A new penalty-free neural network method,PFNN-2,is presented for solving partial differential equations,which is a subsequent improvement of our previously proposed PFNN method[1].PFNN-2 inherits all advantages of PFN...A new penalty-free neural network method,PFNN-2,is presented for solving partial differential equations,which is a subsequent improvement of our previously proposed PFNN method[1].PFNN-2 inherits all advantages of PFNN in handling the smoothness constraints and essential boundary conditions of self-adjoint problems with complex geometries,and extends the application to a broader range of non-self-adjoint time-dependent differential equations.In addition,PFNN-2 introduces an overlapping domain decomposition strategy to substantially improve the training efficiency without sacrificing accuracy.Experiments results on a series of partial differential equations are reported,which demonstrate that PFNN-2 can outperform state-of-the-art neural network methods in various aspects such as numerical accuracy,convergence speed,and parallel scalability.展开更多
We develop and analyze an adaptive hybridized Interior Penalty Discontinuous Galerkin(IPDG-H)method for H(curl)-elliptic boundary value problems in 2D or 3D arising from a semi-discretization of the eddy currents equ...We develop and analyze an adaptive hybridized Interior Penalty Discontinuous Galerkin(IPDG-H)method for H(curl)-elliptic boundary value problems in 2D or 3D arising from a semi-discretization of the eddy currents equations.The method can be derived from a mixed formulation of the given boundary value problem and involves a Lagrange multiplier that is an approximation of the tangential traces of the primal variable on the interfaces of the underlying triangulation of the computational domain.It is shown that the IPDG-H technique can be equivalently formulated and thus implemented as a mortar method.The mesh adaptation is based on a residual-type a posteriori error estimator consisting of element and face residuals.Within a unified framework for adaptive finite element methods,we prove the reliability of the estimator up to a consistency error.The performance of the adaptive symmetric IPDG-H method is documented by numerical results for representative test examples in 2D.展开更多
基金supported in part by National Natural Science Foundation of China(grants#12131002 and#12288101)Huawei Technologies Co.,Ltd.
文摘A new penalty-free neural network method,PFNN-2,is presented for solving partial differential equations,which is a subsequent improvement of our previously proposed PFNN method[1].PFNN-2 inherits all advantages of PFNN in handling the smoothness constraints and essential boundary conditions of self-adjoint problems with complex geometries,and extends the application to a broader range of non-self-adjoint time-dependent differential equations.In addition,PFNN-2 introduces an overlapping domain decomposition strategy to substantially improve the training efficiency without sacrificing accuracy.Experiments results on a series of partial differential equations are reported,which demonstrate that PFNN-2 can outperform state-of-the-art neural network methods in various aspects such as numerical accuracy,convergence speed,and parallel scalability.
基金The work of the first author has been supported by the German Na-tional Science Foundation DFG within the Research Center MATHEON and by the WCU program through KOSEF(R31-2008-000-10049-0).The other authors acknowledge sup-port by the NSF grant DMS-0810176.1
文摘We develop and analyze an adaptive hybridized Interior Penalty Discontinuous Galerkin(IPDG-H)method for H(curl)-elliptic boundary value problems in 2D or 3D arising from a semi-discretization of the eddy currents equations.The method can be derived from a mixed formulation of the given boundary value problem and involves a Lagrange multiplier that is an approximation of the tangential traces of the primal variable on the interfaces of the underlying triangulation of the computational domain.It is shown that the IPDG-H technique can be equivalently formulated and thus implemented as a mortar method.The mesh adaptation is based on a residual-type a posteriori error estimator consisting of element and face residuals.Within a unified framework for adaptive finite element methods,we prove the reliability of the estimator up to a consistency error.The performance of the adaptive symmetric IPDG-H method is documented by numerical results for representative test examples in 2D.
