Cell-average based neural network method for third order and fifth order KdV type equations

Abstract

In this paper, we develop the cell-average based neural network (CANN) method to solve third order and fifth order Korteweg-de Vries (KdV) type equations. The CANN method is based on the weak or integral formulation of the partial differential equations. A simple feedforward network is forced to learn the cell average difference between two consecutive time steps. One solution trajectory corresponding to a generic initial value is used to generate the data set to train the network parameters, which uniquely determine a one-step explicit finite volume based network method. Once well-trained, the CANN method can be generalized to a suitable family of initial value problems. Comparing with conventional explicit methods, where the time step size is restricted as Δt = O(Δx3) or Δt = O(Δx5), the CANN method is able to evolve the solution forward accurately with a much larger time step size of Δt = O(Δx). A large group of numerical tests are carried out to verify the effectiveness, stability and accuracy of the CANN method. Wave propagation is well resolved with indistinguishable dispersion and dissipation errors. The CANN approximations agree well with the exact solution for long time simulation.