Feedforward Neural Network Optimization Research Articles

An optimal feed-forward artificial neural network model and a new empirical correlation for prediction of the relative viscosity of Al2O3-engine oil nanofluid

This study presents the design of an artificial neural network (ANN) to evaluate and predict the viscosity behavior of Al2O3/10W40 nanofluid at different temperatures, shear rates, and volume fraction of nanoparticles. Nanofluid viscosity ({mu }_{nf}) is evaluated at volume fractions (varphi=0.25% to 2%) and temperature range of 5 to 55 °C. For modeling by ANN, a multilayer perceptron (MLP) network with the Levenberg–Marquardt algorithm (LMA) is used. The main purpose of this study is to model and predict the {mu }_{nf} of Al2O3/10W40 nanofluid through ANN, select the best ANN structure from the set of predicted structures and manage time and cost by predicting the ANN with the least error. To model the ANN, varphi, temperature, and shear rate are considered as input variables, and {mu }_{nf} is considered as output variable. From 400 different ANN structures for Al2O3/10W40 nanofluid, the optimal structure consisting of two hidden layers with the optimal structure of 6 neurons in the first layer and 4 neurons in the second layer is selected. Finally, the R regression coefficient and the MSE are 0.995838 and 4.14469E−08 for the optimal structure, respectively. According to all data, the margin of deviation (MOD) is in the range of less than 2% < MOD < + 2%. Comparison of the three data sets, namely laboratory data, correlation output, and ANN output, shows that the ANN estimates laboratory data more accurately.

Blood Glucose Level Prediction of Diabetic Type 1 Patients Using Nonlinear Autoregressive Neural Networks

Diabetes type 1 is a chronic disease which is increasing at an alarming rate throughout the world. Studies reveal that the complications associated with diabetes can be reduced by proper management of the disease by continuously monitoring and forecasting the blood glucose level of patients. Objective. The prior prediction of blood glucose level is necessary to overcome the lag time for insulin absorption in diabetic type 1 patients. Method. In this research, we use continuous glucose monitoring (CGM) data to predict future blood glucose level using the previous data points. We compare two neural network techniques. We apply the optimal feedforward neural network and then propose optimal nonlinear autoregressive neural networks for blood glucose prediction 15–30 minutes earlier for diabetic type 1 patients. We validate the proposed model with 2 virtual subjects using their 24-hour blood glucose level data. These two case studies have been compiled from AIDA, i.e., the freeware mathematical diabetes simulator. Results. In the prediction horizon (PH) of 15 and 30 minutes, improved results have been shown for minimal inputs for blood glucose level of a particular subject. Root mean square error (RMSE) is used for performance calculation. For the optimal feedforward neural network, the RMSE is 0.9984 and 3.78 ml/dl, and for the optimal nonlinear autoregressive neural network, it reduces the RMSE to 0.60 and 1.12 ml/dl for 15 min and 30 min prediction horizons, respectively, for subject 1. Similarly, for subject 2 for the optimal feedforward neural network, RMSE is 1.43 and 3.51 ml/dl which is improved using the optimal autoregressive neural network to 0.7911 and 1.6756 ml/dl for 15 min and 30 min prediction horizons, respectively. Validation. We further validate our proposed model using UCI machine learning datasets (Abalone and Servo), and it shows improved results on that as well. Conclusion and Future Work. The proposed optimal nonlinear autoregressive neural network model performs better than the feedforward neural network model for these time series data. In the future, we intend to investigate a greater collection of AIDA scenarios and data that are real and influence other factors of BGLs.

Detection of Intruder in Cloud Computing Environment using Swarm Inspired based Neural Network

Cloud computing services offered a resource pool with a wide range of storage for large amounts of data. Cloud services are generally used as a demand-driven private or open data forum, and the increase in use has led to security concerns. Therefore, it is necessary to design an accurate Intrusion Detection System (IDS) to identify the suspected node in the cloud computing environment. This is possible by monitoring network traffic so that the quality of service and performance of the system can be maintained. Several researchers have worked on designing valid IDS with the help of a machine learning approach. A single classification algorithm seems to be impossible to detect intruders with high accuracy. Therefore, a hybrid approach is presented. This approach is a combination of Cuckoo Search. CS as an optimization algorithm and Feed Forward Back Propagation Neural Network (FFBPNN) as a multi-class classification approach. The user's request to access cloud data is collected and essential features are selected using CS as an optimization approach. The selected features are used to train FFBPNN with reduced training time and complexity. The experimental analysis has been performed in terms of precision, recall, F-measure, and accuracy. The evaluated value for parameters i.e., precision (85.5%), recall (86.4%), F-measure (85.9%), and accuracy (86.22%) are observed. At last, the parameters are also compared with the existing approach.

Using spotted hyena optimizer for training feedforward neural networks

Spotted hyena optimizer (SHO) is a novel metaheuristic optimization algorithm based on the behavior of spotted hyena and their collaborative behavior in nature. In this paper, we design a spotted hyena optimizer for training feedforward neural network (FNN), which is regarded as a challenging task since it is easy to fall into local optima. Our objective is to apply metaheuristic optimization algorithm to tackle this problem better than the mathematical and deterministic methods. In order to confirm that using SHO to train FNN is more effective, five classification datasets and three function-approximations are applied to benchmark the performance of the proposed method. The experimental results show that the proposed SHO algorithm for optimization FNN has the best comprehensive performance and has more outstanding performance than other the state-of-the-art metaheuristic algorithms in terms of the performance measures.

Neighborhood centroid opposite-based learning Harris Hawks optimization for training neural networks

The Harris Hawks Optimization Algorithm is a new metaheuristic optimization that simulates the process of Harris Hawk hunting prey (rabbit) in nature. The global and local search processes of the algorithm are performed by simulating several stages of cooperative behavior during hunting. To enhance the performance of this algorithm, in this paper we propose a neighborhood centroid opposite-based learning Harris Hawks optimization algorithm (NCOHHO). The mechanism of applying the neighborhood centroid under the premise of using opposite-based learning technology to improve the performance of the algorithm, the neighborhood centroid is used as a reference point for the generation of the opposite particle, while maintaining the diversity of the population and make full use of the swarm search experience to expand the search range of the reverse solution. Enhancing the probability of finding the optimal solution and the improved algorithm is superior to the original Harris Hawks Optimization algorithm in all aspects. We apply NCOHHO to the training of feed-forward neural network (FNN). To confirm that using NCOHHO to train FNN is more effective, five classification datasets are applied to benchmark the performance of the proposed method. Comprehensive comparison and analysis from the three aspects of mean, variance and classification success rate, the experimental results show that the proposed NCOHHO algorithm for optimization FNN has the best comprehensive performance and has more outstanding performance than other metaheuristic algorithms in terms of the performance measures.

Optimal feed forward neural network based automatic moving vehicle detection system in traffic surveillance system

In intelligent transportation system, traffic surveillance is an important topic. One challenging problem for complex urban traffic surveillance system is robust vehicle detection and tracking. Therefore, in this paper, we develop a two-stage approach for moving vehicle detection system. The proposed system mainly consists of two stages such as hypothesis generation (HG) and hypothesis verification (HV). In the first step, we generate the hypotheses using shadows under vehicles is darker than road region concept. In the second step, we verify hypotheses generated in the first step whether correct or not using optimal feedforward neural network (OFFNN). Here, to extract vehicle features, we utilize two types of histogram orientation gradients descriptors (HOG). In training stage, the histogram orientation gradients features are given to the OFFNN classifier. The weights corresponding FFNN is optimally select using improved grasshopper optimization algorithm (IGOA). The experimental results show that the proposed moving vehicle detection system performs better accuracy compare to other methods.

Optimization of Feedforward Neural Networks Using an Improved Flower Pollination Algorithm for Short-Term Wind Speed Prediction

It is well known that the inherent instability of wind speed may jeopardize the safety and operation of wind power generation, consequently affecting the power dispatch efficiency in power systems. Therefore, accurate short-term wind speed prediction can provide valuable information to solve the wind power grid connection problem. For this reason, the optimization of feedforward (FF) neural networks using an improved flower pollination algorithm is proposed. First of all, the empirical mode decomposition method is devoted to decompose the wind speed sequence into components of different frequencies for decreasing the volatility of the wind speed sequence. Secondly, a back propagation neural network is integrated with the improved flower pollination algorithm to predict the changing trend of each decomposed component. Finally, the predicted values of each component can get into an overlay combination process and achieve the purpose of accurate prediction of wind speed. Compared with major existing neural network models, the performance tests confirm that the average absolute error using the proposed algorithm can be reduced up to 3.67%.

Predicting death by suicide using administrative health care system data: Can feedforward neural network models improve upon logistic regression models?

BackgroundSuicide is a leading cause of death worldwide. With the increasing volume of administrative health care data, there is an opportunity to evaluate whether machine learning models can improve upon statistical models for quantifying suicide risk. ObjectiveTo compare the relative performance of logistic regression and single hidden layer feedforward neural network models that quantify suicide risk with predictors available in administrative health care system data. MethodsThe modeling dataset contained 3548 persons that died by suicide and 35,480 persons that did not die by suicide between 2000 and 2016. 101 predictors were selected, and these were assembled for each of the 40 quarters (10 years) prior to the quarter of death, resulting in 4040 predictors in total for each person. Logistic regression and single hidden layer feedforward neural network model configurations were evaluated using 10-fold cross-validation. ResultsThe optimal feedforward neural network model configuration (AUC: 0.8352) outperformed logistic regression (AUC: 0.8179). LimitationsMany important predictors are not available in administrative data and this likely places a limit on how well prediction models developed with administrative data can perform. ConclusionsAlthough the models developed in this study showed promise, further research is needed to determine the performance limits of statistical and machine learning models that quantify suicide risk, and to develop prediction models optimized for implementation in clinical settings.

Optimized Feedforward Neural Network Training for Efficient Brillouin Frequency Shift Retrieval in Fiber

Artificial neural networks (ANNs) can be used to replace traditional methods in various fields, making signal processing more efficient and meeting the real-time processing requirements of the Internet of Things (IoT). As a special type of ANN, recently the feedforward neural network (FNN) has been used to replace the time-consuming Lorentzian curve fitting (LCF) method in Brillouin optical time-domain analysis (BOTDA) to retrieve the Brillouin frequency shift (BFS), which could be used as the indicator in temperature/strain sensing, etc. However, FNN needs to be re-trained if the generalization ability is not satisfactory, or the frequency scanning step is changing in the experiment. This is a cumbersome and inefficient process. In this paper, FNN only needs to be trained once with the proposed method. 150.62 km BOTDA is built to verify the performance of the trained FNN. Simulation and experimental results show that the proposed method is promising in BOTDA because of its high computational efficiency and wide adaptability.

Large-Scale Feedforward Neural Network Optimization by a Self-Adaptive Strategy and Parameter Based Particle Swarm Optimization

Feedforward neural network (FNN) is one of the most widely used and fastest-developed artificial neural networks. Much evolutionary computation (EC) methods have been used to optimize the weights of FNN. However, as the dimension of datasets increases, the number of weights also increases dramatically. On high-dimensional datasets, if EC methods are used directly to optimize the weights of FNN, it is impossible to obtain the optimal weights of the FNN by EC methods in an acceptable time. Feature selection is a method that can effectively reduce the computational complexity of FNN by reducing irrelevant and redundant features. It may be practical to optimize the FNN by EC methods if we first employ the feature selection for the large-scale datasets. In this paper, we present a self-adaptive parameter and strategy-based particle swarm optimization (SPS-PSO) algorithm to optimize FNN with feature selection. First, we propose an optimization model for FNN by transforming the designing of FNN into a weights optimization problem. Simultaneously, we present a feature selection optimization model. Second, we present an SPS-PSO algorithm. In this algorithm, we use the parameter and strategy self-adaptive mechanism. In addition, five candidate solution generating strategies (CSGS) are used. The experiments are divided into two groups. In the first group, SPS-PSO and three other EC methods are used to directly optimize the weights of FNN on eight datasets without any modification. In the second group, we first employ SPS-PSO-based feature selection on the original datasets and obtain eight relatively smaller datasets with the k-nearest neighbor (KNN) which is used as the evaluation function for saving time. Then, we use the new datasets as the inputs for FNN. We optimize the weights of FNN again by SPS-PSO and three other EC methods to investigate whether we can get similar or even better classification accuracy by comparing the results with that of the first group. The experimental results show that SPS-PSO has the advantage in optimizing the weights of FNN compared with the other EC methods. Meanwhile, the SPS-PSO-based feature selection can reduce the solution size and computational complexity while ensuring the classification accuracy when it is used to preprocess the datasets for FNN. In this method, a solution with an originally higher than 700 000 dimensions can be even reduced to hundreds of dimensions.

Metaheuristic design of feedforward neural networks: A review of two decades of research

Over the past two decades, the feedforward neural network (FNN) optimization has been a key interest among the researchers and practitioners of multiple disciplines. The FNN optimization is often viewed from the various perspectives: the optimization of weights, network architecture, activation nodes, learning parameters, learning environment, etc. Researchers adopted such different viewpoints mainly to improve the FNN's generalization ability. The gradient-descent algorithm such as backpropagation has been widely applied to optimize the FNNs. Its success is evident from the FNN's application to numerous real-world problems. However, due to the limitations of the gradient-based optimization methods, the metaheuristic algorithms including the evolutionary algorithms, swarm intelligence, etc., are still being widely explored by the researchers aiming to obtain generalized FNN for a given problem. This article attempts to summarize a broad spectrum of FNN optimization methodologies including conventional and metaheuristic approaches. This article also tries to connect various research directions emerged out of the FNN optimization practices, such as evolving neural network (NN), cooperative coevolution NN, complex-valued NN, deep learning, extreme learning machine, quantum NN, etc. Additionally, it provides interesting research challenges for future research to cope-up with the present information processing era.

QSPR prediction of the hydroxyl radical rate constant of water contaminants

In advanced oxidation processes (AOPs), the aqueous hydroxyl radical (HO) acts as a strong oxidant to react with organic contaminants. The hydroxyl radical rate constant (kHO) is important for evaluating and modelling of the AOPs. In this study, quantitative structure-property relationship (QSPR) method is applied to model the hydroxyl radical rate constant for a diverse dataset of 457 water contaminants from 27 various chemical classes. The constricted binary particle swarm optimization and multiple-linear regression (BPSO-MLR) are used to obtain the best model with eight theoretical descriptors. An optimized feed forward neural network (FFNN) is developed to investigate the complex performance of the selected molecular parameters with kHO. Although the FFNN prediction results are more accurate than those obtained using BPSO-MLR, the application of the latter is much more convenient. Various internal and external validation techniques indicate that the obtained models could predict the logarithmic hydroxyl radical rate constants of a large number of water contaminants with less than 4% absolute relative error. Finally, the above-mentioned proposed models are compared to those reported earlier and the structural factors contributing to the AOP degradation efficiency are discussed.

Flood-routing modeling with neural network optimized by social-based algorithm

Forecasting and operational routing flood requires accurate forecasts on proper feed time, to be able to issue suitable warnings and take suitable emergency actions. Flood-routing problem is one of the most complicated matters in hydraulics of open channels and river engineering. Flood routing is the process of computing the progressive time and shape of a flood wave at successive points along a river. To get an approximate solution of the flood-routing problem, different techniques are used. This paper describes an approach to train artificial neural network (ANN) using social-based algorithm (SBA). The approach illustrates feed-forward neural network optimization for the flood-routing problem of Kheir Abad River called FF-SBA. To this end, the number and effective time lag of input data in ANN models are initially determined by means of linear correlation between input and output time series; subsequently, the weights of the feed-forward network is optimized by SBA. Optimization algorithms and statistical models like Genetic Algorithm and linear regression are compared to FF-SBA. Compared to the results of optimization algorithms and statistical models, the FF-SBA model for the Kheir Abad River in Iran shows more flexibility and accuracy.

Fruit classification using computer vision and feedforward neural network

Fruit classification is a difficult challenge due to the numerous types of fruits. In order to recognize fruits more accurately, we proposed a hybrid classification method based on fitness-scaled chaotic artificial bee colony (FSCABC) algorithm and feedforward neural network (FNN). First, fruits images were acquired by a digital camera, and then the background of each image were removed by split-and-merge algorithm. We used a square window to capture the fruits, and download the square images to 256×256. Second, the color histogram, texture and shape features of each fruit image were extracted to compose a feature space. Third, principal component analysis was used to reduce the dimensions of the feature space. Finally, the reduced features were sent to the FNN, the weights/biases of which were trained by the FSCABC algorithm. We also used a stratified K-fold cross validation technique to enhance the generation ability of FNN. The experimental results of the 1653 color fruit images from the 18 categories demonstrated that the FSCABC–FNN achieved a classification accuracy of 89.1%. The classification accuracy was higher than Genetic Algorithm–FNN (GA–FNN) with 84.8%, Particle Swarm Optimization–FNN (PSO–FNN) with 87.9%, ABC–FNN with 85.4%, and kernel support vector machine with 88.2%. Therefore, the FSCABC–FNN was seen to be effective in classifying fruits.

Performansi Algoritma CODEQ dalam Penyelesaian Vehicle Routing Problem

Genetic Algorithm, Tabu Search, Simulated Annealing, and Ant Colony Optimization showed a good performance in solving vehicle routing problem. However, the generated solution of those algorithms was changeable regarding on the input parameter of each algorithm. CODEQ is a new, parameter free meta-heuristic algorithm that had been successfully used to solve constrained optimization problems, integer programming, and feed-forward neural network. The purpose of this research are improving CODEQ algorithm to solve vehicle routing problem and testing the performance of the improved algorithm. CODEQ algorithm is started with population initiation as initial solution, generated of mutant vector for each parent in every iteration, replacement of parent by mutant when fitness function value of mutant is better than parent’s, generated of new vector for each iteration based on opposition value or chaos principle, replacement of worst solution by new vector when fitness function value of new vector is better, iteration ceasing when stooping criterion is achieved, and sub-tour determination based on vehicle capacity constraint. The result showed that the average deviation of the best-known and the best-test value is 6.35%. Therefore, CODEQ algorithm is good in solving vehicle routing problem.

Artificial neural network weights optimization based on social-based algorithm to realize sediment over the river

Predictions of sediment load are required in a wide spectrum of problems such as design of the dead volume of a dam, sediment transport in the river, design of stable channels, and dredging needs. Researchers have used the regression between sediment concentration and water discharge. Such relationships are obtained through the application of regression analysis in many studies. Unfortunately, in the classical regression approach to determine sediment concentration---water discharge relationships, internal uncertainties are not taken explicitly into consideration. Therefore, researchers look for non-linear methods to estimate sediment load such as artificial neural network (ANN) methods to solve non-linear problems. The main purpose of this paper is to optimize the ANN connection weights with novel social-based algorithm (SBA) to realize the sediment in Maroon river. The SBA tries to capture several people in different types of countries. They try to reach high levels. The approach illustrated feed-forward neural network optimization for sediment estimation of four Maroon river stations in Iran, which was called FF-SBA. Three inputs were presented in each station: length of river, discharge and debit. Sediment parameter on the same station is measured as the output parameter. Results of optimization algorithms such as the genetic algorithm, particle swarm optimization and imperialist competitive algorithm were compared with the SBA results, and it was found that the FF-SBA model exhibited more capability, flexibility, and accuracy in sediment training, testing, and forecasting steps for the Maroon river in Iran.

Phase equilibria modeling of binary systems containing ethanol using optimal feedforward neural network

A comprehensive understanding of vapor liquid equilibrium (VLE) data is one of the most important information for designing and modeling of process equipment. Because, it is not always possible to completely carry out experiments at all of the possible operational temperatures and pressures range, generalized thermodynamic models, e.g. equations of state are constructed for computing of required VLE data. In this work, artificial neural network (ANN) was used to derive predictive models of bubble point pressure and vapor phase composition of binary ethanol (C2H5OH) mixtures. In the neural network model, it is assumed that the considered VLE data depend on critical properties, acentric factor, normal boiling point, liquid phase composition of the solutes, and temperature. The proposed ANN model has been constructed and trained with VLE experimental data of nine different binary systems containing C2H5OH collected from various literatures. Optimal configuration of the ANN model has been determined using minimizing the average absolute relative deviation percent (%AARD), mean square errors (MSE) and the maximizing the correlation coefficient (R2) between observed and predicted VLE data with the ANN model. By using this procedure a two-layer ANN model with twenty-three hidden neuron has been found as an optimal topology. The accuracy of our optimal two layers ANN model has been compared with the Peng–Robinson cubic equation combined with Wong–Sandler (WS) mixing rules including a Van Laar (VL) model for the excess Gibbs free energy. Comparison with available literatures data and Peng–Robinson equation of state confirm that the present ANN model is more accurate and superior than the other published works. The sensitivity errors analysis clarify that our ANN model could predict vapor phase composition and bubble point pressure of all of the nine binary ethanol systems with %AARD of 1.52% and 2.59% respectively. The study demonstrates that the neural network model is a good alternative method for the estimation of VLE properties of the binary system containing C2H5OH.

A neuro-evolutive technique applied for predicting the liquid crystalline property of some organic compounds

A simple self-adaptive version of the differential evolution algorithm was applied for simultaneous architectural and parametric optimization of feed-forward neural networks, used to classify the crystalline liquid property of a series of organic compounds. The developed optimization methodology was called self-adaptive differential evolution neural network (SADE-NN) and has the following characteristics: the base vector used is chosen as the best individual in the current population, two differential terms participate in the mutation process, the crossover type is binomial, a simple self-adaptive mechanism is employed to determine the near-optimal control parameters of the algorithm, and the integration of the neural network into the differential evolution algorithm is performed using a direct encoding scheme. It was found that a network with one hidden layer is able to make accurate predictions, indicating that the proposed methodology is efficient and, owing to its flexibility, it can be applied to a large range of problems.

Input vector optimization of feed-forward neural networks for fitting ab initio potential-energy databases

The variation in the fitting accuracy of neural networks (NNs) when used to fit databases comprising potential energies obtained from ab initio electronic structure calculations is investigated as a function of the number and nature of the elements employed in the input vector to the NN. Ab initio databases for H(2)O(2), HONO, Si(5), and H(2)C[Double Bond]CHBr were employed in the investigations. These systems were chosen so as to include four-, five-, and six-body systems containing first, second, third, and fourth row elements with a wide variety of chemical bonding and whose conformations cover a wide range of structures that occur under high-energy machining conditions and in chemical reactions involving cis-trans isomerizations, six different types of two-center bond ruptures, and two different three-center dissociation reactions. The ab initio databases for these systems were obtained using density functional theory/B3LYP, MP2, and MP4 methods with extended basis sets. A total of 31 input vectors were investigated. In each case, the elements of the input vector were chosen from interatomic distances, inverse powers of the interatomic distance, three-body angles, and dihedral angles. Both redundant and nonredundant input vectors were investigated. The results show that among all the input vectors investigated, the set employed in the Z-matrix specification of the molecular configurations in the electronic structure calculations gave the lowest NN fitting accuracy for both Si(5) and vinyl bromide. The underlying reason for this result appears to be the discontinuity present in the dihedral angle for planar geometries. The use of trigometric functions of the angles as input elements produced significantly improved fitting accuracy as this choice eliminates the discontinuity. The most accurate fitting was obtained when the elements of the input vector were taken to have the form R(ij) (-n), where the R(ij) are the interatomic distances. When the Levenberg-Marquardt procedure was modified to permit error minimization with respect to n as well as the weights and biases of the NN, the optimum powers were all found to lie in the range of 1.625-2.38 for the four systems studied. No statistically significant increase in fitting accuracy was achieved for vinyl bromide when a different value of n was employed and optimized for each bond type. The rate of change in the fitting error with n is found to be very small when n is near its optimum value. Consequently, good fitting accuracy can be achieved by employing a value of n in the middle of the above range. The use of interparticle distances as elements of the input vector rather than the Z-matrix variables employed in the electronic structure calculations is found to reduce the rms fitting errors by factors of 8.86 and 1.67 for Si(5) and vinyl bromide, respectively. If the interparticle distances are replaced with input elements of the form R(ij) (-n) with n optimized, further reductions in the rms error by a factor of 1.31 to 2.83 for the four systems investigated are obtained. A major advantage of using this procedure to increase NN fitting accuracy rather than increasing the number of neurons or the size of the database is that the required increase in computational effort is very small.

Meta-learning approach to neural network optimization

Optimization of neural network topology, weights and neuron transfer functions for given data set and problem is not an easy task. In this article, we focus primarily on building optimal feed-forward neural network classifier for i.i.d. data sets. We apply meta-learning principles to the neural network structure and function optimization. We show that diversity promotion, ensembling, self-organization and induction are beneficial for the problem. We combine several different neuron types trained by various optimization algorithms to build a supervised feed-forward neural network called Group of Adaptive Models Evolution (GAME). The approach was tested on a large number of benchmark data sets. The experiments show that the combination of different optimization algorithms in the network is the best choice when the performance is averaged over several real-world problems.