A comparative investigation of a time-dependent mesh method and physics-informed neural networks to analyze the generalized Kolmogorov-Petrovsky-Piskunov equation

The Kolmogorov-Petrovsky-Piskunov (KPP) partial differential equation (PDE) is solved in this article using the moving mesh finite difference technique (MMFDM) in conjunction with physics-informed neural networks (PINNs). We construct a time-dependent mesh to obtain approximate solutions for the KPP problem. The temporal derivative is discretized using a backward Euler, while the spatial derivatives are discretized using a central implicit difference scheme. Depending on the error measure, several moving mesh partial differential equations (MMPDEs) are employed along the arc-length and curvature mesh density functions (MDF). The proposed strategy has been suggested to yield remarkably precise and consistent results. To find the approximate solution, we additionally employ physics-informed neural networks (PINNs) to compare the outcomes of the adaptive moving mesh approach. It has been observed that solutions obtained using the moving mesh method (MMM) are sufficiently accurate, and the absolute error is also much lower than the PINNs. This work considers the Kolmogorov-Petrovsky-Piskunov (KPP) partial differential equation (PDE), which is solved in this article using the moving mesh finite difference method (MMFDM) together with Physics-informed neural networks (PINNs). The approximate solutions are obtained using the unsteady mesh method for the KPP problem, such that the temporal derivative is discretized using a backward-Euler, while the spatial derivatives are discretized using a central implicit semidiscretization scheme. Depending on the error measures, a number of moving mesh partial differential equations (MMPDEs) are employed along the arc-length and curvature mesh density functions (MDF). To compare the obtained results, Physics-informed neural networks (PINNs) are used to obtain the approximate solution. It has been observed that solutions obtained using the moving mesh method (MMM) are significantly more accurate, and the absolute error is also much lower than the PINNs.image