Solving the nonlinear Schrödinger equation with an unsupervised neural network: estimation of error in solution

Abstract

We present a practical method for estimating the upper error bound in the neural network (NN) solution of the nonlinear Schrödinger equation (NLSE) under different degrees of nonlinearity. The error bound is a function of the nonnegative energy E value that is minimized when the NN is trained to solve the NLSE. The form of E is derived from the NLSE expression and the NN solution becomes identical with the true NLSE solution only when the E value is reduced exactly to zero. In practice, machines with finite floating-point range and accuracy are used for training and E is not decreased exactly to zero. Knowledge of the error bound permits the estimation of the maximum average error in the NN solution without prior knowledge of the true NLSE solution – a crucial factor in the practical applications of the NN technique. The error bound is verified for both the linear time – independent Schrödinger equation for a free particle, and the NLSE. We also discuss the conditions where the error bound formulation is valid.