Matrix Manipulations using Artificial Neural Networks

A new neural network approach for matrix computations is presented. The idea is to construct a feed-forward neural network (FNN) and then train it by matching a desired set of patterns. The solution of the problem is the converged weight of the FNN. Accordingly, unlike the conventional FNN research that concentrates on external properties (mappings) of the networks, this study concentrates on the internal properties (weights) of the network. The present network is linear and its weights are usually strongly constrained; hence, a complicated overlapped network needs to be constructed. It should be noticed, however, that the present approach depends highly on the training algorithm of the FNN. Unfortunately, the available training methods such as the, the original Back-propagation (BP) algorithm, encounter many deficiencies when applied to matrix algebra problems, including slow convergence due to improper choice of learning rates (LR). Thus, this study focused on the development of new efficient and accurate FNN training methods. One improvement suggested to alleviate the problem of LR choice is the use of a line search with steepest descent method, namely, bracketing with golden section method. This provides an optimal LR as training progresses. Another improvement proposed in this study is the use of conjugate gradient (CG) methods to speed up the training process of the neural network. The computational feasibility of these methods is assessed on two matrix problems; namely, the LU-decomposition of both band and square ill-conditioned unsymmetric matrices and the inversion of square ill-conditioned unsymmetric matrices. In this paper, only the first one is reported.