Neural grey system model based on generalized conformable fractional derivatives and its applications

Accurately forecasting the evolution of complex systems with scant observational data demands models that reconcile long-range memory effects with expressive nonlinear mappings. In this study, we introduce a rigorously formulated fractional-order calculus framework-together with its discrete analogue-that enables continuously tunable differentiation orders and thus a refined representation of hereditary dynamics. Building on this theoretical advance, we derive a fractional grey prediction model whose fractional accumulation operator and discrete fractional differential equation extend the descriptive reach of classical grey theory. We further embed this model in a neural architecture by employing Chebyshev-polynomial activation functions, whose orthogonality accelerates functional approximation, and by estimating all parameters through a closed-form least-squares scheme, thereby preserving analytical transparency. Applied to real-world time-series forecasting problems, the resulting neural fractional grey system consistently attains lower forecasting errors than both conventional grey models and standard feed-forward neural networks, underscoring the complementary strengths of fractional calculus, grey-system parsimony, and neural nonlinearity. The proposed framework offers a transferable methodology for robust prediction under data scarcity and enriches the methodological arsenal available for energy, economic, and other application domains characterized by limited yet highly uncertain observations.