Auxiliary Solutions via Dimension Reduction for the (2+1)-Dimensional Calogero–Bogoyavlenskii–Schiff Equation Based on a Bilinear Neural Network Method Coupled with Symbolic Computation
- 1School of Mathematics and Physics, Yibin University, College Street, Yibin 644000, China
- 2School of Electronic Information Engineering, Wuhan Donghu University, Zhifang Street, Wuhan 430212, China
Abstract
Nonlinear partial differential equations formed the mathematical core of nonlinear science. They revealed the intrinsic link between algebraic structures and complex dynamical behaviors. Traditional analytical and numerical methods, however, faced high computational costs, cumbersome derivations, limited accuracy, and poor generality. They especially struggled to solve high-dimensional integrable systems efficiently. To meet these challenges, this study built on traditional neural network methods. We introduced the Hirota bilinear transformation to convert the original nonlinear equation into a bilinear form. We then constructed a bilinear neural network solving procedure deeply coupled with symbolic computation. This strategy effectively suppressed error accumulation caused by random initialization and approximate approximation in traditional numerical iterative methods. Within this framework, this study constructed complete single-hidden-layer and double-hidden-layer neural network models to obtain analytical solutions of the auxiliary equation. By designing network structures with multiple activation functions, we built diverse trial solutions. We also systematically established a coupling adaptation mechanism among activation functions, network depth, and the bilinear transformation. To verify the effectiveness and robustness of the proposed method, we applied it to the Calogero–Bogoyavlenskii–Schiff (CBS) equation. We successfully derived multiple new exact solutions. We further constructed various neural network models and heuristic strategies. This study provided new ideas for neural network solving of integrable systems. It also offered a feasible path for approximation-free exact solutions of high-dimensional nonlinear partial differential equations. The findings held important theoretical value and broad application prospects in fluid dynamics, optical communication, plasma physics, and related engineering fields.
Related
- Contour Computation of Linearized Painlevé II and IV Solutions with Monodromy-Based Error Control
- The patterns of lump solutions for PT-symmetric (2+1)-dimensional nonlocal Mel’nikov equation
- Deep Learning of Hirota τ-Function for Soliton Solutions of a (3+1)-Dimensional Nonlocal Evolution Equation
- Eight-dimensional Manin triples, Yang-Baxter deformations and solutions of Supergravity Equations
- Higher-order semi-rational solutions of the (2+1)-dimensional nonlocal multi-component Yajima--Oikawa system
BibTeX
@article{gao2026auxiliary,
title = {{Auxiliary Solutions via Dimension Reduction for the (2+1)-Dimensional Calogero–Bogoyavlenskii–Schiff Equation Based on a Bilinear Neural Network Method Coupled with Symbolic Computation}},
author = {Yu Gao and Keyan Dai and Hangjing Xiong and Jianglong Shen and Qian Li},
journal = {Mathematics},
year = {2026},
month = oct,
doi = {10.3390/math14203669},
url = {https://www.mdpi.com/2227-7390/14/20/3669},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Auxiliary Solutions via Dimension Reduction for the (2+1)-Dimensional Calogero–Bogoyavlenskii–Schiff Equation Based on a Bilinear Neural Network Method Coupled with Symbolic Computation
AU - Yu Gao
AU - Keyan Dai
AU - Hangjing Xiong
AU - Jianglong Shen
AU - Qian Li
PY - 2026
DA - 2026/10/10
JO - Mathematics
DO - 10.3390/math14203669
UR - https://www.mdpi.com/2227-7390/14/20/3669
AB - Nonlinear partial differential equations formed the mathematical core of nonlinear science. They revealed the intrinsic link between algebraic structures and complex dynamical behaviors. Traditional analytical and numerical methods, however, faced high computational costs, cumbersome derivations, limited accuracy, and poor generality. They especially struggled to solve high-dimensional integrable systems efficiently. To meet these challenges, this study built on traditional neural network methods. We introduced the Hirota bilinear transformation to convert the original nonlinear equation into a bilinear form. We then constructed a bilinear neural network solving procedure deeply coupled with symbolic computation. This strategy effectively suppressed error accumulation caused by random initialization and approximate approximation in traditional numerical iterative methods. Within this framework, this study constructed complete single-hidden-layer and double-hidden-layer neural network models to obtain analytical solutions of the auxiliary equation. By designing network structures with multiple activation functions, we built diverse trial solutions. We also systematically established a coupling adaptation mechanism among activation functions, network depth, and the bilinear transformation. To verify the effectiveness and robustness of the proposed method, we applied it to the Calogero–Bogoyavlenskii–Schiff (CBS) equation. We successfully derived multiple new exact solutions. We further constructed various neural network models and heuristic strategies. This study provided new ideas for neural network solving of integrable systems. It also offered a feasible path for approximation-free exact solutions of high-dimensional nonlinear partial differential equations. The findings held important theoretical value and broad application prospects in fluid dynamics, optical communication, plasma physics, and related engineering fields.
SN - 2227-7390
ER -