Time integration schemes based on neural networks for solving partial
  differential equations on coarse grids
Xinxin Yan Zhideng Zhou Xiaohan Cheng Xiaolei Yang
The accuracy of solving partial differential equations (PDEs) on coarse grids
is greatly affected by the choice of discretization schemes. In this work, we
propose to learn time integration schemes based on neural networks which
satisfy three distinct sets of mathematical constraints, i.e., unconstrained,
semi-constrained with the root condition, and fully-constrained with both root
and consistency conditions. We focus on the learning of 3-step linear multistep
methods, which we subsequently applied to solve three model PDEs, i.e., the
one-dimensional heat equation, the one-dimensional wave equation, and the
one-dimensional Burgers' equation. The results show that the prediction error
of the learned fully-constrained scheme is close to that of the Runge-Kutta
method and Adams-Bashforth method. Compared to the traditional methods, the
learned unconstrained and semi-constrained schemes significantly reduce the
prediction error on coarse grids. On a grid that is 4 times coarser than the
reference grid, the mean square error shows a reduction of up to an order of
magnitude for some of the heat equation cases, and a substantial improvement in
phase prediction for the wave equation. On a 32 times coarser grid, the mean
square error for the Burgers' equation can be reduced by up to 35% to 40%.