  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.
