随机性与不确定性无处不在。建模方面，由于信息的不完整或降维等模型的简化处理，随机性和随机微分方程就会出现，比如在朗之万方程里系统与热浴相互作用的涨落部分就是用白噪声模拟的。算法方面，我们经常用随机算法来降低因为大数据、维数诅咒、非线性等因素导致的高计算复杂度。本项目一方面是研究一些随机模型的数值格式，另一方面是利用随机分析的技巧去理解、进而设计物理学以及机器学习的新算法。具体地，数值格式方面，设计基于高斯混合模型的高精度算法，球面上的投影算法，分数阶随机微分方程弱收敛格式。随机算法方面，利用机器学习里小批的思想去设计物理学里相互作用粒子系统算法，方差降低方法；反之，用物理学里面的诸如朗之万方程，吉布斯分布等概念去理解与设计机器学习里数据处理算法。本项目将推进应用数学、物理、机器学习的融合。

Randomness and uncertainty are ubiquitous. On the modeling side, due to lacking of full information or modeling simplification like dimension reduction, randomness and stochastic differential equations appear in various models. For example, for Langevin equations, one often uses the white noise to describe the fluctuation in the interaction between the system and the heat bath. On the algorithm side, one often uses random algorithms to reduce the complexity due to big data, curse of dimensionality, and nonlinearity. On one hand, this project is to design numerical schemes for some stochastic models. On the other hand, we aim to use the techniques of stochastic analysis to understand and design random algorithms in physics and machine learning. In particular, regarding numerical schemes, we aim to design schemes with high accuracy based on Gaussian mixtures, projection schemes on sphere, and weak convergent schemes for fractional stochastic differential equations. Regarding algorithms, inspired by the mini-batch idea in machine learning, we plan to design numerical algorithms for interacting particle systems, and relevant variance reduction methods; moreover, we also want to make use of some ideas in physics, like Langevin equations, Gibbs distribution etc to understand and design data analysis algorithms in machine learning. This project will promote the interaction between applied math, physics and machine learning.