GEPINN: An innovative hybrid method for a symbolic solution to the Lane–Emden type equation based on grammatical evolution and physics-informed neural networks

In this paper, we present an innovative and powerful combination of grammatical evolution and a physics-informed neural network approach for symbolically solving the Lane–Emden type equation, which is a nonlinear ordinary differential equation. We employ a grammatical evolution algorithm based on a context-free grammar to construct a mathematical expression comprising some parameters. Subsequently, these parameters are determined using the physics-informed neural networks approach. To achieve this, the computational graph of the mathematical expression generated in each iteration of the grammatical evolution is treated as a network. To assess the proposed method, we consider the Lane–Emden type equation. The proposed method demonstrated that it is a capable method for symbolically solving nonlinear ordinary differential equations accurately. •Combining the grammatical evolution and neural networks can solve ODEs symbolically.•The hybrid method is capable of solving the Lane–Emden type equations symbolically.•In general, 10,000 iterations for GE and 1,000 epochs for NN give accurate solutions.•Using rational Chebyshev functions as basis functions enhances accuracy.•In some cases of the Lane–Emden type equation, this method can find the true solution.•This method is also capable of solving a system of nonlinear ODEs.