当下数据科学在各个领域中起着越来越重要的作用，其本质为如何从随机样本出发对潜在的函数或方程进行逼近。而逼近理论尤其是径向基函数方法为多维、散乱数据的处理和学习提供了有力的工具。学习理论作为其新的延伸点成为了其中的核心方法。本项目拟将统计学的方法引入到高维数据的逼近问题中，主要研究内容包括：一方面，对高维随机采样点，利用蒙特卡罗方法，构造随机拟插值逼近算子，并在理论上探讨算子是否具有良好的逼近能力；另一方面，研究在稀疏网格上逼近高维函数的拟插值算子构造策略，探寻缓解维数灾难，提高高维拟插值算子逼近效率的有效路径。最后，结合问题背景，探寻一些未知或者难以求解的（随机）微分方程的自适应稀疏网格数值解法。本项目将从随机性和确定性两个观点出发，结合逼近论、统计学与学习理论等知识，探索随机散乱数据与方程的拟合问题，希望这样的交叉能引起更多数学领域对数据科学的研究，带来更多相关数学工具的产生和发展。

Nowadays, data science which aims to approximate potential functions or equations from random samples, plays an increasingly important role in various fields. The approximation theory, especially the radial basis function method, provides a powerful tool for processing and learning multidimensional and scattered data. As its new extension learning theory has become the key method. This project intends to introduce the statistical methods to the classical approximation theory. The project will focus on the three points: On the one hand, we construct an random quasi-interpolation operator using Monte Carlo method for random sampling data, and discuss the approximation ability theoretically. On the other hand, in order to alleviate dimension coarse problem, we study the construction of quasi-interpolation operators that approximate high-dimensional functions on sparse grids, and explore the method to improve the approximation efficiency of quasi-interpolation operators. Finally, combined with the background of the problem, the adaptive sparse grid numerical solutions of some unknown or difficult (random) differential equations are studied. This project will explore the simulation problems of random scattered data and equations from the perspectives of both randomness and certainty. It is an interdisciplinary research, which combines with the approximation theory, statistical method and learning theory. We hope this project will lead to more research on Data Science in mathematical field, as well as more production and development of relevant mathematical tools.