Predicting beach profile evolution with group method data handling-type neural networks on beaches with seawalls

Abstract

A major goal of coastal engineering is to develop models for the reliable prediction of short- and long-term near shore evolution. The most successful coastal models are numerical models, which allow flexibility in the choice of initial and boundary conditions. In the present study, evolutionary algorithms (EAs) are employed for multi-objective Pareto optimum design of group method data handling (GMDH)-type neural networks that have been used for bed evolution modeling in the surf zone for reflective beaches, based on the irregular wave experiments performed at the Hydraulic Laboratory of Imperial College (London, UK). The input parameters used for such modeling are significant wave height, wave period, wave action duration, reflection coefficient, distance from shoreline and sand size. In this way, EAs with an encoding scheme are presented for evolutionary design of the generalized GMDH-type neural networks, in which the connectivity configurations in such networks are not limited to adjacent layers. Also, multi-objective EAs with a diversity preserving mechanism are used for Pareto optimization of such GMDH-type neural networks. The most important objectives of GMDH-type neural networks that are considered in this study are training error (TE), prediction error (PE), and number of neurons (N). Different pairs of these objective functions are selected for two-objective optimization processes. Therefore, optimal Pareto fronts of such models are obtained in each case, which exhibit the trade-offs between the corresponding pair of the objectives and, thus, provide different non-dominated optimal choices of GMDH-type neural network model for beach profile evolution. The results showed that the present model has been successfully used to optimally prediction of beach profile evolution on beaches with seawalls.

Keywords

beach profile evolution / genetic algorithms / group method of data handling / Pareto / reflective beaches

Introduction

A major goal of coastal engineering is to develop models for the reliable prediction of short- and long-term nearshore evolution. Beach erosion occurred as a result of marine structures construction or changing hydrodynamic characteristics of flow adjacent to the structures. Although beach erosion occurs frequently close to the coastal structures, engineering procedures for predicting scour potential are relatively rare. Beach profile modeling is most often used to evaluate the impact of a major storm with elevated storm tides and wave heights on the beach and the dunes. Another application is to predict the profile evolution of a beach nourishment project that is placed at a slope initially steeper than the equilibrium profile. A third application is in a three-dimensional model that simulates more realistic situations by cross shore sediment transport [1].

Extensive literature reviews originally made by Kraus [2] and Kraus and McDougal [3] and reported by Abedimahzoon et al. [4] show that only a small number of theoretical studies (e.g., [5-8]) consider the beach profile change in front of a seawall and results have been derived for highly simplified conditions. Numerous physical models as well as field investigations have been performed to investigate the interaction between seawalls and the beach (e.g., [9,10]).

Study of hydrodynamics in front of reflective structures shows that seawalls can modify the velocity field if they are located around the active zone [5,11,12]. Therefore, it can be expected that seawalls can contribute in cross-shore sediment transport resulting in beach profile change during storm conditions. The results of profile evolution experiments in the literature are also in good agreement with this fact and clearly show the different patterns in profile change with and without the structure.

System identification techniques are applied in many fields in order to model and predict the behaviors of unknown and/or very complex systems based on given input–output data [13,14]. The group method of data handling (GMDH) algorithm is a self organizing approach by which gradually complicated models are generated based on the evaluation of their performances on a set of multi-input–single-output data pairs. In recent years, however, the use of such self- organizing networks has led to successful application of the GMDH-type algorithm in a broad range of areas in engineering, science and economics [15]. The self-organizing property of gradually increasing complexity networks, in which all the training data set are used for each connection of any two neurons, is the main advantage of such GMDH-type neural networks.

All these methods devised previously have been based on single objective optimization processes in which either training error (TE) or prediction error (PE) are selected to be minimized with no control of other objectives. However, in order to obtain more robust models, it is required to consider all the conflicting objectives, TE, PE, and number of neurons (N) (representing the complexity of the models) be minimized simultaneously [16] in the sense of a multi-objective Pareto optimization process. In this paper, evolutionary algorithms (EAs) with an encoding scheme are used to evolve design the generalized structure GMDH-type (GS-GMDH) neural networks [16], in which the connectivity configuration in such networks is not limited to adjacent layers for modeling and prediction of beach profile evolution .This study is a contribution to compensate for the lack of information concerning the effect of reflective beaches on sediment transport and beach evolution in the surf zone due to random waves attack.

Experiment

Hydraulic Laboratory of Imperial College, London, UK

To determine the velocity field inside the surf zone, laboratory experiments were performed in a large two-dimensional wave tank in the Imperial College Hydraulics Laboratory. Figure. 1 shows the experimental set up. The overall dimensions of the tank were 2.80 m wide (which was divided into three sections), 1.5 m deep and 60 m long. The width of channel used in experiments was 0.87 m. The water depth was 0.9 m for all experiments. A beach profile according to an equilibrium profile equation was built at the end of the tank. An absorbent foam is located behind a porous seawall, made of steel rods, to absorb the energy of incident waves. Previous studies have measured the undertow under regular or random waves on a plane sloping beach [17]. This study is different in that it measures the undertow under random waves on a profile-type of beach.

Using a typical Jonswap spectrum, irregular waves were generated at one end of the tank generator controlled by an electro-hydraulic system which accepted an input voltage from a personal computer and the water particle velocities were measured in the surf zone using a one-component fiber-optic Laser-Doppler Anemometer (LDA). Table 1 summarizes the tests conditions for this experiment [18]. Four horizontal locations were chosen in the surf zone. At these locations measurements were made at several points between the bottom and still water level. The locations for all measured points are given in Table 2.

In Table 1, H_s = significant wave height; T_s = zero-crossing period; L₀ = deep water wavelength; S₀ = deep water wave steepness.

To consider the effect of reflective structures on beach hydrodynamics, the experiment was repeated in front of a partially reflective seawall located in the surf zone. The main objective of this experiment was a quantitative comparison of near-bed velocities in two cases, i.e., with and without the reflective structure [19]. For this purpose, a permeable seawall was built at the end of the beach. The exact distance of seawall from the shoreline was 0.6 m resulting in 0.100 m water depth in front of the structure. A sampling rate of 25 Hz with a record length of 6 min, similar to the first experiment, was selected to provide the suitable conditions in order to compare the velocity field in two different experiments.

Figure 2 contrasts the vertical distribution of undertow velocity on an open beach with that found in front of the reflective structure. Also, in this figure the magnitudes of horizontal mean velocity at one particular position above the bed are compared for two different experiments. As indicated in Fig. 2, the presence of the reflective structure and the addition of reflected waves causes significant changes in the mean flow particularly further offshore of the surf zone.

In the next stage of this experiment, an extensive series of tests was performed in front of a permeable, reflective seawall for a selected range of wave conditions and water depths which cover conditions varying from mainly wave reflection from structure wave breaking in front of structure. In this experiment, horizontal velocities were measured at four horizontal locations (corresponding to L/4, L/2, 3L/4 and L from the seawall where L is the significant wavelength) for three different positions of the seawall with respect to the shoreline (i.e., three different water levels: 0.05, 0.10, and 0.15 m in front of the structure), and for three different wave spectra (S1, S2 and S3). At each location the velocity measurements were made for several vertical points. The experimental conditions are summarized in Table 3.

Profile evolution

The velocity measurements were complemented by beach profile measurements performed with two different sizes of sand (fine sand (D₅₀ = 0.5 mm) and coarse sand (D₅₀ = 1.5 mm)). The experiments were then repeated in front of the partially reflective structure located in the surf zone. The sand sizes were chosen so that the threshold velocity of motion was exceeded for a significant period of time for both sediments.

The results of the open beach experiments show the coarse sediment beach building a berm, while the finer sand beach erodes to form an alongshore bar; the results of the experiments with the seawall show the tendency of the beach to form a berm profile in front of the structure (Figs. 3). These figures also contrast the change in bed elevation from the original profiles for both fine and coarse sediments on an open beach with that measured in front of the seawall.

Modeling using the GMDH type neural networks

By means of the GMDH algorithm, a model can be represented as a set of neurons in which different pairs of them in each layer are connected through a quadratic polynomial and, thus, produce new neurons in the next layer. The formal definition of the identification problem is to find a function \stackrel{︵}{f} that can be approximately used instead of the actual one, f in order to predict output \stackrel{︵}{y} for a given input vector X=(x_{\mathrm{1}},x_{\mathrm{2}},x_{\mathrm{3}},\cdots ,x_n) as close as possible to its actual outputy. Therefore, given M observations of multi-input, single output data pairs so that

(1)y_i=f(x_{i\mathrm{1}},x_{i\mathrm{2}},x_{i\mathrm{3}},\cdots ,x_{in}),i=\mathrm{1},\mathrm{2},\mathrm{3},\cdots ,M.

It is now possible to train a GMDH type neural network to predict the output values {\stackrel{︵}{y}}_i for any given input vectorX=x_{i\mathrm{1}},x_{i\mathrm{2}},x_{i\mathrm{3}},\cdots ,x_{in}, that is

(2){\stackrel{︵}{y}}_i=\stackrel{︵}{f}(x_{i\mathrm{1}},x_{i\mathrm{2}},x_{i\mathrm{3}},\cdots ,x_{in}),i=\mathrm{1},\mathrm{2},\mathrm{3},\cdots ,M.

The problem is now to determine a GMDH type neural network so that the square of the differences between the actual output and the predicted one is minimized, that is

(3){\displaystyle \sum _{i=\mathrm{1}}^M{\left(\stackrel{︵}{f}(x_{i\mathrm{1}},x_{i\mathrm{2}},x_{i\mathrm{3}},\cdots ,x_{in})-y_i\right)}^{\mathrm{2}}}\to \min .

The general connection between the inputs and the output variables can be expressed by a complicated discrete form of the Volterra functional series in the form of

(4)y=a_{\mathrm{0}}+{\displaystyle \sum _{i=\mathrm{1}}^n{a}_i{x}_i+}{\displaystyle \sum _{i=\mathrm{1}}^n{\displaystyle \sum _{j=\mathrm{1}}^n{a}_{ij}{x}_i{x}_j}+}{\displaystyle \sum _{i=\mathrm{1}}^n{\displaystyle \sum _{j=\mathrm{1}}^n{\displaystyle \sum _{k=\mathrm{1}}^n{a}_{ijk}{x}_i{x}_j{x}_k}}+}\cdots ,

where is known as the Kolmogorov-Gabor polynomial. This full form of mathematical description can be represented by a system of partial quadratic polynomials consisting of only two variables (neurons) in the form of

(5)\stackrel{︵}{y}=G(x_i,x_j)=a_{\mathrm{0}}+a_{\mathrm{1}}{x}_i+a_{\mathrm{2}}{x}_j+a_{\mathrm{3}}{x}_i^{\mathrm{2}}+a_{\mathrm{4}}{x}_j^{\mathrm{2}}+a_{\mathrm{5}}{x}_i{x}_j.

In this way, such partial quadratic description is recursively used in a network of connected neurons to build the general mathematical relation of the inputs and output variables given in Eq. (4). The coefficients a_i in Eq. (5) are calculated using regression techniques, so that the difference between the actual output, y and the calculated one, \stackrel{︵}{y} for each pair ofx_i, y_i as input variables is minimized. Indeed, it can be seen that a tree of polynomials is constructed using the quadratic form given in Eq. (5) whose coefficients are obtained in a least squares sense. In this way, the coefficients of each quadratic function G_i are obtained to fit optimally the output in the whole set of input–output data pairs, that is

(6)E={\displaystyle \frac{{\displaystyle \sum _{i=\mathrm{1}}^M{(y_i-G_i())}^{\mathrm{2}}}}{M}}\to \min .

In the basic form of the GMDH algorithm, all the possibilities of two independent variables out of the total n input variables are taken in order to construct the regression polynomial in the form of Eq. (5) that best fits the dependent observations (y_i,i=\mathrm{1},\mathrm{2},\cdots ,M) in a least squares sense. Consequently,\left(\begin{array}{l}n\\ \mathrm{2}\end{array}\right)={\displaystyle \frac{n(n-\mathrm{1})}{\mathrm{2}}} neurons will be built up in the first hidden layer of the feed forward network from the observations

\{(y_i,x_{ip},x_{iq});(i=\mathrm{1},\mathrm{2},\cdots ,M)\}

for differentp,q\in \{\mathrm{1},\mathrm{2},\cdots ,n\}. In other words, it is now possible to construct M data triples \{(y_i,x_{ip},x_{iq});(i=\mathrm{1},\mathrm{2},\cdots ,M)\} from observations using such p,q\in \{\mathrm{1},\mathrm{2},\cdots ,n\} in the form

\left[\begin{array}{lll}{x}_{\mathrm{1}p}& x_{\mathrm{1}p}& y_{\mathrm{1}}\\ x_{\mathrm{2}p}& x_{\mathrm{2}p}& y_{\mathrm{2}}\\ x_{Mp}& x_{Mp}& y_M\end{array}\right]

.

Using the quadratic sub-expression in the form of Eq. (5) for each row of M data triples, the following matrix equation can be readily obtained as

(7)Aa=Y,

where a is the vector of unknown coefficients of the quadratic polynomial in Eq. (5):

(8)a=\{a_{\mathrm{0}},a_{\mathrm{1}},a_{\mathrm{2}},a_{\mathrm{3}},a_{\mathrm{4}},a_{\mathrm{5}}\}

and

(9)Y={\{y_{\mathrm{1}},y_{\mathrm{2}},y_{\mathrm{3}},\cdots ,y_M\}}^{\mathrm{T}}

is the vector of the output’s value from observation. It can be readily seen that

(10)A=\left[\begin{array}{llllll}\mathrm{1}& x_{\mathrm{1}p}& x_{\mathrm{1}q}& x_{\mathrm{1}p}{x}_{\mathrm{1}q}& x_{\mathrm{1}p}^{\mathrm{2}}& x_{\mathrm{1}q}^{\mathrm{2}}\\ \mathrm{1}& x_{\mathrm{2}p}& x_{\mathrm{2}q}& x_{\mathrm{2}p}{x}_{\mathrm{2}q}& x_{\mathrm{2}p}^{\mathrm{2}}& x_{\mathrm{2}q}^{\mathrm{2}}\\ \mathrm{1}& x_{Mp}& x_{Mq}& x_{Mp}{x}_{Mq}& x_{Mp}^{\mathrm{2}}& x_{Mq}^{\mathrm{2}}\end{array}\right].

The least squares technique from multiple regression analysis leads to the solution of the normal equations in the form of

(11)a={(A^{\mathrm{T}}A)}^{-\mathrm{1}}{A}^{\mathrm{T}}Y,

which determines the vector of the best coefficients of the quadratic Eq. (5) for the whole set of M data triples. It should be noted that this procedure is repeated for each neuron of the next hidden layer according to the connectivity topology of the network. However, such a solution directly from normal equations is rather susceptible to round off errors and, more importantly, to the singularity of these Eqs [20].

Multi-objective optimization

Multi-objective optimization, which is also called multi-criteria optimization or vector optimization, has been defined as finding a vector of decision variables satisfying constraints to give optimal values to all objective functions [21]. In multi-objective optimization problems (MOPs), there are several objective or cost functions (a vector of objectives) to be optimized (minimized or maximized) simultaneously. These objectives often conflict with each other so that improving one of them will deteriorate another. Therefore, there is no single optimal solution as the best with respect to all the objective functions. Instead, there is a set of optimal solutions, known as Pareto optimal solutions located on Pareto front [21] for MOPs. The inherent parallelism in EAs makes them suitable for solving MOPs. There has been a growing interest in devising different EAs for MOPs. A very good and comprehensive survey of these methods has been presented in [22].

Without loss of generality, it is assumed that all objective functions are to be minimized. Such multi-objective minimization based on the Pareto approach can be conducted using some definitions. The Pareto dominance of a vector of objective functions is well defined in Jamali et al. [23].

EAs have been widely used for multi-objective optimization because of their natural properties suited for these types of problems. This is mostly because of their parallel or population-based search approach. Therefore, most difficulties and deficiencies within the classical methods in solving MOPs are eliminated. For example, there is no need for either several runs to find the Pareto front or quantification of the importance of each objective using numerical weights.

It is very important in EAs that the genetic diversity within the population be preserved sufficiently [24]. Consequently, the premature convergence of MOEAs is prevented and the solutions are directed and distributed along the true Pareto front if such genetic diversity is well provided. The Pareto-based approach of NSGA-II [25] has been recently used in a wide range of engineering MOPs because of its simple yet efficient non-dominance ranking procedure in yielding different levels of Pareto frontiers. However, the crowding approach in such a state-of-the-art MOEA works efficiently for two-objective optimization problems as a diversity-preserving operator, which is not the case for problems with more than two objective functions. The reason is that the sorting procedure of individuals based on each objective in this algorithm will cause different enclosing hyper-boxes. It must be noted that, in a two-objective Pareto optimization, if the solutions of a Pareto front are sorted in a decreasing order of importance to one objective, these solutions are then automatically ordered in an increasing order of importance to the second objective. Thus, the hyper-boxes surrounding an individual solution remain unchanged in the objective-wise sorting procedure of the crowding distance of NSGA-II in the two-objective Pareto optimization problem. However, in multi-objective Pareto optimization problem with more than two objectives, such sorting procedure of individuals based on each objective in this algorithm will cause different enclosing hyper-boxes.

Multi-objective evolutionary optimization of GMDH-type neural network models of beach profile evolution on beaches with seawalls

The parameters of interest in this multi-input and single-output system are significant wave height H_s, wave period T, wave action duration t_w, reflection coefficient R, distance from shoreline x and sand size D₅₀. Among these parameters, log x, t_w/T, H_s, R, D₅₀ exhibit the best composition of input variables for GMDH modeling of beach profile evolution on beaches with seawalls. Therefore, the input vector BoldItalic is then selected as BoldItalic = {log x, t_w/T, H_s, R, D₅₀}. In this work, the output parameter has been the logarithmic value of beach profile evolution (log z_b). The modified NSGA-II [16] is used for multi-objective optimum design of GMDH-type neural networks for modeling and prediction of beach profile evolution on beaches with seawalls using the input– output experimental data. However, in order to demonstrate the prediction ability of the evolved GMDH-type neural networks, the data have been divided into two different sets, namely training and testing sets. The training set, which consists of 130 out of 204 input–output data pairs, is used for training the neural networks models using the evolutionary method of this paper. The testing set, which consists of 74 unforeseen input–output data samples during the training process, is merely used for testing to show the prediction ability of such evolved GMDH-type neural network models. Population of 100 individuals with a crossover probability of 0.95 and a mutation probability of 0.1 has been used in 400 generations for which no further improvement has been achieved. In such multi-objective optimization design of the GMDH type neural networks, different pairs of conflicting objectives (TE, PE), (TE, N), and (PE, N) are selected for two-objective optimization design process. The obtained Pareto front for each pair of two-objective optimization have been shown through Figs. 4 to 6 for (TE, PE), (TE, N), and (PE, N), respectively. It is clear, from these figures, that all design points representing different GMDH-type neural networks are non-dominated with respect to the corresponding pair of conflicting objectives.

Figure 4 depicts the Pareto front of two-objective optimization of TE and PE representing different non dominated optimum points. In this figure, points C and A stand for the best PE and TE, respectively. It should be noted that N is not an objective function in this case and only TE and PE have been accounted in such two-objective optimum design of GMDH-type neural networks. Similarly, Figs. 5 and 6 depict the Pareto front of two-objective optimization of training error and number of neurons (TE, N) and prediction error and number of neurons (PE, N), respectively. In these figures, points F and I stand for the best optimum values obtained for TE and PE in their corresponding two-objective optimization process with respect to the N. On the other hand, points D and G stand for the simplest structure of GMDH type neural networks (N = 1) with respect to their corresponding values of TE and PE. The values of the objective functions together with their networks structures are given in Table 4 in which a, b, c, d, and e stand for log x, t_w/T, H_s, R, D₅₀, respectively. It is clear from these figures that choosing a better value for any objective function in a Pareto front would cause a worse value for another objective. Clearly, there are some important optimal design facts between the two objective functions which have been discovered by the Pareto optimum design approach of GMDH-type neural networks. Such important design facts could not have been found without the multi-objective Pareto optimization of those GMDH-type neural networks. From Figs. 4 to 6, points B, E, and H are, respectively, the points that demonstrate these important optimal design facts. Point B in the two-objective Pareto optimum design of TE and PE exhibits a very small increase in the value of PE (about 5 percent) in comparison with that point of C except that its TE is about 17 percent better than that of point C. Therefore, point B could be a trade-off optimum choice when considering the minimum values of both PE and TE.

Similarly, points E and H of Figs. 5 and 6 demonstrate some trade-offs between the complexities of networks (N) and TE and PE, respectively. Obviously, optimum design point H exhibits a very small increase in PE in comparison with that of point I while its number neurons is 67 percent less than that, which corresponds to a much simpler structure of neural network.

The structure of the evolved 3-hidden-layer corresponding to optimum point B is shown in Fig. 7 whose good behavior of such GMDH-type neural networks model is shown in Fig. 8.

Figure 8 shows examples of profile evolution predicted by the evolved GMDH model corresponding to optimum point B and their comparison with those found in experimental measurements and analytic model [4] for both coarse and fine sediments. The agreement between the experimental results and predictions is quite good.

Conclusions

Modeling and prediction of beach profile evolution on beaches with seawalls have been accomplished using a multi objective EA for optimization of generalized GMDH type neural networks. Such multi-objective optimization led to the discovery of useful optimal design principles in the space of objective functions. In this work, the important conflicting objective functions of GMDH-type neural networks have been selected as the training error (TE), prediction error (PE), and number of neurons (N) of such neural networks. Different pairs of these objective functions have been considered for various two-objective optimization processes. Therefore, optimal Pareto fronts of such models have been obtained in each case, which exhibit the tradeoffs between the corresponding pair of conflicting objectives and, thus, provide different non-dominated optimal choices of GMDH-type neural networks models for prediction of beach profile evolution. It has been shown that there exist some optimal structures of neural networks (points B, E, and H of the given Pareto fronts) that exhibit a very reasonable compromise between the conflicting objective functions and, thus, can be confidently chosen as optimum polynomial neural networks for modeling of beach profile evolution on beaches with seawalls. The level of reflectivity of the beach is an important parameter to control the magnitude and distribution of the undertow velocity. The results obtained from experiments and theoretical investigations show that as the reflection coefficient of a beach increases, the magnitude of undertow reduces which can affect the offshore sediment transport rate in the surf zone.

Nomenclatures

References

[1] Dean R G, Dalrymple R A. Coastal Processes with Engineering Applications. Cambridge: Cambridge University Press, 2004

[2] Kraus N C. The effects of seawalls on the beach: An extended literature review. Journal of Coastal Research, CERF. Special Issue No., 1988, 4: 1-28

[3] Kraus N C, McDougal W G. The effects of seawalls on the beach: Part I. An updated literature review. Journal of Coastal Research, 1996, 12(3): 691-701

[4] Abedimahzoon N, Molaabasi H, Neshaei M A L, Biklaryan M. Investigation of undertow in reflective beaches using a GMDH-type neural network. Turkish Journal of Engineering and Environmental Sciences, 2010, 34: 201-213

[5] Rakha K A, Kamphuis J W. Wave induced currents in the vicinity of a seawall. Coastal Engineering, 1997, 30(1-2): 23-52

[6] Szmytkiewicz M, Biegowski J, Kaczmarek M L, Okroj T, Ostrowski R, Pruszak Z, Rozynsky G, Skaja M. Coastline changes nearby harbor structures: Comparative analysis of one-line models versus field data. Journal of Coastal Engineering, 2000, 40(2): 119-139

[7] Ruggiero P, McDougal W G. An analytic model for the prediction of wave setup, long-shore currents and sediment transport on beaches with seawalls. Coastal Engineering, 2001, 43(3-4): 161-182

[8] Karim M F, Tingsanchali T. A coupled numerical model for simulation of wave breaking and hydraulic performances of a composite seawall. Ocean Engineering, 2006, 33(5-6): 773-787

[9] El-Bisy S M. Bed changes at toe of inclined seawalls. Ocean Engineering, 2007, 34(3-4): 510-517

[10] Dombusch U, Robinson A D, Williams R B G, Moses C A. Chalk shore platform erosion in the vicinity of sea defense structures and the impact of construction methods. Coastal Engineering, 2007, 54(11): 801-810

[11] Lin P, Liu P L F. Vertical variation of the flow across the surf zone. Coastal Engineering, 2002, 45(3-4): 169-198

[12] Neshaei M A L, Mehrad M A, Veiskarami M. The effect of beach reflection on undertow. Iranian Journal of Science & Technology, 2009, 33(B1): 49-60

[13] Mehrdad M A, Nariman-Zadeh N, Jamali A, Teymoorzadeh A. ANFIS networks design using hybrid genetic and SVD methods for modelling of the level variations of the Caspian Sea. Wseas Transactions on Information Science & Applications, 2005, 2: 121-127

[14] Åström K J, Eykhoff P. System identification, a survey. Automatica, 1971, 7(2): 123-162

[15] Ivakhnenko A G. Polynomial theory of complex systems. IEEE Transactions on Systems, Man, and Cybernetics, 1971, SMC-1(4): 364-378

[16] Jamali A, Nariman-zadeh N, Darvizeh A, Masoumi A, Hamrang S. Multi-objective evolutionary optimization of polynomial neural networks for modeling and prediction of explosive cutting process. Engineering Applications of Artificial Intelligence, 2009, 22(4-5): 676-687

[17] Vellinga P. A tentative description of a universal erosion profile for sandy and rock beaches. Coastal Engineering, 1984, 8(2): 177-188

[18] Holmes P, Baldock T E, Chan R T C, Neshaei M A L. Beach evolution under random waves. In: Proceedings of the 25th International Conference on Coastal Engineering. ASCE, Orlando, 1996, 3006-3019

[19] Holmes P, Neshaei M A L. The effect of seawalls on coastal morphology. In: Proceedings of the Second IAHR Symposium on Habitats Hydraulics. Eco- hydraulics, A, 1996, 525-530

[20] Nariman-Zadeh N, Darvizeh A, Jamali A, Moeini A. Evolutionary design of generalized polynomial neural networks for modelling and prediction of explosive forming process. Journal of Materials Processing Technology, 2005, 164-65: 1561-1571

[21] Fonseca C M, Fleming P J. Genetic algorithms for multi-objective optimization: Formulation, discussion and generalization, In: Forrest S. ed. Proceedings of the Fifth Intern. Conf. on Genetic algorithms. San Mateo, CA, 1993, 416-423

[22] Coello C A C. A comprehensive survey of evolutionary based multi-objective optimization techniques. Knowledge and Information System, 1999, 1(3): 269-308

[23] Jamali A, Ghamati M, Ahmadi B, Nariman-zadeh N. Probability of failure for uncertain control systems using neural networks and multi-objective uniform-diversity genetical gorithms (MUGA). Engineering Applications of Artificial Intelligence, 2013, 26(2): 714-723

[24] Toffolo A, Benini E. Genetic diversity as an objective in multi-objective evolutionary algorithms. Evolutionary Computation, 2003, 11(2): 151-167

[25] Deb K, Agrawal S, Pratap A, Meyarivan T. A fast and elitist multi-objective genetic algorithm: NSGA-II. IEEE Transactions on Evolutionary Computation, 2002, 6(2): 182-197

RIGHTS & PERMISSIONS

Higher Education Press and Springer-Verlag Berlin Heidelberg