Abstract
The solving of the derivative nonlinear Schrodinger equation (DNLS) has attracted considerable attention in theoretical analysis and physical applications. Based on the physics-informed neural network (PINN) which has been put forward to uncover dynamical behaviors of nonlinear partial different equation from spatiotemporal data directly, an improved PINN method with neuron-wise locally adaptive activation function is presented to derive localized wave solutions of the DNLS in complex space. In order to compare the performance of above two methods, we reveal the dynamical behaviors and error analysis for localized wave solutions which include one-rational soliton solution, genuine rational soliton solutions and rogue wave solution of the DNLS by employing two methods, also exhibit vivid diagrams and detailed analysis. The numerical results demonstrate the improved method has faster convergence and better simulation effect. On the bases of the improved method, the effects for different numbers of initial points sampled, residual collocation points sampled, network layers, neurons per hidden layer on the second order genuine rational soliton solution dynamics of the DNLS are considered, and the relevant analysis when the locally adaptive activation function chooses different initial values of scalable parameters are also exhibited in the simulation of the two-order rogue wave solution.
Highlights
The derivative nonlinear Schrodinger equation (DNLS)iqt + qxx + i(q2q∗)x = 0, (1.1)plays a significant role both in the integrable system theory and many physical applications, especially in space plasma physics and nonlinear optics [1,2]
We extend the physics-informed neural network (PINN) based on locally adaptive activation function with slope recovery term which proposed by Jagtap and cooperator [36] to solve the nonlinear integrable equation in complex space, and construct the localized wave solutions which consist of the rational soliton solutions and rogue wave solution of the integrable DNLS
The improved PINN method achieves a better performance of the neural network through such learnable parameters in the activation function
Summary
Plays a significant role both in the integrable system theory and many physical applications, especially in space plasma physics and nonlinear optics [1,2]. Jagtap and collaborators employed adaptive activation functions for regression in PINN to approximate smooth and discontinuous functions as well as solutions of linear and nonlinear partial differential equations, and introduced a scalable parameters in the activation function, which can be optimized to achieve best performance of the network as it changes dynamically the topology of the loss function involved in the optimization process [35]. We extend the PINN based on locally adaptive activation function with slope recovery term which proposed by Jagtap and cooperator [36] to solve the nonlinear integrable equation in complex space, and construct the localized wave solutions which consist of the rational soliton solutions and rogue wave solution of the integrable DNLS.
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