Optuna-Optimized Pythagorean Fuzzy Deep Neural Network: A Novel Framework for Uncertainty-Aware Image Classification

Abstract

By using Geographic Information Systems, satellite imagery from remote sensing techniques provides quantitative and qualitative data about Earth’s natural and human elements. However, the direct use of raw imagery may prevent the accurate identification of the spectral and temporal characteristics of the target objects. To obtain meaningful results from these data, the object and surface features in the image must be classified correctly. In this context, this study develops a new deep learning approach that includes hyperparameter optimization that considers uncertainty factors when classifying satellite imagery. In the proposed approach, a hybrid architecture called CNN-Pythagorean Fuzzy Deep Neural Network (PFDNN) is developed by combining the ability of convolutional neural networks (CNN) to reveal expressive features with the ability of Pythagorean fuzzy set (PFS) theory to predict uncertainty. In addition, to further improve the model’s success, the hyperparameters are automatically optimized using Optuna. In the experiments conducted on the EuroSAT RGB dataset, the CNN+PFDNN+Optuna model achieved 0.9696 ± 0.0037 accuracy and a macro-AUC value of 0.9983, outperforming other methods such as DNN, FDNN, PFDNN and VGG16+PFDNN. Including the Pythagorean fuzzy layer in the system provided about 13.05% higher accuracy than conventional fuzzy systems. These results show that integrating the Pythagorean fuzzy set approach into deep learning models contributes to more effective management of uncertainties in remote sensing data and that hyperparameter optimization significantly impacts model performance.

1. Introduction

Today, remote sensing technologies are among the most effective tools used in both civilian and military applications. They enable information about objects and events on Earth without physical contact. Satellite imagery is obtained by placing mobile platforms equipped with sensors at specific distances from Earth, in the atmosphere, or in space. Unlike other geospatial data sources, they provide up-to-date digital imagery of large geographic areas, including hazardous or inaccessible areas, quickly and cost-effectively, and enable the detection of invisible features of objects. Such imagery is widely used for environmental change monitoring, disaster risk assessment, agricultural planning, and urbanization [1]. In the face of global challenges such as migration, rapid population growth, and climate change, the ability to effectively analyze satellite data has become even more critical. Aerial and satellite imagery are among the main data sources with their ability to provide detailed information about the Earth [2,3,4]. However, complex geometric patterns and spatial patterns in these images can sometimes make the analysis process difficult. Therefore, automatic classification of remote sensing data, in other words, assigning pixels or regions in the image into specific land use types, is critical for decision-making processes [5]. Such data are usually multidimensional, from different sources and may contain noise. However, factors such as unclear boundaries between classes, atmospheric conditions, distortions caused by sensors, and similar spectral characteristics of different classes can cause errors in the classification process [6]. Such challenges cause traditional machine learning methods to be limited and reveal the need for more advanced solutions.

In recent years, deep learning approaches, particularly CNNs, have produced remarkable results in image processing thanks to their ability to automatically extract hierarchical spatial features [7,8]. However, CNNs alone often require large labeled datasets and struggle with high uncertainty [9]. To address these challenges, advanced CNN-based frameworks have been proposed for satellite image classification. For example, the FCIHMRT model demonstrates strong performance by combining feature extraction with hierarchical multi-resolution techniques, underscoring the effectiveness of CNN architectures in remote sensing applications [10]. Fuzzy logic systems, on the other hand, are effective in modeling uncertain class transitions and ambiguous data. Therefore, hybrid approaches combining CNNs for feature extraction with fuzzy logic for uncertainty management offer a promising avenue for improving classification performance in remote sensing.

Pythagorean fuzzy sets (PFS) extend this capability by providing greater flexibility in uncertainty modeling, making them suitable for remote sensing applications. The use of PFS in classification and segmentation applications where uncertainty is high is also becoming increasingly common [11]. To further improve performance, hyperparameters can be systematically optimized with frameworks such as Optuna [12]. While traditional neural networks achieve strong results on nonlinear problems, they are often criticized as “black boxes” due to the opacity of their decision-making [13]. Fuzzy logic systems, in contrast, offer transparency and interpretability but limited learning capacity. By integrating the strengths of both approaches, hybrid models can achieve high performance while improving interpretability.

A growing body of research shows that combining fuzzy set theory with deep learning is effective in various fields. For example, a new model called Fuzzy RBM (FRBM) was proposed by combining the classical Restricted Boltzmann Machines (RBM) model with fuzzy logic [14]. This model performed better than the standard model, especially on noisy data, and experiments on MNIST handwritten digit recognition and Bar-and-Stripe tests proved that FRBM provides a better representation ability. The model was further extended with Pythagorean fuzzy sets to improve uncertainty management [15]. In the domain of natural language processing, ref. [16] combined deep neural networks and fuzzy systems to achieve successful results in automatic text summarization tasks. Similarly, an effective method for image classification was presented by integrating the Takagi–Sugeno–Kang (TSK) fuzzy model with deep learning [17]. Fuzzy layers have also been incorporated into deep learning architectures to improve performance on complex tasks such as image segmentation [18]. Studies in the medical field also emphasize the potential of this hybrid approach. For example, a fuzzy deep neural network model for COVID-19 diagnosis achieved 97% accuracy and 98% recall rates [19]. Similarly, heuristic fuzzy sets have been combined with deep learning to achieve successful results in highly uncertain areas such as medical diagnosis [20]. Recent studies also demonstrate the effectiveness of these methods in different application areas. For instance, a model combining fuzzy deep neural networks with optimization algorithms was proposed for urban traffic management [21]. In addition, a new approach to uncertain data classification was introduced by integrating quantum computing and fuzzy logic [22].

In the field of image analysis and remote sensing, the success of fuzzy logic and deep learning in modeling ambiguous data structures is remarkable [23]. The DCNFIS model combines deep learning and fuzzy logic to offer both high accuracy and interpretability. This model has achieved particularly successful results in image classification tasks. Ref. [24] developed a generalized hierarchical fuzzy logic algorithm that can manage uncertainty and imprecise data structures, especially for processing large image datasets. The proposed approach divides large image datasets into small samples and combines them into hierarchical fuzzy subsystems. Ref. [25] proposed a new model called RS- image segmentation with Fuzzy Convolutional Neural Network (RS-FCNN). Experimental results show that RS-FCNN provides higher segmentation accuracy compared to existing methods. Ref. [26] presented the Interval Type-2 Fuzzy Convolutional Neural Network (IT2FCNN), an end-to-end land cover classification method for high-resolution remote sensing images (HRRSI). The model extends traditional convolution kernels with fuzzy membership functions and adaptive boundary mapping, supported by fuzzy rule libraries and hierarchical optimization. Ref. [27], proposed the integration of fuzzy logic and deep learning for automatic diagnosis of Alzheimer’s disease (AD). This review examines the role of FDL models in AD diagnosis and discusses fuzzy-based image preprocessing, segmentation and classification methods. An FDNN-based model was developed for recognizing human activities of disabled individuals, achieving high accuracy rates [28]. Ref. [29] proposed a basic approach that combines the advantages of fuzzy logic and machine learning methods in their work and targeted its effectiveness in the classification of JAFFE [30,31] facial expressions.

Although the studies on Pythagorean fuzzy deep neural networks are still limited, the current findings show that integrating this method with deep learning in uncertainty management and model transparency can make significant contributions to the literature.

Additionally, several recent surveys and benchmarking studies have highlighted the rapid progress of deep learning in remote sensing. For example, ref. [32] presented a comprehensive review and comparative analysis of satellite image classification models, including widely used architectures such as ResNet and DenseNet. Other studies have introduced and benchmarked large-scale datasets such as EarthNet [33] and SEN12MS [34], which are now widely adopted for evaluating classification methods. More recent surveys have also discussed the application of transformers and their basis models in remote sensing [35]. These studies present the rapidly evolving research landscape and further motivate us to integrate Pythagorean fuzzy logic with CNNs and Optuna optimization to address uncertainty in satellite image classification.

Based on all these mentioned cases, this paper presents an innovative approach for satellite image classification that integrates PFS theory with deep neural networks and is supported by hyperparameter optimization. The proposed method aims to provide significant advantages over existing systems in terms of both uncertainty modeling and classification performance.

The main contributions of this study can be summarized as follows:

• We propose a novel hybrid architecture that integrates Convolutional Neural Networks (CNN) with Pythagorean Fuzzy Deep Neural Networks (PFDNN). To our knowledge, this is the first attempt to apply the PFDNN framework to remote sensing image classification, which requires robust handling of uncertainty arising from spectral overlaps and environmental noise.
• The uncertainty modeling capability of Pythagorean fuzzy sets combined with the feature extraction ability of deep learning provides more reliable classification results in uncertain land cover categories.
• Hyperparameters are systematically optimized using the Optuna framework, which reduces the risk of overfitting while improving the generalization ability of the model. This optimization strategy demonstrates clear advantages over manually tuned or fixed hyperparameters.
• Extensive experiments on the EuroSAT RGB dataset validate the effectiveness of the proposed method, outperforming traditional deep learning and fuzzy-based baselines by a statistically significant margin.

The remainder of the paper, in Section 2, will explain the architecture of the proposed method in detail and present how this approach is applied in the classification process. The results of the experimental studies will be evaluated in Section 3 and Section 4, and the findings will be discussed in general. The contributions of the study will be summarized in Section 5.

2. Materials and Methods

This section presents the proposed methodology, which integrates image preprocessing, CNN-based feature extraction, a Pythagorean fuzzy deep neural network, and hyperparameter optimization using Optuna.

2.1. Dataset

In this study, the EuroSAT dataset was used. The EuroSAT dataset contains 27,500 labeled images of 10 different land use classes (Industrial Buildings, Residential Buildings, Annual Crop, Permanent Crop, River, Sea&Lake, Herbaceous Vegetation, Highway, Pasture, Forest) with a resolution of 64 × 64 pixels [36,37]. Each image was collected from the Sentinel-2 satellite. The patches contained in the dataset are gathered from cities distributed in over 30 different European countries based on the coverage in the European Urban Atlas [37]. Sample images from each class of the EuroSAT dataset are presented in Figure 1.

2.2. Overall Framework

Classifying remote sensing images is a critical challenge that requires the synergy of deep learning and fuzzy logic. While traditional methods suffer performance loss in the face of fuzzy class boundaries or spectral noise, PFS theory provides an extended framework to mathematically encompass these uncertainties. In this study, the success of CNN in spatial feature extraction and the decision-making flexibility of PFS are combined in a parallel hybrid architecture. The model consists of four stages: image preprocessing, CNN-based feature extraction, Pythagorean fuzzy layered classification and Optuna-assisted hyperparameter optimization (Figure 2). In particular, the learning rate, regularization coefficients and layer sizes optimized by Optuna’s dynamic search algorithm enable the model to achieve both high accuracy and generalization.

2.3. Image Processing

The first stage of our approach is image preprocessing. In image classification problems, the success level of the model largely depends on the quality and consistency of the inputs. Real-world images are usually not obtained under ideal conditions. Therefore, it is necessary to perform some preprocessing on the images before classification. Image preprocessing basically involves transformations to make the data more suitable for the model. These operations both improve the data quality and train the model more efficiently [39]. Raw images from satellites may contain various types of noise due to atmospheric conditions, sensor errors or environmental factors. The model may become sensitive to unwanted patterns in classification without noise removal. The pixel values in an image are usually in the range [0, 255]. Scaling these values to [0, 1] enables the model to learn faster and more stably. Especially in deep learning models, this normalization increases the effectiveness of activation functions. When processes such as noise reduction or normalization of pixel values are considered as a whole, it is seen that image preprocessing directly affects the accuracy, speed and generalization success of classification models.

2.4. CNN-Based Feature Extraction

Image classification is essentially a pattern recognition problem. One of the most steps at the center of this process is feature extraction. This is the second stage of our model. This stage aims to transform an image into feature vectors that represent its distinctive, meaningful and learnable aspects, rather than expressing it directly in pixel values. In other words, feature extraction enables the image to be transformed from its “raw state” into a representation suitable for learning [40]. CNN automatically extracts multilayer features from images. These networks learn layered features, from basic edges to complex object shapes [41]. In this study, spatial features of images are extracted using a CNN. CNNs are deep learning architectures widely used in fields such as image processing and remote sensing. The main advantage of CNNs is that they can automatically extract spatial hierarchical features in local areas. This process is realized by successive convolution, activation function (ReLU) and sampling (pooling) layers. Deep layers are better able to learn semantic features (edges, textures, object parts), which improves classification performance. ReLU has further advantages, including the ability to implement the computation on any software or hardware platform, which eliminates the possible loss in quantization and is more evident in deep networks.

Local feature maps are created in the convolution layers by shifting the learned filters (kernels) on the input image. The convolution process is expressed by Equation (1).

$$\left(I\ast K\right)\left(x,y\right)=\sum _{i=1}^a\sum _{j=1}^{b}I\left(x+i,y+j\right)K\left(i,j\right)$$

(1)

where $I$ is the input image and $K$ is the ($x,y$) pixel coordinates of the convolution filter. In deep layers, filters capture semantic features such as edges, textures and object parts. Pooling layers reduce computational complexity and provide invariance by reducing the size of feature maps. Max-pooling prevents significant information loss by preserving the most dominant attribute in a region [42].

$$P_{max}\left(x,y\right)=\underset{(i,j)\in \mathcal{R}}{\max }I\left(x+i,y+j\right)$$

(2)

In Equation (2), R the pooling window is represented. The Global Average Pooling layer facilitates the transition to fully connected layers by converting the output of the last convolution layer into a single vector. This reduces the risk of overfitting the model.

2.5. Pythagorean Fuzzy Deep Neural Network (PFDNN)

For the third stage of our approach, classification, we present the PFDNN model. The PFDNN model integrates Pythagorean Fuzzy Set (PFS) theory into deep learning to model uncertainty and hesitation. Integrating Pythagorean fuzzy logic into DNN architectures goes beyond previous intuitionistic fuzzy approaches in uncertainty modeling. The developed architectures enrich the information flow thanks to the parallel fusion structure (fuzzy pathway + dense pathway). In this case, it is necessary to briefly recall Pythagorean fuzzy set theory and deep learning theory.

To deal with uncertainties in decision making, ref. [43] introduced fuzzy set theory. Later, ref. [44] proposed Intuitionistic Fuzzy Set, which is an extension of fuzzy set theory. Intuitionistic fuzzy sets (IFS) are based on both membership and non-membership degrees of an element in a fuzzy set. In classical fuzzy sets, the sum of the degrees of membership and non-membership is always 1, while in intuitionistic fuzzy sets, this sum should be less than or equal to 1 [45]. However, in solving some problems, the sum of membership and non-membership degrees may be greater than 1. In such cases, the IFS solution is out of the IFS solution. To express this situation, the Pythagorean fuzzy set was proposed by Yager [46,47]. In the PFS theory, the sum of the squares of the degrees of membership and non-membership must be less than or equal to 1. Some basic concepts and mathematical operations related to PFS are defined as Equations (3)–(10).

Definition 1.

A PFS S is defined in the form:

$$S=\left\{<x,{\mu }_S\left(x\right),{\nu }_S\left(x\right)>|xϵU\right\},$$

(3)

where ${\mu }_S$is membership function and ${\nu }_S$is the dual function. And for ∀x ∈ U, the set U is a universe of discourse. Also condition $0\le {{\mu }^2}_S\left(x\right)+{{\nu }^2}_S\left(x\right)\le 1$ is valid.

This representation provides richer semantic information to the deep learning process by modeling ambiguities in the input. This structure increases the generalizability of the model, especially in domains with a lot of ambiguous or incomplete data. The degree of hesitation for PFS can be defined by Equation (4).

$${\pi }_S\left(x\right)=\sqrt{1-{{\mu }^2}_S\left(x\right)-{{\nu }^2}_S\left(x\right).}$$

(4)

The main difference between PFS and IFS is illustrated by Figure 3. Obviously, PFS has a larger space than IFS, which means that PFS has more representational power. Every IFS should be a PFS but not all PFSs are IFS.

Definition 2.

Given two PFNs,$S_1=({\mu }_{S_1},{\upsilon }_{S_1})$ and $S_2=({\mu }_{S_2},{\upsilon }_{S_2})$ the following operations on PFS initially defined as in Equations (5)–(8).

$$S_1\oplus S_2=S\left(\sqrt{{{\mu }_{S_1}}^2+{{\mu }_{S_2}}^2-{{\mu }_{S_1}}^2{{\mu }_{S_2}}^2},{\upsilon }_{S_1}{\upsilon }_{S_2}\right)$$

(5)

$$S_1\otimes S_2=S\left({\mu }_{S_1}{\mu }_{S_2},\sqrt{{{\upsilon }_{S_1}}^2+{{\upsilon }_{S_2}}^2-{{\upsilon }_{S_1}}^2{{\upsilon }_{S_2}}^2}\right)$$

(6)

$$\lambda S=\left(\sqrt{1-{\left(1-{\mu }_S^2\right)}^{\lambda }},{\upsilon }_S^{\lambda }\right),\hspace{1em}\lambda >0$$

(7)

$$S^{\lambda }=\left({\mu }_S^{\lambda },\sqrt{1-{\left(1-{\upsilon }_S^2\right)}^{\lambda }}\right),\hspace{1em}\lambda >0$$

(8)

Definition 3.

There is a PFN $S=\left({\mu }_S,{\upsilon }_S\right)$ and then the score of $S$ is defined as:

$$s\left(S\right)={\mu }_S^2-{\upsilon }_S^2.$$

(9)

Definition 4.

There is a PFN $S=\left({\mu }_S,{\upsilon }_S\right)$ and then the accuracy of $S$ is defined as:

$$a\left(S\right)={\mu }_S^2+{\upsilon }_S^2.$$

(10)

A DNN is a group of neurons arranged in layers [48] and each layer receives data from the neuron in the previous layer and performs a specific operation. The neuron establishes a complex and non-linear relationship from input to output. To perform the transformation from input, a loss feed-forward model is applied that changes the weights of all neurons.

The proposed PFDNN is based on a classical deep neural network architecture. Nonlinear transformations are applied in multiple hidden layers following the input layer. In the model, PFS theory is implemented as a special layer (PythagoreanFuzzyLayer). This layer receives CNN outputs and processes them according to fuzzy logic rules. The function of the membership function is to calculate the degree of membership and determine to which specific fuzzy set the input belongs. We denote $x$ as input, o as output and t as layer number. Accordingly, $x_i^t$ denotes the input of the i-th neuron in the t-th layer and $o_j^t$ denotes the output of the j-th neuron in the t-th layer. Here, we utilize the Gaussian membership function as proposed in [49]:

$${\mu }_i\left(x_i^t\right)=e^{-{\displaystyle \frac{{\left(x_i^t-c_i\right)}^2}{{\sigma }_i^2}}}$$

(11)

where ${\mu }_i$ is the membership function, $c_i$ is the mean and ${\sigma }^2$ denotes the variance. The final output is converted into Pythagorean fuzzy averages presented as follows:

$$o_i^t=\sigma \left({\mu }_i^2\left(x_i^t\right)-v_i^2\left(x_i^t\right)\right)$$

(12)

In the first stage, the weights of the Pythagorean layer are initialized with the Glorot uniform distribution (Xavier uniform), while the other layers are randomly initialized with the Xavier normal initialization method. The Pythagorean layer weights are initialized with the Glorot uniform distribution (in the range $\pm \sqrt{6}/\sqrt{n^{t-1}-n^t}$) to preserve the signal variance during forward/backward propagation. Unlike the simple uniform distribution, this approach balances both the input and output layer sizes.

The PFDNN architecture used in this study provides information flow through two different pathways: (i) fuzzy pathway and (ii) dense pathway. The PFDNN architecture is shown in Figure 4. The features obtained from both pathways are then combined and passed to the final decision maker. The outputs of these pathways are combined in a fusion layer. The fusion process is performed as in Equation (13).

$$h_{fusion}=ReLU(w_f\cdot Concat\left[h_{fuzzy},h_{dense}\right]+b_f$$

(13)

In Equation (13), $w_f$ and $b_f$ are the weight matrix and bias term respectively. The aim of this last section is to develop the trained deep neural network model and then use it for visual classification. The fusion output is fed to a softmax layer to produce the final classification decision as in Equation (14).

$$P\left(y∣x\right)=softmax\left(Ws\cdot h_{fusion}+b_f\right)$$

(14)

In Equation (14), $P\left(y∣x\right)$ is the probability that an image belongs to one of the 10 classes.

During back-propagation, the parameters of the Pythagorean fuzzy layer are updated by partial derivatives calculated by the chain rule. For example, the gradient of $c_{\mu }$ is obtained from the product of the derivative of the loss function with respect to μ and the derivative of $\mu$ with respect to $c_{\mu }$ (Equation (15)). Similarly, the weight matrix $w_f$ of the fusion layer is optimized by gradient descent, as in traditional DNNs (Equation (16)).

$${\displaystyle \frac{\partial L}{\partial c_{\mu }}}={\displaystyle \frac{\partial L}{\partial h_{fuzzy}}}\left(2\mu {\displaystyle \frac{\partial \left({\mu }^2-v^2\right)}{\partial \mu }}\right){\displaystyle \frac{2\left(x-c_{\mu }\right)}{{\sigma }^2}}\mu$$

(15)

$${\displaystyle \frac{\partial L}{\partial w_f}}={\displaystyle \frac{\partial L}{\partial h_{fuzzy}}}{ReLU}^{\prime }\left(w_f\left[h_{fuzzy}\oplus h_{dense}\right]+b_f\right){\left[h_{fuzzy}\oplus h_{dense}\right]}^T$$

(16)

The training phase of the PFDNN algorithm is presented in Algorithm 1. The steps outlined in Algorithm 1 include the dynamic calculation of the membership and hesitation degrees of the Pythagorean fuzzy layer, the fusion of these outputs with traditional convolutional and dense layers, and then training with adaptive optimization techniques.

Algorithm 1. PFDNN training model steps
Step Name	Description
Input	Training data $(X,Y),$ hyperparameters θ
Output	Trained model $M$
Initialization	- Initialize Pythagorean layer weights using Glorot uniform distribution.
	- Initialize other layer weights using Xavier normal initialization.
Forward Propagation	For each $x\in X$:
	- Compute Pythagorean output: $h_{fuzzy}=[Skor,$ Hesitation$,x]$
	- Compute $h_{dense}$
	- Generate fusion output using Equation (13)
Output	- Compute final output using Equation (14)
Backpropagation	- Compute gradients $\nabla \mathsf{\Theta }$ using Equations (15) and (16)
Regularization	- Apply Dropout and L2 norm
Stopping Criterion	- Stop if validation accuracy does not improve for 5 consecutive epochs
Return Model	- Return the trained model $M.$

2.6. Hyperparameter Optimization with Optuna

One of the most important factors affecting the performance of machine learning and deep learning models is the correct selection of the model’s hyperparameters. The last step of our approach is hyper-parameterization. Hyperparameters are parameters such as learning rate, number of layers, activation functions, etc. that are predetermined by the user and remain constant during model training. Each of these parameters play a decisive role on the learning capacity, generalization ability and computational cost of the model. Among traditional hyperparameter optimization methods, grid search, Bayesian optimization, hyperband and random search techniques are the most common. While grid search is a systematic approach based on trying all combinations, random search seeks solutions with less computational cost by randomly sampling these combinations. However, these methods can be inefficient regarding both time and resources, especially in high-dimensional search spaces [50]. Optuna is an open source hyperparameter optimization library developed by [12]. Its main differentiating feature is that it allows the search space to be defined dynamically with a define-by-run philosophy. With this approach, users can flexibly define the search space in the code according to the conditions. Thus, more flexible optimization strategies can be developed by going beyond classical grid-based methods. Optuna mostly uses the Tree-structured Parzen Estimator (TPE) method. In contrast to classical Bayesian optimization methods, instead of directly modeling the expected value of the objective function, TPE constructs two separate density functions on the parameters. These densities aim to maximize the probability difference between the trials with good results and the other trials. This allows for more sampling in more promising regions. Methods based on Bayesian optimization model the trial-and-error process, enabling more informed and predictive search. Optuna, one of the most recent and powerful examples of these approaches, is notable for its early stopping capability. This feature is essential in deep learning models with high computational cost. In addition, Optuna supports multi-objective optimization, enabling applications that want to improve multiple performance criteria simultaneously, such as accuracy and computation time. In such applications, solutions are generated based on Pareto optimality.

3. Experimental Setup

All experiments were conducted using Google Colab’s GPU resources (NVIDIA A100-SXM4 GPU with 40 GB memory, NVIDIA, Santa Clara, CA, USA) under Python (v3.12), with TensorFlow (v2.19.0), Keras (v3.10.0), NumPy (v2.0.2), scikit-learn (v1.6.1), and Optuna (v4.5.0) libraries. The proposed model was trained with the Adam optimizer using a mini-batch strategy. Hyperparameters such as learning rate, dropout rate, L2 regularization coefficient, fuzzy layer units (m), dense layer size, and fusion layer size were optimized using Optuna’s Tree-structured Parzen Estimator (TPE). A total of 30 trials were executed, and the configuration with the highest accuracy was selected. The maximum number of epochs was set to 100, with early stopping and ReduceLROnPlateau strategies applied to prevent overfitting and reduce training time. Each experiment was repeated three times to ensure reproducibility, and the averaged results are reported.

For the dataset preparation, the images were downloaded from the original source and extracted to a local directory. The images were converted to RGB format and pixel values were normalized. To ensure robust evaluation, we implemented a 5-fold cross-validation strategy, where each fold maintained the original class distribution through stratified sampling. This approach provides more reliable performance estimates compared to a single train-test split. To enhance model generalization and prevent overfitting, we applied extensive data augmentation techniques exclusively during the training phase of each fold. The augmentation pipeline included: random rotation (±15°), horizontal/vertical shift (10%), zoom (10%) and flip (10%) operations were applied to the training images to increase the generalization ability of the model.

4. Results and Discussion

4.1. General Comparative Results

Increasing satellite image resolutions and growing data volumes are pushing the limits of traditional classification methods. In particular, the complex spectral signatures of transitions between agricultural areas, urban areas and natural ecosystems clearly demonstrate the need for next generation hybrid models. In this section, the performance of our hybrid model developed with the aforementioned CNN+PFDNN and Optuna optimization on the EuroSAT dataset is examined in detail. In the experimental evaluation process, different machine learning models (DNN, FDNN, PFDNN, transfer learning based VGG16+PFDNN and CNN+PFDNN+Optuna as the final model) are analyzed comparatively. In the experiment, all models are separated by 5-fold cross-validation. Comparative performance results are presented in

Table 1. Overall classification accuracy and kappa.

Model	Accuracy	Kappa
DNN	0.8577 ± 0.0050	0.8529 ± 0.0056
FDNN	0.8161 ± 0.0056	0.7954 ± 0.0062
PFDNN	0.8419 ± 0.0075	0.8013 ± 0.0084
VGG16+PFDNN	0.8905 ± 0.0023	0.8781 ± 0.0026
CNN+PFDNN+Optuna	0.9696 ± 0.0037	0.9661 ± 0.0042

and

Table 2. Detailed performance metrics (Precision, Recall, F1, Training Time).

Model	Precision	Recall	F1-Score	Training Time (min)
DNN	0.8565	0.8547	0.8548	1.40
FDNN	0.8113	0.8561	0.8346	1.18
PFDNN	0.8520	0.8513	0.8466	1.36
VGG16+PFDNN	0.8903	0.8905	0.8900	1.43
CNN+PFDNN+Optuna	0.9692	0.9686	0.9686	1.68

.

According to Table 1, the DNN model with a deeper architecture achieved significantly higher accuracy (85.77%), showing the importance of hierarchical feature learning. The FDNN model with classical fuzzy logic performed slightly worse than DNN, reaching 81.61% accuracy. The main reason for this is believed to be the restricted ability of the traditional fuzzy layers used in the FDNN model. The PFDNN model developed with Pythagorean Fuzzy Layer achieved a performance improvement of about 3.16% compared to the classical FDNN and reached an accuracy of 84.19%. This increase is because Pythagorean fuzzy sets have a wider uncertainty representation capacity than classical intuitionistic fuzzy models. On the other hand, the VGG16+PFDNN model enriched with the transfer learning architecture outperformed PFDNN with an accuracy of 89.05%, but it could not reach the proposed CNN+PFDNN model. This shows that integrating a specially trained CNN architecture with Pythagorean Fuzzy Layer provides more effective results compared to pre-trained general-purpose feature extraction models.

According to Table 2, the deep learning based DNN model set a benchmark with 85.65% precision and 85.48% F1-score. Among the fuzzy logic integrated models, FDNN achieved the highest recall rate with 85.61% precision, while the PFDNN model showed a balanced performance of 85% in all metrics. In the category of hybrid models, the VGG16+PFDNN combination achieved a significant improvement with a metric value of 89%. However, the CNN+PFDNN+Optuna model performed the best with 96.92% precision, 96.86% recall and F1-score, demonstrating the advantage of customized architectures. When evaluated in terms of training times, a non-linear increase in the training times of the models with improved performance was observed. Especially the 1.68 min. training time of CNN+PFDNN+Optuna model is acceptable considering the performance improvement it provides. These findings prove the superiority of deep learning-based hybrid approaches in remote sensing image analysis.

4.2. Statistical Analysis

In our study, the nonparametric Friedman test was applied to statistically compare model performance across cross-validation folds. The Friedman test was chosen because it is specifically designed for repeated-measures experimental designs where the same data folds are evaluated under different models and because it does not require the normality assumption that standard ANOVA relies on. The test results indicate that the performance differences between the models are statistically significant ($\chi$²(4) = 20.000, $p=0.0005$). Following the Friedman test, the Nemenyi post-hoc test was performed. The Nemenyi procedure was chosen because it is the most appropriate nonparametric pairwise comparison method after Friedman and allows us to determine which model pairs exhibit significant performance differences without assuming homogeneity of variances. According to the mean ranking values, the CNN+PFDNN+Optuna model achieved the best overall ranking (1.0), followed by VGG16+PFDNN+Optuna (2.0), and DNN (3.0). The lowest rankings were achieved by FDNN and Plain PFDNN. Nemenyi comparisons confirmed that CNN+PFDNN significantly outperforms FDNN and Plain PFDNN with the largest rank differences (4.0 and 3.0, respectively). These findings support the conclusion that hybrid deep learning models, especially CNN+PFDNN, provide a statistically significant advantage over traditional architectures, demonstrating the effectiveness of the proposed approach.

4.3. Confusion Matrix & Class-Based Analysis

The contribution of Optuna-based hyperparameter optimization to the model is particularly noteworthy. The optimized parameters in the CNN+PFDNN model include learning rate, dropout rate, L2 regularization coefficient, number of fuzzy units and fully connected layer sizes. In this way, the model is optimized efficiently in terms of training time and the risk of overfitting is reduced. The best parameters of the proposed model are presented in

Table 3. Best parameters for CNN+PFDNN model.

Parameters	Values
Learning rate:	0.00057
Fuzzy units (m):	281
Dense layer sizes:	1,580,350,116
Fusion layer size:	494
Dropout rate:	0.25587
L2 regularization:	9.600242615788519 × 10−7

.

When the classification performance of the model is analyzed in more detail on a class basis, the confusion matrix provides remarkable insights. The confusion matrix is presented in Figure 5.

When the normalized confusion matrix is examined, it is seen that the CNN+PFDNN+Optuna model exhibits exceptional accuracy rates in many classes. Almost perfect classification performance was achieved in “Forest” (99.58%), “Residential Buildings” (99.25%) and “Sea&Lake” (99.54%) classes. The model is also very successful in agricultural classes such as “Annual Crop” (96.42%) and “Permanent Crop” (92.30%). The highest confusion rates of the model are seen in the “Industrial Buildings” class, where 2.15% of the samples of this class are confused with the “River” class. This can be explained by the reason that industrial areas are often found close to water sources and display similar light patterns. The 2.50% confusion rate between “Herbaceous Vegetation” and “Grassland” is due to the visual and spectral similarities of these two vegetation types. Another noteworthy finding is that 4.80% of the “Permanent Crop” class is mixed with “Herbaceous Vegetation”. This confusion may be due to the similar appearance of orchards and natural vegetation in seasonal changes. However, the fact that the rates of all these confusions remained below 5% shows that the model successfully learned the subtle distinctive features between the classes. It is also noteworthy that the model achieves very high accuracy rates in natural environment classes such as “Grassland” (95.25%) and “River” (95.75%). These results prove that the CNN+PFDNN+Optuna architecture is very successful in distinguishing both artificial and natural land cover types. The high performance observed in the highway class (96.90%) demonstrates the model’s ability to recognize linear structures.

The classification performance of the CNN+PFDNN+Optuna model is analyzed in detail based on the five-fold cross-validation results (Figure 6). The obtained metrics reveal that the performance of the model varies significantly in different land cover classes. “Forest” and “Sea&Lake” classes stand out as the classes where the model performs the best, with precision, recall and F1-score above 95%. This result demonstrates that the model is highly successful in recognizing natural environments with homogeneous structure and spectrally distinct features. In agricultural classes, it is observed that the performance varies according to the class. Although all metrics are above 90% in the “Annual Crop” class, there is a slight decrease in the “Permanent Crop” class, especially in the recall values (85–88% range). It is noteworthy that the precision values of the model are higher than the recall values in the urban land use classes “Residential Buildings” and “Industrial Buildings” (92% vs. 88%). This finding indicates that the model is more conservative in recognizing urban areas but makes fewer errors in the samples it recognizes. When the classes with relatively lower performance are analyzed, it is seen that the mixtures between “Herbaceous Vegetation” and “Grassland” have achieved F1-scores of 82–85%. The spectral and textural similarities of these two classes can be considered as the main reason for the difficulty in classification. When the standard deviation values are analyzed, variation in the range of 1–3% is observed in all classes. This is an indication of the stable performance of the model on different data subsets. Especially in the “River” class, the standard deviation is lower than the others (1.2%), which proves the consistency of the model in recognizing waterways.

A more detailed examination of the misclassifications highlights certain challenges faced by the proposed model. For example, the confusion between Industrial Buildings and River classes (2.15%) can be attributed to the spatial coexistence of industrial sites near waterways and their similar reflectance patterns. Similarly, the overlap between Herbaceous Vegetation and Grassland (2.5%) and Permanent Crop, Vegetation and Herbaceous Vegetation (4.8%) reflects the strong spectral and seasonal similarities of the vegetation types, especially under varying canopy density. These errors indicate that while the model performs exceptionally well overall, subtle within-class similarities are difficult to capture. However, the fact that all confusion rates remain below 5% indicates that the CNN+PFDNN+Optuna framework successfully discriminates between both natural and artificial land cover types. Such an analysis also provides valuable information for future studies by suggesting that incorporating additional spectral bands or contextual features could further reduce these specific misclassifications.

When the learning dynamics of the model are evaluated through the accuracy and loss curves, it is clear from the graphs (Figure 7) that the model was trained for 35 epochs. The red shaded area around the validation curves represents the standard deviation (or variability) of the validation metrics across different runs, indicating the consistency of the model’s performance. Assuming that the training and validation metrics evolve without deviating much from each other, it can be argued that the model exhibits balanced learning, and the risk of overfitting is low.

Figure 8 illustrates the multi-class ROC analysis of the CNN+PFDNN+Optuna model. Figure 8 also shows the micro- and macro-averaged ROC curves to illustrate the overall evaluation across all classes. The black dashed diagonal line represents the performance of a random classifier (AUC = 0.5) and is used as a baseline for comparing model performance. The overall AUC value of 0.9983 indicates that the model exhibits nearly perfect discriminatory performance and a balanced behavior between recall and specificity.

4.4. Advantages of the Proposed Method

To summarize, the proposed method has several key advantages. Experimental findings highlight several key advantages of the proposed CNN+PFDNN+Optuna model over baseline approaches. First, the model consistently achieved the highest values in terms of accuracy, kappa, precision, recall, and F1-score, demonstrating superior generalization across convolutions. Compared to classical FDNN and plain PFDNN models, the inclusion of a Pythagorean fuzzy layer provided a significantly broader capacity to model uncertainty, resulting in an accuracy improvement of more than 13%. Second, integrating CNN-based feature extraction with fuzzy reasoning enhanced the network’s ability to capture both spatial structures and ambiguous class transitions; this was particularly evident in challenging land cover categories such as “Herbaceous Vegetation” and “Grassland.” Third, Optuna-based hyperparameter optimization not only improved classification accuracy but also reduced the risk of overfitting, as supported by the balanced training/validation dynamics and nearly perfect AUC score. Finally, despite these performance gains, the training time of the proposed model remained comparable to standard architectures, demonstrating a favorable balance between computational cost and predictive power. These advantages demonstrate that the proposed hybrid architecture is not only more accurate but also more robust and efficient, making it a strong candidate for practical remote sensing applications.

On the other hand, to evaluate the success of the proposed approach more precisely, a comparison is also made with recent studies working with similar datasets and band information. For example, the CNN-based model proposed by [51] achieved 88% accuracy on test data using only RGB bands. This study emphasized the generalization capability of the model by pointing out that there is not a big difference between the training and test accuracies. However, our proposed model outperforms this approach by about 8% and provides a more stable classification performance. Similarly, a CNN-based architecture developed by [52] on the EuroSAT dataset reported 96.83% accuracy. Although this work presents high achievements, it is understood that the model has a greater capacity in factors such as data augmentation and network depth. Our proposed approach integrates uncertainty modeling with a simpler architecture, resulting in a more explainable and optimizable structure. In addition, a semi-supervised learning method developed for low-orbit satellites achieved about 91% accuracy [53]. Although the model makes a significant contribution in terms of practical applicability in the field, the training process was conducted with limited data and therefore the accuracy rate remained at a certain threshold. In this context, the proposed model provides higher performance through both supervised training and robust hyperparameter optimization. Moreover, up to 94% accuracy values were obtained in the model that works with Kolmogorov–Arnold Network (KAN) architecture and hybridized with CNN [54]. This model attracts attention by providing high accuracy even in early epochs. However, the complexity of the architecture and its limited interpretability may lead to some limitations in applications. Our proposed PFDNN architecture, on the other hand, integrates Pythagorean fuzzy layers to enhance interpretability, thus dealing with uncertainty and providing a transparent decision process. Ref. [37] presented benchmarks with spectral bands of the EuroSAT dataset using the ResNet-50 algorithm and achieved an overall classification accuracy of 98.57% with the proposed new dataset. Finally, ref. [55] achieved 87.83% accuracy on approximate Eurosat RGB data with basic CNN architectures.

Overall, the proposed approach is quite competitive compared to similar studies in the literature in terms of both classification accuracy and model reliability. Moreover, the model’s ability to effectively represent uncertainties and the hyperparameter optimization with Optuna make the proposed method an attractive option in both academic and applied fields.

5. Conclusions

Image processing and remote sensing technologies enable images from satellite and aerial platforms to be analyzed and transformed into meaningful information. These images help to make critical decisions in many areas such as land use, environmental changes, disaster management and agriculture. However, the complexity, noise and uncertainty of these images require the use of advanced methods for accurate classification.

In this study, a hybrid model combining PFS theory and deep learning architectures is proposed for the classification of satellite images. The proposed CNN+PFDNN+Optuna approach achieved impressive results such as 96.96% accuracy and a macro-AUC score of 0.9983 on the EuroSAT dataset. The success of the model is due to the fact that the Pythagorean fuzzy layer effectively manages the uncertainties and the hyperparameter optimization with Optuna improves the performance. Moreover, this method outperforms similar studies in the literature by providing higher accuracy compared to classical fuzzy systems.

The proposed CNN+PFDNN+Optuna framework has the potential to serve many practical applications in remote sensing. For example, it could support crop classification, yield estimation, and precision farming in agriculture; the detection of land use changes and deforestation in environmental monitoring; and the early detection of natural disasters such as floods or fires in disaster management. In this context, the proposed architecture not only provides academic success but also provides a framework that can directly contribute to critical decision-making processes.

Although the proposed framework has achieved high accuracy and robustness on the EuroSAT RGB dataset, it also has some limitations. The evaluation is limited to a single dataset, which may not fully represent the diversity of real-world remote sensing images. However, the use of 5-fold cross-validation and statistical significance tests (Friedman and Nemenyi) demonstrates the robustness and generalization capability of the proposed method on the EuroSAT dataset. Applying the method to multispectral and hyperspectral datasets will be necessary to validate its generalizability.

Additionally, baseline comparisons are limited to selected deep learning and fuzzy-based models. While the proposed CNN+PFDNN+Optuna framework clearly outperforms these baselines, comparisons with widely used architectures such as ResNet, DenseNet, and EfficientNet would provide a more comprehensive assessment.

This study did not specifically test the model’s robustness to noise and distortions. However, extensive data augmentation (rotation, shift, zoom, flip, brightness changes, etc.) applied during the training process indirectly exposed the model to different types of distortions and increased its generalization ability. Furthermore, the direct modeling of uncertainties in the Pythagorean fuzzy layer contributed to more effective management of noise-induced instabilities. Hyperparameter optimization using Optuna provided a more stable learning process, strengthening the model’s resilience to distortions. Furthermore, future studies should systematically examine different types of noise, particularly Gaussian noise, blur, or atmospheric distortions, to provide a more detailed assessment of the model’s robustness.

Finally, the current work primarily focuses on classification performance. Interpretability is only considered as a future direction, and the integration of explainable AI techniques will contribute to making the framework more transparent and practical for decision-making processes.