Application of a physics-informed neural network to solve the steady-state Bratu equation arising from solid biofuel combustion theory

Abstract

Physic-Informed Neural Networks (PINN) has attracted extensive attention in recent years. This method can use partial differential equations (PDE) or boundary value problems (BVPs) to describe the physics behind the desired problems and provide a solution without utilizing any data-driven techniques. This ability makes PINN applicable to a wide range of physical and engineering complex problems. In this paper, the PINNs is developed for solving Bratu equation arising from solid biofuel combustion theory. The governing BVP, is presented in a steady-state form. Assuming the answer to the problem in the form of a deep multilayer neural network and introducing it into the established BVP, a loss function is defined. Next, a discrete space of variables is developed by discretizing the range of the independent input variables, namely, location, and the loss function is minimized over all collocation points using the optimization algorithm. In this way, the proposed deep neural network is trained to estimate the response to the Bratu differential equation with the least possible error. Moreover, using a package of data collected from the real solutions to the problem, the accuracy of the developed PINN is evaluated. The results show that the trained PINNs could accurately estimate the answer to the problem and present the solution of governing BVPs systematically without any prior observation of the answer.