High-order harmonic generation (HHG) from the interaction of ultra-intense laser pulses with atoms is an important tabletop short-wave coherent light source. Accurate quantum simulations of it present large computational difficulties due to multi-electron multidimensional effects. In this paper, the time-dependent response of hydrogen atoms is calculated using a time-series prediction scheme, the HHG spectrum is reconstructed very accurately. The accuracy of the forecasting is further improved by using a neural network scheme. This scheme is also applied to the simulation of the harmonic emission on multi-electron systems, and the applicability of the scheme is confirmed by the harmonic calculation of complex systems. This method is expected to simulate the nonlinear dynamic process of multi-electron atoms and molecules irradiated by intense laser pulses quickly and accurately.