报告时间：2025年11月15日 8:30-11:30

报告地点：数理楼145

题目：Machine Learning for Solving Inverse Problems of Differential Equations

摘要：This presentation introduces two machine learning frameworks that exploit thermodynamic structures for differential equation problems. EnSIL (Entropy Structure Informed Learning) solves inverse problems by learning entropy balance equations rather than original differential equations, using Legendre polynomial approximation with integral-based regression and thermodynamic constraints. EnSIL achieves mean relative errors below 1% and demonstrates robustness to noise and large time steps compared to SINDy and PDE-FIND across chemical reactions, chaotic systems, and stochastic processes. MEP-Net (Maximum Entropy Principle Neural Network) addresses forward problems by generating probability distributions from limited moment constraints through binomial features and fixed-point iteration entropy loss. MEP-Net successfully reconstructs complex multimodal distributions where VAEs and flow models fail, and effectively solves variational problems including Onsager diffusion and Allen-Cahn phase separation. Both frameworks demonstrate that incorporating fundamental thermodynamic principles into machine learning architectures yields physically plausible, robust, and accurate solutions across ordinary, partial, and stochastic differential equations.

简介：

Dr. Wuyue Yang is an Assistant Research Fellow at the Beijing Institute of Mathematical Sciences and Applications (BIMSA). She received her Ph.D. in Applied Mathematics from Tsinghua University's Yau Mathematical Sciences Center in 2022.Her research focuses on the intersection of artificial intelligence and scientific discovery, specializing in physics-informed machine learning, scientific machine learning, and biomedical AI. She develops interpretable neural network models that incorporate differential equations and physical principles for applications in complex dynamical systems, protein aggregation dynamics, and disease modeling.Dr. Yang has published 17peer-reviewed papers in journals including SIAM Journal on Scientific Computing,Physics of Fluids, Journal of Chemical Physics, and Chaos. She serves as Principal Investigator on research grants from the National Natural Science Foundation of China and COMAC. She also teaches AI coursesandsupervisesthreedoctoral students.

报告题目: Flow Measurement: an inverse problem formulation

报告摘要: This work proposes a new mathematical formulation for flow measurement based on the inverse source problem for wave equations with partial boundary measurement. Inspired by the design of acoustic Doppler current profilers (ADCPs), we formulate an inverse source problem that can recover the flow field from the observation data on boundary receivers. To our knowledge, this is the first mathematical model of flow measurement using partial differential equations. This model is proved well-posed, and the corresponding algorithm is derived to compute the velocity field efficiently. Extensive numerical simulations are performed to demonstrate the accuracy and robustness of our model. The comparison results demonstrate that our model is ten times more accurate than ADCP. Our formulation is capable of simulating a variety of practical measurement scenarios.

报告人简介:

蔚辉，湘潭大学数学与计算科学学院教授。2008年本科毕业于南开大学，2013年在Iowa State University获得博士学位，导师为刘海亮教授。后随Pierre Degond与Michael Herty开展博士后工作，2017年入职清华大学丘成桐数学科学中心，2023年加入湘潭大学。主要研究领域为偏微分方程保物理结构特征的间断有限元数值方法和复杂系统多粒子的自组织集群现象的建模问题。