Optimization of Physics-Informed Neural Networks for Solving the Nolinear Schrödinger Equation

Physics Informed Neural Networks (PINN) is a promising method for solving partial differential equations using machine learning. In this paper we consider the application of PINN to the nonlinear Schrödinger equation to describe the propagation of signal in an optical fibre. The factors determining the convergence of PINN from the physical point of view are investigated. Estimates of the convergence domain of the method in terms of fibre length and pulse energy are obtained. It is shown that the application sinusoidal activation function, as well as the weighting of the loss function terms are able to extend the convergence region of PINN with respect to fibre length and pulse energy. A generalization of the method (meta-PINN) is derived, allowing to solve the equation at its various parameters by using the pre-trained neural network.