Applications of Extended Physics Informed Neural Network Method for Nonlinear Differential Equations

Abstract

Nonlinear partial differential equations featuring nonlinear terms occur in a number of engineering and physical problems, including reaction-diffusion processes, heat conduction, and population dynamics. Classical numerical techniques struggle to efficiently tackle high-dimensional problems and observe intricate nonlinear phenomena. Physics-Informed Neural Networks (PINNs) present a new opportunity in solving PDEs by incorporating physical boundary conditions inside the learning problem. Discontinuities, domain decomposition, and multi-scale phenomena, however, pose a problem to traditional PINNs. Extended PINNs (XPINNs) and conservative PINNs (cPINNs) have been proposed to overcome these drawbacks. In this thesis, we examine the use of PINN, XPINN, and cPINN in solving a reactiondiffusion equation with an exponential nonlinearity. We compare and examine their efficiency, accuracy, and stability in different parameter regimes. We will also examine the dynamics of mean square and absolute error in the given models. We will consider certain specific initial and boundary conditions to train the NN for governing model and will calculate the error of the equations by comparing the result extracted using PINN, cPINN and XPINN with the exact solution. With this, the results depict a sufficiently good agreement in the approximate solution of the governing model, the mean square error of which is of the order of 10−1 to 10−4 in the simulated equations. To the best of our knowledge, we apply this method to this model for the first time and the results are new and novel.