Fractional variable transformation neural networks for analytical solution of nonlinear fractional partial differential equations

Limei Yan; Shanhao Yuan; Yanqin Liu; Runfa Zhang; Qiuping Li; Limei Yan; Shanhao Yuan; Yanqin Liu; Runfa Zhang; Qiuping Li

doi:10.3934/era.2026032

Electronic Research Archive

2026, Volume 34, Issue 2: 694-722. doi: 10.3934/era.2026032

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Fractional variable transformation neural networks for analytical solution of nonlinear fractional partial differential equations

1.
School of Mathematics and Big Data, Dezhou University, Dezhou 253023, China
2.
School of Mathematics and Statistics, Qilu University of Technology (Shandong Academy of Sciences), Jinan 250353, China
3.
School of Automation and Software Engineering, Shanxi University, Taiyuan 030013, China
^‡ The authors contributed equally to this work.

Received: 26 September 2025 Revised: 25 December 2025 Accepted: 09 January 2026 Published: 22 January 2026

In this paper, we propose fractional variable transformation neural networks (fVTNNs, for short), a framework that embeds fractional variable transformation into neural networks (NNs), to systematically derive analytical solutions for nonlinear space-time fractional partial differential equations (fPDEs) via symbolic computation. This approach significantly enhances both the computational speed and the result precision by combining the robust approximation capacity of NNs with the exactness of symbolic computation. The output of fVTNNs, which consists of weights, biases, and activation functions, is taken as a trial function for the considered equation. In order to explain the feasibility of the proposed method, some examples are investigated. Hyperbolic function solutions and exponential function interactive solutions of these equations are obtained. The analytical solutions obtained using this method are accurate and have no calculation errors. To visualize the dynamic characteristics of the solutions, three-dimensional plots, contour plots, and density plots are employed. This research introduces a novel computational framework for obtaining exact solutions to fPDEs, with a broad applicability in science and engineering.
- neural networks,
- symbolic computation,
- nonlinear fractional partial differential equations,
- fractional Burgers-type equation,
- exact solution
Citation: Limei Yan, Shanhao Yuan, Yanqin Liu, Runfa Zhang, Qiuping Li. Fractional variable transformation neural networks for analytical solution of nonlinear fractional partial differential equations[J]. Electronic Research Archive, 2026, 34(2): 694-722. doi: 10.3934/era.2026032

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Abstract

In this paper, we propose fractional variable transformation neural networks (fVTNNs, for short), a framework that embeds fractional variable transformation into neural networks (NNs), to systematically derive analytical solutions for nonlinear space-time fractional partial differential equations (fPDEs) via symbolic computation. This approach significantly enhances both the computational speed and the result precision by combining the robust approximation capacity of NNs with the exactness of symbolic computation. The output of fVTNNs, which consists of weights, biases, and activation functions, is taken as a trial function for the considered equation. In order to explain the feasibility of the proposed method, some examples are investigated. Hyperbolic function solutions and exponential function interactive solutions of these equations are obtained. The analytical solutions obtained using this method are accurate and have no calculation errors. To visualize the dynamic characteristics of the solutions, three-dimensional plots, contour plots, and density plots are employed. This research introduces a novel computational framework for obtaining exact solutions to fPDEs, with a broad applicability in science and engineering.

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