A Novel Approach to Modeling Incommensurate Fractional Order Systems Using Fractional Neural Networks

This research explores the application of the Riemann–Liouville fractional sigmoid, briefly ${}^{RL}F\sigma$, activation function in modeling the chaotic dynamics of Chua’s circuit through Multilayer Perceptron (MLP) architecture. Grounded in the context of chaotic systems, the study aims to address the limitations of conventional activation functions in capturing complex relationships within datasets. Employing a structured approach, the methods involve training MLP models with various activation functions, including ${}^{RL}F\sigma$, sigmoid, swish, and proportional Caputo derivative ${}^{PC}\sigma$, and subjecting them to rigorous comparative analyses. The main findings reveal that the proposed ${}^{RL}F\sigma$ consistently outperforms traditional counterparts, exhibiting superior accuracy, reduced Mean Squared Error, and faster convergence. Notably, the study extends its investigation to scenarios with reduced dataset sizes and network parameter reductions, demonstrating the robustness and adaptability of ${}^{RL}F\sigma$. The results, supported by convergence curves and CPU training times, underscore the efficiency and practical applicability of the proposed activation function. This research contributes a new perspective on enhancing neural network architectures for system modeling, showcasing the potential of ${}^{RL}F\sigma$ in real-world applications.