Convolution Neural Network has become the most popular topic in recent years with the development of hardware technology. The CNN has been adapted to many different disciplines because classical image processing stages can be performed automatically. The CNN method takes pictures as input and produces feature maps by passing them through layers. It then classifies images by feature selection. A CNN architecture consists of an input layer, convolution layer, pooling layer, fully connected (FC) layer, SoftMax, and output layers [30]. The CNN architecture is shown in Fig. 1.

The input layer is where the image data is applied raw to the CNN architecture. The input layer size is determined according to the resolution of the image. The convolution layer extracts the features from the images. In this layer, many features are obtained using filters of different sizes such as 2 × 2, 3 × 3, and 5 × 5. This filter is moved over all the pixels in order, starting from one corner of the picture. A new value is obtained by multiplying and summing the overlapping pixel values on the picture with the filter. In this way, different features are brought to the fore from each filter. Rectified Linear Unit (ReLu) activation function is generally used in the convolution layer.

The pooling layer is used to reduce the load on the hardware while working on the CNN architecture. It may cause some features to be lost but, it also prevents overfitting. It does not have to be used in every CNN architecture. All features extracted in the fully connected layer are smoothed into a single vector. Among the features extracted by this layer, the most contributing and non-contributing features to the classification are determined by updating the weights. The feature selection process is performed with this layer. In the SoftMax layer, the outputs of the FC layer are passed through the activation function and the class predictions are transferred to the output layer. The output layer predicts the class labels.

Grey Wolf Optimization (GWO) algorithm is a meta-heuristic optimization algorithm inspired by Grey wolves’ survival and hunting strategies in nature [31–33]. Grey wolves have a chain in which alpha wolves are at the top, and beta, delta, and omega wolves are listed, respectively (Fig. 2).

Alpha wolves represent the most dominant and commanding wolves, as seen from the chain structure. Beta wolves act as alpha wolves’ assistants and run the communication network between alpha wolves and other wolves. Omega wolves are the lowest level wolves chosen by alpha wolves. They are the ones who eat at last during hunting. Wolves that do not belong to the class of alpha, beta, or omega wolves are called delta wolves. Delta wolves are predominantly selected by alpha and beta wolves from omega wolves [34].

Galaxy images used in this study were created by Galaxy Zoo Project. Galaxy10 Sloan Digital Sky Survey (SDSS) Dataset [37] consists of ten classes, accepted by more than 55% of researchers. This dataset consists of 21785 images, that are 69 × 69 in size. Table 1 presents the classes of the dataset and the number of samples in each class.

In Fig. 3, sample images of the classes in the Galaxy10 SDSS dataset are shown. As seen from Fig. 3, there are 10 galaxy classes in the dataset. Although some classes are similar to each other, they differ from each other with small differences. These similar classes complicate the problem being considered. This situation requires a more complex CNN architecture instead of a simpler CNN architecture.

In this study, a basic CNN architecture was determined. The number of neurons in the layers of this architecture, learning coefficient and batch sizes were optimized. The CNN architecture is presented in Table 2, and ranges of optimized parameters are given in Table 3.

Parameter 1⁓11 given in Table 2 represents the number of neurons to be optimized. In addition to these parameters, Parameters 12 and 13 represent the learning coefficient and batch size, respectively.

The parameters of the CNN architecture are optimized using the GWO algorithm. All operations have been carried out with Python 3.6 and Keras library. In the development of the CNN model, Adam optimizer as optimization function, Mean Squared Error (MSE) and Mean Absolute Error (MAE) as loss function, were used. MSE and MAE calculations are given in Eqs. (8) and (9), respectively. Fig. 4 shows the flow chart of the proposed method. (8)MSE=1n∑i=1n(yi−y~i)2 (9)MAE=1n∑i=1n|yi−y~i|

Here yi denotes the actual class label and y~i denotes the predicted class label.

In addition to the GWO algorithm, I optimized the CNN architecture with the well-known Particle Swarm Optimization (PSO) and Artificial Bee Colony (ABC) algorithms. These algorithms are used successfully in most real-world problems. The PSO and the ABC algorithms are state-of-art metaheuristic algorithms that were compared with GWO [38–43]. Therefore, these algorithms were used for comparison in this study.

The model’s accuracy value was considered as the cost function of the used metaheuristic algorithms. The accuracy formula is given in Eq. (10). It is aimed to determine the optimum parameters that will maximize the accuracy value. (10)Accuracy=NumberofCorrectPredictionsTotalNumberofPredictions

A total of 21785 images were separated as training and test data with the 10-fold cross-validation technique. The optimum parameters of the CNN architecture were determined by metaheuristic algorithms. All algorithms were run with a population size of 30 and 50 iteration steps. In Table 4, the performance metrics of the train and test dataset are given. The best results were obtained with the GWO. To prevent overfitting in the CNN model, the training phase was terminated in cases where the dropout layers and early stop process and the validation accuracy value could not be improved for ten epochs. In addition, a comparison was made with another study using the same dataset in the literature [44]. The results of the reference study are presented in Table 4.

Considering the results given in Table 4, the test accuracy value was found to be 85%, whereas in the training as the accuracy was 93%. The GWO algorithm achieved better results than other metaheuristic algorithms. Compared to the Reference Study, the GWO algorithm is more successful in the training phase. In the test phase, the GWO algorithm achieved a better result by a small difference.

The differences between test and train results were examined, It was found that the model is not overfitting and performs a balanced learning process using GWO. During the model’s training, the accuracy and loss value of the model obtained with the train and test data are given graphically in Fig. 5 for the GWO.

The confusion matrix in Fig. 6 is given to see the classes that our model learned best and could not learn. A confusion matrix is a technique used to summarize the performance of a classification algorithm. If you have an unequal number of samples in each class, or more than two classes in your dataset, the classification accuracy alone can be misleading. A confusion matrix contains more detailed information about the classes in which your classification model fails. In this way, it can be easier to identify the areas to be focused on to improve the algorithm. The values in the matrix represent the classification results as percentages.

If Fig. 6 is examined, my model was able to learn Classes 1, 2, 4, 6 and 7 with an accuracy of over 90%. Except for Class 5, the remaining classes could be learned at a level considered successful. However, during the training phase, it learned very little about Class 5 and confused it with Class 4. As explained in the previous sections, our dataset has very little data belonging to Class 5. For this reason, the desired success in the learning process could not be achieved. Not including Class 5 in the study could have been considered as a solution, as there are very few examples compared to other class sample numbers. Since I assert a hybrid approach, I do not need to simplify my dataset. Learning achievements in the training phase are also seen in the testing phase. In direct proportion to the success in the training phase, classification was also possible in the testing phase. If the examples of all classes are examined in Fig. 3, it is understood that Classes 1 and 2 have a different and rounded shape from all other classes. Similarly, Classes 7 and 8 have a different spiral appearance than other classes. Galaxies belonging to the other six classes are in the form of disks and are likely to be confused with each other. The examples where the model will have the most difficulty learning are the examples of these six galaxy types. Based on the confusion matrix, the classes with the most success are Classes 1 and 2. Even though Class 8 has a helical appearance, it is a disc-like shape. For this reason, a classification as successful as the training results could not be obtained in the test results. In Class 4 and 6, my CNN model could generate a distinctive feature map and distinguish it from other disk-like galaxies. In Table 5, the Precision, Recall and F1Score values obtained by the GWO algorithm for each class are given. The data in Table 5 also supports the results in Fig. 6. The Precision, Recall and F1Score formulas are given in Eqs. (11), (12) and (13), respectively. (11)Precision=TPTP+FP (12)Recall=TPTP+FN (13)F1Score=2∗Precision∗RecallPrecision+Recall

Here, TP, FP and FN represent True Positives, False Positives and False Negatives, respectively. As can be seen from Table 5, the small number of samples in Class 5 caused the Recall value to be low during training. Similarly, during the test, the Recall value of Classes 7, 8 and 9 is low. These classes have been mixed with other classes. This is clearly seen in the confusion matrix. In addition to Table 5, ROC curves of model results according to classes are given in Fig. 7.

In Table 6, the parameters of my optimally obtained CNN architecture are given. These parameters have been determined with the support of the GWO algorithm.