Time Integration Schemes Based on Neural Networks for Solving Partial Differential Equations on Coarse Grids (Communications in Computational Physics, 2024)
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 bothroot and consistency conditions. We focus on the learning of 3-step linear multistepmethods, which we subsequently applied to solve three model PDEs, i.e., the heatequation, the wave equation, and the Burgers’ equation. The results show that the prediction error of the learned fully-constrained scheme is close to that of the Runge-Kuttamethod and Adams-Bashforth method. Compared to the traditional methods, thelearned unconstrained and semi-constrained schemes significantly reduce the prediction error on coarse grids, with an overall better performance for the semi-constrainedmodel. On a grid $4×$ coarser than the reference, the mean square error (MSE) showsa reduction of up to two order of magnitude for some heat equation cases, and a substantial improvement in phase prediction for the wave equation. On a $32×$ coarsergrid, the MSE for the Burgers’ equation can be reduced by up to 40% to 45%. Testson the two-dimensional heat equation and wave equation show a good generalizationcapability for the constant optimized coefficients.
Keywords: Time integration scheme; partial differential equation; numerical simulation on coarse grids; neural networks; mathematical constraints.