Parameter estimation of fractional uncertain differential equations

Fractional uncertain differential equations are used for forecast in this paper. It is crucial to provide an accurate estimation method since fractional uncertain differential equations hold memory effects and parameter estimation strongly depends on the accuracy of numerical schemes. A generalized moment estimation method is developed by the famous \(L_1\) $L_1$ numerical scheme. Adam optimization algorithm is adopted in the minimum optimization problem and the obtained optimal parameters can pass the uncertain hypothesis test. Compared with the uncertain differential equations, the fractional one holds more parameter freedom degrees and captures long-term interactions of observed data. Finally, the applications to aftershock frequency and pharmacokinetics are provided as examples. It can be concluded that fractional uncertain differential equations lead to the higher forecast accuracy which also provides some potential works on “fractional deep learning" under uncertain environments.