Fractional sub-equation neural networks (fSENNs) method for exact solutions of space–time fractional partial differential equations

Analytical solutions of space-time fractional partial differential equations (fPDEs) are crucial for understanding dynamics features in complex systems and their applications. In this paper, fractional sub-equation neural networks (fSENNs) are first proposed to construct exact solutions of space-time fPDEs. The fSENNs embed the solutions of the fractional Riccati equation into neural networks (NNs). The NNs are a multi-layer computational models that are composed of weights and activation functions between neurons in the input, hidden, and output layers. In fSENNs, every neuron of the first hidden layer is assigned to the solutions of the fractional Riccati equation. In this way, the new trial functions are obtained. The exact solutions of space-time fPDEs can be obtained by fSENNs. In order to verify the rationality of this method, space-time fractional telegraph equation, space-time fractional Fisher equation, and space-time fractional CKdV-mKdV equation are investigated, and generalized fractional hyperbolic function solutions, generalized fractional trigonometric function solutions, and generalized fractional rational solutions are obtained. Since the fractional sub-equation is applied to the NNs model for the first time, more and new solutions can be obtained in this paper. The dynamic characteristics of some solutions corresponding to waves have been demonstrated through some diagrams.