Numerical Simulation of the Korteweg-de Vries Equation with Machine Learning

A machine learning procedure is proposed to create numerical schemes for solutions of nonlinear wave equations on coarse grids. This method trains stencil weights of a discretization of the equation, with the truncation error of the scheme as the objective function for training. The method uses centered finite differences to initialize the optimization routine and a second-order implicit-explicit time solver as a framework. Symmetry conditions are enforced on the learned operator to ensure a stable method. The procedure is applied to the Korteweg-de Vries equation. It is observed to be more accurate than finite difference or spectral methods on coarse grids when the initial data is near enough to the training set.