On stability and regularization for data-driven solution of parabolic inverse source problems

讲座时间 Datetime: 2022年9月8日，星期四，16:00-17:00

地点 Venue: 腾讯线上会议：384897653

主持人 Host：张植栋

报告人 Speaker: 张萌萌

单位 Affiliation: 河北工业大学

报告摘要 Abstract:

The diffusion process from some internal source arising in engineering situations can be mathematically described by a parabolic system. We consider an inverse source problem for parabolic system using parametric approximations, where deep neural networks (DNNs) are used to approximate the solution of the inverse problem. First, we prove generalization error estimates depending on training errors and data noise levels by establishing conditional stability of the inverse problem. Following our analysis, we propose a new loss function involving the derivative of the residuals for PDE and measurement data. These extra fit terms adding higher regularity requirements can be understood as the regularizing penalty specified to the solution of inverse problems, dealing with the ill-posedness of the problem. Using these regularization terms, we develop reconstruction schemes and demonstrate the effectiveness of our proposed methodology on a number of test problems. This work is joint with my supervisor Prof. Li Qianxiao and Prof. Liu jijun.