AM-GPINN algorithm and its application in a variable-coefficient resonant nonlinear Schrödinger equation

In this paper, we provide an effective deep learning model to predict the soliton solutions and their interactions of nonlinear partial differential equations (PDEs). Based on gradient information, we investigate the gradient-enhanced physics-informed neural networks (GPINN) method to improve the accuracy and training efficiency of PINN, which embeds the gradient of the residual into the neural network loss function. To further improve the performance of GPINN, we combine the GPINN method with the adaptive mixing sampling (AM) and then propose the AM-GPINN algorithm, which can improve the distribution of training points adaptively. As an example, we use the AM-GPINN algorithm for solving the variable-coefficient resonant nonlinear Schrödinger equation. This is also the first time to solve the variable-coefficient resonant nonlinear Schrödinger equation via deep learning methods. Under different inhomogeneous parameter conditions, the data-driven nonautonomous soliton solutions are discussed. The experimental results demonstrate that the L2 relative error of AM-GPINN algorithm improves the accuracy by one order of magnitude over the original PINN algorithm.