Application of artificial neural network to calculation of solitary wave run-up

You-xing WEI*, Deng-ting WANG, Qing-jun LIU

1. College of Harbour, Coastal and Offshore Engineering, Hohai University, Nanjing 210098, P. R. China

2. River and Harbor Engineering Department, Nanjing Hydraulic Research Institute,Nanjing 210024, P. R. China

1 Introduction

Many researchers have focused on the study of solitary wave run-up because of its important practical significance in ocean engineering. Carrier and Greenspan (1958)first proposed the theoretical results for wave run-up of a non-breaking long wave on a flat sloping beach by solving the shallow water equations. Through the application of the Lagrange equation, Shuto (1967)obtained a run-up formula for a nonlinear long wave on a flat slope.Synolakis (1987)extended the analytical solution of Carrier and Greenspan (1958), and obtained a formula for estimation of solitary wave run-up. Titov and Synolakis (1998)studied the roughness effects on wave run-up through numerical methods. In order to examine the wave dispersion influence on solitary wave run-up, Zhang (1996)proposed a new solitary wave run-up model according to the nonlinear dispersion Boussinesq equation, and their results confirmed that the run-up estimated by the Boussinesq model was slightly larger than that obtained with the shallow water equations. In recent years, Hsiao et al. (2008)put forward a simple formula that can predict the maximum run-up of a breaking wave on a uniform beach over a wide range of the beach slope. Choi et al. (2008)presented that a parabolic cross-slope channel on the plane beach might cause the run-up intensification. Didenkulova and Pelinovsky (2008)proved the universality of a formula for calculation of the maximum run-up of a solitary wave on a beach under different incident wave conditions. They also studied the influence of the incident wave form on the extreme (maximal)characteristics of a wave at a beach, including run-up and run-down heights, run-up and run-down velocities, and the wave breaking parameter. Hsieh et al. (2007)developed a numerical model for experimental research on solitary wave running, shoaling and breaking on a sloping bed. To investigate the propagation and run-up of both non-breaking and breaking solitary waves, Mahdavi and Talebbeydokhti (2009)developed a force-muscle scheme model based on the non-linear shallow water equations. Hwang et al. (2007)investigated the evolution and run-up of breaking solitary waves on plane beaches. Que and Xu (2005)introduced a finite-volume kinetic Bhatnagar-Gross-Krook scheme as well as its applications to the study of roll and solitary waves. Nam and Haeng (2008)carried out numerical simulations of solitary wave propagation along a vertical wall and a sloping wall using the smoothed particle hydrodynamics (SPH)method.

In contrast with the above research, artificial neural network technology has been rapidly developed and is widely used in coastal and marine engineering. Deo et al. (2001)and Mandal and Prabaharan (2010)discussed how to use neural networks in wave forecasting. Mase and Kitano (1999)proposed a neural network model to investigate the wave impact on composite breakwater superstructures. Based on previous studies, this paper presents a further study on the application of artificial neural networks to solitary wave run-up calculation.

2 Back-propagation (BP)network and its Improvement

2.1 Neural network

The basic unit of the nervous system is a nerve cell which is also termed a neuron.Specific networks are composed of several parallel interconnected neuron models, and signal processing is achieved through interaction between neurons. The essence of neural networks demonstrates a functional relationship between input and output variables. Fig. 1 shows a single neuron with r input components.

Fig. 1 Diagram of single neuron with r input components

The neuron output can be expressed as

The activation transfer function f is the core of neurons and networks. The problem-solving ability and effectiveness of the network are not only related to the network structure, but also dependent on the activation transfer function.

2.2 BP network

A BP network is a multi-layer network that generalizes the W-H learning rule and carries out weight training with the nonlinear differentiable function. It gradually approaches the goal through continuous calculation of network weight and bias variations using the gradient descent of error function. Generally, BP networks use an S-type activation transfer function for the hidden layer, while using a linear activation transfer function for the output layer. Fig. 2 shows the structure of a neural network model with r input components and a hidden layer,where W1and B1are, respectively, the weight vector and bias vector between the input layer and hidden layer; W2and B2are, respectively, the weight vector and bias vector between the hidden layer and output layer; f1and f2are the activation transfer functions of the hidden layer and output layer; s1and s2are the neuron numbers of the hidden layer and output layer; A1and A2are the output vectors of the hidden layer and output layer; and P is the input vector.

Fig. 2 Diagram of neural network model with r input components and one hidden layer

2.3 Improvement of BP network

In order to speed up the training process and avoid local minimum values, the additional momentum method and the auto-adjusting learning factor are introduced into the BP network.

2.3.1 Additional momentum method

The additional momentum method is based on the BP method, the essence of which is to transfer the influence of the last weight variation through a momentum factor. It modifies each weight variation by adding an extra value that is proportional to the former one to produce a new weight variation according to the BP method.

In this study, only the weight adjustment for W1was considered. The weight adjustment formula with the additional momentum factor is

where Δωijis the weight variation corresponding to the ith neuron in the hidden layer and the jth input component (i =1, 2, … , s1and j =1, 2, … ,r ), Δbiand δiare, respectively,the bias variation and error term for the ith neuron in the hidden layer, pjis the jth input component, η ( k)is the learning factor for the kth training, and m is the momentum factor,which usually has a value of 0.95.

2.3.2 Auto-adjusting learning factor

It is not easy to determine an appropriate learning factor for a specific problem. A formula for determining the auto-adjusting learning factor (Cong 1998)is given as

where Esis the sum of square errors between the calculation results and experimental values.The selection scope of the initial learning factor η(0)is arbitrary, and the error rate Reis determined as 1.04.

The criterion of adjusting the learning factor is to check whether the weight variation correction value makes the error decrease. If it does, it indicates that the selected learning factor is small and the value can be increased by multipling an increasing factor, such as 1.05 in Eq. (3); if it does not, the value of the learning factor should be reduced by multipling a decreasing factor, such as 0.7 in Eq. (3).

3 Artificial neural network calculation model of solitary wave run-up

3.1 Model establishment

When establishing a neural network model, theoretical analysis should be carried out first to determine the main factors influencing the solitary wave run-up, and then adequate experimental values must be obtained. Secondly, the parameters must be determined, which include the neurons of the input and output layers, the number of hidden layers and associated neurons, the initial weights, the activation transfer functions, the initial learning factor, the multiplication factors for auto-adjusting learning factor, the error rate, the momentum factor,and the anticipation error. The network is trained by inputting the experimental values as input layer and output layer neurons. After the network training, solitary wave run-up can be calculated with the obtained W1, W2, B1, and B2. A comparison between the experimental and calculation results should be carried out. If their correlation coefficient is too low,re-training of the network is needed until the experimental and calculation results achieve a better correlation. Fig. 3 illustrates the process of establishing a neural network model.

Fig. 3 Flow chart of neural network establishment

3.2 Setup of input layer, output layer, and hidden layer

The largest solitary wave run-up is determined by the following equation (Shuto 1967):

where Rmaxis the largest solitary wave run-up, H is the solitary wave height, d is the water depth in front of the slope, and β is the slope gradient.

Based on the analysis, we selected two neurons for the input layer, H d and cotβ,and one neuron for the output layer, a dimensionless number Rmaxd. After many times of model training, we selected one hidden layer and five neurons for the hidden layer.

3.3 Data sources

For this study, 70 sets of experimental values were selected from the literature (Synolakis 1987; Hall and Watts 1953)to establish the neural network model, and 18 sets of experimental values were used to validate the applicability of the model. The experimental conditions are shown in Table 1 and Table 2.

Table 1 Experimental values and conditions for establishment of neural network model

Table 2 Experimental values and conditions for validation of neural network model

3.4 Selection of initial weights and activation transfer functions

Because the system is nonlinear, the training time, as well as whether the training result reaches the local minimum and whether the system converges is directly determined by initial weights. The components of initial weights are usually random values in the range of 0 to 1. They were selected as follows: the components of initial weight of W1and bias of B1were random values obtained using the nwtan.m function of the MATLAB toolbox; the components of initial weight of W2and bias of B2were random values obtained by the random function rands.

The BP network established in this study used an S-type activation transfer function for the hidden layer, while using a linear activation transfer function for the output layer.

3.5 Determination of other parameters

Other parameters determined for the establishment of the neural network model in this paper were as follows: the initial learning factor η(0)= 0.05, the increasing factor for auto-adjusting learning factor Linc= 1.02, the decreasing factor for auto-adjusting learning factor Ldec= 0.75, the error rate Re= 1.05, the momentum factor m= 0.95, and the anticipation error Ea= 0.04.

The Trainbpx.m function within the MATLAB toolbox was used to program, combining the advantages of the additional momentum method and the auto-adjusting learning factor.

4 Comparisons

4.1 Comparison between experimental values and calculation results by Synolakis formula

In the process of derivation, Synolakis (1987)considered the angle β of the flat sloping beach and used the solitary wave theory to solve the basic shallow water equation, obtaining a maximum relative solitary wave run-up formula:

Eq. (5)can be used under the conditions that H d ≫0.288tan βand that the wave does not break.

When wave breaking occurs, the calculation formula is

Fig. 4 shows the comparison between the experimental values of wave run-up and the Synolakis formula calculation results. It can be seen from the figure that they have a good correlation, with a correlation coefficient of 0.963 5. However, there remains a problem when using the Synolakis formula: it is difficult to accurately determine whether solitary waves have broken on the slope or not. The existing numerical calculation and model test have addressed this problem (Grilli et al. 1994, 1997; Tanaka et al. 1987; Zelt 1991). In this study, it was assumed that when H d is greater than 0.5, the solitary wave breaks.

4.2 Comparison between experimental values and neural network model calculation results

Fig. 5 shows the comparison between the experimental data and neural network model calculation results. The training time of the neural network model is 309 265. It can be found from Fig. 5 that the calculation results of the neural network model are highly accurate, and have a good correlation with the experimental results, with a correlation coefficient of 0.996 5.

Fig. 4 Comparison between experimental values of wave run-up and calculation results by Synolakis formula

Fig. 5 Comparison between experimental values of wave run-up and calculation results by neural network model

Using the neural network model, it is unnecessary to determine whether solitary waves have broken on the slope or not. However, it needs complete and accurate data. The model established in this study only considered the variables H d and cotβ, and therefore, was insubstantial. Under specific circumstances, fluid viscosity and bottom friction may have certain effect on solitary wave run-up. Teng et al. (2000)studied the maximum non-breaking solitary wave run-up on smooth and rough flat slopes through experiments. It was found that when the slope exceeded 20°, viscosity and roughness had little effect on the maximum solitary wave run-up, but when the slope did not exceed 20°, both factors had a significant effect on the run-up. According to their experimental results, the measured data of the maximum solitary wave run-up were always smaller than the prediction results without consideration of viscosity. Therefore, comprehensive experiments should be carried out to acquire adequate data, in order to obtain an optimal neural network model.

5 Conclusions

Application of artificial neural networks in solitary wave run-up calculation is presented in this paper. A BP network with one hidden layer was used and the model was improved by introducing the additional momentum method and the auto-adjusting learning factor. There are two neurons in the input layer, H d and cotβ, and five neurons in the hidden layer in the network, and the neuron of the output layer is a dimensionless number Rmaxd. The calculated results of solitary wave run-up by this neural network model and the experimental results have a good correlation; therefore, the neural network model can be used for solitary wave run-up calculation and analysis.

It is necessary to point out that the neural network model is established based on the finite amplitude wave theory. As the offshore wave is random, nonlinear, and clustered, a large number of observations must be conducted to establish a neural network model with greater practical applicability. On the other hand, as a limited number of data were used in this study,considering the limitations of the original trial data, the neural network model established in this paper is not suitable for a complex terrain. The neural network model has the potential to be optimized if additional experimental data are acquired.

Finally, the method of neural network modeling can be further popularized and has the potential to be applied to many other aspects of coastal and ocean engineering, such as tidal prediction and wave force calculation.

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