乔中华 讲座教授（香港理工大学）

刘歆 研究员

A structure-preserving framework for solving PDEs with neural networks

4月30日（周三）14:30-15:30，南楼714

Neural network (NN) solvers for partial differential equations (PDE) have been widely used in simulating complex systems in various scientific and engineering fields. However, most existing NN solvers mainly focus on satisfying the given PDEs, without explicitly considering intrinsic physical properties such as mass conservation or energy dissipation. This limitation can result in unstable or nonphysical solutions, particularly in long-term simulations. To address this issue, we propose Sidecar, a novel framework that enhances the accuracy and physical consistency of existing NN solvers by incorporating structure-preserving knowledge. This framework builds upon our previously proposed TDSR-ETD method for solving gradient flow problems, which satisfies discrete analogues of the energy-dissipation laws by introducing a time-dependent spectral renormalization (TDSR) factor. Inspired by this approach, our Sidecar framework parameterizes the TDSR factor using a small copilot network, which is trained to guide the existing NN solver in preserving physical structure. This design allows flexible integration of the structure- preserving knowledge into various NN solvers and can be easily extended to different types of PDEs. Our experimental results on a set of benchmark PDEs demonstrate that it improves the existing neural network solvers in terms of accuracy and consistency with structure-preserving properties.

报告人简介：乔中华讲座教授于2006年在香港浸会大学获得博士学位，现为香港理工大学应用数学系讲座教授，中科院数学与系统科学研究院——香港理工大学应用数学联合实验室港方副主任，中国工业与应用数学学会理事，中国数学会计算数学分会副理事长。 他主要从事数值微分方程方面算法设计及分析，近年来研究工作集中在相场方程的数值模拟及计算流体力学的高效算法。他至今在SCI期刊上发表论文80余篇，文章被合计引用3000余次。他于2013年获香港研究资助局颁发2013至2014年度杰出青年学者奖，于2018年获得香港数学会青年学者奖，并且于2020年获得香港研究资助局研究学者奖。