Existence, Controllability and Neural Network Modeling for Impulsive Fractional Volterra–Fredholm Integro-Differential Equations

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In this paper, the existence and uniqueness of solutions for a class of impulsive fractional Volterra–Fredholm integro-differential equations (VFIDEs) with Caputo derivative are investigated. The mathematical framework is developed in the piecewise continuous space PC([0,1],X) to properly accommodate impulsive solutions with state jumps. The existence of the solution is obtained based on the application of Krasnoselskii’s fixed-point theorem (FPT), while the uniqueness is established using Banach’s FPT under suitable Lipschitz and boundedness conditions. The main contribution of this work concerns the controllability investigation, where sufficient conditions for the controllability of the impulsive fractional system are derived. A feedback control function is constructed using the controllability Gramian operator, and the existence of a solution to the controlled system is proved via Schauder’s FPT. The controllability criteria are validated through a numerical example with explicit parameter verification, where the corrected integral boundary condition properly accounts for impulse contributions occurring in the interval [0,η]. To bridge theory and applications, we investigate artificial neural networks (ANNs) as surrogate models for fast state prediction. The input features are restricted to quantities computable solely from (ζ,u(ζ)), eliminating target leakage. The dataset is split by complete trajectories to ensure proper generalization assessment. The predictive power of the designed neural network gives an R2≈0.995 score on unseen test trajectories, with additional verification of equation residuals, impulse errors and boundary condition errors. This indicates that the network effectively captures both the non-local memory behavior and the impulsive discontinuities of the system. The hybrid mathematical–AI framework presented in this work provides both theoretical guarantees and practical computational tools for complex fractional dynamical systems with memory effects and impulsive properties.

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