In this work, a new zeroing neural network (ZNN) using a versatile activation function (VAF) is presented and introduced for solving time-dependent matrix inversion. Unlike existing ZNN models, the proposed ZNN model not only converges to zero within a predefined finite time but also tolerates several noises in solving the time-dependent matrix inversion, and thus called new noise-tolerant ZNN (NNTZNN) model. In addition, the convergence and robustness of this model are mathematically analyzed in detail. Two comparative numerical simulations with different dimensions are used to test the efficiency and superiority of the NNTZNN model to the previous ZNN models using other activation functions. In addition, two practical application examples (i.e., a mobile manipulator and a real Kinova JACO2 robot manipulator) are presented to validate the applicability and physical feasibility of the NNTZNN model in a noisy environment. Both simulative and experimental results demonstrate the effectiveness and tolerant-noise ability of the NNTZNN model.