Metasurface-Based Solar Absorption Prediction System Using Artificial Intelligence

Abstract

Solar energy is a significant, environment-friendly source of renewable energy. The solar absorber transforms solar radiation into heat energy as an effective green energy source. Therefore, increasing its absorbing capacity can improve a solar absorber’s effectiveness. This paper proposes a tungsten tantalum alloy with silicon dioxide (WTa-SiO₂) ceramic layer-based solar absorber system with two different metasurfaces to enhance absorptivity and boost the solar absorber efficacy. The absorbance is also improved by adjusting the resonator thickness and material thickness, and the maximum visible light absorption is achieved by the suggested solar filter design. Moreover, Golden Eagle Optimization (GE)-based deep AlexNet algorithm is proposed for predicting the parameter variation and their effect on absorbance. The optimization technique is used to increase the effectiveness of the solar absorber by optimizing the design parameters. The features from the WTa-SiO₂ design are extracted by the proposed Principal Component-Autoencoder (PC-AE) method. Experimental results show that the proposed system can effectively predict absorptivity with a reduced computational time. The proposed method demonstrates superior prediction performance with an absorption prediction efficiency of 99.8% compared to the existing methods. Thus, the proposed WTa-SiO₂ metasurface-based solar absorber can be used for photovoltaic applications.

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Due to rapid industrialization and rising home energy use, the energy demand is increasing endlessly. Renewable energy sources are naturally existing resources and do not pollute the environment [1]. New methods of using these naturally existing resources and turning them into valuable products are constantly being developed via recent studies. The world has abundant naturally regenerating renewable energy sources, including the sun, water, wind, and air [2]. It takes many substantially developed locations to transform one energy source into another using air, wind, or water. On Earth, there is an abundant supply of the sun’s pure, natural, renewable energy [3]. One of the most excellent methods to deal with the rising energy demand is the use of solar energy [4]. There is still room to improve the efficiency of solar energy use. In addition, solar energy is readily transformable into various energy sources that can be used to develop photovoltaic systems, solar cells, and solar absorbers [5].

An ideal absorber is prudently wise for many applications, like solar energy harvesting, light control, and sensing [6]. By absorbing incoming solar energy, the coating layers on the solar cell aid in improving the absorption. Solar energy harvesting also has a significant interest in broadband absorbers that span the whole solar spectrum [7]. Examples of material structures used as high-performance solar absorbers include photonic crystals, dense nanorods, multilayer planar photonic frameworks, and nanotube films [8]. Maximum energy is present in the visible and infrared spectrum when light reaches the earth. The design of broadband absorbers can utilize the metasurface absorber [9]. The metasurface’s two-dimensional planar surface has recently attracted the attention of scientists because of its distinct electro-optical properties and small size as opposed to bulkier optical systems [10]. As a result, they are used as polarizers, detectors, and absorbers that operate at a wide range of frequencies, including microwave, visible, infrared, ultraviolet, terahertz [11].

Due to losses and a complex three-dimensional structure, the design of metamaterial-based solar absorbers is challenging [12]. To reduce mid- and far-IR emissivity, a solar absorber with enhanced absorption bandwidth is highly desired, encompassing the full near- and visible-IR spectrum with a configurable cutoff wavelength [13]. Predicting absorption performance is an effective way to improve performance in future designs. The inverse design approach based on deep learning, one of the most recent and frequently used sophisticated computer algorithms, allows for quick design, and exhibits excellent performance, particularly when used to the global optimization issue [14]. To forecast the optical reactions, however, a massive quantity of data gathered computationally is costly and a blind early simulation is needed. A quick, highly effective, and accurate result synergetic approach for developing photonic applications is described here to address the drawback of the traditional deep-learning-based inverse parametric study [15]. To avoid blind computations, this method uses an implicit deep learning method that unintentionally uses accurate approximation as the guideline [16, 17]. The study of absorbers with unique and challenging spectral patterns serves as a case study for the synergetic paradigm that has been presented. Our method resolves the tricky problems associated with batch models and the intricate and time-consuming procedures that plagued conventional techniques [18–20].

Based on the results from the WTa-SiO₂ ceramic layer, we have engineered the design of a metasurface-solar absorber. In addition, it is recommended to use deep AlexNet with Golden Eagle Optimization (GE-deep AlexNet) to predict absorption values for varying metasurface thickness and resonator thickness with different wavelengths. A standardized optimal deep-learning model that concurrently considers all the essential dimensional factors is needed to forecast the optical response of solar absorbers based on metasurfaces (such as periodicity, height, width, and aspect ratio). The design and modeling section discusses the design parameters for the proposed absorber. The proposed design’s absorption is enhanced by adjusting the resonator and substrate thickness parameters. Furthermore, the Principal Component-Autoencoder (PC-AE) approach is used to extract the features from the preprocessed dataset for effective prediction of performance. This study’s implementation makes use of the MATLAB programming language. Comparisons between the simulation results and traditional approaches demonstrate the effectiveness of the proposed strategy. The rest of the paper is summarized as follows: The recent literature related to this research is detailed in Section 2. The proposed methodology of the research design is explained in Section 3. The result and its performances are analyzed in Section 4. At last, the work is concluded in Section 5.

2. Literature Review

Good numbers of solar absorbers and prediction systems have already been reported in the open literature. For example, in [21], Patel et al. propose a solar absorber that absorbs most of the energy from the available solar spectra, such as the visible spectrum and ultraviolet spectrum emissions. The use of a long short-term memory model to forecast absorption values for various changes in substrate thickness and resonator thickness for upcoming wavelengths is an innovative aspect of this research. The studies’ findings demonstrate that the prediction system can effectively estimate absorption levels using less time and resources during simulation. Patel et al. put out a solar absorber model employing metasurfaces and machine learning for polynomial regression analysis [22]. Here, circular array-based metasurface layout and circular array metasurface architecture are examined in the visible, infrared, and ultraviolet regions with a wavelength range of 0.2 m to 0.8 m. The experimental findings demonstrate that the elevated-degree polynomial regression analysis can produce prediction efficiencies greater than 0.99 R², and the visible zone has the highest median absorption of 89%. A multilayer grating structure made of titanium and gallium arsenide was used by Zhang et al. to build a broadband absorber [23]. When simulating the specified model using the finite difference time domain approach, they discovered that the absorption effectiveness was 99.69% at 867 nm, which is quite close to absolute absorption. This metamaterial grating optimal absorber type is expected to be widely used in optical disciplines such as thermal electronics, optical monitoring, and infrared detection. In [24], Parmar et al. suggested graphene-based solar absorber architecture with two distinct metasurfaces for better absorption and greater solar absorber efficiency. The metasurfaces’ symmetry and asymmetry are considered while choosing them (L and O-shape). A convolutional neural network (CNN) in one dimension is developed to calculate intermediary wavelength absorption estimates for a range of values, and regression is utilized to build a machine learning algorithm. Experimental findings show that a highly accurate prediction of absorption levels is possible. A SiO₂ substrate-based double layer of a gold multipattern swastika (DLMP) resonator was suggested by Patel et al. [25], which achieved a bandwidth of 2516 nm, an absorptanceI spectrum of 0.314 to 2.830 m and an absorption coefficient of 90%. A generalized regression neural network was employed to create regression models that can learn and forecast the behavior of absorbers under various situations. Experimental findings demonstrate that this model has a high degree of accuracy in predicting absorber behavior and may slash simulation resources and time needs by 80%. In [26], Donda et al. present a deep learning-based technique for modeling acoustic metasurface absorbers that drastically cuts the characterization time while preserving accurately. Based on a CNN, the developed network can simulate a broad absorption spectrum reaction in milliseconds. In addition, based on the idea of supercells, they presented a broadband low-frequency metasurface absorber by linking unit cells with various characteristics. In [27], Patel et al. proposed a graphene absorber for the visible and near-infrared (NIR) regions. Here, an MgF₂ substrate is put on top of the gold resonator, which is then covered in a gold base layer. The use of polynomial regression models to forecast the absorption capacity across a range of wavelengths for different angles, substrate thickness, resonator thickness, and needle-point breadth is a new aspect of this study. According to experimental findings, the polynomial regression model can accurately predict the values of absorption capacity (R² score). To boost the absorption in the infrared, visible, and ultraviolet spectrums, a metasurface solar absorber based on Ge₂Sb₂Te₅ substrate was proposed by Patel et al. [28]. The absorber is also examined using a machine learning algorithm to forecast the absorptivity at various wavelengths. Experiments are used to evaluate the K-nearest neighbors (KNN) accuracy and the regressor algorithms for forecasting the absorption with missing wavelength estimates. The experimental results demonstrate that a smaller value of K in a KNN-regressor system can achieve good prediction accuracy (greater than 0.9 estimated (R²)). A summary of some of the previous works is listed in Table 1.

Table 1

Summarization of literature review.

References	Year	Methodology	Parameters/metrics	Merits/limitation
Patel et al. [21]	2022a	Graphene and LSTM	Resonator and substrate thickness	(i) Precisely predict absorption levels, shorten simulation time, and use fewer resources
				(ii) Training takes more time and memory
				(iii) High computation burden
Patel et al. [22]	2022b	SiO2 and regression models	Resonator thickness, SiO2 width, SiO2 height, and chemical potential of graphene	(i) More than 0.99 prediction accuracy
				(ii) It was unable to match complicated datasets accurately
Zhang et al. [23]	2022	Titanium and gallium arsenide metal	Absorber absorption efficiency	(i) The building was effectively angle-insensitive and capable of maintaining strong absorption
				(ii) Yet effective validation is not achieved in this research
Parmar et al. [24]	2022	CNN-regression	Incidence angle, substrate, and resonator thickness	(i) Precision at a high level
				(ii) Lack of spatial invariance with respect to the supplied data
Patel et al. [25]	2022c	General regression neural network	Different structural parameters	(i) Decreased simulation time and increased accuracy
				(ii) It may be quite large, which would make calculation difficult. There is not a perfect way to make things better
Donda et al. [26]	2021	CNN	Absorption spectrum response and low frequency	(i) It appeals to applications that need quick on-demand design
				(ii) However, the performance of the model is very poor
Patel et al. [27]	2022d	Polynomial regression models and MgF2 substrate	Substrate and resonator thickness, thickness, needlepoint, and plus shape width	(i) Ability to absorb with great accuracy
				(ii) The probability of prediction error is higher than in other methods
Patel et al. [28]	2022e	KNN and Ge2Sb2Te5 (GST) substrate	Thickness: metasurface, substrate, and ground plane	(i) Superior prediction accuracy
				(ii) The prediction phase is very slow due to the higher amount of data

3. Proposed Methodology

The proposed network design for the prediction of the metasurface’s solar absorbance from its WTa-SiO₂ ceramic basis is shown in Figure 1.

[figure(s) omitted; refer to PDF]

In the following sections, we go through several aspects of the proposed model and its justification, beginning with how to represent the metasurface, moving on to how to extract pertinent features using the PC-AE approach, and concluding with a discussion of utilizing GE-deep AlexNet to produce optical properties.

3.1. WTa-SiO₂ Experimental Design

This part introduces the metasurface-based solar absorber layout that uses WTa-SiO₂ as the phase transition material. The tungsten’s phase transition reduces heat loss during annealing; cermet changes significantly impact the effectiveness of spectrally selective absorbances. The deployment of the metasurface enhances the absorption of solar absorbers. Cermets were created by cosputtering the appropriate metal along with dielectric objects with various ratios of volume of WTa-SiO₂ of 2 : 3, 1 : 2, and 1 : 3, respectively. Acetone and pure ethyl alcohol were used to sanitize the substrates, and the primary compartment was then emptied to a pressure smaller than 4 × 10⁻⁴ Pa before sparking. Every aspect of the absorbers is placed in an argon atmosphere at a pressure of 0.3 Pa. The complete design diagram is shown in Figure 2. We have taken an O-shaped metasurface element whose inner and outer length is illustrated in Figure 2(a). The 3D view is portrayed in Figure 2(b), while the 2D front view of the proposed metasurface-based solar absorber is shown in Figure 2(c). Figure 2(c) shows that W and Ta metal particles are preferred to form the WTa solid solution alloy for the WTa-SiO₂ ceramic. The annealing process causes the free energy of the entire system to seem to be in the lowest energy state due to the diffusion and aggregation of metal atoms in the ceramic layer. Furthermore, in the case of the binary alloy structure, the atoms with low surface energy often prefer those with bigger atom radii to precipitate from the binary alloy and create surface segregation. To assure the greater stability of absorbers at high working temperatures, SiO₂ is generated, which prevents further migration and diffusion of alloy atoms.

[figure(s) omitted; refer to PDF]

In the case of the WTa bimetallic alloy, Ta’s atomic radius (RTa, 146 pm) is greater than W’s (R_W, 139 pm) [29]. Furthermore, as the Ta metal has lower surface free energies than W, the Ta surface segregates on metal surfaces during the melting process. When there is still oxygen present, it is unavoidable that the relocated Ta atom will be an oxide rather than a W atom. Because Ta₂O₅ has a lower Gibbs free energy than WO₃ during the melting process, it is a durable oxide [30]. In order to guarantee the better stability of absorbers at high working temperatures, the formed stable oxide layer (Ta₂O₅) acts as a protective layer by inhibiting the further migration and dispersion of alloy atoms. The experiment estimates the absorption analysis based on reflectance and transmittance [31].

3.2. Input Design Data

From the experimentally designed model, the parameters of metasurface height, width, aspect ratio, periodicity, physical parameter resonator thickness, substrate thickness, scattering rate, and so on are collected and stored in the dataset for performance prediction.

3.3. Data and Preprocessing

The designed setup parameters and values are collected and formed as a dataset for simulation validation using an optimization-based deep learning method. The developed dataset may have some irreversible and missing data processed by the preprocessing filter methods to normalize the data for further use.

3.4. PC-AE Feature Extraction

The preprocessing work has already been finished in preparation for future improvements on characteristics that may be extracted for classification. We extract certain features using a combination of an autoencoder and principal component analysis. Encoder and decoder submodels make up an autoencoder. The encoder compresses the input, and the decoder tries to reconstruct the input from the encoder’s compressed form. Following training, the decoder model is abandoned, and the encoder model is saved. However, the autoencoder’s performance is deteriorating because key variables are misunderstood. As a result, the performance of effective feature vector extraction is obtained by combining the PC algorithm. To prepare raw data for GE-deep AlexNet model training, the encoder may then be used to extract features from the data. In our study, combining these two algorithms can significantly impact variables’ extraction and significance. Both extraction strategies have their own identities for detecting certain characteristics. The PC-AE combination algorithm is defined as follows:

Initialize the preprocessed data into the PC-AE algorithm in the input layer function. Then, the dense layer function is applied for the feature extraction by multiplying the weight matrix of input features via the activation function using the following equation:$$\begin{array}{}\text{(1)}& y^j={z^j}^T\ast m+d\text{,}\end{array}$$where m is the matrix of weight, z is the input matrix considered for the observation count, feature count for covariance, and the bias is represented as d. The activation function used for this layer is expressed in the following equation:$$\begin{array}{}\text{(2)}& {\widehat{x}}^j=b^j=\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{m}\mathrm{o}\mathrm{i}\mathrm{d}{y}^j\text{.}\end{array}$$

The encoder and decoder function is executed using the following equation:$$\begin{array}{}\text{(3)}& E{y}^j={\widehat{x}}^j{z^j}^T\ast m+d\text{,}\end{array}$$where ${\widehat{x}}^j$ is the sigmoid activation function. The covariance matrix is estimated using the following equation:$$\begin{array}{}\text{(4)}& G=\mathrm{C}\mathrm{o}\mathrm{v}z=Ez-\mu {z-\mu }^T\text{.}\end{array}$$

Estimate the eigenvalue ${\alpha }_j$ and the Eigen vector $e_1,e_2,\dots \dots e_{nj}$, and the value for $G$ covariance is $j=\mathrm{1,2,3}\dots n$, ordering the values of Eigen is the order of the descending function.$$\begin{array}{}\text{(5)}& \alpha I-G=0\text{.}\end{array}$$

Furthermore, the batch normalization function of the autoencoder is applied for sorting the eigenvalues. Data are transformed to have a mean of 0 and a standard deviation of 1 through the process of normalization. First, we must get the mean of this concealed activation given that we have the batch input from layer e in this stage.$$\begin{array}{}\text{(6)}& \delta =\frac{1}{n}{\displaystyle \sum }{e}_j\text{.}\end{array}$$

The neuron count is denoted as $n$ at the $e$ layer. The next step is to determine the standard deviation of the concealed activations after we have activated at the end.$$\begin{array}{}\text{(7)}& \sigma ={\frac{1}{n}{\displaystyle \sum }{e_j-\delta }^2}^{1/2}\text{.}\end{array}$$

Furthermore, the mean and standard deviation are available. Using these numbers, we will normalize the concealed activations. To do this, we will take the average out of each input and divide the total by the smoothing factor $\phi$ plus the sum of the standard deviation. By preventing division by a zero point, the smoothing term $\phi$ ensures numerical stability inside the operation.$$\begin{array}{}\text{(8)}& e_{j\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}=\frac{e_j-\delta }{\sigma +\phi }\text{.}\end{array}$$

The input is resized and offset during the final procedure. Here, the two autoencoder algorithm components gamma and delta enter the data (beta). With the help of these parameters, the vector comprising the results of the preceding operations is rescaled and shifted.$$\begin{array}{}\text{(9)}& e_{j\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}=\gamma e_{j\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}+\beta \text{.}\end{array}$$

These two parameters may be learned, and during training, an autoencoder makes sure that the best values for $\gamma$ and $\beta$ are employed. This will make it possible to accurately normalize each batch. Then, select the first largest eigenvectors. Following is how we determine how much the $j^{\mathrm{t}\mathrm{h}}$ feature component contributed to the outcome of the feature extraction:$$\begin{array}{}\text{(10)}& f_j={\displaystyle \sum }_v^n{e}_{vj}\text{,}\end{array}$$where $j^{\mathrm{t}\mathrm{h}}$ entry of $e_v,j=\mathrm{1,2,3}\dots \dots n,v=\mathrm{1,2,3}\dots n$ is denoted as $e_{vj}$ and the absolute value of $e_{vj}$ is denoted as $e_{vj}$. Furthermore, $f_j$ is sorted as the descending order function, and it is stored in the order of $g_j$. The recommended study’s objective is to gather more valuable characteristics that are perfect for deep learning algorithms’ input to boost accuracy. The method used an autoencoder on a labeled dataset during its training phase. It looks for the variable’s coefficient that best captures the situation, assesses the error value, and attempts to keep it as low as possible for subsequent steps. Then, covariance is applied to the dataset, and the PC analysis is used to predict the optimal feature for the dependent variables with eigenvectors and data matrices.

3.5. GE-Deep AlexNet Architecture

After extracting the features, the optimal absorbance is predicted using the proposed GE-deep AlexNet strategy. Figure 3 depicts the GE-deep AlexNet architecture. Deep AlexNet enables multi-GPU training by dividing the model’s neurons in half and training on two GPUs. This shortens the training time while also enabling the training of a larger model. The vanishing gradient problem was resolved since the gradient values were no longer restricted to a certain range. The GEO algorithm improves the hyperparameters of AlexNet.

[figure(s) omitted; refer to PDF]

3.5.1. Deep AlexNet Algorithm

Eight layers, including three fully connected layers and five convolutional layers, balance out the architecture. Rectified linear units (ReLU) are used by deep AlexNet in place of the standard tanh function. Through the convolutional layer, equation (11) is used to compute the convolution for the featured data.$$\begin{array}{}\text{(11)}& Aa,b=y\ast wa,b={\displaystyle \sum }_i{\displaystyle \sum }_{j}yi,j\times wa-i,b-j\text{,}\end{array}$$where $Aa,b$ is the output of the convolutional layer that passes the data on to the subsequent layer. The symbol denotes the convolution process, $w$ is the kernel or filter matrix, and denotes the input data, which is made up of a collection of data. The input and kernel’s element-by-element product is computed, aggregated, and then expressed as the corresponding point in the next layer. The result of the mathematical functions was carried out through the convolutional layer and then passed on to the nonlinearity layer, which is the next layer. This layer can be utilized to modify or remove the output that was created. The output is saturated or limited with this layer. However, the convolutional layer has a nonlinearity layer permanently included in it. As the following equations show, the rectified linear unit (ReLU) gives simpler explanations of both the functions and gradient.$$\begin{array}{c}\text{(12)}& \mathrm{R}\mathrm{e}\mathrm{L}\mathrm{U}y=\max 0,y\text{,}\frac{d}{\mathrm{d}y}\mathrm{R}\mathrm{e}\mathrm{L}\mathrm{U}y=1\mathrm{i}\mathrm{f}y>0;0\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}\text{.}\end{array}$$

For the prediction, it is first necessary that the output feature map from pooling have a fixed size. For instance, no matter how big the filters are, when max pooling is applied to each of the 256 filters, the output is 256 dimensions. In order to reduce the data dimensionality and reduce the amount of time needed for data training for subsequent layers in the network, downsampling is a crucial step in the layer pooling process. The fully connected layer follows the pooling layer and links and organizes every neuron in a neural network. As a result, every neuron in a completely connected layer is directly coupled to every neuron in the layer above it and the layer below it. The softmax layer, the final layer in the model that is being presented, is used to calculate the probability distribution. The softmax function is described in the following equation:$$\begin{array}{}\text{(13)}& P_j=\frac{e^{y_j}}{{{\displaystyle \sum }}_{j=1}^N{e}^{y_j}}\text{,}\end{array}$$where $y_j$ is the output prior the softmax function and the overall neuron output is denoted as $N$. Consequently, the performance of deep AlexNet is improved by parameter tuning using the GEO algorithm.

3.5.2. Golden Eagle Optimization

In this research, we used the activation function and other hyperparameters, such as the number of layers in a system and the number of each layer’s nodes. The learning rate value is one of the supplementary GE technique parameters that must be given. The recommended deep AlexNet method selects the optimal settings while considering the fitness of the GE algorithm’s circling and hunting behaviors. Each data $q$ start by randomly selecting the traits of another data $h$ and then makes a circle around the best location that data $h$ have so far visited each iteration. Their personal recollections can be circled in the data $h\in \mathrm{1,2},\dots N$. Each data must select a feature at each cycle to carry out the cruise and assault activities. The characteristics employed in this method are based on the top conclusion the data flock has come to so far. Each piece of knowledge can recall the most effective answer it has so far discovered. We propose a random one-to-one mapping approach where each data randomly select their properties for the current iteration from the memories of the other flock members to help data better navigate the terrain. It is critical to recognize that the traits preferred frequently diverge from those of the nearby or distant prey. This strategy assigns or maps each object to a single, unique piece of data.

According to this strategy, a single, distinct piece of data is assigned to or mapped to each memory feature. Each data on the chosen features then carry out the attack and cruise operations. A vector that begins at the data’s current location and ends at the location where the features are stored in the data memory may be used to depict the attack. The data assault vector $q$ may be identified using the following equation:$$\begin{array}{}\text{(14)}& {\stackrel{⟶}{O}}_q={\stackrel{⟶}{Z}}_h^{\ast }-{\stackrel{⟶}{Z}}_h\text{.}\end{array}$$

The exploitation of data $q$ is denoted as ${\stackrel{⟶}{O}}_q$, the best features selected by the data $h$ are denoted as ${\stackrel{⟶}{Z}}_h^{\ast }$, and the present location of the features in the data $k$ are denoted as ${\stackrel{⟶}{Z}}_h$. The exploitation vector directs the data populace to the most well-liked locations. The exploration vector is calculated based on the exploitation. While the exploration vector is perpendicular to the circle, exploitation is parallel to it. Alternately, the exploration might be seen as the linear pace of the data in proportion to the features. Equation (15) is used to determine the tangent hyperplane’s dimension space $u$:$$\begin{array}{}\text{(15)}& w_1{y}_1+w_2{y}_2+\cdots \cdots \cdots w_l{y}_l=g\Rightarrow {\displaystyle \sum }_{i=1}^u{w}_i{y}_i=g\text{,}\end{array}$$where $w_1,w_2\dots w_u$ are represented for the ordinary vector and $Y=y_1,y_2,\dots y_u$ are the changing vector of $i^{\mathrm{t}\mathrm{h}}$ node. The exploration hyperplane’s overall depiction of the destination location is expressed using the following equation:$$\begin{array}{}\text{(16)}& {\stackrel{⟶}{E}}_q=k_1=r,k_2=r,k_v=\frac{g-{{\displaystyle \sum }}_{i;i\ne b}{Q}_i}{Q_b},\dots k_l=r\text{.}\end{array}$$

Once the goal point has been determined, the exploration vector for the data $q$ is now computed $t$ iteratively. These arbitrary numbers between zero and one make up the components of the obtained destination location. It is interesting to note that the data population is pushed outside the memory-stored regions by the exploration vector. Exploration and exploitation are both involved in the data migration process. To create the step vector for data in iteration, we utilize the following equation:$$\begin{array}{}\text{(17)}& ∆Y_q={\stackrel{⟶}{r}}_1{d}_s^0+\frac{t}{T}{d}_s^T-d_s^0\frac{{\stackrel{⟶}{Q}}_q}{{\stackrel{⟶}{Q}}_q}+{\stackrel{⟶}{r}}_2{d}_e^0-\frac{t}{T}{d}_e^T-d_e^0\frac{{\stackrel{⟶}{E}}_q}{{\stackrel{⟶}{E}}_q}\text{,}\end{array}$$where the exploitation coefficient in $T$ iteration is represented as $d_s^T$ and the exploration coefficient in $T$ iteration is considered as $d_e^T$, the random vector is in the limit of [0, 1] is denoted as ${\stackrel{⟶}{r}}_1$ and ${\stackrel{⟶}{r}}_2$, the Euclidean norm of the exploitation and exploration is denoted as ${\stackrel{⟶}{Q}}_q$ and ${\stackrel{⟶}{E}}_q$. Beginning with the initial random value, loop through all iterations before lowering the random value in accordance with alpha. The conditions are terminated if the optimal solution is achieved until it returns. To find the position of the data in iteration $t$, the step vector in iteration $t$ is simply added to the positions in iteration $t+1$.$$\begin{array}{}\text{(18)}& y^{t+1}=y^t+∆y_q^t\text{.}\end{array}$$

The memory of these features is updated to reflect the new location if the new location of the data $k$ is more appropriate than the position that is previously recorded in its memory. Otherwise, while the features are kept at the new place, the memory is unaffected. In the present version, each feature in the population has a randomly chosen position that revolves around it. The step vector and the new position for the subsequent iteration are then decided, followed by the calculation of exploitation and exploration. Up until one or more of the termination requirements are satisfied, this loop is still being executed.

4. Results and Discussion

MATLAB 2019b, windows 7 intel core, 4 GB RAM, 64-bit operating system are used to implement and train using optimized improved deep learning methods in metasurface prediction of solar absorbance in energy harvesting application.

4.1. Evaluation Metrics

4.1.1. Accuracy

Accuracy measures how closely the experimental value matches the actual value.$$\begin{array}{}\text{(19)}& \mathrm{A}\mathrm{c}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{y}=\frac{\mathrm{C}\mathrm{o}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{t}}{\mathrm{O}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{t}}\text{.}\end{array}$$

The performance of the suggested technique, which is taught using the remaining records of simulated data, is assessed using records that were randomly chosen. The prediction accuracy of suggested models is assessed using the R² score as a criterion. Equation (19) is used to compute the R² score$$\begin{array}{c}\text{(20)}& R^2=1-\frac{\mathrm{S}{\mathrm{S}}_r}{\mathrm{S}{\mathrm{S}}_t}\text{,}\mathrm{S}{\mathrm{S}}_r={\displaystyle \sum }_{j=1}^M{\mathrm{E}\mathrm{x}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{T}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}{\mathrm{t}}_j-\mathrm{M}\mathrm{e}\mathrm{a}\mathrm{n}⥂\mathrm{T}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}⥂\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{e}}^2\text{,}\\ \mathrm{S}{\mathrm{S}}_t={\displaystyle \sum }_{j=1}^M{\mathrm{M}\mathrm{e}\mathrm{a}\mathrm{n}⥂\mathrm{T}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}⥂\mathrm{r}\mathrm{a}\mathrm{t}{\mathrm{e}}_j-\mathrm{E}\mathrm{x}\mathrm{a}\mathrm{c}\mathrm{t}⥂⥂⥂\mathrm{T}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}⥂\mathrm{r}\mathrm{a}\mathrm{t}{\mathrm{e}}_j}^2\text{.}\end{array}$$Here, $\mathrm{S}{\mathrm{S}}_r$ is the sum of squares of the residual errors and $\mathrm{S}{\mathrm{S}}_t$ is the total sum of the errors, and $M$ is the number of testing records.

4.1.2. Mean Absolute Percentage Error (MAPE)

It is employed to determine the pattern of variation in absorption values over a difference in wavelength values under various circumstances. The developed pattern is then used to predict future values, in this case absorption for upcoming wavelengths. The accuracy of time-series models is measured using MAPE. In equation (21), the method for calculating MAPE is shown.$$\begin{array}{}\text{(21)}& \mathrm{M}\mathrm{A}\mathrm{P}\mathrm{E}=\frac{1}{n}{\displaystyle \sum }_{j=1}^n{\frac{\mathrm{A}\mathrm{c}\mathrm{t}\mathrm{u}\mathrm{a}\mathrm{l}\mathrm{v}\mathrm{a}\mathrm{l}\mathrm{u}{\mathrm{e}}_j-\mathrm{P}\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{c}\mathrm{t}\mathrm{e}\mathrm{d}\mathrm{v}\mathrm{a}\mathrm{l}\mathrm{u}{\mathrm{e}}_j}{\mathrm{A}\mathrm{c}\mathrm{t}\mathrm{u}\mathrm{a}\mathrm{l}\mathrm{v}\mathrm{a}\mathrm{l}\mathrm{u}{\mathrm{e}}_j}}^2\text{,}\end{array}$$where $n$ is the number of forecasted values by the model after training. The predicted performance is validated via the value of MAPE.

4.2. Performance Analysis

The input of developed WTa-SiO₂ solar absorbance of metasurface is preprocessed for removing disturbance in the data and applied PC-AE algorithm. From there, the optimal features are selected for the prediction algorithm based on the eigenvalues and eigenvectors. The highest eigenvalue is considered as 50. Consequently, the GE-deep AlexNet algorithm is applied for the prediction performance. The optimized hyper parameters learning rate 0.01, batch size 128, and 15 epochs are obtained using GE for deep AlexNet algorithm. The workflow of the proposed prediction model is detailed in Table 2.

Table 2

Workflow of the proposed prediction model.

Name of layers	Description
Input data	Featured metasurface solar absorbance
ConV1	11 × 11 conv/stride 4 and [0 0 0 0] of padding
ReLu1	Activation 1
Pool1	3 × 3 max pooling/stride 2 and [0 0 0 0] of padding
ConV 2	5 × 5 conv/stride 2 and [2 2 2 2] of padding
ReLu 2	Activation 2
Pool 2	3 × 3 max pooling/stride 2 and [0 0 0 0] of padding
ConV 3	3 × 3 conv/stride 1 and [1 1 1 1] of padding
ReLu 3	Activation 3
Pool 3	3 × 3 max pooling/stride 2 and [0 0 0 0] of padding
ConV 4	3 × 3 conv/stride 1 and [1 1 1 1] of padding
ReLu4	Activation 4
Pool 4	3 × 3 max pooling/stride 2 and [0 0 0 0] of padding
ConV 5	3 × 3 conv/stride 1 and [1 1 1 1] of padding
ReLu 5	Activation 5
Pool 5	3 × 3 max pooling/stride 2 and [0 0 0 0] of padding
Fully connected 1	4096FC
Dropout	50%
Fully connected 2	4096FC
Fully connected 3	1000FC
Softmax	Softmax activation
Ouptut	Absorption prediction

By altering several physical factors including the metasurface thickness, resonator thickness, and angle of incidence, a complete study of the high-performing architecture of the O-shaped perforated metamaterial-based solar absorber is conducted. Figures 4(a) and 4(b) displays the dependence of metasurface thickness and absorbance on the corresponding wavelength. The change in the absorption response with regard to the variation in metasurface thickness is shown in Figure 4(a). There is a 0.2 μm step increase in the metasurface thickness, which ranges from 0.1 to 1.0 μm. It is clear that this adjustment has little to no impact on the absorption response. Therefore, we may conclude that the metasurface thickness is fixed at 0.3 μm to keep the solar absorber affordable.

[figure(s) omitted; refer to PDF]

Figure 5 displays the simulation and model predicted results for a few random test specimens for the solar absorber dataset. As can be seen, the model’s prediction of the optical absorption effect and the simulated response accord well. Several very effective absorber unit cells with more than 80% absorbance across the solar spectrum are shown in graphical representation. From Figure 5, it is revealed that a unit cell can exhibit excellent visible-only absorption efficiency and such absorbers are quite helpful in numerous applications. Therefore, we can use the model to determine the efficiency if the designed absorber structure covers the complete spectrum. For five test scenarios, the proposed method’s predicted absorption value is compared to the actual absorption value and is portrayed in Figure 6. In the test, the resonator thickness is either 0.6 μm, 0.8 μm, or 1.0 μm while the incidence angel takes the value of 0°, 10°, and 20°. These findings clearly indicate a possible use for the enhancement of photovoltaic devices as well, where the concentration of solar radiation is high. Since the resonator structure of the shown device is symmetrical, we will able to get the same absorption response.

[figure(s) omitted; refer to PDF]

For the solar absorber to be widely used in nature, it must be insensitive and polarization independent to large incidence angles. The wide-angle and angle insensitive characteristics of the suggested O-shaped perforated metamaterial solar absorber are demonstrated in Figures 6(d)–6(f). The absorption response is the same for most angles, except for the 10° angle of incidence, as shown in Figure 6(e). The mean absorption for the whole area, ranging from 0.6 to 1.0 μm, is still approximately 90% for 10°. And for the remaining incidence angles, the lower absorption response is around 93%. Therefore, we can say that the suggested solar absorber is broad angle sensitive for 0° to 20°. In simulations, the first 80% of simulation data are used to train the GE-deep AlexNet-based prediction model, while the remaining 20% of records are used to evaluate how well the designs predict the future. To predict the absorption value for next wavelengths, simulations are run utilizing various durations of previous inputs.

Figures 7(a)–7(c) show training loss, MAPE, predicted absorption values from suggested approaches, and actual absorption values for a metasurface thickness of 0.6 μm. Similarly, the identical data for metasurface thicknesses of 0.8 m and 1.0 m are shown in Figures 8(a)–8(c) and 9(a)–9(c), respectively. Figures 7(a), 7(b), 8(a), 8(b), 9(a), and 9(b) make it clear that training loss was eliminated after four training sessions and that MAPE during the testing phase was approximately 1.0%. Scattergrams of predicted values of absorption by suggested models versus actual values of absorption in Figures 7(c), 8(c), and 9(c) show that predicted values are extremely similar to real values of absorption. The proposed model is trained for the prediction effectiveness (R2 score) for various values of meta-surface thickness. However, a graph shows that the optimum deep learning mode is only trained using half of the simulation data points. The model can still predict the absorption values for the remaining half of the wavelength values with high accuracy (R² score >0.9998).

[figure(s) omitted; refer to PDF]

Table 3 thoroughly examines several designs, their measurements, and their efficiency based on the absorption area under the curve (AUC) % throughout the full spectrum.

Table 3

Absorption area under the curve (AUC) %.

Test cases	Shape	Dimension (length)	Efficiency (%)
Case 1	O-shape	L = 8, 9, 10, 11, and 12	99.8
Case 2	O-shape	L = 8, 9, 10, 11, and 12	99.1
Case 3	O-shape	L = 8, 9, 10, 11, and 12	98.0
Case 4	O-shape	L = 8, 9, 10, 11, and 12	99.0
Case 5	O-shape	L = 8, 9, 10, 11, and 12	98.5

4.3. Experimental Analysis

A kind of WTa-SiO₂ ceramic-based absorber with transition wavelengths of around 0.6 μm, 0.8 μm, and 1.0 μm was experimentally proven. Using the appropriate metal and dielectric substrates, cosputtering was used to deposit cermets with various WTa and SiO₂ volume ratios. The high-purity W, Ta, and SiO₂ are among the target materials that are commercially accessible. We compare the observed absorption coefficients to those predicted by our enhanced algorithm for deep learning to evaluate our process for the assessment session. We chose to simulate an absorber with the GE-deep AlexNet algorithm working at 66 Hz as the operational frequency bandwidth of our measurement equipment begins at about 50 Hz to allow accurate comparison with the experimental data. At the tube’s end, the attached structure allows us to detect the absorption spectra of the associated metasurface. Figure 10 displays the predicted and experimental absorption curves. Despite some discrepancies, the two results agree well, confirming the suitability of the proposed method for predicting acoustic absorption qualities utilizing the proposed improved AI algorithm program. The results of experimental and simulation studies are illustrated in Table 4.

[figure(s) omitted; refer to PDF]

Table 4

Comparison of the experimental and simulated absorption.

Design	Absorption (%)			Overall absorption (%)
	Metasurface thickness (μm)
	0.6	0.8	1.0
Experimental	97	98	97.9	98
Simulated (PC-AE and GE-deep AlexNet algorithm)	98.2	98.3	97.9	98.1

The absorption efficiency value of the proposed system is estimated individually by varying the metasurface thickness 0.6, 0.8, and 1.0 μm is defined as absorption. For instance, while changing the metasurface thickness to 0.6 μm, the absorption efficiency is 97% for the experimental and 98.2 for PC-AE and GE-deep AlexNet algorithm. Similarly, 0.8 and 1.0 μm is estimated, and the values are shown in Table 4. The overall absorption defines the average of a total individual obtained absorption. The absorption is calculated from the corresponding transmittance and reflectance rates. Moreover, the overall absorption is calculated by the average of these three absorption values. Our proposed method’s average total absorption performance efficiency evaluation also estimates the overall absorption. Both absorption and overall absorption estimation are needed because absorption is the evaluation of individual performance, yet the overall absorption shows the average of the system’s total performance.

4.4. Comparison Study

In Table 5, we have compared the performance of the suggested solar absorber with earlier research outcomes. The frequency-dependent resonant response of the metamaterial absorbers in the infrared, visible, and ultraviolet spectrums make them a topic of current research. In [22], Patel et al. show that the visible regime has the highest median absorption of 89%, which is better than 0.99 prediction efficiency (R²). Titanium and gallium arsenide were used to develop a broadband multilayer grating structure that attained an absorption of 99.69% at 867 nm [23]. In Patel et al. present a SiO₂-based substrate-based DLMP that achieves more significant than 90% absorption. In [24], the authors offer a graphene-based solar absorber design with two distinctive L and O-shaped metasurfaces. The experimental results show that polynomial regression analysis can accurately predict the numbers of absorption capacity (R² score) in [27]. The metasurface solar absorber based on Ge₂Sb₂Te₅ (GST) substrate can still achieve better performance even with a lower value of K in a KNN-regressor system, with excellent prediction accuracy more significant than 0.9 estimated (R²) [28].

Table 5

Comparison of the proposed method with recent literature.

Refs.	Materials	Methods	Limit	% of total mean absorption	Incidence angle (degree)	Resonance thickness (μm)	Accuracy (%)	Execution time (sec)	R2
[21]	Graphene	LSTM	0.2–1.4 μm	92	0 to 70	0.1–0.8	85	32	0.98
[22]	SiO2	Regression models	0.2–1.0 μm	95	20 to 60	0.8	91	35	0.96
[23]	Titanium and gallium arsenide metal		500–4000 nm	99.69	0 to 50	—	86	26	0.97
[24]	Graphene	CNN-regression	0.2–0.4 μm	91	0 to 80	0.2–1	90	20	0.999
[25]	Gold multipattern swastika (DLMP) based SiO2	General regression neural network	0.1–3 μm	95	0 to 60	0.5–1	91	30	0.92
[26]		CNN	5.7 μm	97	—	—	—	17	0.86
[27]	MgF2 substrate	Polynomial regression models	0.2–1.2 μm	95.5	10 to 80	0.6–1.8	92.2	21	0.97
[28]	Ge2Sb2Te5 (GST) substrate	KNN	0.2–1.5 μm	92	—	—	90.2	38	0.9
This work	WTa-SiO2	PC-AE and GE-deep AlexNet	0.1–2.5 μm	98	0 to 20	0.6, 0.8, and 1.0	99.8	15	0.999

The commercial software only examines the performance of the individual parameter because the absorber is made up of randomly arranged parameters rather than computing the overall efficiency of the metamaterial. We emphasized that every material from the WTa-SiO₂ ceramic method is readily available and affordable compared to the precious materials that are often used for the solar absorber. Furthermore, the materials used in the proposed structure are thinner than 1 μm in thickness. Furthermore, compared to other absorbers, our absorber’s absorption efficiency is better at 99.8% for 0.2 μm and 95% for 2.5 μm, making it superior. Moreover, compared to the commercial software-based parametric analysis in metasurface absorbance, the proposed method achieved very less execution time and higher prediction by varying cases. Our findings show that, compared to other current technologies, our absorber has very good average absorption efficiency. Considering everything, it is evident that solar absorbers, with their straightforward construction and superior performance, play a significant role in solar absorption. It is clear that the proposed deep learning model performs significantly faster and uses less memory than the typical commercial simulation software once it has been trained. The proposed model’s average prediction time is 15 seconds, compared to the current approaches’ average prediction time of more than 17 seconds. It is therefore revealed that the optical response predicted by the proposed model is 99.8% faster than by simulations. This model can be scaled up by providing enough data and training structures to completely replace commercial software.

5. Conclusion

This paper presents the design of a WTa-SiO₂ ceramic layer O-shaped metasurface solar absorber. The designed O-shaped absorber achieved an overall absorption rate of 99.8% in the light spectrum. A thorough analysis is also performed by adjusting the physical factors, including resonator thickness, angle of incidence, and metasurface thickness that influence the absorption rate. Experiment findings reveal that the proposed Golden Eagle Optimization (GE)-based deep AlexNet design and Principal Component-Autoencoder (PC-AE) technique can effectively develop the prediction model. It has good precision in predicting absorptivity at middle frequencies (higher than 0.9998 R² score). It is also revealed that a high prediction accuracy is obtained when the model is designed using a greater exponential degree of features. The obtained absorption response is also insensitive for the incidence angle of 0° to 20°, and the absorber consistently exhibits excellent absorption performance even when the incident angle varies from 0° to 50° which makes the proposed absorber more versatile with fewer limitations. The engineered absorber has a more expansive absorbing range and a simple structural footprint when compared to other absorbers of the same type.

Authors’ Contributions

All authors contributed equally to the writing of this paper and approved the final manuscript.

Acknowledgments

This research work was funded by the Institutional Fund Projects under grant no. (IFPIP: 1273-135-1443). The authors gratefully acknowledge the technical and financial support provided by the Ministry of Education and King Abdulaziz University, DSR, Jeddah, Saudi Arabia.

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