm [19-21] is used to optimize the restricted error functions (18). Particle swarm optimization (PSO) is a recently invented high-performance stochastic algorithm introduced in the middle of the 1990s. This algorithm is developed from the theory of artificial life and evolutionary computation and is used to solve
JOURNAL OF NETWORKS, VOL. 8, NO. 11, NOVEMBER 2013
optimization problems. This technique is akin to the other stochastic methods performing a global search in the parameter space without getting trapped in local minima. A key characteristic of the PSO is that the algorithm itself is highly robust. This method may offer different routes through the problem hyperspace than other methods such as evolutionary algorithms, ant colony optimization, simulated annealing, genetic algorithms (GA) [23], and so on. PSO [24] is similar to GA or evolutionary algorithms in several ways, but is much simpler than the rules of GA. This method does not have the “cross” (crossover) and “variability” (mutation) operation of a GA. Thus, this condition implies that PSO requires less computational bookkeeping and generally only a few lines of code. For applications in practical industrial engineering, a simpler algorithm is generally appealing. The calculations for the minimization of the error functions Q in the current study have established that the PSO is an effective optimization method. In PSO for the minimization of Q , positions of particles are candidate solutions to the Nk -dimensional problem, and the moves of the particles are regarded as the search processes of better solutions. The position of the i th particle is represented by xi ( xi 1 , xi 2 , , xi Nk ) , and its velocity is represented by vi (vi 1 , vi 2 , , vi Nk ) . During the search process, the particle successively adjusts its position according to two factors, namely their personal best position and the global best position. The personal best position Pi ( Pi 1 , Pi 2 , , Pi Nk ) is the best previous position of the i th particle that gives the best fitness value. The best particle among all the particles in the population is represented by Pg ( Pg 1 , Pg 2 , , Pg Nk ) . Each particle is updated by the following two best values, Pi and Pg during its iteration. The particle updates its velocity and positions as follows: vi d new wvi d old c1r1 Pi d xi d c2 r2 Pi d xi d
xi d new xi d old vi d new
(19)
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singular value decomposition for W . Let the singular value decomposition of W is W USV T , where U , S are orthogonal and diagonal matrix respectively. Thus combination matrix W US is orthogonal which the minimal error is retained by WW T USV T VSU T WW T . III.
A NUMERICAL EXAMPLE
In this section, the optimal low-dimensional dynamic system of nonlinear PDEs is constructed for a simple PDE model. Note that this example serves an illustration of the proposed methodology rather than a demonstration of the model reduction of a very large and complex system. In order to evaluate the optimal low-dimensional dynamical systems for nonlinear PDEs, the Burgers Equation is studied. The Burgers equation is a onedimensional spatial model of a variety of threedimensional physical phenomena, greatly simplifying the problems while retaining many of the complex behavior characteristics. This equation contains nonlinear convection and diffusion terms and retains many of the interesting features of the Navier-Stokes equation. The governing equation may be written as u x, t t
u
u x, t x
2 u x, t
(21)
x 2
where the viscosity is 0.3 and (21) subjects to the following boundary condition: u 0, t u 1, t 0
(22)
u x,0 sin x
(23)
and initial condition:
The numerical solution of the Burgers equations is obtained using the Runge-Kuta methods with Finite Difference discretization at the space and time domain. The approximation solution of Burgers equations in the example are given in the Fig. 1. 3
(20)
2 1
where: w is an inertia weight, c1 and c2 are two positive learning factors, and r1 and r2 are two random numbers uniformly distributed in the range between 0 and 1. A time parameter in Eq. (20) determines the different flying time for each particle. Eq. (19) is used to update the velocity according to its previous velocity and the distances of its current position from Pi and Pg . Then, the particle flies toward a new position according to Eq. (20).The principle and details of PSO algorithm is given in Ref [19]. For the optimization of (18) by PSO directly, the obtained initial combination matrix W is not orthogonal. To keep the minimization of the error function (18), a satisfied combination matrix W can be derived by
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0 -1 -2 20
150
15
100
10
Discretization locations
50
5 0
0
Simulation time
Figure 1. The approximation solution of the Burgers equations
To obtain a high dimensional ODE system (9), the spatial basis functions and the weighting functions of Burgers equation (21) are selected as
x i
2 sin(i x), i 1, 2,
,
(24)
The first four spatial basis functions given by the Eq. (2) are shown in the following Fig. 2.
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1.5
The 1st mode of initial dynamical ODE system
1
1.2 0.5
1 0
0.8
-0.5
0.6
-1
-1.5
0
0.1
0.2
0.3
0.4
0.5 0.6 Spatial domain
0.7
0.8
0.9
0.4
1
Figure 2. The first four spatial basis functions
0.2
0
0
0.5
1
1.5
2
2.5
3
3
Figure 6. The 1st mode of initial dynamical ODE system
2 1 0
1
-1
The 1st mode of the 2-order optimal dynamical system The 2nd mode of the 2-order optimal dynamical system
0.8
-2 20
150
15
0.6
100
10
Discretization location
50
5 0
0
0.4
Simulation time 0.2
Figure 3. The approximation by 10-order dynamical system
0 -0.2 -0.4 -0.6 -0.8
0.1
-1 0
-0.1
0
0.5
1
1.5
2
2.5
3
Figure 7. The 1st and 2nd modes of the 2-order optimal dynamical system
-0.2 20 15
150 10
100 5
1
50 0
The 1st mode of the initial dynamical ODE system The 2nd mode of the initial dynamical ODE system
0.8
0
0.6
Figure 4. The approximation error of the 10-order dynamical system
0.4 0.2 0
Combining basis functions expansion (5) based on (24) and Galerkin method, an initial 10-order dynamical ODE system has been derived. Synthesis of the temporal output of 10-order dynamical ODE system together with the spatial basis functions (24), the approximation of the Burgers equation is given in Fig. 3. The approximation error of the 10-order dynamical system is given in the Fig. 4. The temporal integral interval is [0,3] and the energy value of the 10-order dynamical ODE system is 0.0811. To evaluate the optimal low dimensional dynamical systems of Burgers equations (21), 10 1 and 10 2 combination matrices are obtained from the optimization for error in (18) by the standard PSO algorithm and singular value decomposition. Thus, 1order and 2-order optimal dynamical systems have been derived by linear combination (12).
-0.2 -0.4 -0.6 -0.8 -1
0
0.5
1
1.5
2
2.5
3
Figure 8. The 1st and 2nd modes of initial dynamical ODE system
0.09 Energy of inital ODE Energy of optimal ODE
0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01
0
0
-0.2 The 1st mode of 1-order optimal dynamical system
1 mode
2 mode
-0.4
-0.6
Figure 9. Comparison for the energy value of the optimal dynamical system and initial ODE system with the same modes
-0.8
-1
-1.2
-1.4
0
0.5
1
1.5
2
2.5
3
Figure 5. The 1st mode of the 1-order optimal dynamical system
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The 1st mode of 1-ordr optimal dynamical system and the 1st mode of initial 10-order dynamical ODE system are shown in Fig. 5 and Fig. 6, respectively. Similarly, the 1st and 2nd modes of 2-order optimal dynamical systems
JOURNAL OF NETWORKS, VOL. 8, NO. 11, NOVEMBER 2013
and that of initial 10-order dynamical ODE system are also shown in Fig. 7 and Fig. 8. As shown in Fig. 9, the energy value of the obtained 1order and 2-order optimal dynamical systems are 0.0809 and 0.0810, where that of the initial ODE systems with the same modes are 0.0312 and 0.0555 respectively. For the purpose of control, energy is the representative of the dynamics of systems. In Fig. 5, the optimal dynamical system with the 1 or 2 modes have almost the all energy of the initial ODE system. In other words, the optimal dynamical system can capture the dynamics of the nonlinear PDE perfectly. IV.
CONCLUSIONS
In this note, a new approach based on spatial basis function expansions for constructing low-dimensional dynamical systems of nonlinear partial differential equations (PDEs) is presented. After choosing the proper spatial basis functions, spatial basis functions expansions combined with weighted residual methods are used for time/space separation and truncation. Optimal dynamical systems are obtained by linear combination from the modes of spectral based ODE systems of nonlinear PDEs, while the combination matrix is obtained by optimization for a energy error function. A simple algorithm based on particle swarm optimization (PSO) to determine the optimal combination matrix is also introduced. The numerical example shows that the optimal dynamical systems require much smaller number of modes to capture the dynamics of nonlinear PDEs. ACKNOWLEDGEMENT Financial support from National Natural Science Foundation of China (11271376) is gratefully acknowledged. REFERENCE [1] J. Liu, M. Li, and F. He (2012), "A Novel Inpainting Model for Partial Differential Equation Based on Curvature Function," Journal of Multimedia, vol. 7(3), pp. 239-246. [2] J. P. Boyd, Chebyshev and Fourier spectral methods: Dover Pubns, 2000. [3] M. Jiang and H. deng (2012), "Optimal combination of spatial basis functions for the model reduction of nonlinear distributed parameter systems," Communications in Nonlinear Science and Numerical Simulation, vol. 17pp. 5240-5248. [4] e. a. Canuto. C., Spectral methods in fluid dynamics. New York: Springer-Verlag, 1988. [5] D. Gottlieb and S. A. Orszag, Numerical analysis of spectral methods: theory and applications: Society for industrial and applied mathematics, 1993. [6] H. Deng, H. X. Li, and G. Chen (2005), "Spectralapproximation-based intelligent modeling for distributed thermal processes," Control Systems Technology, IEEE Transactions on, vol. 13(5), pp. 686-700. [7] H. Deng, M. Jiang, and C.-Q. Huang (2012), "New spatial basis functions for the model reduction of nonlinear distributed parameter systems," Journal of Process Control, vol. 22(2), pp. 404-411.
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[8] A. Adrover, G. Continillo, S. Crescitelli, M. Giona, and L. Russo (2002), "Construction of approximate inertial manifold by decimation of collocation equations of distributed parameter systems," Computers & Chemical Engineering, vol. 26(1), pp. 113-123. [9] P. D. Christofides and P. Daoutidis (1997), "FiniteDimensional Control of Parabolic PDE Systems Using Approximate Inertial Manifolds* 1," Journal of Mathematical Analysis and Applications, vol. 216(2), pp. 398-420. [10] P. D. Christofides (1998), "Robust control of parabolic PDE systems," Chemical Engineering Science, vol. 53(16), pp. 2949-2965. [11] K. Hasselmann (1988), "PIPs and POPs: The reduction of complex dynamical systems using principal interaction and oscillation patterns," J. Geophys. Res, vol. 93pp. 1101511021. [12] F. Kwasniok (1996), "The reduction of complex dynamical systems using principal interaction patterns," Physica D: Nonlinear Phenomena, vol. 92(1-2), pp. 28-60. [13] F. Kwasniok (1997), "Optimal Galerkin approximations of partial differential equations using principal interaction patterns," Physical Review E, vol. 55(5), pp. 5365-5375. [14] F. Kwasniok (2004), "Empirical low-order models of barotropic flow," Journal of the atmospheric sciences, vol. 61(2), pp. 235-245. [15] H. Zhao and C. Wu (1998), "Construction of optimal truncated low-dimensional dynamical systems directly from PDEs," Communications in Nonlinear Science and Numerical Simulation, vol. 3(1), pp. 1-5. [16] Y. Lei, H. Lai, and Q. Li (2012), "Geometric Features of 3D Face and Recognition of It by PCA," Journal of Multimedia, vol. 6(2), pp. 207-216. [17] M. Jiang and H. deng (2013), "Improved Empirical Eigenfunctions Based Model Reduction for Nonlinear Distributed Parameter Systems," Industrial & Engineering Chemistry Research, vol. 52(2), pp. 934-940. [18] I. Bonis and C. Theodoropoulos (2012), "Model reductionbased optimization using large-scale steady-state simulators," Chemical Engineering Science, vol. 69(1), pp. 69-80. [19] N. Yusup, A. M. Zain, and S. Z. M. Hashim (2012), "Overview of PSO for Optimizing Process Parameters of Machining," Procedia Engineering, vol. 29(0), pp. 914923. [20] G. Cui, L. Qin, S. Liu, Y. Wang, X. Zhang, and X. Cao (2008), "Modified PSO algorithm for solving planar graph coloring problem," Progress in Natural Science, vol. 18(3), pp. 353-357. [21] H. Shinzawa, J.-H. Jiang, M. Iwahashi, I. Noda, and Y. Ozaki (2007), "Self-modeling curve resolution (SMCR) by particle swarm optimization (PSO)," Analytica Chimica Acta, vol. 595(1–2), pp. 275-281. [22] D. G. Zill, & Cullen, M. R., Differential equations with boundary-value problems (5th ed.). vol. Brooks/Cole Thomson Learning. Australia: Pacific Grove, CA, 2001. [23] L. D. M. SON J S, KIM I S (2004), "A study on genetic algorithm to select architecture of a optimal neural network in the hot rolling process," Journal of Materials Processing Technology, pp. 153-156. [24] Z. WANG, X. ZHAO, B. WAN, and J. XIE (2013), "Research of BP Neural Network based on Improved Particle Swarm Optimization Algorithm," Journal of Networks, vol. 8(4), pp. 947-954.
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Jun Shuai received the B. Eng. degree from the Academy of Equipment Command & Technology (General Armament Department), Beijing, China, in 1997. Currently, he is pursuing the Ph.D. degree in school of Mathematics and Statistics at the Central South University of China. He is a senior engineer at Software Park in Changsha, China. His research interests include Dynamical systems, Optimization and Signal Analyzing and Processing.
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JOURNAL OF NETWORKS, VOL. 8, NO. 11, NOVEMBER 2013
