Evaluation of automatic algorithm for solving differential equations of plane problems based on BP neural network algorithm

Abstract

Plane problem is a typical combinatorial optimization problem. Aiming at the difference method of plane problem, BP neural network is proposed, the algorithm of solving difference equation is established, and the corresponding program is compiled. By calculating the calculation example, the continuity condition under the condition of modulus abruption is further discussed. The correctness and practicability of the difference equation algorithm are verified. A dynamic model of the parallel difference equation is constructed according to the characteristics of the parallel structure of BP neural network. The study shows that the two groups of differential equations are used to identify and verify the model, and the energy function that satisfies both the linear embedding condition and the correct wiring is given. Furthermore, BP neural network is used to realize the search and routing of the maximum plane subgraph of the planable line and the non-planar plan. The results show that the verification used is effective. Difference equation calculations have the ability to help BP networks get rid of local minima and get better results.

Keywords

BP neural network algorithm Plane problem Difference equation

1 Introduction

With the development of modern science and technology, a large number of time-delay dynamic system problems have been proposed in many fields of natural science and social science [1]. For example, many difference equations are involved in physics, biology, circuit signal system, genetics, automatic control system, chemical cycle system, information system and socioeconomics [2]. Various evolutionary behaviours of the system state starting from all possible initial states, such as equilibrium state, periodic or regressive behavior, long-term behavior, their interrelationship and stability [3]. The purpose of studying the difference equation is to grasp the objective law reflected by it, to actively explain the various phenomena and to predict the possible situation in the future [4]. The grid node code is automatically generated by the computer, and the difference equation is established and solved. All the user needs to do is input several control data and establish the corresponding data file. This model brings great convenience to the teaching demonstration of the difference method of the plane problem of elastic mechanics [5 –7]. BP neural network is a network model that simulates human brain characteristics by constructing a certain abstraction, simplification and simulation of neural networks and connecting them by a large number of neurons [8]. The information processing of BP neural network is realized by the interaction of neurons, and the storage of knowledge and information is represented by the distributed physical connection of network elements. It has a strong ability to describe the nonlinear mapping between input and output [9].

In 1943, the mathematical model of formal neurons was proposed and the era of theoretical research in neuroscience was created [10]. In 1997, a new learning method of multilayer neural network was proposed [11]. In 1999, the stability study of neural network was proposed in [12]. In 2002, a new method for determining the stability of neural networks was proposed. The stability criteria of the network with and without delay are obtained. It is concluded that the discrete system retains the dynamic characteristics of the original continuous neural network [13]. In 2006, the study of parameter dispersion of differential equation load model was proposed [14]. In 2008, the neural network structure was directly determined before the transformation. The algorithm has the advantages of simple operation, less parameters to be adjusted, strong global search ability and fast convergence speed [15]. In 2016, data extraction based on BP neural network methods was proposed [16]. There are many phenomena in the engineering field that can only be described by discrete mathematical models, and the computer itself can only understand the discretized model, which makes the research of discrete dynamic systems attract the attention of scholars [17]. The BP algorithm is improved and a fast error back propagation algorithm is proposed. The effectiveness and superiority of the algorithm are verified by dynamic model and field test data [18]. In recent years, scientists have proposed a number of neural network models with different information processing capabilities, and a variety of models have been implemented and opened [28 –31]. And it has been widely used in the fields of information processing, pattern recognition, automatic control, signal processing, assistant decision making, artificial intelligence, computer technology and so on. Since the neural network model, a very large part is the difference equation model [19].

We know that the plane problem in graph theory is a typical combinatorial optimization problem. Planar problems play an important role in the design of printed circuit boards and the wiring of large scale integrated circuits [20]. Parallel intelligent algorithm of network, but linear embedding method is used in plane embedding [21]. The result is inconsistent when the position of vertex and line corresponds in different order [22]. This will lead to the conclusion that the planar graph is not planar because the network operation results show that it is not straight line embedding, thus it is impossible to realize the planar graph straight line embedding and wiring [23]. We skillfully associate the solution of the equation with the solution of a corresponding ordinary difference equation. By means of comparison and using the properties of solutions of this kind of constant difference equation we have established, we obtain the sufficient condition that all solutions of the equation converge asymptotically to a constant as a special case of the equation [24]. The BP network can only fit the data points of the output sequence one by one, so its biggest advantage lies in its one-step prediction function, which is very suitable for predictive control and is not suitable for comprehensive load modeling [25]. Based on the characteristics of BP neural network structure, a simplified neural network is used to construct a parallel difference equation model for modeling research [26]. The BP network can only fit the data points of the output sequence one by one, so its biggest advantage lies in its one-step prediction function, which is very suitable for predictive control and is not suitable for comprehensive load modeling [27]. Based on the in-depth study of the characteristics of BP neural network structure, this paper uses a simplified neural network to construct a parallel difference equation model for modelingresearch.

2 Materials and methods

The application of BP neural network theory has penetrated into many fields, and has made gratifying progress in computer vision, pattern recognition, intelligent control, non-linear optimization, adaptive filtering and information processing, robots and so on. BP neural network is a network which is widely interconnected by a large number of processing units. It is an abstraction of human brain, simplifies simulation and reflects the basic characteristics of human brain. Since the realization of human brain computing is a completely different way from traditional digital computers. Feedback neural network adds a receiving layer to the hidden layer of feedforward network, feeds back the output of the hidden layer to the receiving layer, and then outputs to the hidden layer after one step delay of the receiving layer. In order to achieve the goal of memory, the system has the ability to adapt to time-varying characteristics, and can directly reflect the characteristics of dynamic process system. Based on these equations and boundary conditions, an applied stress function solution is established. At the same time, an approximate equation for ensuring the main speed condition and the deformation cointegration is presented. The structure of the network is directly related to the performance of the network. The BP network structure consists of the input layer, the hidden layer, the output layer, and the connection weights between the layers. The number of nodes in the input layer and output layer is determined by the actual problem, and there is no theoretical basis for the number of hidden layer nodes to guide. The number of neurons in the hidden layer directly affects the quality of network weight training. Too many hidden layer nodes make the weight connection information redundant, and too few hidden layer nodes make the network training insufficient, and the information is not fully utilized. The selection of BP neural network hidden layer nodes and the processing of network weight redundant information have once become a hot topic in various countries. Take the largest planar subgraph from a non-planar view. The problem of planar embedding is to embed a maximum planable subgraph of a planable view or an unplanar plan onto a plane and implement planar routing on that plane. The variance test for samples 1 and 2 is shown in Table 1 and Fig. 1. For different starting points, the number of times the objective function runs in the algorithm is different, and the relationship between them is shown inTable 2.

Fig.1

Comparison of difference equations.

Table 1

Comparison of difference equations

	Parallel difference equation number	Traditional difference equation number
Sample 1	10.23±6.64	9.51±5.32
Sample 2	7.56±6.01	7.33±5.30

Table 2

Relationship between run times and search times

	Running times	Search times
Starting point	3	6
Final point	5	2

Figure 2 shows that the output of the hidden layer of BP neural network is self-connected to the input of the hidden layer through the delay and storage of the receiving layer, which makes the BP neural network sensitive to the historical data:

Fig.2

Neural network model.

In this paper, BP neural network can be regarded as a computational system. Its state changes nonlinearly with time, so it is a non-linear system. Because it is very difficult to solve all the special solutions of the difference equation, when people study the stability of BP neural network, they usually bypass the way of solving the difference equation, and turn to the general behavior of solving the difference equation qualitatively to form the qualitative theory of the difference equation. The weights expressed by row vectors are superimposed and summed up, thus forming a parallel difference equation. We can avoid this non convergence phenomenon by modifying the rate formula. By enhancing the cooperation between particles. Experimental results show that the training of neural network based on inter particle cooperation can achieve good results. When the transfer function of the network neuron is a S function, the network weight coefficient matrix is symmetric. Then the energy of the network will decrease or not change with time. And only when the output potential changes with time, the energy of the network will not change. The difference equation can be used as a signal model to describe a large class of signals. Its application extends beyond many fields, such as multiresolution analysis, construction of one- or multi-dimensional orthogonal wavelet bases and fractal curves, image generation and compression. A special form of it is the lattice multi-scale equation. In terms of system convergence, we use a suitable transformation, using the function method, we get a sufficient condition for the global asymptotic stability of the unique equilibrium point of the system. In the eigenvalues of the characteristic equations of the linear variational system discussed in the first paragraph, we prove that the sufficient condition for the global asymptotic stability of the unique equilibrium point of the system is also a necessary condition, and proves that under certain conditions, the time lag will lead to the emergence of the system. Unstable state. In the experiment, the parameters selected for different graphs (different number of nodes) are different. We use the three algorithms in Table 3 and Fig. 3 to conduct experiments. The results of running the number of vertices and the number of sides are shown in Table 4 and Fig. 4.

Fig.3

Experiments on three algorithms.

Fig.4

Comparison of operation results between vertex number and edge number.

Table 3

Experiments on three algorithms

	Correspondence between vertex and linear position	Method of escaping local minimum
Algorithm 1	10.23±5.34	9.64±5.22
Algorithm 2	15.35±13.21	17.38±15.05
Algorithm 3	14.89±12.35	16.59±14.52

Table 4

Comparison of operation results between vertex number and edge number

	Vertex number	Edge number
Algorithm 1	19.81	15.90
Algorithm 2	18.23	17.51
Algorithm 3	16.54	16.23

We guide the network design according to the results of theoretical analysis. Before training, for a specific problem and a given training set, the relationship between them and the topological structure of the neural network is analyzed theoretically. What is the minimum size of the network? It can be seen that the choice of hidden layer and hidden node number has certain theoretical guidance. In the experiment, the parameters selected for different graphs (nodes number) are different. The elements in the network are all randomly distributed between 0.25, and the sign of network entry stability is chosen. The variation of each element before and after iteration is less than 0.03. If the network has not entered a stable state after up to 30 iterations (at this time it is considered to have entered the limit cycle), or has run to a stable state but is not satisfied (at this time it is considered that the network has entered a local minimum), then the network is re-initialized and run until the network runs. If it is equal, it is proved to be a planar graph, and its straight line embedding graph and routing are given. The approximate solution is obtained by studying the discrete sample values of functions. The design of corresponding energy functions according to the problems to be solved will make the application of wavelet networks more extensive. Only one-dimensional situations are discussed here, and the corresponding results are easily extended to multidimensional situations. It is known from the theory of delay difference equations that if the zero solution of the system is asymptotically stable, the zero solution of the system is locally asymptotically stable. The necessary and sufficient condition for the system to be stable is that all the characteristic roots of its characteristic equation have a negative real part. The difference equation has easy identification, and the obtained model can better fit the measured data. It reflects the external dynamic characteristics of the load from the input and output as a whole, and the order can be selected preferentially, which can be suitable for the description of the complex components of the load. Therefore, the difference equation model is considered to be a promising dynamic load model. For each of the above cases, the corresponding equations and discriminating conditions can be written. We only need to consider the symbolic conditions of the two initial conditions. When the initial condition is 1, it is obvious that each solution is the final cycle, and it can be observed through experiments that the solution is unbounded if the solution is the final cycle. The same support domain radius is selected for each node on the plane, and the node R is selected for different time to calculate, and the calculation results are analyzed. The error norm analysis is shown in Table 5 and Fig. 5.

Fig.5

Comparison of functions under different time conditions.

Table 5

Comparison of functions under different time conditions

Node R	Spline function	Gauss function
1h	8.02±7.33	15.35±18.23
1.2h	7.03±6.38	13.15±16.26
1.5h	8.51±7.32	15.65±17.26
1.8h	7.80±6.96	13.23±16.26

In this paper, for some simple boundary value problems on simply connected irregular domains, the coordinate transformation can be mapped to regular domains (including the regular domains in polar coordinates), and the centroid interpolation collocation method is used to calculate them. The coordinate transformation method is used to deal with the boundary value problem in irregular region. Although it can be calculated in regular region, the form of the governing difference equation will become complex, which is not conducive to the discretization of the governing equation. Moreover, most irregular regions are difficult to find a suitable mapping to map them into regular regions. When judging whether a graph is planar by whether it is linear embeddable or not, a very critical factor is the order of vertices in the process of linear embedding, which can not be ignored at all. We know that the vertex order of a graph can be numbered arbitrarily, so it is based on variance algorithm. Under the inappropriate vertex number, it can be concluded that the plan view can not be linearly embedded due to the network operation result, so that the straight line embedding and routing of the plan view cannot be realized. For non-planars, it can’t really find the largest planar subgraph, which fails to solve the planar problem completely. In addition, in terms of structural optimization design problems, if BP neural network is already suitable for optimization calculation, there is no need to delete redundant neurons. As long as the extra neurons do not affect the optimization results, they can be left in the network. Because the extra neurons have little effect on the efficiency of the optimization calculation. The constructed model subtly utilizes the parallel structure of the model. Through the parallel difference equation constructed by this parallel structure, the parallel computing ability is achieved in the calculation process, and the active power and reactive power output of the system are organically coupled together. This structure has great advantages in terms of computational power and model expression compared to traditional difference equations. The neuron is the basic processing unit of the BP neural network, and its multi-input, single-output data is shown in Table 6 and Fig. 6.

Fig.6

Neuron multi input and single output data.

Table 6

Neuron multi input and single output data

	Multiple input	Single output
Neuron	8.39	9.63

3 Result analysis and discussion

By establishing the global boundary difference equation of plane linear elastic problems, we can know the governing equation of linear elastic statics problems. According to the weighted residual method, the basic solution of elastic statics problems is introduced through step-by-step integration. When the error accumulates to a certain extent, the global adjustment of network weights and thresholds is replaced by local adjustment, and the output layer weights are adjusted accordingly. In this case, the amount of computation has been greatly reduced, and the learning speed has also been improved. At the same time, for different training samples, only a part of the neurons in the network need to be stimulated to produce effective output data. At the same time, the analytical solution formula is used to calculate the function value of the nodes in the physical area, and the average errors of displacement and stress in the calculated area are calculated, which are called the calculation area errors. Then, the numerical solutions of nodes in irregular domain are extracted, and the average errors of displacement and stress are calculated, which are called physical region errors. It can be seen from the difference equation that the vertex graph is a high order network for plane test problems. Apply the network’s operating equation to run the BP neural network to bring the network close to a minimum. Then simulate the algorithm and get another value. The energy function value difference between two iterations cannot be less than a certain small positive number. The distance between the current optimal value and the target optimal value is simulated by the variation of the cotangent hyperbolic, and the feature of the cosecant hyperbola is that there is a large numerical interval in the vicinity of zero. As the independent variable increases, its value is infinitely close to zero but does not intersect with zero. The interval between the two values decreases very slowly. Think of the independent variable as the number of iterations, and use the variant of the cotangent hyperbolic to indicate the nature of the match. A parameter of the adjustment function image, which affects the maximum value of the change, that is, the maximum distance that allows the deviation from the target optimal value during the optimization process during the iterative process. The position and position of the connection in the plane are as shown in Table 7 and Fig. 7.

Fig.7

Comparison of displacement and stress errors between position and position on a plane.

Table 7

Comparison of displacement and stress errors between position and position on a plane

	Displacement error	Stress error
Position and position connection	10.29	3.21
Location and location connections	9.11	4.50

In order to run the network, it is necessary to derive the difference equation of the network. Under arbitrary initial conditions, the network runs along the direction of energy decline through the difference equation, and eventually enters the minimum energy point, so as to obtain the solution of the problem. According to the energy requirements defined above, according to the formula: $p_{ij} = x_{ij}^{'} / \sum_{i = 1}^{m} x_{ij}^{'}$ (1)

Applying the difference equation of the network in running the BP neural network, the network is very close to the formula: $e_{j} = - k \sum_{i = 1}^{m} (p_{ij} ln p_{ij})$ (2)

In order to solve the plane problem of a vertex completely, the BP neural network is constructed according to the formula: $w_{j} = g_{i} / \sum_{j = 1}^{n} g_{i}$ (3)

The dynamic properties of the BP neural network model with two neurons simulated by the following delay differential systems are considered: $CI = \frac{X + Y + Z}{\sqrt{X^{2} + Y^{2} + Z^{2}}}$ (4) $T = \frac{M}{R} = \frac{i η_{e} M_{e}}{r}$ (5)

According to the dynamic properties of discrete systems, discrete systems are obtained: $F_{N} = Ac + mg tan φ + F^{'}$ (6)

Considering the time value of two neurons with BP neural network model: $F_{q} = F_{r} = F_{1} + F_{2} = mg (f_{1} + f_{2})$ (7) $T (n) = {\begin{matrix} \frac{W * P}{1 - P * [r mod (1 / P)]}, \\ 0, \end{matrix} \begin{matrix} n \in G \\ others \end{matrix}$ (8)

A sufficient condition is established that the zero solution of the system is globally asymptotically stable without delay:

$\begin{matrix} E_{ch} & = & {lE}_{elec} (\frac{N}{k} - 1) + {lE}_{DA} \frac{N}{k} + {lE}_{elec} \\ + l ξ_{amp} d_{toBS}^{4} \end{matrix}$ (9)

Use the property to simplify the right side of the form: $E_{non - CH} = {lE}_{elec} + l ξ_{fs} d_{toCH}^{2}$ (10)

We consider the linear variational system at the origin: $ρ (x, y) = {\begin{matrix} k / (M_{1} * M_{2}) \\ 0 \end{matrix}, \begin{matrix} (x, y) \in S \\ (x, y) \notin S \end{matrix}$ (11)

Then the trivial solution of the system is obtained:

$\begin{matrix} E [d_{toCH}^{2}] & = & \iint (x^{2} + y^{2}) ρ (x, y) dxdy \\ = & \iint r^{2} ρ (r, θ) drd θ \end{matrix}$ (12)

It is unstable and there exists a frequency in the linear variational system: $E [d_{toCH}^{2}] = \frac{1}{2 π} \frac{M_{1} * M_{2}}{k}$ (13)

Obviously, there is a necessary and sufficient condition for the existence of solutions: $E_{non - CH} = {lE}_{elec} + l ξ_{fs} \frac{1}{2 π} \frac{M_{1} M_{2}}{k}$ (14)

We have studied the difference equation. We mainly discuss the case when the parameters and initial conditions are positive. The main conclusion is the theorem. In this case, we will discuss the dynamic properties of the solution of the difference equation when there is at least one parameter and the initial condition is negative. We find and prove that every solution of the difference equation is unbounded in one subcase and the final period in other subcases. The parallel difference equation model is compared with the traditional difference equation model. By using the same optimization algorithm for the traditional difference equation, we can see that the traditional difference equation model has worse descriptive ability and generalization ability than the difference equation under the same optimization calculation. Approximating derivatives with finite difference is a commonly used method to discretize the continuous time model. However, in many cases, when approximated by forward difference, the “composite” or chaotic and pseudo-stable states which are not available in the continuous-time form often occur after discretization. The main idea of chaotic search: When the particle is in a state of stagnation, first use an initial chaotic variable, then use the mapping function to generate the chaotic sequence, and transform the chaotic variable of each dimension into the value interval of the optimization variable. Record the optimal value found until the maximum algebra of chaotic search is reached, and finally replace one particle randomly to increase its diversity. When using parallel algorithms, it is necessary to introduce vertex-position correspondence constraints, which greatly improves the accuracy of experimental results. It is not necessary to calculate the derivative of the shape function for the integral subdomain completely in the solution domain. For the subdomain where the boundary intersects the boundary of the solution domain, the calculation of the derivative of the shape function is only involved when the intersection is on the displacement boundary. Therefore, it shows great advantages in solving linear and nonlinear boundary value problems. Moreover, in the derivation of the meshless local boundary element method, the boundary conditions can be directly introduced, which reduces the complexity of processing boundary conditions. Different simulations of different tissue levels and abstract levels of the nervous system according to the BP neural network are shown in Table 8 and Fig. 8.

Fig.8

Different simulations at different organizational levels and abstract levels.

Table 8

Different simulations at different organizational levels and abstract levels

	Different levels of organization	Abstract level
Combined model	20.25	19.23
Intelligent model	18.33	18.76
Network level model	21.39	18.77
Neuron hierarchical model	19.32	19.22

The numerical examples show that the barycentric regularized region method can solve the problem of elasticity in irregular regions. The barycentric regularized region method is different from the finite element method in solving elastic problems in irregular regions. It does not need to draw grids, thus greatly reducing the workload. The finite element method can also solve the elastic problem of irregular regions, but its preprocessing is rather complicated. The accuracy of computation depends on the mesh refinement and reduces the work efficiency. In each iteration, the individual will first roll over and gradually move to a new position. If the result of the adaptive position in this particular position is better, then the current individual will follow the force of the last rollover. In the process of learning and training of BP neuralnetwork, the algorithm can make reasonable planning of input data and output data. Therefore, BP neural network is an algorithm with high self-learning ability and adaptability. Then, the corresponding calculation program is compiled according to the proposed coupling theory, and the examples of horizontal uniform tension stress at both ends of the cantilever beam and the infinite plate with a central circular hole acting on the concentrated force at the free end are calculated. By comparing the calculated results with the exact solutions, the suggestions for choosing the regional weight function and parameters of the meshless local boundary element method are given, and the results of displacement and stress curves and error norms of each section are obtained. Finally, it is proved that the proposed algorithm can achieve a certain accuracy, is feasible, and can reduce the computational workload. Planar problems have important applications in circuit cabling, bridge erection, and visualization of gene regulatory networks. For the first time, BP neural network was used to solve the planar problem, which opened up a new method for embedding a large class of graphs such as planar problems. It can be estimated that since the BP neural network itself is a graph, it can definitely solve the embedding problem of any spatial dimension graph, and the key is how to represent the network. In the figure, the number of iterations of the algorithm is represented by the abscissa, and the ordinate represents the value of the objective function. The value of the objective function after running the simulation part and the value of the objective function after running the gradient descent method part. The convergence shown in Fig. 9 is obtained by connecting the points.

Fig.9

Convergence trajectories in plane.

The stability of the network is studied. For a network, stability is a major performance index. The stability in serial mode is called serial stability. Similarly, the stability of parallel mode is called parallel stability. When the BP neural network is stable, its state is called steady state. Due to the introduction of energy function, the neural network directly corresponds to the problem optimization. This kind of work is pioneering. Using BP neural network to optimize calculation is to give the initial estimation point, i. e. the initial condition, in the dynamic system of BP neural network. Then, with the propagation of the network, the corresponding minimum points are found. In this way, a large number of optimization problems can be solved by continuous nets. The computational ability of BP neural network is obviously the following two points: the learning ability of large-scale parallel distributed structure neural network and its generalization ability. Generalization means that BP neural network can produce reasonable output for data which is not in the training set. These two information processing capabilities allow BP neural networks to solve complex problems that are currently not handled. But in practice, BP neural networks cannot be answered individually, they need to be integrated into a coordinated system engineering approach. According to the interpretation of exploration and development, this shows that in the trajectory of particles with fluctuations, their exploration ability is enhanced, and the development does not show better ability. There are many reasons for this, which are related to parameter selection. Different parameter selections can result in particle motion trajectories that can be harmonized or Z-shaped. It may also be caused by a fluctuation term function. The ability of particles to explore is strong, which is conducive to the global optimization of particle swarms. The convergence speed is fast and it is not easy to fall into local optimum. The ability of the particle swarm to develop is weak, indicating the ability of the particle to find an exact solution near the optimal value. The common feature of these networks is to establish an energy function by using the penalty function to optimize the objective function and the constraint condition, and then use this energy function to give a BP neural network dynamic evolution difference equation. The curve embedding map obtained by the BP neural network algorithm in the plane is shown in Fig. 10.

Fig.10

The curve embedding of BP neural network algorithm in plane.

4 Conclusions

Experiments in this paper show that the BP network algorithm is applied to the plane problem of graphs. The straight line embedding in the plane is successfully solved. In the implementation of the algorithm, considering the search speed of the simulation algorithm, the method of limiting the temperature sequence is adopted. So that the experiment can get better results in the effective time than the predecessors. The meshless local boundary element method for elastostatics is deduced. The approximate function is constructed by moving least square approximation, and the basic solution and adjoint solution are introduced as trial function. Finally, the difference equation is established. At the same time, the flexibility of network node function selection allows us to select the corresponding equation type according to the characteristics to be solved. Compared with the radial basis function network, since the orthogonal tomb is selected as the node function, the nodes have no redundancy, and the theoretical analysis and simulation results at the time of calculation indicate this point. The differential equation system is transformed to obtain a new class of nonlinear difference equation systems. Then, the dynamic behavior of the solution is discussed separately for the different parameter values of the system. The section analyzes the properties of solutions in several low-order cases of a class of differential equations with maximum values, especially the final periodicity of the solution when the parameters and initial conditions are both negative.

References

Shi

, Xie

and Wang

, A warpage optimization method for injection molding using artificial neural network with parametric sampling evaluation strategy, International Journal of Advanced Manufacturing Technology 65 (2013), 343–353.

Singh

and Walia

B.S.

, Performance evaluation of nature-inspired algorithms for the design of bored pile foundation by artificial neural networks, Neural Computing & Applications 13 (2016), 1–10.

Silva

S.F.

, Anjos

C.A.R.

, Cavalcanti

R.N.

, et al., Evaluation of extra virgin olive oil stability by artificial neural network, Food Chemistry 179 (2015), 35–43.

Zhang

, You

, Gao

, et al., Automatic gap tracking during high power laser welding based on particle filtering method and BP neural network, International Journal of Advanced Manufacturing Technology 96 (2018), 685–696.

Samtaş

, Measurement and evaluation of surface roughness based on optic system using image processing and artificial neural network, The International Journal of Advanced Manufacturing Technology 73 (2014), 353–364.

You

and Cao

, Study of Liquid Lithium Coolant Interaction Based on BP Neural Network Optimized by Genetic Algorithm, Journal of Fusion Energy 34(4) (2015), 918–924.

Silva

A.S.

, Santos

R.C.D.

, Bottura

F.B.

, et al., Development and evaluation of a prototype for remote voltage monitoring based on artificial neural networks, Engineering Applications of Artificial Intelligence 57 (2017), 50–60.

Sodeifian

, Sajadian

S.A.

and Ardestani

N.S.

, Evaluation of the response surface and hybrid artificial neural network-genetic algorithm methodologies to determine extraction yield of Ferulago angulata through supercritical fluid, Journal of the Taiwan Institute of Chemical Engineers 60 (2016), 165–173.

Betiku

, Okunsolawo

S.S.

, Ajala

S.O.

, et al., Performance evaluation of artificial neural network coupled with generic algorithm and response surface methodology in modeling and optimization of biodiesel production process parameters from shea tree (Vitellaria paradoxa) nut butter, Renewable Energy 76 (2015), 408–417.

10.

Tong-Tong

, Jian

, Chuan

T.U.

, et al., Application of IPSO-BP Neural Network in Water Quality Evaluation for Tianshui Section of Wei River, Environmental Science & Technology 36(8) (2013), 175–181.

11.

Monjezi

, Mehrdanesh

, Malek

, et al., Evaluation of effect of blast design parameters on flyrock using artificial neural networks, Neural Computing & Applications 23(2) (2013), 349–356.

12.

Hirota

, Fukui

, Okamoto

, et al., Evaluation of combinations of sensitization test descriptors for the artificial neural network-based risk assessment model of skin sensitization: Evaluation of the descriptor for ANN risk assessment of skin sensitization, Journal of Applied Toxicology 35(11) (2015), 1333–1347.

13.

Fan

, Fei

, He

, et al., Evaluation of driver fatigue with multi-indicators based on Artificial Neural Network, Iet Intelligent Transport Systems 10(8) (2016), 555–561.

14.

Improved shuffled frog leaping algorithm-based BP neural network and its application in bearing early fault diagnosis, Neural Computing and Applications 27(2) (2016), 375–385.

15.

Shafabakhsh

G.A.

, Talebsafa

, Motamedi

, et al., Analytical evaluation of load movement on flexible pavement and selection of optimum neural network algorithm, KSCE Journal of Civil Engineering 19(6) (2015), 1738–1746.

16.

Mansouri

, Golsefid

M.T.

and Nematbakhsh

, A Hybrid Intrusion Detection System Based on Multilayer Artificial Neural Network and Intelligent Feature Selection, Archives of Medical Research 44(4) (2015), 266–272.

17.

Xue

and Xiaoauthor

, Application of genetic algorithm-based support vector machines for prediction of soil liquefaction, Environmental Earth Sciences 75(10) (2016), 1–11.

18.

Gusev

E.M.

, Ayzel

G.V.

and Nasonova

O.N.

, Runoff evaluation for ungauged watersheds by SWAP model. 1. Application of artificial neural networks, Water Resources 44(2) (2017), 169–179.

19.

Ahmed

, El Sayed

, Gadsden

S.A.

, et al., Automotive Internal-Combustion-Engine Fault Detection and Classification Using Artificial Neural Network Techniques, IEEE Transactions on Vehicular Technology 64(1) (2015), 21–33.

20.

Liu

X.H.

, Shan

M.Y.

and Zhang

L.H.

, Low-carbon supply chain resources allocation based on quantum chaos neural network algorithm and learning effect, Natural Hazards 83(1) (2016), 389–409.

21.

Zhao

, Ding

, Zhao

, et al., Modelling of the hot deformation behaviour of a titanium alloy using constitutive equations and artificial neural network, Computational Materials Science 92(5) (2014), 47–56.

22.

Chang

X.L.

, Mi

X.M.

and Muppala

J.K.

, Performance evaluation of artificial intelligence algorithms for virtual network embedding, Engineering Applications of Artificial Intelligence 26(10) (2013), 2540–2550.

23.

Nair

V.V.

, Dhar

, Kumar

, et al., Artificial neural network based modeling to evaluate methane yield from biogas in a laboratory-scale anaerobic bioreactor, , Bioresource Technology 217 (2016), 90–99.

24.

Kowalski

P.A.

and Łukasik

, Training Neural Networks with Krill Herd Algorithm, Neural Processing Letters 44(1) (2016), 5–17.

25.

Liu

, Huang

H.Z.

and Ling

, Reliability prediction for evolutionary product in the conceptual design phase using neural network-based fuzzy synthetic assessment, International Journal of Systems Science 44(3) (2013), 545–555.

26.

Bayram

, Uzlu

, Kankal

, et al., Modeling stream dissolved oxygen concentration using teaching– learning based optimization algorithm, Environmental Earth Sciences 73(10) (2015), 6565–6576.

27.

Roberto

A.C.

, Assessing rainfall erosivity indices through synthetic precipitation series and artificial neural networks, Anais Da Academia Brasileira De Ciencias 85 (2013), 1523–1535.

28.

Krishnaraj

, Elhoseny

Mohamed

, Thenmozhi

, Selim

Mahmoud M.

and Shankar

, “Deep learning model for real-time image compression in Internet of Underwater Things (IoUT)”, Journal of Real-Time Image Processing (2019). In Press. 10.1007/s11554-019-00879-6.

29.

Elhoseny

Mohamed

and Shankar

, “Optimal Bilateral Filter and Convolutional Neural Network based Denoising Method of Medical Image Measurements”, Measurement 143 (2019), 125–135.

30.

Murugan

B.S.

, Elhoseny

Mohamed

, Shankar

and Uthayakumar

, “Region-based scalable smart system for anomaly detection in pedestrian walkways”, Computers and Electrical Engineering 75 (2019), 146–160.

31.

Lakshmanaprabu

S.K.

, Shankar

, Ashish

, Deepak

, Joel

J.P.C.

, Rodrigues , Pinheiro

Plácido R.

and de Albuquerque

Victor Hugo C.

, “Effective Features to Classify Big Data using Social Internet of Things”, IEEE Access 6 (2018), 24196–24204.