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

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.