Abstract
This paper presents a hybrid method of combining the Random Forest (RF) algorithm in machine learning (ML) and the Gaussian process (GP) to design microstrip patch antennas at any frequency from 0.6 to 6.5 GHz. Distinct from many published works, the proposed model is trained and tested with a high quality large dataset obtained from full-wave simulations in CST software, with experimental verification also performed. These factors are vital to assess the ML model’s efficacy, but they haven’t been thoroughly examined in other studies. In this paper, the GP is employed to seek the optimal hyper-parameters for the ML model using the RF algorithm, leading to superior predictive accuracy, as evaluated by the root mean square error (RMSE) of 0.0056. The advantage of this method is that antenna designers can directly use it to design antennas at any desired frequency within the 0.5–6 GHz range with high accuracy. The proposed method offers a transformative approach to antenna design by significantly reducing the optimization time by up to 99%, thereby improving the overall antenna design process.
Subject terms: Engineering, Mathematics and computing
Introduction
Machine Learning (ML) has seen significant growth recently, supporting applications such as image and speech recognition, recommendations, detection, virtual personal assistants, and specifically prediction. Prediction is a fundamental ML task that extracts useful information from data using reliable algorithms. In antenna design, ML helps optimize antenna parameters and predict output response, reducing design time and enhancing performance1. ML can be divided into three main types: supervised learning, unsupervised learning, and reinforcement learning with different learning algorithms2. Supervised learning is a popular algorithm in ML that predicts the outcome of new data based on known pairs of input and outcome. It is the most prevalent category in ML and is commonly used and capable of providing accurate predictions for various inputs after training on a specific dataset2,3. Supervised methods can be classified into two main types, which are classification and regression. In unsupervised learning algorithms, input data is not assigned a label. The unsupervised learning algorithm will use the inherent organization of the data to execute a certain task; it is useful for categorization. Unsupervised learning algorithms include clustering, anomaly detection, and neural networks. Reinforcement learning trains machines by rewarding beneficial behavior and penalizing undesirable behavior. A reinforcement learning agent can sense and comprehend its surroundings, execute actions, and learn via trial and error. This approach is widely applied in fields such as resource management, recommendation systems, and robotics. Generally, supervised machine learning is used to generate predictions based on data. In antenna design, it can substitute conventional optimization methods, including genetic algorithms, particle swarm optimization, and simulated annealing, thereby reducing dependence on electromagnetic simulations. Machine learning techniques may optimize antenna behavior prediction, minimize the need for simulations, promote computational efficiency, and expedite the antenna design process4.
ML has been applied to design and optimize many antenna types, such as Yagi microstrip and Beidou dual-mode antenna5, ring compact microstrip antenna6, fractal, spiral and horn antenna7,8, monopole antenna9, antenna array10,11, planar MIMO antenna12, and microstrip patch antenna. The ML model with Gaussian process (GP) in5, was applied to design Yagi microstrip and Beidou dual-mode antennas. The simulated return loss coefficient (
) from HFSS software and 
result from the proposed model fit together very well. However, the small dataset number, only 36 samples, increases the risk of overfitting the model. When a model is overfitting, the model does not generalize well to a new dataset. The model can obtain a high accuracy with low RMSE (Root Mean Squared Error) on the training set. However, the model has learned too much from the samples in the training dataset instead of learning the unseen data, leading to a poor performance on new data. Therefore, this overfitting reduces the reliability of the proposed model. The resonant frequency of the ring compact microstrip antenna in6 is predicted by the ML which is trained by 100 simulated data samples. In7, the dimensions of the spiral antenna and the horn antenna are designed using a support vector with a low mean absolute percentage error (MAPE) of less than 6
. In8, the antenna output responses of the horn and the fractal antenna are estimated by the k-nearest neighbors ML algorithm, but the proposed method achieved a low accuracy. A monopole antenna in9 is optimized using the ML model with its input dataset of 16 dielectric constant samples. Similarly, the researchers in10,11 have investigated ML techniques and GP to optimize antenna gain and to obtain a maximizing main lobe as well as minimizing side lobes of the antenna array, respectively. These ML models mainly concentrate on solving the problem of predicting the resonance of an arbitrary antenna based on the input dimensions. However, the proposed ML models in5–11 are not validated by measured results and/or are trained with small dataset, which significantly reduces the model reliability. Besides the various types of antenna mentioned above, applying ML to microstrip patch antenna designs has attracted researchers due to the popularity of this antenna type and the feasibility of ML models. Various methods have been proposed in the open literature13–24. The approach in13 uses the regression method to greatly accelerate the process of antenna optimization based on the initial dimensions at a particular resonant frequency of patch antenna. Similarly, the Feedforward Neural Network (FNN) is employed in the PSO-SA algorithm25 to optimize patch antenna dimensions14. This paper14 does not focus on predicting antenna dimensions; instead, it focuses on optimization of rectangular patch antennas for Ku and C band applications. Moreover, 10,000 data samples of antenna dimensions for training the FNN in14 is generated using MATLAB, which are calculated based on the theoretical equation of the patch antenna, so it is a significant deviation from the data generated using professional simulation software for designing antenna and reality.
Another slot patch antenna for C and X frequency bands is also designed using an artificial neural network (ANN)15. Nonetheless, the critical problem of this model is the data set size, which is quite small with 10 simulated and 2 measured samples. In16, the resonant frequency of patch antenna is predicted by ML model with 20480 samples, which are generated by the Computational Electromagnetics Simulator (CEMS), a tool with fast simulation speed but low accuracy. In17–20, the authors proposed several traditional ML models to predict the dimensions of patch antennas. The reliability of such models can be increased with a large dataset size from 95 to 4209 samples. However, the accuracy of these models is limited with RMSE obtained from 0.0087 to 0.45. In21, auto-metric Graph Neural Network is applied to predict antenna dimensions with low RMSE. Moreover, the number of data for the model is only 100 records, that makes overfitting problems. A Deep Neural Network trained by 350 data samples in22 is suggested to predict the S-parameters of the MIMO antenna at a center resonant frequency of 3.5 GHz. Applying the same as this Deep Neural Network, the MIMO antenna in23 is optimized for dual-bands at center frequencies of 3.1 GHz and 7.8 GHz. Deep learning method reduces design time and predicts optimal performance of MIMO antennas compared to using traditional electromagnetic simulations. However, unlike as the publications mentioned above for predicting antenna parameters for any desired resonance frequencies, the authors in22,23 have applied deep learning to determine optimal parameters at the specific frequency of the antenna. It reduces the model’s generalization to other frequency bands. Besides designing antennas, the author in24 presents a deep learning–based design approach for metasurface absorbers. A dataset of 7500 samples is used to train the neural network for absorber development. The absorber’s structure can be designed to meet the desired response at the range of 4 to 16 THz. However, data samples for training this neural network structure have not been validated by measurement. In addition, these works are described in more detail in Section ”Comparison with related works.” Based on the reviews, existing challenges in ML-based patch antenna design include low accuracy and reliability. disadvantages are often caused by the use of traditional ML models or datasets with few samples and limited confirmed measurement data.
In this paper, we focus on applying a hybrid ML model to design microstrip patch antenna at any frequency from 0.6 to 6.5 GHz. The proposed hybrid ML model uses a traditional ML algorithm, combining the GP to optimize its hyper-parameter. This combination enhances the accuracy of the model. The hybrid model implements six different traditional ML algorithms. The compared results demonstrate that the combination of Random Forest (RF) algorithm and GP can provide the highest accuracy with an RMSE of 0.0056. The proposed predictive model takes the desired resonant frequency as input and produces the optimal dimensions for the antenna as output. The antenna gain and efficiency are almost unchanged in the frequency range from 0.6 to 6 GHz, so these parameters cannot be used as a data feature. Our datasets consist of the optimized parameters from 1041 patch antenna samples that operate over a wide frequency range of 0.6 to 6 GHz. The optimized parameters for training and testing the model are obtained from the full-wave simulation CST software, resulting in a high quality large dataset. It is worth noting that the predicted dimensions from the ML model have been verified by measurement, which contributes to significantly increasing the reliability of the proposed ML model.
Antenna design and dataset
Antenna design
This section shows the procedure to design a microstrip patch antenna operating at an arbitrary frequency, 
. The simulation is conducted in CST software and then measurement is implemented to verify the accuracy of the simulation data. In simulation, the radiating patch can be modelled with resonant length of L, width of W, and it is excited through a feeding line with a length of 
. Here, the FR-4 substrate with a relative permittivity of 4.4 and thickness of 1.6 mm is used.
The size of the radiator is calculated using to equations (1, 2)26.
 |
1 |
 |
2 |
where 
, 
, 
, and c respectively denote resonant frequency, relative permittivity, effective permittivity, and light velocity. Using these Equations, a microstrip patch operating at 3.5 GHz is designed. The optimized dimensions are W = 27.74 mm, L = 20.9 mm, and 
=18.83 mm. Figure 1 shows the antenna structure as well as the simulated and measured reflection coefficients (
) of this antenna. The simulated result shows that the antenna operates at the center frequency band of 3.5 GHz with the bandwidth from 3.47 to 3.53 GHz. As seen, there is a small difference between the simulation and measurement. The measured data shows a center resonant frequency at 3.49 GHz, which is about 0.29% deviation from the simulated center frequency (3.5 GHz). In fact, this value is acceptable, and deviation is very common in designing antenna since a tolerance in fabrication and measurement setup is unavoidable. The simulated surface current (Js) distribution in Fig. 2 and the simulated data in Fig. 3 presents the realized gain and efficiency of the antenna, with the gain values from 5 to 6.2 dBi and radiation efficiency values above 80%. Figure 4a,b present the radiation pattern diagrams in terms of co- and cross-polarization in E-plane (elevation plane) and H-plane (azimuth plane) at 3.5 GHz. It is seen that in both planes, E-plane and H-plane, cross-polarization values are lower than co-polarization values. The co-polarization values are from 3.2 to 6.2 dB in radiation region in both E-plane and H-plane, whereas cross polarization values are lower than -11 dB in H-plane and -98 dB in E-plane. Thus, it can be concluded that the ML model trained on high quality data can achieve high reliability.
Fig. 1.
Antenna structure, simulated and measured 
results.
Fig. 2.
Simulated 
distribution on the patch at 3.5 GHz.
Fig. 3.
Simulated gain and radiation efficiency with center resonant frequency of 3.5 GHz.
Fig. 4.
Simulated co-polar and cross-polar radiation pattern of the patch antenna at 3.5 GHz in, (a), elevation (E-plane) and, (b), azimuth (H-plane).
Dataset
We generated a large and reliable dataset including parameters of resonant frequency, width (W) and length (L) of the patch, and length of the feeding line 
. The width and length of the patch are calculated and optimized by CST software to operate at frequencies ranging from 0.6 GHz to 6.5 GHz. The initial antenna size includes three important dimensions, which determine the resonant frequency of the antenna, (W), (L), and 
, and other dimensions. They are calculated by equations 1, 2 at a resonant frequency of 3.5 GHz. Then, the antenna is simulated and optimized based on the Time Domain method in CST Microwave Studio software, using the finite integration technique (FIT). These dimension parameters are used as the base value X. From this value, we use parameter sweep method to optimize for neighboring frequencies within 
GHz (e.g. 3.4, 3.3, 3.6, 3.7 GHz,…). When the desired frequency exceeds 3.5 GHz, the antenna size tends to decrease, and the optimizing range of CST is set from X to 
. Conversely, when the target frequency is lower than 3.5 GHz, the antenna size increases, CST searches optimal parameters in the range from X to 
. At each desired frequency, the CST software automatically determines the optimal set of parameters 
within the preset dimension range, aiming to achieve a minimum return loss. We continue enriching the datasets by sweeping dimension parameter between two adjacent frequencies and collect samples that have return loss 
dB, although the standard only requires 
dB. A microstrip patch antenna with a return loss below 
dB indicates very good impedance matching. After each optimization, the datasets including antenna dimensions and antenna operating parameters including resonant frequency, bandwidth, minimum return loss value are exported directly from CST and saved to a data file. As a result, a total of 1041 records are used as input data for ML models. This approach ensures that the obtained data is always standardized and reliable.
The antenna efficiency and gain remain almost constant across the entire frequency range of data, over 80% and around 6 dBi, respectively, so they are not data features. A record consists of four values separated by commas and stored as a CSV file. This dataset can be easily used for different programming platforms, such as Python, Matlab, and C++. The dataset can be found at https://github.com/KientvEPU/Design-Patch-Antena-with-Machine-Learning.git. The model’s dataset is divided according to the following ratio: 75% of the dataset is used for training, 20% is used for testing, and 5% is used for validating the model and comparison.
Proposed hybrid method for antenna design
Overview of candidate ML algorithms
The data with antenna dimensions and resonant frequencies are suitable for the ML regression model because the dataset is large and high quality with continuous target and input variables. Several popular algorithms for ML regression model include Linear Regression, LASSO, Elastic Net, KNN, Decision Tree, Decision Tree.
Linear regression
The linear regression algorithm1 is a statistical method used to model the relationship between a dependent variable concerning one or more independent variables. The Linear Regression algorithm aims to find a straight line or hyperplane in multidimensional space to predict the value of the dependent variable based on the value of the independent variable.
LASSO
The LASSO27 algorithm (Least Absolute Shrinkage and Selection Operator) is a linear regression method combined with regularization technique, used to improve the accuracy and ability to select attributes in the model. LASSO is especially useful when working with data with many explanatory variables or when it is necessary to select the most important variables. LASSO has the ability to make the coefficients of some attributes equal to 0, thereby eliminating unimportant attributes and helping with attribute selection. Based on regularization technique, LASSO helps reduce overfitting phenomenon, improving the generalization ability of the model. LASSO can create simpler and more explainable models by eliminating unnecessary attributes.
Elastic net
The Elastic Net28 algorithm is a linear regression method used to improve accuracy and attribute selectivity in statistical models. Elastic Net combines the advantages of two popular regression methods, Lasso (Least Absolute Shrinkage and Selection Operator) and Ridge Regression. Elastic Net has the ability to make the coefficients of some attributes equal to 0, making attribute selection efficient. Elastic Net can handle highly correlated attributes well. Combining Lasso and Ridge, Elastic Net provides flexibility and adaptability to many different types of data.
KNN
The K-Nearest Neighbors (KNN)29 algorithm is a machine learning method used for classification and regression problems. KNN is a non-parametric and distance-based algorithm in which the decision about a new data point is made based on the data points near it in the feature space. To determine ”nearest neighbors”, KNN uses a distance function to measure the similarity or closeness between data points. Popular distance functions such as Euclidean Distance, Manhattan Distance, Minkowski Distance, and Cosine Distance. The number of nearest neighbors is considered to determine the classification or prediction.
Decision tree
In the decision tree regression algorithm30, the training dataset is divided into subsets. The output of the model belongs to the subset that contains the input features. The choice of splitting criteria is based on minimizing variation within the subset as we traverse the branches. The node origin points are obtained based on the expected property which increases the accuracy. The standard deviation is expected to decrease after passing through each phase.
Random forest
The RF algorithm31 is a machine learning technique that uses ensemble learning to perform both classification and regression tasks. It is built on combining multiple decision trees to improve the accuracy and generalization ability of the model. The main components of RF include: Decision Trees, Bootstrap Aggregating, Random Subspace Method. In particular, decision tree is a prediction model based on dividing data into subsets based on attributes. Bagging (Bootstrap Aggregating) is the main technique RF uses to create decision trees. Bagging involves generating many subsamples from the original training data set by random selection with return. Each of these subsamples is then used to train a separate decision tree. In addition to using subsamples of the data, RF also randomly selects a subset of attributes to use when building each decision tree. This reduces correlation between decision trees, improving the model’s accuracy and generalization ability.
Proposed hybrid method using GP to optimize hyper-parameter of ML model
The proposed hybrid ML model is shown in Fig. 5. The steps in detail of the algorithm are described as follows.
Fig. 5.
The performing process of the proposed ML model.
1. Input: The input data, a set of simulated antenna parameters of dimensions and resonant frequency exported from CST to a CSV file, is randomly shuffled as raw data and then divided into a 75:20:5 ratio with optimal hyper-parameters obtained by GP-BO. Cross-validation techniques were used to divide the dataset into training sets and test sets with various scales, adjust the model to fit the training set, and evaluate its performance on the test set. A commonly used cross-validation method is to divide the data into k subsets. The model is trained on 
subsets and tested on the remaining subsets. This process is repeated k times, each time using a different subset as the test set.
2. Models: The model uses one of the supervised regression algorithms, including Linear Regression, LASSO, Elastic Net, KNN, Decision Tree and RF algorithms. The goal is to compare performance and identify the best model with optimal hyper-parameters.
3. Hyperparameter Initialization: Each model has different hyper-parameters. Initially, the algorithm will initialize these hyper-parameters with default or random values. The process begins by sampling the objective function f at a few initial points. These points can be chosen randomly in the hyper-parameter space. A GP is described by equation 3.
 |
3 |
where 
is Kernel math function used to evaluate the relation of samples.
4. Bayesian Optimization Loop (BO): The algorithm repeats until convergence is reached. BO works by building a probabilistic model of the objective function and using it to select the most promising points to evaluate next. A GP is a non-parametric model that establishes a distribution across functions and it is often used as the surrogate model in BO.
Steps within the loop: - Train the model with current hyper-parameters: Each model is trained on the dataset with the current set of hyper-parameters. - Build an alternative GP model: Each algorithm is then combined with GP to seek the optimal hyper-parameter set for improving the accuracy of the model based on Bayesian Optimization (BO)32. The algorithm uses GP to simulate the relationship between hyper-parameters and model performance. GP helps predict the performance of untested hyperparameter sets without actual training. - Use Acquisition Function: The acquisition function determines the next set of hyper-parameters to try based on the GP’s prediction. The acquisition function assesses GP’s output to choose a new hyper-parameter set for the next loop to seek an optimal hyper-parameter. The acquisition function is presented in equation 4 .
 |
4 |
where 
is the best target value corresponding to the selected hyper-parameter set. The value of hyper-parameter chosen to optimize the acquisition function and to update the next loop of GP. The hyper-parameter value is chosen to optimize the acquisition function, which is updated for the next GP iteration.
- Evaluate the model with new hyper-parameters: Train the ML model with the new hyperparameter set and calculate the performance. The convergence block decides to stop the seeking process when the value of the acquisition function is not improved significantly after several iterations.
- Update the performance history and hyper-parameters: Store the pair (hyperparameter set, performance) to use for the next iteration in the GP.
5. Output: When the iteration converges, the algorithm outputs the best model along with the optimal hyper-parameters. Based on the optimal hyper-parameters, RF is the most suitable model to combine with GP for the Microstrip Patch Antenna.
The above steps of ML algorithms are summarized in the pseudocode algorithm 1.
Algorithm 1.
Bayesian Optimization Loop
Results and discussion
The effectiveness of the proposed model is evaluated by the RMSE difference between the dataset performed by the model and simulated by CST software. RMSE depends on these selected algorithms as well as the condition and size of the collected data. 
index reflects the influence of independent variables and dependent variables in linear regression models. The 
index ranges from 0 to 1. The closer the 
index is to 1, the better the fit between the model and the dataset. RMSE and 
are calculated by the Equations (5) and (6).
 |
5 |
 |
6 |
where 
and 
refer to the predicted and actual values of antenna dimension, and N denotes the number of samples in the test set.
First, the model is performed without using GP. Then, it is combined with GP to obtain the optimal hyper-parameters for the model. Each ML algorithm has a set of hyper-parameters. However, GP does not search for optimal values of all hyper-parameters because certain settings may reduce the model’s generalization ability and lead to decreased performance33,34. Furthermore, the computational cost of BO increases rapidly with the number of hyper-parameters. In this proposed model, the ML algorithm combined with GP automatically selects the important hyper-parameters and optimal values of these hyper-parameters with minimum RMSE. The set of optimal hyper-parameters for ML algorithms is listed in Table 1. This table also shows the comparison in terms of RMSE of the models with and without GP. It can be seen obviously that hybrid models achieve higher accuracy than traditional models without GP. The combination of traditional ML algorithms and GP significantly enhances the accuracy of the traditional ML model. According to the comparison of six different algorithms, including Decision Tree, RF, Linear Regression, LASSO, KNN, and Elastic Net, it is demonstrated that the hybrid model using RF algorithm and GP is the best, with an RMSE of 0.0056. The model achieves high accuracy thanks to the employment GP to find the optimal hyper-parameters for the ML model using the RF algorithm.
Table 1.
RMSE comparison of the models with and without GP.
| Model | RMSE with GP | RMSE without GP | R2 score | Hyper-parameters |
|---|---|---|---|---|
| Linear regression | 0.186 | 0.187 | 0.56 | fit_intercept = True |
| LASSO | 0.169 | 0.174 | 0.45 | max_iter = 460; alpha = 2 |
| Elastic net | 0.086 | 0.12 | 0.32 | max_iter = 74; alpha = 2 |
| KNN | 0.0061 | 0.012 | 0.994 | n_neighbors = 1; p = 5 |
| Decision tree | 0.0061 | 0.015 | 0.994 | max_depth = 1000; min_samples_split=2 |
| RF | 0.0056 | 0.0059 | 0.995 | max_depth = 89; min_samples_split = 10 |
Figure 6 depicts residual between prediction of the antenna dimensions and test set of six models with GP. In the comparison between the tested and predicted values for antenna dimensions, the highest prediction accuracy is achieved with the RF algorithm. Besides, the models are evaluated by the coefficient of determination, 
, in which the models using Decision Tree, RF, KNN reach 0.99. It shows that the selected data set is completely suitable for these algorithms. However, 
for Linear Regression, LASSO, and Elastic Net is less than 0.6. This is confirmed by the values of RMSE and 
shown in Table 1 for the models using these three algorithms which are greatly higher than RMSE of the others.
Fig. 6.
Accuracy comparison of different algorithms with GP.
Figure 7 shows simulated, measured and predicted results of reflection coefficients at certain operating frequencies to validate the accuracy of the hybrid model. It can be observed that there is a similarity between the simulation, measurement and prediction results. Noted that the discrepancy in high-frequency region might be significant. However, this is not the targeted frequency, and thus high discrepancy is acceptable. Table 2 shows examples of simulated datasets and predicted values of dimension for the RF algorithm with GP. Obviously, the simulated values and the predicted values are similar.
Fig. 7.
The predicted, simulated and measured 
of patch antenna at different frequencies.
Table 2.
Examples of simulated and predicted antenna’s dimensions.
| No. | Frequency (GHz) | Simulated parameters from CST software (mm) | Predicted parameter using proposed ML model (mm) | ||||
|---|---|---|---|---|---|---|---|
| W | L |  |
 |
 |
 |
||
| 1 | 1.2 | 121.018 | 57.450 | 13.160 | 121.32 | 57.452 | 13.224 |
| 2 | 1.5 | 91.169 | 46.664 | 13.304 | 91.755 | 46.878 | 13.313 |
| 3 | 1.8 | 70.340 | 39.424 | 13.293 | 70.238 | 39.400 | 13.297 |
| 4 | 2.4 | 47.962 | 29.537 | 15.698 | 48.370 | 20.892 | 15.744 |
| 5 | 3.5 | 27.737 | 20.899 | 18.834 | 27.543 | 16.89 | 18.836 |
| 6 | 5.8 | 13.889 | 12.759 | 18.345 | 13.692 | 12.776 | 18.346 |
Finally, the effectiveness of any ML model in antenna design should be evaluated based on the time to achieve the optimal design parameters. Here, the time for obtaining antenna dimensions at a desired frequency using the proposed hybrid ML model and CST software is considered. For a fair comparison, both simulations were conducted on an Intel NUCi7 12th-generation computer with a 2.4 GHz CPU and 16GB RAM. The optimization process was performed on CST version 2022 software. The ML models were run on a CPU using Python version 3.10. The total prediction time by using the proposed model is less than 3 seconds with input as frequency and output as antenna dimensions. If optimization is performed on CST software, the time depends on range of optimizing dimensions and initial ones. Hence, it is highly dependent on designers’ experience. In our case study, we optimize three antenna dimensions including W, L, 
with the normal range of ±10% of the initial antenna dimensions, which are calculated using theoretical equations. The total time of the optimization process using CST simulation is around 300 seconds for different particular frequencies. Therefore, using the ML model to predict antenna dimensions can save 99% of the time compared to using traditional CST simulation . We assume that achieving the desired resonance frequency is possible for only one optimization process by CST. In reality, achieving it may require more than one time, so the proposed model can help achieve a time saving of more than 99%. Combining RF and GP can help to take the advantages of both models, which improves RMSE of the proposed model. However, this combination still faces problems such as expensive computation and difficult to manage model complexity compared with using only RF algorithm, especially with large datasets or increasing numbers of optimization iterations. The computational complexity of the RF optimization process using BO is governed by two main components, including the cost of training the RF model at each iteration 
and the cost of interpolating the GP with the complexity of 
. Here, n is number of data samples, K is number of trees in the forest, m is number of features selected at each node, and T is the optimization iteration. The computational complexity of GP cubic increment of T. Overall, the execution time of BO is 
.
Comparison with related works
Table 3 shows the comparison between the other related works and our proposed model. It is obvious that one of the most important factors to demonstrate the effectiveness of the ML model is the validation by measurement. In this paper, the validation is implemented on six fabricated antennas with different predicted parameters by the proposed ML model. Meanwhile, this step is ignored in the other works.
Table 3.
Comparison with other related works.
| Ref. | Antenna type | Sampled freq. band (GHz) | Dataset | Algorithm | RMSE | Measured validation | Summaries | |
|---|---|---|---|---|---|---|---|---|
| Software | No. samples | |||||||
| 17 | Patch |
2.03–2.64 (26.1%) |
FEKO | 95 | GP | 0.45 | No |
High RMSE Small dataset No measurement Low model reliability No public data |
| 18 | Patch |
5.92–25.7 (125%) |
CST | 125 | GP | 0.4002 | No |
High RMSE High RMSE Small dataset No measurement Low model reliability No public data |
| 19 | Patch |
0.48–7.84 (177%) |
CST | 3822 | GP | 0.0087 | Yes |
High RMSE Only one sample is measured Large deviation in No public data |
| 20 | Patch |
3.0–15.0 (133%) |
CEMS | 4209 | Support vector regressor | 0.03 | No |
High RMSE Data generated by CEMS No simulations No measurement Low model reliability No public data |
| 21 | Patch |
1.0–8.0 (155%) |
CST | 100 | Graph neural network |
0.0001 - 0.0007 |
No |
Low RMSE Small data No measurement Low model reliability High model complexity No public data |
| 22 | Monopole |
3.2–3.8 MHz (70%) |
CST | 350 | Deep neural network | NA | One sample |
RMSE: not mentioned Small data Design at one fixed center freq. No prediction antenna parameter at other frequencies. High model complexity No public data |
| 23 | Hexagonal |
2.8–4.2 (40%) |
CST | 350 | Deep neural network | NA | No |
RMSE: not mentioned Small data Design at one fixed center freq. No prediction antenna parameter at other frequencies. High model complexity No public data |
| 24 | None (absorber) |
4000–16000 (120%) |
CST | 7500 | Deep learning | 0.137 | No |
High RMSE No measurement Low model reliability High model complexity No public data |
| Pro. | Patch |
0.6–6.5 (181%) |
CST | 1041 | RF combined GP | 0.0056 | Yes |
Low RMSE High quality large data Validated by measurement High model reliability Public data |
Regarding the dataset, the models in17,18 have a small number of samples, only 95 and 125 samples, which could lead to overfitting, resulting in high accuracy during training but low accuracy during testing. The overfitting problem leads to reducing the model’s reliability. The models in19,20 achieve quite high accuracy of 0.0087 and 0.03. However, only one data sample in19 is measured with large deviation in resonant frequency. The data in20 is generated by CEMS, which is a fast simulation software for antenna design but low accuracy. Moreover, the data have not verified by measurement. As a result, these limitations undermine the reliability of the datasets. Similarly to the drawbacks in17,18, the model in21 is trained with small data, only 100 data samples. The authors in21 use five hidden layers despite having only 100 data records, which leads to an increase in model complexity. Additionally, the absence of validation through dataset measurement significantly reduces the reliability of the model.
Studies22,23 have applied a deep learning model with 350 samples to determine optimal antenna parameters at the fixed operating frequencies, reducing time for antenna design. However, it cannot apply to designing antennas at other frequencies, leading to limiting model generalization. Therefore, it reduces the model’s generalization to other frequency bands. Besides designing antennas, the author in24 applies deep learning to design metasurface absorbers in the 4–16 THz range using 7500 samples, but the training data have not been validated by measurements. Meanwhile, the datasets for the proposed model are the optimized parameters of 1041 patch antenna samples operating in a wide frequency range from 0.6 to 6 GHz with return loss below -20 dB at any resonant frequency. The optimized parameters for training and testing the model are obtained from the full-wave simulation CST software, resulting in a high quality large dataset. The antenna gain and radiation efficiency are almost unchanged in the frequency range from 0.6 to 6 GHz, so these parameters cannot be used as a data feature. It is worth noting that the predicted dimensions from the ML model have been verified by simulation measurement, and there is a good agreement between them. It contributes to significantly increasing the reliability of the proposed ML model. By incorporating GP for searching optimal hyper-parameters, the proposed ML model using RF algorithm reduces the RMSE to 0.0056 compared to without GP, while also being less complex than using Neural Network. The advantage of this method is that antenna designers can directly use it to design antennas at any desired frequency within the 0.5–6 GHz range with high accuracy, without needing additional time to optimize antenna dimensions using traditional electromagnetic simulations . The cost is incurred only once for building the model. As a result, the proposed method achieves at least a 99% reduction in the optimization time.
Conclusion
The hybrid ML model combining RF algorithm with GP is proposed to estimate the dimensions of the microstrip patch antenna at any desired operating frequency ranging from 0.6 to 6.5 GHz. By using GP process to optimize the hyper-parameter of RF algorithm, the proposed hybrid ML model for antenna design achieves a high accuracy with an RMSE value of 0.0056. In additional, the model is trained by a large and reliable dataset of antenna with measurement verification, which increase the reliability of the model. The comparison between simulation and prediction antenna dimensions has been performed with high similarity. The proposed model can significantly improve the process of achieving optimal antenna dimensions with the optimization time reduced by up to 99% compared to traditional CST simulation methods. In future works, the predicted frequency range of patch antennas will be extended, and the proposed ML model will be applied for more complex antenna datasets.
Author contributions
Thao Hoang Thi Phuong wrote the main manuscript text and proposed the method. Kien Tran Vu built the model and prepared figures. All authors reviewed the manuscript.
Funding
This research was funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant Number 102.04-2023.28.
Data availability
https://github.com/KientvEPU/Design-Patch-Antena-with-Machine-Learning/tree/main/Dataset.
Declarations
Competing interests
The authors declare no competing interests.
Footnotes
Publisher’s note
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References
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Associated Data
This section collects any data citations, data availability statements, or supplementary materials included in this article.
Data Availability Statement
https://github.com/KientvEPU/Design-Patch-Antena-with-Machine-Learning/tree/main/Dataset.