A deep learning approach: physics-informed neural networks for solving a nonlinear telegraph equation with different boundary conditions.

Abstract

The nonlinear Telegraph equation appears in a variety of engineering and science problems. This paper presents a deep learning algorithm termed physics-informed neural networks to resolve a hyperbolic nonlinear telegraph equation with Dirichlet, Neumann, and Periodic boundary conditions. To include physical information about the issue, a multi-objective loss function consisting of the residual of the governing partial differential equation and initial conditions and boundary conditions is defined. Using multiple densely connected neural networks, termed feedforward deep neural networks, the proposed scheme has been trained to minimize the total loss results from the multi-objective loss function. Three computational examples are provided to demonstrate the efficacy and applications of our suggested method. Using a Python software package, we conducted several tests for various model optimizations, activation functions, neural network architectures, and hidden layers to choose the best hyper-parameters representing the problem's physics-informed neural network model with the optimal solution. Furthermore, using graphs and tables, the results of the suggested approach are contrasted with the analytical solution in literature based on various relative error analyses and statistical performance measure analyses. According to the results, the suggested computational method is effective in resolving difficult non-linear physical issues with various boundary conditions.