Recent advances in data science are opening up new research fields and broadening the range of applications of stochastic dynamical systems. Considering the complexities in real-world systems (e.g., noisy data sets and high dimensionality) and challenges in mathematical foundation of machine learning, this review presents two perspectives in the interaction between stochastic dynamical systems and data science.

On the one hand, deep learning helps to improve first principle-based methods for stochastic dynamical systems. AI for science, combining machine learning methods with available scientific understanding, is becoming a valuable approach to study stochastic dynamical systems with the help of observation data. On the other hand, a challenge is the theoretical explanations for deep learning. It is crucial to build explainable deep learning structures with the help of stochastic dynamical systems theory in order to demonstrate how and why deep learning works.

In this review, we seek better understanding of the mathematical foundation of the state-of-the-art techniques in data science, with the help of stochastic dynamical systems, and we further apply machine learning tools for studying stochastic dynamical systems. This is achieved through stochastic analysis, algorithm development, and computational implementation. Topics involved with this review include Stochastic Analysis, Dynamical Systems, Inverse Problems, Data Assimilation, Numerical Analysis, Optimization, Nonparametric Statistics, Uncertainty Quantification, Deep Learning, and Deep Reinforcement Learning. Moreover, we emphasize available analytical tools for non-Gaussian fluctuations in scientific and engineering modeling.

数据科学的最新进展正在开辟新的研究领域并扩大随机动力系统的应用范围。考虑到现实世界系统的复杂性（例如，噪声数据集和高维度）和机器学习数学基础的挑战，本综述提出了随机动力系统与数据科学之间相互作用的两种观点。

一方面，深度学习有助于改进随机动力系统的基于第一原理的方法。科学人工智能将机器学习方法与现有的科学理解相结合，正在成为借助观测数据研究随机动力系统的一种有价值的方法。另一方面，挑战是深度学习的理论解释。为了证明深度学习如何以及为何发挥作用，借助随机动力系统理论构建可解释的深度学习结构至关重要。

在这篇综述中，我们寻求借助随机动力系统更好地理解数据科学中最先进技术的数学基础，并进一步应用机器学习工具来研究随机动力系统。这是通过随机分析、算法开发和计算实现来实现的。本次综述涉及的主题包括随机分析、动力系统、反问题、数据同化、数值分析、优化、非参数统计、不确定性量化、深度学习和深度强化学习。此外，我们强调科学和工程建模中非高斯波动的可用分析工具。