Exact solutions of the (2+1)-dimensional BLMP equation via neural network based symbolic computation

  • Jiang-Long Shen orcid

    School of Mathematics and Physics, Yibin University, Yibin 644000, China
  • Run-Fa Zhang orcid

    School of Automation and Software Engineering, Shanxi University, Taiyuan 030013, China
  • Jing-Wen Huang orcid

    School of Mathematics and Physics, Yibin University, Yibin 644000, China
  • Yu Gao orcid

    School of Mathematics and Physics, Yibin University, Yibin 644000, China
  • Jing-Bin Liang orcid

    School of Mathematics and Physics, Yibin University, Yibin 644000, China
Article ID: 3589
DOI: https://doi.org/10.59400/adecp3589
Keywords: (2+1)-dimensional BLMP equation; neural networks; nonlinear partial differential equations; symbolic computation; lump solution

Abstract

This paper introduces a neural network-based symbolic computation framework for deriving exact analytical solutions to nonlinear partial differential equations (NLPDEs). By integrating the expressive capability of neural networks with the interpretability of symbolic methods, the approach offers a flexible, generalizable, and computationally efficient alternative to traditional techniques. Its architecture is straightforward, requiring minimal prior assumptions about the equation structure, thus enabling broad applicability across mathematical physics. As a case study, we investigate the (2+1)-dimensional Boiti-Leon-Manna-Pempinelli (BLMP) equation, a fundamental model describing wave propagation in fluid dynamics, nonlinear optics, and plasma physics. By systematically varying hidden-layer configurations and activation functions, we construct six distinct neural network models—comprising both single- and double-hidden-layer designs. These yield multiple novel exact analytical solutions, revealing diverse dynamic behaviors such as localized wave structures, periodic oscillations, and energy-localized modes. The physical characteristics of the obtained solutions are visualized through three-dimensional surface plots, contour maps, density distributions, and temporal evolution graphs. These illustrations elucidate key features including waveform stability, soliton localization, and nonlinear interactions. Unlike conventional methods such as the Hirota bilinear transformation or inverse scattering, the proposed framework avoids complex algebraic manipulations and does not rely on specialized transformations. Compared to physics-informed neural networks, it achieves higher interpretability and reduced computational dependency. This work expands the solution space of the (2+1)-dimensional BLMP equation and highlights the potential of neural-symbolic computation for discovering exact solutions in complex nonlinear systems.

Published
2025-12-01
How to Cite
Shen, J.-L., Zhang, R.-F., Huang, J.-W., Gao, Y., & Liang, J.-B. (2025). Exact solutions of the (2+1)-dimensional BLMP equation via neural network based symbolic computation. Advances in Differential Equations and Control Processes, 32(4). https://doi.org/10.59400/adecp3589
Issue
Vol. 32 No. 4 (2025)
Section
Article

Copyright (c) 2025 Jiang-Long Shen, Run-Fa Zhang, Jing-Wen Huang, Yu Gao, Jing-Bin Liang

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

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