TY - JOUR
T1 - A Local Deep Learning Method for Solving High Order Partial Differential Equations
AU - Yang , Jiang
AU - Zhu , Quanhui
JO - Numerical Mathematics: Theory, Methods and Applications
VL - 1
SP - 42
EP - 67
PY - 2022
DA - 2022/02
SN - 15
DO - http://doi.org/10.4208/nmtma.OA-2021-0035
UR - https://global-sci.org/intro/article_detail/nmtma/20220.html
KW - Deep learning, deep neural network, high order PDEs, reduction of order, deep Galerkin method.
AB - <p style="text-align: justify;">At present, deep learning based methods are being employed to resolve
the computational challenges of high-dimensional partial differential equations
(PDEs). But the computation of the high order derivatives of neural networks is
costly, and high order derivatives lack robustness for training purposes. We propose
a novel approach to solving PDEs with high order derivatives by simultaneously approximating the function value and derivatives. We introduce intermediate variables
to rewrite the PDEs into a system of low order differential equations as what is done
in the local discontinuous Galerkin method. The intermediate variables and the solutions to the PDEs are simultaneously approximated by a multi-output deep neural
network. By taking the residual of the system as a loss function, we can optimize
the network parameters to approximate the solution. The whole process relies on
low order derivatives. Numerous numerical examples are carried out to demonstrate that our local deep learning is efficient, robust, flexible, and is particularly
well-suited for high-dimensional PDEs with high order derivatives.</p>