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 the traditional Reservoir Computing performance and takes advantage of both approaches. This model-free method successfully predicts the long-term evolution of stochastic dynamical systems and replicates dynamical behaviors. With few assumptions about the underlying stochastic dynamical systems, we deal with Markov/non-Markov and stationary/non-stationary stochastic processes defined by linear/nonlinear stochastic differential equations or stochastic delay differential equations. We verify the effectiveness of the proposed framework in five experiments, including the Ornstein-Uhlenbeck process, Double-Well system, El Ni\~no Southern Oscillation simplified model, and stochastic Lorenz system. Additionally, we explore the noise-induced tipping phenomena and the replication of the strange attractor.

翻译：随机动力系统的预测及动力学行为的捕捉是深层次的问题。本文提出一种将储层计算与归一化流相结合的数据驱动框架来研究该问题，该框架通过模拟误差建模来提升传统储层计算的性能，并充分利用两种方法的优势。这种无模型方法成功预测了随机动力系统的长期演化，并复现了其动力学行为。在关于底层随机动力系统假设较少的前提下，我们处理了由线性/非线性随机微分方程或随机延迟微分方程定义的马尔可夫/非马尔可夫以及平稳/非平稳随机过程。我们通过五个实验验证了所提框架的有效性，包括奥恩斯坦-乌伦贝克过程、双阱系统、厄尔尼诺-南方涛动简化模型以及随机洛伦兹系统。此外，我们还探讨了噪声诱导的跃迁现象及奇异吸引子的复现问题。