A new neural network method for solving Bratu type equations with rational polynomials

The Bratu-type equation is a fundamental differential equation with numerous applications in engineering fields, such as radiative heat transfer, thermal reaction, and nanotechnology. This paper introduces a novel approach known as the rational polynomial neural network. In this approach, rational orthogonal polynomials are utilized within the neural network’s hidden layer. To solve the equation, the initial boundary value conditions of both the differential equation and the rational polynomial neural network are integrated into the construction of the numerical solution. This construction transforms the Bratu-type equation into a set of nonlinear equations, which are subsequently solved using an appropriate optimization technique. Finally, three sets of numerical examples are presented to validate the efficacy and versatility of the proposed rational orthogonal neural network method, with comparisons made across different hyperparameters. Furthermore, the experimental results are juxtaposed against traditional methods such as the Adomian decomposition method, genetic algorithm, Laplace transform method, spectral method, and multilayer perceptron, our method exhibits consistently optimal performance.