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.…