现代应用物理2025，Vol.16Issue(1)：33-58,26.DOI:10.12061/j.issn.2095-6223.202412039
从传统数值方法到神经网络:偏微分方程数值解的演进与展望
From Traditional Numerical Methods to Neural Networks:Evolution and Prospect of Numerical Solution of Partial Differential Equations
摘要
Abstract
Partial differential equations(PDEs)are widely used in physics,engineering,biology,medicine,finance,and other fields.However,PDEs often cannot be solved analytically when solving many practical problems.Therefore,the construction of efficient numerical methods to solve these problems has become an important research direction.Traditional numerical methods,such as finite element methods and finite difference methods,have become the core tools of scientific computing,and play an irreplaceable role in the fields of weather forecasting,oil reservoir exploration,aircraft design and energy development.These methods often depend on partition of domain,but the grid generation is often complex and time-consuming,especially when dealing with problems with complex shapes or large deformations.In addition,for time dependent problem,the time iteration may face the problem of instability and error accumulation,which affects the efficiency and accuracy of the calculation.In solving high-dimensional problems,these traditional methods also face the"curse of dimensionality",which need more computing resources and make the solving process difficult.In order to solve these challenges,deep neural network(DNN),as a new technology,has been widely used to solve various PDEs,and has shown great potential and application prospects.This paper will review the latest research progress in this field,focusing on the following directions:approximation theory of neural networks,neural network methods based on PDEs,neural network methods based on variational form,randomized neural network methods,and data-driven operator learning approaches.关键词
偏微分方程/有限元/有限差分/人工智能/随机神经网络/算子学习Key words
partial differential equations/finite element method/finite difference methods/artificial intelligence/randomized neural network/operator learning分类
数学引用本文复制引用
王飞,党浩宁,尚勇,孙靖博,李泽塬,李云龙..从传统数值方法到神经网络:偏微分方程数值解的演进与展望[J].现代应用物理,2025,16(1):33-58,26.基金项目
国家自然科学基金资助项目(92470115,12426105) （92470115,12426105）
陕西数理基础科学研究资助项目(22JSY027) （22JSY027）
