Data-Driven Modeling of Boussinesq-Burgers Equations: Comparing Physics-Informed Neural Networks (PINNs) with Exact Solutions

Abstract

This paper presents a novel deep neural network-based scheme for generating approximate solutions of the coupled Boussinesq-Burgers equations of integer order. Standard numerical techniques can face challenges in the case of these nonlinearly coupled systems, in particular when dealing with high dimensions. This presented work uses feedforward neural networks to learn the solution functions embed the physics of the problem in the learning. The residuals of the governing
PDEs and initial condition are set up and minimized in the structure of a composite loss function by automatic differentiation. The networks are optimized through gradient based techniques to determine a high accurate and generalizable networks over the solution domain. Results show that the proposed neural network model is able to approximate the complex dynamics and is in very good match with the exact solutions where they exist. The absolute error between PINN and exact solution is in the range of 10−4 to 10−3 , indicating the efficiency of the model.