Nowadays, the nonlinear grey Bernoulli model [NGBM(1,1)] has been successfully applied to various fields. Its main advantage is that the power exponent can better reflect the non-linear characteristics of the original data. However, the parameters of the model (i.e., the order of accumulation, coefficient of background value, and power index) must be optimized to fit the development law of the system. In this study, a fractional non-linear grey Bernoulli model [MFNGBM (1,1)] is proposed to

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... e the perturbation limit of the classical NGBM and further improve the accuracy of the model, which uses mutual fractional operators and a new optimization scheme with a differential evolution (DE) algorithm for forecasting education investment. In the scheme, the power exponent of the Bernoulli differential equation, coefficient of background, and cumulative order of the original sequence are taken as decision variables, and their optimal parameters obtained by iteratively adjusting fitness functions. The experimental evaluation is conducted on two types of open-source data, and the results show that the proposed method can be very competitive with the popular baselines. Finally, MFNGBM(1,1) is used to predict China's education investment in 2020-2025. INDEX TERMS Grey system, grey forecasting model, non-linear grey bernoulli model, fractional order, differential evolution algorithm, educational investment.