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At present, deep learning based methods are being employed to resolve thecomputational 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. Wepropose a novel approach to solving PDEs with high order derivatives bysimultaneously approximating the function value and derivatives. We introduceintermediate variables to rewrite the PDEs into a system of low orderdifferential equations as what is done in the local discontinuous Galerkinmethod. The intermediate variables and the solutions to the PDEs aresimultaneously approximated by a multi-output deep neural network. By takingthe residual of the system as a loss function, we can optimize the networkparameters to approximate the solution. The whole process relies on low orderderivatives. Numerous numerical examples are carried out to demonstrate thatour local deep learning is efficient, robust, flexible, and is particularlywell-suited for high-dimensional PDEs with high order derivatives.

A Local Deep Learning Method for Solving High Order Partial Differential Equations