Data-driven soliton solution implementation based on nonlinear adaptive physics-informed neural networks

In order to reduce the computational performance requirements and improve the nonlinear representation ability of neural networks and the accuracy of solving soliton solutions of partial differential equations, this paper utilizes a deep learning machine to design PINNs containing adaptive nonlinear combined activation functions. In particular, the physical information present in the mathematical physics model is incorporated into the neural network through the use of the neural network’s generalized approximation capabilities. The PINNs with adaptive nonlinear combined activation functions are employed to obtain analytical solutions of partial differential equations, which is predicted by neural networks with high accuracy. By analyzing the spatio-temporal dynamics of the solutions of the (1 + 1)-dimensional Burgers equation, the (1 + 1)-dimensional Schrödinger equation, and the (2 + 1)-dimensional Navier–Stokes equation equations, it has been demonstrated that the proposed approach yields more accurate prediction results. Moreover, deep neural networks have been successfully employed in the solution of soliton problems.