  The prediction of stochastic dynamical systems and the capture of dynamical
behaviors are profound problems. In this article, we propose a data-driven
framework combining Reservoir Computing and Normalizing Flow to study this
issue, which mimics error modeling to improve traditional Reservoir Computing
performance and integrates the virtues of both approaches. With few assumptions
about the underlying stochastic dynamical systems, this model-free method
successfully predicts the long-term evolution of stochastic dynamical systems
and replicates dynamical behaviors. We verify the effectiveness of the proposed
framework in several experiments, including the stochastic Van der Pal
oscillator, El Ni\~no-Southern Oscillation simplified model, and stochastic
Lorenz system. These experiments consist of Markov/non-Markov and
stationary/non-stationary stochastic processes which are defined by
linear/nonlinear stochastic differential equations or stochastic delay
differential equations. Additionally, we explore the noise-induced tipping
phenomenon, relaxation oscillation, stochastic mixed-mode oscillation, and
replication of the strange attractor.
