Optimal control of interactions between invasive alien and native species in a certain time period with the r-PINN approach

The spread of invasive species poses a significant challenge to native biodiversity and ecosystem stability. An optimal control strategies to minimize the negative impacts of invasive species populations on native species and the ecosystem must be done in order to preserve the diversity in the ecosystem. This study proposes an optimal control framework to mitigate the impact of invasive species by enhancing native species preservation through a reaction–diffusion mathematical model. To solve the system efficiently, a restarting Physics-Informed Neural Network (r-PINN) is employed and benchmarked against the basic PINN. Numerical simulations reveal that r-PINN achieves a reduced training duration of 236.17 s compared to 289.18 s for the basic PINN, representing an 18.32% improvement in computational efficiency. Moreover, r-PINN demonstrates enhanced predictive accuracy, reducing the mean absolute error (MAE) by 4.12%, mean squared error (MSE) training loss by 12.04%, and MSE test loss by 5.11%. These results were validated against the Finite Difference Method (FDM), ensuring correctness of the proposed PINN-based approach. The implementation of the optimal control strategy led to a clear increase in native species populations and effective suppression of invasive species across spatial and temporal domains. Overall, the r-PINN framework offers a reliable and computationally efficient tool for solving nonlinear ecological models involving spatiotemporal control of species populations.