Cell-average based convolutional neural network method for nonlinear Schrödinger equations

This paper proposes a novel approach called the cell-average based convolutional neural network (CACNN) method, which combines a convolutional neural network (CNN) with the finite volume scheme to solve nonlinear Schrdinger equations.The novelty of the CACNN method is that it introduces a composite network consisting of a CNN as the basis sub-network and a fully connected feedforward neural network (FNN) as the coefficient sub-network, to handle multi-channel input vectors and to approximate the cell average differences between two consecutive time steps accurately.The training data is generated using a generic initial value of the given equations.The well-trained CACNN method can be implemented as an explicit one-step finite volume method and exhibits exceptional ability in solving a wide range of initial value problems with negligible generalization errors.One notable advantage of the CACNN method is its ability to accommodate large time step size of t = O(x), in contrast to conventional explicit methods that are constrained by stringent CFL restrictions on time step sizes for nonlinear Schrdinger equations.Furthermore, the method incorporates temporal-spatial coordinates as input, enabling accurate predictions starting from different initial times and generalizations to various spatial mesh sizes.Numerical tests are conducted to demonstrate the effectiveness, stability, and accuracy of the proposed method.