Discrete-time dimension-wise adaptive neural dynamics model for solving time-varying Sylvester equations

Time-varying Sylvester equations play an important role in many scientific and engineering applications. Neural dynamics provides an effective approach for solving such problems. For practical implementation on digital platforms, discrete-time neural dynamics is suitable for sampled-data processing and real-time computation. However, existing discrete-time neural dynamics generally lack sufficient adaptivity to handle heterogeneous state evolution across different dimensions. To address these issues, this paper proposes a discrete-time dimension-wise adaptive neural dynamics (DDAND) model. By reformulating the evolution law in the discrete-time domain and introducing a discrete adaptive moment estimation mechanism, the proposed model achieves dimension-wise adaptive regulation according to the evolutionary differences among state variables. The convergence and robustness of the model are established theoretically. Experimental results demonstrate that the proposed model achieves high solution accuracy, favourable real-time tracking performance, and relatively low computational complexity, while also exhibiting improved robustness to noise. Its effectiveness is further validated through representative applications.