Physics-informed neural networks with caputo-fabrizio derivatives for nonlinear fractal-fractional delay equations and chaotic systems.

Complex dynamical systems, ranging from anomalous diffusion and memristor circuits to convective fluid motion in rotating cavities, exhibit non-local memory, long-range interactions, and proportional delays governed by Fractal-Fractional Differential Equations (FFDEs). This study introduces PINN-CF, a Physics-Informed Neural Network framework that integrates Caputo-Fabrizio (CF) Fractional Derivatives with the Adam optimizer. The CF operator is accurately approximated using the L1 finite-difference scheme. Unlike standard PINN that relay on singular kernel fractional operators, PINN-CF employs a non-singular exponential kernel, which avoids initial singularities and better capture smooth memory decay. PINN-CF accurately solves four benchmark problems of FFDEs, Pantograph-type delay equations, Memristor-based neural circuits, and FFDE systems of convective fluid motion in rotating cavities. The suggested PINN-CF method achieves residual errors below [Formula: see text], and as low as [Formula: see text] in near classical regimes. Reproducibility and validation ensured by convergence curves, and tabulated results. Offering superior flexibility, scalability, and computational efficiency over traditional numerical solvers, PINN-CF is readily extendable to nonlinear, multi-delay, distributed-order, or stochastic models.