We present a numerical framework for learning unknown stochastic dynamical systems using measurement data. Termed stochastic flow map learning (sFML), the new framework is an extension of flow map learning (FML) that was developed for learning deterministic dynamical systems. For learning stochastic systems, we define a stochastic flow map that is a superposition of two sub-flow maps: a deterministic sub-map and a stochastic sub-map. The stochastic training data are used to construct the deterministic sub-map first, followed by the stochastic sub-map. The deterministic sub-map takes the form of residual network (ResNet), similar to the work of FML for deterministic systems. For the stochastic sub-map, we employ a generative model, particularly generative adversarial networks (GANs) in this paper. The final constructed stochastic flow map then defines a stochastic evolution model that is a weak approximation, in term of distribution, of the unknown stochastic system. A comprehensive set of numerical examples are presented to demonstrate the flexibility and effectiveness of the proposed sFML method for various types of stochastic systems.

翻译：我们提出了一种利用测量数据学习未知随机动力系统的数值框架。该新框架被称为随机流映射学习（sFML），是用于学习确定性动力系统的流映射学习（FML）方法的扩展。为学习随机系统，我们定义了一个由两个子映射叠加而成的随机流映射：确定性子映射和随机子映射。首先利用随机训练数据构建确定性子映射，随后构建随机子映射。确定性子映射采用残差网络（ResNet）形式，与FML处理确定性系统的方法类似。对于随机子映射，我们使用生成模型，本文中特指生成对抗网络（GANs）。最终构建的随机流映射定义了一个随机演化模型，该模型在分布意义上是对未知随机系统的弱近似。通过一系列全面的数值算例，验证了所提出的sFML方法在各类随机系统中的灵活性与有效性。