Dynamic Systems for Total Least Squares of Time-Dependent Tensor Multilinear Equations

This research is the first study focused on studying the total least squares (TLS) for time-varying (TV) tensor multilinear equations (TE). Obtained results extend existing methods for solving the TLS problem in both constant tensor settings and constant and TV matrix settings. Two models are suggested to solve simultaneous perturbations in both tensor coefficient and right-hand vector inside TV TEs. The first model establishes a continuous-time neural network designed for solving the TLS of TV tensor equations (termed as NNTVTETLS), while the second (termed as discrete neural network for total-least-squares solution of time-varying tensor equations (DNNTVTETLS)) is developed as its discrete counterpart. Unlike existing gradient-based neural networks and least-squares methods, the proposed models address TLS perturbation in a high-order TV multilinear settings. A rigorous theoretical convergence analysis is provided, establishing local convergence for both continuous and discrete models. An efficient iterative scheme based on the DNNTVTETLS scheme is also developed. Extensive numerical experiments demonstrate the efficacy of both continuous and discrete-time models. Numerical tests under various TV, constant, bounded vanishing, and bounded nonvanishing noise, confirm the rapid convergence, stability, and robustness of the proposed methods. The suggested framework is applied to solve the Bellman equation, which emerges in dynamic programming, economics, reinforcement learning, and optimal decision-making problems.