A double-layer Associative Memory System (AMS) based on the Cerebella Model Articulation Controller (CMAC) (CMAC-AMS), owing to its advantages of simple structures, fast searching procedures and strong mapping capability between multidimensional input/output vectors, has been successfully used in such applications as real-time intelligent control, signal processing and pattern recognition. However, it is still suffering from its requirement for a large memory size and relatively low precision. Furthermore, the hash code used in its addressing mechanism for memory size reduction can cause a data-collision problem. In this paper, a new high-order Associative Memory System based on the Newton's forward interpolation formula (NFI-AMS) is proposed. The NFI-AMS is capable of implementing high-precision approximation to multivariable functions with arbitrarily given sampling data. A learning algorithm and a convergence theorem of the NFI-AMS are proposed. The network structure and the scheme of its learning algorithm reveal that the NFI-AMS has advantages over the conventional CMAC-type AMS in terms of high precision of learning, much less required memory size without the data-collision problem, and also has advantages over the multilayer Back Propagation (BP) neural networks in terms of much less computational effort for learning and fast convergence rate. Numerical simulations verify these advantages. The proposed NFI-AMS, therefore, has potential in many application areas as a new kind of associative memory system.