  The recurrent neural network with the long short-term memory cell (LSTM-NN)
is employed to simulate the long-time dynamics of open quantum system. The
bootstrap method is applied in the LSTM-NN construction and prediction, which
provides a Monte-Carlo estimation of forecasting confidence interval. Within
this approach, a large number of LSTM-NNs are constructed by resampling
time-series sequences that were obtained from the early-stage quantum evolution
given by numerically-exact multilayer multiconfigurational time-dependent
Hartree method. The built LSTM-NN ensemble is used for the reliable propagation
of the long-time quantum dynamics and the simulated result is highly consistent
with the exact evolution. The forecasting uncertainty that partially reflects
the reliability of the LSTM-NN prediction is also given. This demonstrates the
bootstrap-based LSTM-NN approach is a practical and powerful tool to propagate
the long-time quantum dynamics of open systems with high accuracy and low
computational cost.
