In many scientific fields, the generation and evolution of data are governed by partial differential equations (PDEs) which are typically informed by established physical laws at the macroscopic level to describe general and predictable dynamics. However, some complex influences may not be fully captured by these laws at the microscopic level due to limited scientific understanding. This work proposes a unified framework to model, estimate, and infer the mechanisms underlying data dynamics. We introduce a general semiparametric PDE (SemiPDE) model that combines interpretable mechanisms based on physical laws with flexible data-driven components to account for unknown effects. The physical mechanisms enhance the SemiPDE model's stability and interpretability, while the data-driven components improve adaptivity to complex real-world scenarios. A deep profiling M-estimation approach is proposed to decouple the solutions of PDEs in the estimation procedure, leveraging both the accuracy of numerical methods for solving PDEs and the expressive power of neural networks. For the first time, we establish a semiparametric inference method and theory for deep M-estimation, considering both training dynamics and complex PDE models. We analyze how the PDE structure affects the convergence rate of the nonparametric estimator, and consequently, the parametric efficiency and inference procedure enable the identification of interpretable mechanisms governing data dynamics. Simulated and real-world examples demonstrate the effectiveness of the proposed  methodology and support the theoretical findings.

报告人简介：姚方，北京大学讲席教授、入选国家高层次人才计划，北大统计科学中心主任、概率统计系主任。国际数理统计学会（IMS）Fellow，美国统计学会（ASA）Fellow。获2014年加拿大CRM-SSC奖、2024年第六届科学探索奖。2000年本科毕业于中国科技大学统计专业，2003获得加利福尼亚大学戴维斯分校统计学博士学位，曾任职于多伦多大学统计科学系长聘正教授。至今担任多个国际统计学核心期刊的主编或编委，包括《加拿大统计学期刊》主编，顶级期刊《Journal of the American Statistical Association》和《Annals of Statistics》编委等。