Parameter Identification of Reaction-Diffusion Information Propagation Model Based on Optimal Control and Turing Pattern Theory

In this study, we examine the extrinsic rumor correction mechanisms and the psychological dynamics of spreaders, thereby developing a reaction–diffusion model for rumor dissemination. The occurrence of Turing instability and the resultant patterns are of paramount importance to the system; hence, we delve into the conditions of Turing instability. Subsequently, we analyze the patterns under diverse scenarios to elucidate the impact of various contributing factors. Concurrently, the endeavor to optimize the control of pattern formations is addressed by delineating the first-order necessary optimality conditions pertinent to the optimal control problem. Consequently, the target patterns are achieved under the influence of optimal control. Furthermore, in conjunction with the aforementioned efforts, the gradient descent approach is utilized to ascertain the unknown parameters, specifically the cross-diffusion coefficients $d_{12}$ and $d_{21}$. The outcomes derived from this method are deemed to be satisfactory. Moreover, two distinct algorithms have been deployed for the purpose of parameter identification, with their efficacy compared through a meticulous analysis of error discrepancies. Finally, an illustrative demonstration is provided, showcasing the application of parameter identification techniques in the biological surface pattern conformation.