Data driven soliton solution of the nonlinear Schrödinger equation with certain PT-symmetric potentials via deep learning

Abstract

We investigate the physics informed neural network method, a deep learning approach, to approximate soliton solution of the nonlinear Schrödinger equation with parity time symmetric potentials. We consider three different parity time symmetric potentials, namely, Gaussian, periodic, and Rosen–Morse potentials. We use the physics informed neural network to solve the considered nonlinear partial differential equation with the above three potentials. We compare the predicted result with the actual result and analyze the ability of deep learning in solving the considered partial differential equation. We check the ability of deep learning in approximating the soliton solution by taking the squared error between real and predicted values. Further, we examine the factors that affect the performance of the considered deep learning method with different activation functions, namely, ReLU, sigmoid, and tanh. We also use a new activation function, namely, sech, which is not used in the field of deep learning, and analyze whether this new activation function is suitable for the prediction of soliton solution of the nonlinear Schrödinger equation for the aforementioned parity time symmetric potentials. In addition to the above, we present how the network’s structure and the size of the training data influence the performance of the physics informed neural network. Our results show that the constructed deep learning model successfully approximates the soliton solution of the considered equation with high accuracy.