Predefined-time convergent neural networks for solving the time-varying nonsingular multi-linear tensor equations

Equation solving is one of the main subjects in various research fields. In this study, we consider tensor equations with time-varying nonsingular coefficient tensors and right-hand side vectors, which generalize the time-invariant tensor equations and also the time-varying or time-invariant linear equations studied by many researchers. Four complex-valued neural networks termed as ZNN-I, ZNN-II, WsbpPTZNN-I and WsbpPTZNN-II are proposed for solving this problem. We show that, under certain conditions, the four models converge to a solution of the original tensor equations. Moreover, we prove that WsbpPTZNN-I and WsbpPTZNN-II models have predefined-time convergence property, which means that they converge to a true solution in a prescribed time. The upper bounds of the convergence time are also given. For comparison, the gradient-based model, termed as GNN, is also introduced. Through many numerical tests, we demonstrate the validity of our theoretical analysis of these models. The simulation results also illustrate the effectiveness and reliability of our proposed neural network models and show that their convergence performance can be improved by using the predefined-time convergent WsbpPTZNN models.