Machine Learning Techniques for Predicting Metamaterial Microwave Absorption Performance: A Comparison

This work presents a metamaterial absorber (MMA) for X- and Ku-bands with a metallic resonating patch on top and a ground plane separated by substrate FR-4 with a thickness of $0.053~\lambda $ at the lowest resonance frequency. The proposed MMA demonstrates perfect absorption of 99.42, 98.48, 98.92, and 99.34 % at 9.948, 13.26, 14.92, and 15.80 GHz, respectively at normal incidence. The proposed MMA demonstrates perfect absorption for a polarization and incident angle over a wide range of angles up to 45°. To understand the fundamental EM behavior of the metamaterial structure, equivalent circuit analysis was carried out, and the circuit outputs accorded with the simulation results. This article also compares various machine learning (ML) methods for optimizing the design and predictive modeling of MMAs, such as decision trees, K-nearest neighbors, random forests, extra trees (ET), bagging, LightGBM, XGBoost, hist gradient boosting, cat boost, and gradient boosting regressors. The primary objective is to assess the usefulness of each regressor technique in estimating the performance of MMAs using multiple tests ranging from TC-40 to TC-80 and performance metrics such as adjusted R-squared score, MSE, RMSE, and MAE, in which the ET regressor excels. Simulation results suggest that ML-based techniques can save simulation resources and time while still being an efficient tool for predicting absorber behavior at intermediate and subsequent frequencies.


I. INTRODUCTION
Due to the unbelievable properties of metamaterial which are not found in nature, metamaterials have been widely used in various applications like invisibility cloaks [1], electro- The associate editor coordinating the review of this manuscript and approving it for publication was Qi Luo .magnetic wave absorbers [2], sensing [3], modulators [4], antennas, and so on.Metamaterial has emerged as an ideal alternative for fulfilling the demands of advanced absorber applications owing to its remarkably thin profile, compact nature, and ability to achieve complete absorption.These exceptional characteristics make MMA promising for wide applications, such as energy harvesting [5], sensing [6], [7], satellite communication [8], photodetector [9], microwave imaging [10], stealth technology [11], IoT applications and many more.
Metamaterial absorbers have been around since 2002, when Engheta first came up with the idea of using metamaterial surfaces on screens to absorb electromagnetic waves [12].In 2008, Landy developed a highly efficient metamaterial absorber (MMA) [2].Subsequently, metamaterial absorbers (MMAs) have garnered significant interest due to their exceptional capacity for electromagnetic (EM) wave absorption [13], [14].MMAs are often structured in periodic configurations consisting of resonators that are fabricated on a substrate, while a ground plane is located at the lowermost position.The metallic resonator and dielectric substrate are excited by the incident electric and magnetic field, respectively [2].These structures can minimize the reflection by matching the impedance of the MMA to free space and the transmission is eliminated by the ground plane to achieve perfect absorption [3].
The majority of MMA modeling analysis depends on EM simulation software like High-frequency Structure Simulator (HFSS) and Computer Simulation Technology (CST), which are based on iterative numerical full wave calculations such as the Finite Element Method (FEM) [22], and Finite-Difference Time-Domain (FDTD) [23].Designing metamaterial absorbers with optimal performance remains a significant challenge due to the involvement of large trial and error searches over geometric dimensions, which necessitates the validation and optimization of each parameter.Extensive modeling techniques are needed to precisely estimate the absorber's performance because of the complex relationship between the structural elements of the absorber and its absorption characteristics.A recent study has demonstrated a significant increase in the application of machine learning regressor techniques to enhance the predictive modeling of metamaterial absorber performance, enabling an improved understanding of patterns and behaviors in photonics devices [6].Regression analysis is a statistical methodology employed to determine the connection between a dependent variable, namely absorption value, and an independent variable, specifically frequency.Because of their ability to capture complex relationships among data, decision tree (DT), random forest (RF), K-nearest neighbors (KNN), extra trees (ET), bagging, light gradient boosting machine (LGBM), extreme gradient boosting (XGB), hist gradient boosting (HGB), cat boost (CBR) and gradient boosting (GBR) regressors techniques have emerged as attractive methods.The regressor methods have demonstrated their efficacy in various applications, such as electromagnetic wave analysis and material characterization.The absorption and reflection response of photonic devices can also be observed and predicted using machine learning [6].Several studies have explored the use of machine learning in various applications related to antenna designs, metamaterials for target adaptation, and manufacturing process optimization [24], [25].ML techniques offer faster simulations while achieving the same levels of precision [26].Since then, MMA structures have been demonstrated with different applicable ML regression algorithms, such as Extra trees [6], XGBoost [27], and k-nearest neighbor regressor [28].
This study presents a compact and ultra-thin metamaterial absorber that exhibits ideal absorption characteristics in X-and Ku-band applications.The MMA exhibits quad absorption peaks i.e. one (9.948GHz) in X-band and three (13.26,14.92, and 15.80 GHz) in Ku-band.The primary objective of this study is to address the existing research gap through a comprehensive evaluation and comparison of different well-known regression techniques for metamaterial absorbers, which represents a novel contribution.The aim is to identify the most efficient approach for optimizing the design and predictive modeling of metamaterial absorbers for microwave applications.The absorption dataset is initially split into separate training and testing sets, followed by the utilization of regression models to effectively forecast absorption.The study conducts a comparative evaluation of multiple regression models, specifically DT, KNN, RF, ET, bagging, XGB, LGBM, HGB, CBR, and GBR models.Various performance metrics, including root mean squared error (RMSE), mean absolute error (MAE), mean squared error (MSE), and Adjusted R-squared, are employed to determine the optimal regressor technique for precise prediction modeling of metamaterial absorber performance in microwave applications.The findings of the comparative analysis indicate that the Extra Trees Regressor demonstrates superior predictive capabilities compared to other models across all test sizes.

II. DESIGN AND RESULTS DISCUSSIONS
The proposed MMA configuration comprises a circular enclosure containing T-shaped resonator that have been altered to achieve complete absorption at four distinct resonant frequencies.As shown in Figure 1(a), the unit cell is made up of a metallic resonator and a copper ground plane that is isolated by a FR4 substrate.The loss tangent and permittivity values for the FR4 substrate are reported as 0.02 and 4.4, respectively.Furthermore, copper has a conductivity (σ ) value of 5.8 × 10 7 S/m.The metallic layer has a thickness of 0.035 mm (h c ), while the dielectric layer has a thickness of 1.6 mm (h s ).The top view of the MMA with optimized parameters are a = 12.2 mm, b = 7.6 mm, c = 2.88 mm, g = 0.8 mm, r i =4.8 mm, and r o = 5.6 mm as shown in Fig. 1(b).At the lowest resonance frequency, the absorber is compact (0.406 λ) and thin (0.053 λ).The occurrence of electric and magnetic fields excites the MMA, resulting in variations to its effective permittivity (ε eff ) and effective permeability (µ eff ).
The absorption phenomenon at different resonant frequencies is observed when the incident wave's electric and magnetic fields act simultaneously, as depicted in Figure 1(c).The expression for the reflection coefficient is given by (Z − Z 0 ) (Z + Z 0 ) 2 , where Z 0 and Z are the free space and MMA impedance.The absorption will be perfect by optimizing the geometrical structure and its dimensions [15].The reflection (R), absorption (A), and transmission (T) characteristics are depicted in Figure 1(c), showcasing peak absorptivity values of 99.42%, 98.48%, 98.92%, and 99.34% at resonating frequencies of 9.94 (f 1 ), 13.26 (f 2 ), 14.92 and 15.80 GHz (f 3 ), respectively.In order to reduce the chance of reflection, it is desirable for the real and imag-inary components of the normalised impedance to converge towards values of one and zero, respectively.The normalized impedance (Z ) is 0.85+j0.01,0.92-j0.22,1.15+j0.16and 1.01+j0.17Commercially available software like as ADS can be utilized to create the equivalent circuit of the MMA, as shown in Fig. 2. Due to the presence of capacitive gaps between the resonators, L and C values must be integrated as parallel transmission lines connected by a capacitive element between them for each resonance frequency.Four parallel R-L-C circuits were needed to accommodate the four resonance frequencies.To match the corresponding absorption values obtained from the HFSS simulation, the R values for each R-L-C circuit (for each resonance frequency) were adjusted using the ADS circuit simulator.Since MMAs exhibit periodicity, they may enable electromagnetic coupling through the presence of inductive and capacitive components within the circuit.Introducing series capacitances (C 5 , C 6 , C 7 , and C 8 ) in the circuit, as presented in Fig. 2 (a), can eliminate this issue.The values of capacitors C 5 , C 6 , C 7 , and C 8 were adjusted through fine-tuning using the ADS circuit simulator to attain optimal isolation within the bands located between resonance frequencies.The resonance frequencies in the output obtained from the equivalent circuit and HFSS simulation exhibit a high level of resemblance suggesting a strong agreement in terms of absorption, as illustrated in Fig. 2 (b).The proposed MMA has been validated for various polarization (φ) and incident (θ) angles for TE polarization as depicted in Figure 3.The X-and Ku-band exhibit peaks at frequencies of 9.94, 13.26, 14.92, and 15.80 GHz under the condition of φ = 0 o .The amplitude of the peaks decreases gradually with an increase in the polarization angle, which can be attributed to the asymmetrical characteristics of the structure, as depicted in Figure 3(a).The polarization-sensitive nature of the structure demonstrates the    practical applications in various fields including polarization detection and sensing [14], [19].Moreover, it can be observed that the absorber exhibits insensitivity to incident angles up to 30 degrees.However, beyond this threshold, the rate of absorption diminishes as the angle increases, as depicted in Figure 3(b).Moreover, an analysis has been conducted on the electric field (E-field), magnetic field (H-field), and distribution of surface current to gain a comprehensive understanding of the absorption peaks.Figs. 4 (a) and (b) show that the simultaneous excitation of E-field and H-field distribution on the top plane resonator creates absorption peaks at 9.94, 13.26, 14.92, and 15.80 GHz.Figs. 5 (a) and (b) demonstrates the surface current on top and ground plane are anti-parallel and responsible for the magnetic coupling [33].Nevertheless, the simultaneous excitation of both E-and H-fields leads to a significant enhancement in absorption performance.Table 1 presents a comparison of the proposed metamaterial absorber (MMA) with respect to the dimensions of the unit cell, specifically the size and thickness.The results indicate that the proposed MMA exhibits a compact and ultra-thin structure, which distinguishes it from previously reported multi-band absorbers.

III. MACHINE LEARNING REGRESSION MODELS
Regression models are powerful tools that can significantly reduce the time and resources needed to simulate complex systems.By using machine learning techniques, regression models can predict critical parameters and identify missing values.The process involves simulating the design, creating a comprehensive dataset, training the regression model, and predicting values for intermediate frequencies.Python is commonly used for implementing these regression models using simulated data.

A. SIMPLE REGRESSION MODELS
The relationships between the input parameters and the desired characteristics of the metamaterial absorbers can be understood with the help of simple regression models.In particular, we employ Decision trees and k-nearest neighbors regression techniques to capture and quantify the linear and nonlinear dependencies within the data.Decision trees are a simple and user-friendly method for ML that may be applied to both regression and classification applications.The process involves the recursive partitioning of data into increasingly smaller subsets, to achieve homogeneity within each subset.This can be a decent choice for regression situations with inadequate information, but it is prone to overfitting if not carefully tuned [34], [35].The k-nearest neighbors (KNN) algorithm is a non-parametric approach in machine learning that can be effectively employed for both classification and regression tasks.The functioning of this method involves identifying the k instances in the training dataset that exhibit the highest similarity to a novel instance.Subsequently, the label of the novel instance is predicted by considering the descriptions of the KNN.This can be an adequate option for regression problems with insufficient information, but it can be computationally expensive for large datasets [36].

B. BAGGING-BASED ENSEMBLE MODELS
Ensemble learning techniques are used to improve the robustness and generalisation of our metamaterial absorber optimisation models.Bagging-based Ensemble Models, such as Random Forest, Extra Trees and Bagging, are especially good at reducing overfitting and improving predictive accuracy.Random forest is an ML ensemble approach that mixes numerous decision trees to increase prediction accuracy.They operate by training several decision trees on distinct random selections of the training data and then averaging the individual tree's predictions.This can be a suitable solution for regression problems with inadequate data because it is less prone to overfitting than single decision trees [37], [38].Extra trees are a random forest variant that employs a more randomized technique to train individual trees [39].This has the potential to improve forecast accuracy, particularly in high-dimensional datasets [6], [40].Bagging regression technique combines many copies of a basic model, each trained on a separate bootstrap sample of the training data.This can help to improve forecast accuracy by reducing the variance of the fundamental model.However, Bagging might be computationally expensive for huge datasets [41].

C. BOOSTING-BASED ENSEMBLE MODELS
Boosting-based Ensemble Models iteratively combine weak learners to create a strong predictive model to improve model performance.The light gradient boosting machine (LightGBM) Regressor is well-known for its speed and effectiveness, particularly in instances with sparse datasets [42].The XGBoost Regressor (Extreme Gradient Boosting) is an iterative procedure that integrates extra trees into the model to correct faults produced by previous trees.XGBoost is well-known for its remarkable computing efficiency and great predicted accuracy, particularly when dealing with large datasets [27].The CatBoost algorithm targets categorical features to use the gradient boosting algorithm and integrates several methodologies to improve the algorithm performance when applied to datasets with categorical variables [43].Histogram-based gradient boosting makes use of histograms to represent the features of a given dataset.When applied to datasets with categorical features, this approach has the potential to improve the algorithm's performance [44].Gradient boosting regressors are specifically developed for regression problems, and they work by iteratively adding new trees into the model to correct errors produced by previous trees [45].
These regression algorithms can be assessed using several performance metrics, including MSE, RMSE, MAE, and Adjusted R-squared score (Adj-R 2 S).These metrics reveal the model's tendency for generalization, goodness-of-fit, and forecasting accuracy for MMAs.The subsequent equations can be employed to ascertain these performance indices: (1) (3) where 'N ' is the total number of samples used in the regression model's validation.Regression models were used to forecast the absorption coefficient value for various test sizes.The regression model was trained with 60% of the records and its prediction performance was tested with 40% of the records for the 40% test size.Similarly, with the 80% test size, the regression model was trained on a sample of 20% of the records before being tested on the remaining 80%.This technique was repeated with test sizes ranging from 50% to 70%.

IV. RESULTS AND DISCUSSIONS
The Adjusted-R 2 scores for various regression algorithms utilizing test sizes ranging from 40% to 80% of the data set are shown in Figure 6.The Adjusted-R 2 score is commonly used for assessing the goodness of fit of regression models, taking into account both the model's ability to explain variance in the data and the number of predictors utilized.Adjusted-R 2 values near 1 imply a stronger link between predicted and actual values.As the test size increased from 40% to 80% of the dataset, the adjusted-R 2 scores for all regression methods decreased suggesting that the percentage of training and testing data affects the model's prediction performance.Across all test sizes, the extra trees regressor consistently achieved the best adjusted-R 2 scores, indicating higher predictive performance.This regressor combines the advantages of randomization and extreme random splitting, resulting in decision trees with a higher amount of randomness and diversity.The enhanced randomization enables the additional trees regressor to capture a greater range of patterns while reducing overfitting.The RF regressor also performed well as it constructs a decision tree ensemble with random subsets of features and data samples, decreasing overfitting and improving generalization.predicted and actual values.Lower values are indicative of superior predictive performance.Between 40% and 80% of the dataset was tested in various test sizes during the evaluation.Table 2 shows ET and RF regressors consistently produced relatively low MAE values across all test sizes, with ET achieving MAE as low as 0.0007 and RF as low as 0.0038 for a test size of 0.4.In contrast, LGBM and HGB regressors exhibited higher MAE values, especially as the test size increased, suggesting that they may be more sensitive to larger test datasets.Table 3 and 4 shows that ET and RF regressors produced consistently low MSE and RMSE values across all test sizes.On the other hand, LGBM and HGB regressors had higher MSE and RMSE values, especially as the test size increased.Across all test sizes, the extra trees regressor consistently recorded the lowest MSE, RMSE, and MAE scores, demonstrating its superior predictive ability.This can be attributed to the extra trees regressor's ability to leverage randomization and extreme random splitting, adding more randomness and variety to the decision trees.Similarly, the KNN and RF regressor also demonstrated competitive results with relatively low error values.These observations emphasize the effectiveness of different regression techniques and the need for careful selection based on the specific dataset characteristics and desired predictive accuracy.
Figure 7 presents scatter plots illustrating the relationship between predicted and simulated absorption values for different regression models including DT, KNN, RF, ET, bagging, XGB, LGBM, HGB, CBR, and GBR regressors with test sizes of 0.4, 0.5, and 0.6.Among these ensemble models, the Extra Tree regressor consistently exhibits the best results in accurately predicting the absorption values.These figures show how well ensemble models and other regression models perform in predicting absorption values across a range of test sizes.The scatter plots provide visual representations of the model's performance, accuracy, and overall trends.The inclusion of multiple test sizes from 0.4 to 0.6, enables an in-depth investigation of the model's performance when training and testing data are in varying proportions.In Figure 8, we present a comparative analysis of computational times for various machine learning regression techniques when applied to a test dataset with a size of 40%.Notably, DT and KNN demonstrate remarkable computational efficiency, with execution times of 0.00485 seconds and 0.00459 seconds, respectively.Extra Trees also exhibited competitive computational performance, with an average time of 0.16044 seconds.These findings suggest that DT, KNN, and ET are well-suited for scenarios where rapid predictions are crucial.The longer computational times of CBR and LGBM, at 0.6938 seconds and 0.6061 seconds, respectively, show that despite their superior predictive performance, these models demand more computational power.These computational trade-offs should be taken into account by researchers when choosing regression techniques for particular applications.

V. CONCLUSION
The ultra-thin MMA at microwave frequencies exhibited four distinct peaks of perfect absorption, occurring at frequencies of 9.948, 13.26, 14.92, and 15.80 GHz.The absorption mechanism can be understood by examining the normalized impedance, E-field, H-field, and surface current distribution.In contrast to absorbers that have been previously reported, the proposed compact and ultrathin absorber exhibits perfect absorption at all resonant frequencies.The analysis of the equivalent circuit was carried out to explain the fundamental electromagnetic behavior of the proposed absorber.This study shows the rapid growth of MMAs but traditional design methods are time-and computationally intensive, making optimization difficult.To address this, this article provides a comprehensive comparative analysis of various ML regressors such as DT, KNN, RF, ET, bagging, XGB, LGBM, HGB, CBR, and GBR regressors in forecasting the performance of metamaterial absorbers.The ET regressor demonstrates outstanding performance in predicting MMA performance for an 80% test size with RMSE of 0.0147, MAE of 0.0062, MSE of 0.003, and an Adjusted R 2 of 0.9964.Overall, this study has significant implications for optimizing metamaterial absorber designs for microwave applications such as polarization imaging, detection, sensing, and stealth technology using ML regressor approaches.

FIGURE 1 .
FIGURE 1.(a) Perspective view, (b) Top View, (c) Absorption (A), reflection (R) and transmission (T) response, and (d) Real and imaginary normalized impedance (e) Absorption response for TM and TE mode of the proposed metamaterial absorber.
at 9.948, 13.26, 14.92, and 15.80 GHz, respectively, as shown in Fig.1(d).The absorption performance of the MMA under TM and TE polarization is depicted in Fig.1 (e), where the absorptivity at resonant frequencies of 9.47 and 14.74 GHz is 76.1 and 68.4%, respectively, for TM polarization, demonstrating the polarisation dependence of the proposed MMA.

FIGURE 5 .
FIGURE 5. Surface current distribution for the top plane and ground plane of the MMA at (a) 9.94, (b) 13.26, (c) 14.92, and (d) 15.80 GHz.

FIGURE 8 .
FIGURE 8. Computational time comparison of various machine learning regression techniques for test size of 40%.

TABLE 1 .
Comparative performance with Multi-Band MMAs reported in prior studies.

TABLE 3 .
Mean squared error (MSE) values.The MAE, MSE, and RMSE performance metrics for various regression techniques are shown in Table2, 3 and 4. The accuracy of regression models is assessed using these metrics, which involve the examination of discrepancies between