Physics-informed neural networks for non-smooth PDE-constrained optimization problems

：宋永存（香港城市大学）
：2025-08-13 11:00
：海韵园行政楼C503

报告人：宋永存（香港城市大学）

时  间：2025年8月13日11:00

地  点：海韵园行政楼C503

内容摘要:

We study the application of well-known physics-informed neural networks (PINNs) for solving non-smooth PDE-constrained optimization problems. First, we consider a class of PDE-constrained optimization problems where additional nonsmooth regularization is employed for constraints on the control or design variables. To solve such problems, we combine the alternating direction method of multipliers (ADMM) and PINNs and propose the ADMM-PINNs algorithmic framework, which unties the PDE constraints and the nonsmooth regularization terms for iterations. Accordingly, at each iteration, one of the resulting subproblems is a smooth PDE-constrained optimization, which PINNs can efficiently solve, and the other is a simple nonsmooth optimization problem that usually has a closed-form solution or can be efficiently solved by various standard optimization algorithms or pre-trained neural networks. Then, we consider the optimal control of PDEs with interfaces. We employ the recently developed discontinuity-capturing neural network to tackle the non-smoothness of the PDEs with interfaces and propose hard-constraint PINNs for solving such optimal control problems. The hard-constraint PINNs ensure both the boundary and interface conditions are satisfied strictly, and meanwhile, they are decoupled from the learning of the PDEs.  All these PINNs methods are mesh-free, easy to implement, and scalable to different PDE settings. Various numerical results are reported to validate the effectiveness and efficiency of the proposed PINNs methods. 

个人简介：

宋永存，香港城市大学数学系助理教授。2021年于香港大学数学系获博士学位，2021-2024年在德国埃尔朗根-纽伦堡大学担任洪堡研究员。主要研究领域为科学计算和机器学习，包括数值优化、算子分裂算法、科学机器学习、最优控制问题等。代表性论文发表于SIAM Review、SIAM Journal on Numerical Analysis、Numerische Mathematik、SIAM Journal on Scientific Computing、Inverse Problems、Journal of Computational Physics等国际一流期刊。2021年获Humboldt Research Fellowship，2022年获EASIAM学生论文奖（一等奖），2023年获香港数学会最佳论文奖。

 

联系人：谭志裕