Tempered fractional neural grey system model with Hermite orthogonal polynomial

Recently, the study of fractional calculus has emerged as a thriving field in academic research. The rise of artificial intelligence technology has fueled researchers’ extensive exploration of neural network models, leading to an abundance of practical applications. Consequently, this paper aims to propose an innovative neural grey system model that integrates neural network technology with fractional calculus theory. By utilizing the remarkable features of Hermite polynomials, we have developed a scalable model. The model parameters are determined directly through the least squares method, simplifying the computational complexity. Moreover, we present a hyperparameter determination method using simulation techniques and intelligent optimization algorithms. To assess the effectiveness of the proposed model, we conduct fitting experiments on 12 real datasets to evaluate its accuracy. Comparative analysis with classical fractional grey models and machine learning models provides substantial evidence of the superior performance of our model. The proposed model can not only improve the ability of sorting nonlinear data by polynomials, but also can greatly improve the prediction accuracy by new fractional order accumulation and difference operators.