一类基于不可压缩流体方程及其耦合问题的深度学习方法研究  

A DEEP LEARNING METHOD FOR INCOMPRESSIBLE FLUID EQUATIONS AND THE COUPLING PROBLEMS

作　　者：李剑 张文 岳靖 彭珂依 陈掌星[4] Li Jian;Zhang Wen;Yue Jing;Peng Keyi;Chen Zhangxing(School of Mathematics and Data Science,Shaanci University of Science and Technology,Xi'an 710021,China;School of Electrical and Control engineering,Shaanri University of Science and Technology,Xi'an 710021,China;School of Mathematics and Statistics,Ningria University,Yinchuan 750021,China;Department of Chemical and Petroleum Engineering,University of Calgary,Calgary T4B 2Z8,Canada)

机构地区：[1]陕西科技大学数学与数据科学学院,西安710021 [2]陕西科技大学电气与控制工程学院,西安710021 [3]宁夏大学数学统计学院,银川750021 [4]卡尔加里大学化学与石油工程学院,卡尔加里T4B2Z8,加拿大

出　　处：《计算数学》2024年第2期232-252,共21页Mathematica Numerica Sinica

基　　金：国家自然科学基金(11771259);陕西省人工智能联合实验室(2022JC-SYS05);陕西省教育厅创新团队项目(21JP013);陕西省自然科学基础研究计划重点项目(2023-JC-ZD-02)资助.

摘　　要：本文通过实现深度前馈人工神经网络求解不可压缩流体偏微分方程,基于方程残差、初边值条件构造合适的损失函数和深度学习求解算法.与传统数值方法相比,该方法只需在内部、边界和初始时刻随机生成样本点作为训练集,因此该方法是无网格的,并且各物理场变量之间并行求解,便于分析复杂多物理场耦合模型中物理量的变化规律.收敛性分析在统一框架下为深度学习方法求解此类不可压缩流体偏微分方程提供了理论支撑,通过求解一类非定常Stokes方程,一类粘性Boussinesq方程和一类Navier-Stokes/Darcy耦合方程说明此方法可以有效求解不可压缩流体偏微分方程并且具有较好的精度.In this paper,we propose to implement deep neural network to solve the incompressible fuid partial differential equations,the loss function is composed of the equation residual,initial condition and boundary conditions.The sample points are randomly generated at the interior,boundary and initial time as training sets.Compared with traditional numerical methods,the method based on deep neural network is meshfree,and each physical field variable is parallel solved,which is convenient for solving the complex multi-physical field coupling partial differential equations model.Besides,the convergence analysis provides theoretical support for deep neural networks solving partial differential equations.The numerical results show that the method can effectively solve incompressible fluid equations with good accuracy by solving a class of unsteady Stokes equations,a class of viscous Boussinesq equations and a class of Navier-Stokes/Darcy equation.

关 键 词：偏微分方程 深度学习 STOKES方程 BOUSSINESQ方程 Navier-Stokes/Darcy方程 

分 类 号：TP18[自动化与计算机技术—控制理论与控制工程] O241.8[自动化与计算机技术—控制科学与工程]