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.

翻译：随机动力系统的预测及其动力学行为的捕捉是深层次的难题。本文提出一种融合储层计算与归一化流的数据驱动框架来研究该问题，该框架通过模拟误差建模提升传统储层计算的性能，并整合了两种方法的优势。在几乎不对底层随机动力系统做任何假设的前提下，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。我们通过多个实验验证了所提框架的有效性，这些实验包括随机范德波尔振荡器、厄尔尼诺-南方涛动简化模型以及随机洛伦兹系统。这些实验涵盖了由线性/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/非马尔可夫过程以及平稳/非平稳随机过程。此外，我们还探索了噪声诱导的临界转变现象、弛豫振荡、随机混合模式振荡以及奇异吸引子的复现。