Integration of quantum derivative into artificial neural network learning algorithms and new algorithms for solving the monotone inclusion problem

In this thesis, the concept of the q-derivative, which renders the learning process more flexible and adaptable, is integrated with artificial neural networks. While the q-derivative has previously been considered in the literature only at the level of activation functions, this study introduces novel approaches by employing it in place of the ordinary derivative within backpropagation-based algorithms (backpropagation, stochastic backpropagation, and resilient backpropagation). The proposed algorithms were evaluated on classification problems using the Digits, MNIST, and Iris datasets, and were compared against their classical counterparts. The results demonstrate that the q-derivative-based algorithms achieved superior performance in terms of accuracy, precision, recall, and F1-score compared to traditional methods. On the other hand, two new algorithms were developed to address monotone inclusion problems defined in real Hilbert spaces. By incorporating double inertial and viscosity terms into the existing forward–backward splitting algorithms, a more general framework was constructed, and the convergence properties of these algorithms were formally established and proven. Both approaches were applied to regression and classification tasks using the California Housing, Diabetes, Iris, Digits, MNIST, Anacalt, Delta Ail, Machine CPU, BCWD, and Fashion MNIST datasets. The findings indicate that the proposed methods ensure stable convergence, attain high accuracy, and exhibit performance competitive with existing techniques.