Metamaterials(MM) are special materials that are not found in nature and are used in many fields with their unusual properties [
1]. Metamaterials have a wide range of uses, including optical filters, medical devices, remote aerospace applications, sensor detection and infrastructure monitoring, smart solar power management, crowd control, high-frequency battlefield communication, and lenses for high-gain antennas, improving ultrasonic sensors, anti-radar devices, and superlenses. The breadth of these usage areas has been the motivation for many studies on developing metamaterials. The studies are generally based on iteration-based studies of optimization algorithms. Optimization algorithms generally proceed iteratively until optimum and satisfactory results are found. The increase in big data and processing power demonstrates that the future of optimization algorithms will be even brighter. In the past few decades, electromagnetic (EM) designers have rapidly and accurately developed models for rapid progress in technology. Moreover, these models have allowed for more variety and modeling complex problems than previously imagined.
In recent decades, although many global optimization techniques have been developed, the most used technique is the genetic algorithm (GA) for designing metamaterials. (GA) [
2]. GAs mimic natural selection and mutation to optimize constrained and unconstrained problems. The GA’s basic operation chooses the best options from the previous generation to breed the next generation; it was popularized and widely used by the design of pixelized metallic structures [
3], which is the basis of many metamaterial designs [
4]. Optical nanoantenna array configurations [
5] and shapes [
6] have been optimized to maximum field enhancement at the selected location. In [
7], the GA has a decreased field enhancement, which is a prohibitive factor for metamaterial usage in high power microwave implementations and thus, has a reflectarray unit cell design based on pixels. In recent applications, coding metasurfaces based on radar cross-section (RCS) reduction and best suitable metamaterials developed for polarization conversion, has been included by the GA in [
8,
9,
10]. Many algorithms are created by nature [
11,
12], such as artificial bee colonies (ABC) [
13] and bat-inspired algorithms [
14]. These algorithms are known as lower branches of swarm intelligence algorithms that try to optimize complex problems by mimicking the collective behavior of decentralized, self-organized systems, natural or artificial [
15]. The optimization of an ant colony (ACO) [
16] is counted as one of the swarm intelligence algorithms, and uses stigmergy to find the best suitable solution. Ant colony optimization is used for graph-based solutions, such as finding the route of the vehicle, job scheduling, and traveling seller issues [
17]. The GA has achieved immense results in electromagnetics and optical materials, but disjointed pixels in solution are a drawback of GAs because these designs are limited to planar configurations. On the other hand, ACO maps the optimal “trace” found by artificial “ants” in the graph topology to an adjacent structure. For that reason, ACO has achieved enormous success in the design of the electromagnetic device, particularly in the meander-line antennas [
18]. For the high performance of frequency selective surfaces (FSS), the ACO algorithm and lazy ants combined and improved the performance of FSS by ZHU [
19,
20]. In the mid-1990s, another branch of swarm intelligence algorithms, particle swarm optimization (PSO), emerged [
21], and has been used immensely for the optimization of electromagnetic devices [
22,
23,
24,
25]. Compared to PSO and GA, PSO offers many more advantages: firstly, PSO uses real values as vectors instead of binary values, and real values were later used in GAs [
26]. Secondly, in PSO, members of the population tend to work more independently and cooperatively. This can be considered as individual members work on solutions of a different part of the space and communicate with each other. PSO gives us better performance than GA for designing negative-index metamaterial [
27]. Moreover, PSO is used in meta-optic device optimizations. PSO has been used to optimize geometrical parameters and spacing-based Yagi-Uda antennas [
28]. PSO is used in beam steering applications by optimizing nanohole array-based metamaterials [
29]. While global optimizers, such as GA and PSO have been successfully applied to a wide variety of design problems in RF and optical regimes, they are often sensitive to internal parameters that require problem-wise adjustment. Optimum control parameter tuning in evolutionary algorithms has been studied by many studies [
30,
31,
32]. Moreover, it is not always clear how best to set these parameters for a particular problem and may require adaptive tuning during optimization to maximize algorithm performance. In this context, the covariance matrix adaptation evolution strategy (CMA-ES) with a self-adaptive nature, emerges as the answer to the adaptive tuning problem [
33]. It was found that CMA-ES performed significantly better than GA in terms of convergence speed and solution quality when the wideband polarization converter was applied to the optimization of metastatic surfaces [
34].
In this article, first, metamaterial design was made by the proposed nested-CNN model with the inverse functioning method. The nested-CNN model consisted of two models: Model-1 was trained to detect the image of MM by using a desired reflection coefficient as the input. Model-2 was trained to detect the reflection coefficient of a given image of MM as the input. Trained Model-2 was used in Model-1’s loss function in the training process of Model-1, which compared the desired reflection coefficient and reflection coefficient of the produced image of MM by Model-1. In nested-CNN, the output of Model-1 was the image of the MM; the output of Model-1 was used as the input to Model-2; Model-2’s output was the reflection coefficient of the produced image of MM by Model-1; and the input of Model-1, which was the desired reflection coefficient and output of Model-2’s obtained reflection coefficient used in loss function to obtain the error of Model-1. In deep learning, gradient-based optimizers are used to update weights and biases by using the loss of the models. Model-1 has been updated using our defined loss function by the gradient-based optimizer. Secondly, imputation is a method to complete the missing values of the dataset before the training process. Imputation was used for the prediction side for non-desired parts of reflection coefficients to decrease the loss of the interested region.
The size of the reflection coefficient produced for training is 1 × 1500 and the size of the binary images is 50 × 50. Collected reflection coefficients are between 6 GHz and 14 GHz. In binary images, black pixels represent FR4 material and white pixels represent copper. The nested-CNN was developed for 2D metamaterial, the thickness was a constant value, and the width and height could take variable values. The microwave CST simulation and Matlab program were used to collect data.