Stochastic dynamics and data science

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.