Physics

Nonlinear system

Bilinear interpolation

Mathematics

Evolutionary biology

Biology

Partial differential equation

Statistics

Machine learning

Mathematical analysis

Function (biology)

Composite material

Materials science

Range (aeronautics)

Computer science

Exact solutions in general relativity

Quantum mechanics

Applied mathematics

Artificial neural network

First-order partial differential equation

Nonlinear Dynamics

A new method named bilinear neural network is introduced in this paper, and the corresponding tensor formula is proposed to obtain the exact analytical solutions of nonlinear partial differential equations (PDEs). This is the first time that the neural network model is used to find the exact analytical solution, and this method covers almost all methods of constructing a function after bilinearization to solve nonlinear PDEs. Furthermore, this method is most likely a universal method to obtain the exact analytical solutions of nonlinear PDEs. Abundant arbitrary functions solutions of the reduced p-gBKP equation are obtained by using this method. Various beautiful plots of the presented solutions, which show diversity of exact solutions to PDEs, are made. By choosing appropriate values and functions, the fractal solitons waves are obtained and the self-similar characteristics of these waves are observed by reducing the observation range and magnifying local images. Via various three-dimensional plots, the evolution characteristics of these waves are exhibited.

Bilinear neural network method to obtain the exact analytical solutions of nonlinear partial differential equations and its application to p-gBKP equation