A deep learning approach: Physics-informed neural networks for solving the 2D nonlinear Sine–Gordon equation

In the current work, we apply a physics-informed neural networks (PINNs), a machine learning approach, for solving the non-linear hyperbolic sine–Gordon problem with two space dimensions. To include all the physical information of a PDE in to the learning process, we considered a multi-objective loss function that takes into account the problem PDE residual, the initial condition residual, and the boundary condition residual. The problem was approximated using PINNs employing a variety of artificial neural network topologies, one of which being feedforward deep neural networks, a densely connected network. To establish the effectiveness, soundness, and practical implications of the suggested technique, we provide three computational illustrations from the nonlinear two-dimensional sine–Gordon equations. We trained the PINNs model and run various tests using Python software as a computational tool. We gave the theoretical error bounds of the proposed approach in approximating the NLSGE. We evaluated the accuracy of the model by comparing it to other standard numerical methods in the literature through root mean square error (RMSE), $L_{\infty }$, and $L_2$ relative errors. The findings suggested that the offered PINN approach is more effective and accurate than the other numerical methods. The method can be directly applied to any problem that involves different boundary conditions without requiring linearization, perturbation, or interpolation techniques. Thus, for the purpose of solving the nonlinear hyperbolic sine–Gordon equation in two dimensions and other difficult nonlinear physical issues across several fields, the PINN model provides an appropriate programming machine learning technique that is both accurate and efficient.