Relaxed conditions and PSO-based optimization for the problem of Mittag-Leffler synchronization and its application in image restoration for fractional-order octonion-valued two-layer neural networks.

Abstract

This study focuses on Mittag-Leffler synchronization of fractional-order octonion-valued two-layer neural networks (FOOVTLNNs) and its application in color image restoration. The non-commutativity and non-associativity of octonion algebra and memory-dependent dynamics of fractional-order systems pose key challenges for low-conservative synchronization criteria and efficient optimization. A unified FOOVTLNNs' model is established via Caputo fractional derivatives and general activation functions. A novel fractional-order quadratic inequality in octonion field is derived to reduce conservatism, and a hybrid framework such as non-decomposed Lyapunov-Krasovskii functional(LKF) and real-component-separated deduction is proposed to handle octonion algebraic constraints. Quadratic coefficients are embedded into LKF, criteria and controller gains for joint constraint relaxation. Particle swarm optimization (PSO) optimizes the quadratic coefficients, and three PSO-based synchronization algorithms are designed. Numerical simulations and image restoration experiments validate the method that the optimized algorithms can reduce conservatism, accelerate convergence, and improve Peak Signal-to-Noise Ratio (PSNR) and so on. The hybrid strategy overcomes some limitations of fractional-order octonion-valued network, while the relaxed criteria and PSO-aided algorithms provide a reliable theoretical and practical basis for FOOVTLNN synchronization and multimedia applications.