To address the fitting limitations of traditional grey models in handling complex non-linear exponential sequences, this study proposes an adaptive grey prediction model driven by a fractional-order polynomial, abbreviated as ${\rm GMFP} (1, 1, N)$. A systematic framework is established, covering the model’s construction mechanism, parameter estimation method, and time response function. Furthermore, the evolution of model properties under specific parameter settings is analyzed to deepen theoretical understanding. To further enhance the performance, the linear decreasing weight particle swarm optimization (LDW-PSO) algorithm is employed to optimize the model’s hyperparameters. Experimental validation is conducted using China’s carbon emission data prediction, demonstrating that the ${\rm GMFP} (1, 1, N)$ model achieves superior accuracy and stability in both fitting and forecasting stages compared to the classical ${\rm GM} (1,1) $ model and other optimized variants. This verifies the effectiveness and applicability of the proposed model in modeling complex carbon emission datasets characterized by nonlinearity and uncertainty, providing a robust tool for high-precision forecasting in this specific domain with limited data samples.