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Numerical solution of the nonlinear Schrödinger equation by feedforward neural networks. (English) Zbl 1221.65315
Summary: We present a method to solve boundary value problems using artificial neural networks (ANN). A trial solution of the differential equation is written as a feed-forward neural network containing adjustable parameters (the weights and biases). From the differential equation and its boundary conditions we prepare the energy function which is used in the back-propagation method with momentum term to update the network parameters. We improved energy function of ANN which is derived from Schrödinger equation and the boundary conditions. With this improvement of energy function we can use unsupervised training method in the ANN for solving the equation. Unsupervised training aims to minimize a non-negative energy function. We used the ANN method to solve Schrödinger equation for few quantum systems. Eigenfunctions and energy eigenvalues are calculated. Our numerical results are in agreement with their corresponding analytical solution and show the efficiency of ANN method for solving eigenvalue problems.

MSC:
65N99 Numerical methods for partial differential equations, boundary value problems
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