Exploring exact wave solutions of the Cahn–Allen equation via a novel Bernoulli sub-equation neural networks method

In this work, a new Bernoulli sub-equation neural network method is proposed to probe the Cahn–Allen equation for the first time. This method combines the Bernoulli sub-equation function with the neural networks (NNs) models to develop the new exact solutions of the considered equation. The solutions of the Bernoulli sub-equation are considered as the activation function for the NNs of the first hidden layer, which can establish a new mathematical link between the equation and deep learning. Two different “2-2-2-1” NNs models are constructed via selecting two different activation functions for the second hidden layer. Then, two different trial functions are obtained to find the diverse exact wave solutions. The shapes of the wave solutions are depicted through the Maple. The proposed method in this work can provide a new way to explore the exact solutions of other NPDEs in physics.