SHoP: A Deep Learning Framework for Solving High-order Partial
  Differential Equations
Tingxiong Xiao Runzhao Yang Yuxiao Cheng Jinli Suo Qionghai Dai
Solving partial differential equations (PDEs) has been a fundamental problem
in computational science and of wide applications for both scientific and
engineering research. Due to its universal approximation property, neural
network is widely used to approximate the solutions of PDEs. However, existing
works are incapable of solving high-order PDEs due to insufficient calculation
accuracy of higher-order derivatives, and the final network is a black box
without explicit explanation. To address these issues, we propose a deep
learning framework to solve high-order PDEs, named SHoP. Specifically, we
derive the high-order derivative rule for neural network, to get the
derivatives quickly and accurately; moreover, we expand the network into a
Taylor series, providing an explicit solution for the PDEs. We conduct
experimental validations four high-order PDEs with different dimensions,
showing that we can solve high-order PDEs efficiently and accurately.