# Metamaterial Design with Nested-CNN and Prediction Improvement with Imputation
## 1 Introduction
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]. The deep neural network (DNN) and convolutional neural network (CNN) move beyond the traditional supervised learning algorithms and optimization methods since computational capacity is improving rapidly day by day [35]. DNN is used in classification and regression problems and has achieved great success. CNN is a DNN algorithm and can take pictures, matrices and signals as input. The purpose of CNN is achieved by extracting the features with the filters, the coefficients of the filters and biases are updated with gradient-based optimizations. In the creation of metamaterials, the shapes were generally optimized by iteration-based experiments, for example, optimization based on reflection coefficients. In such iteration-based experiments, at best, the duration of the simulation program elapses in each iteration while trying the shapes. Instead of finding reflection coefficients from the shape as traditional iteration-based methods do, a model that provided metamaterial pictures corresponding to the desired reflection coefficient was developed by approaching the metamaterial production with DNN in reverse. 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 \(1\times 1500\) and the size of the binary images is \(50\times 50\). Collected reflection coefficients are between 6 GHz and 14 GHz.