Time integration schemes based on neural networks for solving partial differential equations on coarse grids
This work addresses accuracy issues in PDE simulations for computational science and engineering, but it is incremental as it builds on existing neural network and numerical method approaches.
The authors tackled the problem of solving partial differential equations (PDEs) on coarse grids by learning time integration schemes with neural networks, resulting in reduced prediction errors, such as up to an order of magnitude reduction in mean square error for the heat equation and 35-40% reduction for the Burgers' equation on coarser grids.
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%.