Zeroing Neural Networks : an Introduction to Predictive Computations for Time-varying Matrix Problems via ZNN

This paper is designed to bring understanding and computational know-how for time-varying matrix problems and Zeroing Neural Networks in the West. Zeroing Neural Networks (ZNN) were invented for time-varying matrix problems around 2001 in China and almost all of their advances have been made in and most still come from its birthplace. ZNN methods have become a backbone for solving discretized sensor driven time-varying matrix problems in real-time, in theory and in on-chip applications for robots, in control theory and in engineering in China. They have become the method of choice for many time-varying matrix problems that benefit from or require efficient, accurate and predictive real-time computations. The typical discretized ZNN algorithm needs seven distinct steps for its initial set-up. The construction of discretized ZNN algorithms starts from a model with its associated error equation and the stipulation that the error function decrease exponentially fast. The error function differential equation is then mated with a convergent look-ahead finite difference formula to create a distinctly new multi-step style solver that predicts the future state of the system reliably from current and earlier state and solution data. Matlab codes for discretized ZNN matrix algorithms typically consist of one linear equations solve and one recursion of already available data per time step. This makes discretized ZNN based algorithms highly competitive with ordinary differential equation initial value path following or homotopy methods that are designed to work adaptively and gives ZNN different characteristics and applicabilities from multi-step ODE initial value solvers that by design cannot be predictive. Discretized ZNN time-varying matrix methods can solve problems given by sensor data with constant sampling gaps or from functional equations.