A family of high-order accuracy explict difference schemes for solving 3-dimension parabolic P. D. E. is constructed. The stability condition is r = Deltat/Deltax(2) Deltat/Deltay(2) = Deltat/Deltaz(2) < 1/2 ,and t...A family of high-order accuracy explict difference schemes for solving 3-dimension parabolic P. D. E. is constructed. The stability condition is r = Deltat/Deltax(2) Deltat/Deltay(2) = Deltat/Deltaz(2) < 1/2 ,and the truncation error is 0(<Delta>t(2) + Deltax(4)).展开更多
At present, deep learning based methods are being employed to resolvethe computational challenges of high-dimensional partial differential equations(PDEs). But the computation of the high order derivatives of neural n...At present, deep learning based methods are being employed to resolvethe computational challenges of high-dimensional partial differential equations(PDEs). But the computation of the high order derivatives of neural networks iscostly, and high order derivatives lack robustness for training purposes. We proposea novel approach to solving PDEs with high order derivatives by simultaneously approximating the function value and derivatives. We introduce intermediate variablesto rewrite the PDEs into a system of low order differential equations as what is donein the local discontinuous Galerkin method. The intermediate variables and the solutions to the PDEs are simultaneously approximated by a multi-output deep neuralnetwork. By taking the residual of the system as a loss function, we can optimizethe network parameters to approximate the solution. The whole process relies onlow order derivatives. Numerous numerical examples are carried out to demonstrate that our local deep learning is efficient, robust, flexible, and is particularlywell-suited for high-dimensional PDEs with high order derivatives.展开更多
文摘A family of high-order accuracy explict difference schemes for solving 3-dimension parabolic P. D. E. is constructed. The stability condition is r = Deltat/Deltax(2) Deltat/Deltay(2) = Deltat/Deltaz(2) < 1/2 ,and the truncation error is 0(<Delta>t(2) + Deltax(4)).
基金supported by the National Natural Science Foundation of China/Hong Kong RRC Joint Research Scheme(NSFC/RGC 11961160718)the fund of the Guangdong Provincial Key Laboratory of Computational Science and Material Design(No.2019B030301001)+1 种基金supported by the National Science Foundation of China(NSFC-11871264)the Guangdong Basic and Applied Basic Research Foundation(2018A0303130123).
文摘At present, deep learning based methods are being employed to resolvethe computational challenges of high-dimensional partial differential equations(PDEs). But the computation of the high order derivatives of neural networks iscostly, and high order derivatives lack robustness for training purposes. We proposea novel approach to solving PDEs with high order derivatives by simultaneously approximating the function value and derivatives. We introduce intermediate variablesto rewrite the PDEs into a system of low order differential equations as what is donein the local discontinuous Galerkin method. The intermediate variables and the solutions to the PDEs are simultaneously approximated by a multi-output deep neuralnetwork. By taking the residual of the system as a loss function, we can optimizethe network parameters to approximate the solution. The whole process relies onlow order derivatives. Numerous numerical examples are carried out to demonstrate that our local deep learning is efficient, robust, flexible, and is particularlywell-suited for high-dimensional PDEs with high order derivatives.
