In this paper, we introduced the gradient-enhanced fractional physics-informed neural networks (gfPINNs) for solving the forward and inverse problems of the multiterm time-fractional Burger-type equation. The gfPINNs leverage gradient information derived from the residual of the fractional partial differential equation and embed the gradient into the loss function. Since the standard chain rule in integer calculus is invalid in fractional calculus, the automatic differentiation of neural networks does not apply to fractional operators. The automatic differentiation for the integer order operators and the finite difference discretization for the fractional operators were used to construct the residual in the loss function. The numerical results demonstrate the effectiveness of gfPINNs in solving the multiterm time-fractional Burger-type equation. By comparing the experimental results of fractional physics-informed neural networks (fPINNs) and gfPINNs, it can be seen that the training performance of gfPINNs is better than fPINNs.

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.