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%.

翻译：在粗网格上求解偏微分方程（PDE）的精度受离散化格式选择的显著影响。本文提出学习基于神经网络的时间积分格式，该格式满足三组不同的数学约束条件，即无约束、带根条件的半约束以及同时带根条件和相容性条件的全约束。我们重点学习三步线性多步法，并将其应用于求解三个模型偏微分方程：一维热传导方程、一维波动方程和一维伯格斯方程。结果表明，学习得到的全约束格式的预测误差接近龙格-库塔法和亚当斯-巴什福思法。与传统方法相比，学习得到的无约束和半约束格式在粗网格上显著降低了预测误差。在比参考网格粗4倍的网格上，部分热传导方程案例的均方误差降低了一个数量级，波动方程的相位预测也得到显著改善。在粗32倍的网格上，伯格斯方程的均方误差可降低35%至40%。