The design of an antenna requires a careful selection of its parameters to retain the desired performance. However, this task is time-consuming when the traditional approaches are employed, which represents a significant challenge. On the other hand, machine learning presents an effective solution to this challenge through a set of regression models that can robustly assist antenna designers to find out the best set of design parameters to achieve the intended performance. In this paper, we propose a novel approach for accurately predicting the bandwidth of metamaterial antenna. The proposed approach is based on employing the recently emerged guided whale optimization algorithm using adaptive particle swarm optimization to optimize the parameters of the long-short-term memory (LSTM) deep network. This optimized network is used to retrieve the metamaterial bandwidth given a set of features. In addition, the superiority of the proposed approach is examined in terms of a comparison with the traditional multilayer perceptron (ML), K-nearest neighbors (K-NN), and the basic LSTM in terms of several evaluation criteria such as root mean square error (RMSE), mean absolute error (MAE), and mean bias error (MBE). Experimental results show that the proposed approach could achieve RMSE of (0.003018), MAE of (0.001871), and MBE of (0.000205). These values are better than those of the other competing models.

Machine learning is an active research field that is widely used in data analysis for numerous domains. On the other hand, the design of antennas becomes complex every day; which encourages antenna designers to benefit from the powerful capabilities of machine learning to build reliable models to be used for intelligent and fast optimization of antenna designs. Optimization techniques based on machine learning algorithms; such as particle swarm optimization and genetic algorithms are used recently to optimize the structure of antennas while retaining their significant target performance. These optimization techniques are essential especially when the structure of the antenna is complex. In this case, the machine learning algorithms are effective in discovering the best antenna design parameters. In addition, the tools used in the simulation and analysis of electromagnetic (EM), which can be used to measure the fitness of antennas, require a lot of effort and time by antenna designers to get the proper results from the optimization process performed by these tools. The solution to this bottleneck is offered by machine learning that can be used to overcome the limitations of these simulation tools. The advantage of machine learning is embedded in its ability to provide approximate and proper antenna structures without the need to use EM simulations [

Machine learning techniques; such as K-nearest neighbors (KNN), artificial neural network (ANN), and least absolute shrinkage and selection operator (LASSO) were used recently in integrated and unified frameworks to optimize the parameters of antennas and to discover their optimal design parameters [

Many antennas are designed with variable performance and usage. The rectangular antenna is considered the simplest design that can be simply designed based on machine learning. The relevant parameters are the length and width of the rectangular patch, the dielectric, the height of the substrate, and the resonant frequency. A circular patch antenna is another essential type of antenna. The parameters of this design are similar to the rectangular patch antenna but with a radius of the patch antenna instead of the length and width of the rectangular type. An elliptical patch antenna is a special design of a circular antenna, where the shape is an ellipse. Similar parameters to the circular antenna are specified to be optimized using machine learning. In addition, in a fractal antenna, the perimeter (outer structure or inner sections of antenna) can be increased, and the effective length can be maximized based on the self-similar design structures/fractals of the material that can transmit or receive electromagnetic radiation within a given volume or a total surface area. This type of antenna consists of several parameters to be optimized, requiring a suitable dataset with enough samples to predict some of these parameters efficiently. Another type of microstrip antenna is the monopole antenna. The class of monopole antennas consists of a conductor in the form of a straight rod shape. In addition, there is a type of conductive surface, namely, a ground plane used to hold the conductor perpendicularly. When this antenna works as a receiver, the output signal is taken between the ground plane and the lower end of the monopole. Moreover, to the lower end of the monopole, one side of a feedline is attached to the ground plane, and the other side is attached to the antenna. There are other types of microstrip antennas that are designed with particular configurations. These types include substrate integrated waveguide, planar inverted-F antenna, reflect-array, and other special patch designs. Each type of these design has a set of configuration parameters that can be optimized and adjusted using machine learning [

One of the main outcomes of the field of computational electromagnetics is known as metamaterial antenna. This type of antenna has a significant advantage which is the possibility of designing this type of antenna using computations and optimization methods without relying on the modeling of its broader type [

Recently, many optimization approaches have emerged in the literature with promising performance [

Paper | Approach | Input | Output | Results |
---|---|---|---|---|

[ |
RBF + MLP, resilient backpropagation | Length and width, and substrate’s dielectric constant r and thickness | Resonant frequency | Error = 3.5 × 10^{–14} |

[ |
SVR + kernel configurations | Height and width | Resonant frequency | Error = 3 dB |

[ |
Gaussian kernel + SVR | Length and width | Voltage standing wave ratio (VSWR), gain, and resonant frequency | NA |

[ |
SVR | Height and length | Input impedance Rn, bandwidth (BW), and resonant frequency RF. | Error are 1.21% for RF, 2.15% for BW, and 0.2% for Rn |

[ |
RBF | Permittivity, height, and resonance frequency | Width and length of the patch | Error = 0.91% |

[ |
MLP | Permittivity, height and resonance frequency | Width and length of the patch | Error = 3.47% |

Metaheuristic algorithms are capable of handling problems with unexpected behavior. These algorithms are distinguished by their intelligence which is based on the random search methodology. In addition, these algorithms can efficiently avoid local perfections as usually referred to as simple and flexible approaches. This type of algorithm has two main processes namely exploitation and exploration that are running while searching the population for the optimal solution. In the exploration process, the search space is examined thoroughly to find the best solution. In recent decades, many global optimization approaches were developed based on observing natural objects in the real life. One of these global optimization approaches is known as population-based metaheuristic algorithms that can be used in a variety of situations. Two types of metaheuristic approaches are widely spread in the field namely metaphor and non-metaphor base approaches. The metaphor approach is based on observing human behavior or a natural phenomenon in its inherent details [

In this paper, we propose a novel approach for predicting the bandwidth of the metamaterial antenna based on optimized long short-term memory (LSTM). The optimization of LSTM is performed in terms of the recently emerged algorithm referred to as adaptive dynamic particle swarm algorithm (AD-PSO) with a guided whale optimization algorithm (Guided WOA) [

The rest of this paper is organized as follows. The description of the dataset along with a detailed discussion of the proposed methodology are explained in Section 2. The achieved results are comparisons are presented and investigated in Section 3. Then, the conclusions are examined in Section 4.

The proposed methodology used to improve the prediction of the metamaterial antenna bandwidth is discussed in this section. The section starts with presenting the dataset employed in this research, followed by discussing the main components of the proposed approach.

The metamaterial dataset used in this work contains eleven features, which are described in

No. | Feature | Description |
---|---|---|

1 | S11 | Return loss |

2 | Bandwidth | The bandwidth of the antenna |

3 | VSWR | Standing wave ration voltage of the antenna |

4 | Gain | Antenna Gain |

5 | Ya | Split ring resonator cell distance |

6 | Xa | Array-patch antenna distance |

7 | SRR_num | Number of cells in the split ring resonator |

8 | Tm | Rings’ width |

9 | Dm | Rings’ distance |

10 | W0 m | Rings’ gap |

11 | Wm | Dimensions of the split ring resonator |

As the dataset is collected from electromagnetic simulation tools, it has been noted that some samples were recorded in the dataset with missing values. To deal with the missing values (denoted by null in the dataset) in a certain feature column, we replaced these values with the average of the surrounding not-null values in the same column. In addition, it is noted that the values of the features in the dataset do not lay in the same range; which might affect the performance of the predictor. Therefore, we applied the min-max normalization. After applying this step, each feature in the dataset has the lowest value of zero and the greatest value of one.

As explained in [

After the optimization process has been initialized, and for each solution in the population, a fitness value is calculated. To get the highest possible fitness value, the optimization algorithm selects the optimum agent for the situation (solution) [

The Guided WOA refers to a modified version of the original WOA algorithm [

A variety of swarming patterns in nature, such as those shown by birds, are represented by the PSO algorithm, which replicates their social behavior. By shifting their places, the agents in the PSO algorithm look for the optimal solution or meal based on the most recent velocity. The following algorithm shows the detailed steps of the AD-PSO-Guided WOA algorithm employed in this research.

A set of experiments were conducted to evaluate the proposed model and compare it with the other competing models in the literature. These models are the Multilayer Perceptron (MLP), K-Nearest Neighbors (KNN), LSTM, in addition to the proposed AD-PSO-Guided WOA for the LSTM model which is used to select the optimal hyperparameters value of the LSTM network for predicting the bandwidth of metamaterial antenna. The evaluation of these models is based on the criteria listed in

Metric | Value |
---|---|

MAPE | |

RMSE | |

RRMSE | |

R | |

WI |

It was necessary to utilize around 80 percent of the dataset obtained at each site for training, with the remaining 20 percent being used to test for the dependability of the models that had been constructed. A uniform distribution was used to provide random sampling and assignment of data to the two subgroups, rather than using a chronological partitioning method. This is done to lessen the reliance of the produced models on the particular data that was used in the fitting process and to guarantee that the models behave in the same manner when dealinsg with other datasets. In addition, the calculation of validation metrics is performed to present non-dimensional error estimates.

After preparing the training and testing sets, a set of values were selected for the configuration parameters of the adaptive PSO-guided WOA algorithm to train the LSTM model. The settings of these values are presented in

Parameter | Value |
---|---|

Number of iterations | 50 |

Number of runs | 20 |

Number of whales | 20 |

Dimension | Number of features |

Inertia Wmin, Wmax | [0.6, 0.9] |

Acceleration constants C1, C2 | [2, 2] |

α for Fn | 0.99 |

β for Fn | 0.01 |

The parameters listed in ^{th} iteration. During the optimization process, the exploration and exploitation operations vary depending on the recorded optimized values until reaching the saturation level, then the recorded values of the optimized LSTM parameters are saved to be used in predicting the metamaterial antenna bandwidth.

The prediction results of the metamaterial antenna bandwidth using the proposed optimized model are shown in

The plots depicted in

Moreover, the plots of

The evaluation criteria listed in

MLP | KNN | LSTM | Proposed model | |
---|---|---|---|---|

RMSE | 0.051860371 | 0.027929868 | 0.020169502 | 0.003108 |

MAE | 0.045904674 | 0.023111993 | 0.015496072 | 0.001871 |

MBE | −0.042390451 | 0.000227761 | 0.004493794 | 0.000205 |

R | 0.991067787 | 0.992886219 | 0.995676454 | 0.999888 |

R^{2} |
0.982215359 | 0.985823043 | 0.991371601 | 0.999776 |

RRMSE | 8.723646595 | 4.698198039 | 11.57581043 | 0.522864 |

NSE | 0.936988554 | 0.981723798 | 0.990468998 | 0.999774 |

WI | 0.873977428 | 0.93655041 | 0.957458475 | 0.994862 |

The ANOVA test, using the prediction results achieved by the proposed model, is presented in

ANOVA table | SS | DF | MS | F (DFn, DFd) | |
---|---|---|---|---|---|

Treatment (between columns) | 0.0177 | 3 | 0.005901 | F (3, 52) = 203.5 | |

Residual (within columns) | 0.001508 | 52 | 0.000029 | ||

Total | 0.01921 | 55 |

Descriptive statistics | MLP | KNN | LSTM | Proposed model |
---|---|---|---|---|

Number of values | 14 | 14 | 14 | 14 |

Minimum | 0.03186 | 0.02093 | 0.01817 | 0.003071 |

25% Percentile | 0.05186 | 0.02793 | 0.02017 | 0.003108 |

Median | 0.05186 | 0.02793 | 0.02017 | 0.003108 |

75% Percentile | 0.05186 | 0.02793 | 0.02017 | 0.003108 |

Maximum | 0.08186 | 0.03793 | 0.02817 | 0.003138 |

Range | 0.05 | 0.017 | 0.01 | 6.72E−05 |

10% Percentile | 0.04186 | 0.02443 | 0.01917 | 0.003081 |

90% Percentile | 0.06686 | 0.03361 | 0.02417 | 0.003128 |

95% CI of median | ||||

Actual confidence level | 98.71% | 98.71% | 98.71% | 98.71% |

Lower confidence limit | 0.05186 | 0.02793 | 0.02017 | 0.003108 |

Upper confidence limit | 0.05186 | 0.02793 | 0.02017 | 0.003108 |

Mean | 0.05257 | 0.02824 | 0.02060 | 0.003107 |

Std. Deviation | 0.00997 | 0.00339 | 0.00224 | 1.44E−05 |

Std. Error of Mean | 0.002665 | 0.000906 | 0.0006 | 3.84E−06 |

Lower 95% CI of mean | 0.04682 | 0.02628 | 0.0193 | 0.003099 |

Upper 95% CI of mean | 0.05833 | 0.0302 | 0.02189 | 0.003116 |

Coefficient of variation | 18.97% | 12.01% | 10.89% | 0.4619% |

Geometric mean | 0.05175 | 0.02806 | 0.0205 | 0.003107 |

Geometric SD factor | 1.203 | 1.124 | 1.1 | 1.005 |

Lower 95% CI of geo. mean | 0.0465 | 0.02622 | 0.0194 | 0.003099 |

Upper 95% CI of geo. mean | 0.05758 | 0.03003 | 0.02167 | 0.003116 |

Harmonic mean | 0.05091 | 0.02788 | 0.02042 | 0.003107 |

Lower 95% CI of harm. mean | 0.04571 | 0.0261 | 0.01947 | 0.003099 |

Upper 95% CI of harm. mean | 0.05744 | 0.02993 | 0.02147 | 0.003116 |

Quadratic mean | 0.05345 | 0.02843 | 0.02071 | 0.003107 |

Lower 95% CI of quad. mean | 0.04656 | 0.02627 | 0.01916 | 0.003099 |

Upper 95% CI of quad. mean | 0.05954 | 0.03044 | 0.02215 | 0.003116 |

Skewness | 1.468 | 1.195 | 3.329 | −0.6237 |

Kurtosis | 7.539 | 6.955 | 12.21 | 4.044 |

Sum | 0.736 | 0.3954 | 0.2884 | 0.0435 |

The Wilcoxon signed-rank test is performed on the metamaterial test set and the results are recorded in

MLP | KNN | LSTM | Proposed LSTM | |
---|---|---|---|---|

Theoretical mean | 0 | 0 | 0 | 0 |

Actual mean | 0.05257 | 0.02824 | 0.0206 | 0.003107 |

Number of values | 14 | 14 | 14 | 14 |

One sample t-test | ||||

t, df | t = 19.73, df = 13 | t = 31.16, df = 13 | t = 34.35, df = 13 | t = 810.1, df = 13 |

<0.0001 | <0.0001 | <0.0001 | <0.0001 | |

**** | **** | **** | **** | |

Significant (alpha = 0.05)? | Yes | Yes | Yes | Yes |

How big is the discrepancy? | ||||

Discrepancy | 0.05257 | 0.02824 | 0.0206 | 0.003107 |

SD of discrepancy | 0.009972 | 0.003391 | 0.002243 | 0.00001435 |

SEM of discrepancy | 0.002665 | 0.000906 | 0.0005996 | 0.000003836 |

95% confidence interval | 0.0468:0.0583 | 0.0263:0.0302 | 0.0193:0.0219 | 0.00310:0.00312 |

R squared (partial eta squared) | 0.9677 | 0.9868 | 0.9891 | 1 |

Machine learning is considered the backbone of the ongoing research efforts which contributes significantly to many fields of today’s technology. The choice of a machine learning model usually affects the accuracy of the predictions of the task results. In this paper, we adopted the application of the recently emerged optimization algorithm, which is referred to as Adaptive PSO-Guided WOA, to search for the best parameters of the LSTM deep network. This network is then used to predict the bandwidth of the metamaterial antenna. The choice of metamaterial antenna in this work was due to its capability to overcome the bandwidth constraints imposed by tiny sizes of antennas. On the other hand, the interesting features and capabilities of deep learning allow it to be widely used in almost all fields of science. LSTM is one of the most significant types of deep networks which is optimized using the adaptive guided whale algorithm to fit the task of predicting the bandwidth. To emphasize the superiority of the proposed approach, other competing models were incorporated in the conducted experiments. The findings of this work indicate that the prediction accuracy using the proposed approach outperforms the standard LSTM, MLP, and KNN models.

Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2022R308), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.