# A Comprehensive Comparison of the Performance of Metaheuristic Algorithms in Neural Network Training for Nonlinear System Identification

## Abstract

**:**

## 1. Introduction

- In this study, the success cases of metaheuristic algorithms are revealed. As it is known, the identification of nonlinear systems is one of the difficult problems. Which metaheuristic algorithm is more effective in solving this problem? There is no clear answer to this in the literature. Therefore, the performances of sixteen metaheuristic algorithms are compared in this study. Although this comparison is valid only for nonlinear systems, it will be a reference for different types of problems. It is thought that this study will guide the future studies of many researchers in different fields. It is an important innovation that it is one of the first studies to compare the aforementioned sixteen metaheuristic algorithms. At the same time, the results make a significant contribution to the literature.
- The success of ANNs is directly related to the training process. In particular, metaheuristic algorithms have been used in ANN training. Some metaheuristic algorithms are heavily used, while others are more limited. However, there is no clear information about what the most effective metaheuristic-based ANN training algorithms are. This study is one of the first studies to identify the most effective metaheuristic-based ANN training algorithms. Therefore, it is innovative. The use of a training algorithm without relying on any analysis in solving a problem may be insufficient for success. This study gives an idea to the literature about which metaheuristic algorithms can be used in ANN training.
- The importance of nonlinear systems has been emphasized above. This study is one of the most comprehensive studies in terms of the identification of nonlinear systems. It is innovative in terms of the technique used. This study was carried out on nonlinear test systems. Many systems in the real world exhibit nonlinear behavior. Therefore, this study will be a guide for the solution of many problems and will make important contributions to the literature.

## 2. Related Works

## 3. Artificial Neural Networks

## 4. Simulation Results

## 5. Discussion

## 6. Conclusions

- The performances of metaheuristic algorithms were examined in three groups. BBO, MFO, ABC, TLBO, and MVO were in Group 1. The most effective results were obtained with these algorithms. The algorithms CS, SSA, PSO, FPA, and SCA were in Group 2. Compared to Group 3, acceptable results were achieved in Group 2. The algorithms WOA, BAT, HS, BSA, BA, and JAYA were included in Group 3. It was seen that these algorithms were ineffective in solving the related problem. All rankings were valid within the stated limitations of the study.
- The type of nonlinear systems, network structures, and training/testing processes affected the performance of the algorithms.
- Nonlinear system identification is a difficult problem due to its structure. It was determined that most algorithms that were successful in solving numerical optimization problems cannot show the same resistance in system identification.
- The speed of convergence is also an important criterion. The speed of convergence was very good, as was the solution quality of BBO.

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

ABC | Artificial bee colony |

BAT | Bat algorithm |

CS | Cuckoo search |

FPA | Flower pollination algorithm |

PSO | Particle swarm optimization |

TLBO | Teaching–learning-based optimization |

JAYA | Jaya algorithm |

SCA | Sine-cosine algorithm |

BBO | Biogeography-based optimization |

WOA | Whale optimization algorithm |

BSA | Bird swarm algorithm |

HS | Harmony search |

SSA | Salp swarm algorithm |

BA | Bee algorithm |

MFO | Moth-flame optimization |

MVO | Multi-verse optimizer |

ANNs | Artificial neural networks |

FFNN | Feedforward neural network |

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**Figure 2.**A block diagram of the identification of stationary dynamic systems with the proposed approaches.

**Figure 3.**A block diagram of the identification of non-stationary dynamic systems with the proposed approaches.

System | Equation | Inputs | Output | Number of Training/Test Data |
---|---|---|---|---|

${S}_{1}$ | $y=2sin\left(\pi {x}_{1}\right)$ | ${x}_{1}$ | y | 80\20 |

${S}_{2}$ | $y=10.391\left\{\left({x}_{1}-0.4\right)\left({x}_{2}-0.6\right)+0.36\right\}$ | ${x}_{1},{x}_{2}$ | y | 80\20 |

${S}_{3}$ | $y=tanh\left({x}_{1}+{x}_{2}-11\right)$ | ${x}_{1},{x}_{2}$ | y | 80\20 |

${S}_{4}$ | $y=1+{x}_{1}^{0.5}+{x}_{2}^{-1}+{x}_{3}^{-1.5}$ | ${x}_{1},{x}_{2},{x}_{3}$ | y | 173\43 |

${D}_{1}$ | $y\left(k+1\right)=\frac{y\left(k\right)y\left(k-1\right)\left[y\left(k\right)+2.5\right]}{1+{\left[y\left(k\right)\right]}^{2}+{\left[y\left(k-1\right)\right]}^{2}}+u\left(k\right)$ | $y\left(k\right),y\left(k-1\right),u\left(k\right)$ | $y\left(k+1\right)$ | 200\50 |

${D}_{2}$ | $y\left(k+1\right)=\frac{y\left(k\right)}{1+y\left(k-1\right)}+u{\left(k\right)}^{3}$ | $y\left(k\right),y\left(k-1\right),u\left(k\right)$ | $y\left(k+1\right)$ | 200\50 |

Algorithm | Network Structure | Train | Test | ||
---|---|---|---|---|---|

Mean | SD. | Mean | SD. | ||

ABC | 1-5-1 | $1.6\times {10}^{-3}$ | $4.2\times {10}^{-4}$ | $2.4\times {10}^{-3}$ | $6.2\times {10}^{-4}$ |

1-10-1 | $8.2\times {10}^{-4}$ | $3.4\times {10}^{-4}$ | $1.6\times {10}^{-3}$ | $6.8\times {10}^{-4}$ | |

1-15-1 | $7.1\times {10}^{-4}$ | $2.4\times {10}^{-4}$ | $1.5\times {10}^{-3}$ | $5.3\times {10}^{-4}$ | |

BAT | 1-5-1 | $2.6\times {10}^{-2}$ | $3.6\times {10}^{-2}$ | $3.0\times {10}^{-2}$ | $3.8\times {10}^{-2}$ |

1-10-1 | $2.2\times {10}^{-2}$ | $3.9\times {10}^{-2}$ | $2.3\times {10}^{-2}$ | $4.0\times {10}^{-2}$ | |

1-15-1 | $2.5\times {10}^{-2}$ | $4.6\times {10}^{-2}$ | $2.7\times {10}^{-2}$ | $5.0\times {10}^{-2}$ | |

CS | 1-5-1 | $1.3\times {10}^{-3}$ | $2.2\times {10}^{-4}$ | $2.2\times {10}^{-3}$ | $5.1\times {10}^{-4}$ |

1-10-1 | $8.0\times {10}^{-4}$ | $2.4\times {10}^{-4}$ | $1.6\times {10}^{-3}$ | $5.5\times {10}^{-4}$ | |

1-15-1 | $7.3\times {10}^{-4}$ | $1.7\times {10}^{-4}$ | $1.8\times {10}^{-3}$ | $5.8\times {10}^{-4}$ | |

FPA | 1-5-1 | $1.4\times {10}^{-3}$ | $2.6\times {10}^{-4}$ | $2.3\times {10}^{-3}$ | $5.6\times {10}^{-4}$ |

1-10-1 | $9.2\times {10}^{-4}$ | $2.9\times {10}^{-4}$ | $1.7\times {10}^{-3}$ | $5.3\times {10}^{-4}$ | |

1-15-1 | $8.0\times {10}^{-4}$ | $2.7\times {10}^{-4}$ | $1.7\times {10}^{-3}$ | $7.1\times {10}^{-4}$ | |

PSO | 1-5-1 | $1.7\times {10}^{-3}$ | $4.6\times {10}^{-4}$ | $1.7\times {10}^{-3}$ | $4.6\times {10}^{-4}$ |

1-10-1 | $2.1\times {10}^{-3}$ | $6.2\times {10}^{-4}$ | $2.1\times {10}^{-3}$ | $6.2\times {10}^{-4}$ | |

1-15-1 | $2.1\times {10}^{-3}$ | $6.0\times {10}^{-4}$ | $2.1\times {10}^{-3}$ | $6.0\times {10}^{-4}$ | |

JAYA | 1-5-1 | $1.5\times {10}^{-2}$ | $8.3\times {10}^{-3}$ | $1.6\times {10}^{-2}$ | $9.0\times {10}^{-3}$ |

1-10-1 | $1.3\times {10}^{-2}$ | $6.5\times {10}^{-3}$ | $1.4\times {10}^{-2}$ | $7.4\times {10}^{-3}$ | |

1-15-1 | $1.4\times {10}^{-2}$ | $8.3\times {10}^{-3}$ | $1.4\times {10}^{-2}$ | $9.3\times {10}^{-3}$ | |

TLBO | 1-5-1 | $1.1\times {10}^{-3}$ | $6.7\times {10}^{-4}$ | $2.0\times {10}^{-3}$ | $1.1\times {10}^{-3}$ |

1-10-1 | $9.7\times {10}^{-4}$ | $6.5\times {10}^{-4}$ | $1.6\times {10}^{-3}$ | $7.7\times {10}^{-4}$ | |

1-15-1 | $9.9\times {10}^{-4}$ | $5.1\times {10}^{-4}$ | $1.6\times {10}^{-3}$ | $6.5\times {10}^{-4}$ | |

SCA | 1-5-1 | $4.5\times {10}^{-3}$ | $8.2\times {10}^{-4}$ | $5.6\times {10}^{-3}$ | $1.5\times {10}^{-3}$ |

1-10-1 | $5.0\times {10}^{-3}$ | $1.6\times {10}^{-3}$ | $5.9\times {10}^{-3}$ | $2.0\times {10}^{-3}$ | |

1-15-1 | $4.9\times {10}^{-3}$ | $1.5\times {10}^{-3}$ | $5.9\times {10}^{-3}$ | $2.8\times {10}^{-3}$ | |

BBO | 1-5-1 | $9.2\times {10}^{-4}$ | $3.9\times {10}^{-4}$ | $1.9\times {10}^{-3}$ | $6.5\times {10}^{-4}$ |

1-10-1 | $6.9\times {10}^{-4}$ | $2.5\times {10}^{-4}$ | $1.6\times {10}^{-3}$ | $4.9\times {10}^{-4}$ | |

1-15-1 | $5.3\times {10}^{-4}$ | $2.3\times {10}^{-4}$ | $1.5\times {10}^{-3}$ | $5.2\times {10}^{-4}$ | |

WOA | 1-5-1 | $7.8\times {10}^{-3}$ | $9.4\times {10}^{-3}$ | $8.3\times {10}^{-3}$ | $6.9\times {10}^{-3}$ |

1-10-1 | $3.8\times {10}^{-3}$ | $2.1\times {10}^{-3}$ | $5.4\times {10}^{-3}$ | $3.2\times {10}^{-3}$ | |

1-15-1 | $7.4\times {10}^{-3}$ | $9.7\times {10}^{-3}$ | $8.4\times {10}^{-3}$ | $8.6\times {10}^{-3}$ | |

BSA | 1-5-1 | $3.0\times {10}^{-2}$ | $2.6\times {10}^{-2}$ | $3.3\times {10}^{-2}$ | $2.9\times {10}^{-2}$ |

1-10-1 | $9.2\times {10}^{-3}$ | $1.8\times {10}^{-2}$ | $1.0\times {10}^{-2}$ | $1.7\times {10}^{-2}$ | |

1-15-1 | $2.7\times {10}^{-2}$ | $3.5\times {10}^{-2}$ | $2.8\times {10}^{-2}$ | $3.5\times {10}^{-2}$ | |

HS | 1-5-1 | $1.1\times {10}^{-2}$ | $1.2\times {10}^{-2}$ | $1.2\times {10}^{-2}$ | $1.2\times {10}^{-2}$ |

1-10-1 | $2.4\times {10}^{-2}$ | $1.6\times {10}^{-2}$ | $2.7\times {10}^{-2}$ | $2.0\times {10}^{-2}$ | |

1-15-1 | $2.7\times {10}^{-2}$ | $1.2\times {10}^{-2}$ | $3.0\times {10}^{-2}$ | $1.3\times {10}^{-2}$ | |

BA | 1-5-1 | $9.0\times {10}^{-3}$ | $3.5\times {10}^{-3}$ | $1.1\times {10}^{-2}$ | $4.8\times {10}^{-3}$ |

1-10-1 | $6.4\times {10}^{-3}$ | $2.3\times {10}^{-3}$ | $7.2\times {10}^{-3}$ | $2.8\times {10}^{-3}$ | |

1-15-1 | $6.2\times {10}^{-3}$ | $2.3\times {10}^{-3}$ | $7.8\times {10}^{-3}$ | $3.6\times {10}^{-3}$ | |

MVO | 1-5-1 | $1.2\times {10}^{-3}$ | $5.6\times {10}^{-4}$ | $2.4\times {10}^{-3}$ | $9.7\times {10}^{-4}$ |

1-10-1 | $9.0\times {10}^{-4}$ | $5.9\times {10}^{-4}$ | $2.0\times {10}^{-3}$ | $8.7\times {10}^{-4}$ | |

1-15-1 | $4.9\times {10}^{-4}$ | $2.6\times {10}^{-4}$ | $1.5\times {10}^{-3}$ | $5.5\times {10}^{-4}$ | |

MFO | 1-5-1 | $1.8\times {10}^{-3}$ | $1.4\times {10}^{-3}$ | $2.6\times {10}^{-3}$ | $1.5\times {10}^{-3}$ |

1-10-1 | $8.2\times {10}^{-4}$ | $8.4\times {10}^{-4}$ | $1.7\times {10}^{-3}$ | $1.1\times {10}^{-3}$ | |

1-15-1 | $3.5\times {10}^{-4}$ | $2.7\times {10}^{-4}$ | $1.4\times {10}^{-3}$ | $6.9\times {10}^{-4}$ | |

SSA | 1-5-1 | $1.4\times {10}^{-3}$ | $5.8\times {10}^{-4}$ | $2.7\times {10}^{-3}$ | $9.0\times {10}^{-4}$ |

1-10-1 | $9.7\times {10}^{-4}$ | $6.1\times {10}^{-4}$ | $2.0\times {10}^{-3}$ | $8.3\times {10}^{-4}$ | |

1-15-1 | $7.9\times {10}^{-4}$ | $5.4\times {10}^{-4}$ | $1.8\times {10}^{-3}$ | $6.7\times {10}^{-4}$ |

Algorithm | Network Structure | Train | Test | ||
---|---|---|---|---|---|

Mean | SD. | Mean | SD. | ||

ABC | 2-5-1 | $9.1\times {10}^{-4}$ | $2.9\times {10}^{-4}$ | $5.1\times {10}^{-3}$ | $2.3\times {10}^{-3}$ |

2-10-1 | $5.4\times {10}^{-4}$ | $1.7\times {10}^{-4}$ | $4.1\times {10}^{-3}$ | $2.5\times {10}^{-3}$ | |

2-15-1 | $7.1\times {10}^{-4}$ | $3.9\times {10}^{-4}$ | $5.7\times {10}^{-3}$ | $4.1\times {10}^{-3}$ | |

BAT | 2-5-1 | $9.9\times {10}^{-3}$ | $1.3\times {10}^{-2}$ | $2.2\times {10}^{-2}$ | $1.9\times {10}^{-2}$ |

2-10-1 | $1.2\times {10}^{-2}$ | $3.2\times {10}^{-2}$ | $2.1\times {10}^{-2}$ | $3.4\times {10}^{-2}$ | |

2-15-1 | $8.9\times {10}^{-3}$ | $2.8\times {10}^{-2}$ | $1.7\times {10}^{-2}$ | $3.3\times {10}^{-2}$ | |

CS | 2-5-1 | $1.3\times {10}^{-3}$ | $3.8\times {10}^{-4}$ | $6.0\times {10}^{-3}$ | $3.3\times {10}^{-3}$ |

2-10-1 | $1.0\times {10}^{-3}$ | $2.6\times {10}^{-4}$ | $4.4\times {10}^{-3}$ | $2.1\times {10}^{-3}$ | |

2-15-1 | $9.8\times {10}^{-4}$ | $2.5\times {10}^{-4}$ | $5.1\times {10}^{-3}$ | $2.8\times {10}^{-3}$ | |

FPA | 2-5-1 | $1.6\times {10}^{-3}$ | $5.7\times {10}^{-4}$ | $6.4\times {10}^{-3}$ | $3.1\times {10}^{-3}$ |

2-10-1 | $1.2\times {10}^{-3}$ | $3.2\times {10}^{-4}$ | $5.8\times {10}^{-3}$ | $2.9\times {10}^{-3}$ | |

2-15-1 | $1.1\times {10}^{-3}$ | $2.8\times {10}^{-4}$ | $6.0\times {10}^{-3}$ | $3.1\times {10}^{-3}$ | |

PSO | 2-5-1 | $1.8\times {10}^{-3}$ | $1.3\times {10}^{-3}$ | $1.8\times {10}^{-3}$ | $1.3\times {10}^{-3}$ |

2-10-1 | $2.6\times {10}^{-3}$ | $1.2\times {10}^{-3}$ | $2.6\times {10}^{-3}$ | $1.2\times {10}^{-3}$ | |

2-15-1 | $3.5\times {10}^{-3}$ | $1.4\times {10}^{-3}$ | $3.5\times {10}^{-3}$ | $1.4\times {10}^{-3}$ | |

JAYA | 2-5-1 | $1.6\times {10}^{-2}$ | $5.8\times {10}^{-3}$ | $2.8\times {10}^{-2}$ | $1.0\times {10}^{-2}$ |

2-10-1 | $2.3\times {10}^{-2}$ | $8.9\times {10}^{-3}$ | $4.4\times {10}^{-2}$ | $2.3\times {10}^{-2}$ | |

2-15-1 | $3.2\times {10}^{-2}$ | $1.3\times {10}^{-2}$ | $5.6\times {10}^{-2}$ | $2.8\times {10}^{-2}$ | |

TLBO | 2-5-1 | $1.2\times {10}^{-3}$ | $9.7\times {10}^{-4}$ | $5.8\times {10}^{-3}$ | $3.8\times {10}^{-3}$ |

2-10-1 | $6.2\times {10}^{-4}$ | $5.6\times {10}^{-4}$ | $4.0\times {10}^{-3}$ | $2.2\times {10}^{-3}$ | |

2-15-1 | $8.0\times {10}^{-4}$ | $5.8\times {10}^{-4}$ | $4.5\times {10}^{-3}$ | $2.8\times {10}^{-3}$ | |

SCA | 2-5-1 | $7.5\times {10}^{-3}$ | $1.3\times {10}^{-3}$ | $1.7\times {10}^{-2}$ | $6.7\times {10}^{-3}$ |

2-10-1 | $9.2\times {10}^{-3}$ | $2.4\times {10}^{-3}$ | $2.2\times {10}^{-2}$ | $6.2\times {10}^{-3}$ | |

2-15-1 | $1.1\times {10}^{-2}$ | $3.3\times {10}^{-3}$ | $2.2\times {10}^{-2}$ | $7.3\times {10}^{-3}$ | |

BBO | 2-5-1 | $1.0\times {10}^{-3}$ | $1.2\times {10}^{-3}$ | $6.6\times {10}^{-3}$ | $4.5\times {10}^{-3}$ |

2-10-1 | $5.7\times {10}^{-4}$ | $4.9\times {10}^{-4}$ | $4.3\times {10}^{-3}$ | $2.4\times {10}^{-3}$ | |

2-15-1 | $4.7\times {10}^{-4}$ | $2.8\times {10}^{-4}$ | $4.3\times {10}^{-3}$ | $2.1\times {10}^{-3}$ | |

WOA | 2-5-1 | $9.1\times {10}^{-3}$ | $5.3\times {10}^{-3}$ | $2.4\times {10}^{-2}$ | $7.6\times {10}^{-3}$ |

2-10-1 | $9.9\times {10}^{-3}$ | $5.5\times {10}^{-3}$ | $2.2\times {10}^{-2}$ | $7.7\times {10}^{-3}$ | |

2-15-1 | $9.7\times {10}^{-3}$ | $5.9\times {10}^{-3}$ | $2.1\times {10}^{-2}$ | $6.4\times {10}^{-3}$ | |

BSA | 2-5-1 | $1.3\times {10}^{-2}$ | $6.9\times {10}^{-3}$ | $2.6\times {10}^{-2}$ | $8.6\times {10}^{-3}$ |

2-10-1 | $8.6\times {10}^{-3}$ | $6.0\times {10}^{-3}$ | $1.8\times {10}^{-2}$ | $9.2\times {10}^{-3}$ | |

2-15-1 | $1.4\times {10}^{-2}$ | $8.1\times {10}^{-3}$ | $2.6\times {10}^{-2}$ | $9.5\times {10}^{-3}$ | |

HS | 2-5-1 | $4.9\times {10}^{-3}$ | $3.7\times {10}^{-3}$ | $1.2\times {10}^{-2}$ | $5.6\times {10}^{-3}$ |

2-10-1 | $2.9\times {10}^{-2}$ | $1.5\times {10}^{-2}$ | $4.8\times {10}^{-2}$ | $2.9\times {10}^{-2}$ | |

2-15-1 | $5.1\times {10}^{-2}$ | $1.6\times {10}^{-2}$ | $6.7\times {10}^{-2}$ | $3.1\times {10}^{-2}$ | |

BA | 2-5-1 | $1.0\times {10}^{-2}$ | $2.4\times {10}^{-3}$ | $2.5\times {10}^{-2}$ | $9.8\times {10}^{-3}$ |

2-10-1 | $1.2\times {10}^{-2}$ | $3.7\times {10}^{-3}$ | $2.3\times {10}^{-2}$ | $1.1\times {10}^{-2}$ | |

2-15-1 | $1.7\times {10}^{-2}$ | $7.0\times {10}^{-3}$ | $3.0\times {10}^{-2}$ | $1.5\times {10}^{-2}$ | |

MVO | 2-5-1 | $2.5\times {10}^{-3}$ | $2.1\times {10}^{-3}$ | $1.0\times {10}^{-2}$ | $6.9\times {10}^{-3}$ |

2-10-1 | $9.2\times {10}^{-4}$ | $6.7\times {10}^{-4}$ | $5.5\times {10}^{-3}$ | $4.5\times {10}^{-3}$ | |

2-15-1 | $5.5\times {10}^{-4}$ | $4.0\times {10}^{-4}$ | $3.8\times {10}^{-3}$ | $2.6\times {10}^{-3}$ | |

MFO | 2-5-1 | $2.8\times {10}^{-3}$ | $2.4\times {10}^{-3}$ | $8.9\times {10}^{-3}$ | $7.5\times {10}^{-3}$ |

2-10-1 | $8.8\times {10}^{-4}$ | $8.4\times {10}^{-4}$ | $5.9\times {10}^{-3}$ | $5.4\times {10}^{-3}$ | |

2-15-1 | $5.2\times {10}^{-4}$ | $5.5\times {10}^{-4}$ | $3.7\times {10}^{-3}$ | $2.2\times {10}^{-3}$ | |

SSA | 2-5-1 | $4.0\times {10}^{-3}$ | $2.0\times {10}^{-3}$ | $1.2\times {10}^{-2}$ | $7.3\times {10}^{-3}$ |

2-10-1 | $1.9\times {10}^{-3}$ | $1.5\times {10}^{-3}$ | $7.9\times {10}^{-3}$ | $5.5\times {10}^{-3}$ | |

2-15-1 | $1.2\times {10}^{-3}$ | $1.0\times {10}^{-3}$ | $6.7\times {10}^{-3}$ | $3.9\times {10}^{-3}$ |

Algorithm | Network Structure | Train | Test | ||
---|---|---|---|---|---|

Mean | SD. | Mean | SD. | ||

ABC | 2-5-1 | $2.7\times {10}^{-4}$ | $1.4\times {10}^{-4}$ | $3.4\times {10}^{-3}$ | $2.6\times {10}^{-3}$ |

2-10-1 | $2.3\times {10}^{-4}$ | $1.2\times {10}^{-4}$ | $3.8\times {10}^{-3}$ | $2.8\times {10}^{-3}$ | |

2-15-1 | $2.9\times {10}^{-4}$ | $1.5\times {10}^{-4}$ | $4.1\times {10}^{-3}$ | $2.7\times {10}^{-3}$ | |

BAT | 2-5-1 | $1.6\times {10}^{-3}$ | $2.1\times {10}^{-3}$ | $9.5\times {10}^{-3}$ | $6.8\times {10}^{-3}$ |

2-10-1 | $3.2\times {10}^{-3}$ | $8.4\times {10}^{-3}$ | $1.5\times {10}^{-2}$ | $2.4\times {10}^{-2}$ | |

2-15-1 | $1.6\times {10}^{-3}$ | $4.7\times {10}^{-3}$ | $9.2\times {10}^{-3}$ | $7.5\times {10}^{-3}$ | |

CS | 2-5-1 | $1.4\times {10}^{-4}$ | $4.2\times {10}^{-5}$ | $3.5\times {10}^{-3}$ | $1.3\times {10}^{-3}$ |

2-10-1 | $2.1\times {10}^{-4}$ | $7.9\times {10}^{-5}$ | $4.0\times {10}^{-3}$ | $1.3\times {10}^{-3}$ | |

2-15-1 | $3.0\times {10}^{-4}$ | $9.0\times {10}^{-5}$ | $5.0\times {10}^{-3}$ | $2.0\times {10}^{-3}$ | |

FPA | 2-5-1 | $2.0\times {10}^{-4}$ | $8.5\times {10}^{-5}$ | $4.0\times {10}^{-3}$ | $2.0\times {10}^{-3}$ |

2-10-1 | $2.4\times {10}^{-4}$ | $8.4\times {10}^{-5}$ | $4.2\times {10}^{-3}$ | $1.9\times {10}^{-3}$ | |

2-15-1 | $3.2\times {10}^{-4}$ | $1.2\times {10}^{-4}$ | $5.0\times {10}^{-3}$ | $2.2\times {10}^{-3}$ | |

PSO | 2-5-1 | $2.1\times {10}^{-4}$ | $1.6\times {10}^{-4}$ | $2.1\times {10}^{-4}$ | $1.6\times {10}^{-4}$ |

2-10-1 | $4.1\times {10}^{-4}$ | $2.8\times {10}^{-4}$ | $4.1\times {10}^{-4}$ | $2.8\times {10}^{-4}$ | |

2-15-1 | $7.1\times {10}^{-4}$ | $3.1\times {10}^{-4}$ | $7.1\times {10}^{-4}$ | $3.1\times {10}^{-4}$ | |

JAYA | 2-5-1 | $6.2\times {10}^{-3}$ | $3.1\times {10}^{-3}$ | $2.4\times {10}^{-2}$ | $1.6\times {10}^{-2}$ |

2-10-1 | $8.2\times {10}^{-3}$ | $3.1\times {10}^{-3}$ | $2.8\times {10}^{-2}$ | $1.8\times {10}^{-2}$ | |

2-15-1 | $1.1\times {10}^{-2}$ | $3.6\times {10}^{-3}$ | $2.9\times {10}^{-2}$ | $1.9\times {10}^{-2}$ | |

TLBO | 2-5-1 | $5.6\times {10}^{-5}$ | $4.1\times {10}^{-5}$ | $2.9\times {10}^{-3}$ | $9.2\times {10}^{-4}$ |

2-10-1 | $7.1\times {10}^{-5}$ | $7.9\times {10}^{-5}$ | $2.8\times {10}^{-3}$ | $1.1\times {10}^{-3}$ | |

2-15-1 | $1.3\times {10}^{-4}$ | $1.0\times {10}^{-4}$ | $4.1\times {10}^{-3}$ | $1.5\times {10}^{-3}$ | |

SCA | 2-5-1 | $1.2\times {10}^{-3}$ | $4.8\times {10}^{-4}$ | $7.6\times {10}^{-3}$ | $4.8\times {10}^{-3}$ |

2-10-1 | $1.7\times {10}^{-3}$ | $7.1\times {10}^{-4}$ | $1.0\times {10}^{-2}$ | $5.4\times {10}^{-3}$ | |

2-15-1 | $2.1\times {10}^{-3}$ | $9.3\times {10}^{-4}$ | $9.7\times {10}^{-3}$ | $4.2\times {10}^{-3}$ | |

BBO | 2-5-1 | $1.1\times {10}^{-4}$ | $8.9\times {10}^{-5}$ | $4.0\times {10}^{-3}$ | $1.6\times {10}^{-3}$ |

2-10-1 | $1.0\times {10}^{-4}$ | $7.5\times {10}^{-5}$ | $3.5\times {10}^{-3}$ | $1.7\times {10}^{-3}$ | |

2-15-1 | $2.0\times {10}^{-4}$ | $2.1\times {10}^{-4}$ | $4.7\times {10}^{-3}$ | $1.8\times {10}^{-3}$ | |

WOA | 2-5-1 | $1.6\times {10}^{-3}$ | $1.5\times {10}^{-3}$ | $9.8\times {10}^{-3}$ | $4.2\times {10}^{-3}$ |

2-10-1 | $1.6\times {10}^{-3}$ | $2.1\times {10}^{-3}$ | $1.0\times {10}^{-2}$ | $7.5\times {10}^{-3}$ | |

2-15-1 | $1.9\times {10}^{-3}$ | $2.4\times {10}^{-3}$ | $1.2\times {10}^{-2}$ | $1.1\times {10}^{-2}$ | |

BSA | 2-5-1 | $4.2\times {10}^{-3}$ | $6.8\times {10}^{-3}$ | $1.9\times {10}^{-2}$ | $1.8\times {10}^{-2}$ |

2-10-1 | $2.3\times {10}^{-3}$ | $2.0\times {10}^{-3}$ | $1.4\times {10}^{-2}$ | $8.5\times {10}^{-3}$ | |

2-15-1 | $2.2\times {10}^{-3}$ | $1.9\times {10}^{-3}$ | $1.4\times {10}^{-2}$ | $8.4\times {10}^{-3}$ | |

HS | 2-5-1 | $2.2\times {10}^{-3}$ | $1.7\times {10}^{-3}$ | $1.0\times {10}^{-2}$ | $7.5\times {10}^{-3}$ |

2-10-1 | $9.5\times {10}^{-3}$ | $2.9\times {10}^{-3}$ | $2.6\times {10}^{-2}$ | $1.3\times {10}^{-2}$ | |

2-15-1 | $1.4\times {10}^{-2}$ | $4.0\times {10}^{-3}$ | $3.0\times {10}^{-2}$ | $1.6\times {10}^{-2}$ | |

BA | 2-5-1 | $2.7\times {10}^{-3}$ | $1.1\times {10}^{-3}$ | $9.8\times {10}^{-3}$ | $5.4\times {10}^{-3}$ |

2-10-1 | $3.5\times {10}^{-3}$ | $1.3\times {10}^{-3}$ | $1.3\times {10}^{-2}$ | $5.3\times {10}^{-3}$ | |

2-15-1 | $4.6\times {10}^{-3}$ | $1.5\times {10}^{-3}$ | $1.4\times {10}^{-2}$ | $9.8\times {10}^{-3}$ | |

MVO | 2-5-1 | $1.6\times {10}^{-4}$ | $1.2\times {10}^{-4}$ | $3.8\times {10}^{-3}$ | $1.4\times {10}^{-3}$ |

2-10-1 | $1.2\times {10}^{-4}$ | $8.6\times {10}^{-5}$ | $3.7\times {10}^{-3}$ | $1.6\times {10}^{-3}$ | |

2-15-1 | $1.3\times {10}^{-4}$ | $7.8\times {10}^{-5}$ | $3.6\times {10}^{-3}$ | $1.5\times {10}^{-3}$ | |

MFO | 2-5-1 | $5.1\times {10}^{-4}$ | $9.8\times {10}^{-4}$ | $4.5\times {10}^{-3}$ | $4.7\times {10}^{-3}$ |

2-10-1 | $2.0\times {10}^{-4}$ | $2.7\times {10}^{-4}$ | $4.5\times {10}^{-3}$ | $4.2\times {10}^{-3}$ | |

2-15-1 | $3.2\times {10}^{-4}$ | $4.0\times {10}^{-4}$ | $6.3\times {10}^{-3}$ | $6.7\times {10}^{-3}$ | |

SSA | 2-5-1 | $1.6\times {10}^{-4}$ | $1.2\times {10}^{-4}$ | $3.8\times {10}^{-3}$ | $1.3\times {10}^{-3}$ |

2-10-1 | $1.6\times {10}^{-4}$ | $1.3\times {10}^{-4}$ | $3.8\times {10}^{-3}$ | $1.6\times {10}^{-3}$ | |

2-15-1 | $1.1\times {10}^{-4}$ | $8.0\times {10}^{-5}$ | $3.6\times {10}^{-3}$ | $1.3\times {10}^{-3}$ |

Algorithm | Network Structure | Train | Test | ||
---|---|---|---|---|---|

Mean | SD. | Mean | SD. | ||

ABC | 3-5-1 | $1.9\times {10}^{-3}$ | $6.0\times {10}^{-4}$ | $2.3\times {10}^{-3}$ | $8.1\times {10}^{-4}$ |

3-10-1 | $1.5\times {10}^{-3}$ | $4.6\times {10}^{-4}$ | $2.1\times {10}^{-3}$ | $7.3\times {10}^{-4}$ | |

3-15-1 | $2.0\times {10}^{-3}$ | $1.0\times {10}^{-3}$ | $3.1\times {10}^{-3}$ | $1.4\times {10}^{-3}$ | |

BAT | 3-5-1 | $1.5\times {10}^{-2}$ | $1.1\times {10}^{-2}$ | $1.5\times {10}^{-2}$ | $1.1\times {10}^{-2}$ |

3-10-1 | $1.6\times {10}^{-2}$ | $2.3\times {10}^{-2}$ | $1.7\times {10}^{-2}$ | $2.5\times {10}^{-2}$ | |

3-15-1 | $1.8\times {10}^{-2}$ | $3.1\times {10}^{-2}$ | $1.8\times {10}^{-2}$ | $3.3\times {10}^{-2}$ | |

CS | 3-5-1 | $2.6\times {10}^{-3}$ | $5.7\times {10}^{-4}$ | $2.9\times {10}^{-3}$ | $7.9\times {10}^{-4}$ |

3-10-1 | $2.5\times {10}^{-3}$ | $4.5\times {10}^{-4}$ | $3.0\times {10}^{-3}$ | $7.6\times {10}^{-4}$ | |

3-15-1 | $3.0\times {10}^{-3}$ | $5.0\times {10}^{-4}$ | $3.3\times {10}^{-3}$ | $7.4\times {10}^{-4}$ | |

FPA | 3-5-1 | $3.1\times {10}^{-3}$ | $9.1\times {10}^{-4}$ | $3.5\times {10}^{-3}$ | $1.1\times {10}^{-3}$ |

3-10-1 | $3.2\times {10}^{-3}$ | $6.7\times {10}^{-4}$ | $3.5\times {10}^{-3}$ | $1.0\times {10}^{-3}$ | |

3-15-1 | $3.3\times {10}^{-3}$ | $1.0\times {10}^{-3}$ | $3.4\times {10}^{-3}$ | $1.2\times {10}^{-3}$ | |

PSO | 3-5-1 | $2.1\times {10}^{-3}$ | $8.7\times {10}^{-4}$ | $2.1\times {10}^{-3}$ | $8.7\times {10}^{-4}$ |

3-10-1 | $3.6\times {10}^{-3}$ | $1.6\times {10}^{-3}$ | $3.6\times {10}^{-3}$ | $1.6\times {10}^{-3}$ | |

3-15-1 | $5.2\times {10}^{-3}$ | $2.2\times {10}^{-3}$ | $5.2\times {10}^{-3}$ | $2.2\times {10}^{-3}$ | |

JAYA | 3-5-1 | $3.0\times {10}^{-2}$ | $1.0\times {10}^{-2}$ | $2.9\times {10}^{-2}$ | $1.2\times {10}^{-2}$ |

3-10-1 | $4.8\times {10}^{-2}$ | $1.8\times {10}^{-2}$ | $4.9\times {10}^{-2}$ | $1.9\times {10}^{-2}$ | |

3-15-1 | $5.5\times {10}^{-2}$ | $1.6\times {10}^{-2}$ | $5.4\times {10}^{-2}$ | $1.9\times {10}^{-2}$ | |

TLBO | 3-5-1 | $1.5\times {10}^{-3}$ | $1.2\times {10}^{-3}$ | $1.9\times {10}^{-3}$ | $1.1\times {10}^{-3}$ |

3-10-1 | $4.8\times {10}^{-4}$ | $2.3\times {10}^{-4}$ | $9.7\times {10}^{-4}$ | $3.9\times {10}^{-4}$ | |

3-15-1 | $6.9\times {10}^{-4}$ | $6.4\times {10}^{-4}$ | $1.2\times {10}^{-3}$ | $5.6\times {10}^{-4}$ | |

SCA | 3-5-1 | $7.5\times {10}^{-3}$ | $2.4\times {10}^{-3}$ | $7.6\times {10}^{-3}$ | $2.5\times {10}^{-3}$ |

3-10-1 | $9.2\times {10}^{-3}$ | $2.7\times {10}^{-3}$ | $9.1\times {10}^{-3}$ | $3.0\times {10}^{-3}$ | |

3-15-1 | $9.6\times {10}^{-3}$ | $3.9\times {10}^{-3}$ | $9.4\times {10}^{-3}$ | $3.6\times {10}^{-3}$ | |

BBO | 3-5-1 | $8.8\times {10}^{-4}$ | $5.5\times {10}^{-4}$ | $1.4\times {10}^{-3}$ | $7.6\times {10}^{-4}$ |

3-10-1 | $8.1\times {10}^{-4}$ | $6.3\times {10}^{-4}$ | $1.5\times {10}^{-3}$ | $8.7\times {10}^{-4}$ | |

3-15-1 | $1.2\times {10}^{-3}$ | $5.8\times {10}^{-4}$ | $1.8\times {10}^{-3}$ | $9.0\times {10}^{-4}$ | |

WOA | 3-5-1 | $8.2\times {10}^{-3}$ | $4.8\times {10}^{-3}$ | $8.2\times {10}^{-3}$ | $5.1\times {10}^{-3}$ |

3-10-1 | $9.4\times {10}^{-3}$ | $7.0\times {10}^{-3}$ | $9.4\times {10}^{-3}$ | $7.1\times {10}^{-3}$ | |

3-15-1 | $9.5\times {10}^{-3}$ | $6.5\times {10}^{-3}$ | $9.3\times {10}^{-3}$ | $7.4\times {10}^{-3}$ | |

BSA | 3-5-1 | $1.5\times {10}^{-2}$ | $6.5\times {10}^{-3}$ | $1.4\times {10}^{-2}$ | $6.8\times {10}^{-3}$ |

3-10-1 | $1.5\times {10}^{-2}$ | $8.0\times {10}^{-3}$ | $1.5\times {10}^{-2}$ | $8.8\times {10}^{-3}$ | |

3-15-1 | $1.4\times {10}^{-2}$ | $7.0\times {10}^{-3}$ | $1.4\times {10}^{-2}$ | $7.8\times {10}^{-3}$ | |

HS | 3-5-1 | $8.2\times {10}^{-3}$ | $4.2\times {10}^{-3}$ | $7.9\times {10}^{-3}$ | $4.5\times {10}^{-3}$ |

3-10-1 | $3.2\times {10}^{-2}$ | $1.5\times {10}^{-2}$ | $2.7\times {10}^{-1}$ | $1.0\times {10}^{-1}$ | |

3-15-1 | $6.2\times {10}^{-2}$ | $1.4\times {10}^{-2}$ | $6.0\times {10}^{-2}$ | $1.5\times {10}^{-2}$ | |

BA | 3-5-1 | $1.7\times {10}^{-2}$ | $2.9\times {10}^{-3}$ | $1.7\times {10}^{-2}$ | $3.5\times {10}^{-3}$ |

3-10-1 | $2.4\times {10}^{-2}$ | $5.5\times {10}^{-3}$ | $2.3\times {10}^{-2}$ | $5.1\times {10}^{-3}$ | |

3-15-1 | $3.7\times {10}^{-2}$ | $9.0\times {10}^{-3}$ | $3.6\times {10}^{-2}$ | $9.3\times {10}^{-3}$ | |

MVO | 3-5-1 | $2.5\times {10}^{-3}$ | $1.3\times {10}^{-3}$ | $3.2\times {10}^{-3}$ | $1.4\times {10}^{-3}$ |

3-10-1 | $2.1\times {10}^{-3}$ | $1.3\times {10}^{-3}$ | $2.7\times {10}^{-3}$ | $1.4\times {10}^{-3}$ | |

3-15-1 | $2.0\times {10}^{-3}$ | $1.2\times {10}^{-3}$ | $2.5\times {10}^{-3}$ | $1.4\times {10}^{-3}$ | |

MFO | 3-5-1 | $4.3\times {10}^{-3}$ | $2.9\times {10}^{-3}$ | $5.1\times {10}^{-3}$ | $3.4\times {10}^{-3}$ |

3-10-1 | $3.8\times {10}^{-3}$ | $2.8\times {10}^{-3}$ | $3.2\times {10}^{-1}$ | $1.0\times {10}^{-1}$ | |

3-15-1 | $2.2\times {10}^{-3}$ | $1.2\times {10}^{-3}$ | $3.4\times {10}^{-3}$ | $1.6\times {10}^{-3}$ | |

SSA | 3-5-1 | $3.6\times {10}^{-3}$ | $1.9\times {10}^{-3}$ | $3.8\times {10}^{-3}$ | $1.7\times {10}^{-3}$ |

3-10-1 | $2.4\times {10}^{-3}$ | $1.5\times {10}^{-3}$ | $2.8\times {10}^{-1}$ | $9.8\times {10}^{-2}$ | |

3-15-1 | $1.7\times {10}^{-3}$ | $6.4\times {10}^{-4}$ | $2.2\times {10}^{-3}$ | $7.3\times {10}^{-4}$ |

Algorithm | Network Structure | Train | Test | ||
---|---|---|---|---|---|

Mean | SD. | Mean | SD. | ||

ABC | 3-5-1 | $8.7\times {10}^{-4}$ | $2.1\times {10}^{-4}$ | $9.9\times {10}^{-4}$ | $3.8\times {10}^{-4}$ |

3-10-1 | $6.1\times {10}^{-4}$ | $1.8\times {10}^{-4}$ | $7.5\times {10}^{-4}$ | $2.8\times {10}^{-4}$ | |

3-15-1 | $5.3\times {10}^{-4}$ | $1.7\times {10}^{-4}$ | $6.6\times {10}^{-4}$ | $3.7\times {10}^{-4}$ | |

BAT | 3-5-1 | $7.0\times {10}^{-3}$ | $1.4\times {10}^{-2}$ | $7.3\times {10}^{-3}$ | $1.4\times {10}^{-3}$ |

3-10-1 | $5.9\times {10}^{-3}$ | $1.5\times {10}^{-2}$ | $6.1\times {10}^{-3}$ | $1.5\times {10}^{-2}$ | |

3-15-1 | $5.2\times {10}^{-3}$ | $1.7\times {10}^{-2}$ | $5.3\times {10}^{-3}$ | $1.6\times {10}^{-2}$ | |

CS | 3-5-1 | $8.7\times {10}^{-4}$ | $1.5\times {10}^{-4}$ | $9.3\times {10}^{-4}$ | $2.4\times {10}^{-4}$ |

3-10-1 | $8.6\times {10}^{-4}$ | $1.6\times {10}^{-4}$ | $9.4\times {10}^{-4}$ | $2.1\times {10}^{-4}$ | |

3-15-1 | $9.3\times {10}^{-4}$ | $1.4\times {10}^{-4}$ | $1.0\times {10}^{-3}$ | $2.2\times {10}^{-4}$ | |

FPA | 3-5-1 | $1.0\times {10}^{-3}$ | $3.0\times {10}^{-4}$ | $1.2\times {10}^{-3}$ | $3.6\times {10}^{-4}$ |

3-10-1 | $9.9\times {10}^{-4}$ | $1.9\times {10}^{-4}$ | $1.1\times {10}^{-3}$ | $3.3\times {10}^{-4}$ | |

3-15-1 | $1.0\times {10}^{-3}$ | $2.5\times {10}^{-4}$ | $1.1\times {10}^{-3}$ | $3.6\times {10}^{-4}$ | |

PSO | 3-5-1 | $1.3\times {10}^{-3}$ | $3.9\times {10}^{-4}$ | $1.3\times {10}^{-3}$ | $3.9\times {10}^{-4}$ |

3-10-1 | $1.7\times {10}^{-3}$ | $5.6\times {10}^{-4}$ | $1.7\times {10}^{-3}$ | $5.6\times {10}^{-4}$ | |

3-15-1 | $2.3\times {10}^{-3}$ | $6.6\times {10}^{-4}$ | $2.3\times {10}^{-3}$ | $6.6\times {10}^{-4}$ | |

JAYA | 3-5-1 | $1.3\times {10}^{-2}$ | $4.9\times {10}^{-3}$ | $1.3\times {10}^{-2}$ | $4.8\times {10}^{-3}$ |

3-10-1 | $1.3\times {10}^{-2}$ | $4.3\times {10}^{-3}$ | $1.3\times {10}^{-2}$ | $4.4\times {10}^{-3}$ | |

3-15-1 | $1.7\times {10}^{-2}$ | $7.4\times {10}^{-3}$ | $1.7\times {10}^{-2}$ | $7.4\times {10}^{-3}$ | |

TLBO | 3-5-1 | $6.2\times {10}^{-4}$ | $2.8\times {10}^{-4}$ | $7.4\times {10}^{-4}$ | $3.8\times {10}^{-4}$ |

3-10-1 | $6.2\times {10}^{-4}$ | $2.4\times {10}^{-4}$ | $6.7\times {10}^{-4}$ | $2.8\times {10}^{-4}$ | |

3-15-1 | $9.3\times {10}^{-4}$ | $4.0\times {10}^{-4}$ | $9.9\times {10}^{-4}$ | $4.5\times {10}^{-4}$ | |

SCA | 3-5-1 | $3.5\times {10}^{-3}$ | $1.2\times {10}^{-3}$ | $3.9\times {10}^{-3}$ | $1.3\times {10}^{-3}$ |

3-10-1 | $4.2\times {10}^{-3}$ | $1.7\times {10}^{-3}$ | $4.5\times {10}^{-3}$ | $1.7\times {10}^{-3}$ | |

3-15-1 | $5.0\times {10}^{-3}$ | $1.6\times {10}^{-3}$ | $5.5\times {10}^{-3}$ | $1.8\times {10}^{-3}$ | |

BBO | 3-5-1 | $5.2\times {10}^{-4}$ | $2.1\times {10}^{-4}$ | $6.1\times {10}^{-4}$ | $3.2\times {10}^{-4}$ |

3-10-1 | $5.3\times {10}^{-4}$ | $1.8\times {10}^{-4}$ | $6.8\times {10}^{-4}$ | $4.1\times {10}^{-4}$ | |

3-15-1 | $5.5\times {10}^{-4}$ | $2.0\times {10}^{-4}$ | $6.2\times {10}^{-4}$ | $3.1\times {10}^{-4}$ | |

WOA | 3-5-1 | $5.3\times {10}^{-3}$ | $4.3\times {10}^{-3}$ | $5.7\times {10}^{-3}$ | $4.2\times {10}^{-3}$ |

3-10-1 | $6.2\times {10}^{-3}$ | $3.8\times {10}^{-3}$ | $6.4\times {10}^{-3}$ | $3.7\times {10}^{-3}$ | |

3-15-1 | $5.9\times {10}^{-3}$ | $4.1\times {10}^{-3}$ | $6.3\times {10}^{-3}$ | $3.9\times {10}^{-3}$ | |

BSA | 3-5-1 | $1.1\times {10}^{-2}$ | $1.2\times {10}^{-2}$ | $1.1\times {10}^{-2}$ | $1.2\times {10}^{-2}$ |

3-10-1 | $7.2\times {10}^{-3}$ | $6.8\times {10}^{-3}$ | $7.4\times {10}^{-3}$ | $7.0\times {10}^{-3}$ | |

3-15-1 | $6.5\times {10}^{-3}$ | $3.6\times {10}^{-3}$ | $6.8\times {10}^{-3}$ | $3.7\times {10}^{-3}$ | |

HS | 3-5-1 | $5.7\times {10}^{-3}$ | $3.3\times {10}^{-3}$ | $5.8\times {10}^{-3}$ | $3.3\times {10}^{-3}$ |

3-10-1 | $1.7\times {10}^{-2}$ | $5.7\times {10}^{-3}$ | $1.7\times {10}^{-2}$ | $5.7\times {10}^{-3}$ | |

3-15-1 | $2.2\times {10}^{-2}$ | $5.8\times {10}^{-3}$ | $2.2\times {10}^{-2}$ | $5.8\times {10}^{-3}$ | |

BA | 3-5-1 | $6.9\times {10}^{-3}$ | $1.8\times {10}^{-3}$ | $7.3\times {10}^{-3}$ | $1.8\times {10}^{-3}$ |

3-10-1 | $6.7\times {10}^{-3}$ | $2.1\times {10}^{-3}$ | $7.0\times {10}^{-3}$ | $2.2\times {10}^{-3}$ | |

3-15-1 | $8.8\times {10}^{-3}$ | $2.3\times {10}^{-3}$ | $9.1\times {10}^{-3}$ | $2.4\times {10}^{-3}$ | |

MVO | 3-5-1 | $1.1\times {10}^{-3}$ | $5.1\times {10}^{-4}$ | $1.3\times {10}^{-3}$ | $6.7\times {10}^{-4}$ |

3-10-1 | $7.0\times {10}^{-4}$ | $1.9\times {10}^{-4}$ | $8.2\times {10}^{-4}$ | $3.0\times {10}^{-4}$ | |

3-15-1 | $6.7\times {10}^{-4}$ | $3.2\times {10}^{-4}$ | $7.6\times {10}^{-4}$ | $4.2\times {10}^{-4}$ | |

MFO | 3-5-1 | $1.4\times {10}^{-3}$ | $1.3\times {10}^{-3}$ | $1.5\times {10}^{-3}$ | $1.3\times {10}^{-3}$ |

3-10-1 | $5.8\times {10}^{-4}$ | $3.7\times {10}^{-4}$ | $6.9\times {10}^{-4}$ | $5.5\times {10}^{-4}$ | |

3-15-1 | $5.4\times {10}^{-4}$ | $5.8\times {10}^{-4}$ | $6.3\times {10}^{-4}$ | $6.4\times {10}^{-4}$ | |

SSA | 3-5-1 | $1.3\times {10}^{-3}$ | $5.1\times {10}^{-4}$ | $1.8\times {10}^{-3}$ | $9.3\times {10}^{-4}$ |

3-10-1 | $9.3\times {10}^{-4}$ | $3.3\times {10}^{-4}$ | $1.2\times {10}^{-3}$ | $5.7\times {10}^{-4}$ | |

3-15-1 | $6.9\times {10}^{-4}$ | $2.4\times {10}^{-4}$ | $8.1\times {10}^{-4}$ | $3.0\times {10}^{-4}$ |

Algorithm | Network Structure | Train | Test | ||
---|---|---|---|---|---|

Mean | SD. | Mean | SD. | ||

ABC | 3-5-1 | $3.8\times {10}^{-3}$ | $4.4\times {10}^{-4}$ | $3.9\times {10}^{-3}$ | $4.9\times {10}^{-4}$ |

3-10-1 | $3.1\times {10}^{-3}$ | $3.0\times {10}^{-4}$ | $3.2\times {10}^{-3}$ | $3.5\times {10}^{-4}$ | |

3-15-1 | $2.9\times {10}^{-3}$ | $3.0\times {10}^{-4}$ | $3.0\times {10}^{-3}$ | $5.1\times {10}^{-4}$ | |

BAT | 3-5-1 | $1.1\times {10}^{-2}$ | $5.7\times {10}^{-3}$ | $1.3\times {10}^{-2}$ | $1.5\times {10}^{-2}$ |

3-10-1 | $9.1\times {10}^{-3}$ | $5.9\times {10}^{-3}$ | $9.4\times {10}^{-3}$ | $5.9\times {10}^{-3}$ | |

3-15-1 | $8.6\times {10}^{-3}$ | $1.0\times {10}^{-2}$ | $9.0\times {10}^{-3}$ | $1.0\times {10}^{-2}$ | |

CS | 3-5-1 | $3.8\times {10}^{-3}$ | $3.3\times {10}^{-4}$ | $3.9\times {10}^{-3}$ | $3.4\times {10}^{-4}$ |

3-10-1 | $3.6\times {10}^{-3}$ | $2.7\times {10}^{-4}$ | $3.7\times {10}^{-3}$ | $4.6\times {10}^{-4}$ | |

3-15-1 | $3.4\times {10}^{-3}$ | $2.7\times {10}^{-4}$ | $3.5\times {10}^{-3}$ | $3.3\times {10}^{-4}$ | |

FPA | 3-5-1 | $4.5\times {10}^{-3}$ | $4.9\times {10}^{-4}$ | $4.7\times {10}^{-3}$ | $6.9\times {10}^{-4}$ |

3-10-1 | $3.9\times {10}^{-3}$ | $3.8\times {10}^{-4}$ | $4.1\times {10}^{-3}$ | $4.4\times {10}^{-4}$ | |

3-15-1 | $3.7\times {10}^{-3}$ | $3.5\times {10}^{-4}$ | $3.9\times {10}^{-3}$ | $5.1\times {10}^{-4}$ | |

PSO | 3-5-1 | $4.1\times {10}^{-3}$ | $6.0\times {10}^{-4}$ | $4.1\times {10}^{-3}$ | $6.0\times {10}^{-4}$ |

3-10-1 | $4.7\times {10}^{-3}$ | $5.7\times {10}^{-4}$ | $4.7\times {10}^{-3}$ | $5.7\times {10}^{-4}$ | |

3-15-1 | $5.1\times {10}^{-3}$ | $8.0\times {10}^{-4}$ | $5.1\times {10}^{-3}$ | $8.0\times {10}^{-4}$ | |

JAYA | 3-5-1 | $1.4\times {10}^{-2}$ | $6.4\times {10}^{-3}$ | $1.4\times {10}^{-2}$ | $6.3\times {10}^{-3}$ |

3-10-1 | $1.4\times {10}^{-2}$ | $4.1\times {10}^{-3}$ | $1.4\times {10}^{-2}$ | $4.0\times {10}^{-3}$ | |

3-15-1 | $1.5\times {10}^{-2}$ | $5.7\times {10}^{-3}$ | $1.5\times {10}^{-2}$ | $5.7\times {10}^{-3}$ | |

TLBO | 3-5-1 | $3.2\times {10}^{-3}$ | $5.9\times {10}^{-4}$ | $3.3\times {10}^{-3}$ | $8.3\times {10}^{-4}$ |

3-10-1 | $3.0\times {10}^{-3}$ | $7.8\times {10}^{-4}$ | $3.2\times {10}^{-3}$ | $9.3\times {10}^{-4}$ | |

3-15-1 | $3.1\times {10}^{-3}$ | $6.8\times {10}^{-4}$ | $3.4\times {10}^{-3}$ | $8.8\times {10}^{-4}$ | |

SCA | 3-5-1 | $6.2\times {10}^{-3}$ | $1.1\times {10}^{-3}$ | $6.6\times {10}^{-3}$ | $1.3\times {10}^{-3}$ |

3-10-1 | $7.2\times {10}^{-3}$ | $1.3\times {10}^{-3}$ | $7.5\times {10}^{-3}$ | $1.5\times {10}^{-3}$ | |

3-15-1 | $7.6\times {10}^{-3}$ | $1.3\times {10}^{-3}$ | $8.0\times {10}^{-3}$ | $1.4\times {10}^{-3}$ | |

BBO | 3-5-1 | $3.1\times {10}^{-3}$ | $6.0\times {10}^{-4}$ | $3.2\times {10}^{-3}$ | $6.2\times {10}^{-4}$ |

3-10-1 | $2.4\times {10}^{-3}$ | $4.1\times {10}^{-4}$ | $2.5\times {10}^{-3}$ | $4.3\times {10}^{-4}$ | |

3-15-1 | $2.5\times {10}^{-3}$ | $6.4\times {10}^{-4}$ | $2.6\times {10}^{-3}$ | $6.6\times {10}^{-3}$ | |

WOA | 3-5-1 | $8.8\times {10}^{-3}$ | $3.1\times {10}^{-3}$ | $9.1\times {10}^{-3}$ | $3.0\times {10}^{-3}$ |

3-10-1 | $9.1\times {10}^{-3}$ | $3.3\times {10}^{-3}$ | $9.4\times {10}^{-3}$ | $3.2\times {10}^{-3}$ | |

3-15-1 | $9.3\times {10}^{-3}$ | $4.3\times {10}^{-3}$ | $9.8\times {10}^{-3}$ | $4.3\times {10}^{-3}$ | |

BSA | 3-5-1 | $1.3\times {10}^{-2}$ | $4.6\times {10}^{-3}$ | $1.4\times {10}^{-2}$ | $4.5\times {10}^{-3}$ |

3-10-1 | $1.1\times {10}^{-2}$ | $4.4\times {10}^{-3}$ | $1.1\times {10}^{-2}$ | $4.4\times {10}^{-3}$ | |

3-15-1 | $1.2\times {10}^{-2}$ | $4.2\times {10}^{-3}$ | $1.2\times {10}^{-2}$ | $4.3\times {10}^{-3}$ | |

HS | 3-5-1 | $9.8\times {10}^{-3}$ | $3.5\times {10}^{-3}$ | $9.8\times {10}^{-3}$ | $3.5\times {10}^{-3}$ |

3-10-1 | $1.8\times {10}^{-2}$ | $3.8\times {10}^{-3}$ | $1.8\times {10}^{-2}$ | $3.9\times {10}^{-3}$ | |

3-15-1 | $2.3\times {10}^{-2}$ | $4.7\times {10}^{-3}$ | $2.3\times {10}^{-2}$ | $4.6\times {10}^{-3}$ | |

BA | 3-5-1 | $1.0\times {10}^{-2}$ | $1.4\times {10}^{-3}$ | $1.0\times {10}^{-2}$ | $1.5\times {10}^{-3}$ |

3-10-1 | $9.7\times {10}^{-3}$ | $1.7\times {10}^{-3}$ | $1.0\times {10}^{-2}$ | $1.6\times {10}^{-3}$ | |

3-15-1 | $1.0\times {10}^{-2}$ | $1.8\times {10}^{-3}$ | $1.1\times {10}^{-2}$ | $2.0\times {10}^{-3}$ | |

MVO | 3-5-1 | $3.9\times {10}^{-3}$ | $8.4\times {10}^{-4}$ | $4.1\times {10}^{-3}$ | $1.0\times {10}^{-3}$ |

3-10-1 | $3.3\times {10}^{-3}$ | $6.1\times {10}^{-4}$ | $3.5\times {10}^{-3}$ | $7.9\times {10}^{-4}$ | |

3-15-1 | $3.1\times {10}^{-3}$ | $5.7\times {10}^{-4}$ | $3.2\times {10}^{-3}$ | $6.1\times {10}^{-4}$ | |

MFO | 3-5-1 | $4.9\times {10}^{-3}$ | $2.6\times {10}^{-3}$ | $5.0\times {10}^{-3}$ | $2.6\times {10}^{-3}$ |

3-10-1 | $3.1\times {10}^{-3}$ | $8.0\times {10}^{-4}$ | $3.2\times {10}^{-3}$ | $1.0\times {10}^{-3}$ | |

3-15-1 | $2.4\times {10}^{-3}$ | $4.9\times {10}^{-4}$ | $2.5\times {10}^{-3}$ | $5.4\times {10}^{-4}$ | |

SSA | 3-5-1 | $4.5\times {10}^{-3}$ | $9.0\times {10}^{-4}$ | $4.8\times {10}^{-3}$ | $1.1\times {10}^{-3}$ |

3-10-1 | $4.3\times {10}^{-3}$ | $1.3\times {10}^{-3}$ | $4.5\times {10}^{-3}$ | $1.4\times {10}^{-3}$ | |

3-15-1 | $3.6\times {10}^{-3}$ | $5.3\times {10}^{-4}$ | $3.8\times {10}^{-3}$ | $6.1\times {10}^{-4}$ |

**Table 8.**The best training results obtained with the relevant metaheuristic algorithms for ${S}_{1}$, ${S}_{2}$, ${S}_{3}$, and ${S}_{4}$ (NS: network structure).

Algorithm | ${\mathit{S}}_{1}$ | ${\mathit{S}}_{2}$ | ${\mathit{S}}_{3}$ | ${\mathit{S}}_{4}$ | ||||
---|---|---|---|---|---|---|---|---|

NS | Mean | NS | Mean | NS | Mean | NS | Mean | |

ABC | 1-15-1 | $7.1\times {10}^{-4}$ | 2-10-1 | $5.4\times {10}^{-4}$ | 2-10-1 | $2.3\times {10}^{-4}$ | 3-10-1 | $1.5\times {10}^{-3}$ |

BAT | 1-10-1 | $2.2\times {10}^{-2}$ | 2-15-1 | $8.9\times {10}^{-3}$ | 2-15-1 | $1.6\times {10}^{-3}$ | 3-5-1 | $1.5\times {10}^{-2}$ |

CS | 1-15-1 | $7.3\times {10}^{-4}$ | 2-15-1 | $9.8\times {10}^{-4}$ | 2-5-1 | $1.4\times {10}^{-4}$ | 3-10-1 | $2.5\times {10}^{-3}$ |

FPA | 1-15-1 | $8.0\times {10}^{-4}$ | 2-15-1 | $1.1\times {10}^{-3}$ | 2-5-1 | $2.0\times {10}^{-4}$ | 3-5-1 | $3.1\times {10}^{-3}$ |

PSO | 1-5-1 | $1.7\times {10}^{-3}$ | 2-5-1 | $1.8\times {10}^{-3}$ | 2-5-1 | $2.1\times {10}^{-4}$ | 3-5-1 | $2.1\times {10}^{-3}$ |

JAYA | 1-10-1 | $1.3\times {10}^{-2}$ | 2-5-1 | $1.6\times {10}^{-2}$ | 2-5-1 | $6.2\times {10}^{-}$ |