Symbolic computation of analytical solutions for nonlinear partial differential equations based on bilinear neural network method

The integration of neural networks with the Hirota bilinear form represents an innovative approach that focuses on solving nonlinear partial differential equations (NPDEs). In this paper, we introduce a novel Maple algorithm tailored for bilinear neural network methods (BNNM). This algorithm progresses from the initial NPDE to the final solutions, illustrating a new method for automatically constructing analytical solutions to NPDEs. The foundational principles of this algorithm are derived from the Hirota bilinear method and neural network techniques, with significant improvements made to the original BNNM algorithm to enhance its applicability across a broader range of scenarios. By combining the strengths of the Hirota bilinear form with the adaptability of neural networks, our approach not only simplifies the solution process but also expands the potential for discovering new solutions and streamlines the derivation of analytical solutions. To showcase the algorithm’s capability, we applied it to several equations, including the classical Kadomtsev-Petviashvili equation, the \((3+1)\) $(3+1)$ -dimensional generalized KP equation, and the \((3+1)\) $(3+1)$ -dimensional generalization of the BKP equation.