随机算法与应用随机分析

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.

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。

随机模型广泛出现在物理、化学、生物及数据科学等很多领域，随机模型数值格式的分析及设计随机算法非常重要。本项目在随机模型数值分析方面，研究了一些随机微分方程数值格式的弱收敛性，针对过阻尼广义朗之万方程对应的分数阶随机微分方程提出了一些数值格式并进行了理论分析；在随机算法方面，对随机梯度下降算法的长时间行为进行了研究，针对交互作用粒子系统发展了一系列基于随机分批的算法、进行了相关的误差估计与渐近行为分析等，并应用到了分子动力学模拟、等离子体模拟中。项目的成果能在理论上加深对数据科学中算法、随机数值格式的理解，设计的算法有望提升电化学、分子动力学模拟的效率。特别地，系列基于随机分批的算法如Random Batch Method, Random Batch Ewald等在同行内已经产生了一定的影响，并且在一些软件包上实现，相关文章也在多个渠道被推广宣传。