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Neural network-based symbolic calculation approach for solving the Korteweg–de Vries equation

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  • Xie, Xiao-Ran
  • Zhang, Run-Fa

Abstract

In this study, we propose an innovative method that combines neural network models and symbolic computing to quickly solve exact solutions to nonlinear partial differential equations (NLPDEs). By combining the high accuracy of symbolic computing with the strong adaptability of neural networks, this method significantly improves the efficiency and accuracy of decomputation. As an application, this paper uses the neural network symbol calculation method to successfully obtain multiple sets of new analytical solutions of the Korteweg–de Vries equation, and constructs a variety of new neural network models and their heuristic functions. This study provides a new computational paradigm for solving the exact solution of NLPDEs, and has the potential for a wide range of scientific and engineering applications.

Suggested Citation

  • Xie, Xiao-Ran & Zhang, Run-Fa, 2025. "Neural network-based symbolic calculation approach for solving the Korteweg–de Vries equation," Chaos, Solitons & Fractals, Elsevier, vol. 194(C).
    • Handle: RePEc:eee:chsofr:v:194:y:2025:i:c:s0960077925002450
    DOI: 10.1016/j.chaos.2025.116232
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    References listed on IDEAS

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    1. Zhang, Run-Fa & Li, Ming-Chu & Gan, Jian-Yuan & Li, Qing & Lan, Zhong-Zhou, 2022. "Novel trial functions and rogue waves of generalized breaking soliton equation via bilinear neural network method," Chaos, Solitons & Fractals, Elsevier, vol. 154(C).
    2. Gu, Yongyi & Peng, Liudi & Huang, Zhishang & Lai, Yongkang, 2024. "Soliton, breather, lump, interaction solutions and chaotic behavior for the (2+1)-dimensional KPSKR equation," Chaos, Solitons & Fractals, Elsevier, vol. 187(C).
    3. Ma, Wen-Xiu & Lee, Jyh-Hao, 2009. "A transformed rational function method and exact solutions to the 3+1 dimensional Jimbo–Miwa equation," Chaos, Solitons & Fractals, Elsevier, vol. 42(3), pages 1356-1363.
    4. Gupta, A.K. & Ray, S. Saha, 2018. "On the solution of time-fractional KdV–Burgers equation using Petrov–Galerkin method for propagation of long wave in shallow water," Chaos, Solitons & Fractals, Elsevier, vol. 116(C), pages 376-380.
    5. Xin Li & Qunxi Zhu & Chengli Zhao & Xiaojun Duan & Bolin Zhao & Xue Zhang & Huanfei Ma & Jie Sun & Wei Lin, 2024. "Higher-order Granger reservoir computing: simultaneously achieving scalable complex structures inference and accurate dynamics prediction," Nature Communications, Nature, vol. 15(1), pages 1-13, December.
    6. Chen, Yu & Lü, Xing, 2025. "PINN-wf: A PINN-based algorithm for data-driven solution and parameter discovery of the Hirota equation appearing in communications and finance," Chaos, Solitons & Fractals, Elsevier, vol. 190(C).
    7. Wazwaz, Abdul-Majid, 2024. "Study on a (3+1)-dimensional B-type Kadomtsev–Petviashvili equation in nonlinear physics: Multiple soliton solutions, lump solutions, and breather wave solutions," Chaos, Solitons & Fractals, Elsevier, vol. 189(P1).
    8. Kohnesara, Sima Molaei & Firoozjaee, Ali Rahmani, 2023. "Numerical solution of Korteweg–de Vries equation using discrete least squares meshless method," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 206(C), pages 65-76.
    9. Liu, Ye & Li, Jie-Ying & Zhang, Li-Sheng & Guo, Lei-Lei & Zhang, Zhi-Yong, 2024. "Symmetry group based domain decomposition to enhance physics-informed neural networks for solving partial differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 189(P1).
    10. Zheng, Hang & Xia, Yonghui, 2024. "Persistence of solitary wave solutions for the delayed regularized long wave equation under Kuramoto–Sivashinsky perturbation and Marangoni effect," Chaos, Solitons & Fractals, Elsevier, vol. 184(C).
    11. Qin, Chun-Yan & Zhang, Run-Fa & Li, Yao-Hong, 2024. "Various exact solutions of the (4+1)-dimensional Boiti–Leon–Manna–Pempinelli-like equation by using bilinear neural network method," Chaos, Solitons & Fractals, Elsevier, vol. 187(C).
    12. Wang, Xiaoli & Wu, Zekang & Song, Jin & Han, Wenjing & Yan, Zhenya, 2024. "Data-driven soliton solutions and parameters discovery of the coupled nonlinear wave equations via a deep learning method," Chaos, Solitons & Fractals, Elsevier, vol. 180(C).
    13. Yang, Liu & Gao, Ben, 2024. "The nondegenerate solitons solutions for the generalized coupled higher-order nonlinear Schrödinger equations with variable coefficients via the Hirota bilinear method," Chaos, Solitons & Fractals, Elsevier, vol. 184(C).
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