Discrete fractional neural networks within the framework of octonions: A preliminary exploration

Conventional neural networks constructed on real or complex domains have limitations in capturing multi-dimensional data with memory effects. This work is a preliminary exploration of discrete fractional neural network modeling within the framework of octonions. Initially, by introducing the discrete fractional Caputo difference operator into the octonion domain, we establish a novel system of discrete fractional delayed octonion-valued neural networks (DFDOVNNs). The new system provides a theoretical support for developing neural network algorithms that are useful for solving complex, multi-dimensional problems with memory effects in the real world. We then use the Cayley–Dickson technique to divide the system into four discrete fractional complex-valued neural networks to deal with the non-commutative and non-associative properties of the hyper-complex domain. Next, we establish the existence and uniqueness of the equilibrium point to the system based on the homeomorphism theory. Furthermore, by employing the Lyapunov theory, we establish some straightforward and verifiable linear matrix inequality (LMI) criteria to ensure global Mittag-Leffler stability of the system. In addition, an effective feedback controller is developed to achieve the system's drive-response synchronization in the Mittag-Leffler sense. Finally, two numerical examples support the theoretical analysis. This research introduces a novel direction in neural network studies that promises to significantly advance the fields of signal processing, control systems, and artificial intelligence.