The data-driven solutions and inverse problems of some nonlinear diffusion convection-reaction equations based on Physics-Informed Neural Network

The Physics-Informed Neural Network (PINN) has achieved remarkable results in solving partial differential equations (PDEs). This paper aims to solve the forward and inverse problems of some specific nonlinear diffusion convection-reaction equations, thereby validating the practical efficacy and accuracy of data-driven approaches in tackling such equations. In the forward problems, four different solutions of the studied equations are reproduced effectively and the approximation errors can be reduced to 10-5. Experiments indicate that the PINNs method based on adaptive activation functions (PINN-AAF), outperforms the standard PINNs in dealing with inverse problems. The unknown parameters are estimated effectively and the approximation errors can lower to 10-4. Additionally, training rules for both PINN and PINN-AAF are summarized. The results of this study validate the exceptional performance of the data-driven approach in solving the complex nonlinear diffusion convection-reaction equation problems, and provide an effective mechanism for dealing with analogous, intricate nonlinear problems.