A new efficient symbolic computation fusion neural network method: Solving exact solutions of (3+1)-dimensional nonlinear partial differential equations

Yu Gao; Jingwen Huang; Baoying Du; Jianglong Shen; Yu Gao; Jingwen Huang; Baoying Du; Jianglong Shen

doi:10.3934/math.2026435

AIMS Mathematics

2026, Volume 11, Issue 4: 10566-10588. doi: 10.3934/math.2026435

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A new efficient symbolic computation fusion neural network method: Solving exact solutions of (3+1)-dimensional nonlinear partial differential equations

1.
School of Mathematics and Physics, Yibin University, College Street, Yibin 644000, Sichuan, China
2.
Key Laboratory of Computational Physics of Sichuan Province, Yibin University, College Street, Yibin 644000, Sichuan, China

Received: 12 February 2026 Revised: 01 April 2026 Accepted: 14 April 2026 Published: 17 April 2026
MSC : 35A25, 35Q51

This study proposed a novel symbolic computing algorithm based on neural networks for solving the (3+1)–dimensional Jimbo-Miwa equation. By constructing a direct neural network model and integrating neural networks with symbolic computing, activation functions were assigned to the neurons in the hidden layer of the neural network. After deriving the trial function, symbolic computing using Maple was employed to obtain the exact analytical solution of the equation. Our innovative method effectively avoids the reliance on large datasets and low computational efficiency of traditional methods. Based on this improved method, we have constructed single-hidden-layer and double-hidden-layer neural network models to solve the equation's exact solutions, and successfully obtained breather solutions, shock wave solutions, and lump solutions. The successful solution of the equation in this study fully demonstrates the efficiency of the constructed framework and indicates its promising application prospects in other important nonlinear partial differential equation fields.
- (3+1)–dimensional Jimbo–Miwa equation,
- symbolic–neural integration,
- exact solution,
- neural network,
- nonlinear partial differential equation
Citation: Yu Gao, Jingwen Huang, Baoying Du, Jianglong Shen. A new efficient symbolic computation fusion neural network method: Solving exact solutions of (3+1)-dimensional nonlinear partial differential equations[J]. AIMS Mathematics, 2026, 11(4): 10566-10588. doi: 10.3934/math.2026435

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Abstract

This study proposed a novel symbolic computing algorithm based on neural networks for solving the (3+1)–dimensional Jimbo-Miwa equation. By constructing a direct neural network model and integrating neural networks with symbolic computing, activation functions were assigned to the neurons in the hidden layer of the neural network. After deriving the trial function, symbolic computing using Maple was employed to obtain the exact analytical solution of the equation. Our innovative method effectively avoids the reliance on large datasets and low computational efficiency of traditional methods. Based on this improved method, we have constructed single-hidden-layer and double-hidden-layer neural network models to solve the equation's exact solutions, and successfully obtained breather solutions, shock wave solutions, and lump solutions. The successful solution of the equation in this study fully demonstrates the efficiency of the constructed framework and indicates its promising application prospects in other important nonlinear partial differential equation fields.

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