北京雁栖湖应用数学研究院 北京雁栖湖应用数学研究院

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关于我们
院长致辞
理事会
协作机构
参观来访
人员
管理层
科研人员
博士后
来访学者
行政团队
学术研究
研究团队
公开课
讨论班
招生招聘
教研人员
博士后
学生
会议
学术会议
工作坊
论坛
学院生活
住宿
交通
配套设施
周边旅游
新闻
新闻动态
通知公告
资料下载
清华大学 "求真书院"
清华大学丘成桐数学科学中心
清华三亚国际数学论坛
上海数学与交叉学科研究院
BIMSA > 控制理论和非线性滤波讨论班 Finite Expression Method for Solving High-Dimensional PDEs
Finite Expression Method for Solving High-Dimensional PDEs
组织者
丘成栋
演讲者
康家熠
时间
2024年01月05日 21:30 至 22:00
地点
Online
摘要
Learning high-dimensional functions (e.g., solving high-dimensional partial differential equations (PDEs) and discovering governing PDEs) is fundamental in scientific fields such as diffusion, fluid dynamics, and quantum mechanics, and optimal control, etc. Developing efficient and accurate solvers for this task remains an important and challenging topic. Traditional solvers (e.g., finite element method (FEM) and finite difference) are usually limited to low-dimensional domains since the computational cost increases exponentially in the dimension as the curse of dimensionality. Neural networks (NNs) as mesh-free parameterization are widely employed in solving regression problems and high-dimensional PDEs. Yet the highly non-convex optimization objective function in NN optimization makes it difficult to achieve high accuracy. The errors of NN-based solvers would still grow with the dimension. Besides, NN parametrization may still require large memory and high computation cost for high-dimensional problems. Finally, numerical solutions provided by traditional solvers and NN-based solvers are not interpretable, e.g., the dependence of the solution on variables cannot be readily seen from numerical solutions. The key to tackle these issues is to develop symbolic learning to discover the low-complexity structures of a high-dimensional problem. Low-complexity structures are applied to transform a high-dimensional task into a low-dimensional learning problem.
北京雁栖湖应用数学研究院
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北京雁栖湖应用数学研究院 101408

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