New adaptive memory stochastic fractional operators and their applications in computational intelligence

This paper introduces a new family of adaptive memory fractional integral and derivative operators designed for second-order stochastic processes in the mean-square (m.s.) sense. The proposed operators incorporate a probabilistically weighted, time-increment-dependent kernel that enables flexible modeling of random dynamical systems with nonlocal memory effects. We define left- and right-sided adaptive memory fractional integrals for square-integrable stochastic processes and derive their corresponding Cauchy-type formulations. The associated adaptive memory fractional derivatives are also developed through a Gamma-function-based generalization adapted to the m.s. framework. Fundamental properties, including linearity, boundedness, continuity, semigroup property, and m.s. convergence, are rigorously established, and the underlying operator spaces are shown to form complete m.s. Banach structures. Several limiting cases recover well-known stochastic analogues of classical operators such as the Riemann–Liouville, Hadamard, and Katugampola forms. A comparative analysis provides that the new adaptive memory operators possess enhanced flexibility in capturing memory, increment growth, and correlation structures inherent to random processes. To illustrate computational relevance, these newly-framed fractional operators are incorporated into a stochastic neural network setting, where adaptive memory-based fractional gradients improve training dynamics under randomness and uncertainty. The framework offers strong potential for applications in stochastic modeling, probabilistic fractional dynamics, statistical signal processing, and machine-learning systems driven by second-order random processes.