The time-varying linear equations resolved by the Zeroing neural network (ZNN) models play very important roles in many areas. The traditional ZNN can only achieve the simple finite-time convergence problem, which makes ZNN very easy to be disturbed in the noisy environment. Since the initial state of ZNN is unknown, the finite time convergence depends on its initial state, so the ZNN models activated by the traditional nonlinear activations cannot deal with the different noise interference. To solve the above problems, a novel Power plus Sign activation function (PpSAF) is designed to activate the ZNN model. The predefined-time robust ZNN (PTR-ZNN) model activated by the simple and special nonlinear PpSAF is used to solve the time-variant linear system equations. Through theoretical analysis and simulation comparison of the system expected convergence time and noise tolerance performance, the numerical experiments present that the PTR-ZNN model possesses the predefined-time convergence and strong noise-tolerance performance, and also prove its superiority in solving time-varying linear system equations.