This paper proposes a symbolic computational method using neural networks to solve Boussinesq equations. By combining activation functions such as sine, hyperbolic tangent, hyperbolic secant, and exponential, we constructed a single-hidden-layer model [3-3-1] and a double-hidden-layer model [3-3-2-1] to solve the (2+1)-dimensional and (1+1)-dimensional Boussinesq equations, respectively. This yielded multiple sets of novel, exact analytical solutions, which were visualized using 3D plots, contour plots, heatmaps, and evolution plots. To address the issues of insufficient direct neural network feature representation and poor generalization ability, this paper introduces a feature-enhanced neural network method. Model [(2-5)-3-1] was constructed for the (1+1)-dimensional equation, expanding the number of input neurons from 2 to 5; Model [(3-9)-3-1] was constructed for the (2+1)-dimensional equation, expanding the number of input neurons from 3 to 9. Both models retain 3 hidden-layer neurons and optimize the network structure based on the equation's dimension. By reconstructing the physical, spatiotemporal, and nonlinear characteristics of the equations and coupling them with inherent constraints, this method offers the advantages of richer solution results, higher accuracy, and significantly improved model generalization. It does not require bilinear transformations, features a simple structure, and has low computational cost, providing a new, viable solution for solving strongly nonlinear water wave equations.
Citation: Hang-Jing Xiong, Ke-Yan Dai, Jiang-Long Shen, Jing-Bin Liang, Yu Gao. Symbolic trial-function method inspired by neural network architectures[J]. Networks and Heterogeneous Media, 2026, 21(5): 1800-1821. doi: 10.3934/nhm.2026067
This paper proposes a symbolic computational method using neural networks to solve Boussinesq equations. By combining activation functions such as sine, hyperbolic tangent, hyperbolic secant, and exponential, we constructed a single-hidden-layer model [3-3-1] and a double-hidden-layer model [3-3-2-1] to solve the (2+1)-dimensional and (1+1)-dimensional Boussinesq equations, respectively. This yielded multiple sets of novel, exact analytical solutions, which were visualized using 3D plots, contour plots, heatmaps, and evolution plots. To address the issues of insufficient direct neural network feature representation and poor generalization ability, this paper introduces a feature-enhanced neural network method. Model [(2-5)-3-1] was constructed for the (1+1)-dimensional equation, expanding the number of input neurons from 2 to 5; Model [(3-9)-3-1] was constructed for the (2+1)-dimensional equation, expanding the number of input neurons from 3 to 9. Both models retain 3 hidden-layer neurons and optimize the network structure based on the equation's dimension. By reconstructing the physical, spatiotemporal, and nonlinear characteristics of the equations and coupling them with inherent constraints, this method offers the advantages of richer solution results, higher accuracy, and significantly improved model generalization. It does not require bilinear transformations, features a simple structure, and has low computational cost, providing a new, viable solution for solving strongly nonlinear water wave equations.
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Hang-Jing Xiong, Ke-Yan Dai, Jiang-Long Shen, Jing-Bin Liang, Yu Gao. Symbolic trial-function method inspired by neural network architectures[J]. Networks and Heterogeneous Media, 2026, 21(5): 1800-1821. doi: 10.3934/nhm.2026067
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