A Deep Learning Algorithm for Solving Generalized Burgers–Fisher and Burger’s Equations

In this paper, a deep-learning method called Deep-Galerkin-Method is presented to find the approximate solution of the generalized Burgers–Fisher equation (gBFE) and generalized Burger’s equation (gBE). The gBFE models various phenomena such as gas dynamics, fluid dynamics, heat transfers, etc. However, the gBE describes the far field of wave propagation in non-linear dissipative systems. In this method, a deep neural network (DNN) is used to approximate the solution, satisfying differential operator, initial conditions, and boundary conditions. This method is mesh-free, making it particularly valuable for higher-dimensional problems where mesh construction becomes impractical. Instead of constructing a mesh, the DNN is trained on batches of randomly chosen time and space points. The optimization of DNN parameters are done using Adam optimizer. To assess the effectiveness of the proposed method, a comparative analysis is conducted with existing methods such as the Adomain-decomposition method, Compact finite difference method, optimal homotopy asymptotic method, Reduced differential transform method, and others. The results obtained through this approach are highly promising, showcasing the method’s superiority in terms of accuracy and computational efficiency.