Establishing appropriate mathematical models for complex systems in natural
phenomena not only helps deepen our understanding of nature but can also be
used for state estimation and prediction. However, the extreme complexity of
natural phenomena makes it extremely challenging to develop full-order models
(FOMs) and apply them to studying many quantities of interest. In contrast,
appropriate reduced-order models (ROMs) are favored due to their high
computational efficiency and ability to describe the key dynamics and
statistical characteristics of natural phenomena. Taking the viscous Burgers
equation as an example, this paper constructs a Convolutional
Autoencoder-Reservoir Computing-Normalizing Flow algorithm framework, where the
Convolutional Autoencoder is used to construct latent space representations,
and the Reservoir Computing-Normalizing Flow framework is used to characterize
the evolution of latent state variables. In this way, a data-driven stochastic
parameter reduced-order model is constructed to describe the complex system and
its dynamic behavior.

建立适当的数学模型来研究自然现象中的复杂系统不仅有助于加深对自然的理解，还可以用于状态估计和预测。然而，自然现象的极端复杂性使得发展全阶模型并将其应用于研究多个感兴趣的量非常具有挑战性。相反，适当的降阶模型由于其高计算效率和描述自然现象的关键动态和统计特性的能力而备受青睐。以粘性 Burgers 方程为例，本文构建了一个卷积自编码器 - 蓄积计算 - 归一化流算法框架，其中卷积自编码器用于构建潜空间表示，蓄积计算 - 归一化流框架用于描述潜状态变量的演化。通过这种方式，构建了一个数据驱动的随机参数降阶模型来描述复杂系统及其动态行为。