The simulation for particle or soliton propagation based on linear or nonlinear Schrödinger equations on unbounded domains requires the computational domain to be bounded, and therefore, a special boundary treatment such as an absorbing boundary condition (ABC) or a perfectly matched layer (PML) is needed so that the reflections of outgoing waves at the boundary can be minimized in order to prevent the destruction of the simulation. This article presents a new artificial neural network (ANN) method for solving linear and nonlinear Schrödinger equations on unbounded domains. In particular, this method randomly selects training points only from the bounded computational space-time domain, and the loss function involves only the initial condition and the Schrödinger equation itself in the computational domain without any boundary conditions. Moreover, unlike standard ANN methods that calculate gradients using expensive automatic differentiation, this method uses accurate finite-difference approximations for the physical gradients in the Schrödinger equation. In addition, a Metropolis-Hastings algorithm is implemented for preferentially selecting regions of high loss in the computational domain allowing for the use of fewer training points in each batch. As such, the present training method uses fewer training points and less computation time for convergence of the loss function as compared with the standard ANN methods. This new ANN method is illustrated using three examples.