A novel new iterative method–physics-informed neural networks approach for accurate and efficient solutions of nonlinear damped Burgers’ equations

Abstract

In this work, a hybrid semi-analytical and deep learning framework is proposed for solving nonlinear damped Burgers equations by combining the New Iterative Method (NIM) with Physics-Informed Neural Networks (PINNs). The developed methodology consists of the use of NIM to build an analytical baseline approximation, and the neural network is used to learn the rest of the corrective term. This decomposition greatly enhances the convergence properties as well as the computational cost of the learning process. The proposed framework is applied to two nonlinear damped Burgers equations in order to check its accuracy and stability. Numerical simulations are shown to agree very well with the exact solutions throughout the computational domain for the hybrid solutions obtained. The training histories, contour plots, PDE residuals, and error distributions demonstrate the stability and efficiency of the proposed method. It can also be seen from the results that the hybrid NIM–PINN solution framework has significantly reduced the approximation error compared to the baseline analytical solution. The proposed methodology is capable of integrating the analytical iterative methods, which converge very rapidly, and the good approximation power of physics-informed neural networks. The proposed framework is thus accurate, stable, and efficient for solving nonlinear partial differential equations, and can be generalized to more complex nonlinear and fractional-order models in the future.