为准确求解双曲守恒律,得到高分辨率数值结果,将数据驱动与三阶CCWENO(Compact Central Weighted Essentially Non-Oscillatory)格式相结合,提出了一种基于数据驱动的CCWENO-ANN高分辨率格式求解双曲守恒律。通过构建人工神经网络的归...为准确求解双曲守恒律,得到高分辨率数值结果,将数据驱动与三阶CCWENO(Compact Central Weighted Essentially Non-Oscillatory)格式相结合,提出了一种基于数据驱动的CCWENO-ANN高分辨率格式求解双曲守恒律。通过构建人工神经网络的归一化校准层和稀疏化层,引入适当的先验知识,加快收敛速度;同时,损失函数动态地调整神经网络输出与理想权重之间的偏差,并在合适的数据集上采用监督学习策略进行离线训练,以提高神经网络性能。通过一维无粘Burgers方程、一维Euler方程、二维无粘Burgers方程以及二维Euler方程验证算法性能,结果表明本文提出的CCWENO-ANN继承了传统CCWENO格式的收敛性,能够准确捕捉激波和接触间断,具有鲁棒性强、低耗散和高分辨率的优点。展开更多
In this paper,we propose a systematic approach for accelerating finite element-type methods by machine learning for the numerical solution of partial differential equations(PDEs).The main idea is to use a neural netwo...In this paper,we propose a systematic approach for accelerating finite element-type methods by machine learning for the numerical solution of partial differential equations(PDEs).The main idea is to use a neural network to learn the solution map of the PDEs and to do so in an element-wise fashion.This map takes input of the element geometry and the PDE’s parameters on that element,and gives output of two operators:(1)the in2out operator for inter-element communication,and(2)the in2sol operator(Green’s function)for element-wise solution recovery.A significant advantage of this approach is that,once trained,this network can be used for the numerical solution of the PDE for any domain geometry and any parameter distribution without retraining.Also,the training is significantly simpler since it is done on the element level instead on the entire domain.We call this approach element learning.This method is closely related to hybridizable discontinuous Galerkin(HDG)methods in the sense that the local solvers of HDG are replaced by machine learning approaches.Numerical tests are presented for an example PDE,the radiative transfer or radiation transport equation,in a variety of scenarios with idealized or realistic cloud fields,with smooth or sharp gradient in the cloud boundary transition.Under a fixed accuracy level of 10^(−3) in the relative L^(2) error,and polynomial degree p=6 in each element,we observe an approximately 5 to 10 times speed-up by element learning compared to a classical finite element-type method.展开更多
A new quadrilateral edge element method is proposed and analyzed for Maxwell equations.This proposed method is based on Duan-Liang quadrilateral element(Math.Comp.73(2004),pp.1–18).When applied to the eigenvalue prob...A new quadrilateral edge element method is proposed and analyzed for Maxwell equations.This proposed method is based on Duan-Liang quadrilateral element(Math.Comp.73(2004),pp.1–18).When applied to the eigenvalue problem,the method is spectral-correct and spurious-free.Stability and error estimates are obtained,including the interpolation error estimates and the error estimates between the finite element solution and the exact solution.The method is suitable for singular solution as well as smooth solution,and consequently,the method is valid for nonconvex domains which may have a number of reentrant corners.Of course,the method is suitable for arbitrary quadrilaterals(under the usual shape-regular condition).展开更多
In this paper,we propose and analyze two second-order accurate finite difference schemes for the one-dimensional heat equation with concentrated capacity on a computa-tional domain=[a,b].We first transform the target ...In this paper,we propose and analyze two second-order accurate finite difference schemes for the one-dimensional heat equation with concentrated capacity on a computa-tional domain=[a,b].We first transform the target equation into the standard heat equation on the domain excluding the singular point equipped with an inner interface matching(IIM)condition on the singular point x=ξ∈(a,b),then adopt Taylor’s ex-pansion to approximate the IIM condition at the singular point and apply second-order finite difference method to approximate the standard heat equation at the nonsingular points.This discrete procedure allows us to choose different grid sizes to partition the two sub-domains[a,ξ]and[ξ,b],which ensures that x=ξ is a grid point,and hence the pro-posed schemes can be generalized to the heat equation with more than one concentrated capacities.We prove that the two proposed schemes are uniquely solvable.And through in-depth analysis of the local truncation errors,we rigorously prove that the two schemes are second-order accurate both in temporal and spatial directions in the maximum norm without any constraint on the grid ratio.Numerical experiments are carried out to verify our theoretical conclusions.展开更多
文摘为准确求解双曲守恒律,得到高分辨率数值结果,将数据驱动与三阶CCWENO(Compact Central Weighted Essentially Non-Oscillatory)格式相结合,提出了一种基于数据驱动的CCWENO-ANN高分辨率格式求解双曲守恒律。通过构建人工神经网络的归一化校准层和稀疏化层,引入适当的先验知识,加快收敛速度;同时,损失函数动态地调整神经网络输出与理想权重之间的偏差,并在合适的数据集上采用监督学习策略进行离线训练,以提高神经网络性能。通过一维无粘Burgers方程、一维Euler方程、二维无粘Burgers方程以及二维Euler方程验证算法性能,结果表明本文提出的CCWENO-ANN继承了传统CCWENO格式的收敛性,能够准确捕捉激波和接触间断,具有鲁棒性强、低耗散和高分辨率的优点。
基金partially supported by the NSF(Grant No.DMS 2324368)by the Office of the Vice Chancellor for Research and Graduate Education at the University of Wisconsin-Madison with funding from the Wisconsin Alumni Research Foundation.
文摘In this paper,we propose a systematic approach for accelerating finite element-type methods by machine learning for the numerical solution of partial differential equations(PDEs).The main idea is to use a neural network to learn the solution map of the PDEs and to do so in an element-wise fashion.This map takes input of the element geometry and the PDE’s parameters on that element,and gives output of two operators:(1)the in2out operator for inter-element communication,and(2)the in2sol operator(Green’s function)for element-wise solution recovery.A significant advantage of this approach is that,once trained,this network can be used for the numerical solution of the PDE for any domain geometry and any parameter distribution without retraining.Also,the training is significantly simpler since it is done on the element level instead on the entire domain.We call this approach element learning.This method is closely related to hybridizable discontinuous Galerkin(HDG)methods in the sense that the local solvers of HDG are replaced by machine learning approaches.Numerical tests are presented for an example PDE,the radiative transfer or radiation transport equation,in a variety of scenarios with idealized or realistic cloud fields,with smooth or sharp gradient in the cloud boundary transition.Under a fixed accuracy level of 10^(−3) in the relative L^(2) error,and polynomial degree p=6 in each element,we observe an approximately 5 to 10 times speed-up by element learning compared to a classical finite element-type method.
基金supported by the National Natural Science Foundation of China(12401482)the second author was supported by the National Natural Science Foundation of China(12371371,12261160361,11971366)supported by the Open Research Fund of Hubei Key Laboratory of Computational Science,Wuhan University.
文摘A new quadrilateral edge element method is proposed and analyzed for Maxwell equations.This proposed method is based on Duan-Liang quadrilateral element(Math.Comp.73(2004),pp.1–18).When applied to the eigenvalue problem,the method is spectral-correct and spurious-free.Stability and error estimates are obtained,including the interpolation error estimates and the error estimates between the finite element solution and the exact solution.The method is suitable for singular solution as well as smooth solution,and consequently,the method is valid for nonconvex domains which may have a number of reentrant corners.Of course,the method is suitable for arbitrary quadrilaterals(under the usual shape-regular condition).
基金supported by the National Natural Science Foundation of China(Grant No.11571181)by the Natural Science Foundation of Jiangsu Province(Grant No.BK20171454).
文摘In this paper,we propose and analyze two second-order accurate finite difference schemes for the one-dimensional heat equation with concentrated capacity on a computa-tional domain=[a,b].We first transform the target equation into the standard heat equation on the domain excluding the singular point equipped with an inner interface matching(IIM)condition on the singular point x=ξ∈(a,b),then adopt Taylor’s ex-pansion to approximate the IIM condition at the singular point and apply second-order finite difference method to approximate the standard heat equation at the nonsingular points.This discrete procedure allows us to choose different grid sizes to partition the two sub-domains[a,ξ]and[ξ,b],which ensures that x=ξ is a grid point,and hence the pro-posed schemes can be generalized to the heat equation with more than one concentrated capacities.We prove that the two proposed schemes are uniquely solvable.And through in-depth analysis of the local truncation errors,we rigorously prove that the two schemes are second-order accurate both in temporal and spatial directions in the maximum norm without any constraint on the grid ratio.Numerical experiments are carried out to verify our theoretical conclusions.
